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arXiv:physics/0003080v1 [physics.gen-ph] 27 Mar 2000ICRC1999 Evening Workshop Session talk (updated version) DeformedLorentzSymmetry andHigh-EnergyAstrophysics(I) L.Gonzalez-Mestres1,2 1Laboratoire de Physique Corpusculaire, Coll `ege de France, 75231 Paris Cedex 05, France 2L.A.P.P.,B.P.110, 74941 Annecy-le-Vieux Cedex, France Abstract Lorentz symmetry violation (LSV)can be generated at the Pla nck scale, or at some other fundamental length scale,andnevertheless naturallypreserve Lorentzsymmet ryasalow-energy limit(deformedLorentzsymme- try,DLS).Deformedrelativistic kinematics (DRK)wouldth enbeconsistent withspecial relativity inthelimit k(wave vector) →0and allow for a deformed version of general relativity and gr avitation. If LSV is a very high-energy, very low-distance phenomenon, it is expected to be driven by energy-dependent parameters and, ifatalldetectable, toproduce thecleanest signatures att hehighest attainable energies. Wepresent anupdated discussion of the possible implications of this pattern for high-energy cosmic-ray physics, focusing mainly on an approach where the LSV parameter varies like the square of the energy scale (quadratically deformed relativistic kinematics, QDRK). It turns out that a ≈10−6LSV at Planck scale, leading to a DLS pattern, would potentially be enough to produce very important obser vable effects on the properties of cosmic rays at the≈1020eVscale (absence of GZKcutoff, stability of unstable particl es, lower interaction rates, kinemat- ical failure of any parton model as well as of standard formul ae for Lorentz contraction and time dilation...). Possible neutrino pulses at energies up to ≈1019eVfrom≈1020eVprotons accelerated by gamma-ray bursts, withneutrinoarrivaltimesimplyingenergy-depen dent delaysfromDRK,arealsodiscussed. Although ultra-high energy cosmicrays(UHECR)appear tobethemosta ppropriate probetotest Planck-scale LSV,we also discuss suggestions to explore it at the observed gamma -ray burst (GRB) photon energies using models withaLSVparameter proportional to the energy scale (linea rly deformed relativistic kinematics, LDRK). This paper updates and further develops contributions H.E. 1.3.16 and OG 3.1.10 published in the Pro- ceedings of the ICRC 1999 Conference, Salt-Lake City August 1999 (Gonzalez-Mestres, 1999a and 1999b). 1 What isrelativity? Arguments used in the priority debate: ”who was (were) the au thor(s) of the special relativity theory?” tacitly involve, without really addressing it, a fundament al physics issue. It is implicitly assumed that the basic physics behind special relativity is exactly what sta ndard textbooks have been teaching in the last eight decades: Lorentz symmetry would be an abstract, intrinsic p roperty of space-time that matter cannot escape. In this approach, all particles are compelled to move inside the minkowskian space-time. However, such a non-trivial interpretation of special relativity isnot a w ell-established physical law and there is no proof of its absolute validity. It does not correspond to the initial for mulation of the Poincar´ e relativity principle (1895- 1905), and current particle theory suggests that Lorentz sy mmetry can be violated. A close look reveals that historical arguments are biased byphysical prejudices and arbitrary interpretations. TheFrenchmathematician Henri Poincar´ e wasthe firstautho r toconsistently formulate therelativity prin- ciple, stating (Poincar´ e, 1895): ”Absolute motion of matter, or, to be more precise, the relat ive motion of weighable matter and ether, cannot be disclosed. All that ca n be done is to reveal the motion of weighable matter with respect to weighable matter” . Such a revolutionary claim was not easily accepted. Poinca r´ e was fightingforadecadetoconvinceotherscientists aswellast hepublicopinion. Hefurther emphasizedthedeep content of this new and, at the time, unconventional law of Na ture when he wrote (Poincar´ e, 1901): ”This principle will be confirmed withincreasing precision, as me asurements become more and moreaccurate” . 1Although textbooks andpressusuallypresent special relat ivity ashaving beenformulated inthecelebrated Einstein’s 1905 paper, several authors have emphasized the actual role of H. Poincar´ e in building relativity theory previous to Einstein and the relevance of Poincar´ e’ s thought (Logunov, 1995 and 1997; Feynmann, Leighton, & Sands, 1964). In his June 1905 paper (Poincar´ e, 1905), published before Einsteins’s article (Ein- stein, 1905) arrived (on June 30) to the editor, Henri Poinca r´ e explicitly wrote the relativistic transformation lawforthecharge density andvelocity of motionandapplied togravity the”Lorentz group”, assumed tohold for ”forces of whatever origin”. All the ingredients of spec ial relativity, as well as its basic original concepts, are clearly formulated in this work which, furthermore, emp hasizes the need of a new, relativistic, theory of gravitation. But Poincar´ e’s priority is sometimes denied on the scientific grounds that ”Einstein essentially announced the failure of all ether-drift experiments past a nd future as a foregone conclusion, contrary to Poincar´e’s empirical bias” (Miller, 1996), that Poincar´ e did never ”disavow the ether” (Miller, 1996) or that ”Poincar´e never challenges... the absolute time of newtonian mechan ics... the ether is not only the absolute space of mechanics... but a dynamical entity” (Paty, 1996). Is this argumentation correct, is it based on w ell- established physical evidence? We do not think so. In fact, t hese authors implicitly assume that A. Einstein wasrightin1905when ”reducingethertotheabsolutespaceofmechanics” (Paty,1996)andthatH.Poincar´ e was wrong because ”the ether fits quite nicely into Poincar ´e’s view of physical reality: the ether is real...” (Miller, 1996). But, with the present status of particle phy sics and cosmology, as well as of condensed-matter physics and of the theory of dynamical systems, there is no sc ientific evidence for Einstein’s 1905 absolute, ”geometric” view of relativity. The existence of a physical ”ether”, playing an important dynamical role, would not be incompatible withthe existing (low-energy) ex perimental evidence for the relativity principle. 1.1 Particlephysicspoint ofview: Modern particle physics has brought back the concept of a non -empty vacuum where free particles propa- gate: without suchan”ether”wherefieldscancondense, thes tandard modelofelectroweak interactions could not be written and quark confinement could not be understood. The mechanism producing the masses of the W and Z bosons is close to the Meissner effect, where the conde nsed Cooper pairs (equivalent to the Higgs field) prevent the magnetic field (virtual photons) from prop agationg beyond a certain distance (the London length) inside a superconductor: in the case of the standard model, the effect is mainly observed through the masses (inverse London lengths) of the intermediate bosons propagating in the (vacuum) Higgs field conden- sate. Furthermore, modern cosmology is not incompatible wi th the idea of an ”absolute local frame” close to that suggested by the study of cosmic microwave background r adiation (see f.i. Peebbles, 1993). Therefore, the ”ether” may well turn out to be a real entity in theXXI-thcentury physics and astrophysics. Then, the relativity principle would become a symmetry of ph ysics, another revolutionary concept whose paternity was attributed to H. Poincar´ e by R.P. Feynman (as quoted by Logunov, 1995): ”Precisely Poincar ´e proposed investigating what could bedone withthe equation s without altering their form. Itwasprecisely his idea to pay attention tothe symmetry properties of the laws o f physics” . Actually, Poincar´ e did even more: in all hispapers since 1895, he emphasized another deep concep t: thedynamical origin of special relativity. Dynamics is, by definition, a scale-dependent property of ma tter. In a global view of physics, adynamical property of matter would be the opposite concept to an intrin sic, geometric property of space-time. A basic, unanswered question for particle physics is therefore: was Poincar´ e right when, in his papers since 1895 and inparticular inhis note of June 1905 strongly premonitory o f nowadays grand-unified theories, heconsidered the relativity principle as a dynamical phenomenon, related toacommon origin of all the existing fo rces? Assymmetriesinparticlephysicsareingeneralviolated,L orentzsymmetrymaybebrokenandanabsolute local rest frame may be detectable through experiments perf ormed beyond some critical scale or close to that scale. Poincar´ e’s special relativity (a symmetry applyin g to physical processes) could live with this situation, but not Einstein’s approach such asit wasformulated in1905 (anabsolute geometry of space-time that matter cannot escape). But,howtocheckwhether Lorentzsymmetryi sactually broken? Wediscuss heretwoissues: 2a) the scale where we may expect Lorentz symmetry to be violat ed and the scale(s) at which the effect may be observable; b) the physical phenomena and experiments po tentially able to uncover Lorentz symmetry violation (LSV).Previous papers on the subject are (Gonzal ez-Mestres, 1998a, 1998b, 1998c and 1999a) and referencestherein. WehaveproposedthatLorentzsymmetry bealow-energylimit,brokenfollowinga k2-law (k= wave vector) between the low-energy region and some fundam ental energy (length) scale. 1.2 Condensed-matter point ofview: It seems obviously justified to examine what could be a ”conde nsed-matter point of view”, as particle physics and cosmology have used many condensed-matter anal ogues in the last five decades (e.g. the pattern ofspontaneoussymmetrybreaking)andtheKlein-Gordonequ ationisatypicalequationforwavepropagation. Particlephysics usesnowadays concepts suchasstrings, vo rtices, monopoles, topological defects... whichare closely related to condensed-matter phenomena and makeref erence to the”vacuum” as amaterial medium. Lorentz symmetry, viewed as a property of dynamics, implies no reference to absolute properties of space and time(Gonzalez-Mestres, 1995). In atwo-dimensional ga lilean space-time, the wave equation: α ∂2φ/∂t2−∂2φ/∂x2=F(φ) (1) withα=1/c2 oandco= critical speed, remains unchanged under ”Lorentz” transf ormations leaving invariant the squared interval ds2=dx2−c2 odt2. Any form of matter made with solutions of equation (1) , buil t in the laboratory in a set-up at rest, would feel a relativistic space-time even if the real space-time is galilean and if anabsolute rest frame exists inthe underlying dynami cs beyond the wave equation. The solitons of the sine-Gordon equation are obtained taking in(1) : F(φ) =−(ω/co)2sin φ (2) ωbeingacharacteristic frequency ofthedynamical system. T hetwo-dimensional universemadeofsuchsine- Gordon solitons would indeed behave like a two-dimensional minkowskian world with the laws of special relativity. The actual structure of space and time can only b e found by going to deeper levels of resolution where the equation fails, similar to the way high-energy acc elerator experiments explore the inner structure of ”elementary” particles (but cosmic rays have the highest attainable energies). As modern particle physics views ”elementary” particles as excitations of vacuum, the re would be no inconsistency in assumig that the space-time felt bysuch particles issimilar to the soliton a nalogy. and does not have and absolute meaning. In such a scenario, that cannot be ruled out by any present exper iment, superluminal sectors of matter can exist and even be its ultimate building blocks. This clearly makes sense, as: a) in a perfectly transparent crystal, at least two critical speeds can be identified, those of light and sound; b) the potential approach to lattice dynamics in solid-sate physics is precisely the form of elec tromagnetism in the limit csc−1→0, where cs isthe speed of sound and cthat of light. See,for instance, (Gonzalez-Mestres, 1999a ) and references therein. Atthefundamental lenght scale,andtakingasimplifiedtwo- dimensional illustration, gravitation mayeven beacomposite phenomenon (Gonzalez-Mestres, 1997a), rela ted for instance tofluctuations of theparameters of equations like: A d2/dt2[φ(n)] +H d/dt [φ(n+ 1)−φ(n−1)]−Φn[φ] = 0 (3) where wehave quantized space to schematically account for t he existence of the fundamental length a,φis a wavefunction, ndesigns byaninteger latticesitesspaced byadistance a,AandHarecoeficients and Φn[φ] isdefined by: Φn[φ] =Kfl[2φ(n)−φ(n−1)−φ(n+ 1)] + ω2 restφ (4) Kflbeing acoefficient and (2π)−1ωrestarest frequency. 3Inthecontinuum limit,thecoefficients A=g00,H=g01=g10and−Kfl=g11canberegarded as thematrixelementsofaspace-timebilinearmetricwithequ ilibrium values: A= 1,H= 0andKfl=K. Then, asmall local fluctuation: A= 1 + γ (5) Kfl=K(1−γ) (6) withγ≪1would beequivalent to asmall, static gravitational fieldcr eated byafar awaysource. In conclusion, our present knowledge of condensed-matter p hysics does not plead in favour of Einstein’s 1905 , intrinsically geometric, approach to relativity. It instead suggests that Poincar´ e was right in not ruling out the Ether and in developing instead the concept of physic al symmetry (the ”Lorentz group”). 2 Lorentz SymmetryAs a Low-EnergyLimit Low-energy tests of special relativity have confirmed itsva lidity toan extremely good accuracy ( ≈10−21 from nuclear magnetic resonance experiments), but the situ ation at very high energy remains unclear. Not only high-energy measurements are less precise, but the vio lation of the relativity principle can be driven by energy-dependent parameters. To discuss possible Lorentz symmetry violation, the hypothesis of a preferred refrence frame seems necessary. In what follows, all discus sions are performed in this frame, that weassume tobe close tothe natural cosmological one defined by cosmic b ackground radiation (see, f.i. Peebles 1993). If Lorentz symmetry violation (LSV)follows a E2law (E= energy), similar to the effective gravitational coupling, itcanbe ≈1atE≈1021eV(just abovethehighest observed cosmic-ray energies) and ≈10−26 atE≈100MeV(corresponding to the highest momentum scale involved in nu clear magnetic resonance experiments). Such a pattern of LSV(deformed Lorentz symme try, DLS)will escape all existing low-energy bounds. If LSVisof order 1at Planckscale ( E≈1028eV),and following asimilar law,it willbe ≈10−40 atE≈100MeV. Our suggestion isnot in contradiction with Einstein’s tho ught such asit became after he haddeveloped general relativity. In1921,A.Einstein wrot ein”Geometry andExperiment” (Einstein, 1921): ”Theinterpretation of geometry advocated here cannot be di rectly applied to submolecular spaces... it might turnoutthatsuchanextrapolationisjustasincorrectasan extensionoftheconceptoftemperaturetoparticles of asolid of molecular dimensions” . Theabsoluteness of theminkowskian space-time wasclearl y abandoned throughthisstatement. Itisinitselfremarkablethatspec ialrelativityholdsattheattainedaccelerator energies, but there isno fundamental reason for this tobe thecase abov e Planck scale. 2.1 Deformed relativistickinematics A typical example of patterns violating Lorentz symmetry at very short distance is provided by nonlocal modelswhereanabsolutelocalrestframeexistsandnon-loc ality inspaceisintroducedthroughafundamental lengthscale awherenewphysicsisexpectedtooccur (Gonzalez-Mestres, 1 997a). Suchmodelsnaturally lead toa deformed relativistic kinematics (DRK)of the form (Gon zalez-Mestres, 1997a and 1997b): E= (2π)−1h c a−1e(k a) (7) where his the Planck constant, cthe speed of light, kthe wave vector, and [e(k a)]2is a convex function of (k a)2obtained from vacuum dynamics. Such an expression is equiva lent to special relativity in the small k limit. Expanding equation (1) for k a≪1,wecan write(Gonzalez-Mestres, 1997a and 1997c): e(k a)≃[(k a)2−α(k a)4+ (2π a)2h−2m2c2]1/2(8) αbeing amodel-dependent constant, intherange 0.1−0.01for full-strength violation of Lorentzsymmetry at the fundamental length scale, and mthe massof the particle. For momentum p≫mc, weget: E≃p c+m2c3(2p)−1−p c α(k a)2/2 (9) 4The”deformation” approximated by ∆E=−p c α(k a)2/2intheright-hand sideof(9)impliesaLorentz symmetry violation in the ratio E p−1varying like Γ (k)≃Γ0k2where Γ0=−α a2/2. Ifcis a universal parameter for all particles, the DRK defined by (7) - (9) preserves Lorentz symmetry in the limit k→0, contrary to the standard THǫµmodel (Will, 1993). If, besides c,αis also universal, LSV does not lead (Gonzalez-Mestres, 1997a, c and e) to the spontaneous d ecays predicted in (Coleman, & Glashow, 1997 and subsequent papers) at ultra-high energy using a THǫµ-type approach. On more general grounds, as we also pointed out, the existence of very high-energy cosmic r ays can by no means be regarded as an evidence against LSV,astherelevant kinematical balances canbesen sitivetomanysmallparameters. Inparticular, any form of LSV should be considered (Gonzalez-Mestres, 1997d a nd 1997e) and not only (like in the papers by Coleman and Glashow) models driven bylow-energy constant p arameters. As previously emphasized, the above non-locality may actua lly be an approximation to an underlying dynamics involving superluminal particles (Gonzalez-Mes tres, 1996, 1997b, 1997f and 1997g), just as elec- tromagnetismlooksnonlocalinthepotential approximatio ntolatticedynamicsinsolid-statephysics: itwould then correspond to the limit c c−1 i→0where ciis the superluminal critical speed. Contrary to the THǫµ- type scenario considered by Coleman and Glashow, where LSV o ccurs explicitly in the hamiltonian already atk= 0through a non-universality of the critical speed in vacuum ( the rest masses of charged particles are nolonger given bytherelation E=m c2,cbeing thespeed of light inthe k→0limit), our DLSapproach can preserve standard gravitation and general relativity a slow-energy limits. Furthermore, phenomenological estimates by Coleman and Glashow do not consider possible de formations of the relativistic kinematics: they use undeformed kinematics for single particles at ultra-hi gh cosmic-ray energies. More recent (1998) papers by these authors bring no new result as compared to our 1997 pa pers and present the same fundamental limi- tation astheir 1997 article. Physically, the twoapproache s are really different: bychoosing a THǫµscenario, Coleman and Glashow implicitly assume that Lorentz symmetr y is broken in an ”external” way, by a small macroscopic effect. On the contrary, as explained above, our pattern att ributes LSV to an ”internal” very high-energy, very low-distance phenomenon completely dis appearing at macroscopic scale. A fundamental question is whether candαare universal. This may be the case for all ”elementary” particles, i.e. quarks, leptons, gauge bosons..., but thes ituation is less obvious for hadrons, nuclei and heavier objects. From a naive soliton model (Gonzalez-Mestres, 199 7b and 1997f), we inferred that: a) cis expected to be universal up to extremely small corrections ( ∼10−40,far below the values considered by Coleman and Glashow) escaping all existing bounds; b) an approximate ru le can be to take αuniversal for leptons, gauge bosons and light hadrons (pions, nucleons...) and assume a α∝m−2law for nuclei and heavier objects, the nucleonmasssettingthescale. Withthisrule,DRKintroduc esnoanomalyintherelationbetweeninertial and gravitational massesatlargescale(Gonzalez-Mestres, 19 98c). Basically, the α∝m−2lawmakescompatible DRKfor alarge bodywithasimilar DRKfor smaller partsofitw hich,otherwise, couldnot travel atthesame speed asthe whole body if the relation v=dE/dp(Gonzalez-Mestres, 1997a ,1997d and 1997f) is used. The main effect of DRK can be decribed as follows. The deforma tion term increases with energy roughly like≈E3, whereas the ”mass term” in (9) decreases like ≈E−1. The ratio between the two terms varies like ≈E4and, above some energy depending on the parameters involved , the deformation becomes dominantascomparedtothemassterm. Very-highenergykine maticsinthelaboratoryrestframeis,basically, dominated by longitudinal momentum: as everything else bec omes ”small”, and longitudinal momentum has to be exactly conserved, the real kinematical balances occu r entirely between ”small” terms. Therefore, a ”small”violationoftherelativityprinciplecanpotentia llyplayacrucialroleinthesebalances. If cisuniversal, αmust be positive to avoid the spontaneous decay of UHE (ultra -high energy) particles (unless the effect is not observable below 3.1020eV). Theuniversality of αuptosmall corrections isimposed bytherequirement that elementary particles be able to reach very high energie s (again, if the effect is to be obervable below 3.1020eV). Otherwise, particles with smaller positive values of αwould decay into those with larger α (Gonzalez-Mestres, 1997e) and the effect would manifest it self inhigh-energy cosmic-ray events. 52.2 Alternativemodels The above model isnot the only possible waytodeform relativ istic kinematics. Alternatives are: Mixing with superluminal sectors (MSLS). A form of DRKwas predicted in our papers since 1995, where weattributed (Gonzalez-Mestres, 1995) Lorentz symmetry v iolation to a very high-energy, very low-distance phenomenon whichwouldmodifythepropagators of”ordinary ” particles (thosewithcritical speedinvacuum equal to c, the speed of light): the dynamics driving LSV was expected t o be generated at Planck scale or at some other fundamental length scale. The energy-dependenc e of LSV was claimed to be the explanation to the apparent validity of the Poincar´ e relativity principl e, as inferred from low-energy tests. We also pointed out (see, e.g. Gonzalez-Mestres, 1996) that ultra-high-en ergy cosmic rays would be a natural experimental framework to explore possible Lorentz symmetry violation p henomena. A LSV scenario suggested in all these papers was mixing between ”ordinary” and superlumina l particles directly deforming propagators (see also Gonzalez-Mestres, 1997d where energy-dependent mixi ng parameters were explicitly used, preserving Lorentz symmetryinthe k→0limit). Using suchmodels, counterexamples totheclaims ma dein(Coleman and Glashow, 1997) were presented, based on the energy-depe ndence of LSV effective parameters. Through the parametrizations thus considered, it was also pointed o ut that, besides a fundamental length scale, masses of heavy superluminal particles can play asignificant role i nkilling low-energy LSV. Thebasic ideaofour superluminal particle ( superbradyons , seee.g. Gonzalez-Mestres, 1997g) model was that several sectors of matter are generated at the fundamen tal length scale(s), each sector possibly satisfy- ing a ”sectorial” Lorentz invariance vith a ”sectorial” cri tical speed in vacuum ( cifor the i-th superluminal sector). Superbradyons would have positive mass and energy , and satisfy sectorial motion equations (e.g. Klein-Gordon) with critical speed ci: thay are not tachyons. Dynamical mixing between two sector s would break both Lorentz invariances, and mixing withsuperlumin al sectors (MSLS)would beenough toproduce a consistent DRKfor”ordinary” matter. But,ifLSVisgenerat ed inthisway,wealsoexpect moreconventional deformations of particle propagators tooccur, other than d irect mixing between different sectors of matter. Linear deformation. When building (1997) the DRKapproach given by (7)-(9), wher e the effective defor- mation parameter depends quadratically on energy (quadrat ically deformed relativistic kinematics, QDRK), we were also naturally led to consider models where this depe ndence is linear (linearly deformed relativistic kinematics, LDRK),i.e. where e(k a)isa function of k aand, for k a≪1: e(k a)≃[(k a)2−β(k a)3+ (2π a)2h−2m2c2]1/2(10) βbeing amodel-dependent constant. Formomentum p≫mc: E≃p c+m2c3(2p)−1−p c β(k a)/2 (11) the deformation ∆E=−p c β (k a)/2being now driven by an effective parameter proportional to momentum, Γ (k) = Γl 0kwhere Γl 0=−β a/2. QDRKnaturallyemergeswhenafundamental lengthscaleisin troducedtodeformtheKlein-Gordonequa- tions. It is typical, for instance, of phonons in condensed- matter physics. Asrecently pointed out (Ellis et al., 1999aandb)usingaclassofstringmodels,LDRKcanbegenera tedbyintroducingabackgroundgravitational field in the propagation equations of free particles. In the fi rst case, the Planck scale is an internal parameter ofthebasicwaveequations generating the”elementary” par ticles asvacuumexcitations. Inthesecondcase,it manifests itself only asaparameter of the background gravi tational field, similar to arefraction phenomenon. By discriminating between the two parametrizations, or exc luding both approaches, feasible experiments at availableenergiescanpotentiallyprovideveryvaluablei nformationonfundamentalPlanck-scalephysics. Our choice in 1997 was to concentrate on the QDRK model and disreg ard LDRK, for phenomenological reasons which seem to remain valid if thenew physics is expected to be generated not too far from Planck scale. If existing bounds on LSV from nuclear magnetic resonance ex periments are to be intepreted as setting a bound of ≈10−21on relative LSVat the momentum scale p∼100MeV, this implies β a < 10−34cm. 6However,asitwillbeexplainedlater,itturnsoutthatLDRK canleadtomanyinconsistencies withcosmic-ray experiments unless β ais much smaller. Concepts and formulae presented in our prev ious papers for QDRK (see next section) canbe readily extended toLDRK,using sim ilar techniques. Inparticular: - the linear deformation term −p c β(k a)/2and the mass term m2c3(2p)−1become of the same order at the energy scale Etrans≈π−1/3h1/3(2β)−1/3a−1/3m2/3c5/3; - the linear deformation term and the target energy ETbecome of the same order at the energy scale Elim≈(2π)−1/2(ETa−1β−1h c)1/2. - if the same philosophy as for QDRK is to be followed, candβwould be universal for all ”elementary” particles including light hadrons, whereas for larger obje ctsβ∝m−1,the nucleon mass setting the scale. Modifications of QDRK. Our conjecture that αhas the same value for light hadrons as for the photon and leptons derives from the result (Gonzalez-Mestres, 199 7f) that the elementary soliton solution on a one- dimesional space lattice obeys the same deformed kinematic s as plane waves on the same lattice and with the same dalembertian operator (discretized in space), if the s oliton size scale is basically the quantum inverse of itsmassscale. Asthehighest-energy observedcosmic-raye ventsseemtobehadronicandnotelectromagnetic, theconjectureseemssensibleonpracticalgrounds. Butitc analsobearguedthatquarksaretherealelementary particles and that, to be consistent with quark propagation , the value of αshould be divided by a factor ≈4for mesons and ≈9for baryons. Actually, the parton picture seems impossible to implement when the deformation becomes important, as in the conventional p arton model the constituents can carry arbitrary fractions ofenergyandmomentumatveryhighenergy andthed eformation energy dependscrucially onthese fractions (Gonzalez-Mestres, 1997f): wetherefore expect the failure of any parton model at these energies. A phenomenological discussion of the latter hypothesis is n evertheless worth attempting: for instance, if alpha is to be divided by a factor of 9 for the proton and 4 for th e pion, and the highest-energy cosmic-ray eventsareduetoprotons, therequirement that thespontane ous decay p→p+γdoesnotoccur wouldthen imply α a2<2.10−73cm2. This would exclude LSV with strong coupling at the Planck sc ale. A similar kinematical analysis seems to hold for nuclei unless dynami cs prevents the decay (see next section). With α a2= 2.10−73cm2, a proton with E≈1021eVcould also emit pions and, above E≈1020eV, pion lifetimeswouldbecomemuchshorter thanpredicted byspeci al relativity andcharged pionscanemitphotons. 3 QDRK andUltra-High EnergyCosmic-Ray (UHECR) Physics IfLorentzsymmetryisbrokenatPlanckscaleoratsomeother fundamentallengthscale,theeffectsofLSV maybeaccessibletoexperimentswellbelowthisenergy: inp articular,theycanproducedetectablephenomena atthehighestobservedcosmicrayenergies. DRK(Gonzalez- Mestres 1997a,1997b,1997c,1997hand1998a) plays a crucial role. Taking the quadratic deformation (QDR K) version of DRK, it is found that, at energies above Etrans≈π−1/2h1/2(2α)−1/4a−1/2m1/2c3/2, the very small deformation ∆Edominates over the mass term m2c3(2p)−1in (3) and modifies all kinematical balances: physics gets th us closer to Planck scale than to electroweak scale (this is actually the case in a logarithmic plot of energy scales) and UHECR become an efficient probe of Planck-scale physics. Because o f the negative value of ∆E, it costs more and more energy, as energy increases above Etrans, to split the incoming logitudinal momentum in the laborato ry rest frame. Astheratio m2c3(2p∆E)−1varies like ∼E−4, the transition at Etransis very sharp. With such a LSV pattern, we also inferred (Gonzalez-Mestres , 1997f) from a toy soliton model that the parton picture (in any version), as well as standard relativ istic formulae for Lorentz contraction and time dilation, are expected to fail above this energy (Gonzalez- Mestres, 1997b and 1997f) which corresponds to E≈1020eVform= proton mass and α a2≈10−72cm2(f.i.α≈10−6anda= Planck length), and to E≈1018eVform=pionmassand α a2≈10−67cm2(f.i.α≈0.1anda=Plancklength). Sucheffects are in principle detectable. A phenomenological study of th e implications of DRK allowed to draw several important conclusions for UHECRexperiments (Gonzalez-Me stres, 1997-99). 73.1 Ourprevious predictions withQDRK Assumingthattheearthmovesslowlywithrespect totheabso lute restframe(the”vacuumrest frame”),so thattheapproximation(9)remainsvalidinthelaboratoryr estframe,QDRKcanleadtoobservablephenomena infuture experiments devoted to the highest-energy cosmic rays: a) For α a2>10−72cm2, assuming universal values of αandc, there is no Greisen-Zatsepin-Kuzmin (GZK)cutoff (Greisen 1966; Zatsepin &Kuzmin, 1966) for the particles under consideration. Duetothenew kinematics, interactions with cosmic microwave backgroun d (CMB)photons are strongly inhibited or forbid- den, andultra-high energy cosmicrays(e.g. protons) froma nywhereinthepresently observable Universecan reach the earth (Gonzalez-Mestres, 1997a and 1997c). In par ticular, for an incoming UHE nucleon hitting a CMB photon, the ∆resonance can no longer be formed due to the deformation term . Proton deceleration in astrophysical objects (e.g. gamma-ray bursters) can be inh ibited inasimilar way. b) With the same hypothesis, unstable particles with at leas t two stable particles in the final states of all their decay channels become stable at very high energy. Abov eEtrans, the lifetimes of all unstable particles (e.g. the π0in cascades) become much longer than predicted by relativis tic kinematics (Gonzalez-Mestres, 1997a, 1997band1997c). Then,forinstance, theneutronore venthe ∆++canbecandidates fortheprimaries of the highest-energy cosmic ray events. If candαare not exactly universal, many different scenarios can happen concerning the stability of ultra-high-energy part icles (Gonzalez-Mestres, 1997a, 1997b and 1997c). c) In astrophysical processes at very high energy, similar m echanisms can inhibit radiation under external forces (e.g. synchrotron-like, where the interactions occ ur with virtual photons), GZK-like cutoffs, decays, photodisintegration of nuclei, momentum loss trough colli sions (e.g. with a photon wind in reverse shocks), production of lower-energy secondaries... potentially co ntributing to solve all basic problems raised by the highest-energy cosmic rays (Gonzalez-Mestres, 1997e), in cluding acceleration mechanisms. d)Withthesamehypothesis, theallowedfinal-state phasesp aceoftwo-bodycollisions isstrongly reduced at very high energy, leading (Gonzalez-Mestres, 1997e) to a sharp fall of partial and total cross-sections for incoming cosmic ray energies above Elim≈(2π)−2/3(ETa−2α−1h2c2)1/3, where ETis the energy of the target. As a consequence, and with the previous figures for Lorentz symmetry violation parameters, above some energy Elimbetween 1022and1024eVa cosmic ray will not deposit most of its energy in the atmosphere and can possibly fake anexotic event withmuch le ss energy (Gonzalez-Mestres, 1997e). e) Actually, requiring simultaneously the absence of GZK cu toff in the region E≈1020eV, and that cosmic rays with energies below ≈3.1020eVdeposit most of their energy in the atmosphere, leads in the DRK scenario to the constraint: 10−72cm2< α a2<10−61cm2, equivalent to 10−20< α < 10−9 fora≈10−26cm(≈1021GeVscale). Remarkably enough, assuming full-strength LSV for cesato be in the range 10−36cm < a < 10−30cm. But a≈10−6LSV at Planck scale can still explain the data. Thus, the simplest version of QDRK naturally fits, on phenomenolog ical grounds, with the expected potential role of Planck scale in generating the standard ”elementary” par ticles and opening the door to new physics. f) Effects a) to e) are obtained applying DRK to single partic les and collisions. If further dynamical anomalies are added (failure, at very small distance scales , of the parton model and of the standard relativis- tic formulae for Lorentz contraction and time dilation...) , we can expect much stronger effects in the early cascade development profiles of cosmic-ray events (Gonzale z-Mestres, 1997b, 1997f and 1998a). Detailed phenomenology and data analysis in next-generation experi ments may uncover spectacular new physics and provide apowerful microscope directly focused onthe funda mental length (Planck?) scale. g) Cosmic superluminal particles would produce atypical ev ents with very small total momentum (due to the high E/pratio), isotropic or involving several jets (Gonzalez-Mes tres, 1996, 1997b, 1997d, 1997 and 1998b). In the atmosphere (f.i. AUGER or satellite-based ex periments), such events would generate excep- tional cascade development profiles and muon spectra, as wil l bediscussed in aforthcoming paper. 8*************************************************** ************************************ It should be noticed that our description of all these phenom ena and, in particular, of points a) and b) , on the grounds of DRKwasprior to anysimilar claim by Coleman an d Glashow from THǫµ-like models. *************************************************** ************************************ It follows from b) and f) that early cascade development is a c rucial point for possible tests of special relativity in UHCR cosmic-ray events. Unfortunately, this part of the interactions induced by the incoming cosmic ray is not easily detectable: it occurs early in the at mosphere, and it takes a few collisions before energy isdegraded intoalargeenoughnumber ofparticles to produce anobservable fluorescence signal. Data on cascade development start well below the first interactio n of the primary with the atmosphere whereas we expect LSV, if present, to manifest itself only in the first fe w collisions. But the effect will propagate to later multiparticle production and be observable: for instance, if the π0does not decay at very high energy, we expect a smaller electromagetic component developing late r and more muons produced. A serious drawback is the present ambiguousness of phenomenological air showe r models, but combined data from AUGER and satellite-based experiments should help to clarify this si tuation. A recent fit to the UHECR spectrum with a model close toQDRK,reproducing the absence of GZKcutoff, c anbe found in(Chechin & Vavilov, 1999). 3.2 Oncosmic-ray composition Cosmic-ray composition above 1017eVis a crucial question. Protons are often preferred as candid ates to UHECR events (Bird et al., 1993), but detailed analysis are n ot yet conclusive (f.i. AGASA Collaboration, 1999) and there are claims in favor of light nuclei up to 2.1019eV(f.i. Wolfendale & Wibig, 1999). It is commonly agreed that the UHECR composition becomes lighter above 5.1017eV, as energy increases, and tends to be protons at the highest observed energies. It may b e interesting to compare this phenomenon with QDRK predictions, keeping in mind the proposed α∝m−2law. This law naturally allows, kinematically, spontaneous N→N+γdecays ( N= nucleus) at very high energy. If α a2≈10−67cm2, the threshold for spontaneous gammaemission would be ≈3.1019eVforFeand≈1019eVforHe4. It should be noticed, however, that although spontaneous ga mma emission will be kinematically allowed for large objects because of the α∝m−2law, it will become more and more unnatural dynamically as th eir sizeincreases. Suchanoutgoingphotonwouldcarryanenerg ymuchlargerthanthatofanyincomingnucleon and its wavelength would be smaller, by orders of magnitude, than the size of the composite object. 3.3 Neutrinos from gamma-raybursts The basic idea (Waxman and Bahcall, 1999) is that, in reverse shocks, protons are accelerated to energies ∼1020eVand collide with ambient photons producing, in the kinemati cal region where the center of mass energy corresponds to the ∆+resonance, charged pions with about 20 %of the initial proton energy. These pions subsequently decay into muons and muon neutrinos ( π+→µ++νµ), and the muons decay later into electrons, electron neutrinos and muon antineutrinos (µ+→e++νe+νµ) . It should be noticed that, beacuse of Lorentz dilation, the lifetime for the first decay at pion energy ∼2.1019eVis∼3000s whereas the muon lifetime at ∼1019eVis∼105s. These time scales are already longer than those of gamma-ray bursts. But the situation, for UHE neutrino produ ction, may be considerably worsened by LSV. Forα a2>10−72cm2, the lifetime of a ∼1020eV∆+is modified by QDRK and gets much longer. If α a2>10−70cm2, the resonance becomes stable at the same energy with respec t to the n e+νµνµνe decaychannel. Also,asprevioulsystated,muchhigherphot onenergiesarerequiredtoforma ∆resonance. In both cases, pion photoproduction isstrongly inhibited and the calculation presented in(Waxman and Bahcall, 1999) is to be modified leading to a lower neutrino flux. Simila rly, the lifetimes of UHE charged pions and muons become much longer and lower the neutrino flux further. At the same time, QDRK may inhibit UHE proton syncrotron radiation and allow the proton tobe accel erated to energies above ∼1020eV. 9As DRK makes velocity, v=dE/dp, energy-dependent (Gonzalez-Mestres, 1997a and 1997d), t he arrival time on earth of particles produced in a single burst is expected to dependent on the particle energy. But, contrary to the claim presented in (Ellis et al., 1999a) , it follows from the above considerations that observing UHE neutrinos from GRB bursts for QDRK with α a2>10−72cm2will most likely be much more difficult (if at all possible) than naively expected fro m the Waxman-Bahcall model. 4 LDRK, gamma-raybursts andTeVphysics Considering vacuum as a medium similar to an electromagneti c plasma, it was suggested in (Amelino- Camelia et al., 1998) that quantum-gravitational fluctuati ons may lead to a correction, linear in energy, to the velocity of light. This isequivalent toa LDRKthat, for k a≪1, can beparameterized as: E≃p c−p c β(k a)/2 =p c−p2M−1(12) where Mis an effective mass scale. Possible tests of this model thro ugh gamma-ray bursts, measuring the delays inthearrival timeofphotons of different energies, have beenconsidered in(Norris etal.,1999) having inmindtheGamma-rayLargeAreaSpaceTelescope(GLAST)and in(Ellisetal.,1999b) withamoregeneral scope. Biller et al. (1998) claim a lower bound on Mslightly above ≈1016GeV. However, from the same considerations already developed in our 1997-99 papers taking QDRK as an exemple, stringent bounds on LDRK can be derived. Assume tha t LDRK applies only to photons, and not to charged particles, so that at high energy wecan writefor ach arged particle, ch, the dispersion relation: Ech≃pchc+m2 chc3(2pch)−1(13) where the chsubscript stands for the charged particle under considerat ion. Then, it can be readily checked that the decay ch→ch+γwould be allowed for pabove≃(2m2 chM c3)1/3, i.e: - for an electron, above E≈2TeVifM= 1016GeVand≈20TeVifM= 1019GeV -foramuonorchargedpion,above E≈80TeVifM= 1016GeVand≈800TeVifM= 1019GeV - for aproton, above E≈240TeVifM= 1016GeVand≈2.4PeVifM= 1019GeV - for a τlepton, above E≈400TeVifM= 1016GeVand≈4PeVifM= 1019GeV sothatnoneoftheseparticleswouldbeobervedabovesuchen ergies, apartfromveryshortpaths. Suchdecays seem to be in obvious contradiction with cosmic ray data, but avoiding them forces the charged particles to have the same kind of propagators as the photon, with the same effective value of Mup to small differences. Similar conditions are readily derived for all ”elementary ” particles, leading for all of them, up to small devations, to aLDRKgiven by the universal dispersion relat ion: E≃p c+m2c3(2p)−1−p2M−1(14) Forinstance, π0productionwouldotherwisebeinhibited. Butif,asitseems compulsory, the π0kinematics follows a similar law, then the decay time for π0→γ γwill become much longer than predicted by special relativity at energies above ≈50TeVifM= 1016GeVand≈500TeVifM= 1019GeV. Again, this seemstobeincontradiction withcosmic-ray data. Requirin g that the π0lifetime agreeswithspecial relativity atE≈1017eVwouldforce Mtobeabove ≈1026GeV,far awayfrom the values tobetested at GLAST. Another bound is obtained from the condition that there are 3.1020eVcosmic-ray events. Setting Elimto this value, and taking oxygen to be the target, yields M≈3.1021GeV. In view of these bounds, it appears very difficult to make LDRK , with Mreasonably close to Planck scale, compatible with experime ntal data. It therefore seems necessary to reconsider the models to bet ested at GLAST. 105 Conclusions and comments If, as conjectured by Poincar´ e, special relativity is a sym metry of dynamical origin, Lorentz symmetry violation at very high energy would not be unnatural. But che cking it appears to be a difficult task. If LSV is generated at some fundamental length (e.g. Planck) scale , we expect it to be driven by energy-dependent parametersbecomingverysmallinthelow-energyregionwhe reimpressivetestsofthevalidityoftherelativity principle have been performed (Lamoreaux et al., 1986; Hill s & Hall, 1990). It therefore seems natural, contrary to the THǫµmodel, to preserve Lorentz symmetry as a low-energy limit. I n deformed Lorentz symmetrymodels,leadinginparticulartovariousversions ofdeformedrelativistickinematics,thedeformation disappears in the limit k→0. DRK is far from being unique, as it can be generated inmany di fferent ways, and its predictions are strongly model-dependent. Unlike a previous attempt (Kirzhnits & Chechin, 1972), we have pursued the idea that: a) the fundamental length scal e is at the origin of LSV and should be taken seriously in all respects; b) physics can vary smoothly down to this scale where a new dynamics manifests itself; c) LSV is not related to any other fundamental symmet ry generalizing special relativity. It then turns out that, at energies well below the fundamental length scal e, a very small LSV can generate observable leading-order effects and even allow to discriminate betwe en models of vacuum at Planck scale. Ambitious prospects, based on LDRK (linearly deformed rela tivistic kinematics, obtained from Planck- scale ”vacuum recoil” models), to measure a possible system atic energy-dependence in time delays from gamma-ray bursts are confronted to apparent incompatibili ties between cosmic-ray data and the orders of magnitude of LDRK parameters considered. The basic reason i s that LSV, if realized in this way, would manifest itself in many other phenomena already accessible to experiment (cosmic rays in the TeV - PeV range). However, the fact that the study of LDRKhas led to con sider such comparatively low energies pleads in favour of precision tests of special relativity at the hig hest-energy accelerators (LHC, VLHC and beyond). AlthoughtheexistenceofobservableeffectsofLDRKattheg amma-rayburstphotonenergiesseemsunlikely, systematictestsofspecialrelativityatenergiesbetween 1TeVand1PeVaremissingandshouldbeperformed. On phenomenological grounds, QDRK (quadratically deforme d relativistic kinematics, obtained when a fundamental length scale is introduced in the dynamics gene rating ”elementary” particles) seems to be the most performant model and naturally fits with possible new Pl anck-scale physics. UHE (ultra-high energy) CR (cosmic-ray) physics appears to be the safest laboratory to test LSV, allowing for unconventional phe- nomena in early cascade development. If Lorentz symmetry is violated, the study of UHECR events may be a powerful tool to get direct information on fundamental p hysics at Planck scale. At the highest oberved cosmic-ray energies, the effect of LSV at Planck scale can al ready lead to spectacular signatures: absence of the GZK cutoff; drastic modifications of lifetimes, as wel l as of total and partial cross-sections; failure of any parton picture as well as of standard formulae for time di lation and Lorentz contraction... But further work is required to clearly translate the physical signatur es into measurable data (cascade development pro- file, muons, electromagnetic yield...). Models of air showe r formation should beimproved in order to remove manyinterpretation uncertainties. Abasicdifficultyisth at,although theprimaryinteraction occursquiteearly intheatmosphere, adetectable fluorescence yieldisemitte donlyafter afewinteractions, whenmanycharged particles have been produced and the atmosphere is denser. C ombined information from future experiments, such as AUGERand satellite-based measurements, will hopef ully make this task easier. The existence of superluminal particles ( superbradyons ) is not excluded if LSV occurs at Planck scale: they mayeven bethe ultimate constituents of matter. Thissu bject will bediscussed at length elsewhere. References AGASA Collaboration, 1999, Proceedings of the 26thInternational Cosmic Ray Conference (ICRC 1999), Salt Lake CityAugust 1999, paper HE2.3.02. 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arXiv:physics/0003081v1 [physics.atom-ph] 27 Mar 2000THEORY OF COMPLEX SCATTERING LENGTHS M.S. Hussein Instituto de F´ ısica, Universidade de S˜ ao Paulo, C.P. 66318, S˜ ao Paulo, 05315-970, Brazil (February 20, 2014) Abstract We derive a generalized Low equation for the T-matrix approp riate for com- plex atom-molecule interaction. The properties of this new equation at very low enegies are studied and the complex scattering length an d effective range are derived. 1The recent realization of Bose-Einstein condensation (BEC ) of ultracold atoms with the accompanying upsurge of theoretical activities have re kindled interest in low energy collisions of atoms and molecules. The subsequent proposal s for the creation of ultracold molecular [1-5] and hybrid atomic-molecular BEC [6,7] inte nsified the above mentioned interest Of particular importance in the above recent devel opments is the idea of decay of the condensates. In a series of papers, Dalgarno and collabo rators [8-12] have looked into the idea of using a complex scattering length to represent th e low-energy atom-molecule scattering. Implicit in this is the multichannel nature of t he collision process: an atom hits a vibrationally excited molecule at extremely low energies . The open inelastic channels are those where the molecule is excited into lower vibrational s tates. In this sense one has a depletion of the elastic channel. In Ref. [8], the quenching ofH2molecules in collisions with Hwas considered. It was found that, the inelastic cross-sect ions and the corresponding depletion rate coefficients were very large for high vibratio nal levels of H2. In the above studies, the following form of low-energy S-wav e scattering amplitude is used f(k) =1 g(k2)−i k, (1) where k is the wave number related to the center of mass energy of the colliding partners, E, byℏk2 2µ=E, with µbeing the reduced mass of the system. The function g(k2) is even in kand is given by the effective range formula. g/parenleftbig k2/parenrightbig =−1 a+1 2rok2, (2) where a is the scattering length and r◦the effective range, both directly related to the inter- action. When applied to atom-molecule scattering at very lo w energies, with the molecules suffering inelastic transitions to lower vibrational state s, the scattering length a is taken to be complex, a=α−iβ, with βrelated to the total inelastic cross-section. 2The question we raise here is the validity of Eq. (1) with a and eventually r◦taken as complex in the case of the elastic scattering with strong c oupling to inelastic channels. Of course, an equivalent one-channel description of the ela stic scattering can be formulated with the introduction of an appropriate complex optical pot ential as described by Feshbach [13]. It is therefore legitimate to inquire about the validi ty of Eq. (1), originally obtained for real potential, if a complex interaction is used [14]. The ge neral structure of the low energy scattering amplitude is also of potentially fundamental im portance to very low energy matter interferometry. This method for the obtention of ffor molecule-molecule scattering has been quite successful at room temperatures [15,16]. Extension t o very low temperatures of this method seems natural and would welcome studies of the type re ported here. For the above purpose, it is useful to summarize the elegant d erivation of Eq. (1) given by Weinberg [17]. If we denote the interaction by Vand the free Green’s function by G(+) o(E) = (E−Ho+iε)−1, then the T-matrix given by the Lippmann-Schwinger equatio n T(+)=V+V G(+) oT(+), can be written as T(+)=V+V G(+)V, with the full Green’s function G(+)= (E+iε−Ho−V)−1. Using the spectral expansion of G(+), with the complete set of bound and scattering states/braceleftbig |B/angbracketright,/vextendsingle/vextendsingleΨ(+)/angbracketrightbig/bracerightbig , we obtain the Low-equation /angbracketleftBig /vectork′/vextendsingle/vextendsingleT(+)(E)/vextendsingle/vextendsingle/vectork/angbracketrightBig =/angbracketleftBig /vectork′ |V|/vectork/angbracketrightBig +/summationdisplay B/angbracketleftBig /vectork′ |V|B/angbracketrightBig /angbracketleftBig B|V|/vectork/angbracketrightBig E+EB+/integraldisplay d/vectork′′T(+) /vectork′k′′(Ek′′)/parenleftBig T(+) /vectork′′k(Ek′′)/parenrightBig∗ E−Ek′′+iε. (3) At very low energies relevant for BEC, we seek a solution T/vectork′k(E)≡T(E) and writing V/vectork′k≡t(+)we have t(+)=¯V+/summationdisplay B|gB|2 E+EB+/integraldisplay d/vectork′′/vextendsingle/vextendsingle/vextendsinglet(+) /vectork′k′′(Ek′′)/vextendsingle/vextendsingle/vextendsingle2 E−Ek′′+iε. (4) Calculating now t(+)(E)−1−t(−)(E)−1, we find t(+)(E)−1−t(−)(E)−1=t(−)(E)−t(+)(E) t(−)(E)t(+)(E). (5) 3Since t(−)(E) =T(E−iε) =/parenleftbig T(+)(E+iε)/parenrightbig∗, ifVis real, we have, with the change of t(+)(E)−1−t(−)(E)−1=−2ik2π2µ /planckover2pi12which is just the discontinuity across the positive energy cut in the complex energy plane. Besides the poles in t,/parenleftbig zeros in ( t)−1/parenrightbig , the only other terms in/parenleftbig t(+)/parenrightbig−1are entire functions of W≡E+iε. Accordingly, with the identification f= −1 2π2µ /planckover2pi12t, Eq. (1) follows. We turn next to a complex interaction V/negationslash=V†. The completeness relation now reads /summationtext B|B/angbracketright /angbracketleftB|+/integraltext d/vectork′′/vextendsingle/vextendsingle/vextendsingleΨ(+) /vectork′′/angbracketright /angbracketleft˜Ψ(+) k′′/vextendsingle/vextendsingle/vextendsinglewhere/vextendsingle/vextendsingle/vextendsingle˜Ψ(+) /vectork′′/angbracketrightis the dual scattering state which is a so- lution of the Schr¨ odinger equation with Vreplaced by V†[18,19]. Another form of the completeness relation may also be used,/summationtext B|B/angbracketright /angbracketleftB|+/integraltext d/vectork′′/vextendsingle/vextendsingle/vextendsingle˜Ψ(−) /vectork′′/angbracketright /angbracketleftΨ(−) k′′/vextendsingle/vextendsingle/vextendsingle,with/vextendsingle/vextendsingle/vextendsingleΨ(−) /vectork′′/angbracketrightBig being the physical scattering state with incoming wave boundary c ondition/parenleftbig V†,−iε/parenrightbig and/vextendsingle/vextendsingle/vextendsingle˜Ψ(−) /vectork′′/angbracketrightBig its corresponding dual state ( V,−iε). Thus, the full Green’s function now has the spectral form G(+)(E) =/summationdisplay B|B/angbracketright /angbracketleftB| E+EB+/integraldisplay d/vectork′′/vextendsingle/vextendsingle/vextendsingleΨ(+) /vectork/angbracketright /angbracketleft˜Ψ(+) /vectork′′/vextendsingle/vextendsingle/vextendsingle E−Ek′′+iε. (6) Accordingly, Eq. (3) now reads /angbracketleftBig /vectork′ |T|/vectork/angbracketrightBig =/angbracketleftBig /vectork′ |V|/vectork/angbracketrightBig +/summationdisplay B/angbracketleftBig /vectork′ |V|B/angbracketrightBig /angbracketleftBig B|V|/vectork/angbracketrightBig E+EB+iε+/integraldisplay d/vectork′′/angbracketleftBig /vectork′ |V|Ψ(+) /vectork′′/angbracketrightBig /angbracketleftBig ˜Ψ(+) /vectork′′|V|/vectork/angbracketrightBig E−Ek′′+iε. (7) It is clear that the Low equation, Eq. (3), is not valid anymor e. However, as we show below Eq. (1) is still valid, with the appropriate generaliz ation of the real function g(k2) to a complex one [16]. To see this we analyze the matrix elemen t/angbracketleftBig ˜Ψ(+) /vectork′′|V|/vectork/angbracketrightBig . From the L−Sequation for/angbracketleftBig ˜Ψ(+) /vectork′′/vextendsingle/vextendsingle/vextendsingle, /angbracketleftBig ˜Ψ(+) /vectork′′/vextendsingle/vextendsingle/vextendsingle=/angbracketleftBig /vectork′/vextendsingle/vextendsingle/vextendsingle+/angbracketleftBig /vectork′/vextendsingle/vextendsingle/vextendsingleV1 Ek′′−H◦−V−iε≡/angbracketleftBig /vectork′/vextendsingle/vextendsingle/vextendsingle/bracketleftbig 1 +V G(−)(Ek′′)/bracketrightbig . (8) Thus/angbracketleftBig ˜Ψ(+) /vectork′′|V|/vectork/angbracketrightBig =/angbracketleftBig /vectork′/vextendsingle/vextendsingle/vextendsingle˜T(Ek′′−iε)/vextendsingle/vextendsingle/vextendsingle/vectork/angbracketrightBig , where the unphysical T-matrix ˜Tis given by 4˜T=V+V G(−)V. (9) Accordingly the T-matrix equation, Eq (7), may be written as /angbracketleftBig /vectork′ |T(E)|/vectork/angbracketrightBig =/angbracketleftBig /vectork|V|/vectork/angbracketrightBig +/summationdisplay B/angbracketleftBig /vectork′ |V|B/angbracketrightBig /angbracketleftBig B|V|/vectork/angbracketrightBig E+EB+iε +/integraldisplay d/vectork′′/angbracketleftBig /vectork′ |T(E′′)|/vectork′′/angbracketrightBig /angbracketleftBig /vectork′′/vextendsingle/vextendsingle/vextendsingle˜T(E′′)/vextendsingle/vextendsingle/vextendsingle/vectork/angbracketrightBig E−E′′+iε. (10) A similar equation holds for/angbracketleftBig /vectork′/vextendsingle/vextendsingle/vextendsingle˜T(E)/vextendsingle/vextendsingle/vextendsingle/vectork/angbracketrightBig withiεreplaced by −iε. It is interesting at this point to show the relation between the physical T-matri x element/angbracketleftBig /vectork′ |T(E)|/vectork/angbracketrightBig and /angbracketleftBig /vectork′/vextendsingle/vextendsingle/vextendsingle˜T(E)/vextendsingle/vextendsingle/vextendsingle/vectork/angbracketrightBig . This can be done easily following operator manipulations o f [18], and using the relation/angbracketleftBig ˜Ψ(+) /vectork′′/vextendsingle/vextendsingle/vextendsingle=/angbracketleftBig Ψ(+) /vectork′′/vextendsingle/vextendsingle/vextendsingle+/angbracketleftBig Ψ(+) /vectork′′/vextendsingle/vextendsingle/vextendsingle/parenleftbig V−V†/parenrightbig G(−)(Ek′′), Eq. (8), /angbracketleftBig /vectork′/vextendsingle/vextendsingle/vextendsingle˜T(E)/vextendsingle/vextendsingle/vextendsingle/vectork/angbracketrightBig =/angbracketleftBig /vectork′ |T(E)|/vectork/angbracketrightBig∗ +/integraldisplay d/vectork′′/angbracketleftBig Ψ(+) /vectork′/vextendsingle/vextendsingle/parenleftbig V−V+/parenrightbig/vextendsingle/vextendsingleΨ(+) /vectork′′/angbracketrightBig S−1 /vectork′′/vectork, (11) where S−1is the inverse S-matrix in the elastic channel, S−1 /vectork′′/vectork=/angbracketleftBig ˜Ψ(+) /vectork′′/vextendsingle/vextendsingle/vextendsingle˜Ψ(−) /vectork/angbracketrightBig , and the diagonal part of the matrix element/angbracketleftBig Ψ(+) /vectork′|(V−V+)|Ψ(+) /vectork′′/angbracketrightBig is directly related to the total inelastic scattering cross-section, σin, viz [18] /angbracketleftBig Ψ(+) /vectork/vextendsingle/vextendsingle/parenleftbig V−V+/parenrightbig/vextendsingle/vextendsingleΨ(+) /vectork/angbracketrightBig =−2iE kσin(E) . (12) Eq. (11) explicitly exhibits the connection between ˜TandTthrough the absorptive part of the effective interaction. Now we seek the low energy solution/angbracketleftBig /vectork′ |T|/vectork/angbracketrightBig ≡t+(E) and/angbracketleftBig /vectork′/vextendsingle/vextendsingle/vextendsingle˜T/vextendsingle/vextendsingle/vextendsingle/vectork/angbracketrightBig ≡t−(E) and following the same steps as Weinberg’s [17], we find immediat ely, with f±=−1 2π2µ /planckover2pi12t±, f−1 +=gc/parenleftbig k2/parenrightbig −ik; (13) 5f−1 −=gc/parenleftbig k2/parenrightbig +ik, (14) where gc(k2) is the complex generalization of g(k2) of Eq. (1). We turn now to the connection between gc(k2) and the low-energy observables. This is most conveniently accomplished by employing the generaliz ed optical theorem 4π kImf+=σel+σin, (15) where σelis the total elastic scattering cross section 4 π|f+|2andσin, the total inelastic cross-section. Using (12), we find −Imgc(k2) (Regc(k2))2+ (Im gc(k2)−k)2=k 4πσin. (16) Atk= 0,gc(0) = −1 a, where a is the complex scattering length written as [8] α−iβ. Thus the imaginary part of a, β, is found to be β=(k σin)k=0 4π, (17) an expression also derived in Ref. [8]. Eq. (16) clearly impl ies that σinshould go as k−1as kis lowered, in accordance with Wigner’s law. We go a bit beyond Refs. [8-12] and derive a relation between βand the imaginary part of the effective potential. Since σinis given by (for S-wave scattering), Eq. (12) σin=4π kE∞/integraldisplay 0|u(r)|2(ImV)dr, (18) where u(r) is the S-wave elastic radial wave function, we find 6β= 1 E∞/integraldisplay 0|u(r)|2|ImV|dr  E→0. (19) An equation for the complex effective range, r◦, can also be easily derived Equation (18) is the principle result of this work. It summarizes the follo wing: 1) The coupled-channels calculation aimed to describe the m olecular quenching can be recast as an effective one-channel calculation with a comple x interaction whose imag- inary part account for flux loss. 2) The low-energy behaviour of the scattering amplitude wit h the complex interaction alluded to above can be conveniently parametrized in terms o f complex scattering length and effective range. The message this work conveys is the potential usefulness of constructing the effective complex (optical) interaction for the scattering of ro-vib rational molecules from atoms at low energies. The calculation of a and r◦from knowledge of this potential can be done in a direct and unambiguous way. Acknowledgement Part of this work was done while the author was visiting ITAMP -Harvard. He wishes to thank Prof. Kate Kirby and Dr. H. Sadeghpour for hospitali ty. He also thanks Drs. N. Balakrishnan and V. Kharchenko for useful discussion. Partial support form the ITAMP-NSF grant and from FAPESP and CNPq is acknowl- edged. 7REFERENCES [1] R. Wynar, R. Preelond, D.J. Han, C. Ryh and D.J. Henzen, Sc ience, 287 , 1016 (2000). [2] J.M. Doyle, B. Friedrich, J. Kim and D. Patterson, Phys. R ev. A52 , R2515 (1995). [3] Y.B. Band and P.S. Juliene, Phys. Rev. A51 , R4317 (1995). [4] B. Friedrich and D.R. Herschbach, Phys. Rev. Lett. 74 , 4623 (1995). [5] J.t. Bahns, W.C. Stwalley, P.L. Gould, J. Chem. Phys. 104 , 9689 (1996). [6] E. Timmermans, P. Tommasini, R. Cˆ ote, M.S. Hussein and A .K. Kerman, Phys. Rev. Lett. 83 , 2691 (1999). [7] E. Timmermans, P. Tommasini, M.S. Hussein and A.K. Kerma n, Phys. Reports 315 , 199 (1999). [8] N. Balakrishnan, R.C. Forrey and A. Dalgarno, Chem. Phys . Lett. 280 , 5 (1997). [9] N. Balakrishnan, R.C. Forrey and A. Dalgarno, Phys. Rev. Lett. 80 , 3224 (1998). [10] R.C. Forrey, N. Balakrishnan, V. Kharchenko and a. Dalg arno, Phys. Rev. A58 , R2645 (1998). [11] R.C. Forrey, V. Kharchenko, N. Balakrishnan and A. Dalg arno, Phys. Rev. A59 , 2146 (1999). [12] R.C. Rorrey, N. Balakrishnan, A. Dalgarno, M.R. Haggar ty and E.J. Heller, Phys. Rev. Lett. 82 , 2657 (1999). [13] H. Feshbach, Ann. Phys. (NY) 15 , 357 (1958); Ann. Phys. (NY) 19 , 287 (1962). [14] L.D. Landau and E. Lifchitz, Quantum Mechanics (1966). [15] J. Schmiedmayer, M.S. Chapman, C.P. Ekstrom, T.D. Hamm od, S.Wehinger and D.E. Pritchard, Phys. Rev. Lett. 74 , 1043 (1995). 8[16] M.S. Chapman, C.R. Ekstrom, T.D. Hammond, R. Rubenskin , J. Schmiedmayer, S. Hehinger and D.E. Pritchar, Phys. Rev. Lett. 74 , 4783 (1995). [17] S. Weinberg, Brandeis Summer Institute in Theoretical Physcis, S. Desser and K.W. Ford, Editors (Prentice Hall Inc.) (1965), vol. II, p. 291-4 03. [18] M.S. Hussein, Ann. Phys. 175, 197 (1987); H. Feshbach, A nn. Phys. (NY) 164 , 398 (1985). [19] M.S. Hussein and E.J. Moniz, Phys. Rev. C29 , 2054 (1984). 9

arXiv:physics/0003083v1 [physics.plasm-ph] 27 Mar 2000MAGNETIC GEOMETRY AND THE CONFINEMENT OF ELECTRICALLY CONDUCTING PLASMAS L. Faddeev∗♯and Antti J. Niemi∗∗♯ ∗St.Petersburg Branch of Steklov Mathematical Institute Russian Academy of Sciences, Fontanka 27 , St.Petersburg, R ussia‡ ∗∗Department of Theoretical Physics, Uppsala University P.O. Box 803, S-75108, Uppsala, Sweden‡ and ♯Helsinki Institute of Physics P.O. Box 9, FIN-00014 University of Helsinki, Finland We develop an effective field theory approach to inspect the el ectromagnetic inter- actions in an electrically neutral plasma, with an equal num ber of negative and positive charge carriers. We argue that the static equilibrium config urations within the plasma are topologically stable solitons, that describe knotted a nd linked fluxtubes of helical magnetic fields. ‡permanent address ∗E-mail: FADDEEV@PDMI.RAS.RU and FADDEEV@PHCU.HELSINKI.FI ∗∗E-mail: NIEMI@TEORFYS.UU.SEPlasma comprises over 99.9 per cent of known matter in the Uni verse. However, among the different states of matter its physical properties are th e least understood. This is largely due to a highly complex and nonlinear behaviour, w hich makes theoretical investigations quite difficult. Particularly notorious are the instabilities that hamper plasma confinement in thermonuclear fusion energy experime nts [1]. In the present Letter we consider the electromagnetic inter actions within a charge neutral plasma, with an equal number of negative and positiv e charge carriers. We propose a first principles field theory model to describe the fl uid dynamical properties of this plasma, and find results that challenge certain widel y held views on plasma behaviour. In particular, we argue that stable self-confini ng plasma filaments can exist, and are described by topologically nontrivial knotted soli tons. In magnetohydrodynamics [1] the geometrical properties of an electrically neutral plasma are conventionally described using a single-fluid ap proximation. The individual charged particles contribution is described collectively by the hydrostatic pressure p, which according to standard kinetic theory relates to the ki netic energies of the individual particlesp∝mv2. The equation of motion then follows from the properties of t he pertinent energy-momentum tensor Tµν, the spatial part of its divergence coincides with the external dissipative force which leads to the Navier-St okes equation ρd/vectorU dt=−∇p+ (∇ ×/vectorB)×/vectorB+η∇2/vectorU2(1) Here/vectorUis the bulk (center of mass) velocity of the plasma, and ηis the coefficient of viscosity. The plasma evolves according to (1), dissipatin g its kinetic energy by the viscous force. This force is present whenever the plasma is i n motion but ceases when the plasma reaches a magnetostatic equilibrium configurati on. In that limit the Navier- Stokes equation reduces to a balance relation between the gr adient of the hydrostatic pressure and the magnetic force, ∇p= (∇ ×/vectorB)×/vectorB Ideally, one might expect that under proper conditions a pla sma in isolation becomes self-confined due to the currents that flow entirely within th e plasma itself. But this appears to be excluded by a simple virial theorem [1] which su ggests that any static plasma configuration in isolation is dissipative. As a conse quence of such apparently inborn instabilities, strong external currents are then co mmonly introduced to confine a plasma in laboratory experiments. We now argue that there are important non-linear effects whic h are not accounted for by a structureless mean field variable such as the pressure p. These nonlinearities have their origin in the electromagnetic interactions between t he charged particles within the plasma. They remain hidden when the energy-momentum tensor relates to the kinetic energies of the individual particles, but become visible on ce we recall the familiar but 1nontrivial relation between the kinetic momentum m/vectorvand the canonical momentum /vectorp of a charged point particle, m/vectorv=/vectorp−e/vectorA where/vectorAis the electromagnetic vector potential. We propose that wh en these electro- magnetic forces within the plasma are properly accounted fo r, the ensuing field theory model has the potential of supporting stable soliton-like c onfigurations which describe helical, self-confined structures within the plasma medium . Our starting point is a natural kinetic field theory model of a two-component plasma of electromagnetically interacting charged point particl es such as electrons and deuterons. In natural units the classical action is S=/integraldisplay dtd3x/bracketleftbigg iψe∗(∂t+ieAt)ψe+iψi∗(∂t−ieAt)ψi−1 2m|(∂k+ieAk)ψe|2 −1 2M|(∂k−ieAk)ψi|2−1 4F2 µν/bracketrightbigg (2) As usualFµν=∂µAν−∂νAµ. Theψeandψiare two (complex) non-relativistic fields for electrons and ions with masses mandMand electric charges ±e, respectively. Notice that we describe both charged fields by macroscopic (Hartree -Fock) wave functions. This is adequate in the classical Bolzmannian limit which is relevant in conventional plasma scenarios [1]. The action (2) determines our first pri nciples description of a non- relativistic plasma. Its magnetohydrodynamical properti es are governed by the pertinent energy-momentum tensor Tµν, which can be constructed from (2) in a standard manner. When we include the contributions that account for the bulk m otion of the plasma medium, this leads to an appropriate version of the Navier-S tokes equation (1). Here we are interested in the ensuing static equilibrium configur ations. These configurations are local minima of the internal energy E, which is determined by the temporal T00 component of the energy-momentum tensor. For a stationary p lasma fluid (2) we get from (2) E=/integraldisplay d3x/bracketleftbigg1 2µ/braceleftbigg sin2α|(∂k+ieAk)ψe|2+ cos2α|(∂k−ieAk)ψi|2/bracerightbigg +1 2B2 i+g(ψe∗ψe−ψi∗ψi)2/bracketrightbigg (3) Hereµ=m·sin2α=M·cos2αis the reduced mass and Bi=1 2ǫijkFjkis the magnetic field. The quartic potential is the remnant of the Coulomb int eraction with gan effective coupling constant. It emerges when we first use Gauss’ law to e liminate the electric field, and then recall that in any realistic plasma the Debye s creening radius is small in comparison to any characteristic length scale of interest. The free energy (3) is subjected to the conditions that the pl asma is electrically neutral with an equal (large) number neof electrons and niof ions,ne=niand the 2total number of charge carriers in the volume Vremains intact ne+ni=N. These conditions can be implemented by adding appropriate chemic al potential terms to (3) in the usual fashion. But for simplicity we here account for the m as constraints, imposed by appropriate boundary conditions. Besides the terms that we have displayed in (3) there can also be additional interaction terms for the charg ed fields. Such terms are usually induced by thermal fluctuations and finite density eff ects, or by gravitational interactions. However, according to standard universalit y arguments we expect the main features of (3) to persist at temperatures and distance scales which are relevant in conventional plasma scenarios. We propose that (3) yields an adequate approximation for a no n-relativistic plasma in a kinetic regime where the thermal energy is sufficiently high to prevent the formation of charge neutral bound states, which correspond to hydrogen a toms in the case of electrons and deuterons. Such bound states are present at lower temper atures, and their presence can be accounted for by terms of the form Ebs=/integraldisplay d3x/bracketleftbigg1 2·1 m+M(∂kΦ)2+λ·Φψeψi+¯λ·Φψ∗ eψ∗ i/bracketrightbigg Here Φ a real scalar field that describes a charge neutral boun d state ofψeandψi. At a sufficiently high temperature this bound state degree of fre edom decouples, and (3) becomes adequate for describing the bulk properties of the p lasma. Sincene=niwe have overall charge neutrality. However, there can be loc al charge density fluctuations that should not be ignored. Indeed, we n ow proceed to argue that static charge density fluctuations are naturally present in (3). These fluctuations accom- pany stable, static solitons which describe filamental self -confined structures within the plasma. For this we first note that the different contribution s in (3) respond differently to a scaling /vectorx→λ/vectorx. The kinetic terms scale in proportion to λand the Coulomb potential in proportion to λ3, but the magnetic energy scales like λ−1. Consequently the existence of nontrivial, non-dissipative plasma configura tions in (3) can not be excluded by simple virial arguments, quantitative investigations b ecome necessary. We start by observing that the vector potential Akenters at most quadratically. Consequently it can be eliminated: We vary (3) w.r.t.Akand get Ak=1 2e·1 sin2α|ψe|2+ cos2α|ψi|2/bracketleftbigg isin2α·(ψe∗∂kψe−∂kψe∗ψe) −icos2α·(ψi∗∂kψi−∂kψi∗ψi)−2µ e·ǫkij∂iBj/bracketrightbigg (4) which determines Akin terms of an iterative gradient expansion, in powers of der ivatives in the charged fields. We introduce new variables by (ψe,ψi) =ρ·( cosα·sinθ 2eiϕ,sinα·cosθ 2eiχ) (5) 3For reasons that will soon become obvious we have chosen thes e variables so that they are natural for describing tubular field configurations, with ϕandχrelated to the toroidal and poloidal angles and θa shape function that measures the distance away from the centerline of the tube. We compute the free energy (3) to the l eading order in a self- consistent gradient expansion, where we keep only terms whi ch are at most fourth order in the derivatives of the variables (5). This approximation is adequate in conventional plasma scenarios where the fields are relatively slowly vary ing. We start by determining Akfrom (4) iteratively in the variables (5). We substitute the result in (3), and by defining a three-component unit vector /vectorn= (cos(χ+ϕ) sinθ,sin(χ+ϕ) sinθ,cosθ) we finally get for the free energy E=/integraldisplay d3x/bracketleftbigg1 2·1 m+M·/braceleftbigg (∂kρ)2+ρ2·|∂k/vectorn|2/bracerightbigg +1 4e2(/vectorn·∂i/vectorn×∂j/vectorn)2+gρ4 4(n3−cos 2α)2/bracketrightbigg (6) We note that since mandMare both nonvanishing, overall charge neutrality implies that asymptotically θ→2α∝negationslash=nπ. Sinceρ→const. ∝negationslash= 0 asymptotically (see below), the Coulomb interaction then yields a mass term for the varia bleθ. We also note that (6) naturally embodies a helical structure, described by th e Hopf invariant [2]. To the relevant order in our gradient expansion QH=−1 e24π2/integraldisplay d3x/vectorB·/vectorA=−/integraldisplay d3x∇cosθ·∇ϕ 2π×∇χ 2π= ∆ϕ·∆χ(7) Here ∆ϕresp.∆χdenotes the (2 π) change in the pertinent variable over the (would-be) tube, when we cover it once in the toroidal and poloidal direc tions over a magnetic flux surface with constant θ. The fieldρis a measure of the particle density in the bulk of the plasma. If its average (asymptotic) value <ρ2>=ρ2 0becomes too small, the collective behaviour of the plasma will be lost and instead we have an individual-partic le behaviour of the charged constituents, interacting via Coulomb collisions. Conseq uently we select the average ρ2 0 so that it acquires a sufficiently large value in the medium. Lo cal charge fluctuations then occur in regions where the unit vector /vectornbecomes a variable so that θ∝negationslash= 2α. According to our adiabatic approximation |∂k/vectorn|is a slowly varying bounded function over the entire charge fluctuation region, and in particular it vanishes outside of the fluctuation region. When we inspect the ρ-equation of motion that follows from (6) we find that it can be related to a Schroedinger equation for the l owest energy scattering state in an external potential ∝ |∂k/vectorn|2. From this we then conclude that |ρ(/vectorx)|never vanishes; it is bounded from below by a non-vanishing positi ve value which is related to the ensuing scattering length. This implies that if we avera ge the free energy (6) over ρ(/vectorx), to the relevant order in our gradient expansion the result can be related to the universality class determined by the Hamiltonian H=/integraldisplay d3x/bracketleftbigg γ· |∂k/vectorn|2+1 4e2(/vectorn·∂i/vectorn×∂j/vectorn)2+λ·(n3−cos 2α)2/bracketrightbigg (8) 4whereγ,λare nonvanishing positive constants, proportional to the s cattering length of our Schroedinger equation. This Hamiltonian is known to s upport stable knotlike solitons [2]. In particular, since the third (Coulomb) term is positive it does not interfere with the lower bound estimate derived in [4]. This estimate s tates that the first two terms in (8) are bounded from below by the fractional power |QH|3/4of the Hopf invariant. Even though we do not expect that in the case of (6) this lower b ound estimate remains valid as such, we nevertheless conclude that when QH∝negationslash= 0 the energy (6) admits a nontrivial lower bound; the conclusions from the virial the orem in [1] should not be adapted too hastily. The properties of (8) with λ= 0 have been studied in [2]-[8]. In particular, the numerical simulations in [7], [8] clearly confirm the existe nce of stable, knotted and linked solitons with a nontrivial Hopf invariant [2]. The pr esent considerations firmly suggest that the conclusions in [2]-[8] prevail also in the c ase of (6). Indeed, we have tentatively verified that similar solitons are present in (6 ), by numerically constructing a line vortex soliton in this model; we describe our solution in figure 1. These soli- tons then become natural candidates for describing filament al and toroidal structures in the plasma, including coronal loops above the solar photo sphere and the design of magnetic geometries in thermonuclear fusion energy experi ments. The numerical sim- ulations reported in [7]-[9] are very extensive, and clearl y reveal the complexity of the problem. Accordingly the interest has thus far mainly conce ntrated on the identification of soliton geometries, very little is still known about the s olitons detailed physical prop- erties. Consequently at this time we are not in a position to p resent definite physical predictions in the context of actual applications, high pre cision numerical methods still remain under active development [7], [8] and we have to limit ourselves to a few general remarks: In the numerical simulations that have been comple ted thus far, it has been found that for generic integer values (∆ ϕ,∆χ) = (n,m) in (7) the λ= 0 solitons of (8) form involved knotted and linked structures. Such complex g eometries might be natural in a number of applications, for example when modelling coro nal loops. But they might not be of any immediate practical interest for the design of p lasma geometries in fusion energy experiments, where planar toroidal configurations a re preferable. Indeed, there are also a few torus-shaped solitons which are essentially p lanar. These occur for values (n,m) = (1,1),(2,1),(1,2),(2,2) [8]. The simplest one is (1 ,1) but it appears to have an energy density that peaks at the toroidal symmetry axis. A s such this may be an advantage in designing actual fusion reactors. But it could also become problematic, as it may interfere with the construction of an external torus- shaped coiling system which should be needed to create the soliton. On the other hand, the (2,1) soliton seems to have a torus-shaped energy density distribution which vani shes at the symmetry axis and peaks at the centerline of the torus (see [8]). Since this soliton is also quite sturdy [8], it is a natural candidate e.g.for designing magnetic geometries for thermonuclear fusion energy purposes. In particular, this configuration s trongly suggests that for a sta- ble, toroidal planar geometry the safety factor [1] in the bu lk of the plasma should not 5exceedq≈2. A configuration with a higher value for qtends to adjust itself towards a geometrical shape which is not planar; see the computer anim ations in the www-address of reference [8]. In conclusion, we have argued that an electrically neutral c onducting plasma can form stable, self-confining structures. This is due to soliton-l ike solutions, which we have shown will appear when we properly account for the nontrivia l electromagnetic interac- tions within the plasma. We have proposed that our solitons c an become relevant in a number of practical scenarios, including coronal loops and the design of magnetic geome- tries in thermonuclear fusion energy experiments. However , in order to assess the impact of our findings, detailed numerical investigations are nece ssary. Unfortunately the simu- lations remain highly complex, even with the present day sup ercomputers. Consequently we have not been able to reliably confirm that parameters such as the asymptotic density ρ0and the coupling gcan indeed be selected appropriately for the solitons to hav e direct technological relevance for example in the design of magnet ic geometries for energy pro- ducing thermonuclear fusion reactors. But since over 99.9 p er cent of all known matter in the Universe exists in the plasma state, there are no doubt numerous scenarios where our results can become important. Besides astrophysical ap plications or quark-gluon plasma experiments, these might include even an explanatio n to the highly elusive ball lightning. We thank A. Alekseev, E. Babaev, A. Bondeson, H. Hansson, E. L angmann, V. Maslov, H.K. Moffatt, S. Nasir, A. Polychronakos, R. Ricca an d G. Semenoff for discus- sions. We are particularly indebted to M. L¨ ubcke for his hel p, and to J. Hietarinta for communicating the results in [8] prior to publication. We th ank the Center for Scientific Computing in Espoo, Finland for the use of their computers. T he work of L.F. has been supported by grants RFFR 99-01-00101 and INTAS 9606, and the work of A.J.N. has been supported by NFR Grant F-AA/FU 06821-308. 6References [1] J. P. Freidberg, Ideal Magnetohydrodynamics Plenum Press, New York and London 1987; D. Biskamp, Nonlinear Magnetohydrodynamics Cambridge University Press, Cambridge 1993 [2] L. Faddeev, A.J. Niemi, Nature 387(1997) 58; and Phys. Rev. Lett. 82(1999) 1624 [3] L. Faddeev, Quantisation of Solitons , preprint IAS Print-75-QS70, 1975; and in Einstein and Several Contemporary Tendencies in the Field T heory of Elementary Particles in Relativity, Quanta and Cosmology vol. 1, M. Pantaleo, F. D e Finis (eds.), Johnson Reprint, 1979 [4] A.F. Vakulenko, L.V. Kapitanski, Dokl. Akad. Nauk USSR 248810 (1979) [5] A.J. Niemi Knots in interaction (Physical Review D, in press) hep-th/9902140 [6] J. Gladikowski, M. Hellmund, Phys. Rev. D56(1997) 5194 [7] R. Battye, P. Sutcliffe, Phys. Rev. Lett. 81(1998) 4798; and Proc. R. Soc. Lond. A455 (1999) 4305 [8] J. Hietarinta, P. Salo, Phys. Lett. B451 (1999) 60; and The ground state in the Faddeev-Skyrme model , University of Turku preprint, 1999; For video animations, seehttp://users.utu.fi/hietarin/knots/index.html [9] M. Miettinen, A.J. Niemi, Yu. Stroganoff, Aspects of duality and confining strings (Physics Letters B, in press) hep-th/9908178 7Figure Caption figure 1: An example of a numerically constructed tubular line vortex solution of (6), with energy density plotted as a function of the distanc e from the tubular center- line. We use standard cylindrical coordinates ( r,φ,z) so that the tubular center-line coincides with the z-axis. For simplicity we have taken a limit of large ion mass w hich sends 2α→π. All numerical parameters in (6) are O(1) and the helical structure is characterized by ϕ+χ=φ+ 0.6z. 800.10.2 10distance (r)figure 1

arXiv:physics/0003084v1 [physics.bio-ph] 28 Mar 2000Spatial-temporal correlations in the process to self-orga nized criticality C.B. Yang1, X. Cai1and Z.M. Zhou2 1Institute of Particle Physics, Hua-Zhong Normal Universit y, Wuhan 430079, China 2Physics Department, Hua-Zhong University of Science and Te chnology, Wuhan 430074, China (January 10, 2014) A new type of spatial-temporal correlation in the process ap proaching to the self-organized criticality is investigated for the two simple models for bi ological evolution. The change behaviors of the position with minimum barrier are shown to be quantita tively different in the two models. Different results of the correlation are given for the two mod els. We argue that the correlation can be used, together with the power-law distributions, as crit eria for self-organized criticality. The phenomenon of “self-organized criticality” (SOC), wit h potential applications ranging from the behavior of sandpile and the description of the growth of surfaces to gen eric description of biological evolution, has become as a topic of considerable interest [1–8]. It is observed tha t the dynamics of complex systems in nature does not follow a smooth, gradual path, instead it often occurs in ter ms of punctuations, or “avalanches” in other word. The appearance of the spatial-temporal complexity in nature, c ontaining information over a wide range of length and time scale, presents a fascinating but longstanding puzzle. Suc h complexity also shows up in simple mathematical models for biological evolution and growth phenomena far from equi librium. In former studies, power-law distributions for the spatial size and lifetime of the “avalanches” have been o bserved in various complex systems and are regarded as “fingerprints” for SOC. It seems that there is no general agre ement on a suitable definition of SOC [9,10], although a minimal definition was given in [11]. Because there is no uni versally accepted “black-box” tests for the presence or absence of SOC based solely on observables, systems with a wi de range of characteristics have all been designated as “self-organized critical”. While numerous numerical studies have claimed SOC to occur i n specific models, and although the transition to the SOC state was studied in [12–14], a question has never bee n answered: How is the process approaching to the final dynamical SOC attractor characterized? One may even as k whether the phenomenon SOC can be adequately characterized by such power-law distributions. The answer to the latter question seems to be negative, as concluded in [15]. In Ref. [15] were pointed out “some striking observa ble differences between two ‘self-organized critical’ models which have a remarkable structural similarity”. The two models, as called the Bak-Sneppen (B-S) models, are introduced in [16–18] and are used to mimic biological ev olution. The models involve a one-dimensional random array on Lsites. Each site represents a species in the “food-chain”. T he random number (or barrier) assigned to each site is a measure of the “survivability” of the species. Initially, the random number for each species is drawn uniformly from the interval (0, 1). In each update, the least survivable species (the update center) and some others undergo mutations and obtain new random numbers which are al so drawn uniformly from (0, 1). In the first version of the model (the local or nearest-neighbor model), only the update center and its two nearest neighbors participate the mutations. In the second version, K−1 other sites chosen randomly besides the update center are i nvolved in the update and assigned new random survivabilities (so th is version is called random neighbor model). Periodic boundary conditions are adopted in the first model. As shown i n [18–20], the second version is analytically solvable. Investigation in [15] shows that some behaviors of the local and random neighbor models are qualitatively identical. They both have a nontrivial distribution of barrier heights of minimum barriers, and each has a power-law avalanche distribution. But the spatial and temporal correlations be tween the minimum barriers show different behaviors in the two models and thus can be used to distinguish them. In all the studies mentioned above, spatial and/or temporal distributions of the “avalanches” and correlations between positions with minimum of barriers are investigate d separately. As shown in many studies, however, spatial and/or temporal distribution of the “avalanches” alone can not be used as a criterion for SOC, nor can the spatial or temporal correlation do. In this paper, it is attempted to st udy a new kind of correlation between minimum barriers in the process of the updating in the two models for biologica l evolution. The correlation between the positions with minimum barriers at time (or update) sands+ 1 is investigated. Since the new correlation involves two sites at different times, it is of spatial-temporal type. Thus it may be suitabl e for the study of spatial-temporal complexity. Consider the update process of the local neighbor model. Ini tially, each site is assigned a random number. All the random numbers are drawn uniformly from interval (0,1). Den oteX(s) the site number with minimum barrier after supdates. The sites can be numbered such that 1 ≤X(s)≤L. To see how X(s) changes in updating process in the model X(s) is shown in Fig. 1 as a function of sfor an arbitrary update process for lattice size L=200 with s from 1 to 2000. The lower part of Fig. 1 is a zoomed part of the up per one for small s. It is clear that X(s) seems 1to be random when sis small. With the going-on of updating, X(s) becomes more and more likely to be in the neighborhood of last update center, X(s−1). So there appear some plateau like parts in Fig. 1. In other word, there appears some correlation between X(s) when the system is self-organized to approach the critical state. So, it may be fruitful to study the self-correlation of X(s) in searching quantities characterizing the process to SOC . For this purpose, one can define a quantity C(s) =/angbracketleftX(s)X(s+ 1) /angbracketright − /angbracketleftX(s)/angbracketright/angbracketleftX(s+ 1) /angbracketright, (1) with average over different events of updating. Obviously, i f there is no correlation between the sites with minimum barrier at time sands+ 1, or /angbracketleftX(s)X(s+ 1) /angbracketright=/angbracketleftX(s)/angbracketright/angbracketleftX(s+ 1) /angbracketright,C(s) will be zero. Thus, C(s) can show whether there is correlation between X(s) and also give a measure of the strength of the correlation. B ecause of the randomness of the survivability at each site, X(s) can be 1, 2, · · ·,Lwith equal probability, 1 /L. Thus, /angbracketleftX(s)/angbracketright= (L+ 1)/2 for every time s. It should be pointed out that /angbracketleftX(s)/angbracketright= (L+ 1)/2 does not mean any privilege of sites with numbering about (L+1)/2. In fact, all sites can be the update center wit h equal chance at time sif the update process is repeated many times from the initial state. Due to the randomness of th e updated survivability X(s+ 1) can also take any integer from 1 to L. However, the distribution of X(s+ 1) is peaked at X(s) when sis large, see [13] for detail. With the update going on, the width of the distribution becom es more and more narrower. When the width becomes narrow enough, /angbracketleftX(s)X(s+ 1) /angbracketrightwill turn out to be /angbracketleftX2(s)/angbracketright= (2L2+ 3L+ 1)/6. So, C(s) will approach ( L2−1)/12 for large s. In above definition for C(s), however, the neighboring relation between X(s) and X(s+ 1) cannot be realized once the numbering for the sites is given. Due to the periodic boundary conditions adopted in the model, one of the nearest neighbors of the site with numbering 1 is the on e numbered L. To overcome this shortcoming, one can introduce an orientational shorter distance ∆(s) between X(s) and X(s+ 1). Imagine the Lsites with numbering 1,2,· · ·, Lare placed on a circle in clockwise order. Then |∆(s)|is the shorter distance between the two sites on the circle. If X(s+ 1) is reached along the shorter curve from X(s) in clockwise direction, ∆( s) is positive. Otherwise ∆(s) is negative. For definiteness, one can assume −L/2≤∆(s)< L/2. With ∆( s), one can use X′(s+ 1) = X(s) + ∆( s) (2) in place of X(s+1) in the definition of C(s). Since X′(s) can cross the (non-existing) boundary between 1 and Land reflect the neighboring relation with X(s), the effect of periodic boundary conditions on the correlat ion can be taken into account. (In the simulation of the B-S model numbering t heLsites with integer numbers 1 ,2,· · ·, Lis necessary, but the start position can be arbitrary. Different numbering scheme will give the same results for C(s), as physically demanded. This in return is also an indication of the equival ence of all sites in the presence of periodic boundary conditions.) To normalize the dependence of C(s) on the size of the one-dimensional array, we can renormaliz eC(s) by (L2−1)/12. In the following, we use a normalized definition of C(s) as C(s) =/angbracketleftX(s)X′(s+ 1) /angbracketright − /angbracketleftX(s)/angbracketright/angbracketleftX′(s+ 1) /angbracketright (L2−1)/12. (3) In current study X(s) and ∆( s) are determined from Monte Carlo simulations, and 500,000 s imulation events are used to determine the averages involved. For each event, 200 0 updates are performed from an initial state with random barriers on the sites uniformly distributed in (0, 1) . The normalized correlation function C(s) is shown as a function of sin Fig. 2 for L= 50,100,and 200. One can see that C(s) is a monotonously increasing function of times. As in our naive consideration, C(s) is very small in the early stage of updates and becomes large r and larger for larger s, indicating the increase of the strength of correlation bet ween the sites with minimum barrier at different times. The behavior of C(s) with sexhibits different characteristics for small and large s.C(s) increases with svery quickly for small s, but the rate becomes quite slow after a knee point. The knee p oint appears earlier for smaller L, showing the existence of a finite-size effect. Also, the seeml y saturating value of C(s) depends on the size Lof the lattice, or more clearly, it increases with the lattice size L. Since only 500,000 simulation events are used in current study, there shows the effect of fluctuations in the figure. The correlation between X(s) can be investigated for the random neighbor model for biolo gical evolution in the same way. For simplicity only the case with K= 3 is taken into account. The generalization to other cases i s straight forward. First, one can have a look on how X(s) changes with update. X(s) is shown as a function of sin the upper part of Fig. 3. This plot may look as a random scatter of points at first sight. But it is not. A close look reveals correlations: X(s) often has almost same value for several consecutive or almo st consecutive svalues. However, no obvious plateau like part can be seen in the figure, showing th e difference between the two versions of B-S model. C(s) is also studied and shown in the lower part of Fig. 3 as a funct ion of sfor the lattice size L= 200. In the random neighbor version of the B-S model, sites numbered with 1 and Lare no longer neighbors. So, in the calculation of C(s) from Eq. (3), X(s+ 1) is used instead of X′(s+ 1). The counterpart for the nearest neighbor model is also 2drawn in the figure for comparison. One can see that the satura ting value is much smaller than in the case of the local neighbor version of the model. From the discussions above one can see that the correlation b etween the sites with minimum barrier may play an important role in investigating SOC. The power-law distrib utions for the size and lifetime of the “avalanches” togethe r with the new kind of correlation may be used as criteria for SO C. This work was supported in part by the NNSF in China and NSF in H ubei, China. One of the authors (C.B.Yang) would like to thank Alexander von Humboldt Foundation of Ger many for the Research Fellowship granted to him. [1] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A38, (1987) 364; Phys. Rev. Lett. 59, (1987) 381. [2] K. Chen, P. Bak, and S.P. Obukhov, Phys. Rev. A43, (1991) 625. [3] P. Bak, K. Chen, and M. Creuts, Nature 342, (1989) 780. [4] K. Sneppen, Phys. Rev. Lett. 69, (1992) 3539; K. Sneppen and M.H. Jensen, ibid. 70, (1993) 3833; 71, (1993) 101. [5] P. Bak and K. Chen, Sci. Am. 264(1), (1991) 46. [6] K. Chen and P. Bak, Phys. Lett. A140 , (1989) 46. [7] P. Bak, K. Chen, and C. Tang, Phys. Lett. A147 , (1990) 297. [8] A. Sornette and D. Sornette, Europhys. Lett. 9, (1989) 197. [9] D. Sornette, Phys. Rev. Lett. 72, (1994) 2306. [10] G. Canelli, R. Cantelli, and F. Cordero, Phys. Rev. Lett .72, (1994) 2307. [11] H. Flyvbjerg, Phys. Rev. Lett. 76, (1996) 940. [12] M. Paczuski, S. Maslov and P. Bak, Europhys. Lett. 27, (1994) 97. [13] M. Paczuski, S. Maslov and P. Bak, Phys. Rev. E53, (1996) 414. [14] A. Corral and M. Paczuski, Phys. Rev. Lett. 83, (1999) 572. [15] J. de Boer, A.D. Jackson, and Tilo Wetig, Phys. Rev. E51, (1995) 1059. [16] P. Bak and K. Sneppen, Phys. Rev. Lett. 71, (1993) 4083. [17] H. Flyvbjerg, P. Bak, and K. Sneppen, Phys. Rev. Lett. 71, (1993) 4087. [18] J. de Boer, B. Derrida, H. Flyvbjerg, A.D. Jackson, and T . Wettig, Phys. Rev. Lett. 73, (1994) 906. [19] Yu.M. Pis’mak, J. Phys. A: Math. Gen. 28, (1995) 3109. [20] Yu.M. Pis’mak, Phys. Rev. E56, (1997) R1326. Figure Captions Fig. 1 The change of site X(s) with time sfor an arbitrary event in the nearest neighbor version of the B-S model for biological evolution. Fig.2 The correlation function C(s) as a function of sfor lattice size L=50, 100, and 200 for the same model as in Fig. 1. Fig. 3 Upper part: The change of site X(s) with sfor the random neighbor version of the B-S model for biologic al evolution; Lower part: The correlation function C(s) for the two versions as functions of sforL=200. 3050100150200 0 500 1000 1500 2000 050100150200 0 50 100 150 20000.20.40.60.81 0 500 1000 1500 2000050100150200 0 500 1000 1500 2000 00.20.40.60.81 0 500 1000 1500 2000

arXiv:physics/0003085v1 [physics.atom-ph] 28 Mar 2000Search for correlation effects in linear chains of trapped io ns C. J. S. Donald, D. M. Lucas, P. A. Barton∗, M. J. McDonnell, J. P. Stacey, D. A. Stevens†, D. N. Stacey, and A. M. Steane ∗Institut f¨ ur Experimentalphysik, Universit¨ at Innsbruc k, Technikerstr. 25, A-6020 Innsbruck, Austria †Institute d’Optique, 91403 Orsay, France Centre for Quantum Computation, Department of Atomic and La ser Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, U.K. (February 20, 2014) We report a precise search for correlation effects in linear chains of 2 and 3 trapped Ca+ions. Unexplained correlations in photon emission times within a linear chain of trapped ion s have been reported, which, if genuine, cast doubt on the po- tential of an ion trap to realize quantum information proces s- ing. We observe quantum jumps from the metastable 3d2D5/2 level for several hours, searching for correlations betwee n the decay times of the different ions. We find no evidence for cor- relations: the number of quantum jumps with separations of less than 10 ms is consistent with statistics to within error s of 0.05%; the lifetime of the metastable level derived from the data is consistent with that derived from independent single-ion data at the level of the experimental errors (1%) ; and no rank correlations between the decay times were found with sensitivity to rank correlation coefficients at the leve l of |R|= 0.024. 42.50.Lc, 42.50.Fx, 32.80.Pj, 32.70.Cs The drive to realise the potential of quantum informa- tion processing [1,2] has led to the investigation of variou s experimental systems; among these is the ion trap, which has several advantages including the capability to gener- ate entanglement actively with existing technology [3]. Following the proposal of an ion-trap quantum processor by Cirac and Zoller [4], several groups have carried out pioneering experiments [5–9]. In a recent review [10], the view was expressed that “the ion trap proposal for realiz- ing a practical quantum computer offers the best chance of long term success.” One of the attractive features of the trap is that the various interactions and processes which govern its behaviour have been exhaustively stud- ied and are in principle well-understood. However, 14 years ago unexplained collective behaviour when several ions were present was reported [11], and recently a further account of such effects has appeared [12]. They manifest themselves as an enhanced rate of coincident quantum jumps. Sauter et al. [11] measured two- and three-fold coinci- dent quantum jumps in a system of three trapped Ba+ ions to occur two orders of magnitude more frequently than expected on the basis of statistics. This observa- tion led to proposals that the ions were undergoing a collective interaction with the light field [11,13]. More recently, Block et al. [12] have observed an en-hanced rate of two- and three-fold coincidences in a linear chain of ten Ca+ions, where the coincidences were not confined to adjacent ions. This led them to suggest an unexplained long range interaction between ions in the linear crystal. They also found that measurements of the lifetime τof the 3D 5/2level (shelved state) from the 10-ion string produced discrepancies of as much as 6 σbe- tween runs under nominally identical conditions, where σis the standard deviation for each run. Since only the electromagnetic interaction is involved, it is extremely unlikely that these observations indicate new physics; nevertheless, they raise serious doubt about the suitability of the ion trap as a quantum information processing device. We have therefore undertaken a new and precise search for such effects in linear chains of 2 and 3 trapped Ca+ions. Our data were taken under conditions such that cor- relation effects would be expected on the basis of the results of [11] and [12], and are significantly more precise than either. We find no evidence at all for correlations, as described below. The experimental method is very similar to that re- ported in our measurement of the lifetime of the 3d2D5/2 level [14], which was originally adopted by Block et al. [12]. Linear crystals of a small number, N, of40Ca+ions separated by about 15 µm are obtained by trapping in a linear Paul trap in vacuo (<2×10−11mbar), and laser-cooling the ions to a few mK. The transitions of in- terest are shown in figure 1. Laser beams at 397 nm and 866 nm continuously illuminate the ions, and the fluores- cence at 397 nm is detected by a photomultiplier. The photon count signal is accumulated for bins of duration tb= 10.01 ms (of which the last 2.002 ms is dead time), and logged. A laser at 850 nm drives the 3D 3/2−4P3/2 transition. The most probable decay route from 4P 3/2is to the 4S 1/2ground state; alternatively, an ion can re- turn to 3D 3/2. However, about 1 decay in 18 occurs to 3D5/2, the metastable “shelving” level. At this point the fluorescence from the ion that has been shelved disap- pears. A shutter on the 850 nm laser beam remains open for 100 ms before it is closed, which gives ample time for shelving of all Nions. Between 5 and 10 ms after the shutter is closed we start to record the photomultiplier count signal in the 10 ms bins. We keep observing the photon count until it abruptly increases to a level above 1a threshold. This is set between the levels observed when 1 and 0 ions remain shelved. The signature for all Nions having decayed is taken to be ten consecutive bins above this threshold. After this we re-open the shutter on the 850 nm laser. This process is repeated for several hours, which constitutes one run. The data from a given run were analysed as follows. The raw data consists of counts indicating the average fluorescence level in each bin of duration tb(see figure 2).Nthresholds λmare set, the mththreshold being set between the levels observed when mand (m−1) ions remain shelved. The number of bins observed below λNgives the decay time, tN, of the first of Nshelved ions to decay. The number of bins observed between λm+1andλmbeing exceeded gives the decay time, tm, of the next ion to decay leaving ( m−1) ions shelved. The large number of tmobtained are then gathered into separate histograms and the expected exponential dis- tribution Aexp (−γmt) is fitted to each, in order to de- rive the decay rate γmof the next ion to decay leaving (m−1) ions shelved (see figure 3). It is appropriate to use a Poissonian fitting method (described in [14]), rather than least-squares, because of the small numbers involved in part of the distribution (at large t). If the Nions are acting independently, each one will have a decay rate γ= 1/τ, where τis the lifetime of the 3D 5/2state. Since we do not distinguish between the fluorescence signals from the different ions, then with m ions remaining shelved the next decay is characterised by the increased rate γm=m/τ. Figure 3 shows the histogram of the decay times, t1, of the second ion of two to decay obtained from a 3.2 hour run. The expected exponential decay fits the data very well. Events in the first bin of the histogram cor- respond to both ions being detected as decaying in the same bin, t1= 0. These quantum jumps, coincident within our time resolution, certainly do not occur two orders of magnitude more frequently than expected by random coincidence as was observed by Sauter et al.[11]. In fact, they are observed to occur less frequently than predicted by the fitted exponential to the histogram data. However, this is an artefact of our finite time resolution. The fitted exponential to the histogram data has value f1in the first bin, which gives the number of second ion decays that are expected to occur within tbof the first ion decaying by random coincidence. However, for both ions to decay within a single bin, the second ion has an average time of less than tbin which to decay. The exact details depend upon the analysis thresholds, λm, and the detector dead time. In the 2-ion case, one can show that, to first order in tb/τ, the first bin width is modified to Ftbwhere: F= 0.98−0.8λ′ 1+0.16λ′ 12+0.16λ′ 22+1.44λ′ 2−0.64λ′ 1λ′ 2 with normalized thresholds:λ′ m=λm−SN SN−1−SN where Smis the mean photon count with mions shelved (soSNis the mean background count level). This ex- pression was verified using real and simulated data. The expected number of coincidences is therefore Ff1. For the histogram shown, the 2-ion data was analyzed with the thresholds λ′ 1= 1.4 and λ′ 2= 0.40 (these are chosen to optimize the discrimination of the fluorescence lev- elsSm), which gives F= 0.42. The expected number of coincidences is Ff1= 24 ±5, assuming√nerrors, which agrees with the observed number of coincidences, 26. The second bin of the histogram is the only other bin expected to have a modified width, which is by a negligible amount. Note that, to ensure the number of coincidences is properly normalized, it is important that only events where at least ( m+1) ions were shelved at the start of an observation are included in the tmhistogram (form/negationslash=N). Table I shows that the observed number of 2-fold co- incidences in the 2- and 3-ion data agree with the ex- pected value within√nerrors. The total expected num- ber of 2-fold coincidences in all the data was 66.3 out of 16132 quantum jumps observed to start with at least 2 ions shelved. We are therefore sensitive to changes in the proportion of 2-fold coincidences at the level of√ 66/16132 = 0 .05% or about 1 event per hour. The expected number of 3-fold coincidences depends on the threshold settings in a more complex way than in the 2-fold case, and here we simply use simulated 3- ion data to provide the predicted number of 3-fold co- incidences shown in table I. The total number of ex- pected 3-fold coincidences is 0.05 in both 3-ion data runs, which have a combined duration of 2.8 hrs. In fact, this predicted value is significantly lower than effects in our trap which can perturb the system sufficiently to cause de-shelving (such as collisions with residual background gas), as discussed in [14]. We observe at most one event, depending on the exact choice of threshold settings, and this does not constitute evidence for correlation. The decay rates obtained from the 2- and 3-ion data are shown in figure 4, where the horizontal lines are the expected rates γm=m/τassuming the ions to act in- dependently. Combining all the γmderived from the 2- and 3-ion data as estimates of m/τyields a value τ= 1177 ±10 ms, where we include a 2 ms allowance for systematic error [14]. This is consistent with the value derived from single-ion data, τ= 1168 ±7 ms [14]. We are therefore sensitive to changes in the apparent value ofτdue to multiple ion effects at the level of 1%. Super- fluorescence and subfluorescence as observed in a two ion crystal [15] are calculated to be negligible with the large interionic distance of about 15 µm in the chain. In order to look for more general forms of correlation between the decay times of each ion, rank correlation 2tests were performed. Table II gives the results; they show no significant correlations. The 2-ion data is the most sensitive, allowing underlying rank-correlation co- efficients to be ruled out at the level of |R12|= 0.024. In summary, we have presented results that are consis- tent with no correlations of spontaneous decay within lin- ear chains of 2 and 3 trapped Ca+ions, contrary to pre- vious studies. First, the number of coincident quantum jumps were found to be consistent with those expected from random coincidence at the level of 0 .05%. Second, the exponential decay expected assuming the ions to act independently fitted the histogram of decay times tmob- tained from the 2- and 3-ion data well. Third, the decay rates from these fits were combined to estimate the life- time of the shelved state, giving a result consistent with our previous precise measurement performed on a single ion [14]. Fourth, rank correlation tests were performed on the decay times obtained from the 2- and 3-ion data; no evidence for rank correlation was found. We suggest therefore that the correlations which have been reported are likely to be due not to interactions between the ions themselves, but to external time- dependent perturbations. In our own trap, we have in- vestigated and reduced such perturbations to a negligible level [14], and the present work demonstrates that when this is done there is no evidence that an ion trap is subject to unexplained effects which would make it unsuitable for quantum information processing. We are grateful to G.R.K. Quelch for technical assis- tance, and to S. Siller for useful discussions. This work was supported by EPSRC (GR/L95373), the Royal Soci- ety, Oxford University (B(RE) 9772) and Christ Church, Oxford. [1] Phil. Trans. R. Soc. Lond. A 356, (1998), special edition. [2] A. M. Steane, Rep. Prog. Phys. 61, 117 (1998). [3] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Mayatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, Nature 404, 256 (2000). [4] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [5] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 75, 4714 (1995). [6] B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 81, 1525 (1998). [7] H. C. N¨ agerl, D. Leibfried, H. Rohde, G. Thalhammer, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. A60, 145 (1999). [8] C. Roos, T. Zeiger, H. Rohde, H. C. N¨ agerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett.83, 4713 (1999).[9] E. Peik, J. Abel, T. Becker, J. von Zanthier, and H. Walther, Phys. Rev. A 60, 439 (1999). [10] R. J. Hughes et al., Fortschr. Phys. 46, 329 (1998). [11] T. Sauter, R. Blatt, W. Neuhauser, and P. E. Toschek, Opt. Commun. 60, 287 (1986). [12] M. Block, O. Rehm, P. Seibert, and G. Werth, Eur. Phys. J. D7, 461 (1999). [13] R. Blatt and P. Zoller, Eur. J. Phys. 9, 250 (1988). [14] P. Barton, C. Donald, D. Lucas, D. Stevens, A. Steane, and D. Stacey, arXiv:physics/0002026 (2000), to be pub- lished in Phys. Rev. A. [15] R. G. DeVoe and R. G. Brewer, Phys. Rev. Lett. 76, 2049 (1996). [16] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C , 2nd ed. (CUP, Cam- bridge, UK, 1992). 4S1/23D3/23D5/24P1/2 866□nm850□nm 397□nm4P3/2 1168 /c1777□ms 1200 /c17710□ms/c54/c46/c57/c50 /c177/c32/c48/c46/c48/c50 ns /c55/c46/c49/c48 /c177/c32/c48/c46/c48/c50 ns FIG. 1. Low-lying energy levels of40Ca+, with their life- times. Lasers at 397 nm, 866 nm and 850 nm drive the cor- responding transitions in the experiments. 2000 4000 6000 8000 10000 120000100200300400500600700 321 321 32 SS10 SSt t tλ λ λ Fluorescence per 8ms Time (ms) FIG. 2. Observed fluorescence signals from a linear 3-ion crystal. The vertical axis is the number of counts given by the photomultiplier during one 10 ms counting bin (2 ms dead time). The grey bars indicate re-shelving periods, when the shutter on the 850 nm laser was open. The de-shelving times, tm, are labelled for one observation of the 3 ions decaying from the shelved state, where mis the number of ions remaining to decay. The dotted horizontal lines show the threshold setti ngs λmfor the data analysis; the dashed horizontal lines show the mean count levels Sm. 30 1000 2000 3000 4000 5000-4-2024 1 Residuals t (ms)010203040506070 Frequency FIG. 3. The histogram of the decay times, t1, of the last ion of 2 to decay, obtained from a 3.2 hour run, with an exponential Aexp (−γ1t) fitted to all bins but the first two. In this case, the analysis gave A= 57 ±1, γ1= 0.860±0.012 s−1, which agrees with the expected rate γ1= 1/τ= 0.856±0.005 s−1, where τis the lifetime de- rived from single-ion data [14]. The residuals are shown on an expanded scale, in the form (data −fit)/√ fit. The first bin gives the number of 2-ion jumps observed to be coincident within one counting bin and has a modified bin width (see text), which reduces the expected number in the first bin to beF= 0.42 of the value, f1= 57, predicted by the fitted exponential. The expected number, Ff1= 24±5 (marked by a cross), agrees with the observed number, 26 (indicated wit h an arrow). A B C0.51.01.52.02.5-13 2 1γγγ 1829 1829 1926 29722802 2802 66036603 Rate (s ) RunFIG. 4. Measured de-shelving rates γmof the the next ion to decay from the state where mions are shelved; errors are purely statistical. The horizontal lines are the expected r ates γm=m/τif the ions are acting independently, where τis the lifetime derived from single-ion data [14] and have negligi ble error on this scale. Runs A and B were conducted with 3 ions, run C with 2 ions. The number below each point gives the number of decay times in the corresponding histogram. Run N Time (hrs) mi→mf NQJ ncnobs 2→0 1926 7.0 10 A 3 1.1 3 →1 1829 9.5 9 3→0 1829 0 .02 0 2→0 2972 10.5 13 B 3 1.7 3 →1 2802 15.4 13 3→0 2802 0 .03 0 C 2 3.2 2 →0 6603 23.9 26 total 2-fold 6.0 (2 ,3)→(0,1) 16132 66.3 71 total 3-fold 2 .8 3 →0 4631 0 .05 0 TABLE I. Two-fold and three-fold (bold type) coincident quantum jumps, with Nions. Coincident quantum jumps occur with miions initially shelved, leaving mfions shelved. NQJis the total number of quantum jumps observed with miions initially shelved. For independent ions, ncof these jumps are predicted to be coincident, taking into account th e modified bin width. nobsgives the number of coincidences observed. The third column gives the total amount of time that one or more ions spent shelved in each run. Run N R 12 R23 R13 R95% A 3 −0.025 −0.010 −0.018 0.046 B 3 −0.019 +0.010 +0.008 0.037 C 2 +0.008 — — 0.024 TABLE II. Results of the rank correlation tests, with N ions. Rnmis the Spearman rank-order correlation coefficient for the decay times tnandtm.|Rnm|would have to be greater thanR95%for 95% significance [16]. 4

arXiv:physics/0003086v1 [physics.data-an] 28 Mar 2000XAFS spectroscopy. I. Extracting the fine structure from the absorption spectra K. V. Klementev Moscow State Engineering Physics Institute, 115409 Kashrs koe sh. 31, Moscow, Russia e-mail: klmn@htsc.mephi.ru (February 2, 2008) Three independent techniques are used to separate fine struc ture from the absorption spectra, the background function in which is approximated by (i) smoothing spline. We propose a new reliable criterion fo r determination of smoothing parameter and the method for raising of stability with respect to kminvariation; (ii) interpolation spline with the varied knots; (iii) the line obtained from bayesian smoothing. This metho ds considers various prior information and includes a natural way to determine the errors of XAFS ext raction. Particular attention has been given to the estimation of unc ertainties in XAFS data. Experimental noise is shown to be essentially smaller than the errors of th e background approximation, and it is the latter that determines the variances of structural para meters in subsequent fitting. 61.10.Ht I. INTRODUCTION X-ray-absorption fine-structure (XAFS), χ, is determined by [1]: χ(E) = [µ(E)−µ0(E)]/[µ0(E)−µb(E)], (1) where µis the measured absorption, µ0is the “atomic” absorption due to electrons of considered at omic level, µb is the absorption of other processes. Since the electronic s tate of an embedded atom is, in general, different from its state in gaseous phase, µ0is not the same as for isolated atom and cannot be found experi mentally. Therefore a demand arises for an artificial construction of µ0. Usually, µbis approximated by a Victoreen polynomial P=aE−3+bE−4[1] or by a more general polynomial P, coefficients of which are found by the least squares method fro mµ(E) =P(E) at energies lower than the edge. Further, energy dependence is transformed to the photoelec tron wave number dependence: k=/radicalbig 2me(E−E0)//planckover2pi1, where E0is the energy of the corresponding absorption edge. Usually , to the E0the energy at half the step is assigned or the energy of inflection point of µ(E). In most practical works the deviation of E0from true value, ∆ E0, is one of the fitting parameters. The most difficult procedure in extracting of XAFS from the mea sured absorption is the construction of µ0since one cannot definitely distinguish the environmental-born p art of absorption from the atomic-like one. All methods for determination of the post-edge background are based on the a ssumption of its smoothness, and the only criterion for its validity is the absence of low-frequncy structure in χ(k)·kw, i. e. the small absolute value of the Fourier transform (FT) ρ(r) at low r. The review of existing post-edge background methods and th e propositions of some new is the main purpose of the article. Special attention must be paid to the estimation of noise and uncertainties in XAFS data. Experimental noise is shown to be essentially smaller than the errors of the backgr ound approximation, and it is the latter that determines the variances of structural parameters in subsequent fittin g. The corresponding section of the present article is close ly related with the next article devoted to the determining the errors of structural parameters [2]. All described in the article methods for background removal , its error estimations, and XAFS-function corrections are realized in the freeware program viper [3] which allows one to vary several parameters by hand and wa tch the results simultaneously. II. METHODS OF µ0CONSTRUCTION A. Smoothing spline Owing to fast algorithm and easy program realization, the ap proximation of µ0by the smoothing spline has become widespread. Let N+ 1 experimental values of µiare defined on the mesh Ei. The smoothing spline µ0minimizes the functional 1J(µ0, µ) =/integraldisplayEmax Emin[µ′′ 0(E)]2dE+1 αN/summationdisplay i=0(µ0i−µi)2. (2) The smoothing parameter (or regularizer) αis the measure of compromise between smoothness of µ0and its deviation fromµ. Atα= 0 the smoothing spline exactly coincides with µ, atα→ ∞ it degenerates to µ0=const. Optimal regularizer should lead to µ0containing only low-frequency oscillations and, hence, to χcontaining only structural oscillations. The formulation of a new criterion for optima lαwe shall consider below. First, we address another problem, the well-known spline in stability with respect to the small variations of input parameters: number of nodes, nodal values of the processed f unction, and limits on integral. In our case the spline is most sensitive to Emindue to fast growth of µin the edge. To raise the stability the method was put forward in viper program which lies in the use of a prior information specifyi ng the shape of µ0(E) dependence. It is known in advance that the absorption edge without so-called white line constitutes nearly smooth step; the white line, if presents, is added to the step. Denote this prior function as p(E). Now we will tend the second derivative of the sought µ0(E) not to zero (at the specified deviation of µ0fromµ) but to the second derivative of p(E). The sought µ0(E) is now minimizes the functional J∗(µ0, µ) =/integraldisplayEmax Emin[µ′′ 0(E)−p′′(E)]2dE+1 αN+1/summationdisplay i=0[µ0i−µi]2. (3) As seen, in fact there is no need to know p(E) itself, its second derivative is sufficient. The explicit pr esence of p(E) in the following formulas should be taken as a consequence of the technical trick applied: at first p(E) is subtracted from the data, then it is added to the found spline. Represent the second derivatives in finite-difference appro ximation, introduce ˜ µ0i=µ0i−pi, and denote ∆ i= Ei+1−Ei: /G14 /G16 /G17/G13 /G13 /G14 /G16 /G1B/G13 /G13 /G14 /G17 /G15/G13 /G13 /G28 /G0F /G03 /G48 /G39/G13/G11 /G13/G13/G11 /G17/G13/G11 /G1B/G14/G11 /G15/G50 /G5B/G14 /G16 /G17/G13 /G13 /G14 /G16 /G17/G18 /G13 /G13 /G17 /G1B /G14 /G15 /G14 /G19 /G4E /G0F /G03 /G63/G10 /G14/G10 /G14/G13/G14 /G46 /G11 /G4E/G15 /G13 /G15 /G17 /G19 /G55 /G0F /G03 /G63/G13/G13/G11 /G18/G14/G14/G11 /G18 /G5F /G29 /G37 /G0B /G46 /G11 /G4E/G15/G0C/G5F/G0B /G44 /G0C /G0B /G45 /G0C /G0B /G46 /G0C /G28 /G13 FIG. 1. Extraction of XAFS from the measured absorption usin g the smoothing spline. Prior function p(E) for the atomic-like absorption is drawn by dots. Solid lines — µ0(E),χ(k)·k2, and ρ(r) obtained with use of the prior function; dashed lines — dittos without prior function. The regularizer αis the same for both cases. 2/G55 /G48 /G4A /G58 /G4F /G44 /G55 /G4C /G5D /G48 /G55 /G03 /G44/G10/G13/G11 /G14/G13/G13/G11 /G14 /G77 /G03/G15/G2B/G15/G12 /G77 /G03 /G0B /G4F/G51 /G03 /G44 /G0C/G15/G0F/G03/G44 /G55/G45/G11 /G03/G58/G51/G4C/G57/G56 /G13/G14/G15/G2B/G15/G0F /G03/G44/G55 /G45/G11/G03 /G58/G51/G4C /G57 /G56/G14 /G13 /GB1 /G15 /G13 /G63 /G03 /G0B /GEE /G18 /G13 /G13 /G0C/G16 /GB1 /G18 /G63 /G14 /GB1 /G15 /G11 /G1B /G63 /G44 /G59 /G4A /G48 /G03 /G13 /GB1 /G13 /G11 /G18 /G63 /G03 /G0B /GEE /G18 /G0C /G14 /G13/G16/G14 /G13/G17/G14 /G13/G18/G14 /G13/G19/G14 /G13/G1A/G14 /G13/G1BFIG. 2. The ρ(r) peak heihts squared, H2, maximal in the indicated areas and average over the range 0 < r < 0.5˚A, as functions ofα. To the right axis relates the second derivative of the first peak height squared with respect to ln α. J∗(µ0, µ) =N/summationdisplay i=1[˜µ0i−1∆−1 i−1−˜µ0i(∆−1 i−1+ ∆−1 i) + ˜µ0i+1∆−1 i]2+1 αN+1/summationdisplay i=0[˜µ0i−(µi−pi)]2=J(˜µ0, µi−pi).(4) Thus, the problem is reduced to the preceding one in which ins tead of initial data µithe difference µi−piis appeared. The sought µ0is found from the smooth ˜ µ0asµ0i= ˜µ0i+pi. In Fig. 1 is shown an example of the atomic-like absorption approximation by the smoothing spline with and w ithout the use of prior function1. Energy E0was determined at half the step height. Here, we constructed p(E) in the following manner. Found the average value ¯µofµ(E) in region 20 ≤E≤70eV above the absorptance maximum. Moving from the beginni ng of spectrum, assign p=µuntilµ >¯µ, further p= ¯µ. Then p(E) was smoothed 5 times on 3 points. To perform the Fourier transform, χ(k)k2was brought into the uniform scale with δk= 0.03˚A−1and multiplied by a Kaiser-Bessel window with parameter A= 1.5. As seen, the use of p(E) has led to disappearance of the spurious peak on the absolut e value of FT at r∼0.5˚A. So far we have considered the atomic-like absorption µ0to be a smooth function with no peculiarities. However, in some spectra µ0itself has a fine structure [4,5] originating from resonance scattering within absorbing atom or from multi-electron transitions. If in these cases, based on the oretical calculations, experimental information, or empi rical considerations, one can nearly indicate the location of pec uliarities, their width and weight relatively to the step height, then one would readily construct the prior function p(E) and find the correct µ0. Instead of constant value above absorption edge, the prior function would have corres ponding valleys and/or peaks. Let us now define the criterion for determination of smoothin g parameter. An attempt to solve the problem was made in Ref. [6], where the requirement was proposed: HR−HN≥0.05HM, where HRis the average value of the weighted Fourier transform magnitude between 0 and 0.25 ˚A,HMis the maximum value in the transform magnitude between 1 and 5 ˚A,HNis the average value of the transform magnitude between 9 and 10˚A attributed to the noise. Obviously, that this criterion cannot pretend to the genera lity since depends on the weighting (op. cit., k3) and the relative contribution of noise and the first coordination sh ell into spectra. In the program viper we have proposed another approach to the problem based on the consideration of heights of FT peaks as functions of regularizer α(see Fig. 2). On increasing αfrom zero, µ0starts to deviate from the experimental absorption µ,ρ(r) is growing and then saturates, the peaks at larger rbeing saturated earlier. Clearly, thatαshould be determined by the first peak height since it is the la st to saturate. Define the start of saturation on the minimum of second derivative of the first peak squared w ith respect to ln α. Declare the value of αin the minimum to be optimal. It is seen that the increase of αfrom the optimal leads to unwanted rapid growth of ρat lowr. In the example in Fig. 1 the regularizer is optimized followi ng our new criterion. 1Here and hereafter for examples is used the spectrum at Bi L3absorption edge in Ba 0.6K0.4BiO3at 50 K recorded in transmission mode at D-21 line (XAS-13) of DCI (LURE,Orsay, France) at positron beam energy 1.85 GeV and the average current ∼250 mA. Energy step — 1eV, counting time — 1s. Energy resoluti on of the double-crystal Si [311] monochromator (detuned to reject 50% of the incident signal in order to mini mise harmonic contamination) with a 0.4 mm slit was about 2–3 eV at 13keV. 3Unfortunately, the method of smoothing spline does not incl ude any approach to the estimations of uncertainties in the µ0obtained, in contrast to the following two methods. B. Interpolation spline drawn through the varied knots The method was put forward in Ref. [7]. Nknots are equally spaced in kspace, through them an interpolation spline is drawn. The ordinates of the knots are varied to mini mizeρor|ρ−ρst|in the chosen low- rregion 0 ≤r≤r0, where ρstis the absolute value of the FT of a “standard” χst(k)·kw, calculated or experimental. The number of knots must not exceed the value Nmax= 2r0∆k/π+ 1, [8] where ∆ kis the krange of useful data. In the Ref. [7] was asserted that one need to know the “standard” χst(k)·kwmerely approximately since it used only to get an estimate of the leakage from the first shell to the region minimized. Th e strange thing is that having omitted the question on the accuracy of found knots (as we show below, rather poor), t he authors of the cited work made a fine comparison between several theoretical models for χ(k) calculations. In Fig. 1 is shown an example of the method application. Ordin ates of the 13 knots ( Nmax= 13.2) were varied to minimize the difference ρ−ρstat 0≤r≤1.05˚A. The function χst(k) was calculated using feff6 program [9] (as was pointed above, a crude estimate is sufficient, so details o mitted). In the minimized region the ρ(r) is somewhat better than that obtained by the previous method. However, a tk >15˚A−1one can distinguish the obviously wrong behavior of χ(k)·k2, and the first peak on ρ(r) becomes quite distorted. Consider now the problem of the accuracy of knot positions yj,j= 1, . . ., N in fitting ρ(r) toρst(r). As a figure of merit, the χ2-statistics appears: χ2=Nmax MM/summationdisplay m=1[ρ(rm)−ρst(rm)]2 σ2m, (5) where σmare the errors of ρ(rm). It can be shown (detailed analysis see in the next article [ 2]) that under the /G14 /G16 /G17/G13 /G13 /G14 /G16 /G1B/G13 /G13 /G14 /G17 /G15/G13 /G13 /G28 /G0F /G03 /G48 /G39/G13/G11 /G13/G13/G11 /G17/G13/G11 /G1B/G14/G11 /G15/G50 /G5B /G13 /G17 /G1B /G14 /G15 /G14 /G19 /G4E /G0F /G03 /G63/G10 /G14/G10 /G14/G13/G14 /G46 /G11 /G4E/G15 /G13 /G15 /G17 /G19 /G55 /G0F /G03 /G63/G13/G13/G11 /G18/G14/G14/G11 /G18 /G5F /G29 /G37 /G0B /G46 /G11 /G4E/G15/G0C/G5F/G14 /G16 /G17/G13 /G13 /G14 /G16 /G17/G18 /G13/G0B /G44 /G0C /G0B /G45 /G0C /G0B /G46 /G0C /G28 /G13/G55 /G13/G47 /G0B /G5C /G4D /G0C FIG. 3. Extraction of XAFS from the measured absorption usin g the interpolation spline through the knots with varied ordinates. On (c) the Fourier transforms are shown for sough tχ(k)·k2(solid), “standard” (dots), and obtained by the previous method (dashed). 4/G13 /G17 /G1B /G14 /G15 /G14 /G19 /G4E /G0F /G03 /G63/G10 /G14/G13/G11/G13/G13 /G13/G13/G11/G13/G13 /G17/G13/G11/G13/G13 /G1B/G13/G11/G13/G14 /G15/G13/G11/G13/G14 /G19/G13/G11/G13/G15 /G13/G48/G48 /G0B /G4E /G20 /G13 /G0C /G61 /G13 /G11 /G17 /G1B /G44 /G45 /G46/G51 /G33 /G51 /G4E/G48 /G0B /G31 /G0C /G61 /G13 /G11 /G13 /G19 FIG. 4. Errors of χ(k) extraction. Solid line with open circles — by the method of interpolation spline drawn through the varied knots. Dots — by the method of bayesian smoothing without ( a) and with ( b c) prior information specifying the second derivative. Besides, ( c) uses additional information thatµ0(E) passes through a point immediately be- foreE0. Solid line with filling — the envelope of χ(k) (not weighted). Dashed lines — the noise es- timates from FT ( nk) and from Poisson counting statistics ( nP) (see Sec. III). assumption of uncorrelated knot positions, the mean-squar e deviation of yjfrom the obtained through the fit optimal value ˆ yjequals δ(yj) = (1 2∂2χ2/∂y2 j)−1/2, where the partial derivatives are calculated in the fitting procedure at the minimum. σmare assumed to be constant and equal to the root-mean-square average of ρ(r) between 15 and 25 ˚A, where solely the noise is present. The errors εj=δ(yj)/[µ0(Ej)−µb(Ej)] found under such assumptions are shown in Fig. 4 as open circles with the solid line. Notice, that the ussumption that the knot positions are not correlated gives quite optimistic εj. Actually, several first knot positions appear to be highly c orrelated; the proper taking into account of the correlations (here we do not present these cal culations) raises εjat the least as twice. But even these underestimated εjare appreciably larger than those given by the following met hod. C. Bayesian smooth curve Ideologically similar to the smoothing spline method is the method of bayesian smoothing (see Appendix on p. 11) proposed in the program viper . This method also finds the regularized function µ0, the regularizer αis the measure of compromise between smoothness of µ0and its deviation from µ. In comparison with smoothing spline method, this method has some advantages. (i) Various prior informat ion on µ0can be considered. (ii) In this method the posterior distributions of all µ0jare sought for. From those distributions one can find not only average values but also any desirable momenta, which appears to be an additiona l difficulty for other methods. (iii) In the framework of the method it is possible also to deconvolute µwith the monochromator rocking curve. The weakness of the me thod is its low speed (comparing with method II A, not with II B!). O n a modern PC the curve drawn through N∼500 points is smoothed for a few minutes. In Fig. 5 the bayesian smoothing was done on the mesh of 536 exp erimental points above E0, without and with the prior function (its construction is described in Sec. II A). Besides, in the last case another information was used: the atomic-like absorption must coincide with the tot al absorption (minus pre-edge background) at energies E < E 0. Therefore, we demanded from the bayesian curve to pass thro ugh a point nearest (at left) to E0. The values ¯ µ0jandδ2(µ0j) were found by formulas (A31) and (A33). Since the smoothed v alues do not lie within the limits of ±δ(µ0j) from µj, we did not look for the most probable smoothness (see. Appen dix), instead we considered the regularizer to be known and equal to the optimal one found in the method II A. The introduction of the prior information has significantly diminished the errors of χ(k) extraction (see dotted curves in Fig. 4) which were defined asεj=δ(µ0j)/(¯µ0j−µbj). This is quite natural: any decrease of our ignorance about µ0should narrow the posterior distribution of µ0jfor all j. Of course, this concerns the experimental information as w ell: errors εjare the less the more measured points Nthe spectrum has. Comparing Fig. 1() and Fig. 5(c), it is seen practically perfect coincidence of the results of bayesian smoothing and smoothing spline. F rom this one can assume the equality of the errors which both methods give. Could we take into account possible systematic errors in the framework of the method? Yes, if we have the information on their nature and are able to translate it into the mathematics language; such a translation might be rather non-trivial. In any case, now we have the tool to extra ct from the prior and experimental information not only the sought values but their errors as well. 5/G14 /G16 /G17/G13 /G13 /G14 /G16 /G1B/G13 /G13 /G14 /G17 /G15/G13 /G13 /G28 /G0F /G03 /G48 /G39/G13/G11 /G13/G13/G11 /G17/G13/G11 /G1B/G14/G11 /G15/G50 /G5B/G14 /G16 /G17/G13 /G13 /G14 /G16 /G17/G18 /G13 /G13 /G17 /G1B /G14 /G15 /G14 /G19 /G4E /G0F /G03 /G63/G10 /G14/G10 /G14/G13/G14 /G46 /G11 /G4E/G15 /G13 /G15 /G17 /G19 /G55 /G0F /G03 /G63/G13/G13/G11 /G18/G14/G14/G11 /G18 /G5F /G29 /G37 /G0B /G46 /G11 /G4E/G15/G0C/G5F/G0B /G44 /G0C /G0B /G45 /G0C /G0B /G46 /G0C /G28 /G13 FIG. 5. Extraction of XAFS from the measured absorption usin g the bayesian smoothing. Prior function p(E) for the atomic-like absorption is drawn by dots. Solid lines — µ0(E),χ(k)·k2, and ρ(r) obtained with the use of the prior function; dashed lines — dittos without prior function. The dot line on (c) is obtained without additional requirement for µ0(E) to pass through a point immediately before E0. The regularizer αis the same for all cases and equals to the optimal one found fo r the smoothing spline. D. Other methods Consider briefly the methods for µ0construction not included into the viper program. A rich variety of computer programs for XAS spectra processi ng is collected on the International XAFS Society Web-site [10]. The vast majority of them use as an approximat ion for the atomic-like absorption a smoothing spline or more general piecewise-polynomial representation. For example, in the method of Ref. [11], the construction of µ0is divided into several stages: µ0is approximated by a low-degree polynomial, obtained χ(k) is multiplied by kw, additional µ′ 0is drawn again as a low-degree polynomial and subtracted, a s moothing spline then approximates one more additional µ′′ 0. The sum of all µ0’s gives the total atomic-like background. The necessity of the preliminary stages was not discussed op. cit., however, clearly it was ca used by the instability of spline with respect to the small variations of input parameters. And the point is not that the preliminary stages make the process stable, but that for each specific spectrum, auxiliary parameters (degrees o f polynomials) could provide an acceptable construction of the atomic-like background. Above (in Sec. II A) we propos ed the way to rise the stability of spline making the preliminary stages to be redundant. In Ref. [12] an iterative approach to “atomic background” re moval was developed. First a spline is used to obtain a rough estimate of the background; this alone is enough to ha ve a reliable χatk >5−6˚A−1. Over that range the χ obtained is fitted to the theoretical χthinr-space. The resulting fit parameters are used then to generat eχth(k) that extends down to low k. This function is transformed back into e-space and µ0is obtained as µ0=µ/(χth+ 1) that need be a little smoothed or fitted by an additional spline. Si nce the logic of reasoning was inversed: not “find µ0 to find χ,” but “find χto find µ0,” the method is suited for the quest of peculiarities on µ0curve, not for structural XAFS-researches. Besides, the range of accuracy of the mode l appears to be unknown in principle: all, that is not described by the model, is included in µ0; the errors of the background approximation are also undefin ed. In the old work [13] for the determination of the background a bsorption µ0was considered the damping of the XAFS amplitude resulting from the measurements with low res olutions (with a large slit width). The superimposition 6/G13 /G15 /G17 /G19/G55 /G0F /G03 /G63/G13/G13/G11 /G17/G13/G11 /G1B/G14/G11 /G15/G14/G11 /G19/G5F /G29 /G37 /G0B /G49 /G03 /G0A/G0A /G11 /G4E/G15/G0C /G12 /G0B/G15 /G4C/G55 /G0C/G15/G5F/G49 /G03 /G20 /G50 /G0B /G4E /G0C /G49 /G03 /G20 /G50 /G13 /G0B /G4E /G0C FIG. 6. On the method of Ref. [15]. Solid curve is the FT of “unnormalized χ(k)·k2”, dotted curve is the contribution from the atomic-like absorption. of two spectra measured with different energy resolutions gi ves the intersection points, the part of which belong to theµ0. Then through the obtained nodal points a smoothing spline i s drawn. As the authors of Ref. [13] noted, the measurements of the spectra with worsened resolution ar e not necessary; the spectra could be damped by the convolution with a “rocking curve,” approximated by a Gauss ian function. Of course, the method is correctly works only with a small variation of the Gaussian curve width since for the large width not only the XAFS amplitude is damped but the very edge is washed out. Because of this only th e extended part of a spectrum could be reliably determined. The damping of the XAFS amplitude can be due to other reasons. For instance, as was pointed in Ref. [13], the nodal points may be obtained from the variable-temperat ure study. This idea was realized in Ref. [14] and is more sound since the atomic-like background is really indep endent of temperature and with temperature the XAFS amplitude is changed, not the shape of the edge. But for all th at it is important that the phase difference between XAFS of different temperatures was negligible, which is true only for low wave numbers. Unfortunately, the method is suitable only for some particular cases (to say nothing of need for measured temperature series of spectra). Op. cit. it was demonstrated for the x-ray-absorption data for t heL3edge of solid Pb. In those spectra the first crossing ofµandµ0occurs already at ∼15eV above edge. In our sample spectra the first crossing occu rs only at ∼30eV, which allows one to find at most 2–3 points and the first of them b eing situated at k/greaterorsimilar2.5˚A−1. An interesting approach to the problem of µ0determination was reported in Ref. [15]. It is based on the si mple identity that relates the FT of some function with the FT of it sn-th derivative: FT[f(n)(k)] = (2 ir)nFT[f(k)], (6) where the conjugate variables are kand 2r. Since the atomic-like background is smooth enough, the hig her derivatives µ(n)(k) (n≥2) are oscillatory near zero. Performing the FT of µ(n)(k)·kwand using Eq. (6), one obtains the FT of unnormalized would-be χ(k)·kw(see Fig. 6). Op. cit. the low- rpart (which in our example is 0 ≤r/lessorsimilar1.1) was cut off, and then the back FT was done. As a result, one has the un normalized χ(k)·kwand, having subtracted it from the µ(k), the atomic-like background on which some peculiarities d ue to multi-electron excitations can be distinguished. Like the method of Ref. [12], this method is s uited for the quest of peculiarities on the µ0curve, not for structural XAFS-researches because of evident distort ion of the first peak on the FT by the contribution from the atomic-like background. To illustrate this assertion, in F ig. 6 we show the FT of the second derivative of the µ0(k) that was found by the present method. As seen, this contribut ion is not as small. If the electronic states of an absorbing atom in gaseous phas e and in the compound of interest may be considered as equivalent, µ0can be set equal to the measured absorption in gas, as was done in Ref [16] for solid, liquid, and gaseous Kr. Some differences in energy positions and relative weight s of double-electron excitation channels were taken into account by a model using simple empirical functions which we re transferred then to the spectra of liquid and solid Kr. Notice that the proposed in the present paper prior funct ion for the methods of smoothing spline and bayesian smoothing can include additional items corresponding to th e multi-electron contributions. III. ERRORS IN µ0CONSTRUCTING, NOISE, AND CHOICE OF LIMITS kminkmax For what we need to know the errors of XAFS-function extracti on? First, without knowing of these values one cannot in principle aim at their minimization. Second, they are used in the definition of χ2-statistics in the fitting 7problems; their underestimation is a source of unjustified o ptimistic errors of fitting parameters. Third, along with analysis of the noise, the errors of µ0construction allow us to choose the limits of reliable EXAFS signal, kminand kmax. Unfortunately, the issue of quality of XAFS extraction from the measured absorption has not been addressed properly. We see several reasons for that. On the one hand, no t having a correctly developed approach to the estimation of the errors of final results (interatomic dista nces, Debye-Waller factors etc. found via fitting), the erro rs of EXAFS extraction are useful. On the other hand, only a few m ethods include approaches to their estimations. Easily one can compare the errors of different methods (see Fi g. 4) and then choose the most reliable one. The problem of plausible limitations on the absolute value of th e errors is more difficult. Define “signal” as the envelope ofχ(k) (solid line with gray filling in Fig. 4). It is quite reasonab le to demand that the errors of µ0construction were less than XAFS signal. For the method of the interpolati on spline drawn through the varied knots to meet this requirement leads to the restriction on the photoelectron w ave numbers: 2 /lessorsimilark/lessorsimilar14˚A−1. For the bayesian curve a this range is 0 ≤k/lessorsimilar14˚A−1, for the bayesian curves bandcthis range is wider: 0 ≤k/lessorsimilar16˚A−1. Another factor that limitates the spectrum length is the pre sence of noise. To determine the noise is a straightforward task for r-space, where XAFS signals at high rhave clearly noise character. By Parseval’s identity the no ise in r-space is related with the noise in k-space [17]: /integraldisplaykmax kmin|nkkw|2dk= 2/integraldisplayπ/2dk 0|nr|2dr. (7) Substitute the mean value over the range 15 < r < 25˚A of the FT magnitude squared for |nr|2. Then n2 k=∝angbracketleft|n2 r|∝angbracketrightπ dk2w+ 1 k2w+1max−k2w+1 min. (8) As seen from the formula, nkdepends on dk, the size of evenly-spaced k-grid. Although above we already have used the Fourier transform, the question of choice of dkwas not raised yet. The algorithm of fast FT needs the transformed function to be set on a uniform grid. Having chos en a small dk, we artificially obtain the large number of “experimental” values. Naturally, this trick would not g ive more information than we have, and the errors nk must be large at the small dk. In our example the choice of dk(0.03˚A−1) was based on the equality of numbers of experimental points and the nodes of the grid. The signal-to -noise ratio obtained is greater than unity for all the spectrum (see Fig. 4). There was no doubt in that: the signal i s visually distinguished even for the very extended end of the spectrum (see Fig. 1(b) and Fig. 5(b)). The noise can be estimated based on the bayesian considerati ons [18]. Let after measurements we have the values of counts from the solid-state or gas-filled detectors and le t there is a positive real number λsuch that the probability that a single count occurs in the time interval dtis P(1|λ) =λdt. (9) It can be shown [19] that merely from this assumption follows that the counts obey the Poisson distribution law: P(N|λ, T) =(λT)Nexp(−λT) N!, (10) where Tis the sampling time. The problem is to find the intensity λand its variance. Using Bayes theorem and introducing prior probabilities P(N) = 1/NandP(λ) = 1/λ[20], one obtains: P(λ|N, T) =P(N|λ, T)P(λ) P(N)=T(λT)N−1exp(−λT) (N−1)!, (11) that is after measurement the variate 2 Tλfollows the χ2-distribution with 2 Ndegrees of freedom. It is easy to find that¯λ=N/T,λ2=N(N+ 1)/T2, and the variance of intensity is δλ=√ N/T. Denote counts from detectors measuring i0andi1asI0andI1. By definition the variate ξ=i0/2I0 i1/2I1follows Fisher’s F-distribution with (2 I0,2I1) degrees of freedom. Its expected value and variance are kno wn:¯ξ=I1/(I1−1), δ2ξ=I2 1(I0+I1−1)/((I1−1)2(I1−2)I0), from where we find for the absorption in the fluorescence mod e (µx=i0/i1): i0/i1=I0 I1−1, δ2(i0/i1) =I0(I0+I1−1) (I1−1)2(I1−2). (12) 8Further, the variate η=1 2lnξfollows z-distribution (Fisher’s distribution of variance ratio) w ith (2I0,2I1) degrees of freedom. Its expected value and variance are known: ¯ η= 0,δ2η=1 4(I1+I1)/(I0I1), from where we find for the absorption in the transmission mode ( µx= ln(i0/i1)): ln(i0/i1) = lnI0 I1, δ2ln(i0/i1) =1 I0+1 I1. (13) The noise of XAFS-function is nP=δµ µ0−µb=/parenleftbigg1 I0+1 I1/parenrightbigg1/21 µ0−µb. (14) In our example this noise at k/greaterorsimilar15˚A−1becomes greater than signal (see Fig. 4). What is the reason f or such significant difference between the really present noise nkand its statistical estimate nP? Of course, the reason is in the false premise (9). In practice this condition is realize d as: P(c|λ) =λdt. For example, the photocurrent in an ion-chamber depends on gas pressure, potential applied etc .; these dependencies are contained in c. In other words, the amplification path works in such a way that one photon give s birth to ccounts. There is no difficulty in writing the posterior distribution for the generalized premise: P(λ|N, T) =T(λT)N c−1exp(−λT) Γ(N/c), (15) with¯λ=N/(cT) and δλ=/radicalbig N/c/T . Thus, having unknown c(and implicitly assigning c= 1), we got wrong variances for i0/i1and ln( i0/i1). Unfortunately, in the most of real experiments the associ ation between the probability of a single count event and the radiation intensity (via c) is unknown. In spite of this, the Poisson counting statisti cs is traditionally used for a long time. For example, in Ref. [2 1] signal-to-noise ratios are evaluated (assuming c= 1) for the different detection schemes. Practically all programs for XAFS spectra processing [10] t o estimate the noise use the Fourier analysis. But then it is the noise that they use as uncertainties εiofχ(k) determination in definition of χ2-statistics: χ2=Nmax MM/summationdisplay i=1[(χexp)i−(χmod)i]2 ε2 i. (16) It would be more correct to consider as εithelarger from the two: the noise and the errors of the construction of µ0. In our case (and as a rule) the latter are essentially greate r (especially in the method II B) than the noise. In the following paper [2] we shall show how the understated εilead to optimistic errors of structural parameters. IV. XAFS-FUNCTION CORRECTION Because of one reason or another the experimental XAFS might be distorted. Consider some of them. (i) Let the counts ( I) from detectors are associated with the intensities ( i) asi0=κ0I0andi1=κ1I1. Then the absorption (in the transmission mode) equals: µx= ln(i0/i1) = ln( I0/I1) + ln( κ0/κ1). (17) The second term is a slightly varied function of energy and ca n be taken into account in independent experiments. Such a distortion appears, for instance, if the absorptance of the gas in ion-chamber detectors depends on energy. (ii) If some part of incident radiation is not attenuated in t he sample as much as expected (due to the pinholes in the sample, harmonics in the incoming beam etc.), that is i0=κ0I0+b, then the real absorption is connected with the measured I0andI1in a complicated way. In Ref. [22] the possible decrease of XA FS amplitude shown to be essential even at low b/(κ0I0) but thick samples. At known ratio b/(κ0I0), the correcting factor can be easily obtained. (iii) In the fluorescence mode, due to absorptance of the fluor escent signal in the sample itself XAFS spectra strongly depend on the detection geometry. In Ref. [23] the correctin g functions are found explicitly. (iv) The problem of glitches is widespread in the XAFS analys is. The glitches are due to multiple Bragg reflection being satisfied simultaneously and for each given monochrom ator are manifested in the strictly determined spectral positions. In most cases the glitches seen on curves I0(E) and I1(E), vanish on I0/I1ratio. If not, one can easily 9/G13 /G17 /G1B /G14 /G15 /G14 /G19 /G4E /G0F /G03 /G63/G10 /G14/G10 /G14/G13/G14/G46 /G11 /G4E/G15/G16 /G11 /G18/G17 /G11 /G13/G17 /G11 /G18/G18 /G11 /G13/G46 /G52/G58/G51/G57 /G56 /G11 /G14/G13/G18/G14/G16/G17/G13 /G13 /G14/G16/G1B/G13 /G13 /G14/G17/G15/G13 /G13/G28 /G0F /G03 /G48 /G39 /G14/G15/G16 /G46 /G52/G58 /G51 /G57 /G56 /G11 /G14/G13/G18 /G0B /G4F /G51 /G2C /G13 /G0C /G0A/G46 /G20 /G14 /G11 /G1A /G1A /G48 /G10 /G16 /G03/G48 /G39/G10 /G14/GA2 /G0B /G4F /G51 /G2C /G13 /G0C /G0A /GB2 /G20 /G1C /G11 /G17 /G13 /G48 /G10 /G17 /G03/G48 /G39/G10 /G14 /G44 /G45 /G46 /G47/G44 /G45 /G46/G47 /G46 /G52 /G55 /G55 /G48 /G46 /G57 /G48 /G47FIG. 7. Energy dependence of experimen- tal counts from ion-chambers. Curve I0(E) relates to the left axis, I1(E) — to the right one. In glitch areas the absolute value of the derivative is greater than the critical level specified. On χ(k)·k2only the glitch b is manifested. The displaced fragment is χ(k)·k2after correction. get rid of them. For instance, the glitch area, usually extre mely thin, is smoothed or, with fixed ends, replaced by a straight-line segment. The main thing in the correct analys is of glitches is their detection. To detect a glitch on curves µorχkwis practically impossible. For this, one needs the primary d ataI0(E) and I1(E), not ln( I0/I1) norI0/I1. Out of glitches the intensity of incident radiation smooth ly, ignoring the noise, depends on energy (see Fig. 7). The idea of detection of glitches via c ritical level for the derivative |dlnI0/dE|cis self-evident. For the presented in Fig. 7 I0(E) curve the absolute value of the derivative in the glitch are as is greater than the critical value chosen to be equal to 1 .77·10−3−1. Having extracted the XAFS, one can see that the first (paired ) glitch ais not manifested on χ(k)·k2, the last two ( c d) are obscured by the noise, solely glitch bis clearly pronounced. Now, being in the firm belief that this is not a part of the XAFS, one can eliminate the glitch with ease. Here, we fixed its ends on µ(E), replaced it by the straight-line segment, and constructe dχ(k)·k2again. V. CONCLUSION In this paper we have considered all stages of XAFS function e xtraction from the measured absorption. We focused our attention on the most important stage, construction of t he atomic-like absorption µ0. For the wide-spread method of approximation of µ0by a smoothing spline we have proposed the way to raise the stability by including the prior information about absorpt ion edge shape (“nearly step” or “nearly step with a white line”). Besides we have propose a new reliable criterion for determination of the smoothing regularizer. A new method for approximation of µ0is proposed, the method of bayesian smoothing. It can includ e various prior information, which raises the accuracy of XAFS determinati on. Following this method one finds the distributions of µ0in each experimental point, from which one can find not only av erage values but also any desirable momenta, which appears to be an additional difficulty for other methods. This method was shown to give more accurate atomic-like background than that obtained by the method of Ref [7]. Particular attention has been given to the analysis of noise . We have discussed the difficulties of its estimates on the basis of statistical approach. More reliable is the determi nation of noise from the Fourier transform. We have shown that the experimental noise is essentially less than the err ors of µ0construction, and the use of values of noise in the χ2-statistics definition appears to be erroneous since leads t o the unjustified optimistic errors of structural parameter s inferred in fitting procedures. For detailed consideration of the accuracy of fitting parameters see the following paper [2]. 10APPENDIX: BAYESIAN SMOOTHING AND DECONVOLUTION 1. Posterior distribution for smoothed data Consider general linear problem of data smoothing with the u se of statistical methods (for introduction see review by Turchin et al.[24] and the articles from Web-site bayes.wustl.edu ). Let data dare defined on the mesh x1, . . . , x N and consist of the true values tand the additive noise n: di=ti+ni, i = 1, . . ., N. (A1) The problem of smoothing is to find the best estimates of t. For an arbitrary node j, find the probability density function for tjgiven the data d: P(tj|d) =/integraldisplay ···dti/negationslash=j···P(t|d), (A2) where P(t|d) is the joint probability density function for all values t, and the integration is done over all ti/negationslash=j. According to Bayes theorem, P(t|d) =P(d|t)P(t) P(d), (A3) P(t) being the joint prior probability for all ti,P(d) is a normalization constant. Assuming that the values niare independent in different nodes and normally distributed wit h zero expected values, the probability P(d|t), so-called likelihood function, is given by P(d|t, σ) = (2 πσ2)−N/2exp/parenleftBig −1 2σ2N/summationdisplay k=1(dk−tk)2/parenrightBig , (A4) where the standard deviation of the noise, σ, appears as a known value. Later, we apply the rules of probab ility theory to remove σfrom the problem. Now define prior probability P(t). Let we know in advance that the function t(x) is smooth enough. To specify this information, introduce the norm of the second derivati ve and indicate its expected approximate value: Ω(t(x)) =/integraldisplay/parenleftbiggd2t dx2/parenrightbigg2 dx≈ω. (A5) Denote ∆ i=xi+1−xi,i= 1, . . ., N −1 and represent the second derivative in the finit-difference form: Ω(t(x))≡Ω(t) =N−1/summationdisplay i=2[ti−1∆−1 i−1−ti(∆−1 i−1+ ∆−1 i) +ti+1∆−1 i]2≡N/summationdisplay k,l=1Ωkltktl. (A6) Ωklis a five-diagonal symmetric matrix with the following non-z ero elements: Ω11= ∆−1 1∆−2 2,Ω22= ∆−1 2(∆−1 1+ ∆−1 2)2+ ∆−2 2∆−1 3,Ω12=−(∆1∆2)−1(∆−1 1+ ∆−1 2), (A7) ············ Ωii= ∆−1 i(∆−1 i+ ∆−1 i−1)2+ ∆−2 i∆−1 i+1+ ∆−3 i−1, Ωi−1,i=−∆−2 i−1(∆−1 i−1+ ∆−1 i−2)−(∆i−1∆i)−1(∆−1 i−1+ ∆−1 i), Ωi−2,i= ∆−1 i−2∆−2 i−1, ············ ΩNN= ∆−3 N−1,ΩN−1,N−1= ∆−1 N−1(∆−1 N−1+ ∆−1 N−2)2+ ∆−3 N−2,ΩN−1,N=−∆−2 N−1(∆−1 N−1+ ∆−1 N−2). In order to introduce the minimum information in addition to that contained in (A6), from all normalized to unity functions P(t) which satisfy the condition (A6) we choose a single one that contains minimum information about t that is minimizes the functional 11I[P(t)] =/integraldisplay P(t)lnP(t)dt+β/bracketleftBig 1−/integraldisplay P(t)dt/bracketrightBig +γ/bracketleftBig ω−/integraldisplay Ω(t)P(t)dt/bracketrightBig , (A8) where βandγare the Larrange multipliers. In minimizing I[P(t)], one obtains the equation set lnP(t) + 1−β−γΩ(t) = 0 (A9)/integraldisplay P(t)dt= 1 /integraldisplay Ω(t)P(t)dt=ω, that has a solution: P(t) = (λ1···λN)−1/2/parenleftBig2πσ2 α/parenrightBig−N/2 exp/parenleftBig −α 2σ2Ω(t)/parenrightBig , (A10) where α/2σ2=γ=N/2ω, and λ1, . . . , λ Nare the eigenvalues of the matrix Ω kl. The regularizer αwill be used to control the smoothness of t. The prior distribution obtained is a “soft” one, that is doe s not demand from the solution to have a strictly prescribed form. Thus, we have for the probability density function: P(tj|d, σ, α)∝/integraldisplay ···dti/negationslash=j···σ−2NαN/2exp/parenleftBig −α 2σ2N/summationdisplay k,l=1Ωkltktl/parenrightBig exp/parenleftBig −1 2σ2N/summationdisplay k=1(dk−tk)2/parenrightBig =/integraldisplay ···dti/negationslash=j···σ−2NαN/2exp/parenleftBig −1 2σ2/bracketleftbig d2−2N/summationdisplay k=1dktk+N/summationdisplay k,l=1gkltktl/bracketrightbig/parenrightBig , (A11) where gkl=αΩkl+δkl,d2=N/summationdisplay k=1d2 k. (A12) Since there is no integral over tj, separate it from the other integration variables: P(tj|d, σ, α)∝σ−2NαN/2exp/parenleftBig −1 2σ2[d2−2djtj+gjjt2 j]/parenrightBig ×/integraldisplay ···dti/negationslash=j···exp/parenleftBig −1 2σ2/bracketleftbigN/summationdisplayj k,l=1gkltktl−2N/summationdisplayj k=1[dk−gkjtj]tk/bracketrightbig/parenrightBig , (A13) Here, the symbol jnear the summation signs denotes the absence of j-th item. Further, find the eigenvalues λ′ i and corresponding eigenvectors eiof the matrix gklin which the j-th row and column are deleted, and change the variables: bi=/radicalbig λ′ iN/summationdisplayj k=1tkeik, t k=N/summationdisplayj i=1bieik/radicalbig λ′ i(i, k∝negationslash=j). (A14) Using the properties of eigenvectors: N/summationdisplayj k=1glkeik=λ′ ieil,N/summationdisplayj k=1elkeik=δli (l, i∝negationslash=j), (A15) one obtains: P(tj|d, σ, α)∝σ−2NαN/2exp/parenleftBig −1 2σ2[(d2−h2)−2tj(dj−hu) +t2 j(gjj−u2)]/parenrightBig ×/integraldisplay ···dbl/negationslash=j···exp/parenleftBig −1 2σ2N/summationdisplayj i=1[bi−hi+uitj]2/parenrightBig , (A16) 12where new quantities were introduced: hi=1/radicalbig λ′ iN/summationdisplayj k=1dkeik, u i=1/radicalbig λ′ iN/summationdisplayj k=1gkjeik, h2=N/summationdisplayj i=1h2 i,u2=N/summationdisplayj i=1u2 i,hu=N/summationdisplayj i=1hiui. (A17) Evaluating the N−1 integrals in (A16), one finally obtains the posterior proba bility for j-th node: P(tj|d, σ, α)∝σ−(N+1)αN/2exp/parenleftBig −1 2σ2[(d2−h2)−2tj(dj−hu) +t2 j(gjj−u2)]/parenrightBig . (A18) 2. Eliminating nuisance parameters In most real problems σandαare not known. To eliminate σis a quite straightforward problem: P(tj|d, α) =/integraldisplay dσP(tj, σ|d, α) =/integraldisplay dσP(σ)P(tj|d, σ, α), (A19) one needs only to know a prior probability P(σ). Having no specific information about σ, a Jeffreys prior P(σ) = 1/σ is assigned [20]. Then P(tj|d, α)∝/integraldisplay∞ 0dσσ−(N+2)exp/parenleftBig −1 2σ2[(d2−h2)−2tj(dj−hu) +t2 j(gjj−u2)]/parenrightBig (A20) ∝[(d2−h2)−2tj(dj−hu) +t2 j(gjj−u2)]−(N+1)/2. Introducing the substitution w2 j=N(gjj−u2)2 (d2−h2)(gjj−u2)−(dj−hu)2/parenleftbigg tj−dj−hu gjj−u2/parenrightbigg2 , (A21) one obtains the Student t-distribution with Ndegrees of freedom: P(wj|d, α)∝/parenleftBig 1 +w2 j N/parenrightBig−(N+1)/2 (A22) with zero average and the variance N/(N−2). From where one finds for tj: ¯tj=dj−hu gjj−u2, δ2(tj) =(d2−h2)(gjj−u2)−(dj−hu)2 (gjj−u2)21 N−2. (A23) Thus, we have got rid of unknown σand found the expressions for mean values tjand their dispersions at known regularizer α. To eliminate the latter is more difficult. The idea is not to fin d the smoothest solution, but the solution of the most probable smoothness. For that we will find the post erior probability: P(α|d) =/integraldisplay dtdσP(α, σ,t|d) =/integraldisplay dtdσP(α, σ)P(t|α, σ,d). (A24) Assuming that αandσare independent and using Bayes theorem (A3), one obtains: P(α|d)∝/integraldisplay dtdσP(α)P(σ)P(t|α, σ)P(d|t, α, σ). (A25) Substituting (A10) for the prior probability P(t|α, σ), (A4) for the likelihood, and a Jeffreys prior P(σ) = 1/σand P(α) = 1/α, one obtains the posterior distribution for the regularize rα: 13P(α|d)∝/integraldisplay dtdσσ−2N−1αN/2−1exp/parenleftBig −α 2σ2N/summationdisplay k,l=1Ωkltktl/parenrightBig exp/parenleftBig −1 2σ2N/summationdisplay k=1(dk−tk)2/parenrightBig =/integraldisplay dtdσσ−2N−1αN/2−1exp/parenleftBig −1 2σ2/bracketleftbig d2−2N/summationdisplay k=1dktk+N/summationdisplay k,l=1gkltktl/bracketrightbig/parenrightBig , (A26) where matrix gklwas defined in (A12). After its diagonalization, analogousl y to what was done above, finally one obtains: P(α|d)∝(λ′ 1···λ′ N)−1/2αN/2−1[d2−h2]−N/2, (A27) where h2is given by h2=N/summationdisplay i=1h2 i, h i=1/radicalbig λ′ iN/summationdisplay k=1dkeik, (A28) andλ′ iandeiare the eigenvalues and eigenvectors of gkl. Having found the maximum of the posterior probability (A27) or having averaged over it the expression (A23), one ha s the sought twith the most probable smoothness. However it is necessary to point out that this procedure narr ows the applicability of the bayesian smoothing down to the class of tasks where the smoothed values lie in most withi n the limits ±σfrom the most probable. In practice, there possible other tasks where the condition (A1) is treat ed more wider and the smoothed values exceed the bounds of noise. 3. Expressions for smoothed values and their variances The formulas (A23) appear useless in practice since require to find the eigenvalues and eigenvectors for the matrix of rank N−1 on each node. Those formulas have merely a methodological v alue: the explicit expressions for posterior probabilities enable one to find the average of arbitrary function of tj. However, ¯tjandδ2(tj) could be found significantly easier. Using (A19) and (A11), represent ¯tjas: P(σ)P(tj|d, σ, α)dtj ∝/integraldisplay dtdσσ−2N−1tjexp/parenleftBig −1 2σ2/bracketleftbig d2−2N/summationdisplay k=1dktk+N/summationdisplay k,l=1gkltktl/bracketrightbig/parenrightBig . (A29) Performing the diagonalization, one obtains: ¯tj∝/integraldisplay dbdσσ−2N−1exp/parenleftBig −1 2σ2[d2−h2]/parenrightBig/parenleftBigN/summationdisplay i=1bieij/radicalbig λ′ i/parenrightBig exp/parenleftBig −1 2σ2N/summationdisplay i=1[bi−hi]2/parenrightBig ∝N/summationdisplay i=1hieij/radicalbig λ′ i/integraldisplay dσσ−N−1exp/parenleftBig −1 2σ2[d2−h2]/parenrightBig , (A30) from where ¯tj=N/summationdisplay i=1hieij/radicalbig λ′ i. (A31) Analogously, for the variance δ(tj) one has: δ2(tj)∝/integraldisplay dbdσσ−2N−1exp/parenleftBig −1 2σ2[d2−h2]/parenrightBig/parenleftBigN/summationdisplay i=1(bi−hi)eij/radicalbig λ′ i/parenrightBig2 exp/parenleftBig −1 2σ2N/summationdisplay i=1[bi−hi]2/parenrightBig ∝N/summationdisplay i=1e2 ij λ′ i/integraldisplay dσσ−N−1exp/parenleftBig −1 2σ2[d2−h2]/parenrightBig σ2. (A32) 14Normalizing, one finally obtains: δ2(tj) =/integraltext dσσ−N+1exp/parenleftbig −[d2−h2]/2σ2/parenrightbig /integraltext dσσ−N−1exp (−[d2−h2]/2σ2)N/summationdisplay i=1e2 ij λ′ i =Γ/parenleftbigN 2−1/parenrightbig ([d2−h2]/2)N/2−1/parenleftbig [d2−h2]/2/parenrightbigN/2 Γ/parenleftbigN 2/parenrightbigN/summationdisplay i=1e2 ij λ′ i=[d2−h2] N−2N/summationdisplay i=1e2 ij λ′ i. (A33) Now we got the usable formulas, which require to find the eigen values and eigenvectors for the matrix of rank Njust one time . 4. Addenda to the bayesian smoothing (i) Let the curvature of the function t(x) is approximately known in advance. To specify this informa tion, introduce the norm of the difference between d2t/dx2and approximately known second derivative d2f/dx2: Ω(t(x)) =/integraldisplay/parenleftbiggd2t dx2−d2f dx2/parenrightbigg2 dx≈ω. (A34) Notice, that there is no need to know f(x) itself, its second derivative is sufficient. The explicit pr esence of f(x) in the following formulas should be taken as a consequence of th e technical trick applied: at first f(x) is subtracted from the data, then it is added to the found solution. Everywhere in formulas (A6–A33) make the substitutions: ˜ti=ti−fi, ˜di=di−fi, i = 1, . . ., N. (A35) Performing the described above procedure for smoothing, on e finds ˜ti, from which by inverse transformation the sought vector is given by t=˜t+f. (ii) In some tasks the value on the starting (zero) node is kno wn without measurement. This sort of prior information represents a “hard” one, that is it restricted the class of po ssible solutions; in the given case the solution must pass through the known zero node. The quadratic form Ω( t) (or Ω( ˜t) in the case of approximately known second derivative) in the expression for the prior probability has changed: Ω(t) =N−1/summationdisplay i=1[ti−1∆−1 i−1−ti(∆−1 i−1+ ∆−1 i) +ti+1∆−1 i]2≡N/summationdisplay k,l=1Ωkltktl+ Ω00t2 0+ 2Ω 01t0t1+ 2Ω 02t0t2, (A36) the first few matrix elements of Ω klnow are: Ω00= ∆−2 0∆−1 1,Ω01=−(∆0∆1)−1(∆−1 0+ ∆−1 1),Ω02= ∆−1 0∆−2 1, (A37) Ω11= ∆−1 1(∆−1 0+ ∆−1 1)2+ ∆−1 1∆−2 2,Ω12=−∆−2 1(∆−1 1+ ∆−1 0)−(∆1∆2)−1(∆−1 1+ ∆−1 2). Ift0= 0 (or ˜t0= 0), none further changes to the formulas of smoothing (A6–A 33) are needed; at t0∝negationslash= 0 the changes are evident: instead of the scalar product dtin (A11) will be ( d−ˆd)t, where ˆd1=αt0Ω01,ˆd2=αt0Ω02, all remaining ˆdi= 0; to the d2the term αt2 0Ω00will be added. (iii) Making some changes in the considered above problem of smoothing allows one to solve the problem of deconvolution. If the experimental value djon some node jis determined not only by tjbut also by the values of some neighboring nodes, then instead of (A1) we have: di=N/summationdisplay j=1rijtj+ni, i = 1, . . ., N, (A38) where rijis the grid representation of the impulse response function . Instead of expression (A4), for the likelihood now we have: P(d|t, σ) = (2 πσ2)−N/2exp/parenleftBig −1 2σ2N/summationdisplay k=1/bracketleftBig dk−N/summationdisplay i=1rikti/bracketrightBig2/parenrightBig , (A39) 15and instead of (A11), the posterior probability for tjis now expressed as: P(tj|d, σ, α)∝/integraldisplay ···dti/negationslash=j···σ−2NαN/2exp/parenleftBig −1 2σ2/bracketleftbig d2−2N/summationdisplay k=1Dktk+N/summationdisplay k,l=1Gkltktl/bracketrightbig/parenrightBig , (A40) where Gkl=αΩkl+N/summationdisplay i=1rikril, D k=N/summationdisplay i=1rikdi. 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arXiv:physics/0003087v1 [physics.data-an] 28 Mar 2000XAFS spectroscopy. II. Statistical evaluations in the fitti ng problems K. V. Klementev Moscow State Engineering Physics Institute, 115409 Kashrs koe sh. 31, Moscow, Russia e-mail: klmn@htsc.mephi.ru (December 2, 2013) The problem of error analysis is addressed in stages beginni ng with the case of uncorrelated parameters and proceeding to the Bayesian problem that take s into account all possible correla- tions when a great deal of prior information about the access ible parametr space is available. The formulas for the standard deviations and deviations with ar bitrary confidence levels are derived. Underestimation of the errors of XAFS-function extraction is shown to be a source of unjustified optimistic errors of fitting parameters. The applications o f statistical χ2- and F-tests to the fitting problems are also discussed. 61.10.Ht I. INTRODUCTION In the Open Letter to the XAFS Community [1] Young and Dent, th e leaders of the UK XAFS User Group, expressed their concern over the persistence of lingering c ommon opinion that XAFS is a “sporting technique” and it is possible to obtain the “answer you want”. Some way out they see in a special attention to the publishing XAFS data (first of all, to XAFS spectra) and have formulated several re commendations for editors and referees. Undoubtedly, in the matter of extraction of the real, not invented, inform ation from XAFS experiments the quality of spectra is of great importance. We see here another problem as well. Not having some necessary elements of XAFS analysis (some values and the procedures for their determination), o ne has a quite natural desire to turn those values to advantage. Principally we mean the inability of the standar d methods to find the errors of the atomic-like background µ0. Traditionally, the noise is assigned to these errors. Howe ver, as was shown in Ref. [2], the noise is essentially lower than the errors of the µ0construction. Below, we will show that the underestimation of the errors of XAFS-function extraction is a source of the unreasonable optimistic error s of fitting parameters. Practically all known programs for XAFS modeling [3] in some way calculate confidence limits of fitting parameters. However, since there is no standardized technique for that a nd since most published XAFS works do not contain any mention of methods for estimation of the errors of fitting par ameters, the accuracy of the XAFS results remains to be field for trickery. In the present article we derive the expressions for the erro rs of fitting parameters under different assumptions on the degree of their correlation. Besides, the prior informa tion about parameters is possible to take into account in the framework of Bayesian approach. Moreover one can find the mos t probable weight of the prior information relative to the experimental information. We also discuss the grounds and usage of the statistical test s. The special attention was focused on that where and how one can embellish the results and artificially facilitat e the statistical tests to be passed. All methods and tests described in the paper are realized in t he program viper [6]. II. ERRORS IN DETERMINATION OF FITTING PARAMETERS Let for the experimental curve ddefined on the mesh x1, . . . , x Mthere exists a model mthat depends on N parameters p. In XAFS fitting problems as dmay serve both χ(k) (not weighted by kw) and χ(r). The problem is to find the parameter vector ˆpthat gives the best coincidence of the experimental and mode l curves. Introduce the figure of merit, the χ2-statistics (do not confuse with the symbol of XAFS function ): χ2=M/summationdisplay i=1(di−mi)2 ε2 i, (1) where εiis the error of di. The variate χ2obeys the χ2-distribution law with M−Ndegrees of freedom. Of course, for the given spectrum dand the given model mthe value of χ2is fully determined; we call it “variate” bearing in mind its possible dispersion under different possible realizati ons of the noise and the experimental errors of diextraction. 1Often a preliminary processing (before fitting) is needed: s moothing, filtration etc. Naturally, during the pre- processing some part of the experimental information is los t, and on the variates ξi= (di−mi)/εiadditional dependencies are imposed (before, they were bound solely by the model m). It is necessary to determine the number ofindependent experimental points Nind. For the commonly used in XAFS spectroscopy Fourier filterin g technique the number of independent points is given by [4]: Nind= 2∆k∆r/π+ 2, (2) where ∆ k=kmax−kminand ∆ r=rmax−rminare the ranges in k-r-spaces used for the analysis, and rmin>0. If rmin= 0 then Nind= 2∆k∆r/π+ 1. (3) Instead of keeping in the sum (1) only Ninditems which are equidistantly spaced on the grid x1, . . ., x M, it is more convenient to introduce the scale factor Nind/M: χ2=Nind MM/summationdisplay i=1(di−mi)2 ε2 i. (4) Now the variate χ2follows the χ2-distribution with Nind−Ndegrees of freedom. It can be easily verified that with the use of all available data ( rmin= 0 and rmax=π/2dk) the definition (4) turns into (1). Let us now derive the expression for the posterior distribut ion for an arbitrary fitting parameter pj: P(pj|d) =/integraldisplay ···dpi/negationslash=j···P(p|d), (5) where P(p|d) is the joint probability density function for all values p, and the integration is done over all pi/negationslash=j. According to Bayes theorem, P(p|d) =P(d|p)P(p) P(d), (6) P(p) being the joint prior probability for all pi,P(d) is a normalization constant. Assuming that Nindvalues in d are independent and normally distributed with zero expecte d values and the standard deviations εi, the probability P(d|t), so-called likelihood function, is given by P(d|p)∝exp/parenleftbig −χ2/2/parenrightbig , (7) where χ2was defined above by (4). Its expansion in terms of pnear the minimum ( ∇pχ2= 0) which is reached at p=ˆ pyilds: P(d|p)∝exp/parenleftBig −1 4(p−ˆ p)T·H↔·(p−ˆ p)/parenrightBig ≡/parenleftBig −1 4N/summationdisplay k,l=1∂2χ2 ∂pk∂pl∆pk∆pl/parenrightBig , (8) where ∆ pk=pk−ˆpk, and the Hessian H↔components (the second derivatives) are calculated in the fi tting program at the minimum of χ2. The sufficient conditions for the minimum are H↔ kk>0 and H↔ kkH↔ ll−H↔2 kl>0, for any k, l. Hence, the surfaces of constant level of P(d|p) are ellipsoids. A. Simplest cases If one ignores the prior then the posterior probability dens ity function P(p|d) coincides with the likelihood P(d|p). Let us consider here two widely used approaches. (a)Parameters are perfectly uncorrelated . In this case the Hessian is diagonal and P(pj|d)∝exp/parenleftBig −1 4H↔ jj∆p2 j/parenrightBig . (9) The standard deviation of pjis just 2δ(a)pj= (2/H↔ jj)1/2. (10) (b)Parameter pjessentially correlates solely with pi. In this case P(pj|d)∝/integraldisplay dpiP(pipj|d)∝/integraldisplay dpiexp/parenleftBig −1 4H↔ jj(∆pj)2−1 2H↔ ij∆pj∆pi−1 4H↔ ii(∆pi)2/parenrightBig ∝exp/parenleftBig −1 4/bracketleftBig H↔ jj−H↔2 ij/H↔ ii/bracketrightBig (∆pj)2/parenrightBig , (11) from where one finds ¯ pj= ˆpjand the mean-square deviation δ(b)pj=/parenleftBigg 2H↔ ii H↔ jjH↔ ii−H↔2 ij/parenrightBigg1/2 . (12) In practice, to find the strongly correlated pairs of paramet ers, one finds the pair-correlation coefficients: rij=∝angb∇acketleft∆pi∆pj∝angb∇acket∇ight − ∝angb∇acketleft∆pi∝angb∇acket∇ight∝angb∇acketleft∆pj∝angb∇acket∇ight δ(∆pi)δ(∆pj)(13) taking on the values from -1 to 1. Two parameters are uncorrel ated if their correlation coefficient is close to zero. It is easy to calculate the average values over the distributio n (11): ∝angb∇acketleft∆p2 i∝angb∇acket∇ight= 2H↔ jj/Det,∝angb∇acketleft∆p2 j∝angb∇acket∇ight= 2H↔ ii/Det,∝angb∇acketleft∆pi∆pj∝angb∇acket∇ight= −2H↔ ij/Det, where Det = H↔ jjH↔ ii−H↔2 ij. Notice, by the way, that these are the elements of the invers e matrix of H↔/2. Now the pair-correlation coefficients are given by: rij=−H↔ ij/radicalBig H↔ iiH↔ jj. (14) Via the correlation coefficient the mean-square deviations, found for the cases (a) and (b), are simply related: δ(a)pj=δ(b)pj/radicalBig 1−r2 ij. (15) Consider an example of the error analysis. For L3Pb absorption spectrum1for BaPbO 3compound the average error of the XAFS extraction from the measured absorption was εi= 0.007. For the filtered over the range 1 .0< r < 2.1˚A (the signal from the octahedral oxygen environment of lead a tom) XAFS (see Fig. 1), the model function was calculated as follows. For one-dimensional the Hamiltonian of the lead -oxygen atomic pair with potential U=a/2·(r−r0)2we found the energy levels and corresponding to them wave funct ions. Then, averaging over the Gibbs distribution, the pair radial distribution function (PRDF) normalized to the coordination number Nwas found as: g(r) =N/summationdisplay n|Ψn(r)|2e−En/kT/slashBig/summationdisplay ne−En/kT, N =/integraldisplay g(r)dr, (16) and the XAFS function as: χ(k) =1 kF(k)rmax/integraldisplay rming(r)sin[2 kr+φ(k)]/r2dr. (17) The phase shift φ(k) and the scattering amplitude F(k) were calculated using feff6 program [5]. By variation of the parameters r0,a,N(where Nincludes the factor S2 0), and E0, the shift of the origin for the wave number k, one search for the best accordance between the model and experim ental curves. Here for the fitting, the viper program was used which, in particular, calculates the Hessian of χ2(defind by (4) with Nind= 11.8) at the minimum. The correlation coefficients are listed in the Table I. 1The spectrum was recorded at 50K in transmission mode at D-21 line (XAS-13) of DCI (LURE,Orsay, France) at positron beam energy 1.85 GeV and the average current ∼250 mA. Energy step — 2 eV, counting time — 1s. Energy resoluti on of the double-crystal Si [311] monochromator (detuned to re ject 50% of the incident signal in order to minimize harmonic contamination) with a 0.4 mm slit was about 2–3 eV at 13keV. 3/G0B /G44 /G0C /G0B /G45 /G0C/G48 /G5B/G53/G48 /G55/G4C /G50/G48 /G51 /G57 /G50/G52/G47 /G48 /G4F /G10/G13/G11 /G1B/G10/G13/G11 /G17/G13/G11 /G13/G13/G11 /G17/G13/G11 /G1B/G46 /G0B /G4E /G0C /G4E/G15 /G15 /G11 /G13 /G15 /G11 /G14 /G15 /G11 /G15 /G15 /G11 /G16 /G55 /G0F /G03 /G63/G13/G14 /G13/G15 /G13/G16 /G13 /G4A /G0B /G55 /G0C /G0F /G03/G63/G10 /G14 /G13/G14 /G13 /G13 /G13/G15 /G13 /G13 /G13/G38 /G0B /G55 /G0C /G0F /G03 /G2E/G15 /G17 /G19 /G1B /G14 /G13 /G14 /G15 /G14 /G17 /G14 /G19/G4E /G0F /G03 /G63/G10 /G14 FIG. 1. Experimental and model filtered XAFS χ(k)·k2(first coordination sphere) for BaPbO 3(a) and the model potential with corresponding PRDF and energy levels (b). TABLE I. Pair-correlation coefficients rijfor the example fitting. N a r 0 E0 N 1 −0.286 0.092 0.041 a −0.286 1 −0.044 0.048 r0 0.092 −0.044 1 0.727 E0 0.041 0.048 0.727 1 TABLE II. Mean values and mean-square deviations of the fitti ng parameters. δpare the mean-square deviations calculated: for perfectly uncorrelated parameters (a), trough the maxi mum pair correlations (b), from the bayesian technique with out prior information (maximum likelihood) (c), from the posterior p robability that considers the most probable contribution o f the prior information. Spare the sizes of the parameter space accessible for variatio n (±around the mean value). p ˆp δ(a)p δ(b)p δ(c)p S p δ(d)p N 4.05 0.090 0.094 0.096 ˆN 0.070 a, K/˚A22.28·1054.7·1044.9·1044.9·104ˆa 6.2·103 r0,˚A 2.1456 2.7 ·10−33.9·10−34.0·10−3ˆr0 3.6·10−3 E0, eV 4.42 0.23 0.34 0.35 10 0.21 We now turn our attention to the errors of fitting parameters. In ignoring the correlations, the errors δ(a)pare rather small (see Table II). However, we know that the parameters r0andE0are highly correlated, and their real errors must be larger. In the traditional XAFS-analysis two-dimen sional contour maps have long been used [7] for estimates of the correlation score and the error bars. Notice, that to d o this requires, first, the definition and determination of the correct statistical function χ2(but not a proportionate to it), and, second, a criterion to c hoose the critical value ofχ2(depending on the chosen confidence level). For the most correlated pair, r0andE0, find the joint probability density function P(r0E0|d) using the Hessian elements found at the minimum of the χ2: P(r0E0|d)∝exp/parenleftBig −1 4H↔ r0r0(∆r0)2−1 2H↔ r0E0∆r0∆E0−1 4H↔ E0E0(∆E0)2/parenrightBig (18) 4/G13 /G11 /G15/G13 /G11 /G19 /G13 /G11 /G1B/G13 /G11 /G1C /G47/G0B/G44 /G0C /G55/G13/G03 /G47/G0B/G45 /G0C /G55/G13/G03 /G13 /G11 /G16 /G1C /G16 /G18 /G13 /G11 /G1C /G18 FIG. 2. The joint probability density function P(r0E0|d) calculated via the expansion (8) (solid lines) and using th e exact χ2function (on the right, dashed lines). Also shown the graphi cal interpretation of the mean-square deviations δ(a)r0and δ(b)r0given by (10) and (12). The ellipse of the standard deviation is drawn by the thick line. which is depicted in Fig. 2 as a surface graph and as a contour m ap. The ellipses of the equal probability are described by: H↔ r0r0(∆r0)2+ 2H↔ r0E0∆r0∆E0+ H↔ E0E0(∆E0)2= 4λ. (19) In Fig. 2 they limit such areas that the probability for the ra ndom vector ( r0,E0) to find itself in them is equal to ℓ= 1−e−λ= 0.2, 0.6, 0.8, 0.9 and 0.95. By the thick line is drawn the ellips e corresponding to the standard deviation: λ= 1/2 and ℓ= 1−e−1/2≈0.3935. For this ellipse the point of intersection with the lin e ∆E0= 0 and the point of maximum distance from the line ∆ r0= 0 give the standard mean-square deviations δ(a)r0andδ(b)r0that coincide with the expressions (10) and (12). To find the mean-square de viation δ(b)for an arbitrary confidence level ℓ, one should multiply the standard deviation by/radicalbig −2 ln(1 −ℓ). In Table II the errors in the column δ(b)pwere found as the largest errors among all those calculated f rom the pair correlations. For the parameters Nandaall pair correlations are weak, so their δ(a)andδ(b)are hardly differ. For the parameters r0andE0these mean-square deviations differ remarkable. Finally, we put the question, how much is rightful the expans ion (8) for the likelihood function? In Fig. 2, on the right, the dashed ellipses of equal probability are found fo r the exact χ2that was calculated by the viper program as well. Mainly, just-noticeable difference is caused by the realization of the fitting algorithm or to be more precise, by the values of the variations of the fitting parameters whic h determine the accuracy of the minimum itself and the accuracy of the derivatives at the minimum. Of course, this d ifference can be neglected. B. General case Often, a particular fitting parameter significantly correla tes not with a one, but with several other parameters (in our example this is not so, but, for instance, the problem of approximation of the atomic-like background by interpolation spline drawn through the varied knots [8,2] i s that very case). Now, the consideration of the two- dimensional probability density functions is not correct n o more, one should search for the totaljoint posterior probability P(p|d). For that, first of all, one is to find the prior probability P(p). Let we approximately know in advance the size Sk of the variation range of the parameter pk. Then the prior probability can be expressed as: P(p|α)∝αN/2exp/parenleftBig −α 2N/summationdisplay k=1∆p2 k S2 k/parenrightBig , (20) 5where the regularizer αspecifies the relative weight of the prior probability; at α= 0 there is no prior information, atα→ ∞ the fitting procedure gives nothing and the posterior distri bution coincides with the prior one. In the expression (20) αappears as a known value. Later, we apply the rules of probabi lity theory to remove it from the problem. So, for the sought probability density functions we have: P(pj|d, α)∝/integraldisplay ···dpi/negationslash=j···αN/2exp/parenleftBig −1 2N/summationdisplay k,l=1gkl∆pk∆pl/parenrightBig , (21) where gkl=α S2 kδkl+H↔ kl 2. (22) Since there is no integral over pj, separate it from the other integration variables: P(pj|d, α)∝αN/2exp/parenleftBig −1 2gjj∆p2 j/parenrightBig/integraldisplay ···dpi/negationslash=j···exp/parenleftBig −1 2N/summationdisplayj k,l=1gkl∆pk∆pl−∆pjN/summationdisplayj k=1gkj∆pk/parenrightBig , (23) Here, the symbol jnear the summation signs denotes the absence of j-th item. Further, find the eigenvalues λi and corresponding eigenvectors eiof the matrix gklin which the j-th row and column are deleted, and change the variables: bi=/radicalbig λiN/summationdisplayj k=1∆pkeik,∆pk=N/summationdisplayj i=1bieik√λi(i, k∝negationslash=j). (24) Using the properties of eigenvectors: N/summationdisplayj k=1glkeik=λieil,N/summationdisplayj k=1elkeik=δli (l, i∝negationslash=j), (25) one obtains: P(pj|d, α)∝αN/2exp/parenleftBig −1 2[gjj−u2]∆p2 j/parenrightBig/integraldisplay ···dbl/negationslash=j···exp/parenleftBig −1 2N/summationdisplayj i=1[bi+ui∆pj]2/parenrightBig ∝αN/2exp/parenleftBig −1 2[gjj−u2]∆p2 j/parenrightBig , (26) where new quantities were introduced: ui=1√λiN/summationdisplayj k=1gkjeik,u2=N/summationdisplayj i=1u2 i. (27) Thus, we have found the explicit expression for the posterio r distribution of an arbitrary fitting parameter. This is a Gaussian distribution with the mean ¯ pj= ˆpjand the standard deviation δ(c)pj= (gjj−u2)−1/2. (28) The formulas (26)–(28) require to find the eigenvalues and ei genvectors for the matrix of rank N−1 for each parameter. Those formulas have merely a methodological val ue: the explicit expressions for posterior probabilities enables one to find the average of arbitrary function of pj. However, the standard deviations could be calculated significantly easier, having found the eigenvalues and eige nvectors for the matrix of rank None time. (δ(c)pj)2=/integraltext ∆p2 jP(pj|d, α)dpj/integraltext P(pj|d, α)dpj=/integraltext ∆p2 jexp/parenleftBig −1 2/summationtextN k,l=1gkl∆pk∆pl/parenrightBig dp /integraltext exp/parenleftBig −1 2/summationtextN k,l=1gkl∆pk∆pl/parenrightBig dp. (29) 6Analogously to what was done above, performing the diagonal ization of gkl, one obtains: (δ(c)pj)2=/integraltext db/parenleftBig/summationtextN i=1bieij/√λi/parenrightBig2 exp/parenleftBig −1 2/summationtextN i=1b2 i/parenrightBig /integraltext dbexp/parenleftBig −1 2/summationtextN i=1b2 i/parenrightBig =N/summationdisplay i=1e2 ij λi, (30) where the eigenvalues ( λi) and eigenvectors ( ei) correspond to the full matrix gkl. One can give another interpretation of the δ(c)p-finding process. It is easy to verify that H↔/2 and the covariance matrix Cof the vector pare mutually inverse. Therefore (δ(c)pj)2=Cjj= 2(H↔−1)jj, (31) and the variate ( p−ˆ p)T·C−1·(p−ˆ p) =1 2(p−ˆ p)T·H↔·(p−ˆ p) isχ2-distributed with Ndegrees of freedom if pis theN-dimensional normally distributed vector (by Eq. (26) this condition is met). The ellipsoid that determines the standard deviation is: (p−ˆ p)T·H↔·(p−ˆ p) =N. (32) For an arbitrary confidence level ℓ, on the r.h.s. would be ( χ2 N)ℓ, the critical value of the χ2-distribution with N degrees of freedom. The error δ(c)pkis equal to the half the ellipsoid size along the k-th axis. In our example fitting, the errors found in the absence of any p rior information ( α= 0) from the formula (30) are listed in Table II in the column δ(c)p. Due to every one parameter correlates at the most with one ot her parameter, allδ(c)pare practically coincide with δ(b)p. Generally, this may be not so. Finally, let us find the most probable value of α. Its posterior distribution is given by: P(α|d) =/integraldisplay dpP(α,p|d) =/integraldisplay dpP(α)P(p|α,d). (33) Using a Jeffreys prior P(α) = 1/α[9], one obtains for the posterior distribution: P(α|d)∝/integraldisplay dpαN/2−1exp/parenleftBig −1 2N/summationdisplay k,l=1gkl∆pk∆pl/parenrightBig ∝(λ1···λN)−1/2αN/2−1. (34) In our example we have set the variation range of the paramete rpkto be equal to Sk=±ˆpk(this means that pk∈[0,2ˆpk]) for all parameters except for E0; since it varies near zero, we have chosen SE0=±10eV. For the mentioned variation ranges, the distribution P(α|d) has its mode at α= 2.64·103(see Fig. 3). The bayesian errors found for this regularizer are listed in the column δ(d)pof Table II. As a result, we have got the mean-square errors that for some parameters are significantly lower than even δ(a)p. There is nothing surprising in that: any additional information narrows the posterior distribution. If we woul d choose Skto be less, δ(d)pkwould be yet lower. For instance, XAFS is quite accurate in distance determination , and for many cases one can assume distances to be known within ±0.2˚A. In our case this leads to δ(d)r0= 3.4·10−3˚A. /G44/G33 /G0B /G44 /G5F /G47 /G0C /G14 /G13/G10 /G14/G14 /G13 /G14 /G13/G16/G14 /G13/G18/G14 /G13/G1A/G44 /G50 /G20 /G15 /G11 /G19 /G17 /G11 /G14 /G13/G16 FIG. 3. The posterior distribution for the regularizer αfound from Eq. (34). 7C. Important note Having obtained the expressions (10), (12) and (30) for the e rrors of fitting parameters, we are able now to draw an important conclusion. If in the definition (4) one substit utes for εithe values that are smaller by a factor of β than the real ones, the χ2and its Hessian’s elements are exaggerated by a factor of β2, and from (10), (12) and (30) follows that the errors of fitting parameters are understate d by a factor of β! In the preceding paper [2] it was shown that the errors of the a tomic-like absorption construction are essentially larger than the experimental noise, and therefore it is the f ormer that should determine the εivalues. However, these values are traditionally assumed to be equal to the noi se, or one uses unjustified approximations for them, also understated (like 1 /ε2 i=kw[10]). It is here where we see the main source of the groundles s optimistic errors. III. STATISTICAL TESTS IN FITTING PROBLEMS A.χ2-test Introducing the statistical function χ2, we assumed that it follows the χ2distribution with ν=M−Ndegrees of freedom. However for this would be really so, one should achi eve a sufficient fitting quality. This “sufficient quality” could be defined as such that the variate (4) obeys the χ2distribution law, that is this variate does not fall within the tail of this distribution. Strictly speaking, the follo wing condition must be met: χ2<(χ2 ν)ℓ, (35) where the critical value ( χ2 ν)ℓfor the specified significance level ℓmay be calculated exactly (for even ν) or approxi- mately (for odd ν) using the known formulas [11]. Notice, that the choice of the true εihere also plays a cardinal role. However, it is important her e that one would not use the overestimated values which facilitate to meet the requirement (35). As we h ave shown in [2], one could obtain the overestimated εi, having assumed the Poisson destribution law for the detect ors counts when the actual association between the probability of a single count event and the radiation intensity is unknown. Thus, the exaggerated values εitell about a quality fitting, but give the large errors of fitti ng parameters. The understated εilead to the would-be small errors, but make difficult to pass th eχ2-test (i. e. to meet the condition (35)). We are aware of many works the authors of which do not de scribe explicitly the evaluation process for the errors of XAFS-function extraction and do not report their e xplicit values. However, by implication it is seen that εi were chosen (not calculated!) as low as possible to scarcely (with ℓ= 0.9−0.95) pass the χ2-test; as a result, very impressive errors of the structural parameters were obtain ed. In such approach no wander that the difference of 0.01 ˚A between the diffraction data and the XAFS-result that was fou nd within 0.002 ˚A was attributed to the “suggested presence of a small systematic error” [10]. B.F-test Let there is a possibility to choose between two physical mod els depending on different numbers of parameters N1 andN2(N2> N1). Which one of them is more statistically important? For ins tance one wish to decide whether a single coordination sphere is split into two. Let for the two models the functions χ2 1andχ2 2obey the χ2-distribution law with ν1=Nind−N1andν2=Nind−N2 degrees of freedom, correspondingly. From the linear regre ssion problem (near the minimum of χ2, the likelihood function is expressed by (8) and is identical in form to that o f the linear regression problem) it is known that the value f=(χ2 1−χ2 2)/(ν1−ν2) χ2 2/ν2(36) obeys the Fisher’s F-distribution law with ( ν1−ν2, ν2) degrees of freedom if exactly r=ν1−ν2parameters in the second model are linearly dependent, that is if exist the r×N2matrix Cof rank rand the vector cof the dimension rsuch that Cp=c. In order for the linear restrictions on the second model par ameters to be absent, the value f should notfollow the F-distribution, that is it should be greater than the critica l value ( Fν1−ν2,ν2)ℓfor the specified significance level ℓ: 8/G15 /G17 /G19 /G1B /G14 /G13 /G14 /G15 /G14 /G17 /G14 /G19 /G4E /G0F /G03 /G63/G10 /G14/G10/G13/G11 /G1B/G10/G13/G11 /G17/G13/G11 /G13/G13/G11 /G17/G13/G11 /G1B/G46 /G4E/G15 /G49 /G4C /G4F /G57 /G48 /G55 /G48 /G47/G03 /G47 /G44 /G57 /G44 /G50/G52/G47 /G48 /G4F/G03 /G14 /G50/G52/G47 /G48 /G4F/G03 /G15FIG. 4. On the choice between two different models on statistical grounds. Cited from Ref. [12]. f >(Fν1−ν2,ν2)ℓ (37) or χ2 2< χ2 1/parenleftbigg (Fν1−ν2,ν2)ℓν1−ν2 ν2+ 1/parenrightbigg−1 . (38) Notice, that the expression (38) means the absence of exactl yrlinear restrictions on the second model parameters. Even if (38) is realized, the less number of linear dependenc ies are possible. If, for instance, the splitting of a single coordination sphere into two does not contradict to the F-test (38), some of the parameters of these two spheres may be dependent, but not all. This justifies the introduction of a new sphere into the model XAFS function. Thus, having specified the significance level ℓ, one can answer the question “what decrease of χ2must be achieved to increase the number of parameters from N1toN2?” or, inside out, “what is the probability that the model 2 is better than the model 1 at specified ( N1, χ2 1) and ( N2, χ2 2)?” Notice, that since in the definition for fthe ratio χ2 1/χ2 2appears, the actual values of εibecome not important for theF-test (only if they all are taken equal to a single value). Consider an example of the statistical tests in the fitting pr oblem. In Fig. 4 are shown the experimental curve withNind= 11.8 and two model curves with N1= 4 and N2= 7. The underlying physical models were described in Ref. [12]; here only the number of parameters is of importanc e. Let us apply the statistical tests. Through the fitting procedure for the model 1 we have: ν1= 11−4 = 7, χ2 1= 16.8>14.1 = (χ2 7)0.95, for the model 2: ν2= 11−7 = 4, χ2 1= 5.3<9.5 = (χ2 4)0.95. That is the first model does not pass the χ2-test. Further, f= 2.89 = ( F3,4)0.84, from where with the probability of 84% we can assert that the model 2 is better than the model 1. In the XAFS analysis the F-test has long been in use [7]. However, the words substantia ting the test are often wrong. The authors of Refs. [10,13], for example, even claim ed that the value f(36)mustfollow the F-distribution, although then in Ref. [13] there appears a really correct ine quality (38). IV. CONCLUSION The solution of the main task of the XAFS spectroscopy, deter mination of the structural parameters, becomes worthless if the confidence in this solution is unknown. Here we mean not only the confidence in the obtained fitting parameters that is their mean-square deviations, but also t he credence to the very methods of the error analysis. It is excessive optimistic errors evaluations lead to the susp icious attitude to the XAFS results. To improve the situation could the development of the reliab le and well-grounded techniques that do not allow one to treat the data in an arbitrary way. First of all, this is a technique for determination of the real errors of the atomic-like absorption construction. Second, we regard as necessary to standardize the method for the correct taking into account of allpair correlation between fitting parameters. And third, (we have not raised this question here) programs for scattering phase and amplitude calculations s hould report on the confidence limits for the calculated values, that is report how sensitive the calculated values a re to the choice of the parameters of scattering potentials. 9[1] N. A. Young, A. J. Dent, Open Letter to the XAFS Comunity. Maintaining and improving the quality of published XAFS data: a view from the UK XAFS user group . J. Synchrotron Rad. 6, 799 (1999), (Proc. of Int. Conf. XAFS X). [2] K. V. Klementev, XAFS analysis. I. Extracting the fine structure from the abso rption spectra . The preceding article , (2000). [3] Catalog of XAFS Analysis Programs, http://ixs.csrri.iit.edu/catalog/XAFS_Programs . [4] E. A. Stern, Number of relevant independent points in x-ray-absorption fine-structure spectra . Phys. Rev. B 48(13), 9825– 9827 (1993). [5] J. J. Rehr, J. Mustre de Leon, S. I. Zabinsky, R. C. Albers, Theoretical X-ray Absorption Fine Structure Standards . J. Am. Chem. Soc. 113, 5135–5140 (1991). [6] K. V. Klementev, VIPER for Windows (Visual Processing in EXAFS Researches) , freeware, http://www.crosswinds.net/~klmn/viper.html . [7] R. W. Joyner, K. J. Martin, P. Meehan, Some applications of statistical tests in analysis of EXAFS and SEXAFS data . J. Phys. C: Solid State Phys. 20, 4005–4012 (1987). [8] M. Newville, P. L¯ ıvi¸ nˇ s, Y. Yacoby, J. J. Rehr, E. A. Ste rn,Near-edge x-ray-absorption fine structure of Pb: A comparis on of theory and experiment . Phys. Rev. B 47(21), 14126–14131 (1993). [9] H. Jeffreys, Theory of Probability (Oxford University Pr ess, London, 1939), later editions: 1948, 1961, 1983. [10] A. Filipponi, A. Di Chicco, X-ray-absorption spectroscopy and n-body distribution functions in condensed matter. II. Data analysis and applications . Phys. Rev. B 52, 15135–15149 (1995). [11] Handbook of mathematical functions with formulas, gra phs and mathematical tables, edited by M. Abramowitz, I. Ste gun (Applied mathematical series, 55, National bureau of stand ards, 1964). [12] A. P. Menushenkov, K. V. Klementev, EXAFS indication of double-well potential for oxygen vibra tion in Ba 1−xKxBiO3. J. Phys.: Condens. Matter 12, (2000), (accepted). [13] A. Michalowicz, K. Provost, S. Laruelle, A. Mimouni, F-test in EXAFS fitting of structural models . J. Synchrotron Rad. 6, 233–235 (1999), (Proc. of Int. Conf. XAFS X). 10

arXiv:physics/0003088v1 [physics.atom-ph] 28 Mar 2000Radiative recombination of bare Bi83+: Experiment versus theory A. Hoffknecht, C. Brandau, T. Bartsch, C. B¨ ohme, H. Knopp, S. Schippers and A. M¨ uller Institut f¨ ur Kernphysik, Universit¨ at Giessen, D-35392 G iessen, Germany C. Kozhuharov, K. Beckert, F. Bosch, B. Franzke, A. Kr¨ amer, P.H. Mokler, F. Nolden, M. Steck and Th. St¨ ohlker Gesellschaft f¨ ur Schwerionenforschung (GSI), D-64291 Da rmstadt, Germany Z. Stachura Institute for Nuclear Physics, 31-342 Krak´ ow, Poland (November 24, 2013) Electron-ion recombination of completely stripped Bi83+ was investigated at the Experimental Storage Ring (ESR) of the GSI in Darmstadt. It was the first experiment of this kind with a bare ion heavier than argon. Absolute recombi- nation rate coefficients have been measured for relative ener - gies between ions and electrons from 0 up to about 125 eV. In the energy range from 15 meV to 125 eV a very good agree- ment is found between the experimental result and theory for radiative recombination (RR). However, below 15 meV the experimental rate increasingly exceeds the RR calcula- tion and at E rel= 0 eV it is a factor of 5.2 above the ex- pected value. For further investigation of this enhancemen t phenomenon the electron density in the interaction region w as set to 1.6 ×106cm−3, 3.2×106cm−3and 4.7 ×106cm−3. This variation had no significant influence on the recombination rate. An additional variation of the magnetic guiding field o f the electrons from 70 mT to 150 mT in steps of 1 mT resulted in periodic oscillations of the rate which are accompanied b y considerable changes of the transverse electron temperatu re. 34.80.Lx I. INTRODUCTION Recombination between electrons and highly charged ions plays an important role in different areas of modern physics. The basic two- and three-body recombination processes are of very fundamental nature and thus pro- vide an excellent testing ground for collision theory and atomic structure calculations. Cross sections and rate coefficients of these processes are needed for the under- standing of astrophysical and fusion plasmas and also provide useful information for applications in accelerato r physics [1]. In particular, beam losses in ion storage rings by electron-ion recombination during electron cooling can post harsh limits to the handling and the availability of ions for further experiments. During the last decade electron-ion recombination has been extensively investigated. Merged-beams experi- ments using storage ring coolers and single-pass electron targets at accelerators have provided excellent condition s for a new generation of recombination measurements. Inthese experiments an incident ion beam is merged with a cold beam of electrons over a distance of typically 50 to 250 cm depending on the specific electron beam device. By choosing the appropriate accelerator facility, ions of most elements in all possible charge states can be inves- tigated nowadays. Free electrons can be captured by ions via several dif- ferent mechanisms. The main recombination channel for a bare ion is radiative recombination (RR) e+Aq+→A(q−1)++hν. (1) RR is the direct capture of a free electron by an ion Aq+ where the excess energy and momentum are carried away by a photon. After the capture, which is inverse to pho- toionization, the electron can be in an excited state and there will be further radiative transitions within the ion until the electron has reached the lowest accessible en- ergy level. RR is a non-resonant process with a diverging cross section at zero center-of-mass (c.m.) energy which continuously decreases towards higher c.m. energies. Another recombination mechanism possible for a bare ion is three-body recombination (TBR) Aq++e+e→A(q−1)++e′(2) where the excess energy and momentum are carried away by a second electron. This process is important at high electron densities and very low center-of-mass energies between electrons and ions. The pioneering experiment on radiative recombination of bare ions was performed by Andersen et al. [2] in 1990. Absolute rate coefficients were measured for C6+with a merged-beam technique finding a reasonably good agree- ment between experiment and theory in the investigated energy range from E rel= 0 to 1 eV where E relis the rel- ative energy between the ensembles of electrons and ions in the interaction region (for the definition of E relsee section III). In a number of consecutive measurements different bare ions (D+,He2+,C6+,N7+,Ne10+,Si14+,Ar18+) [3–6] have been investigated at several facilities. The mea- sured rate coefficients have been in accordance with a theoretical approach to RR by Bethe and Salpeter [7] 1for relative energies Erel≥0.01 eV. The experiments were limited by counting statistics to an energy range of only a few eV. Recently an experiment with Cl17+-ions revealed excellent agreement between the measured rate and RR theory for relative energies from 0.01 eV up to 47 eV [8]. However, in all of these measurements strong deviations of the experimental findings from the theo- retical predictions were found at very low electron-ion relative energies ( Erel≤0.01 meV) . The measured rate coefficient typically shows an additional increase towards lower energies resulting in a so-called rate enhancement factor ǫ=αexp/αRRatErel= 0 eV of 1.6 (He2+) to 10 (Ar18+) for bare ions (no enhancement observed with D+) and up to 365 for a multicharged multi-electron sys- tem like Au25+[9]. It should be noted that in contrast to the RR cross section which diverges at Erel= 0 eV the measured rate coefficient obtains a finite value due to the experimental electron and ion velocity spreads (see section 3). After the first observation of the so called enhancement phenomenon in an experiment with U28+-ions at the GSI in Darmstadt in 1989 [10] and a later experimental con- firmation of that same result [11] this effect has been observed over and over again in many different experi- ments at different facilities. Whereas the very high rate enhancement factors of multicharged complex ions like Au25+, Au50+, Pb53+and U28+could be partly traced back to the presence of additional recombination chan- nels such as dielectronic recombination (DR) [9,12–14] or polarization recombination (PIR) [15] the origin of the observed discrepancies between experiment and theory for bare ions is still unknown. Theoretical calculations applying MolecularDynamics (MD) computer simula- tions [16,17] have been performed by Spreiter et al. [18]. Taking into account the experimental conditions at the heavy ion storage ring TSR in Heidelberg they found an increase of the local electron density around the ion. This could possibly lead to a higher number of recombination processes but to date the recombination of the electrons with the ion is not included in their theoretical descrip- tion. For the clarification of this phenomenon dependences of the enhancement on external experimental parameters have been studied at different storage rings. Variations of the electron density [6,8] within a total range from about 106cm−3up to almost 1010cm−3showed little effect on the enhancement. The dependence of the to- tal recombination rate followed the T−1/2 ⊥dependence on the mean transverse electron energy spread T⊥[19] as expected for RR alone. A systematic study of the ion charge-state dependence of the excess rate coefficient for a number of bare ions [3] yielded roughly a Z2.8scaling for atomic numbers 1 ≤Z≤14. The first external parame- ter observed to influence the enhancement has been the magnetic field strength in the interaction region of elec- trons and ions [9]. The experimental results show a clear increase of the recombination rate maximum of Au25+ions at Erel= 0 with increasing magnetic field strength. Recently this dependence has also been observed in ex- periments with lithium-like F6+and bare C6+-ions at the TSR heavy ion storage ring in Heidelberg [20,21]. In the present measurement, which was carried out at the Experimental Storage Ring of the GSI in Darmstadt, we extended the range of investigated bare ions to the high-Z ion Bi83+. It was the first experiment of this kind with such a heavy bare ion providing information on RR in addition to studies of the radiative electron capture in ion-atom collisions using x-ray spectroscopy [22–24]. Be- side a comparison of measured absolute rate coefficients with calculated RR rates for relative energies from Erel = 0 eV to 125 eV dependences of the recombination rate on the electron density and the magnetic guiding field of the electron beam in the cooler have been investigated. The present paper is organized as follows. After a short description of the theoretical approaches, the experimen- tal setup and the evaluation of recombination rates are described. The experimental results are presented and compared with RR calculations. II. THEORY In order to describe RR theoretically Kramers devel- oped a semi-classical theory already in 1923 [25]. A full quantum mechanical treatment was performed by Sto- bbe seven years later [26]. In 1957, Bethe and Salpeter [7] derived an approximate formula for the RR cross sec- tion that is identical to Kramers result σRR(n, E cm) = (2 .1×10−22cm2)E2 0 nEcm(E0+n2Ecm). (3) The capture of an electron by a bare ion produces a hy- drogenic state with principal quantum number n. In this case E 0is the binding energy of the ground state electron in the hydrogenic ion or atom and E cmis the kinetic en- ergy in the electron-ion center-of-mass frame. The total cross section for this process is obtained by summing up the contributions of all accessible Rydberg states: σRR(Ecm) =nmax/summationdisplay n=1σRR(n, E cm) (4) where nmaxis the maximum principal quantum number that can contribute. This number is generally limited by experimental conditions. The approach of Bethe and Salpeter clearly shows the typical features of RR cross sections: the divergence at zero electron energy and a monotonic decrease for increasing electron energy. How- ever, as a semi-classical approximation Eq. 3 is only valid in the limit of high quantum numbers and low electron energies, i.e. for n≫1 and Ecm≪Z2/n2Ry. Since the quantum mechanical treatment of Stobbe involves the 2rather tedious evaluation of hydrogenic dipole matrix el- ements one often applies correction factors Gn(Ecm), the so called Gaunt factors, to Eq. 3 to account for deviations from the correct quantum result at low nand high Ecm. The use of Gaunt factors is convenient because they are either tabulated [27] or given in an easy parametrization [28]. We here apply tabulated [29] values kn=Gn(0) using σRR(Ecm) =nmax/summationdisplay n=1(2.1×10−22cm2)kn× (Z2Ry)2 nEcm(Z2Ry+n2Ecm). (5) where Zis the nuclear charge of the ion and Ry= 13.6 eV is the ground-state energy of the hydrogen atom. Eq. 5 is exact for bare ions at zero relative energy and does not deviate by more than about 5% from the quantum mechanically correct hydrogenic result at the highest en- ergies considered in this paper. For high-Z ions and high relative energies the non- relativistic dipole approximation presented above is not valid. In general an exact relativistic calculation with the inclusion of higher multipoles is required [30]. Nev- ertheless experiments investigating the Radiative Elec- tron Capture (REC) into the K-shell of high-Z ions have shown that the experimental results are in very good overall agreement with the non-relativistic dipole approx - imation [31] as long as Erelis below the electron rest en- ergymec2. Because of this observed accordance Eq. 5 is used for the calculations presented in this paper. III. BASIC RELATIONS In the experiment cross sections at very low center-of- mass energies Ecmare not accessible due to the finite velocity spread of the colliding particles. In the present work the lowest limit of Ecmis at about 1 ×10−5eV. The measured quantity is a rate coefficient αwhich theoreti- cally results from a convolution of the cross section with the velocity distribution function f(/vector v) α(vrel) =/integraldisplay σ(v)vf(vrel,/vector v)d3v. (6) In our experiment the ion velocity distribution is neg- ligibly narrow compared to that of the electrons due to the cooling of the ion beam to a relative momentum spread below 10−4and due to the large mass difference between electrons and ions. Therefore the distribution function f(/vector v) is dominated by the electron velocity dis- tribution. Considering the axial symmetry of the merged beams experiment two velocity coordinates are sufficient to describe the distribution: v/bardblthe velocity componentin beam direction and v⊥the velocity component perpen- dicular to the beam. The appropriate velocity (or energy) spreads are characterized by two corresponding temper- atures T/bardblandT⊥. Due to the acceleration of the elec- trons these temperatures are quite different ( T/bardbl≪T⊥) resulting in a highly anisotropic velocity distribution f(/vector v) which is therefore often called ‘flattened’. Its mathemat- ical form is given by f(vrel,/vector v) =me 2πkT ⊥exp(−mev2 ⊥ 2kT⊥)× /radicalbiggme 2πkT ||exp(−me(v||−vrel)2 2kT||) (7) withvrelbeing the average longitudinal center-of-mass velocity vrel=|ve,||−vi,|||/(1−vi,||ve,||/c2). (8) where ve,/bardblandvi,/bardblare the longitudinal velocity com- ponents of the electron and ion beam in the laboratory frame, respectively. The relative energy of electrons and ions is Erel= (γrel−1)mec2(9) with γrel=/bracketleftBig 1−(vrel/c)2/bracketrightBig−1/2 . (10) Under cooling conditions vrelis zero and, hence, also Erelis zero then. For comparisons of experimental results with theory the cross sections resulting from Eq. 5 have to be convoluted with the experimental velocity distribution according to Eq. 6 using the experimental temperatures T||andT⊥. IV. EXPERIMENT The measurements have been performed at the Ex- perimental Storage Ring (ESR) of the Gesellschaft f¨ ur Schwerionenforschung (GSI) in Darmstadt [32]. 295.3 MeV/u Bi83+-ions supplied by the GSI linear accelera- tor UNILAC and the heavy ion synchrotron SIS were injected into the ESR. Only one shot of ions was suffi- cient to provide an ion current of typically 400-800 µA at the beginning of a measurement. In the storage ring the circulating Bi83+ions were merged with the magnet- ically guided electron beam of the electron cooler with an electron energy of 162 kV (Fig. 1). Before starting a measurement, the ion beam was cooled for several sec- onds until the beam profiles reached their equilibrium widths. During the measurement the electron energy was stepped through a preset range of values different from the cooling energy thus introducing non-zero mean rela- tive velocities vrelbetween ions and electrons. After each voltage step a cooling interval was inserted. The whole 3scheme of energy scan measurements was realized by ap- plying voltages from -5 kV to 5 kV to the two drift tubes surrounding the electron and the ion beam in the inter- action region. The voltages were supplied by a system of sixteen individual power supplies controlled by very fast high-voltage switches. This instrument has specially been constructed for the recombination experiments [33]. Only 2 ms are needed by this device to switch to and set a certain voltage with a relative precision of 10−4. The repetition rate of voltage settings with this precision is limited to 40 per second. Recombined Bi82+ions were counted as a function of the electron energy on a scintillator detector located be- hind the first dipole magnet downstream of the electron cooler. The dipole magnet bends the circulating Bi83+ ion beam onto a closed orbit and separates the recom- bined Bi82+ions from that orbit. In between two mea- surement steps of 40 ms duration, the electron energy was always set to the cooling energy for 20 ms in order to maintain good ion beam quality. The experimental data stream was continuously collected and stored after each ms. The time-resolved measurements allowed us to observe and eliminate drag force effects from the data in a detailed off-line analysis. Such effects are a result of the cooling force exerted by the electrons on the ion beam which can lead to a time-dependent shift of the ion velocity towards the electron velocity. The friction force s are particularly effective at relative energies close to zer o. The kinetic energy Eeof the electrons is defined by the cathode voltage Ugun, the drift tube voltage Udriftand the space charge potential Uspin the interaction region. Assuming coaxial beams it is calculated as Ee=−eUgun+eUdrift−eUsp =−eUgun+eUdrift−Ieremec2 eve[1 + 2 ln( b/a)],(11) where reis the classical electron radius. The quantities b = 10 cm and a= 2.54 cm are the radii of the drift tube and the electron beam, respectively. The ion beam diam- eter is only of the order of a millimeter and, hence, the electron energy distribution probed by the ions is rather flat across the ion beam. In the present experiment we performed measurements with different electron currents Iewhich produced space charge potentials ranging from 17.2 V to 51.7 V. The space charge corrected electron energy Eeand the ion energy Eiare used to calculate the relative energy of electrons and ions in the center-of-mass frame. A rela- tivistic transformation yields Erel=/radicalbig (mic2+mec2)2+ 2 (EiEe+Eimec2+Eemic2−A)−B, A=/radicalbig Ei(Ei+ 2mic2)/radicalbig Ee(Ee+ 2mec2)cos(φ) B= (mic2+mec2) (12)where φis the angle between the electron and the ion beam directions. According to Eq. 12 the minimum rela- tive energy Erel= 0 eV cannot be reached if an angle φ/ne}ationslash= 0 is present. Therefore the alignment of the beams was optimized before the recombination experiments in order to achieve φ= 0 mrad with an uncertainty of 0.1 mrad. The counting rate measured at the scanning energy Emeasis given by R(Emeas) =α(Emeas)ηLn e(Emeas)Ni Cγ2+Rback (13) withαdenoting the electron-ion recombination rate co- efficient, ηthe detection efficiency of the scintillator de- tector which is very close to unity, L= 2.5 m the nominal length of the interaction zone, ne(E) the electron density at energy E,Nithe number of stored ions, C= 108 .36 m the ring circumference and γthe relativistic Lorentz fac- tor for the transformation between the c. m. and the laboratory frames. Rbackdenotes the measured back- ground rate due to collisions with residual gas molecules. In order to extract an absolute rate coefficient from the experimental data the background has to be subtracted by taking into account the counting rate R(Eref) =α(Eref)ηLn e(Eref)Ni Cγ2+Rback,(14) at a reference energy Eref. Combining Eqs. 13 and 14 α atEmeashas to be calculated from α(Emeas) =(R(Emeas)−R(Eref))γ2 ηLne(Emeas)Ni/C +α(Eref)ne(Eref) ne(Emeas). (15) Because of RR one always has a non-zero recombination rate coefficient α(Eref) at the reference point. Usually Erefis chosen such that α(Eref) practically equals zero, but in general one has to re-add the rate which has been neglected by subtracting R(Eref) from R(Emeas). In the present experiment the reference rate has been measured at the maximum accessible scan energy Eref= 125 eV. According to RR theory the rate coefficient at this energy isα(Eref)= 3.6 ×10−10cm3s−1. This correction leads to a modification of the measured rate coefficient at Erel= 0 eV by only 0.2 %. Electron and ion beams are merged and demerged by bending the electron beam in a toroidal magnetic field with a bending radius of 120 cm. An electron beam of 2.54 cm radius is still overlapping the ion beam for 25 cm before and after the straight overlap section of 250 cm. The merging and demerging sections therefore contribute to the measured counting rate. As one can see in the left panel of Fig. 2 where the calculated potential distribu- tion of the drift tubes for an applied voltage of 1 V is plotted against the position inside the cooler (along the central axis of ion and electron beam) the influence of the voltage applied to the drift tubes is restricted to the 4straight overlap section of the cooler. Thus, the electron energy in the toroidal sections is always the same inde- pendent of the drift tube potential. This results in a con- stant contribution to the measured counting rate which is considered by the background subtraction procedure described above. The right panel of Fig. 2 shows the distribution of an- glesφbetween electron trajectories and the ion beam direction along the geometrical cooler axis. The elec- trons strictly following the magnetic field lines, the non- zero angles result from the measured transverse magnetic guiding field components in this section. The measure- ment was restricted to a length of 2.26 m (90 % of the straight overlap section) fully including the drift tube area. As one can see φincreases rapidly at the edges of the measured range. Using steerer magnets the electron beam can be shifted to a position minimizing the influ- ence of angular deviations. Both the distributions of the electric potential and angle (Fig. 2) can be combined via Eq. 12 into a distribution of relative energies along the ion beam axis. Fig. 3 shows the relative energies along the straight overlap section for different voltages applied to the drift tubes. Obviously the desired relative ener- gies are only realized along a certain fraction of the whole interaction length which, in addition, depends on the en- ergy. The measured rate coefficient at a given relative energy Emeasalways contains contributions from other relative energies according to: α(Emeas) =1 L/integraldisplayL 0dl α(Erel(l)) (16) withErel(l) being the relative energy at the position lin- side the cooler. According to Eq. 16 the correct rate co- efficient can be obtained by a deconvolution which is per- formed iteratively. In a first iteration step the measured rate coefficient α(Emeas) =α(0)is inserted as α(Erel(l)) in Eq. 16. Then the difference ∆ α(0)between the ob- tained result and the measured rate is subtracted from α(0). In a next step the new α(1)=α(0)−∆α(0)is likewise inserted in Eq. 16 and the difference ∆ α(1)is calculated. The procedure is carried on until in a step k the rela- tive difference ∆ α(k)(Emeas)/α(Emeas) is below 10−3at all measured energies. This is the case after only a few iteration steps. Although the relative statistical errors of the results presented below amount to less than 1% in the rate co- efficient maximum, the systematic uncertainty in the ab- solute recombination rate coefficient has been estimated to be±23%. V. EXPERIMENTAL RESULTS AND DISCUSSION In Fig. 4 the measured absolute rate coefficient of Bi83+ with free electrons is plotted versus the relative energyfrom 0 eV to 125 eV. The spectrum shows the typical shape of an RR rate coefficient with a maximum at Erel = 0 eV and a continuous decrease for increasing relative energies. For the comparison of the measured rates with RR theory one has to know the maximum principal quan- tum number nmaxand the temperatures T||andT⊥(see sections II and III). For the temperature T⊥character- izing the transverse motion of the electrons one has to assume kT⊥= 120 meV which corresponds to the cath- ode temperature. For the longitudinal electron motion kT||= 0.1 meV is inferred from the analysis of resonance shapes in dielectronic recombination (DR) measurements with lithium-like Bi80+-ions performed during the same beamtime. nmaxis determined by field ionization in the dipole magnet which separates the parent beam and the recombined ions. A first approximation of this value is the field ionization limit nF, which is obtained from nF=/parenleftbigg 7.3×1010Vm−1q3 F/parenrightbigg1/4 (17) [34], where q is the charge state of the ion and F= vi,||B⊥the motional electric field seen by the ions with velocity vi,||in the transverse magnetic field B⊥of the charge-analyzing magnet. In this context one also has to account for the possibility that high Rydberg states can decay to states below nF, provided the ions have some time between the recombination process and the arrival at the ionizing electric field F. Therefore, a realistic es- timate for the cut-off is given by nmax= max( nγ, nF), (18) where nγdenotes the maximum principal quantum num- ber of Rydberg states which decay before the recombined ions arrive at the analyzing magnet and thus are saved from field ionization. A crude estimate of nγbased on a number of assumptions on the population and the decay of excited states is obtained from a numerical solution of the following equation [35] nγ=Z4/5{2.142×1010s−1tF[κ(nγ)]2 [−0.04 + ln( nγ)−lnκ(nγ)]}1/5, κ(nγ) = 1.6 + 0.018nγ. (19) The time of flight tFof the ions between the recombi- nation act and the arrival at the analyzing magnet in- fluences the survival probability of ions in high Rydberg states. In the derivation of Eq. 19 cascade processes were neglected. They should not be of large influence consid- ering the decay rates in undisturbed hydrogenic systems. A more problematic assumption underlying Eq. 19 is that the fields seen by the ions during their flight time are con- sidered not to change the nldistribution of states as it results from the initial population by radiative recombi- nation. Changes of decay rates by Stark mixing in the 5fields are not accounted for but probably deserve further attention. Also, instead of a sharp cut-off at nγornFa distribution of field ionization probabilities [34] around nγornF, respectively, would be more realistic, how- ever, these cannot easily be calculated without further assumptions. In addition, the influences of the experi- mental conditions providing electric fields via /vector v×/vectorBcon- tributions inside the interaction area, the toroidal fields and the dipole correction magnets after the cooler have to be considered. In the present experiment nFandnγhave been calcu- lated to be 116 and 442, respectively. A comparison of both RR rate curves resulting from Eqs. 5 and 6 is shown in Fig. 5. As expected the theoretical rate for nmax=nγ = 442 (dotted line) lies above the curve with nmax=nF = 116 (solid line). Comparing the experimental and the theoretical rate coefficients the curve calculated for nF= 116 shows a very good agreement with the experimental data for relative energies from Erel= 15 meV to 125 eV (Fig. 4). However, one can also obtain a good agree- ment for nγ= 442 if one assumes a higher transverse electron temperature kT⊥= 250 meV. Such an interde- pendency between nmaxandkT⊥has also been found in experiments with bare Cl17+and C6+-ions at the TSR in Heidelberg [8,21]. Because of the impossibility to accu- rately obtain both parameters from a fit of the theoreti- cal RR curve to the experimental spectrum and in view of the rather crude estimation of nmaxwe deliberately choose nmax=nF= 116 throughout the rest of this pa- per. This implies a reasonable choice for the transverse temperature of kT⊥= 120 meV, which is in accordance with the cathode temperature. That same temperature is also suggested by the accompanying DR measurements with Bi80+-ions. In this discussion the longitudinal tem- perature T/bardbldoes not play a role since it has very little influence on the RR rate coefficient. In Fig. 6 the experimental and theoretical data of Fig. 4 are shown again using a logarithmic energy scale in order to focus on a comparison at very low energies. The shape of the experimental spectrum with an additional increase towards low energies ≤15 meV is typical for low energy recombination measurements. At E rel= 0 eV we ob- tained a maximum rate coefficient of 1.5 ×10−7cm3s−1 exceeding the theoretical rate of 2.9 ×10−8cm3s−1by a factor of 5.2. As already mentioned in the introduction this rate enhancement phenomenon has also been ob- served at other facilities, however, the Bi83+-experiment provides the first quantitative determination of the rate enhancement factor ǫfor a bare ion with Z≥18. As mentioned above, for the light ions He2+, N7+, Ne10+ and Si14+aZ2.8dependence of ∆ α=αexp−αtheowas found in Stockholm [3]. This scaling cannot be directly confirmed for Bi83+. In order to obtain an agreement with the Z2.8-scaling the measured rate coefficient for Bi83+would have to be more than 400 % higher which is beyond the experimental uncertainty. However, one has to be careful with comparing results from differentfacilities since the experimental conditions vary drasti- cally. Apart from the extremely high electron energy of 162 keV and the extremely high nuclear charge in the present case the influence of the experimental param- eters kT⊥,kT/bardbland the magnetic guiding field B has to be considered. It is known from previous experi- ments with F6+and C6+ions [21] that the excess rate ∆αscales as ( kT||)−1/2and as ( kT⊥)−1/2. Using the present temperatures in comparison with the Stockholm conditions the excess rate found in the present experi- ment has to be multiplied by a factor of approximately/radicalbig 120mev/10meV·/radicalbig 0.1meV/ 0.12meV≈3.2 in or- der to normalize it to the Stockholm conditions. This removes part of the discrepancy mentioned above. The magnetic field dependence of ∆ αfound in similar exper- iments with other ions would, however, reduce the nor- malization factor again. To little is known to date about the exact numbers for such a normalization. As mentioned already in section IV the alignment of the beams has been carefully optimized before starting the recombination experiment. During the measurement we artificially introduced an angle φbetween the beams in order to check the obtained settings. This was im- plemented by superimposing in the interaction region a defined transverse (horizontal) magnetic field Bxin ad- dition to the unchanged longitudinal field Bzalong the ion beam direction. Fig. 7 shows the maximum recom- bination rate at Erel= 0 eV for different angles φfrom -0.6 mrad to 0.6 mrad in the horizontal plane. At φ= 0 mrad the maximum recombination rate is obtained. The open circles in Fig. 7 denote the expected rates for the selected angles. They have been determined by tak- ing recombination rates from the ( φ= 0 mrad)-spectrum at the minimum relative energies possible at the corre- sponding angle (see eq. 12). The squares in Fig. 7 repre- sent the measured rate coefficients at the minimum rel- ative energies Erelaccessible at the selected angles. All of them are lying above the expected value. Such a be- haviour is already known from other experiments at the ESR. Obviously the ion beam reacts on the introduction of the transverse magnetic field component, Lorentz and cooling forces appear to minimize the effect of the change. In any case, the distribution shows a rather symmetric progression with the maximum at φ= 0 mrad confirming the accurate adjustment of the cooler. In order to investigate the influence of the electron density on the recombination rate we performed recom- bination measurements for three different densities ne= 1.6×106cm−3,ne= 3.2×106cm−3and 4.7 ×106cm−3. In Fig. 8 the rate coefficient at E rel= 0 eV is plot- ted against the electron density n e. The solid line rep- resents the theoretical rate coefficient calculated with kT||= 0.1 meV and kT ⊥= 120 meV at E rel= 0 eV. There is a small difference between the maximum rate coeffi- cientαmax= 1.4×10−7cm3s−1forne= 1.6×106cm−3 andαmax= 1.5×10−7cm3s−1for the two higher den- sities but this deviation is within the experimental un- certainty. In addition Fig. 9 shows that the shape of 6the spectra is equal for all densities indicating identical temperatures in the measurements. Therefore it can be concluded that there is no significant influence of the elec- tron density on the recombination rate. This observation is in accordance with findings at the CRYRING [6], the TSR [8] and the GSI single pass electron target [9]. The lack of any density dependence rules out TBR (eq. 2) as a possible mechanism leading to enhanced recombination rates at low energies. With a significant contribution of TBR to the observed rates one would expect an increase of the recombination rate with increasing electron den- sity in contrast to all experimental observations. In the context of storage rings TBR has been discussed in some detail by Pajek und Schuch [37]. They found theoreti- cally that TBR effectively populates high Rydberg states of the ion where the electrons are very weakly bound. As mentioned above such ions are reionized in the dipole magnet and therefore do not contribute to the measured recombination rate. In a next stage of our experiment we also varied the magnetic guiding field of the electron beam between 70 mT and 150 mT in steps of 1 mT. The standard field strength used for the previous measurements was 110 mT. In contrast to the more careful adjustments of the magnetic field at the TSR [20,21] which were ac- companied by measurements of the cooling force and the beam profiles in order to preserve the beam quality no other cooler setting beside the magnetic field was changed at the ESR. This procedure was motivated by an earlier experiment of the ESR cooler group [38] showing an os- cillation of the recombination rate for 310 MeV/u U92+- ions induced by small changes of the magnetic field. The new results obtained for 295.3 MeV/u Bi83+are shown in Fig. 10 where the maximum recombination rate at Erel= 0 eV is plotted versus the magnetic field strength B||. Since the measurement of a complete recombination spectrum is very time-consuming only the recombination rate at cooling has been recorded for each magnetic field setting. For these data points a background subtraction and corrections due to the potential and angle distribu- tions inside the cooler were not possible. Therefore these recombination rates represented by the open circles in Fig. 10 can only display the qualitative dependance on the magnetic field. A complementary method of determining the recom- bination rate at cooling can be applied by analyzing the lifetime of the Bi83+beam in the ring. Fitting an expo- nential curve I(t) =I0·exp(−t/τ)+IOffto the decay of the ion current stored in the ring with I0being the ion current at the beginning and with IOffrepresenting the offset of the ion current transformer, one can extract the storage lifetime τof the Bi83+-beam in the ring. Assum- ing that electron-ion recombination in the cooler is the only loss mechanism for stored ions one can calculate the corresponding recombination rate αl= 1/(τneff).neff is the effective electron density which is netimes the ra- tio of the interaction length and the ring circumference. Fig. 11 shows a comparison of the rates obtained withthe different methods resulting in a good agreement of the data. A Fourier analysis of the experimental data reveals an ”oscillation period” of the recombination rate of 7.6 mT. Recent measurements of the ESR cooler group with 300 MeV/u Kr36+,35+-ions show that this value can be influ- enced by a variation of the magnetic fields of the toroids (see Fig. 1) moving the electron beam into and out of the interaction area . Due to the Lorentz force the electrons are moving on helical trajectories through the cooler. It has been observed by the ESR group that the ”oscilla- tion period” corresponds to a change of the magnetic field that allows the electrons one more turn inside the toroid. In order to find an explanation for these observations one has to take a precise look at the measured recombination spectra. Therefore we performed complete recombina- tion measurements with all corrections for 12 selected field strengths. The results are represented by the full triangles in Fig. 10. The maximum recombination rates atErel= 0 eV obtained in these measurements show the same progression as the open circles. In Fig. 12 recombination spectra for different magnetic fields between B = 109 mT and 114 mT are shown in the energy range from Erel= 0 eV to 125 eV. For Erel≥ 1 eV the measured rate coefficients are practically iden- tical. At energies below 1 eV the recombination rates show significant differences depending on the magnetic field strength. However, the rate enhancement is always there which one can see in Fig. 13 where the recombi- nation rate for the oscillation minimum at B = 114 mT is compared with RR theory. In order to describe the experimental data a transverse electron temperature of kT⊥= 450 meV was applied which is nearly a factor of 3 higher than the one obtained by a fit of RR theory to the experimental data at the standard magnetic field strength of B = 110 mT. From the measured rate at Erel = 0 eV of αexp= 6.6×10−8cm3s−1and the theoretical value at Erel= 0 eV of αtheo= 1.5×10−8cm3s−1one calculates a rate enhancement factor ǫ=αexp/αtheo= 4.4. Analysing all the complete recombination spectra mea- sured for different magnetic fields a connection between the maximum rate coefficient at Erel= 0 eV and the adapted transverse electron temperature kT⊥becomes obvious which is documented in Fig. 14. In the left panel representing the data for magnetic fields between 76 mT and 80 mT the measured maximum rate coefficient at Erel= 0 eV (full triangles) increases with increasing mag- netic field whereas the transverse electron temperature (open circles) adapted to the corresponding recombina- tion spectra decreases in this region. In the right spec- trum monitoring the same data for magnetic fields be- tween 109 mT and 114 mT one can see a decrease of the maximum rate coefficient at Erel= 0 eV accompanied by an increase of the transverse electron temperature kT⊥. Therefore there seems to be a relationship between the rate coefficient and the transverse electron temperature (or: the electron beam quality) which would easily ex- 7plain the periodic reductions of the recombination rate atErel= 0 eV. Nevertheless it should be pointed out that this interpretation does not explain the observed general enhancement of the rate coefficient at Erel= 0 eV. A further theoretical analysis should especially focus on th e possible influence of the electron energy since in exper- iments with lithium-like 97.2 MeV/u Bi80+ions (corre- sponding to 53.31 kV cooling voltage instead of 162 kV for 295.3 MeV/u) during the same beam-time the os- cillations did not appear. This is in agreement with the results obtained at the ESR and other storage rings in ex- periments at low ion energies. There, oscillations of the recombination rate at cooling have not been observed. VI. CONCLUSIONS The recombination of bare Bi83+ions with free elec- trons has been studied at the GSI Experimental Stor- age Ring (ESR) in Darmstadt. Within the experimen- tal uncertainty we found very good agreement between the measured rate coefficient and the theory for radia- tive recombination (RR) for energies from E rel= 15 meV to 125 eV. At very low center-of-mass energies between ions and electrons, however, the measured rate exceeds the theoretical predictions by a factor of 5.2. This first rate enhancement result for a bare ion with Zvery much greater than 18 does not follow a Z2.8dependence of the enhancement found in an experiment with the light ions He2+, N7+, Ne10+and Si14+. Therefore additional mea- surements in the mid-Z range are necessary in order to obtain more information about the influence of Zon the recombination enhancement phenomenon. The increase of the electron density in the interaction area from ne = 1.6×106cm−3to 4.7×106cm−3appears to have no significant effect on the recombination rate. A variation of the magnetic field from 70 mT to 150 mT revealed a strong dependence of the recombination rate at low ener- gies on this parameter. The observed oscillations of the maximum recombination rate at Erel= 0 eV confirmed previous observations with bare U92+ions. Compar- ing the recombination spectra with RR theory one finds strong variations of the transverse electron temperature connected to the oscillations of the recombination rate. In future experimental and theoretical studies this rela- tionship has to be investigated in more detail. Finally, we want to emphasize, that the observed recombination rate enhancement significantly reduces the lifetime of ion beams in storage rings during the electron cooling pro- cedure. At the present ion energies, recombination in the cooler by far dominates over all factors influencing the beam lifetime. The recombination rate enhancement hence reduces the beam lifetime by approximately a fac- tor 5 as compared to the assumption of pure RR.VII. ACKNOWLEGMENTS The Giessen group gratefully acknowledges support for this work through contract GI M ¨UL S with the Gesellschaft f¨ ur Schwerionenforschung (GSI), Darmstadt , and by a research grant (number 06 GI 848) from the Bundesministerium f¨ ur Bildung und Forschung (BMBF), Bonn. [1] W.G. Graham et al. (eds), Recombination of Atomic Ions, NATO ASI Series B: Physics vol 296 (Plenum Press,New York, 1992) [2] L.H. Andersen, J. Bolko, P. Kvistgaard, Phys. Rev. Lett. 64, 729 (1990) [3] H. Gao, R. Schuch, W. Zong, E. Justiniano, D.R. De- Witt, H. Lebius, W. Spies, J.Phys. B 30, L499 (1997) [4] A. Wolf, J. Berger, M. Bock, D. Habs, B. Hochadel, G. Kilgus, G. Neureither, U. Schramm, D. Schwalm, W. Szmola, A. M¨ uller, M. 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July 1997 ion- beam A(q-1)+ q+Adrift□tubes -5kV...+5kVdipole□magnet scintillatore -beam-gun toroid toroidcollector magnetic□field FIG. 1. Schematic view of the ESR electron cooler and the experimental set-up for recombination measurements. The cold electron beam produced in the gun is guided by the mag- netic field and merged with the ion beam over a distance of 2.5 m. The electron beam is then separated from the ion beam by the magnetic guiding field and transferred to the collecto r. Recombined and parent ions which leave the cooler together are separated from each other in the first dipole magnet af- ter the cooler. A scintillator detector is used to count the recombined particles. 0 50 100 150 200 2500.00.20.40.60.81.0 0 50 100 150 200 2500.00.20.40.60.81.01.21.41.61.8 Drift Tube Potential (V) Position inside the Cooler (cm) Angle (mrad) 9FIG. 2. Electric potential and angle φbetween electron and ion trajectories along the cooler axis. The dependences are displayed along the 2.5 m long straight section between the toroids. The position of the drift tubes covering 2 m of the interaction region is also indicated. The left panel shows t he calculated electric potential along the cooler axis if a vol tage of 1 V is applied to the drift tubes. In the right panel the angl e φis shown corresponding to a measurement of the magnetic guiding field. Here, perfect alignment of the ion beam with the geometrical cooler axis is assumed which is attainable b y using the steerer magnets. 0 50 100 150 200 25010-510-410-310-210-1100101102103 5000 V 1000 V 500 V 100 V 50 V 10 V Relative Energy (eV) Position inside the cooler (cm) FIG. 3. Relative energies between electrons and ions along the straight overlap section inside the cooler. The energie s have been calculated according to Eq. 12 taking into account the distributions of the electric potential and the angle φ (from Fig. 2). The voltages of 10 V, 50 V, 100 V, 500 V, 1000 V and 5000 V applied to the drift tubes correspond to Erel= 1.7×10−4eV, 0.003 eV, 0.013 eV, 0.33 eV, 1.3 eV and 32.7 eV. 0 20 40 60 80 100 120024681012141618 Experiment RR-Theory (kT|| = 0.1 meV, kT⊥ = 120 meV, nmax = 116)Recombination rate (10-8 cm3/s) Relative energy (eV)FIG. 4. Measured absolute recombination rate of Bi83+ (circles) with free electrons plotted against the relative en- ergy between electrons and ions. The spectrum shows the typical shape of the RR peak with a continuous decrease of the rate with increasing energy. There is a very good agree- ment between the experimental data and RR theory from Erel = 15 meV to 125 eV. The theoretical curve follows from Eqs. 5 and 6 using kT||= 0.1 meV, kT⊥= 120 meV and nmax= 116 in the distribution function (Eq. 7). 10-510-410-310-210-11001011020.00.51.01.52.02.53.03.54.0 nγ = 442 nF = 116Recombination rate (10-8 cm3/s) Relative energy (eV) FIG. 5. Calculated rate coefficients for nmax=nγ= 442 andnmax=nF= 116. For both curves electron beam tem- peratures kT||= 0.1 meV and kT⊥= 120 meV have been used. 10-61x10-51x10-410-310-210-1100101102024681012141618 Experiment RR-Theory (kT|| = 0.1 meV, kT⊥ = 120 meV, nmax = 116)Recombination rate (10-8 cm3/s) Relative energy (eV) FIG. 6. Comparison between the measured rate coefficient for Bi83+and RR theory. At Erel= 0 eV a theoretical rate coefficient of 2.8 ×10−8cm3s−1has been obtained using kT|| = 0.1 meV, kT⊥= 120 meV and nmax= 116. The experimen- tal rate of 1.5 ×10−7cm3s−1exceeds this value by a factor of 5.2. 10-0.6-0.4-0.2 0 .0 0 .2 0 .4 0 .60246810121416 Recombination rate (10-8 cm3/s) angle φ (mrad) FIG. 7. Maximum recombination rates at Erel= 0 eV for different angles φset between electron and ion beam. The open circles denote the values expected on the basis of the (φ= 0 mrad)-spectrum. The measured rate coefficients are represented by the squares. The differences at angles larger than 0 mrad a partially due to the small misalignments in the magnetic field (Fig. 2) as well as to the betatron oscillation s. 1.5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .00246810121416 αexp at Erel = 0 eV αtheo at Erel = 0 eV Recombination rate α (10-8 cm3/s) Electron density ne (106 cm-3) FIG. 8. Influence of the electron density on the recombi- nation of Bi83+. The squares represent the measured rate coefficient at Erel= 0 eV plotted against the electron den- sity. The solid line shows the theoretical RR rate coefficient atErel= 0 eV calculated with kT||= 0.1 meV and kT⊥= 120 meV.10-610-510-410-310-210-1100101102024681012141618 ne = 1.6 x 106 cm-3 ne = 3.2 x 106 cm-3 ne = 4.7 x 106 cm-3Rekombination rate α (10-8 cm3/s) Relative energy (eV) FIG. 9. Measured absolute rate coefficients for Bi83+for different electron densities. The shapes of the spectra are practically identical. Greater fluctuations of the recombi na- tion rate for ne= 4.7×106cm−3are due to the measurement procedure. For this electron density a random distribution of the drift tube voltages has been used instead of a continu- ous increase of the voltage for each measurement step. Com- parisons of spectra measured with both procedures show the same progressions but the spectra obtained with the random distribution are not so smooth. 60 80 100 120 140 160024681012141618 Recombination rate (10-8 cm3/s) Magnetic field (mT) FIG. 10. Measured maximum recombination rate at Erel = 0 eV versus the magnetic field. The open circles represent the uncorrected recombination rate (see also Fig. 11). Com- plete recombination measurements with all corrections wer e performed for 12 selected field strengths indicated by the fu ll triangles. 1160 80 100 120 140 1600123456789 Recombination rate (10-8 cm3/s) Magnetic field (mT) FIG. 11. Comparison of recombination rates at Erel= 0 eV obtained via two different methods. The open circles show the recombination rate calculated from the counting rate of recombined particles. These data do not include the back- ground subtraction and corrections due to the potential and angle distribution inside the cooler. Therefore it can only give a qualitative overview. The full triangles represent rate c o- efficients obtained via the storage lifetime of the Bi83+beam in the ring. There is a good overall agreement between the different approaches. 10-61x10-51x10-410-310-210-11001011020246810121416 114 mT 113 mT 112 mT 111 mT 109 mTRecombination rate (10-8 cm3/s) Relative energy (eV) FIG. 12. Comparison of Bi83+recombination spectra for magnetic fields between B = 109 mT and 114 mT. For rela- tive energies Erel≥1 eV the shapes of the spectra are prac- tically identical. At energies below 1 eV there are significa nt differences depending on the magnetic field.10-610-510-410-310-210-110010110201234567 Experiment RR-Theory (nmax = 116, kT|| = 0.1 meV, kT⊥ = 450 meV)Recombination rate (10-8 cm3/s) Relative energy (eV) FIG. 13. Comparison between the measured absolute rate coefficient for Bi83+at the oscillation minimum at B = 114 mT and RR theory. In order to describe the experimental progression the theoretical curve was calculated with a tra ns- verse temperature kT⊥= 450 meV. The longitudinal temper- ature kT||= 0.1 meV and the maximum principal quantum number nmax= 116 were the same as before. 76 77 78 79 804.04.55.05.56.06.57.07.5Transverse temperature kT⊥ (meV) Recombination rate α (10-8 cm3/s) 300350400450500 108 110 112 1146810121416 Magnetic field B (mT)150200250300350400450 FIG. 14. Comparison of the measured maximum rate co- efficients (full triangles) for different magnetic fields and t he transverse electron temperatures (open circles) inferred from fits to the corresponding recombination spectra at energies be- yond 0.1 eV. The left panel represents the data for magnetic fields between B = 76 mT and 80 mT. As already displayed in Fig. 10 the rate coefficient increases with increasing magnet ic field. The transverse electron temperature shows the opposi te behaviour and decreases with increasing magnetic field. Thu s the magnitude of the maximum rate coefficient at Erel= 0 eV is apparently strongly related to the transverse electron t em- perature. This relationship is also present in the right pan el in which the data for magnetic fields between B = 109 mT and 114 mT is presented. 12

arXiv:physics/0003089v1 [physics.plasm-ph] 28 Mar 2000Resistive axisymmetric equilibria with arbitrary flow M. P. Bora∗ Institute for Plasma Research, Bhatt, Gandhinagar 382 428, India. An analysis of axisymmetric equilibria with arbitrary inco mpressible flow and finite resistivity is presented. It is shown that with large aspect ratio approx imation or vanishing poloidal cur- rent, a uniform conductivity profile is consistent with equi librium flows. Also a comment made on coexistence of both toroidal and poloidal flows in an axisymm etric field-reversed configuration. PACS numbers : 52.30.Bt, 52.30.Jb, 52.55.-s Calculating the equilibrium is one of the fundamental problems of magnetically confined plasmas. Most stud- ies are directed toward finding an ideal (i.e. infinitely conducting) magnetohydrodynamic (MHD) equilibria in an axisymmetric plasma. The earliest calculations are those of Grad1and Shafranov2, leading to the famous Grad-Shafranov equation1–3. The ideal and static Grad- Shafranov equation is an elliptic differential equation in the magnetic flux function ψwith two arbitrary surface quantities as the pressure p(ψ) and the poloidal current I(ψ). Consequently, there have been attempts to include var- ious effects e.g. mass flow into the equilibrium equations. An equivalent of Grad-Shafranov equation in an axisym- metric ideal plasma with arbitrary flow has been given by Hameiri4. In some recent works, Steinhauer5deals with a generalization of Grad-Shafranov equilibria in a multi-fluid with flow and Throumoulopoulos and Tasso6 consider a helically symmetric equilibria with flow. The situation with flow becomes more realistic when one re- alizes the existence of equilibrium flows both in toroidal and poloidal directions in tokamaks following momentum deposition through heating by neutral beam injection7–9. With equilibrium flows, the resultant governing differen- tial equation does not remain always elliptic10,11. The in- vestigation of a general MHD equilibrium becomes much more complicated when one tries to include the effects of other important factors, say of viscous stress tensor. Recently Ren et al.12have studied the deformation of magnetic island by including the effect of sheared flow and viscosity into an ideal two-dimensional MHD equi- librium configuration. However, there is an element of inconsistency, whether an ideal equilibrium is realistic13. Heuristically, one ignores the resistivity in the Ohm’s law while calculating the equilibrium, but then a resistive stability analysis based on a stationary equilibrium re- mains questionable as long as the field diffusion is not taken into account14. Montgomery et al.13have investi- gated the problem on non-ideal static axisymmetric equi- librium. There have also been attempts to calculate resis-tive axisymmetric equilibrium with only toroidal flow10. It has been further argued that tokamak equilibrium flow is either purely toroidal7,9or the poloidal component is small8and quickly damped by magnetic pumping15. So there is a natural tendency to exclude the poloidal flow while calculating an equilibrium. But when one consid- ers finite conductivity with purely equilibrium toroidal flow, the conductivity (hence the resistivity) becomes a function of space. In general, the resistivity is not a flux function10,13irrespective of equilibrium flow. In this re- port, we ask the very pertinent question, whether the sit- uation changes in presence of poloidal flow. As we show that a uniform resistivity profile is consistent in presence of poloidal flow, whereas it has been shown that a scalar pressure equilibrium can not have uniform resistivity16. Further we show that in a field-reversed (FRC) axisym- metric configuration with no toroidal magnetic field, both toroidal and poloidal equilibrium flows can coexist with finite resistivity, which is not found to be the case with ideal equilibrium17. We consider the equilibrium resistive MHD equations with plasma flow. The equations are ∇ ·(ρv) = 0, (1) ∇ ·B= 0,∇ ×E= 0,∇ ×B= 0, (2) E+v×B=j/σ, (3) ρ(v· ∇)v=j×B− ∇p, (4) where the symbols have their usual meanings. We use a right handed cylindrical system ( r,θ,z) withzas the axis of symmetry, θas the toroidal angle, and ralong the major radius of an axisymmetric device. We assume the plasma flow to be arbitrary (toroidal and poloidal) and axisymmetry is assumed i.e. ∂/∂θ= 0. The plasma resis- tivityη=σ−1is assumed to be an unspecified function ofrandz. We have further assumed here that the equi- librium is maintained in a steady-state through resistive diffusion. The magnetic induction equation allows us to write the magnetic field as B=1 r∇ψ׈eθ+I rˆeθ, (5) ∗Permanently at Physics Department, Gauhati University, Guwahati, Assam 7 81 014, India. Electronic mail : mbora@gw1.dot.net.in 1whereψis the magnetic flux function which is the az- imuthal component of the vector potential AandIis the current function. Similarly, following the continuity equation, Eq.(1), we can express the plasma equilibrium velocity as v=1 ρr∇ϕ׈eθ+ωrˆeθ, (6) whereϕis the velocity stream function and ω=vθ/ris the toroidal angular velocity. We also assume that the flow is incompressible i.e. ∇ ·v= 0. Because the flow is now in both toroidal and poloidal direction, the poloidal component of current, jpneed not vanish. In general the current can be expressed as j=−1 r∆∗ψˆeθ+1 r∇I׈eθ, (7) where ∆∗is the elliptic operator defined by ∆∗ψ= r2∇ ·/parenleftbig1 r2∇ψ/parenrightbig .Taking curl of the Ohm’s law Eq.(3), we have ∇ ×(v×B) =∇ ×(j/σ) (8) with v×B=1 ρr2∇ϕ× ∇ψ−I ρr2∇ϕ+ω∇ψ. (9) Theˆeθcomponent of Eq.(8) can be now written as ˆeθ· ∇ω× ∇ψ−ˆeθ· ∇/parenleftbiggI ρr2/parenrightbigg × ∇ϕ =1 σr/parenleftbigg1 σ∇σ· ∇I+2 r∂I ∂r− ∇2I/parenrightbigg . (10) We now invoke the large aspect ratio expansion and assume that toroidal magnetic field, to the first approx- imation can be written Bθ≃B0r0/r. Here,B0is the value of the toroidal magnetic field at center of the cylin- drical cross section of the torus at distance r0from the axis of symmetry. To this effect we have the current functionI≈B0r0= const. Under these assumption, the above equation can be written as ˆeθ· ∇ω× ∇ψ=−I ρ2r2ˆeθ· ∇ρ× ∇ϕ −2I ρr3ˆeθ· ∇r× ∇ϕ. (11) The first term on the right hand side of the above equa- tion vanishes by virtue of the incompressibility condition and the continuity equation. We neglect the second term because of its 1 /r3dependence and find that the toroidal angular velocity vθ/r=ω(ψ) becomes a surface quantity. This also further means that j=jθˆeθwithjp= 0. We, however, note that the condition ω≡ω(ψ) is identically satisfied in a field-reversed configuration (FRC) whereI= 0 and large aspect ratio approximation is not re- quired. It can be noted here that without any approxima- tion,ωbecomes a flux function when one considers ideal equilibrium4or resistive equilibrium with only toroidal flow10. Now, we consider the momentum equation Eq.(4) and itsˆeθcomponent. With the above approximations, we can write Eq.(4) as jθˆeθ×Bp=∇P+ρ∇/bracketleftbigg1 2ρ2r2(∇ϕ)2/bracketrightbigg − ∇ ·/parenleftbigg1 ρr2∇ϕ/parenrightbigg ∇ϕ−ω′(∇ϕ× ∇ψ) −2ω r∂ϕ ∂zˆeθ−ω2r∇r (12) In the above equation Bpis the poloidal component of the magnetic field and (′) denotes derivative with respect toψ. Taking the ˆeθcomponent of the above equation, we have, ˆeθ· ∇ϕ× ∇(ωr2) = 0, (13) which means that ϕ≡ϕ(ωr2). We take the simplest situation of ϕ∝ωr2which yields another surface quan- tity,ϕ/r2=ζ(ψ). However, it is important to note that, ζ(ψ) is not an arbitrary function in the sense that it is proportional to the toroidal velocity ω(ψ) i.e. the toroidal and poloidal flows are no longer independent. Physically, one can understand this by noting that finite resistivity allows plasma motion across the the flux surfaces. Equiv- alently, toroidal flow, in a resistive axisymmetric plasma, is always associated with poloidal flow. Because of equilibrium flow, however, plasma pressure pis no longer a flux function now. Taking the Bpcom- ponent of the momentum equation (Eq.(12)), we have Bp·/bracketleftbigg∇p ρ+∇/braceleftbigg1 2ρ2r2(∇ϕ)2−1 2ω2r2/bracerightbigg/bracketrightbigg =1 ρ∇ ·/parenleftbigg1 ρr2∇(ζr2)/parenrightbigg Bp· ∇(ζr2). (14) Depending upon the equation of state, now, several options are possible. However, we note that density, in general, is not a flux function in presence of arbitrary plasma flow. this can be easily seen from the equation of continuity Eq.(1), after applying the incompressibilit y condition, ˆeθ· ∇ϕ× ∇ρ= 0. (15) It can be seen from the above expression that ρis not a surface quantity. We note here that axisymmetric equi- libria with incompressible equilibrium flows are generally associated with constant density magnetic surfaces18–20. One is also free to choose density as a flux function in case of resistive axisymmetric equilibrium with only in- compressible toroidal flow10. 2Taking the ˆeθcomponent of Ohm’s law Eq.(3) along with Eq.(9), we have an expression for plasma conduc- tivity, σ/parenleftbigg E0r0+2 ρrζBr/parenrightbigg + ∆∗ψ= 0, (16) whereE0is the longitudinal externally applied electric field at major radius r=r0. We immediately see from the above expression that conductivity, in general, is a space dependent quantity. In what follows, we shall consider two cases with (i) uniform and constant density and ( ii) a nonuniform density. In the second case, we consider isentropic mag- netic surfaces. We now assume that plasma density is uniform and constant i.e. ρ= const.and normalize our equations to ρ= 1. We can now write Eq.(14) as, Bp· ∇/bracketleftbigg p+1 2r2(∇ϕ)2−1 2ω2r2/bracketrightbigg =1 r2∆∗(ζr2)Bp· ∇(ζr2). (17) Integration of the above equation yields the equivalent Bernoulli’s equation, p+(∇ϕ)2 2r2−ω2r2 2=/integraldisplaydl Bp1 r2∆∗(ζr2)Bp· ∇(ζr2) +χ(ψ), (18) where the integration is along a magnetic field line and χ(ψ) is an arbitrary surface quantity. The solubility con- dition further requires that /contintegraldisplaydl Bp1 r2∆∗(ζr2)Bp· ∇(ζr2) = 0. (19) We further assume that part of the pressure gradient that varies within a magnetic flux tube has no ∇ψ component17i.e. ∇ψ· ∇/contintegraldisplaydl Bp1 r2∆∗(ζr2)Bp· ∇(ζr2) = 0. (20) Together with Eq.(18) and the above assumption, the ∇ψ component of the momentum equation yields the equiv- alent Grad-Shafranov equation, ∆∗ψ+r2(χ′+ω′ω2r) +∆∗(ζr2)∇ψ· ∇(ζr2) |∇ψ|2= 0,(21) with two arbitrary flux functions χ(ψ) andω(ψ). The primes refer derivative with respect to ψ. We now consider the second case where we consider a nonuniform density. With incompressible flow, mag- netic surfaces with constant entropy is quite a reasonable approximation in ideal MHD4,21. However, considering long resistive diffusion time, the right hand side of Eq.(3) can be neglected and we can continue to proceed withisentropic magnetic surfaces10. The equation of state can now be written as, p=Sργ, where,S(ψ) is the entropy which is a flux function and γis the ratio of specific heats. We now write Bp· ∇p/ρasBp· ∇[γSργ−1/(γ−1)], so that equivalent Bernoulli’s equation can be written as, Θ(ψ) +/integraldisplaydl Bp1 ρ∇ ·/parenleftbigg1 ρr2∇(ζr2)/parenrightbigg Bp· ∇(ζr2) =γ γ−1Sργ−1+1 2ρ2r2(∇ϕ)2−1 2ω2r2,(22) where Θ(ψ) is arbitrary. As we have assumed previously, it requires a solubility condition and the equivalent to the assumption (20). We can then continue to write the equivalent Grad-Shafranov equation by taking the ∇ψ component of the momentum equation and applying the Bernoulli’s law Eq.(22), ∆∗ψ+r2/parenleftbigg Θ′+ω′ω2r−S′ργ−1 γ−1/parenrightbigg =r2 ρ|∇ψ|2∇ ·/bracketleftbigg1 ρr2∇(ζr2)/bracketrightbigg ∇ψ· ∇(ζr2). (23) In the above equation we have four arbitrary surface quantities i.e. Θ( ψ),ω(ψ), andS(ψ) and the primes de- note derivative with respect to ψ. We have derived the differential equations, equivalent to the Grad-Shafranov equation, for resistive axisymmet- ric plasma with arbitrary equilibrium flows. These equi- librium equations Eqs.(21, 23) have to be solved subject to conductivity constraint Eq.(16). Further, in a field- reversed configuration (FRC) with no toroidal magnetic field, it can be seen from Eq.(13) that both poloidal and toroidal flow can coexist. We now show that a uniform conductivity profile is consistent with resistive axisymmetric equilibria with ar - bitrary flow. A simple examination of Eq.(4), though reveals that uniform conductivity may be possible with scalar pressure equilibrium in presence of flow, it how- ever provides no easier way of proving it. We note that the usual procedure for solving Eqs.(21) and (23) requires specifying a priori dependence of the respective arbitrary functions on ψ. However, in the presence of finite resis- tivity, the resistivity constraint Eq.(16) can be used to solve forψ, which is uniquely determined if the right hand side of Eq.(16) is specified16. It should be noted here with caution whether the resultant solution for ψ corresponds to realistic profiles for other physical quan- tities such as pressure, density, velocity etc. However, our sole aim, here, is to demonstrate the existence of a solution consistent with uniform resistivity in presence o f flows. From Eq.(15) we know that ρ≡ρ(ϕ), and assume that ρ∝ϕ. We now assume that conductivity is uniform in space so that the resulting Eq.(16) can be written as, ∆∗ψ+α r2∂ψ ∂z=β, (24) 31 1.25 1.5 r-0.2500.25 z 1 1.25 1.5 r-0.2500.25 z FIG. 1. Constant flux ( ψ) contours for conducting circular boundary with ( a)σ= const.and (b)σ∝r2. The major radius isr0= 1.25. whereαandβare arbitrary constants. It is worthwhile mentioning at this point that Eq.(24) can not be used in case of very small resistivity. In the limit of vanishing resistivity (large βin the above equation), the solution of Eq.(24) contains short scale spatial dependence (bound- ary layers), not present in case of ideal equilibrium and may lead to unphysical results. Note that the above equation is a elliptic equation and can be treated as boundary value problem. Following Zheng et al.22, we assume a solution of the form ψh(r,z) =/summationdisplay n=0,1,2,...fn(r)zn(25) for the homogeneous part of Eq.(24). We however re- tain the odd terms in the summation to take care of the asymmetric-term in Eq.(24). For simplicity we as- sume that fn(r) = 0 forn≥3, which, however, can be extended up to any number of terms if required, about which we shall make a comment later. Substi- tuting Eq.(25) in the equivalent homogeneous equation for Eq.(24) we can solve for the functions f0,1,2(r). The homogeneous solution of Eq.(24) is then given by, ψh(r,z) =a1r2{4r2+ 16z2−α2[4(lnr)2−4 lnr+ 2 −r2] + 8αz(r2−2 lnr)}+a2[2r2(2 lnr −1)−α2lnr(lnr+ 1) + 4αzlnr+ 4z2] +a3r2[α(2 lnr−1) + 4z] +a4r2, (26) whereais are arbitrary constants to be determined from the boundary conditions. A particular solution of Eq.(24) isψp=βr2(2 lnr−1)/4. So the complete solution of Eq.(24) is ψ=ψh+ψp, (27) which can be verified by direct substitution. For a con- ducting circular boundary of a toroidal axisymmetric de- vice, the constant flux ( ψ) contours are shown in Fig.1 (a) which shows a scaler pressure equilibrium. The solu- tion forσ∝r2is shown in Fig.1 ( b). Note that σ∝r2 is the only possible solution for resistive axisymmetric equilibrium without flow16.In principle the expansion in Eq.(26) should be re- tained with a large number of terms which will result a equally large number of arbitrary constants for the so- lution inψ. These constants can then be used to shape any arbitrarily shaped plasma boundary. In passing, we would like to note that resistive field diffusion (∂B/∂t ∝ne}ationslash= 0) is intrinsically involved with non- stationary equilibria ( v∝ne}ationslash= 0). However, a series of ideal quasi-stationary equilibrium states can be built up with ∂B/∂t = 0 in which, the effect of finite resistivity is only to slowly evolve the equilibrium in a diffusive time scale23. It is a pleasure to thank A. Sen for the kind hospitality at IPR where part of this work has been completed. 1H. Grad and H. Rubin, in Proceedings of the Second United Nations Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958 , edited by United Nations (United Nations Publications, Geneva, 1958), Vol. 31, pp. 190. 2V. D. Shafranov, Sov. Phys. JETP 6, 545 (1958). 3R. L¨ ust and A. Schl¨ uter, Z. Naturforsch 12a, 850 (1957). 4E. Hameiri, Phys. Fluids 26, 230 (1983). 5L. C. Steinhauer, Phys. Plasmas 6, 2734 (1999). 6G. N. Throumoulopoulos and H. Tasso, “Ideal magnetohy- drodynamic equilibria with helical symmetry and incom- pressible flows”, e-print physics/9907004 (to be published in J. Plasma Phys.). 7S. Suckewer, H. P. Eubank, R. J. Goldston et al., Phys. Rev. Lett. 43, 207 (1979). 8K. Brau, M. Bitter, R. J. Goldston et al., Nucl. Fusion 23, 1643 (1983). 9S. D. Scott, M. Bitter, H. Hsuan et al., inProceedings of the 14st European Conference on Controlled Fusion and Plasma Physics, Madrid, 1987 (European Physical Soci- ety, Geneva 1987), Vol. 11D, pp. 65. 10G. N. Throumoulopoulos, J. Plasma Phys. 59, 303 (1998). 11S. Semenzato, R. Gruber, and H. P. Zehrfeld, Comp. Phys. Rep.1, 389 (1984). 412C. Ren, M. S. Chu, and J. D. Callen, Phys. Plasmas 6, 1203 (1999). 13D. Montgomery, J. W. Bates, and H. R. Lewis, J. Plasma Phys. , (1997). 14D. Dobrott, S. C. Prager, and J. B. Taylor, Phys. of Fluids 20, 1850 (1977). 15A. B. Hassam and R. M. Kulsrud, Phys. Fluids 21, 2271 (1987). 16J. W. Bates and H. R. Lewis, Phys. Plasmas 3, 2395 (1996). 17R. A. Clemente and R. L. Viana, Plasma Phys. Control. Fusion 41, 567 (1999). 18K. Avinash, S. N. Bhattacharyya, and B. J. Green, PlasmaPhys. Control. Fusion 34, 465 (1992). 19G. N. Throumoulopoulos and G. Pantis, Plasma Phys. Control. Fusion 38, 1817 (1996). 20G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 4, 1492 (1997). 21A. I. Morozov and L. S. Solov´ ev, in Reviews of Plasma Physics , edited by M. A. Leontovich (Consultants Bureau, New York, 1980), Vol. 8, pp. 1. 22S. B. Zheng, A. J. Wootton, and R. Solano, Phys. Plasmas 3, 1176 (1996). 23H. Grad and J. Hogan, Phys. Rev. Lett. 24, 1337 (1979). 5

arXiv:physics/0003090v1 [physics.comp-ph] 28 Mar 2000Comparison of direct and Fourier space techniques in time-dependent density functional theory G.F. Bertsch(a)∗, Angel Rubio(b)†, and K. Yabana(c)‡ (a)Department of Physics and National Institute for Nuclear Th eory, University of Washington Seattle, WA 98195 b)Departamento de F´ ısica Te´ orica, Universidad de Valladol id, E-47011 Valladolid, Spain and Donostia International Physics Center, San Sebastian, Spain (c)Institute of Physics, University of Tsukuba, Tsukuba 305-8571, Japan Abstract Several techniques have appeared in the literature to solve the equations of time-dependent density functional theory. We compare the e fficiency of dif- ferent methods based on mesh representations of the wave fun ctions (direct and Fourier space), taking as a test case the calculation of t he surface plas- mon in the cluster Na 8. For smaller systems, the methods have comparable efficiency, but for large systems the direct space and time met hods have a decided advantage. ∗E:mail: bertsch@phys.washington.edu †E:mail: arubio@mileto.fam.cie.uva.es ‡E:mail: yabana@nucl.ph.tsukuba.ac.jp 1I. INTRODUCTION The time-dependent local density approximation has proven to be a useful tool to cal- culate the optical properties of finite systems such as atoms , molecules, and atomic clusters [1–8]. The basic equation to be solved is conceptually very s imple, little more than the time-dependent Schr¨ odinger equation for a particle in a ti me-varying external field. Many numerical methods are in use to solve the equations. On the on e side there are quantum chemistry methods based on atomic orbital representation f or the wave function, and on another side there are methods based on mesh representation s. We only consider the latter here, but even in this category there are a number of publishe d techniques. Most funda- mentally, the time evolution can be calculated directly or i n Fourier space, i.e. in terms of frequencies. The former method is practically a necessit y for dealing with very strong external fields [10,11] and has been applied by two of us (K.Y. and G.B.) for the weak-field response as well [4]. We shall call this approach the “nuclea r physics”(NP) method, since the algorithms were originally developed in that field for de scribing nuclear reactions [12]. The other methods we will consider [3,8] solve equations in f requency space. The method described in ref. [3] had its origins in condensed matter the ory and uses Fourier represen- tation for both space and time; we shall call this the “conden sed matter” (CMP) method. We also comment on ref. [8] which uses Fourier space for the ti me but a real space mesh for the spatial dependence [13]. Here the problem is cast into a m atrix diagonalization in the particle-hole representation; we shall call it the diagona lization method. In this work we will compare the CMP code and the NP code for a sp ecific system and present arguments for the scaling properties of the respect ive algorithms for larger systems. The system we choose to study is the atomic cluster Na 8, and in particular the surface plasmon excitation which is seen as a strong peak at 2.5 eV exc itation. The TDLDA is not an exact theory and it predicts a excitation energy at about 2 .7 eV. We shall demand of both methods that they achieve within 0.1 eV of the converged value. It makes little sense to calculate to higher precision in view of the intrinsic lim itations of the theory. 2We shall now describe the various methods from a computation al point of view. We shall use the symbol Nwith subscripts for quantities that scale roughly as the siz e of the physical system under study, and Mfor quantities that may be large but are independent of the si ze of the system. Important quantities common to the two codes a re the number of electrons Ne and the number of mesh points, NGandNRfor real space and reciprocal space, respectively. Additional quantities that play a role are the number of freq uencies to be calculated Mω, and the number of time steps to evolute the wave function in the re al-time method, MT. Also, in methods that rely on sparse matrix multiplication, we need t he number of nonzero entries in a row of the Hamiltonian, MH, and in iterative methods to solve large matrix equations we need the number of iterations for convergence, Mit. Finally, the response function method usually requires a sum over unoccupied states, Nc. This notation is summarized in Table I. We will use same energy functional for all methods, so the cho ice of specific functional is not an issue in comparing the methods. As is commonly done, we calculate only the dynamics of the valence electrons. The core electrons are frozen and t heir presence is treated by using a pseudopotential to describe the ionic potential. We use th e pseudopotential construction of Troullier and Martins [15], taking the nonlocal part by th e method of Kleinman and Bylander [16] and including partial core corrections for th e exchange-correlation energy [17]. In this method, the local pseudopotential is fixed to th e value in a particular angular momentum channel, and a nonlocal correction is made for othe r channels. Here we use thel= 1 potential as the local potential, and apply the nonlocal c orrection to the l= 0 andl= 2 channels. The electron-electron interaction is taken in the simple local-density approximation (LDA) given by Perdew and Zunger [18]. More co mplicated functionals have better predictive power for ground state properties [6,7], but give only small improvement to the optical response of neutral molecules. The proper des cription of the 1/r asymptotic behavior of the potential is going to be very important to des cribe charged systems, however for the aim of the present work this LDA deficiency is not relev ant. The geometry of the Na 8cluster was computed in ref. [6], and the lowest energy struc - ture found to be the bicapped octahedron (D 2dsymmetry). We use this structure in our 3comparison here. It has an average Na-Na bond length of 3.38 ˚A and a slight deviation from the spherical symmetry. This leads to a polarizability tens or with two different components and two close-lying peaks are obtained in the photoabsorpti on cross section. II. THEORETICAL METHODS Before describing in detail each of the two methods for repre senting the wave functions (direct and Fourier space), we need to comment on the choice o f the spatial cell size and mesh size as well as the time/frequency parameters (all are s ummarized in Table I). Since the wave functions are sensitive to boundaries, the ca lculations must be made in a volume several Angstroms larger than the size of the molecu le or cluster. Using both methods, we determined how large a volume is needed to achiev ed 0.1 eV accuracy on the various excitation energies of interest in the system. We fo und that this is achieved in a spherical volume of radius R= 8˚A using the NP code, and in a simple cubic supercell of side 12.7 ˚A using the CMP code. These have nearly the same volume, and th us the same average distance from the cluster to the boundary. We have ch ecked the convergence of the results by increasing the volume to a sphere of 12 ˚A radius. The value of the plasma frequency is reduced by a maximum of 0.1 eV, that is, within th e required accuracy. We have used an uniform spatial grid with ∆ x= 0.5˚A spacing. This corresponds to a plane-wave cutoff energy of 6 Hartrees in the Fourier space m ethod (see below). Within this parameters, a stable time-step to perform the time-evo lution in the NP method is ∆t= 0.003¯h/eV << ¯h(∆x)2/m. The required 0.1eV accuracy in energy is obtained for total simulation times of 10 ¯ h/eV. Similarly in the Fourier space method we have taken a uni form frequency grid of Mω= 100 between 0 and 5 eV. Note that if the response is required f or larger frequencies we need to increase the number of points. The whole response is obtained at once in the time evolution method (unless up to energies of the order of (∆ t)−1). This is a great advantage when the whole response is needed. 4A. NP method This method uses a direct solution of the time-dependent sin gle-electron Schr¨ odinger equation, i¯h∂φi(r,t) ∂t=HKS(t)φi(r,t) (i = 1...occ.) (1) whereHKSis the Kohn-Sham Hamiltonian operator HKS(t) =−¯h2 2m∇2+Vion(r) +e2/integraldisplay d3r′n(r′,t) |r−r′|+Vxc(r,t) (2) andnis the time-dependent electron density n(r,t) =/summationtextocc i=1φ∗ i(r,t)φi(r,t). In the solution of this equation in the spatial and time variables following the algorithm of ref. [4], there are two time-consuming operations. One is multiplying the s ingle-electron Hamiltonian operator by the vector representing the wave function. The d imensionality of the vector is the number of mesh points NRtimes the numbers of electron orbitals Ne. The operator is a sparse matrix with MHnonzero elements per row. Thus the basic operation requires about NeNRMHcomplex floating point operations. The time evolution opera tor in the NP code is implements by a power series expansion of the exponential op erator exp( −iH∆t) to fourth order. A predictor-corrector cycle requires two such opera tions. Thus the method requires 8 Hamiltonian multiplications per time step. Thus for MTtime steps the total number of floating point operations is given by NPFPO : 10NeNRMHMT The sparseness of the Hamiltonian matrix in a real space form ulation is determined by the finite difference formula for kinetic energy (nine-point formula in our case); and by the nonlocal-projection parts of the potential. In total we hav e a number of non-zero elements of the each Hamiltonian row MH≈100 for the grid parameters used for Na 8. The other time-consuming part of the NP algorithm is solving the Poisson equation, which must be done twice at each time step. The NP code uses a mu ltipole expansion 5combined with a relaxation method to deal with the higher mul tipoles. It is hard to estimate the scaling properties of this part, but in the present study this part of the computation takes 1.5 times as many operations as the Hamiltonian multiplicat ion operation. We shall assume the same factor for estimating the scaling properties of the algorithm. In principle, the Poisson equation can be solved by methods that are of order NRorNRlogNR, as multigrid or fast-Fourier transformation, so this part should not dom inate for large system. Storage requirements are small: the vector wave function pl usVHartree andVionlocal potentials in Hamiltonian, charge densities and some inter mediate arrays. VHartree requires a slightly larger volume because of the way the Poisson equat ion is solved. NP storage : NR(Ne+ 4.5) This NP method is ideal to be combined with molecular dynamic s simulations for the ions because it uses only ground-state occupied informatio n and would scale roughly linearly with the number of atoms in the system. There is not so much boo k-keeping as in the usual perturbative formalism (no need for storing the large set of unoccupied wave-functions and the large dielectric matrices). B. CMP method Here the basic object of the calculation is the linear respon se to an external field of some frequencyω. The linear response matrix χis constructed in momentum space with the following matrix inversion χ= (1−χ0K)−1χ0 (3) where the independent particle response χ0and the interaction Kare matrices defined as follows. The χ0has elements G,G′given by [19] χ0(G,G′,ω) =1 Ω/summationdisplay kj(fk−fj)/angb∇acketleftk|e−iG·r|i/angb∇acket∇ight/angb∇acketlefti|eiG′·r|k/angb∇acket∇ight ω−ǫj+ǫi+iη(4) 6where Ω denotes the unit-cell volume, i,klabel Kohn-Sham eigenfunctions and ǫkandfk are the corresponding eigenenergies and occupancy factors . The sum goes over Neoccupied orbitals and Ncempty orbitals. The interaction Kis the Fourier transform of the electron- electron interaction in the Kohn-Sham equation, which is gi ven in coordinate space by K(r,r′) =e2 |r−r′|+δVxc(r) δn(r′)(5) We now describe the computation starting with the Kohn-Sham wave functions and ener- gies in a momentum space representation. To evaluate the ind ependent particle response χ0 in eq.( 4), one first calculates the particle-hole matrix ele ments of the momentum operator and stores them in a table (or in disk). This computational eff ort is of the order of NeNcN2 G operations, and the table size to be stored is NeNcNGcomplex numbers. Then the evalu- ation of eq. (4) requires N2 Gmatrix elements to be calculated, each requiring particle- hole summation, to give ≈2N2 GNeNcoperations for each frequency. If one were to make full space calculation, the number of empty orbitals summed in eq. (4) w ould be of the same order as the dimensionality of the space. However, the number of em pty orbitals can be severely truncated without effecting the long-wavelength dipole res ponse. In the example, we find Nc= 320 is adequate, which is more than an order of magnitude sma ller than the size of the space and corresponds to include unoccupied states up to 20 e V above the highest-occupied orbital. This is a reasonable approximation as we are intere sted only in getting the optical spectra for excitation energies below 10 eV. This approxima tion is an important saving in building up the response matrix. One also truncates the calculation of the response matrix in another way. We have also assumed that the off-diagonal elements of the response funct ion are zero for G-vectors outside an sphere of 1.25 ˚A (that is to consider ≈3200 points in the G-space). This corresponds to reducing the number of matrix elements to be computed and sto red toNG(NG+4)/18. Note that the necessity to store the N2 Gmatrix puts a higher demand on the computer memory than the NP method. The memory required to store the N2 Gcomplex, double-precision numbers in the example problem is 164Mb. 7There are now three steps to evaluate eq. (3), two matrix mult iplications and a matrix inversion. The matrices are not sparse, so the matrix multip lications each cost 2( NG/3)3 arithmetical operations1. The matrix inversion is of the same order, requiring ( NG/3)3 operations. The total is ≈5(NG/3)3.These represent the most computationally demand- ing steps in the CMP method, given the truncation in the Nc. The computed χis next transformed to the coordinate space representation. Using the fast Fourier transform, this takes ≈N2 GlogNGoperations. The dynamical polarizability can be now comput ed from α(ω) =VextχVextas a matrix times vector multiplication. From this one can ea sily extract the photoabsorption cross section σ(ω) =4πω cImα(ω). Then the total computational effort in the CM method is: CM FPO : Mω(NcNe(NG/3)2+ 5(NG/3)3) with the last term dominant. The storage requirements for al l the occupied and unoccupied wave functions plus the whole complex response matrix is CM storage : ( Ne+Nc)NG+ 2(NG+N2 G/9) +NcNeNG/3 To achieve the targeted energy convergence with this algori thm, the momentum space mesh was chosen to correspond to a simple cubic supercell of L= 12.7˚A on a side. This implies that the mesh spacing in momentum space is δk= 2π/L= 0.137˚A. The momentum space representation takes all the points within a sphere of radiuskmax= 1.83˚A (that corresponds to a plane-wave cutoff energy of 12 Ry). The size o f the vector in the momentum representation is thus NG= 4π(kmax/∆k)3/3≈10,000. Note that this is slightly smaller than the number required for the coordinate space represent ation, however we need to stress that a larger number of G-vectors are needed to describe the a ction of the potential on a 1A small technical point should be mentioned, associated wit h the divergence of the Coulomb interaction at G=G′. This is dealt with [19] by taking a numerical limit as |G−G′| →0, and this adds about 10% to the number of operations for computing the matrix product. 8wave-function ( Vψcorresponds to a convolution in Fourier space). Finally, an additional numerical parameter is the imaginary part of the frequency η, which we have taken as η= 0.05 eV to produce a resolution of 0.1 eV in the spectral feature s. In the discussion below we have not include the computationa l requirements to perform the ground state calculations, occupied and unoccupied orb itals. This could be a major storage bottle-neck for very large systems as the calculati on of a large set of unoccupied wave functions has a cubic scaling of the number of atoms in me mory and computing time. In the present calculation this initialization process tak es 10% of the total computational time. C. Other methods We mention here two other methods from a computational point of view. Since we have not carried out numerically computations on our test pr oblem with these methods, the discussion will be brief. 1. Modified Sternheimer method The modified Sternheimer method was first applied to the time- dependent Kohn-Sham equation for atomic excitations [20], and has since been app lied to the dielectric response of crystals using the momentum space representation [21] an d to the finite system C 60[22] using the coordinate space representation. Here one solves an inhomogeneous equation for the perturbed wave functions φ± iusing an iterative method. The perturbation is a sinusoidal potential field combining the external field Vextand the internal field from the time-varying electron density. The equations are (ǫI−H0 KS±ω+iη)φ± i=ˆPVi (6) where Vi= (Vext+Kδn)φi 9and δn= Re/summationdisplay iφi(φ+ i+φ− i). (7) ˆPis a projection operator removing occupied orbitals. In ref . [22], the two equations are constructed in coordinate space and solved with a double ite ration. One makes a guess for the density δn, and solves eq. (6) by the conjugate gradient method. δnis refined from the resulting φ± iagain with the conjugate gradient method, and the process is repeated to convergence. The numerical cost will thus depend largely on the cost of the Hamiltonian operation which is ≈MHNRNein coordinate space, and the number of iterations Mitre- quired to get a converged solution. Remembering also that fr equency space methods need Mω, the number of frequencies to be examined, the computationa l cost of this method is Modified Sternheimer (real space) : MωMitMHNRNe (8) The method can be used in this form for nonresonant frequenci es, but near the eigenfrequen- cies the nearby singularities in eq. (6) must be removed for t he conjugate gradient method to converge. Thus this method would be similar to methods uti lizing the particle-hole rep- resentation in needing a considerable number of the wave fun ctions and eigenenergies of unoccupied states. The singularities are removed by projec ting on the unoccupied wave function subspace the right hand side of eq. (6), V′ i=Vi−/summationdisplay jφi(φi,Vi). The desired wave functions φ± iare obtained from the projected solutions φ′± iby φ± i=φ′± i+/summationdisplay jφi(φi,Vi) ǫj−ǫi−ω−iη. It is difficult to give an a priori estimate of Mitor its size-scaling properties (although with our notation we have assumed that it does not grow with N). Unfortunately, our implementation of eq. (7) still left the convergence somewh at erratic. Typically it takes of the order of Mit≈1000 iterations of the double loop to get convergence. Thus i t would require some improvement of the algorithm to make it attract ive to apply to large systems. 10The momentum space implementation of the modified Sternheim er method is similar. This method also needs the conditioning step for convergenc e of the CG iteration. The main difference is in the Hamiltonian multiplication, which here requires ≈2(NG/3)3operations as discussed in Sect. IIB. Thus the total is Modified Sternheimer (momentum space) : 2 MωMit(NG/3)3(9) Because the Hamiltonian operation is more costly in momentu m space, this method is prob- ably not competitive to the others, unless it were the case th at the convergence of the iteration were intrinsically much more reliable. 2. Diagonalization method The frequency-space methods discussed so far have relied in some way on operator inver- sion. It is also possible to cast the problem as one of matrix d iagonalization. This method was applied to cluster excitations in the TDLDA by Vasiliev e t al. [8]. The authors start from a basis in coordinate space and construct Kohn-Sham orb itals for both occupied and empty states as is done in the CMP method, but representing th e orbitals in coordinate space mesh, as in the NP method. The storage requirement for the orb itals is ≈(Nc+Ne)NR, which is larger than in the NP method but smaller than in the CM P method. The next step of the calculation is to construct the matrix to be diagonalized. The eigenvalue equation to be solved is RFn=ω2 nFn (10) where Fnare the eigenvectors and Ris a matrix. Its elements are Rα,α′= (ǫi−ǫj)2δα,α′+ 2/radicalBig (ǫi−ǫj)(ǫ′ i−ǫ′ j)Kα,α′ (11) where the indices α= (ij),α= (i′,j′) label combinations of unoccupied orbitals iand occupied orbitals j. The interaction matrix elements Kα,α′are simply the particle-hole matrix elements of the residual interaction, eq.(5). There is a substantial computational 11cost in construct the interaction matrix K. A straightforward transformation from the coordinate space to the particle-hole representation requ ires≈N2 RN2 eN2 coperations for the Coulomb interaction. However, this is reduced considerabl y by using an efficient method to solve the Poisson equation [9]. For example, using the fast F ourier transform one may find the Coulomb field for a given particle-hole state taking only NRlogNRoperations. Saving the Coulomb field in the coordinate representation, the matr ix element to a given final state takes∼NRoperations. The effort of solving the Poisson equation is thu s distributed over the number of final states, and the operations to construct th e full matrix has a leading dependence NRN2 eN2 c, the scaling appropriate for the local part of the interacti on2. Once the matrix is constructed, the diagonalization requires ≈(NcNe)3operations. However, taking the Nvalues from Table I, the matrix diagonalization effort is sma ll compared to that needed to construct the matrix. We have therefore taken that step to assign this method’s size scaling in Table II. III. NUMERICAL RESULTS We will discuss in detail the physical quantities computed i n the NP and CMP methods and refer to [8] for the results using the diagonalization me thod. We want to stress that the three approaches must give the same values if the numerical p arameters are chosen with fine enough grids and large enough cutoffs to get converged result s. With the parameter sets chosen for the two methods, the resul ts are quite similar. In Table IV we show calculated Kohn-Sham energies and the surfa ce plasmon energy. The first entryǫ1is the Kohn-Sham eigenvalue of the most bound orbital. The ab solute energies have no significance in the supercell method, because the abs olute Coulomb potential is undefined. Therefore, for this entry we give the value from th e NP code and set the scale of the CMP energies at that value. The next three rows corresp ond to the other bound 2However, in the implementation of ref. [8], the Poisson solv er in fact is the most costly operation. 12orbitals use the G= 0 point of the Brillioun zone for the CMP values. We can see th at the methods agree to within less than 0.1 eV. The next entry is the lowest unoccupied orbital. This is significantly different for the two methods. This orbi tal has sufficient extension to have its energy sensitive to the boundary, which of course is different for the two methods. We confirm the boundary sensitivity in the CMP code by calcula ting the energies at other points in the Brillouin zone. Differences are less than 0.1 eV for occupied orbitals, but reach 0.2 eV for the lowest unoccupied orbital. This last point ind icates the fact that the empty orbitals are more sensitive to the boundary conditions and i n the periodic supercell they feel the potential from the other clusters. We have also presented in Table IV the results of the NP method3. We have also checked the convergence of the plasmon frequency with respect to the cell size and found that this value is converged to less than 0.01eV for a sphere of R=12 ˚A. The fully converged value in the NP method is 2.65 eV. The difference with the experiment al value of 2.53 eV can be attributed to deficiencies in the LDA approximation as well a s for finite temperature effects in the experiments [6]. In Table V we summarize the results for the static averaged el ectrical polarizability of Na8obtained by the different methods. The agreement among the di fferent approaches is very good and the remaining difference with experiments ca n be again assigned to core polarization, exchange-correlation and temperature effec ts. These effects tends to increase the polarizability bringing the computed values close to th e experiments [6]. 3The plasmon frequency is sensitive to the core-exchange cor rection at the level of 0.1 eV. We have included that correction in HKSit improves the description of the structural properties of Na metal. We note that the result without core corrections (2 .89 eV) it is very close to the jellium value (2.9 eV). 13IV. CONCLUSIONS In the theory of electronic excitations of finite many-elect ron systems, the time-dependent Kohn-Sham equation with an adiabatic local density approxi mation for the interaction en- ergy function offers an attractive compromise towards the go als of accuracy and computa- tional practicality. But even within the TDLDA scheme there are several methods in use, and our purpose was to compare them on the same footing by appl ying them to the same physical problem, and demanding the same accuracy. The goal is to gain a general under- standing of the numerical resources (total numbers of arith metic operations and computer memory) required by the different methods. One can then extra polate to large systems and make a judgment on which methods offer the best prospects. We have only considered methods based on a grid representati on of the electron wave functions, and have concentrated on two algorithms, the NP m ethod in real time and real space, and the CMP method in Fourier transformed time and spa ce. We chose to study the response of the Na 8cluster around the surface plasmon excitation energy. The two methods turned out to have comparable requir ement on arithmetic oper- ations. However, it should also be noted that the computatio nal work increases with the range of frequencies that one studies in the CMP method, but n ot in the NP method. With latter, the entire response is obtained from a single calcul ation. In comparing the two methods to ascertain their scaling with the size of the system N, we have deliberately ignored the first task in either method, th e construction of the eigenstates of the static Kohn-Sham operator. In the NP method only the oc cupied orbitals are needed, but in the CMP method one also needs a large number of unoccupi ed orbitals as well. Their calculation scales like N3 ein principle, but in practice this phase of the computation i s short compared to the dynamic calculation and so we ignore it. Let u s now compare the scalings by taking the expressions in Table II, dropping the subscrip ts on theNquantities. The NP method thus scales as N2. This behavior was also found studying the excitations of long carbon molecules [5]. The CMP method has a poorer scalin g behavior, namely N3. 14We also considered two other methods without however examin ing them in as much detail. In principle, the modified Sternheimer method in coordinate space can achieve N2scaling without the cost of the large MTfactor of the real-time method. However, we did not find a reliably converging iteration procedure to solve the basi c inhomogeneous linear equation set. The final method we discussed, the diagonalization meth od using real space and Fourier time, seems to have a poorer N-scaling than the others, but may be advantageous in some circumstances (see below). Besides arithmetic operations, storage can play a role in th e practicality of the different algorithms for large systems. Here we find that the storage re quirements are grossly different for the NP and CMP methods, favoring the NP approach. From Tab le II, it has a N2scaling while the CMP method has an N3behavior. This is already significant in the Na 8system we studied, as may be seen from Table III. Thus our results favor the real-time and real-space methods , offering economy in both storage and arithmetic operations. However, there are a num ber of caveats. We have not considered the suitability of the different algorithms for p arallel computing. In a parallel computing environment, the frequency-space methods gain f avor because the Mωfactor can be trivially absorbed in the parallel processing. In additi on, the diagonalization method can benefit from the parallel computation of different rows of the matrix. Also the sparseness of the Hamiltonian matrix is important for the real space met hod; this would be lost if for example the energy functional used the full Fock exchange in teraction. Finally, we mention two nonnumerical benefits of the real-ti me method: as was said earlier, it is nonperturbative and therefore allows effects of large fields to be calculated with the same effort. And it uses the same energy functional (permi tting the program to call the same subroutine) for the dynamic calculation as for the s tatic calculation to prepare the ground state. 15V. ACKNOWLEDGMENT We are grateful to J. Chelikowsky and I. Vasiliev for communi cations and providing us with their computer code. This work was supported by the De partment of Energy un- der Grant FG06-90ER-40561, by the DGES (PB98-0345) and JCyL (VA28/99), and by he Grant-in-Aid for Scientific Research from the Ministry of Ed ucation, Science and Culture (Japan), No. 11640372. AR acknowledges the hospitality of t he Institute for Nuclear The- ory where this work was started and the computer time provide d by the C4(Centre de Computaci´ o i Comunicacions de Catalunya). 16REFERENCES [1] M. Petersilka, U. J. Gossmann and E. K. U. Gross, Phys. Rev . Lett. 76, 1212 (1996). [2] E.K.U. Gross, J.F. Dobson, and M. Petersilka, in Density Functional Theory II , edited by R.F. Nalewajski, “Topics in Current Chemistry”, Vol 181 ( Springer, Berlin, 1996) p.81 [3] A. Rubio, et al., Phys. Rev. Lett. 77247 (1996). [4] K. Yabana and G.F. Bertsch, Phys. Rev. B 54(1996) 4484. [5] K. Yabana and G.F. Bertsch, Int. J. Quantum Chemistry 75(1999) 55. [6] A. Rubio, J.A. Alonso, X. Blase, and S.G. Louie, Int. J. Mo d. Phys. B 11, 2727 (1997), and reference therein. [7] M.E. Casida, et al., J. Chem. Phys. 1084439 (1998). [8] I. Vasiliev, S. Ogut, and J. Chelikowsky, Phys. Rev. Lett .821919 (1999). [9] I. Vasiliev, private communication. [10] C.A. Ullrich, P.-G. Reinhard and E. Suraud, J. Phys. B: A t. Mol. Opt. Phys. 305043 (1997). [11] R. Nagano, K. Yabana, T. Tazawa, Y. Abe, J. Phys. B: At. Mo l. Opt. Phys. 32L65 (1999). [12] H. Flocard, et al., Phys. Rev. C 171682 (1978). [13] Different representations of the wave function based on linear combination of atomic or- bitals (AO) have been used in the literature to address the li near and nonlinear response of molecules [7,14] with quite a good success. As compare to g rid or plane-wave-based representations, the main advantage of the AO representati on stems from the small num- ber of basis needed to expand the wave functions and Hamilton ian matrix elements. On 17the other hand, the matrix elements in the response function cannot be calculated as easily as in a plane-wave reresentation and the check of conv ergence with respect to the size of the AO-basis set is rather difficult (usually the numbe r of AO’s is much less than NG). [14] S.J.A. van Gisbergen, et al, Phys. Rev. Lett. 83, 694 (1999); Phys. Rev. Lett. 78, 3097 (1997) [15] N. Troullier and J.L. Martins, Phys. Rev. B 431993 (1991) [16] L. Kleinman and D. Bylander, Phys. Rev. Lett. 481425 (1982). [17] S.G. Louie, S. Froyen, and M.L. Cohen, Phys. Rev. B 26, 1738 (1982) [18] J. Perdew and A. Zunger, Phys. Rev. B 235048 (1981). [19] M.S. Hybertsen and S.G. Louie, Phys. Rev. B 35, 5585 (1987). [20] M. Stener, P. Decleva and A. Lisini, J. Phys. B 284973 (1995). [21] A. Dal Corso, F. Mauri and A. Rubio, Phys. Rev. B 53, 15638 (1996). [22] J.-I. Iwata, K. Yabana and G.F. Bertsch, Nonlinear Opti cs, to be published. [23] W.A. de Heer, et al, Phys. Rev. Lett. 59, 1805 (1987); C.R.C. Wang, et al, J. Chem. Phys,93, 3789 (1990); W.A. de Heer, Rev. Mod. Phys. 65, 611 (1993). [24] W.D. Knight, K. Clemenger, W.A. de Heer, and W.A. Saunde rs, Phys. Rev. B 312539 (1985). [25] P. Calaminici and A. M. K¨ oster, J. Chem. Phys. 1114613 (1999). 18TABLES TABLE I. Symbol definitions for quantities pertaining to the computational effort required by the various algorithms discussed in the main text, and their values. Symbol Meaning NP method CMP method MT time steps 104- Mω number of frequencies - 10 MH nonzero elements in H matrix row 100 - Mit iterations in conjugate gradient method - - NR real-space points 17,000 - NG reciprocal-space points - 9,771 Ne number of electron orbitals (occupied states) 4 4 Nc unoccupied states - 320 TABLE II. Leading-order for the size scaling of various algo rithms for TDLDA–general com- parison: floating point operations (FPO) and memory require ments. Method FPO Memory NP NeNRMHMT NR(Ne+ 4.5) CMP 5Mω(NG/3)3N3 G/9 Modified Sternheimer MωMitMHNeNR NR(Ne+Nc) Diagonalization N2 cN2 eNR (NcNe)2 TABLE III. Comparison of computational difficulty of NP and CM P methods for Na 8 Resource NP CMP Memory (MBy) 7 350 Floating point operations 1.5×10121.7×1012 19TABLE IV. Orbital energies ǫiand surface plasmon energy ωMin Na 8. For comparison in parenthesis we show the result of a calculation within the NP method without including partial core corrections in the pseudopotential generation and tim e evolution. Energy NP CMP Exp. (eV) ǫ1 -4.63 -4.63 ǫ2 -3.41 -3.35 ǫ3 -3.00 -2.97 ǫ4 3.00 -2.97 ǫ5(LUMO ) -1.88 -2.01 ωM 2.77 2.6 2.53 [23] TABLE V. Static polarizability of Na 8(˚A3) Exp. NP Atomic CMP All-electron 128.7 [24] 103 117 [8] 119 114.9 [25] 20

arXiv:physics/0003091v1 [physics.atom-ph] 28 Mar 2000A waveguide atom beamsplitter for laser-cooled neutral ato ms Dirk M¨ uller, Eric A. Cornell,∗Marco Prevedelli,†Peter D. D. Schwindt, Alex Zozulya,‡ Dana Z. Anderson Department of Physics, and JILA, University of Colorado and NIST, Boulder, CO 80309-0440 July 13, 2011 Abstract A laser-cooled neutral-atom beam from a low-velocity inten se source is split into two beams while guided by a magnetic-field potential. We gene rate our multimode- beamsplitter potential with two current-carrying wires on a glass substrate combined with an external transverse bias field. The atoms bend around several curves over a 10-cm distance. A maximum integrated flux of 1 .5·105atoms /s is achieved with a current density of 5 ·104Ampere /cm2in the 100- µm diameter wires. The initial beam can be split into two beams with a 50/50 splitting ratio. Like their optical counterpart, atom beamsplitters are the pivotal element of atom-optical interferometers. While the original beamsplitter was perh aps the Stern-Gerlach apparatus1, modern free-space beamsplitters are based on mechanical or light-based refractive elements. ∗Also at Quantum Physics Division, National Institute of Sta ndards and Technology, Boulder, CO 80309 †Permanent address: Dipartimento di Fisica, Dell’ Universi t` a di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy ‡Present address: Department of Physics, Worchester Polyte chnic Institute, Worchester, MA 01609 1Such beamsplitters have been used with good success in Mach- Zehnder interferometers to measure the Sagnac-effect with high sensitivity2,3. Free-space beamsplitters are generally characterized by small splitting angles because the effecti ve grating spacing is large compared to the atomic de Broglie wavelength. A waveguide-based beamsplitter has the potential to provid e arbitrary splitting angles. Furthermore, the confining potential of a waveguide also sup presses the beam divergence and gravitational sag to which free-space interferometers are subjected. Several atom-guiding schemes using magnetic forces have been proposed and demons trated4–10. We recently demonstrated guiding a beam of laser-cooled atoms around a c urve11using magnetic forces from photolithographically patterned current-carrying w ires. The multimode-atom beam- splitter reported on here is a natural extension of our previ ously demonstrated guiding scheme. Like its fiber and integrated optical counterparts, our waveguide beamsplitter merges and then diverges two guiding regions. We guide87Rb atoms in a weak-field-seeking state along a magnetic-field minimum. This magnetic guide leads atoms around several curves to a beamsp litter region. Our beamsplitter region consists of two such magnetic-field minima that merge to one field minimum and separate again into two minima. A variable fraction of atoms initially launched into one of the two magnetic field minima are guided and transferred in to the second magnetic field minimum at the beamsplitter region. For our atom source we prepare a laser-cooled beam of87Rb atoms with a modified vapor- cell magneto-optical trap (MOT)12. To generate our low-velocity intense source (LVIS)13 we drill a 500- µm hole in the center of a retro-reflecting mirror placed insid e our vacuum chamber [Fig. 1(d)]. We couple LVIS atoms into our magnetic g uide by positioning the guide opening directly behind the mirror hole. The atom’s interna l state and velocity distributions are as measured previously11,14. We generate our one-dimensional guiding potential by addin g an external transverse bias field to the magnetic field generated by a 100 ×100-micron current-carrying wire on a glass substrate7,8,10. The vector sum of the transverse bias field /vectorBbiasand the wire’s magnetic field 2/vectorBwirebecomes zero at a position outside the wire, if the bias field i s smaller than the field generated by the wire at its surface [Fig. 1(a)]. As the bias fi eld is increased (decreased) the position of the magnetic field minimum moves linearly tow ard (away from) the wire and the potential depth increases (decreases). Furthermor e, when the wire’s magnetic-field maximum is twice the transverse bias field, the magnetic field zero is 50- µm above the current-carrying-wire surface and the field magnitude incr eases linearly with displacement in the transverse directions. We generate the transverse bi as field for the guide with an electromagnet placed near the substrate [Fig. 1(d)]. An add itional ∼14-G longitudinal bias field is applied to prevent the magnetic-field magnitude from vanishing at the field minimum. As the wire current and bias field are increased proportional ly the magnetic-potential depth and gradient increase linearly, but the field-minimum posit ion remains unchanged. With current only in the wire positioned right behind the mir ror hole, wire 1, and a 86-G external bias field applied we guide atoms and measure at om flux versus wire current for different bias fields. In this guiding experiment we run 35 -msec-long current pulses of up to 5.5 A at a 1 sec repetition rate through wire 1. We choose s hort current pulses to prevent the glass substrate from overheating, allowing us t o run larger wire currents than a continuous current would allow. After the atoms exit the gui de, they are ionized by a hot wire and the subsequent ions are then detected by a channeltr on. For each external bias-field value there is an optimum track current that maximizes the gu ided-atom flux [Fig. 2(a)]. For wire currents too large the magnetic-field minimum is shi fted far away from the wire resulting in a reduced field gradient. This field-gradient re duction helps to couple atoms into the guide opening, but also leads to guiding losses as atoms c an no longer be guided around the curves of the guide. When the track current is too low the g enerated magnetic-field gradient is sufficient to bend the atoms around the curve, but t he field minimum is close to the wire surface and atom-surface interactions as well as a tighter guide opening result in a lower flux. Our guiding flux peaks when the condition for mo de matching atoms into the guide opening and maintaining a sufficient gradient to ben d atoms around the curves is 3optimized. We measure the heating of guided atoms by comparing the trans verse velocities before and after the guiding process. At a wire current of 5.0 A and a t ransverse bias field of ∼86 G we measure the guided-atoms’ transverse-velocity profi le by translating the hot wire transverse to the propagation direction to map out the s patial extent of the atom beam as it diverges from the guide exit [Fig. 2(b)]. The 70- µm-diameter hot wire is placed ∼2.5 cm from the output of the magnetic guide. We calculate that t he atoms’ emergence from the confining fields of the guide is almost completely non -adiabatic—the transverse kinetic energy of the emerging beam should thus be a faithful reflection of the transverse kinetic energy in the guide. From the width of the fit in Figure 2(b) we determine that the transverse-velocity distribution of the guided atoms is vt= 17.2±3.5 cm/sec, in contrast to an initial transverse velocity of vt= 10.0±1.5 cm/s of LVIS. We attribute the observed heating to the non-adiabatic loading of the LVIS atoms into t he guide. An atom that enters the guide displaced from the magnetic field minimum experien ces a sudden increase in its potential energy because it is not mode-matched to the guide . The additional potential energy increases its total energy inside the guide, which is converted into transverse kinetic energy once the atom leaves the guide. We believe this heatin g effect can be ameliorated by adiabatically loading the atoms into a tapered guide. Once atoms are guided along the one-dimensional magnetic-fi eld minimum we turn on our beamsplitter. As we increase the current running throug h wire 2 we observe that the flux from guide 2 increases and the flux out of guide 1 decreases . Figure 3(a) shows the flux of guided atoms coming out of each guide versus the current ra tio between the two wires. As we change this current ratio we can tune the splitting rati o of our beamsplitter. We observe a dynamic range of the splitting ratio from 100/0 to 1 5/85. A 50/50 beamsplitter is achieved when the current in wire 2 is 85% of the current in wir e 1. Aside from the current ratio there are two other parameters that determine the spli tting ratio of our beamsplitter. First, as we change the applied transverse bias field we can va ry the degree of overlap of the two magnetic field minima. For very large bias fields the tw o minima remain close to 4their respective wires and their overlap is small in the beam splitter region. Second, the curvature of the guides in the beamsplitter region determin es the manner in which the two magnetic-field minima merge. In our design the atoms are pref erentially switched over into the secondary guide due to the wire curvature in the beamspli tter region. The guides bend with a radius of curvature of ∼30 cm into the splitter region and curve away with a radius of∼70 cm. A calculation of our potential-minima trajectories s hows that for this curvature the field minima follow straight lines that cross (Fig. 4). Th is means that atoms launched into guide 1, the primary guide of our beamsplitter, are more likely to switch over into guide 2, the secondary guide, than to continue along guide 1. This geometric feature of our beamsplitter is responsible for its bias toward coupling at oms into guide 2. Our experimental data shows that at a current ratio of 0.85 between guide 2 and g uide 1 we compensate for the geometric bias of our beamsplitter and achieve 50/50 bea msplitting. As we increase the current in wire 2 to a current larger than in wire 1 the flux out o f guide 2 peaks at a current ratioIwire2/Iwire1= 1.1 and decreases again beyond this value. This decrease is ana logous to the observation in 2(a) where we found a decrease in guided -atom flux for wire currents in a non-optimum Bwire/Bbias-ratio regime. We sum the flux from the output of both guides and find the total fl ux to remain roughly constant. This observed flux conservation of our beamsplitt er as shown in Figure 3(a) is a strong function of the transverse bias field applied. For a cu rrent in wire 1 of 5.0 A and a longitudinal bias field of 14 G a transverse bias field of ∼86 G is necessary to achieve a constant total flux as the current ratio is varied. This is 35 % larger than the optimum bias field measured for maximum guiding efficiency when only on e wire carries current. We postulate that the increased bias field separates the two m agnetic field minima in the beamsplitter region making it easier for atoms to follow gui de 1 instead of guide 2. We model our beamsplitter design with a numerical simulatio n and find qualitative agree- ment with our measured data. The simulation calculates the c lassical trajectories of 2 .5×106 atoms through our beamsplitter potential and records which guide each atom exits [Fig. 3(b)]. The test atoms’ spatial initial conditions are sprea d over a 100 ×100-µm aperture 5centered 60 µm above the wire and their initial velocities match the LVIS- velocity distribu- tion at the guide opening. We use constant bias fields of 100 G i n the transverse and 14 G in the longitudinal direction for the simulation shown. In o ur experiment these values are not constant along the entire guiding region, which limits t he accuracy of our simulation. The simulation shows that more atoms are coupled into guide 2 than remain in guide 1 when the currents in both wires are equal, which confirms our i dea that the beamsplitter is biased toward guide 2 by its geometric design. Further, a 5 0/50 beamsplitting is shown at a current ratio of 0.75, which compares well within the err ors of the simulation to the measured current ratio of 0.85. Our simulation also shows a d ecrease in flux out of both guides at the largest current ratios as observed in our exper iment. In summary, we have split a beam of laser-cooled atoms while g uided in a magnetic-field potential. We are able to vary the ratio of atoms coupled into the secondary guide from 0 to∼85% by varying the current in wire 2. We demonstrate that mode rate currents give guiding potential depths of several millidegrees Kelvin. I n the future the question of coherent beamsplitting will have to be addressed. While coherence ha s been well established for free- space atom interferometers it has yet to be demonstrated tha t our waveguide beamsplitter preserves coherence. The authors would like to thank Carl Wieman and Randal Grow fo r helpful discussions. This work was made possible by funding from the Office of Naval R esearch (Grant No. N00014-94-1-0375) and the National Science Foundation (Gr ant No. Phy-95-12150). 6REFERENCES 1.W. Stern and O. Gerlach, Z. Phys. 8, 10 (1921). 2.A. Lenef, T. D. Hammond, E. T. Smith, M. S. Chapman, R. A Rubens tein, and D. E. Pritchard, Phys. Rev. Lett. 78, 760 (1997). 3.T. L. Gustavson, P. Bouyer, and M. A. Kasevich, Phys. Rev. Let t.78, 2046 (1997). 4.J. Schmiedmayer, Phys. Rev. A 52, R13 (1995). 5.M. Key, I. G. Hughes, W. Rooijakkers, B. E. Sauer, and E. A. Hin ds, Phys. Rev. Lett. 84, 1371 (2000). 6.J. Denschlag, D. Cassettari, A. Chenet, S. Schneider, and J. Schmiedmayer, Appl. Phys. B 69, 291 (1999) 7.J. Denschlag, D. Cassettari, and J. Schmiedmayer, Phys. Rev . Lett. 82, 2014 (1999). 8.N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith , M. Drndic, R. M. Westervelt, and M. Prentiss, Phys. Rev. Lett. 84, 1124 (2000). 9.J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. H¨ ansch, P hys. Rev. Lett. 81, 5310 (1998). 10.J. Reichel, W. H¨ ansel, and T. W. H¨ ansch, Phys. Rev. Lett. 83, 3398 (1999). 11.D. M¨ uller, D. Z. Anderson, R. J. Grow, P. D. D. Schwindt, and E . A. Cornell, Phys. Rev. Lett. 83, 5194 (1999). 12.E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard P hys. Rev. Lett. 59, 2631 (1987); C. Monroe, W. Swann, H. Robinson, and C. E. Wiema nibid65, 1571 (1990). 13.Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell , and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996). 714.D. M¨ uller E. A. Cornell, D. Z. Anderson, and E. R. I. Abraham, Phys. Rev. A 61, 033411 (2000). 8FIGURES d) Electromagnet for transverse bias□field Wires on glass□substrateLVIS□mirrorBbias da) b) c) Fig. 1. Contour lines of magnetic-field potential and guide s chematic. (a-c) We show a cross-sectional cut across the wires. When a bias field is app lied transverse to the wires the mag- netic field becomes zero just above the wire surface (a). For l arge track separations (d=300 µm) the magnetic field minima do not merge (b). As the track spacing is reduced (d=100 µm) the magnetic field minima merge to form one field minimum (c). The transvers e bias field is generated with an electromagnet near the wire substrate (d). The LVIS mirror h ole is aligned with one of the wires to couple the LVIS atoms into the guide. 94.5 5.0 5.5 6.0 6.5-4-2024681012 FWHM = 0.87 mmb)Guide 1 flux Hot Wire Position (mm)1 2 3 4 5 6-1012345678a)Gauss14 21 28 42 56Guided Atom Flux (103/s) Wire Current (A) Fig. 2. Guided atom flux. We measure the guided atom flux versus the wire current at different transverse bias fields. For each bias field we observe an optim um wire current where the guided flux peaks (a). Figure 2(b) shows the guided-atom beam profile. We move the hot wire perpendicular to the propagation direction to map out the transverse-veloci ty distribution. We use the width of the Gaussian fit to determine the RMS transverse velocity to be vt= 17.2±3.5cm/sec. The dip in flux around 6 mm is an artifact due to some stray LVIS atoms that con taminate the guided-atom-beam profile. Those data points appear negative due to a backgroun d-subtraction procedure. 100.0 0.2 0.4 0.6 0.8 1.0 1.2-20246810121416 total (1+2)b) a) Guided Atom Flux (arb. units) Iguide2/Iguide1guide 1 guide 2Guided Atom Flux (103/s) Iguide2/Iguide10.0 0.2 0.4 0.6 0.8 1.0 1.20.00.10.20.30.40.50.60.70.8 Fig. 3. Flux versus wire-current ratio. Figure 3(a) shows th e experimental data and figure 3(b) a simulation of our beamsplitter. With no current in wire 2 al l atoms coupled into guide 1 exit the same guide 1. As the current in wire 2 is increased the beam splitter is turned on and atoms are transferred to guide 2. For the experimental (simulatio n) data the transverse-bias field, the longitudinal-bias field, and Iguide1are held constant at values 86 G (100 G), 14 G (14 G), and 5.0 A (5.0 A) respectively. The simulation shows good qualitati ve agreement with our experimental data. 11/c97=0.7O/c97=1.1O RC=70 cmRC=30 cmCurrent-carrying□wires Magnetic-field□minima Fig. 4. Magnetic-field-minima crossing. For our curvature ( Rc) and spacing the magnetic field minima cross in the center region with an angle αbetween the two trajectories. Atoms coupled into one guide are more likely to be transferred into the othe r guide when both wires carry equal currents, because of the tendency to shoot straight across t he intersection. 12

arXiv:physics/0003092v1 [physics.optics] 29 Mar 2000Ultrahigh sensitivity of slow–light gyroscope U. Leonhardt1,2and P. Piwnicki2 1School of Physics and Astronomy, University of St Andrews, N orth Haugh, St Andrews, Fife, KY16 9SS, Scotland 2Physics Department, Royal Institute of Technology (KTH), L indstedtsv¨ agen 24, S-10044 Stockholm, Sweden Slow light generated by Electromagnetically Induced Trans parency is extremely susceptible with respect to Doppler detuning. Consequently, slow–light gyr oscopes should have ultrahigh sensitivity. 42.50.Gy In recent experiments [1] light has been slowed down dramatically to just a few meters per second. Traveling at this incredibly low speed, light is sensitive enough to serve in detections of subtle motion effects such as the optical Aharonov–Bohm effect of quantum fluids [2]. A moving medium is able to drag light, and this dragging phenomenon gives rise to the sensitivity with respect to motion. Analyzed more carefully [2–5], a moving medium appears as an effective change of the space–time metric (as an effective gravitational field). When the medium outruns the light traveling inside, suitable flows may ap- pear as optical black holes [2]. Apart from detecting quantum flows and from creating artificial black holes, is there a practical application for the incredible motion sensitivity of slow light? Optical gyroscopes are regularly employed to sense mo- tion. What would be the advantage of a slow–light gyro- scope? How do passive dielectric gyroscopes work? Imag- ine for simplicity a solid block of uniform dielectric ma- terial that is rotating at angular velocity Ω 0. (In a real fiber–gyroscope light travels in a multitude of coils, for enhancing the effect.) We assign cylindrical coordinates to the block where the zdirection coincides with the ro- tation axis. First, assume an essentially non–dispersive material such as glass that is characterized by the re- fractive index n. In this material, light experiences the space–time metric [3] ds2=c2dt2 n2+ 2αΩ0r dt dϕ−dr2−r2dϕ2−dz2(1) in the limit of low rotation velocities Ω 0rcompared with the speed of light in vacuum, c. Fresnel’s dragging coef- ficient, α= 1−1 n2, (2) quantifies the degree to which light is forced to move along with the medium. Clearly, αvanishes in the ab- sence of a medium ( n= 1) and αapproaches unity in the limit of a very strong medium ( n→ ∞). Now, imagine that the block of dielectric material is at rest and instead of the medium the laboratory frame is rotating at angu- lar velocity Ω. We obtain from the space–time metric of light in the medium frame, ds2=c2dt2 n2−dr2−r2dϕ′2−dz2, (3)the metric in the rotating laboratory frame by the simple transformation ϕ′=ϕ−Ωt . (4) To leading order in Ω r/cthe transformed metric (3) co- incides with the metric (1) of light in the rotating block if we put Ω =αΩ0. (5) A rotating dielectric and a dielectric in a rotating frame are practically equivalent, yet they appear to rotate at different angular velocities. Fresnel’s dragging coefficien t (2) quantifies the ratio between the actual and the ap- parent angular velocity of the rotating dielectric body, Ω 0 and Ω, respectively. The coefficient αcharacterizes the degree to which light follows the actual rotation and, con- sequently, the degree to which rotation can be detected by optical interference. Slow light [1] has been generated using Electromag- netically Induced Transparency (EIT) [6]. EIT takes ad- vantage of a quantum–interference effect in multi–level atoms, bought about by dressing the atoms with the light of an appropriate auxiliary beam. A probe beam at reso- nance frequency can travel though the EIT medium that would be totally opaque without the assistance of the auxiliary light. Exactly on resonance a continuous probe wave travels at a phase velocity of c, i.e. the medium has a refractive index of unity. On the other hand, the EIT resonance is very sharp and is ultrasensitive with respect to frequency detuning. An EIT medium is thus extremely dispersive and, in turn, short light pulses hav- ing an extended spectrum travel at a very low group ve- locity vg. Another aspect of the extreme dispersion of an EIT medium is the ultrasensitivity with respect to the Doppler detuning due to motion. So, what happens if we replace the rotating block of non–dispersive material by an EIT medium of extreme dispersion? How large is the dragging coefficient? Let us determine the space–time line element, ds2=gµνdxµdxν, dxµ= (c dt, dx), (6) i.e. the covariant metric tensor gµν. For this we invert the contravariant metric of slow light [2], gµν, to lowest order in the ratio between the medium velocity and the velocity 1of light, c. In the limit of slow rotations, the line element turns out to have the same structure as the metric (1) of non–dispersive media with, however, a refractive index of unity and a modified dragging coefficient of α=c vg−1. (7) Compared with passive dielectric gyroscopes, a slow light gyroscope operated with light of a modest group velocity of kilometers per second increases the rotation sensitiv- ity by five orders of magnitude, and an ambitious group velocity of meters per second amounts to a fantastic im- provement by eight orders of magnitude. Of course, for this one would need to create slow light in a solid block of material in order to attach the gyroscope to the rotating body one is interested in. Solid–state media tend to de- stroy the quantum–interference conditions of EIT much more rapidly than gases or Bose–Einstein condensates [1]. However, first demonstrations of EIT in solids have been already reported [7] and an interesting proposal of EIT in semiconducters has been recently published [8]. It would be desirable to demonstrate unambiguously slow light in solids, stimulated perhaps by the potential advantage of slow light gyroscopes: ultrahigh motion sensitivity. ACKNOWLEDGEMENTS We are grateful to Malcolm Dunn and Stig Stenholm for valuable discussions. U.L. gratefully acknowledges the support of the Alexander von Humboldt Foundation and of the G¨ oran Gustafsson Stiftelse. [1] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999); M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, H. Welch, M. D. Lukin, Y. Ros- tovsev, E. S. Fry, and M. O. Scully, Phys. Rev. Lett. 82, 5229 (1999); D. Budiker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, ibid.83, 1767 (1999). [2] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 (2000). [3] U. Leonhardt and P. Piwnicki, Phys. Rev. A 60, 4301 (1999). [4] W. Gordon, Ann. Phys. (Leipzig) 72, 421 (1923). [5] Pham Mau Quan, C. R. Acad. Sci. (Paris) 242, 465 (1956); Archive for Rational Mechanics and Analysis 1, 54 (1957/58). [6] P. L. Knight, B. Stoicheff, and D. Walls (eds.), Phil. Trans. R. Soc. Lond. A 355, 2215 (1997); S. E. Har- ris, Phys. Today 50(7), 36 (1997); M. O. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997). [7] Y. Zhao, C. Wu, B.-S. Ham, M. K. Kim, and E. Awad, Phys. Rev. Lett. 79, 641 (1997); B.-S. Ham, M. S. Shariar,and P. R. Hemmer, Opt. Lett. 22, 1138 (1997); K. Ichimura, K. Yamamoto, and N. Gemma, Phys. Rev. A 58, 4116 (1998). [8] M. Artoni, G. C. La Rocca, and F. Bassani, Europhys. Lett.49, 445 (2000). 2

arXiv:physics/0003093v1 [physics.gen-ph] 29 Mar 2000Hierarchic Theory of Condensed Matter: Role of water in protein dynamics, function & cancer emergency Alex Kaivarainen JBL, University of Turku, FIN-20520, Turku, Finland http://www.karelia.ru/˜alexk H2o@karelia.ru Materials, presented in this original article are based on following publications: [1]. A. Kaivarainen. Book: Hierarchic Concept of Mat- ter and Field. Water, biosystems and elementary particles. New York, NY, 1995, ISBN 0-9642557-0-7 [2]. A. Kaivarainen. New Hierarchic Theory of Mat- ter General for Liquids and Solids: dynamics, thermody- namics and mesoscopic structure of water and ice (see: http://www.karelia.ru/˜alexk [New articles] ). [3]. A. Kaivarainen. Hierarchic Concept of Condensed Matter and its Interaction with Light: New Theories of Light Refraction, Brillouin Scattering and M¨ ossbauer effe ct (see: http://www.karelia.ru/˜alexk [New articles]). [4]. A. Kaivarainen. Hierarchic Concept of Condensed Matter : Interrelation between mesoscopic and macroscopic properties (see: http://www.karelia.ru/˜alexk [New arti - cles]). [5]. A. Kaivarainen. Hierarchic Theory of Complex Sys- tems (see URL: http://www.karelia.ru/˜alexk [New arti- cles]). See also papers at Los Alamos archives at: http://arXiv.org/find/physics/1/au:+Kaivarainen CONTENTS OF ARTICLE: Introduction to new Hierarchic theory of condensed matter 1. Role of inter-domain water clusters in large-scale dynam ics of proteins 2. Description of large-scale dynamics of proteins, based o n gen- eralized. Stokes-Einstein and Eyring-Polany equation 3. Dynamic model of protein-ligand complexes formation 4. The life-time of quasiparticles and frequencies of their excitation 5. Mesoscopic mechanism of enzyme catalysis 6. The mechanism of ATP hydrolysis energy utilization in mus cle contraction and protein polymerization 17. Water activity as a regulative factor in the intra- and int er-cell processes 8. Water and cancer Computerized verification of described here new models and theories has been presented, using special computer program, based on our new Hierarchic Theory of Con- densed Matter (copyright, 1997, A. Kaivarainen). =================================================== ========== Introduction to new Hierarchic theory of condensed matter (http://arXiv.org/abs/physics/00030044) A basically new hierarchic quantitative theory, general fo r solids and liquids, has been developed. It is assumed, that unharmonic oscillations of particles in any con- densed matter lead to emergence of three-dimensional (3D) s uperpo- sition of standing de Broglie waves of molecules, electroma gnetic and acoustic waves. Consequently, any condensed matter could b e con- sidered as a gas of 3D standing waves of corresponding nature . Our approach unifies and develops strongly the Einstein’s and De bye’s models. Collective excitations, like 3D standing de Broglie waves o f molecules, representing at certain conditions the mesoscopic molecul ar Bose con- densate, were analyzed, as a background of hierarchic model of con- densed matter. The most probable de Broglie wave (wave B) length is deter- mined by the ratio of Plank constant to the most probable impu lse of molecules, or by ratio of its most probable phase velocity to fre- quency. The waves B are related to molecular translations (t r) and librations (lb). As the quantum dynamics of condensed matter does not follow i n general case the classical Maxwell-Boltzmann distribution, the re al most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular Bose condensat ion in solids and liquids at temperatures, below boiling point. It is one o f the most important results of new theory, which we have confirmed by computer sim ulations on examples of water and ice. Four strongly interrelated new types of quasiparticles (collective excita- tions) were introduced in our hierarchic model: 21.Effectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the effectons (flickering clusters); 3.Transitons are the intermediate [ a⇋b] transition states of the trandlb effectons; 4.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons andconvertons. Primary effectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating ions, atoms or molecules. The volume of effectons (tr and lb) may contain fro m less than one, to tens and even thousands of molecules. The first condition m eans validity ofclassical approximation in description of the subsystems of the effect ons. The second one points to quantum properties of coherent clusters due to molecular Bose condensation . The liquids are semiclassical systems because their primar y (tr) effectons contain less than one molecule and primary (lb) effectons - mo re than one molecule. The solids are quantum systems totally because both kind of t heir primary effectons (tr and lb) are molecular Bose condensates .These conse- quences of our theory are confirmed by computer calculations . The 1st order [ gas→liquid ] transition is accompanied by strong decreasing of rotational (librational) degrees of freedom due to emerg ence of primary (lb) effectons and [ liquid →solid] transition - by decreasing of translational degrees of freedom due to Bose-condensation of primary (tr) effecton s. In the general case the effecton can be approximated by par- allelepiped with edges corresponding to de Broglie waves le ngth in three selected directions (1, 2, 3), related to the symmetry of the molecular dynamics. In the case of isotropic molecular moti on the effectons’ shape may be approximated by cube. The edge-length of primary effectons (tr and lb) can be consid ered as the ”parameter of order”. The in-phase oscillations of molecules in the effectons corr espond to the effecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the effectons differ in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state of the effectons has a common feature with Fr¨ olich’s polar mode. The(a→b)or(b→a)transition states of the primary effectons (tr and lb), defined as primary transitons, are accompanied b y a change in molecule polarizability and dipole moment withou t density fluctuations. At this case they lead to absorption or radiati on of IR photons, respectively. Superposition (interception) of three internal standing I R pho- tons of different directions (1,2,3) - forms primary electro magnetic deformons (tr and lb). 3On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density fluctuations, leading to absorption or radiation of phonons . Superposition resulting from interception of standing phonons in three direc- tions (1,2,3), forms secondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondary effectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromag- netic deformons ,lead to origination of macroeffectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeffec tonsin the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereffectons. In turn, the coherent excitation of both: tr andlb macrodeformons and macroconvertons in the same volume means creation of superdeformons. Su- perdeformons are the biggest (cavitational) fluctuations, leading to microbub- bles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, reflects all of possible combinations of the four basic ones [ 1-4], intro- duced above. This set of collective excitations in the form o f ”gas” of 3D standing waves of three types: de Broglie, acoustic and el ectro- magnetic - is shown to be able to explain virtually all the pro perties of all condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need the semi-empiric intermolecular potentials f or calculations, which are unavoidable in existing theories of many body systems. T he potential energy of intermolecular interaction is involved indirectly in di mensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into ac- count their different properties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra; [2]- Sound velocity; [3]- Density; [4]- Refraction index (extrapolated to the infinitive wave leng th of photon ). The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, to ev aluate more than 300 important characteristics of any condensed matter. Amo ng them are such as: total internal energy, kinetic and potential energies, heat-capacity and ther- mal conductivity, surface tension, vapor pressure, viscos ity, coefficient of self- diffusion, osmotic pressure, solvent activity, etc. Most of calculated parameters are hidden, i.e. inaccessible to direct experimental measu rement. The new interpretation and evaluation of Brillouin light sc attering and M¨ ossbauer effect parameters may also be done on the basis of h ierarchic the- ory. Mesoscopic scenarios of turbulence, superconductivi ty and superfluity are 4elaborated. Some original aspects of water in organization and large-sc ale dynamics of biosystems - such as proteins, DNA, microtubules, membrane s and regulative role of water in cytoplasm, cancer development, quantum neu rodynamics, etc. have been analyzed in the framework of Hierarchic theory. Computerized verification of our Hierarchic concept of matt er on examples of water and ice is performed, using special comput er pro- gram: Comprehensive Analyzer of Matter Properties (CAMP, c opy- right, 1997, Kaivarainen). The new optoacoustic device, ba sed on this program, with possibilities much wider, than that of IR, Ram an and Brillouin spectrometers, has been proposed (http://www.k arelia.ru/˜alexk [CAMP]). This is the first theory able to predict all known experimenta l temperature anomalies for water and ice. The conformity bet ween theory and experiment is very good even without any adjustab le pa- rameters. The hierarchic concept creates a bridge between micro- and m acro- phenomena, dynamics and thermodynamics, liquids and solid s in terms of quantum physics. =================================================== =========== 1. Role of inter-domain water clusters in large-scale dynam ics of proteins The functioning of proteins, namely antibodies, enzymes, i s caused by the physicochemical properties, geometry and dynamics o f their active sites. The mobility of an active site is related to the dynamics of the residual part of a protein molecule, its hydration she ll and the properties of a free solvent. The dynamic model of a protein proposed in 1975 and supported nowadays with numerous data (K¨ aiv¨ ar¨ ainen, 1985,1989b), is based on the following statements: 1. A protein molecule contains one or more cavities or clefts ca- pable to large scale fluctuations - pulsations between two st ates: ”closed” (A) and ”open” (B) with lesser and bigger accessibility to water. The frequency of pulsations (νA⇔B): 104s−1≤νA⇔B≤107s−1 depends on the structure of protein, its ligand state, tempe rature and solvent viscosity. Transitions between A and B states are th e result of 5the relative displacements of protein domains and subunits forming the cavities; 2. The water, interacting with protein, consists of two main frac- tions. The 1st major fraction, which solvates the outer surface reg ions of protein has less apparent cooperative properties than the 2 nd minor fraction confined to ”open” cavities. Water molecules, inte racting with the cavity in the ”open” (B)-state form a cooperative cl uster, whose lifetime (≥10−10s−1).Properties of clusters are determined by the geometry, mobility and polarity of the cavity, as well as by tem- perature and pressure. It is seen from X-ray structural data that the protein caviti es: active sites (AS), other interdomain clefts, the space between subunits of oligomeric pro- teins, have a high nonpolar residues content. In contrast to the small intra domain holes isolated from the outer medium, which sometime s contain several H2Omolecules, the interdomain and intersubunit cavities can c ontain several dozens of molecules (Fig. 1), exchanging with bulk water. The development of the above dynamic model has lead us to the following classification of dynamics in the native globular proteins. 1. Small-scale (SS) dynamics: low amplitude ( ≤1˚A) thermal fluctuations of atoms, aminoacids residues, and displacements of α-helixes and β-structures within domains and subunits, atwhich the effective Stokes radius of domains does not change. This type of motion, related to domain stability, can differ in the content of A and B conformers (Fig. 1, dashed line). The range of characteris tic times at SS dynamics is (10−4−10−11)s, determined by activation energies of corresponding transitions. 2. Large-scale (LS) dynamics: is subdivided into LS-pulsations and LS-librations with a c haracter of limited diffusion of domains and subunits of proteins: LS- pulsations are represented by relative translational-rotational dis place- ments of domains and subunits at distances ≥3˚A. Thus, the cavities, which are formed by domains, fluctuate between states with less (A) and more (B) water- accessibility. The life-times of these states depending on protein structure and external conditions are in the limits of/parenleftbig 10−4−10−7/parenrightbig s. In accordance to our model, one of contributions to this time is determined by frequency of excitations of [ lb/tr]macroconvertons. The frequency of macroconvertons excitation at normal conditions is about 1 07(1/s). The pulsation frequency of big multi-subunit oligomeric pr oteins of about 104(1/s) could be related to stronger fluctuations of water cluster i n their central cavity like macrodeformons or even superdeformons (Fig.3c,d). The life-times of (A) and (B) conformer markedly exceeds the time of tran- sitions between them ≃(10−9−10−11) s. The (A⇔B) pulsations of various cavities in proteins could be correl ated. The corresponding A and B conformers have different Stokes ra dii and effective 6volume. The geometrical deformation of the inter-subunits large ce ntral cavity of oligomeric proteins and the destabilization of the water cl uster located in it lead to relaxational change of ( A⇔B) equilibrium constant: KA⇔B= exp/parenleftbigg −GA−GB RT/parenrightbigg . The dashed line means that the stability and the small-scale dynamics of do- mains and subunits in the content of A and B conformers can diff er from each other. The [ A⇔B] pulsations are accompanied by reversible sorption- desorption of (20 −50)H2Omolecules from the cavities. Structural domains are space-separated formations with a m ass of (1 −2)· 103D. Protein subunits ( MM≥2·103D), as a rule, consist of 2 or more domains. The domains can consist only of αor only of β-structure or have no like secondary structure at all (Schulz, Schirmer, 1979). The shift of A⇔Bequilibrium of central cavity of oligomeric proteins determines their cooperative properties during consecuti ve ligand binding in the active sites. Signal transmission from the active sites to the remote regions of macromolecules is also dependent on ( A⇔B) equilibrium. Fig. 1. Examples of large-scale (LS) protein dynamics: A⇔B pulsations and librations with correlation times ( τB lb< τA lb) (K¨ aiv¨ ar¨ ainen, 1985, 1989): a) mobility of domains connected by flexible hinge or contact region, like in the light chains of immunoglobulins; b) mobility of domains that form the active sites of proteins , like in hexokinase, papain, pepsin, lysozyme etc. due to flexibil ity of contacts; 7c) mobility of subunits forming the oligomeric proteins lik e hemoglobin. Besides transitions of the active sites of each subunit, the (A⇔B) pulsations with frequencies of (104−106)s−1are pertinent to the common central cavity. b)librations represent the relative rotational - translational mo- tions of domains and subunits in composition of A and B confor mers with correlation times τM≃(1−5)·10−8s. LS - librations of domains are accompanied by ”flickering” of water cluster in the open cavity between domains or subunits. The p rocess of water cluster ”flickering”, i.e. [dissociation ⇋association] is close to the reversible first-order phase transition, when: ∆GH2O= ∆HH2O−T∆SH2O≈0 Such type of transitions in water-macromolecular systems c ould be responsible for so called ”enthalpy-entropy compensati on effects” (Lumry and Biltonen, 1969). The ”flickering clusters” means excitation of [ lb/tr]conversions be- tween librational and translational primary water effecton s, accom- panied by [association/dissociation] of coherent water cl uster (see dif- ference in dimensions of lb and tr effectons on Fig. 18a, bof [1] ). The water cluster (primary lb effecton) association and diss ocia- tion in protein cavities in terms of mesoscopic model repres ent the (ac)- convertons or (bc)- convertons. These excitations stimulate the LS- librations of domains in composition of B-conformer. Th e fre- quencies of (ac) and (bc) convertons, has the order of about 108c−1. This value coincides well with experimental characteristi c times for protein domains librations. The (ac) and (bc) convertons represent transitions between similar states of primary librational and translational effectons: [alb⇋atr]and [blb⇋btr](see Introduction to[1, 2]). For the other hand, the Macroconvertons, representing simu ltane- ous excitation of (ac+bc)convertons, are responsible for [B⇋A]large- scale pulsations of proteins. The librational mobility of domains and subunits is reveale d by the fact that the experimental value of τMis less than the theoretical one ( τt M) calculated on the Stokes-Einstein formula: τt M= (V/k)·η/T This formula is based on the assumption that the whole protei n can be approximated by a rigid sphere. It means, that the large-sca le dynamics can be 8characterized by the ”flexibility factor”, in the absence of aggregation equal to ratio: fl= (τM/τt M)≤1 Antonchenko (1986) has demonstrated, using the Monte-Carl o method for sim- ulations, that the disjoining pressure of a liquid in the pores onto the walls changes periodically depending on the distance ( L) between the limiting sur- faces. If the water molecules are approximated by rigid glob es, then the maxima of the wedging pressure lie on the values of distance L: 9.8; 7; and 3 .3˚A. It points, that small changes in the geometry of cavities can lead to sig nificant changes in theirA⇔Bequilibrium constant ( KA⇔B). According to our model the large-scale transition of the pro tein cavity from the ”open” B-state to the ”closed” A-state consists of the fo llowing stages: 1. Small reorientation (libration) of domains or subunits, which form an ”open” cavity (B-state). This process is induced by ( ac) or (bc) convertons of water librational effecton, localized in cavity (flickering of water cluster); 2. Cavitational fluctuation of water cluster, containing (2 0−50)H2Omolecules and the destabilization of the B-state of cavity as a result o f [lb⇋tr]macro- converton excitation; 3. Collapsing of a cavity during the time about 10−10s, dependent on pre- vious stage and concomitant rapid structural change in the h inge region of interdomain and intersubunit contacts: [ B→A] transition. Theb→atransition of one of the protein cavities can be followed by similar or the opposite A→Btransition of the other cavity in the macromolecule. It should be noted that the collapsing time of a cavitation bu bble with the radius: r≃(10−15)˚Ainbulk water and collapsing time of interdomain cavity are of the same order: ∆ t˜10−10sunder normal conditions (Shutilov, 1980). If configurational changes of macromolecules at B→AandA→Btransi- tions are sufficiently quick and occur as a jumps of the effectiv e volume, they accompanied by appearance of the shock acoustic waves in the bulk medium. When the cavitational fluctuation of water in the ”open” cavi ty does not occur, then ( b→a) or (B→A) transitions are slower processes, determined by continuous diffusion of domains and subunits. This happen s when [ lb→tr] macroconvertons are not excited. In their review, Karplus and McCammon (1986) analyzed data o n alco- holdehydrogenase, myoglobin and ribonuclease, which have been obtained using molecular dynamics approach. It has been shown that large-scale reorienta- tion of domains occur together with their deformation and mo tions of αandβ structures. It has been shown also (Karplus and McCammon, 1986) that acti vation free energies, necessary for [ A⇔B] transitions and the reorganization of hinge region between domains, do not exceed (3-4) kcal/mole. Such low values were obtained for proteins with even rather dense interdomain re gion, as seen from X-ray data. The authors explain such low values of activatio n energy by the 9fact that the displacement of atoms, necessary for such tran sition, does not exceed 0.5 ˚A, i.e. they are comparable with the usual amplitudes of atom ic oscillation at temperatures 20 −300C. It means that they occur very quickly within times of 10−12s, i.e. much less than the times of [ A→B] or [B→A] domain displacements (10−9−10−10s). Therefore, the high frequency small- scale dynamics of hinge is responsible for the quick adaptat ion of hinge geometry to the changing distance between the domains and for decreas ing the total activation energy of [ A⇔B] pulsations of proteins,. Recent calculations by means of molecular dynamics reveal t hat the oscilla- tions in proteins are harmonic at the low temperature (T <220K) only. At the physiological temperatures the oscillations are strongly unharmonic, collective, global and their amplitude increases with hydr ation (Stein- back et al., 1996). Water is a ”catalyzer” of protein unharmonic dynamics. It is obvious, that both small-scale (SS) and large-scale (L S) dy- namics, introduced in our model, are necessary for protein f unction. To characterize quantitatively the LS dynamics of proteins , we pro- posed the unified Stokes-Einstein and Eyring-Polany equati on. 2. Description of large-scale dynamics of proteins, based o n generalized Stokes-Einstein and Eyring-Polany equation In the case of the continuous Brownian diffusion of a particle, the rate constant of diffusion is determined by the Stokes-Einstein l aw: k=1 τ=kBT V η(1) where: τis correlation time, i.e. the time, necessary for rotation o f a particle by the mean angle determined as ¯ ϕ≈0.5 of the turn or the characteristic time for the translational movement of a particle with the radius (a) on the distance (¯∆x)1/2≃0.6a(Einstein, 1965); V= 4πa3/3 is the volume of the spherical particle; kBis the Boltzmann constant, T and ηare the absolute temperature and bulk viscosity of the solve nt. On the other hand, the rate constant of [ A→B] reaction for a molecule in gas phase, which is related to passing through the activatio n barrier GA→B, is described with the Eyring-Polany equation: kA→B=kTB hexp/parenleftbigg −GA→B RT/parenrightbigg (2) To describe the large-scale dynamics of macromolecules in s olution related to fluctuations of domains and subunits (librations and pulsat ions), an equation is needed which takes into account the diffusion and activation processes simulta- neously. 10The rate constant for the rotational- translational diffusi on of the parti- cle (kc) forming a macromolecule (continuous LS-dynamics) is expr essed with the generalized Stokes-Einstein and Eyring-Polany equati on (K¨ aiv¨ ar¨ ainen and Goryunov, 1987): Kc=kBT ηVexp/parenleftbigg −Gst RT/parenrightbigg =τ−1 c (3) where: V is the effective volume of domain or subunit, which ar e capable to the Brownian mobility independently from the rest part of th e macromolecule, with the probability: Plb= exp/parenleftbigg −Gst RT/parenrightbigg , (4) where: G stis the activation energy of structural change in the contact (hinge) region of a macromolecule, necessary for independent mobil ity of domain or subunit; τcis the effective correlation time for the continuous diffusio n of this relatively independent particle. The effective volume V can be changed under the influence of temperature, perturbants and ligands. The generalized Stokes-Einstein and Eyring-Polany equati on (3) is applicable also to describing the diffusion of the whole (i nteger) particle, dependent on the surrounding medium fluctuations with ac- tivation energy (Ga). The ligand diffusion in the active site cavity of proteins is such a type of processes. To describe noncontinuous process, the formula for rate constant ( kjump) of the jump-like translations of particle, related to emerg ency of cavitational fluctuations (holes) near the particle was proposed (K¨ aiv¨ ar¨ ainen and Goryunov, 1987): kjump=1 τmin jumpexp/parenleftbigg −W RT/parenrightbigg =1 τjump, (5) where: W=σS+ns(µout−µin) (6) is the work of cavitation fluctuation with the cavity surface S, at which ns molecules of the solvent (water) change its effective chemic al potential from µin toµout. The dimensions of cavity fluctuation near particle must be co mparable to corresponding particles. 11In a homogeneous phase (i.e. pure water) under equilibrium c onditions we have: µin=µout. With an increase of particle sizes, surface of cavitationa l fluctuation (S) and its work (W), the corresponding probabil ity of cavitation fluctuations: Pjump= exp( −W/RT ) will fall. The notion of the surface energy ( σ) retains its meaning even at very small ”holes” because of its molecular nature (see Section 11.4 of [1] and [4]). τmin jumpin eq. (5) is the minimal possible jump-time of a particle wit h mass (m) over the distance λwith the mean velocity: vmax= (2kT/m )1/2(7) Hence, we derive for the maximal jump-rate at W=0: kmax jump=1 τmin jump=Vmax λ=1 λ/parenleftbigg2kT m/parenrightbigg1/2 (8) In the case of hinged domains, forming macromolecules their relative A⇋B displacements (pulsations) are related not only to possibl e holes forming in the interdomain (intersubunit) cavities or near their oute r surfaces, but to the structural change of hinge regions as well. If the activation energy of necessary structure changes is e qual to GA⇋B st, then eq. (5), with regard for (8), is transformed into kA⇔B jump=1 λ/parenleftbigg2kT m/parenrightbigg1/2 exp/parenleftbigg −WA,B+GA⇔B st RT/parenrightbigg (9) where: W A,Bis the work required for cavitational fluctuations of water; this work can be different in two directions: ( WB) is necessary for nonmonotonic B→Atransition; ( WA) is necessary for jump-way A→Btransition. Under certain conditions A⇋Btransitions between protein conformers (LS- pulsations) can be realized owing to the jump-way and continuous types of relative diffusion of domains or subunits as two stage reac tion. In this case, the resulting rate constant of the process will be expressed through (9) and (3) as: kA⇔B res=kA⇔B jump=kA⇔B c=1 λ/parenleftbig2kT m/parenrightbig1/2exp/parenleftBig −WA,B+GA⇔B st RT/parenrightBig + +kT λVexp/parenleftBig −GA⇔B st RT/parenrightBig(10) 12The interaction between two domains in A-conformer can be described using microscopic Hamaker - de Bour theory. One of the contri butions into GA⇔B stis the energy of dispersion interactions between domains of theradius (a)(K¨ aiv¨ ar¨ ainen, 1989b): [Gst∼UH≈ −A∗a/12H]A,B (11) where A∗≈(A1/2 s−A1/2 c)2≈3 2πhνs 0[αsNs−αcNc]2(12) is the complex Hamaker constant; H is the slit thickness betw een domains in A- state;AcandAsare simple Hamaker constants, characterizing the properti es of water in the A-state of the cavity and in the bulk solvent, correspondingly. They depend on the concentration of water molecules ( Nc≃Ns) and their polarizability ( αc/ne}ationslash=αs): Ac=3 2πhνc 0α2 cN2 cand As=3 2πhνs 0α2 sN2 s; where: hνc 0≈hνs 0are the ionization potentials of H2Omolecules in a cavity and in a free solvent. In the ”closed” A-state of a cavity the water layer between do mains has a more compact packing as compared with the ice-like structur e of a water cluster in the B-state of a cavity, or with a free solvent. As far HA< H Bthe dispersion interaction (11) between domains in A- state of cavity is str onger, than that in B-state: UA H> UB H. Disjoining pressure of water in the cavities Π =−A∗/6πH3(12a) decreases with the increase of the complex Hamaker constant (A∗) that corre- sponds to the increase of the attraction energy ( UH) between domains. Cooperative properties of clusters in open (B)-states of th e cavities are more pronounced as compared to that in bulk water. That results in the greater changes of αcNcthan that of αsNsinduced by temperature. The elevation of temperature decreasing the dimensions of interdomain wate r clusters leads to the strengthening of interdomain interaction, while the lo wering temperature leads to opposite effect. We can judge about the changes of αsNsin the experiment on measuring the solvent refraction index, as far from our theory of refra ction index (eq. 8.14 of [1] or paper [3]): (n2 s−1)/n2 s=4 3παsNsor:αsNs=3 4π·n2 s−1 n2s(12b) 13In theclosed cavities the effect of temperature on water properties is lower as compared to that in bulk water. It follows that thermoinduced non- monotonic transition in the solvent refraction index must b e accom- panied by in-phase nonmonotonic changes of the [A⇔B]equilibrium constant (KA⇔B).As far (A) and (B) conformers usually have different sta- bility and flexibility, the changes of KA⇔Bwill be manifested in the changes of protein large-scale and small-scale dynamics. It has bee n shown before that viscosity itself has nonmonotonic temperature dependence due to the nonmono- tonic dependence of n2(t) (eq.11.44,11.45 and 11.48 of [1] or paper [4]). Thus, thermoinduced non-denaturational transitions of ma cromolecules and supramolecular systems located in the aqueous environm ent are caused by nonmonotonic changes in solvent properties, incl uding its refraction index. The influence of D2Oand other perturbants on protein dynamics is explained in a similar way. The effect of deuterium oxide ( D2O) is a result of substitution ofH2Ofrom protein cavities and corresponding change of complex H amaker constant (12). Generalized equation (3) is applicable not only for evaluat ing the frequency of macromolecules transition between A and B conformers but also for the frequency of the dumped librations of domains and subunits w ithin A and B conformers. Judging by various data (K¨ aiv¨ ar¨ ainen, 1985 ,1989b), the interval of A⇋Bpulsation frequency is: νA⇔B=1 tA+tB≈kA→B= (104−107)s−1(13) where: tAandtBare the lifetimes of A and B conformers. The corresponding interval of the total activation energy o f the jump-way A⇔Bpulsations can be evaluated from the eq. (9). We assume for th is end that the pre-exponential multiplier is about 1010s−1as a frequency of cavita- tional fluctuations in water with the radius ˜(10-15)o A. Taking a logarithm of (9) we derive: GA⇔B res= (WA,B+GA⇔B st)≈RT(ln1010−lnνA⇔B) (14) At physiological temperatures the following region of ener gy corresponds to the frequency range of pulsations (13) GA⇔B res≈(4−8) kcal/mole (15) Such a region of energies is pertinent to a wide range of bioch emical processes. The quick jump-way pulsations of macromolecules can cause a coustic shock- waves in the solvent and its structure destabilization. Con comitant increase in water activity leads to distant interaction between diffe rent proteins as well 14as proteins and cells. Such solvent-mediated phenomena wer e discovered and studied in our laboratory by set of specially elaborated met hods (K¨ aiv¨ ar¨ ainen, 1985, 1986, 1987; K¨ aiv¨ ar¨ ainen et al., 1990, K¨ aiv¨ ar¨ ai nen et al., 1993). When the [ A⇔B] transitions in proteins are related to continuous diffusion only, then the Gstvalues calculated using eq.(3) for the same frequency inter val (104−107)s, is about (3 - 7) kcal/mole. TheKramers equation (1940), which has earlier been widely used for de- scribing diffusion processes, has the form: k=A ηexp/parenleftbigg −H∗ RT/parenrightbigg (16) where A is a constant and η- solvent viscosity. The pre-exponential factor in our generalized equation (3) con- tains not only the viscosity, but also the temperature and th e effec- tive volume of a particle. It was shown in our experiments tha t eq. (3) describes the dynamic processes, which occur in macromo lecules, solutions much better than the Kramer’s equation (16). 3. Dynamic model of protein-ligand complexes formation According to our model of specific complexes formation the fo l- lowing order of events is assumed (Fig. 2): 1. Ligand (L) collides with the active site (AS), formed usua lly by two domains, in its open (b) state: the structure of water clu ster in AS is being perturbed and water is forced out of AS cavity tota lly or partially; 2. Transition of AS from the open (b) to the closed (a) state oc curs due to strong shift of [a⇔b]equilibrium to the left, i.e. to the AS domains large scale dynamics; 3. A process of dynamic adaptation of complex [L+AS] begins, accompanied by the directed ligand diffusion in AS cavity due to its domains small-scale dynamics and deformation of their t ertiary structure; 4. If the protein is oligomeric with few AS, then the above eve nts cause changes in the geometry of the central cavity between s ubunits in the open state leading to the destabilization of the large central water cluster and the shift of the A⇋B,corresponding to R⇋Tequi- librium of quaternary structure leftward. Water is partial ly forced out from central cavity. Due to the feedback mechanism this shift can influence the [a⇔b] equilibrium of the remaining free AS and promotes its reacti on with the next ligand. Every new ligand stimulates this process, p romoting 15the positive cooperativity. The negative cooperativity al so could be resulted from the interaction between central cavity and ac tive sites; 5. The terminal [protein −ligand ]complex is formed as a conse- quence of the relaxation process, representing deformatio n of domains and subunits tertiary structure. This stage could be much sl ower than the initial ones [1-3]. As a result of it, the stability of the complex grows up. Dissociation of specific complex is a set of reverse processes to that described above which starts from the [ a∗→b] fluctuation of the AS cavity. In multidomain proteins like antibodies, which consist of 1 2 do- mains, and in oligomeric proteins, the cooperative propert ies of H2O clusters in the cavities can determine the mechanism of sign al trans- mission from AS to the remote effector regions and allosteric protein properties. The stability of a librational water effecton as coherent clu ster strongly de- pends on its sizes and geometry. This means that very small de formations of protein cavity, which violate the [cavity-cluster] comp lementary condition, induce a cooperative shift of [ A⇔B] equilibrium leftward. The clusterphilic interaction, introduced by us (see section 13.3 of [1] or pap er [4]) turns to hy- drophobic one due to [ lb/tr] conversion. This process can be developed step by step . For example, the reori- entation of variable domains, which form the antibodies act ive site (AS) after reaction with the antigen determinant or hapten deforms the next cavity be- tween pairs of variable and constant domains forming F absubunits (Fig.2). The leftward shift of [ A⇔B] equilibrium of this cavity, in turn, changes the ge- ometry of the big central cavity between FabandFcsubunits, perturbing the structure of the latter. Therefore, the signal transmission from the AS to the effector sites of Fcsubunits occurs due to the balance shift between clusterphilic and hydrophobic interactions. This signal is re- sponsible for complement- binding sites activation and tri ggering the receptors function on the lymphocyte membranes, in accordance to our m odel. The leftward shift of [ A⇔B] equilibrium in a number of cavities in the elongated multidomain proteins can lead to the significant d ecrease of their linear size and dehydration. The mechanism of muscular cont raction is probably based on such phenomena and clusterphilic interactions (se e next section). For such a nonlinear system the energy is necessary for reori entation of the first couple of domains only. The process then goes on spontan eously with decreasing the averaged protein chemical potential. The chemical potential of the A- conformer is usually lower t han that of B- conformer ( ¯GA<¯GB) and the relaxation of protein is accompanied by the leftward A⇔Bequilibrium shift of cavities. It is predictable, that hydration of proteins will decrease , when clusterphilic [water-cavity] interaction turns to hydrophobic one. 16Fig. 2. The schematic picture of the protein association (Fab subunits of antibody with a ligand), which is accompanied by the destabilization of water clusters in cavities, according t o the dynamic model (K¨ aiv¨ ar¨ ainen, 1985). The dotted line denotes the p erturba- tion of the tertiary structure of the domains forming the act ive site. Antibodies of IgG type contain usually two such Fab subunit a nd one Fc subunit, conjugated with 2Fab by flexible hinge, formi ng the general Y-like structure. Our dynamic model of protein behavior and signal transmissi on, described above, is an alternative to solitonic mechanism of non dissi pative signal trans- mission in proteins and in other biosystems proposed by Davi dov (1973). Prop- agation of solitonic wave is a well known nonlinear process i n the ordered ho- mogeneous mediums. The solitons can originate, when the nonlinear effe cts are compensated by the wave dispersion effects. Dispersion i s reflected in fact that the longer waves spreads in medium with higher velocity than the shorter ones. However, biosystems of nonregular, fluctuating structure a re not the mediums, good for solitons emergency and propagation. Our dynamic model takes into account the real multidomain an d multiglobular structure of a proteins and properties of the ir hydration shell fractions. In contrast to Davidov’s solitonic model, the dissipa- tion processes like reversible ”melting” of water clusters , accompa- nied by large-scale dynamics of proteins, are the necessary stages of our [hydrophobic ⇋clusterphilic] mechanism of signal transmission in biosystems. 17The evolution of the ideas of the protein-ligand complex for mation proceeded in the following sequence: 1. ”Key-lock” or the rigid conformity between the geometry o f an active site and that of a ligand (Fisher, 1894); 2. ”Hand-glove” or the so-called principle of induced confo rmity (Koshland, 1962); 3. At the current stage of complex-formation process unders tand- ing, the crucial role of protein dynamics gets clearer. Our m odel allows us to put forward the ”Principle of Stabilized Confor mity (PSC)” instead that of ”induced conformity” in protein-lig and spe- cific reaction. Principle of Stabilized Conformity (PSC) means that the geo me- try of the active site (AS), optimal from energetic and stere ochem- ical conditions, is already existing BEFORE reaction with l igand. The optimal geometry of AS is to be the only one selected among the number of others and stabilized by ligand, but not induce d ”de nova”. For example, the [a⇔b]large-scale pulsations of the active sites due to domain fluctuations and stabilization of the closed (a ) state by ligand are necessary for the initial stages of reaction. S uch active site pulsations decreases the total activation energy nece ssary for the terminal complex formation as multistage process. 4. The life-time of quasiparticles and frequencies of their excitation The set of formula, describing the dynamic properties of qua sipar- ticles, introduced in mesoscopic theory was presented at Ch apter 4 of book [1] and paper [2]: The frequency of c- Macrotransitons or Macroconvertons exc itation, repre- senting [dissociation/association] of primary libration al effectons - ”flickering clusters ”as a result of interconversions between primary [lb] and [tr ] effectons is: FcM=1 τMc·PMc/Z (17) where: PMc=Pac·Pbcis a probability of macroconverton excitation; Zis a total partition function (see eq.4.2 of [1, 2]); the life-time of macroconverton is: τMc= (τac·τbc)1/2(18) 18The cycle-period of (ac) and (bc) convertons are determined by the sum of life-times of intermediate states of primary translationa l and librational effec- tons: τac= (τa)tr+ (τa)lb; τbc= (τb)tr+ (τb)lb;(19) The life-times of primary and secondary effectons (lb and tr) ina- and b- states are the reciprocal values of corresponding state fre quencies: [τa= 1/νa;τa= 1/νa]tr,lb; [ τb= 1/νb;τb= 1/νb]tr,lb (20) [(νa) and ( νb)]tr,lbcorrespond to eqs. 4.8 and 4.9 of [1, 2]; [(νa) and ( νb)]tr,lbcould be calculated using eqs.4.16; 4.17 [1, 2]. The frequency of (ac)and(bc) convertons excitation [lb/tr]: Fac=1 τac·Pac/Z (21) Fbc=1 τbc·Pbc/Z (22) where: PacandPbcare probabilities of corresponding convertons excitation s (see eq.4.29a of [1, 2]). The frequency of Supereffectons and Superdeformons (bigges t fluc- tuations) excitation is: FSD=1 (τA∗+τB∗+τD∗)·PD∗ S/Z (23) It is dependent on cycle-period of Supereffectons: τSD=τA∗+τB∗+τD∗ and probability of Superdeformon activation ( PD∗ S),like the limiting stage of this cycle. The averaged life-times of Supereffectons in A∗andB∗state are dependent on similar states of translational and librational macroeff ectons : τA∗= [(τA)tr·(τA)lb] = [(τaτa)tr·(τaτa)lb]1/2(24) and that in B state: τB∗= [(τB)tr·(τB)lb] = [(τbτb)tr·(τbτb)lb]1/2(25) 19The life-time of Superdeformons excitation is determined by frequency of beats between A∗and B∗states of Supereffectons as: τD∗= 1/|(1/τA∗)−(1/τB∗)| (26) The frequency of translational and librational macroeffect onsA⇋ Bcycle excitations could be defined in a similar way: /bracketleftbigg FM=1 (τA+τB+τD)·PD M/Z/bracketrightbigg tr,lb(27) where: (τA)tr,lb= [(τa·τa)tr,lb]1/2(28) and (τB)tr,lb= [(τb·τb)tr,lb]1/2(29) (τD)tr,lb= 1/|(1/τA)−(1/τB)|tr,lb(30) The frequency of primary translational effectons (a⇋b)trtransi- tions could be expressed like: Ftr=1/Z (τa+τb+τt)tr·(Pd)tr (31) where: ( Pd)tris a probability of primary translational deformons excita tion (eq. 4.25 of [1, 2]); [τa;τb]trare the life-times of (a) and (b) states of primary translational ef- fectons (eq. 20). The frequency of primary librational effectons as ( a⇋b)lbcycles excitations is : Flb=1/Z (τa+τb+τt)lb·(Pd)lb (32) where: ( Pd)lbis a probability of primary librational deformons excitati on;τaand τbare the life-times of (a) and (b) states of primary libration al effectons defined as (20). The life-time of primary transitons (tr and lb) as a result of quantum beats between (a) and (b) states of primary effectons could be intro duced as: 20[τt=|1/τa−1/τb|−1]tr,lb (33) For the case of ( a⇔b)1,2,3transitions of primary and secondary effectons (tr and lb), their life-times in (a) and (b) states are the reciprocal val ue of corresponding frequencies: [ τa= 1/νaandτb= 1/νb]1,2,3 tr,lb. These parameters and the resulting ones could be calculated from eqs.(2.27; 2 .28 of [1]) for primary effectons and (2.54; 2.55 of [1]) for secondary ones. The results of calculations, using eqs. (31, 32) for frequen cy of excitations of primary tr and lb effectons are plotted on Fig. 3a,b. The frequencies of Macroconvertons and Superdeformons wer e calculated using eqs.(17 and 23). Fig. 3. (a) - Frequency of primary [tr] effectons excitations, calculated from eq.(31); (b) - Frequency of primary [lb] effectons excitations, calcu lated from eq.(32); (c) - Frequency of [ lb/tr] Macroconvertons (flickering clusters) excitations, calculated from eq.(17); (d) - Frequency of Superdeformons excitations, calculated from eq.(28). At the temperature interval (0-100)0Cthe frequencies of translational and librational macrodeformons (tr and lb) are in the interval o f (1.3-2.8) ·109s−1and 21(0.2-13) ·106s−1correspondingly. The frequencies of (ac) and (bc) converto ns could be defined also using our software and formulae, presen ted at the end of Chapter 4 of [1, 2]. The frequency of primary translational effectons [ a⇔b]trexcitations at 200C, calculated from eq.(31) is ν∼7·1010(1/s). It corresponds to electromag- netic wave length in water with refraction index ( n= 1.33) of: λ= (cn)/ν∼6mm (34) For the other hand, there are a lot of evidence, that irradiat ion of very different biological systems with such coherent electromagnetic fiel d exert great influences on their properties (Grundler and Keilman, 1983). Between the dynamics/function of proteins, membranes, etc . and dynamics of their aqueous environment the strong interrela tion is existing. The frequency of macroconvertons, representing big densit y fluctuation in the volume of primary librational effecton at 37C is about 107(1/s) (Fig 3c). The frequency of librational macrodeformons at the same tem - perature is about 106s−1,i.e. coincides with frequency of large-scale protein cavities pulsations between open and closed to wate r states (see Fig.2). This confirm our hypothesis that the clusterphi lic in- teraction is responsible for stabilization of the proteins cavities open state and that transition from the open state to the closed on e is induced by coherent water cluster dissociation. The frequency of Superdeformons excitation (Fig.3d) is muc h lower: νs∼(104−105)s−1(35) Superdeformons are responsible for cavitational fluctuati ons in liq- uids and origination of defects in solids. Dissociation of o ligomeric proteins, like hemoglobin or disassembly (peptization)of actin and microtubules could be also related with such big fluctuation s. 5. Mesoscopic mechanism of enzyme catalysis The mechanism of enzyme catalysis is one of the most intrigui ng and unresolved yet problems of molecular biology. It become s clear, that it is interrelated not only with a spatial, but as well wi th hier- archical complicated dynamic properties of proteins (see b ook: ”The Fluctuating Enzyme” , Ed. by G.R.Welch, 1986). The [proteins + solvent] system should be considered as a coo perative one with feedback links (Kaivarainen, 1985, 1992). Somogyi and Damjanovich (1986) proposed a similar idea that collective excitations of protein structure are interrelated with surrounded water molecules oscillat ions. 22The enzymatic reaction can be represented in accordance wit h our dynamic model as a consequence of the following stages (K¨ ai v¨ ar¨ ainen, 1989; [1] ). The first stage: (I)Eb+S⇋EbS (36) - the collision of the substrate (S) with the open (b) state of the active site [AS] cavity of enzyme (E). The frequency of collisions between the enzyme and the subst rate, whose concentrations are [ CE] and [ CS], respectively, is expressed with the known formula (Cantor and Schimmel, 1980): νcol= 4πr0(DE+DS)·N0[CE]·[CS] (37) where: r0=aE+aSis the sum of the enzyme’s and substrate’s molecular radii; N0is the Avogadro number; DE=kT 6πηaEand DS=kT 6πηaS(38) - are the diffusion coefficients of the enzyme and substrate; k i s the Boltzmann constant; T is absolute temperature; ηis a solvent viscosity. The probability of collision of ( b) state of the active site with substrate is proportional to the ratio of the b-state outer cross section area to the whole enzyme surface area: Pb=∅b ∅E·Fb (39) where: Fb=fb fb+fais a fraction of time, the active site [AS] spend in the open (b) - state. So, the frequency of collision between the substrate and ( b) state of the active site (AS) with account for (40), meaning the first stag e of reaction is: νb col=νcol·Pb=kI (40) The second stage of enzymatic reaction is a formation of the primary enzyme- substrate complex: (II)EbS⇔[Ea∗S](1)(41) 23It corresponds to transition of the active site cavity from t he open ( b) state to the closed ( a) one and stabilization the latter state by a ligand. The rate constant of the [ b→a] transitions is derived with the Stokes- Einstein and Eyring -Polany generalized equation (3): kb→a∗ II=kT ηVexp/parenleftbigg −Gb→a st RT/parenrightbigg (42) where: ηis the solvent viscosity; V is the effective volume of the enzy me domain, whose diffusional reorientation accompanies the ( b→a) transition of the active site [AS]. The leftward shift of the [ a⇔b] equilibrium between two states of the active site is concomitant with this stage of the reaction. It reflec ts the principle of stabilized conformity, related to AS domains movements that we have put forward in the previous section. The third stage: (III) [EaS]⇔[Ea∗S∗] (43) represents the formation of the secondary specific complex. This process is related to directed ligand diffusion into the active site cav ity and the dynamic adaptation of its geometry to the geometry of the active site . Here the Principle of Stabilized microscopic Conformity is real- ized, when the AS change its geometry from (a) to (a∗)without do- mains reorientation. The rate constant of this stage is determined by the rate constant of substrate diffusion in the closed ( a) state of the active site cavity. It is also expressed by generalized kinetic equation (42), but with other values of variables: k1→2∗ S=1 τ∗sexp/parenleftbigg −Ga s RT/parenrightbigg =kIII (44) where: τs= (vs/k)ηin/T (45) is the correlation time of substrate of volume ( vs) in the ( a) state of the active site;ηinis the internal effective viscosity; Ga Sis the activation energy of thermal fluctuations of groups, representing small-scale dynamics (SS), which determine the directed diffusion of a substrate in the active site [AS] c losed cavity. The directed character of ligand diffusion in AS can be determ ined by the relaxation of a protein structure, due to perturbation of AS domains by ligand. The relaxation changes were observed in many reactions of sp ecific protein- ligand complexes formation (K¨ aiv¨ ar¨ ainen, 1985, 1989). 24The complex formation [pair of domains forming the AS + subst rate], fol- lowed by these domain immobilization can be considered as an emergency of a new enlarged protein primary effecton from two smaller ones, corresponding to less independent AS domains or their compact ”nodes”. We assume that at this important stage, the waves B of the atta ck- ing catalytic atoms (λc B)and the attacked substrate atoms (λS B)start to overlap and interfere in such a way that conditions for qua ntum- mechanical beats between them become possible. Let us consider these conditions in more detail. According to classical statistics, every degree of freedom gets the energy, which is equal to kT/2. This condition corresponds to harmonic approximation when the mean potential and kinetic energies of particles ar e the same: V=Tk=mv2/2≈kT/2 (46) The corresponding to such ideal case the de Broglie wave (wav e B) length is equal to: λB=h mv=h (mkT)1/2(47) For such condition the wave B length of proton at room tempera ture is nearly 2.5˚A, for a carbon atom it is about three times smaller and for oxy gen - four times as small. In the latter two cases, the wave B lengths are comparable and even less than the sizes of the atoms itself. So, their waves can not ove rlap and, beats between them are not possible. However, in real condensed systems with quantum properties , in- cluding the active sites of enzymes, the harmonic approxima tion is not valid because (Tk/V)≪1 (see Fig. 5 of [1]). Consequently, the kinetic energy of atoms of AS: Tk≪(1/2)kT and λB≫h/(mkT)1/2(47a) It must be taken into account that librations, in a general ca se are presented by rotational-translational unharmonic osc illations of atoms and molecules, but not by their rotational motions onl y (Coffey et al., 1984). The length of waves B of atoms caused by a small translational component of most probable impulse, related to librations i s bigger than that related to pure translations: [λlb= (h/Plb)]>[λtr= (h/Ptr)] (47b) 25Even in pure water the linear sizes of primary librational eff ectons are several times bigger than that of translational effecton s and the size of one H2Omolecule (see Fig. 7 of [1] or Fig.4 of [2]). In composition of the active site rigid core the librational waves B of atoms can significantly exceed the sizes of the atoms thems elves. In this case their superposition, leading to quantum beats o f waves B in the [active site - substrate] complex, accelerating the enzyme reaction is quite possible. In accordance to our model, periodic energy exchange result ing from such beats occurs between waves B of the substrate and ac tive site atoms. The reaction [S∗→P], accelerating by these quantum beats, is the next 4th stage of enzymatic process - the chemical transform ation of a substrate into product: (IV) [ Ea∗S∗]→[Ea∗P] (48) The angular wave B frequency of the attacked atoms of substrate with mass mS and the amplitude AScan be expressed by eq.(2.20): ωS=/planckover2pi1/2mSA2 S (49) The wave B frequency of the attacking catalytic atom (or a gro up of atoms) in the active site is equal to: ωcat=/planckover2pi1 2mcAc(50) The frequency of quantum beats which appear between waves B of catalytic and substrate atoms is: ω∗=ωcat−ωS=/planckover2pi1/parenleftbigg1 2mcAc−1 2mSAS/parenrightbigg (51) The corresponding energy of beats: E∗=Ecat−ES=/planckover2pi1ω∗(52) It is seen from these formulae that the smaller the wave B mass of the cat- alytic atom ( mc) and its amplitude (A c), the more frequently these beats occur at constant parameters of substrate (m SandAS).The energy of beats is transmitted to the wave B of the attacked substrate atom from the catalytic atom, accelerating the reaction. 26According to our model, the perturbations in the region of th e active site are accompanied by the appearance of phonons and acoustic defor mons in a form of small-scale dynamics of protein structure. They provide th e signal transmission in oligomeric proteins to the central cavity and other activ e sites leading to allosteric effects. It is known from the theory of oscillations (Grawford, 1973) that the effect of beats is maximal, if the amplitudes of the interacting osc illators are equal: A2 c≈A2 S (53) The [substrate →product] transformation can be considered as a result of the substrate wave B transition from the main [S] state to excite d [P] state. The rate constant of such a reaction in the absence of the catalys t (kS→P) can be presented by the modified Eyring-Polany formulae, leading f rom eq.(2.27 of [1]) at condition: exp( hνp/kT)>>1 νS A=kS→P=EP hexp/parenleftbigg −EP−ES kT/parenrightbigg =νP Bexp/bracketleftbigg −h(νP B−νS B) kT/bracketrightbigg (54) where: ES=hνSandEP=hνPare the main and excited - transitional to product energetic states of substrate, correspondingly ;νSandνPare the substrate wave B frequencies in the main and excited states, respectively. If catalyst is present, which acts by the above described mechanism, then the energy of the substrate ESis increased by the magnitude Ecatwith the quantum beats frequency ω∗(51) and gets equal to: ESc=ES+Ecat(55) Substituting EP=hνP Band ESc=h(νS B+νcat B) in (54), we derive the rate constants for the catalytic reaction in the moment of be ats (kSc→P).This corresponds to the 4th stage of enzymatic reaction: (IV): kSc→P=νPexp/bracketleftbigg −h(νP−νS−νcat) kT/bracketrightbigg =kIV (56) where: νP, νSandνcare the most probable B wave frequencies of the transition [S→P] state, of the substrate and of the catalyst atoms, correspo ndingly. Hence, in the presence of the catalyst the coefficient of accel eration (q) is equal to: qcat=kSc→P kS→P= exp/parenleftbigghνcat kT/parenrightbigg (57) 27For example, at hνc/kT≈10; qcat=2.2·104(58) At room temperatures this condition corresponds to Ecat=hνcat≃6 kcal/mole. The beating acts, followed by transitions of a substrate mol ecule to activated by catalyst excited state ( S→Sc), can be accompanied by the absorption of phonons or photons with the frequency ω∗(51). In this case, the insolation of a [substrate - catalyst] system with ultrasound or electr omagnetic field of the frequency ω∗should strongly accelerate the reaction when the resonance conditions are satisfied. It is possible that the resonance effects of this type can acco unt for the ex- perimentally revealed response of various biological syst ems to electromagnetic field radiation with the frequency about 6 ·1010Hz(Deviatkov et al., 1973). We have to point out that just such frequency is close to the freq uency of (a ⇋b)tr transitions excitation of primary translational water effe ctons. The strict cor- relation between the dynamics of water and that of biosystem s should exist on each hierarchic level of time and space. Changes in the volume, geometry and electronic properties o f the substrate molecule, resulting from its transition to the product, cha nge its interaction energy with the active site by the magnitude: ∆Ea∗ SP= (Ea∗ S−Ea∗ P) (59) It must destabilize the closed state of the active site and in crease the probability of its reverse [ a∗→b∗] transition. Such transition promotes the last 5th stage of the catalytic cycle - the dissociation of the enzyme-product complex: (V) [ Ea∗P]⇔[Eb PP]⇔Eb+P (60) The resulting rate constant of this stage, like stage (II), i s described by the generalized Stokes-Einstein and Eyring-Polany equation ( 42), but with different activation energy Ga∗→b stvalid for the [ a∗→b] transition: ka∗→b P=kT ηVexp/parenleftbigg −Ga∗→b st kT/parenrightbigg = 1/τa∗→b P (61) If the lifetime of the ( a∗) state is sufficiently long, then the desorption of the product can occur irrespective of [ a∗→b] transition, but with a longer char- acteristic time as a consequence of its diffusion out of the ac tive site’s ”closed 28state”. The rate constant of this process ( k∗ p) is determined by small-scale dy- namics. It practically does not depend on solvent viscosity , but can grows up with rising temperature, like at the 3d stage, described by ( 44): k∗ P=kT ηin a∗·vPexp/parenleftbigg −Ga∗ P kT/parenrightbigg = 1/τ∗ P (62) where ηin a∗is active site interior viscosity in the closed ( a∗) state; vPis the effective volume of product molecules; Pa∗= exp( −Ga∗ P/RT) is the probability of small-scale, functionally important motions necessary for the desorption of the product from the ( a∗) state of the active site; Ga∗ pis the free energy of activation of such motions. Because the processes, described by the eq.(61) and (62), ar e in- dependent, the resulting product desorption rate constant is equal to: kV=ka∗→b P+k∗ P= 1/τa∗→b P+ 1/τ∗ P (63) The characteristic time of this final stage of enzymatic reac tion is: τV= 1/kV=τa∗→b P·τ∗ P τa∗→b P+τ∗ P(64) This stage is accompanied by the relaxation of the perturbed AS domain and remnant protein structure to the initial state. After the whole reaction cycle is completed the enzyme gets r eady for the next cycle. The number of cycles (catalytic acts) in the majo rity of enzymes is within the limits of (102−104)s−1. It means that the [ a⇋b] pulsations of the active site cavities must occur with higher frequency as far it is only one of the five stages of enzymatic reaction cycle. In experiments, where various sucrose concentrations were used at constant temperature, the dependence of enzymatic catalysis rate on solvent viscosity (T/η) was demonstrated (Gavish and Weber, 1979). The amendment f or chang- ing the dielectric penetrability of the solvent by sucrose w as taken into account. There are reasons to consider stages (II) and/or (V) in the mo del described above as the limiting ones of enzyme catalysis. According to eqs.(43) and (61), these stages depend on ( T/η), indeed. The resulting rate constant of the enzyme reaction could be e x- pressed as the reciprocal sum of life times of all its separat e stages (I-V): kres= 1/τres= 1/(τI+τII+τIII+τIV+τV) (65) where 29τI= 1/kI=1 νcol·Pb;τII= 1/kII=ηV kTexp/parenleftbiggGb→a st RT/parenrightbigg ; (65a) τIII= 1/kIII=ηinvS kTexp/parenleftbiggGa SS RT/parenrightbigg ; (65b) τIV= 1/kIV=1 (νp)exp/bracketleftBig h(νs−νp−νc kT/bracketrightBig ; τVcorresponds to eq.(64). The slowest stages of the reaction seem to be stages (II), (V) , and stage (III). The latter is dependent only on the small- scale dynam ics in the region of the active site. Sometimes product desorption goes on much more slowly than other stages of the enzymatic process, i.e. kV≪kIII≪kII Then the resulting rate of the process ( kres) is represented by its limiting stage (eq. 63): kres≈kT ηVexp/parenleftbigg −Ga∗→b st RT/parenrightbigg +k∗ P= 1/τres (66) The corresponding period of enzyme turnover: τres≈τV(eq64). The internal medium viscosity ( ηin) in the protein regions, which are far from the periphery, is 2-3 orders higher than the viscosity of a water -saline solvent (0.001 P) under standard conditions: (ηin/η)≥103 Therefore, the changes of sucrose concentration in the limi ts of 0-40% at con- stant temperature can not influence markedly internal small - scale dynamics in proteins, its activation energy ( Ga∗) and internal microviscosity (K¨ aiv¨ ar¨ ainen, 1989b). This fact was revealed using the spin-label method. It is in accordance with viscosity dependencies of tryptophan fluorescence que nching in proteins and model systems related to acrylamide diffusion in protein matrix (Eftink and Hagaman, 1986). In the examples of parvalbumin and ribon uclease T1it has been shown that the dynamics of internal residues is prac tically insensitive to changing solvent viscosity by glycerol over the range of 0 .01 to 1 P. It follows from the above data that the moderate changes in so lvent viscosity (η) at constant temperature do not influence markedly the k∗ Pvalue in eq.(66). 30Therefore, the isothermal dependencies of k reson (T/η)Twith changing sucrose or glycerol concentration must represent straight lines with the slope: tgα=∆kres ∆(T/η)T=k Vexp/parenleftbigg −Ga∗→b st RT/parenrightbigg T(67) The interception of isotherms at extrapolation to ( T/η→0) yields (62): (kres)(T/η)→0=k∗ P=kT ηin a∗vPexp/parenleftbigg −Ga∗ RT/parenrightbigg (68) The volume of the Brownian particle (V) in eq.(67) correspon ds to the effective volume of one of the domains, which reorientation is respons ible for (a ⇋b) transitions of the enzyme active site. Under conditions when the activation energy Ga∗→b stweakly depends on tem- perature, it is possible to investigate the temperature dep endence of the effective volume V, using eq.(67), analyzing a slopes of set of isother ms (67). Our model predicts the increasing of V with temperature risi ng. This reflects the dumping of the large-scale dynamics of proteins due to wa ter clusters melting and enhancement the Van der Waals interactions between prot ein domains and subunits (K¨ aiv¨ ar¨ ainen, 1985;1989b, K¨ aiv¨ ar¨ ainen et al., 1993). The contribution of the small-scale dynamics ( k∗) tokresmust grow due to its thermoactivation and the decrease in ηinandGa∗(eq.32). The diffusion trajectory of ligands, substrates and product s of en- zyme reactions in ”closed” (a) states of active sites is prob ably deter- mined by the spatial gradient of minimum wave B length (maxim um impulses) values of atoms, forming the active site cavity. We suppose, that functionally important motions (FIM) , introduced by H. Frauenfelder et al., (1985, 1988), are determined by spec ific geometry of the impulse space characterizing the distribution of small-sc ale dynamics of domains in the region of protein’s active site. The analysis of the impulse distribution in the active site a rea and energy of quantum beats between de Broglie waves of the at oms of substrate and active sites, modulated by solvent-depend ent large- scale dynamics, should lead to complete understanding of th e physical background of enzyme catalysis. 6. The mechanism of ATP hydrolysis energy utilization in mus cle contraction and protein polymerization A great number of biochemical reactions are endothermic, i. e. they need additional thermal energy in contrast to exothermic on es. The most universal and common source of this additional energy i s a re- action of adenosinetriphosphate (ATP) hydrolysis: 31ATPk1 ⇔ k−1ADP + P (69) The reaction products are adenosinediphosphate (ADP) and i norganic phos- phate (P). The equilibrium constant of the reaction depends on the conc entration of the substrate [ATP] and products [ADP] and [P] like: K=k1 k−1=[ADP] ·[P] [ATP](70) The equilibrium constant and temperature determine the rea ction free energy change: ∆G=−RTlnK= ∆H−T∆S (71) where: ∆H and ∆S are changes in enthalpy and entropy, respect ively. Under the real conditions in cell the reaction of ATP hydroly sis is highly favorable energetically as is accompanied by strong free energy decrease: ∆G=−(11÷13)kcal/M. It follows from (71) that ∆G <0, when T∆S >∆H (72) and the entropy and enthalpy changes are positive (∆S >0and∆H > 0).However, the specific molecular mechanism of these changes i n different biochemical reactions, including muscle contrac tion, remains unclear. Acceleration of actin polymerization and tubulin self-ass embly to the microtubules as a result of the ATP and nucleotide GTP spl itting, respectively, is still obscure as well. Using our model of water-macromolecule interaction [6], we can explain these processes by the ”melting” of the water cluste rs - libra- tional effectons in cavities between neighboring domains an d subunits of proteins. This melting is induced by absorption of energy of ATP or GTP hydrolysis and represents [lb/tr]conversion of primary libra- tional effectons to translational ones. It leads to the parti al dehydra- tion and rapprochement of domains and subunits. The concomi tant transition of interdomain/subunit cavities from the ”open ” B-state to the ”closed” A-state should be accompanied by decreasing of lin- ear dimensions of a macromolecule. This process is usually r eversible and responsible for the large-scale dynamics. 32In the case when disjoining clusterphilic interactions tha t shift the [A⇔B]equilibrium to the right are stronger than Van der Waals interactions stabilizing A-state, the expansion of the mac romolecule can induce a mechanical ”pushing” force. In accordance to our model, this ”swelling driving force” is re- sponsible for shifting of myosin ”heads” as respect to the ac tin fila- ments and muscle contraction. Such FIRST relaxation ”swelling working step” is accompanied by dissociation of products of ATP hydrolysis from the active s ites of myosin heads (heavy meromyosin). The SECOND stage of reaction, the dissociation of the complex: [myosin ”head” + actin], is related to the absorption of ATP at the myo sin active site. At this stage the A⇔Bequilibrium between the heavy meromyosin conformers is strongly shifted to the right, i.e. to an expanded form of t he protein. The THIRD stage is represented by the ATP hydrolysis, (ATP → ADP + P) and expelling of (P) from the active site. The concomi tant local enthalpy and entropy jump leads to the melting of the wa ter clusters in the cavities, B →A transitions and the contraction of free meromyosin heads. The energy of the clusterphilic interaction at this stage is accumulated in myosin like in a squeezed spring. After this 3d stage is over the complex [myosin head + actin] f orms again. We assume here that the interaction between myosin head and actin induces the releasing of the product (ADP) from myosin active site. It is important to stress that the driving force of ”swe lling working stage ” : [A→B]transition of myosin cavities - is represented by our clusterphilic interactions (see Section 13.3 of [1] a nd paper [5]). A repetition of such a cycle results in the relative shift of m yosin filaments with respect to actin ones and finally in muscle cont raction. The mechanism proposed does not need the hypothesis of Davyd ov’s soliton propagation (Davydov, 1984) along a myosin macromolecule. It seems that this nondissipative process scarcely takes place in strong ly fluctuating biological systems. Soliton model does not take into account the real me soscopic structure of macromolecules and their interaction with water as well. Polymerization of actin, tubulin and other globular protei ns com- posing cytoplasmic and extracell filaments due to hydrophob ic inter- action can be accelerated as a result of their selected dehyd ration due to local temperature jumps in mesoscopic volumes where t he ATP and GTP hydrolysis takes place. The [assembly ⇔disassembly] equilibrium is shifted as a result of such mesophase transition to the left in the case when [ protein −protein ] interface Van-der-Waals interactions are stronger than a clusterphi lic one. The latter is 33mediated by librational water effecton stabilization in int erdomain or intersub- unit cavities. It looks that the clusterphilic interactions play an extrem ely im- portant role on mesoscale in the self-organization and dyna mics of biological systems. 7. Water activity as a regulative factor in the intra- and int er-cell processes Three most important factors can be responsible for the spat ial processes in living cells: 1. Self-organization of supramolecular systems in the form of mem- branes, oligomeric proteins and filaments. Such processes c an be me- diated by water-dependent hydrophilic, hydrophobic and in troduced by us clusterphilic interactions [1, 5]; 2. Compartmentalization of cell volume by semipermeable li pid- bilayer membranes and due to different cell’s organelles for mation; 3. Changes in the volumes of different cell compartments by os - motic process correlated in space and time. These changes ar e depen- dent on water activity regulated, in turn, mainly by [assemb ly⇋disassembly] equilibrium shift of microtubules and actin’s filaments. Disassembly of filaments leads to water activity decreasing due to increas- ing the fraction of vicinal water, representing a developed system of enlarged primary librational effectons near the surface of proteins (see section 13.5 of [1] and [5]). The thickness of vicinal water l ayer is about 50 ˚A, depending on temperature and mobility of intra-cell compon ents. The vicinal water with more ordered and cooperative structu re than that of bulk water represents the dominant fraction of intra-cell w ater. A lot of experi- mental evidences, pointing to important role of vicinal wat er in cell physiology were presented in reviews of Drost-Hansen and Singleton (19 92) and Clegg and Drost-Hansen (1991). The dynamic equilibrium: {I⇔II⇔III}between three stages of macro- molecular self-organization, discussed in Section 13.5 of [1] (Table 2) and in paper [5], has to play an important role in biosystems: blood , lymph as well as inter- and intra-cell media. This equilibrium is dependent on the water activ- ity (inorganic ions, pH), temperature, concentration and s urface properties of macromolecules. Large-scale protein dynamics, decreasing the fraction of v icinal water (K¨ aiv¨ ar¨ ainen, 1986, K¨ aiv¨ ar¨ ainen et al., 1990) is dependent on the prote in’s active site ligand state. These factors may play a regulative role in [ coagulation ⇔peptization ] and [gel⇔sol] transitions in the cytoplasm of mobile cells, necessary fo r their migration. A lot of spatial cellular processes such as the increase or de crease in the length of microtubules or actin filaments are dependent also on their self- assembly from corresponding subunits ( α, βtubulins and actin). 34The self-assembly of such superpolymers is dependent on the [association (A)⇔dissociation (B)] equilibrium constant ( KA⇔B=K−1 B⇔A). In turn, this constant is dependent on water activity ( aH2O), as was shown earlier ( eq.13.11a and 13.12 of [1] and [5]). The double helix of actin filaments , responsible for the spatial organization and cell’s shape dynamics, is composed of the monomers of glo bular protein - actin (MM 42.000). The rate of actin filament polymerization or depolymer- ization, responsible for cells shape adaptation to environ ment, is very high and strongly depends on ionic strength (concentration of NaCl ,Ca++, Mg++). For example, the increasing of NaCl concentration and corresponding decreasing of aH2Ostimulate the actin polymerization. The same is true of αandβtubulin polymerization in the form of microtubules. The activity of water in cells and cell compartments can be re gulated by [Na+−K+]ATP−dependent pumps. Even the equal concentrations of Na+ andK+decrease water activity aH2Odifferently due to their different interaction with bulk and, especially with ordered vicinal water (Wiggi ns 1971, 1973). Regulation of pH by proton pumps, incorporated in membranes , also can be of great importance for intra-cell aH2Ochanging. Cell division is strongly correlated with dynamic equilibr ium: [assembly ⇔ disassembly] of microtubules of centrioles.. Inhibition o f tubulin subunits disso- ciation (disassembly) by addition of D2O, orstimulation this process by decreas- ing temperature or increasing hydrostatic pressure stops cell mitosis - division (Alberts et al., 1983). The above mentioned factors enable to affect the A⇔Bequilibrium of cav- ity between α and β tubulins, composing microtubules. These factors action confirm our hypothesis, that microtubules assembly are medi ated by cluster- philic interaction (see section 17.5 of [1] and paper [5]). The decrease in temperature and increase in intra-microtub ules pressure lead to the increased dimensions of librational water effectons, clustrons and finally this induce disassembly of microtubules. Microtubules are responsible for the coordination of intra -cell space organi- zation and movements, including chromosome movement at the mitotic cycle, coordinated by centrioles. The communication between different cells by means of channe ls can regulate the ionic concentration and correspondent aH2Ogradients in the embryo. In accordance with our hypothesis, the gradient of water act ivity, regulated by change of vicinal water fraction in different co mpart- ments of cell can play a role of so-called morphogenic factor necessary for differentiation of embryo cells. 8. Water and cancer We put forward a hypothesis that unlimited cancer cell divis ion is related to partial disassembly of cytoskeleton’s actin- like filaments due to some genetically controlled mistakes in biosynthesi s and in- creasing the osmotic diffusion of water into transformed cel l. 35Decreasing of the intra-cell concentration of any types of i ons (Na+, K+, H+, Mg2+etc.), as the result of corresponding ionic pump destructio n, incorporated in biomembranes, also may lead to disassembly of filaments. The shift of equilibrium: [assembly ⇔disassembly] of microtubules (MTs) and actin filaments to the right increases the amount of intra -cell water, involved in hydration shells of protein and decreases water activity . As a consequence of concomitant osmotic process, cells tend to swell and acquir e a ball-like shape. The number of direct contacts between transformed cells dec rease and the water activity in the intercell space increases also. We suppose that certain decline in the external inter-cell w ater activity could be a triggering signal for the inhibition of n ormal cell division. The shape of normal cells under control of cell’s fi lament is a specific one, providing good dense intercell contacts wi th limited amount of water, in contrast to transformed cells. If this idea is true, the absence of contact inhibition in the case of cancer cells is a result of insufficient decreasing of inter cell water activity due to loose [cell-cell] contacts. If our model of cancer emergency is correct, then the problem of tumor inhibition is related to the problem of inter - and intr a-cell water activity regulation by means of chemical and physical factors. Another approach for cancer healing we can propose here is th e IR laser treatment of transformed cells with IR photons freq uencies, stimulating superdeformons excitation and collective dis assembly of MTs in composition of centrioles. This will prevent cells di vision and should have a good therapeutic effect. This approach is ba sed on assumption that stability of MTs in transformed cells is w eaker and/or resonant frequency of their superdeformons excitat ion differs from that of normal cells. =================================================== =========== REFERENCES Alberts B., Bray D., Lewis J., Ruff M., Roberts K. and Watson J.D. Molecular Biology of Cell. Chapter 10. Gar- land Publishing, Inc.New York, London, 1983. Antonchenko V.Ya. Physics of water. Naukova dumka, Kiev, 1986. Cantor C.R., Schimmel P.R. Biophysical Chemistry. W.H.Fre emen and Company, San Francisco, 1980. 36Clegg J. S. On the physical properties and potential roles of intracellular water. Proc.NATO Adv.Res.Work Shop. 1985. Clegg J.S. and Drost-Hansen W. On the biochemistry and cell physiology of water. In: Hochachka and Momm- sen (eds.). Biochemistry and molecular biology of fishes. Elsevier Science Publ. vol.1, Ch.1, pp.1-23, 1991. 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Oxford University Press, Oxford, 1969. Gavish B., Weber M. Viscosity-dependent structural fluc- tuations in enzyme catalysis. Biochemistry 18(1979)1269. Gavish B. in book: The fluctuating enzyme. Ed. by G.R.Welch. Wiley- Interscience Publication, 1986, p.264- 339. Grawford F.S. Waves. Berkley Physics Course. Vol.3. McGraw- Hill Book Co., N.Y., 1973. Grundler W. and Keilmann F. Sharp resonance in Yeast growth proved nonthermal sensitivity to microwaves. Phys. Rev.Letts., 1983,51,1214−1216. K¨ aiv¨ ar¨ ainen A.I. Solvent-dependent flexibility of pro- teins and principles of their function. D.Reidel Publ.Co., Dordrecht, Boston, Lancaster, 1985,pp.290. K¨ aiv¨ ar¨ ainen A.I. The noncontact interaction between macromolecules revealed by modified spin-label method. Biofizika (USSR ) 1987 ,32,536. 37K¨ aiv¨ ar¨ ainen A.I. Thermodynamic analysis of the sys- tem: water-ions-macromolecules. Biofizika (USSR ),1988,33,549. K¨ aiv¨ ar¨ ainen A.I. Theory of condensed state as a hier- archical system of quasiparticles formed by phonons and three-dimensional de Broglie waves of molecules. Applica- tion of theory to thermodynamics of water and ice. J.Mol.Liq . 1989a,41,53−60. K¨ aiv¨ ar¨ ainen A.I. Mesoscopic theory of matter and its interaction with light. Principles of selforganization in ice, water and biosystems. University of Turku, Finland 1992, pp.275. Kaivarainen A. Dynamic model of wave-particle dual- ity and Grand unification. University of Joensuu, Finland 1993. pp.118. Kaivarainen A. Mesoscopic model of elementary act of perception and braining. Abstracts of conference: Toward a Science of Consciousness 1996, p.74. Tucson, USA. K¨ aiv¨ ar¨ ainen A., Fradkova L., Korpela T. Separate con- tributions of large- and small-scale dynamics to the heat ca - pacity of proteins. A new viscosity approach. Acta Chem.Sca nd. 1993,47,456−460. Karplus M., McCammon J.A. Scientific American, April 1986, p.42. Koshland D.E. 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arXiv:physics/0003094v1 [physics.med-ph] 29 Mar 2000Generalized Optimal Current Patterns and Electrical Safety in EIT William R.B. Lionheart∗Jari Kaipio† Christopher N. McLeod‡ February 20, 2014 Abstract There are a number of constraints which limit the current and volt- ages which can be applied on a multiple drive electrical imag ing system. One obvious constraint is to limit the maximum Ohmic power di ssipated in the body. Current patterns optimising distinguishabili ty with respect to this constraint are singular functions of the difference o f transconduc- tance matrices with respect to the power norm. (the optimal c urrents of Isaacson). If one constrains the total current ( L1norm) the optimal pat- terns are pair drives. On the other hand if one constrains the maximum current on each drive electrode (an L∞norm), the optimal patterns have each drive channel set to the maximum source or sink current v alue. In this paper we consider appropriate safety constraints and d iscuss how to find the optimal current patterns with those constraints. 1 Introduction The problem of optimizing the drive patterns in EIT was first c onsidered by Seagar [1] who calculated the optimal placing of a pair of poi nt drive electrodes on a disk to maximize the voltage differences between the meas urement of a homogeneous background and an offset circular anomaly. Giss er, Isaacson and Newell [2] argued that one should maximize the L2norm of the voltage differ- ence between the measured and calculated voltages constrai ning the L2norm of the current patterns in a multiple drive system. Later [3] th ey used a constraint on the maximum dissipated power in the test object. Eyuboglu and Pilking- ton [4] argued that medical safety legislation demanded tha t one restrict the maximum total current entering the body, and if this constra int was used the distinguishability is maximized by pair drives. Yet anothe r approach [5] is to find a current pattern maximizing the voltage difference for a single differential voltage measurement. ∗Department of Mathematics, UMIST, UK †Department of Applied Physics, University of Kuopio, Finla nd ‡School of Engineering, Oxford Brookes University,UK 12 Medical Electrical Safety Regulations We will review the current safety regulations here, but noti ce that they were not designed with multiple drive EIT systems in mind and we ho pe to stimulate a debate about what would be appropriate safety standards. For the purposes of this discussion the equipment current (“ Earth Leakage Current” and “Enclosure Leakage Current”) will be ignored a s the emphasis is on the patient currents. These will be assessed with the assu mption that the equipment has been designed such that the applied parts, tha t is the electronic circuits and connections which are attached to the patient f or the delivery of current and the measurement of voltage, are fully isolated f rom the protective earth (at least 50 MΩ). IEC601 and the equivalent BS5724 specify a safe limit of 100 µA for current flow to protective earth (“Patient Leakage Current”) throug h electrodes at- tached to the skin surface (Type BF) of patients under normal conditions. This is designed to ensure that the equipment will not put the pati ent at risk even when malfunctioning. The standards also specify that the eq uipment should allow a return path to protective earth for less than 5 mA if so me other equip- ment attached to the patient malfunctions and applies full m ains voltage to the patient. Lower limits of 10 µA (normal) and 50 µA (mains applied to the patient) are set for internal connections, particularly to the heart (Type CF), but that is not at present an issue for EIT researchers. The currents used in EIT flow between electrodes and are descr ibed in the standards as “Patient Auxiliary Currents” (PAC). The limit for any PAC is a function of frequency, 100 microamps from 0.1Hz to 1 kHz; the n 100f µA from 1 kHz to 100 kHz where fis the frequency in kHz; then 10 mA above 100 kHz. The testing conditions for PAC cover 4 configurations; the wo rst case of each should be examined. 1. Normal conditions. The design of single or multiple curre nt source tomo- graphs should ensure that each current source is unable to ap ply more than the maximum values given. 2. The PAC should be measured between any single connection a nd all the other connections tied together. a) if the tomograph uses a s ingle current source then the situation is similar to normal conditions (above) b ) if the tomograph uses multiple current sources then as far as the patient is co ncerned the situation is the same as normal conditions. The design of the sources sh ould be such that they will not be harmed by this test. 3. The PAC should be measured when one or more electrodes are d iscon- nected from the patient. This raises issues for multiple-so urce tomographs : a) if an isolated-earth electrode is used then the current in it will be the sum of the currents which should have flowed in the disconnected electr odes; they could all be of the same polarity. The isolated-earth electrode shoul d therefore include an over-current sensing circuit which will turn down/off all the current sources. b) If no isolated-earth electrode is used then the situation is similar to normal conditions. 4. The PAC should be measured when the disconnected electrod es are con- nected to protective earth. This introduces no new constrai nts given the tomo- graph is fully isolated. 23 Constrained Optimization LetV= (V1, . . . , V K)Tbe the vector of potentials measured on electrodes when a pattern of currents I= (I1, . . . , I K)Tis applied. These are related linearly by Rthe transfer impedance matrix: V=RI. For simplicity we will assume the same system of electrodes is used for current injection and v oltage measurement. We will also assume that the conductivity is real and the curr ents in-phase to simplify the exposition. A model of the body is used with our p resent best estimate for the conductivity and from this we calculate vol tagesVcfor the same current pattern. Our aim is to maximize the distinguishabil ity/bardblV−Vc/bardbl2= /bardbl(R−Rc)I/bardbl2. The use of the L2norm here corresponds to the assumption that the noise on each measurement channel is independent an d identically distributed. If there were no constraints on the currents th e distinguishability would be unbounded. The simplest idea [2] is to maximize /bardbl(R−Rc)I/bardbl2subject to /bardblI/bardbl2≤Mfor some fixed value of M. The solution of this problem that Iis the eigenvector of R−Rccorresponding to the largest (in absolute value) eigenvalu e. One problem is this is that the 2-norm of the current has no particular phy sical meaning. In a later paper [3] it was proposed that the dissipated power be constrained, that isI·V=ITRI. The optimal current is the eigenvector of ( R−Rc)R−1/2. (The inverse implied in the expression R−1/2has to be understood in the generalized sense, that is one projects on to the space orthogonal to (1 , . . .,1)Tand then calculates the power 1 /2.) In practical situations in medical EIT the total dissipated power is unlikely to be an active constraint, alt hough local heating effects in areas of high current density may be an issue. Even i n industrial applications of EIT, the limitations of voltages and curren ts handled by normal electronic devices mean that one is unlikely to see total pow er as a constraint. One exception might be in EIT applied to very small objects. As we have seen a reasonable interpretation of the safety reg ulations is to limit the current on each electrode to some safe level Imax. We will refer to this as an L∞constraint. This corresponds to a convex system of linear co nstraints −Imax≤Ik≤Imax. When we maximize the square of the distinguishabilty, which is a positive definite quadratic function of I, with respect to this set of constraints it is easy to see that the maximum must be a vertex of the convex polytope {I: max k{|Ik|}=Imax,/summationtext kIk= 0}. For example, for an even number 2nof electrodes the2nCnvertices are the currents with each Ik=±Imax, and an equal number with each sign. If one wanted to be safe under the multiple fault condition th at all the electrodes driving a current with the same sign became disco nnected, and the safety mechanism on the isolated-earth failed, one would em ploy the L1con- straint/summationtext k|Ik| ≤2Imax. Again this gives a convex feasible set. In this case a polyhedron with vertices Isuch that all but two Ikare zero, and those two areImaxand−Imax. These are the pair drives as considered by Seagar, and by Eyuboglu and Pilkington. Another constraint which may be important in practice is tha t the current sources are only able to deliver a certain maximum voltage Vmaxclose to their power supply voltage. If the EIT system is connected to a body with transfer impedance within its design specification then the constrai nts−Vmax≤Vk≤ Vmaxwill not be active. If they do become active then the addition al linear constraints in Ispace −Vmax≤R−1I≤Vmax(here R−1is to be interpreted as 3the generalized inverse), will still result in a convex feas ible region. When any of the linear constraints are combined with quadrat ic constraints such as maximum power dissipation the feasible set of curren ts is still convex but its surface is no longer a polytope. 4 Numerical Results Although we can easily find the vertices of the feasible regio n there are too many for it to be wise to search exhaustively for a maximum of the di stinguishability. For 32 electrodes for example there are32C16>6×108. Instead we use a discrete steepest ascent search method of the feasible vert ices. That is from a given vertex we calculate the objective function for all ve rtices obtained by changing a pair of signs, and move to which ever vertex has the greatest value of the objective function. For comparison we also calculate d the L2optimal currents, the optimal currents for the power constraint, an d the optimal pair drive ( L1optimal). We used a circular disk for the forward problem, and the EIDOR S Matlab toolbox [6] for mesh generation and forward solution. The me sh and conduc- tivity targets can be seen in Figure 5. Our results are intere sting in that for the cases we have studied so far the L∞optimal currents have only two sign changes. The distinguishabilies given in Table 5 should be r ead with caution, as it is unfair somewhat unfair to compare for example power c onstrained with L∞patterns. They are designed to optimise different criteria. However the contrast between pair drive and L∞is worth noting as the majority of existing EIT systems can only drive pairs of electrodes. 5 Conclusions If using optimal current patterns it one sure be sure to use th e right constraints. We suggest that in many situations the L∞constraint may be the correct one. We have demonstrated that it is simple to compute these optim al patterns, and the instrumentation required to apply these patterns is muc h simpler than the L2or power norm patterns. While still requiring multiple curr ent sources, they need only be able to switch between sinking and sourcing the s ame current. References [1] A.D. Seagar, Probing with low frequency electric curren t, PhD Thesis, University of Canterbury, Christchurch, NZ, 1983. [2] G. Gisser, D. Isaacson and J.C. Newell, Current topics in impedance imag- ing, Clin. Phys. Physiol. Meas., 8 Suppl A, pp39–46, 1987. [3] G. Gisser, D. Isaacson and J.C. Newell, Electric Current computed- tomography and eigenvalues, SIAM J. Appl. Math., 50 pp.1623 -1634, 1990. [4] Eyuboglu B.M and Pilkington T.C. Comment on Distinguish ability in Electrical-Impedance Imaging, IEEE Trans. Biomed. Eng.,4 0, pp.1328- 1330,1993 4Single anomaly Two anomalies Constraint L2-norm of voltage differences L1Best pair drive 353.2179 347.3579 L2546.7841 571.9161 Power 303.6108 311.9452 L∞1199.9447 968.2656 Table 1: Voltage difference for one and two anomalies with a va riety of con- straints. The constraint levels have been chosen so that the maximum electrode current is the same on each 5 10 15−1−0.500.51 5 10 15−1−0.500.51 Figure 1: Optimal current patterns. Continuous line is the L∞norm, − ◦ − is theL2optimal, · · ·power norm optimal and −−isL1optimal (pair drive). [5] Breckon W.R., Measurement and reconstruction in electr ical impedance tomography, in ‘Inverse problems and imaging’, Ed. G.F. Roa ch, Pitman Res. Notes in Math., 245 , pp1-19, 1991. [6] Vauhkonen M. et al, A Matlab Toolbox for the EIDORS project to recon- struct two- and three-dimensional EIT images, these proceedings . 55 10 15−600−400−2000200400600 5 10 15−400−2000200400600 Figure 2: Voltage difference measurements for one and two ano malies. For key see figure 5 Figure 3: Mesh and conductivity anomalies. 6

arXiv:physics/0003095v1 [physics.acc-ph] 29 Mar 2000Variational–Wavelet Approach to RMS Envelope Equations ANTONINA N. FEDOROVA and MICHAEL G. ZEITLIN Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 199178, Russia, St. Petersbur g, V.O., Bolshoj pr., 61, E-mail: zeitlin@math.ipme.ru, http://www.ipme.ru/zeitlin.html http://www.ipme.nw.ru/zeitlin.html February 21, 2000 Abstract We present applications of variational–wavelet approach t o nonlinear (ra- tional) rms envelope equations. We have the solution as a mul tiresolution (multiscales) expansion in the base of compactly supported wavelet basis. We give extension of our results to the cases of periodic beam mo tion and arbi- trary variable coefficients. Also we consider more flexible va riational method which is based on biorthogonal wavelet approach. Paper presented at: Second ICFA Advanced Accelerator Workshop THE PHYSYCS OF HIGH BRIGHTNESS BEAMS UCLA Faculty Center, Los Angeles November 9-12, 19991 Introduction In this paper we consider the applications of a new numerical -analytical tech- nique which is based on the methods of local nonlinear Fourie r analysis or Wavelet analysis to the nonlinear beam/accelerator physic s problems related to root-mean-square (rms) envelope dynamics [1]. Such appr oach may be use- ful in all models in which it is possible and reasonable to red uce all complicated problems related with statistical distributions to the pro blems described by systems of nonlinear ordinary/partial differential equati ons. In this paper we consider approach based on the second moments of the distrib ution functions for the calculation of evolution of rms envelope of a beam. The rms envelope equations are the most useful for analysis o f the beam self–forces (space–charge) effects and also allow to consid er both transverse and longitudinal dynamics of space-charge-dominated rela tivistic high–bright ness axisymmetric/asymmetric beams, which under short las er pulse–driven radio-frequency photoinjectors have fast transition from nonrelativistic to rel- ativistic regime [2]-[3]. From the formal point of view we may consider rms envelope equ ations after straightforward transformations to standard Cauchy form as a system of nonlinear differential equations which are not more than rat ional (in dynam- ical variables). Such rational type of nonlinearities allo w us to consider some extension of results from [4]-[12], which are based on appli cation of wavelet analysis technique to variational formulation of initial n onlinear problem. Wavelet analysis is a relatively novel set of mathematical m ethods, which gives us a possibility to work with well-localized bases in f unctional spaces and give for the general type of operators (differential, integr al, pseudodifferential) in such bases the maximum sparse forms. An example of such type of basis is demonstrated on Fig. 1. Our approach in this paper is based on the generalization [13 ] of variatio- nal-wavelet approach from [4]-[12], which allows us to cons ider not only poly- nomial but rational type of nonlinearities. So, our variational-multiresolution approach gives us pos sibility to con- struct explicit numerical-analytical solution for the fol lowing systems of non- linear differential equations ˙z=R(z,t) orQ(z,t) ˙z=P(z,t), (1) wherez(t) = (z1(t),...,z n(t)) is the vector of dynamical variables zi(t), R(z,t) is not more than rational function of z, P(z,t),Q(z,t) are not more than polynomial functions of z and P,Q,R have arbitrary dependence of time. 2The solution has the following form z(t) =zslow N(t) +/summationdisplay j≥Nzj(ωjt), ω j∼2j(2) which corresponds to the full multiresolution expansion in all time scales. Formula (2) gives us expansion into a slow part zslow Nand fast oscillating parts for arbitrary N. So, we may move from coarse scales of re solution to the finest one for obtaining more detailed information about our dynamical process. The first term in the RHS of equation (2) corresponds on the global level of function space decomposition to resolution space a nd the second one to detail space. In this way we give contribution to our full s olution from each scale of resolution or each time scale (detailed descriptio n we give in part 3.2 and numerical illustration in part 7 below). The same is c orrect for the contribution to power spectral density (energy spectrum): we can take into account contributions from each level/scale of resolution . In part 2 we describe the different forms of rms equations. Sta rting in part 3.1 from variational formulation of initial dynamical prob lem we construct via multiresolution analysis (3.2) explicit representation f or all dynamical vari- ables in the base of compactly supported (Daubechies) wavel ets. Our solu- tions (3.3) are parametrized by solutions of a number of redu ced algebraical problems one from which is nonlinear with the same degree of n onlinearity and the rest are the linear problems which correspond to part icular method of calculation of scalar products of functions from wavelet bases and their derivatives. Then we consider further extension of our prev ious results. In part 4 we consider modification of our construction to the per iodic case, in part 5 we consider generalization of our approach to variati onal formulation in the biorthogonal bases of compactly supported wavelets a nd in part 6 to the case of variable coefficients. In part 7 we consider result s of numerical calculations. 2 RMS Equations Below we consider a number of different forms of RMS envelope e quations, which are from the formal point of view not more than nonlinea r differential equations with rational nonlinearities and variable coeffic ients. Letf(x1,x2) be the distribution function which gives full information a bout noninteract- ing ensemble of beam particles regarding to trace space or tr ansverse phase coordinates ( x1,x2). Then (n,m) moments are: /integraldisplay /integraldisplay xn 1xm 2f(x1,x2)dx1dx2 (3) 3Figure 1: Wavelets at different scales and locations. The (0,0) moment gives normalization condition on the distr ibution. The (1,0) and (0,1) moments vanish when a beam is aligned to its ax is. Then we may extract the first nontrivial bit of ‘dynamical informati on’ from the second moments σ2 x1=<x2 1>=/integraldisplay /integraldisplay x2 1f(x1,x2)dx1dx2 σ2 x2=<x2 2>=/integraldisplay /integraldisplay x2 2f(x1,x2)dx1dx2 (4) σ2 x1x2=<x1x2>=/integraldisplay /integraldisplay x1x2f(x1,x2)dx1dx2 RMS emittance ellipse is given by ε2 x,rms=<x2 1><x2 2>−<x1x2>2(5) Expressions for twiss parameters are also based on the secon d moments. We will consider the following particular cases of rms envel ope equations, which described evolution of the moments (4) ([1]-[3] for fu ll designation): for asymmetric beams we have the system of two envelope equat ions of the second order for σx1andσx2: σ′′ x1+σ′ x1γ′ γ+ Ω2 x1/parenleftbiggγ′ γ/parenrightbigg2 σx1=I I0(σx1+σx2)γ3+ε2 nx1 σ3x1γ2, σ′′ x2+σ′ x2γ′ γ+ Ω2 x2/parenleftbiggγ′ γ/parenrightbigg2 σx2=I I0(σx1+σx2)γ3+ε2 nx2 σ3x2γ2(6) 4the envelope equation for an axisymmetric beam is σ′′+σ′γ′ γ+ Ω2/parenleftbiggγ′ γ/parenrightbigg2 σ=ks σγ3+ε2 n,th σ3γ2(7) Also we have related Lawson’s equation for evolution of the r ms envelope in the paraxial limit, which governs evolution of cylindrical symmetric envelope under external linear focusing channel of strenghts Kr: σ′′+σ′/parenleftbiggγ′ β2γ/parenrightbigg +Krσ=ks σβ3γ3+ε2 n σ3β2γ2, (8) where Kr≡ −Fr/rβ2γmc2, β≡νb/c=/radicalBig 1−γ−2 (9) After transformations to Cauchy form we can see that all this equations from the formal point of view are not more than ordinary differ ential equa- tions with rational nonlinearities and variable coefficient s and correspond to the form (1) (also,we may consider regimes in which γ,γ′are not fixed func- tions/constants but satisfy some additional differential c onstraint/equation, but this case does not change our general approach). 3 Rational Dynamics The first main part of our consideration is some variational a pproach to this problem, which reduces initial problem to the problem of sol ution of functional equations at the first stage and some algebraical problems at the second stage. We have the solution in a compactly supported wavelet basis. Multiresolu- tion expansion is the second main part of our construction. T he solution is parameterized by solutions of two reduced algebraical prob lems, one is nonlin- ear and the second are some linear problems, which are obtain ed from one of the next wavelet constructions: the method of Connection Co efficients (CC), Stationary Subdivision Schemes (SSS). 3.1 Variational Method Our problems may be formulated as the systems of ordinary diff erential equa- tions Qi(x)dxi dt=Pi(x,t), x= (x1,...,x n), (10) i= 1,...,n, max ideg P i=p,max ideg Q i=q 5with fixed initial conditions xi(0), wherePi,Qiare not more than polynomial functions of dynamical variables xjand have arbitrary dependence of time. Because of time dilation we can consider only next time inter val: 0 ≤t≤1. Let us consider a set of functions Φi(t) =xid dt(Qiyi) +Piyi (11) and a set of functionals Fi(x) =/integraldisplay1 0Φi(t)dt−Qixiyi|1 0, (12) whereyi(t) (yi(0) = 0) are dual (variational) variables. It is obvious that the initial system and the system Fi(x) = 0 (13) are equivalent. Of course, we consider such Qi(x) which do not lead to the singular problem with Qi(x), whent= 0 ort= 1, i.e.Qi(x(0)),Qi(x(1))/ne}ationslash=∞. In part 5 we consider more general approach, which is based on possibility taking into account underlying symplectic structure and on more useful and flexible analytical approach, related to bilinear structur e of initial functional. Now we consider formal expansions for xi,yi: xi(t) =xi(0) +/summationdisplay kλk iϕk(t)yj(t) =/summationdisplay rηr jϕr(t), (14) whereϕk(t) are useful basis functions of some functional space ( L2,Lp, Sobolev, etc) corresponding to concrete problem and because of initi al conditions we need onlyϕk(0) = 0. λ={λi}={λr i}= (λ1 i,λ2 i,...,λN i), r= 1,...,N, i = 1,...,n, (15) where the lower index i corresponds to expansion of dynamica l variable with index i, i.e. xiand the upper index rcorresponds to the numbers of terms in the expansion of dynamical variables in the formal series . Then we put (14) into the functional equations (13) and as result we have the following reduced algebraical system of equations on the set of unknow n coefficients λk i of expansions (14): L(Qij,λ,α I) =M(Pij,λ,β J), (16) where operators L and M are algebraization of RHS and LHS of in itial problem (10), where λ(15) are unknowns of reduced system of algebraical equation s (RSAE)(16). 6Qijare coefficients (with possible time dependence) of LHS of ini tial sys- tem of differential equations (10) and as consequence are coe fficients of RSAE. Pijare coefficients (with possible time dependence) of RHS of ini tial sys- tem of differential equations (10) and as consequence are coe fficients of RSAE. I= (i1,...,i q+2), J= (j1,...,j p+1) are multiindexes, by which are labelled αIandβI— other coefficients of RSAE (16): βJ={βj1...jp+1}=/integraldisplay/productdisplay 1≤jk≤p+1ϕjk, (17) where p is the degree of polinomial operator P (10) αI={αi1...αiq+2}=/summationdisplay i1,...,iq+2/integraldisplay ϕi1...˙ϕis...ϕiq+2, (18) where q is the degree of polynomial operator Q (10), iℓ= (1,...,q+ 2), ˙ϕis= dϕis/dt. Now, when we solve RSAE (16) and determine unknown coefficient s from formal expansion (14) we therefore obtain the solution of ou r initial problem. It should be noted if we consider only truncated expansion (1 4) with N terms then we have from (16) the system of N×nalgebraical equations with degree ℓ=max{p,q}and the degree of this algebraical system coincides with deg ree of initial differential system. So, we have the solution of th e initial nonlinear (rational) problem in the form xi(t) =xi(0) +N/summationdisplay k=1λk iXk(t), (19) where coefficients λk iare roots of the corresponding reduced algebraical (poly- nomial) problem RSAE (16). Consequently, we have a parametr ization of so- lution of initial problem by solution of reduced algebraica l problem (16). The first main problem is a problem of computations of coefficients αI(18),βJ (17) of reduced algebraical system. As we will see, these pro blems may be explicitly solved in wavelet approach. Next we consider the construction of explicit time solution for our problem. The obtained solutions are given in the form (19), where Xk(t) are basis func- tions andλi kare roots of reduced system of equations. In our first wavelet caseXk(t) are obtained via multiresolution expansions and represen ted by compactly supported wavelets and λi kare the roots of corresponding general polynomial system (16) with coefficients, which are given by C C or SSS con- structions. According to the variational method to give the reduction from differential to algebraical system of equations we need comp ute the objects αIandβJ. 73.2 Wavelet Framework Our constructions are based on multiresolution approach. B ecause affine group of translation and dilations is inside the approach, t his method re- sembles the action of a microscope. We have contribution to fi nal result from each scale of resolution from the whole infinite scale of spac es. More exactly, the closed subspace Vj(j∈Z) corresponds to level j of resolution, or to scale j. We consider a r-regular multiresolution analysis of L2(Rn) (of course, we may consider any different functional space) which is a seque nce of increasing closed subspaces Vj: ...V−2⊂V−1⊂V0⊂V1⊂V2⊂... (20) satisfying the following properties: /intersectiondisplay j∈ZVj= 0,/uniondisplay j∈ZVj=L2(Rn), f(x)∈Vj<=>f(2x)∈Vj+1, f(x)∈V0<=>f(x−k)∈V0, ,∀k∈Zn. (21) There exists a function ϕ∈V0such that {ϕ0,k(x) =ϕ(x−k),k∈Zn}forms a Riesz basis for V0. The function ϕis regular and localized: ϕisCr−1, ϕ(r−1)is almost everywhere differentiable and for almost every x∈Rn, for every integer α≤r and for all integer p there exists constant Cpsuch that |∂αϕ(x)|≤Cp(1 +|x|)−p(22) Letϕ(x) be a scaling function, ψ(x) is a wavelet function and ϕi(x) = ϕ(x−i). Scaling relations that define ϕ,ψare ϕ(x) =N−1/summationdisplay k=0akϕ(2x−k) =N−1/summationdisplay k=0akϕk(2x), (23) ψ(x) =N−2/summationdisplay k=−1(−1)kak+1ϕ(2x+k). (24) Let indices ℓ,jrepresent translation and scaling, respectively and ϕjl(x) = 2j/2ϕ(2jx−ℓ) (25) then the set {ϕj,k},k∈Znforms a Riesz basis for Vj. The wavelet function ψ is used to encode the details between two successive levels o f approximation. LetWjbe the orthonormal complement of Vjwith respect to Vj+1: Vj+1=Vj/circleplusdisplay Wj. (26) 8Then just as Vjis spanned by dilation and translations of the scaling funct ion, so areWjspanned by translations and dilation of the mother wavelet ψjk(x), where ψjk(x) = 2j/2ψ(2jx−k). (27) All expansions which we used are based on the following prope rties: {ψjk}, j,k ∈Zis a Hilbertian basis of L2(R) {ϕjk}j≥0,k∈Zis an orthonormal basis for L2(R), L2(R) =V0∞/circleplusdisplay j=0Wj, (28) or {ϕ0,k,ψj,k}j≥0,k∈Zis an orthonormal basis for L2(R). 3.3 Wavelet Computations Now we give construction for computations of objects (17),( 18) in the wavelet case. We use compactly supported wavelet basis: orthonorma l basis for func- tions inL2(R). Let bef:R−→Cand the wavelet expansion is f(x) =/summationdisplay ℓ∈Zcℓϕℓ(x) +∞/summationdisplay j=0/summationdisplay k∈Zcjkψjk(x) (29) If in formulae (29) cjk= 0 forj≥J, thenf(x) has an alternative ex- pansion in terms of dilated scaling functions only f(x) =/summationtext ℓ∈ZcJℓϕJℓ(x). This is a finite wavelet expansion, it can be written solely in term s of translated scaling functions. Also we have the shortest possible suppo rt: scaling function DN(whereNis even integer) will have support [0 ,N−1] andN/2 vanishing moments. There exists λ >0 such that DNhasλNcontinuous derivatives; for smallN,λ≥0.55. To solve our second associated linear problem we need to evaluate derivatives of f(x) in terms of ϕ(x). Let beϕn ℓ= dnϕℓ(x)/dxn. We consider computation of the wavelet - Galerkin integrals . Letfd(x) be d-derivative of function f(x), then we have fd(x) =/summationtext ℓclϕd ℓ(x), and values ϕd ℓ(x) can be expanded in terms of ϕ(x) φd ℓ(x) =/summationdisplay mλmϕm(x), (30) λm=∞/integraldisplay −∞ϕd ℓ(x)ϕm(x)dx, 9whereλmare wavelet-Galerkin integrals. The coefficients λmare 2-term con- nection coefficients. In general we need to find ( di≥0) Λd1d2...dn ℓ1ℓ2...ℓn=∞/integraldisplay −∞/productdisplay ϕdi ℓi(x)dx (31) For Riccati case we need to evaluate two and three connection coefficients Λd1d2 ℓ=/integraldisplay∞ −∞ϕd1(x)ϕd2 ℓ(x)dx, Λd1d2d3=∞/integraldisplay −∞ϕd1(x)ϕd2 ℓ(x)ϕd3m(x)dx(32) According to CC method [14] we use the next construction. Whe nNin scaling equation is a finite even positive integer the function ϕ(x) has compact support contained in [0 ,N−1]. For a fixed triple ( d1,d2,d3) only some Λd1d2d3 ℓmare nonzero: 2 −N≤ℓ≤N−2,2−N≤m≤N−2,|ℓ−m| ≤N−2. There areM= 3N2−9N+ 7 such pairs ( ℓ,m). Let Λd1d2d3be an M-vector, whose components are numbers Λd1d2d3 ℓm. Then we have the first reduced algebraical system : Λ satisfy the system of equations ( d=d1+d2+d3) AΛd1d2d3= 21−dΛd1d2d3, A ℓ,m;q,r=/summationdisplay papaq−2ℓ+par−2m+p (33) By moment equations we have created a system of M+d+ 1 equations in M unknowns. It has rank Mand we can obtain unique solution by combination of LU decomposition and QR algorithm. The second reduced alg ebraical system gives us the 2-term connection coefficients: AΛd1d2= 21−dΛd1d2, d=d1+d2, A ℓ,q=/summationdisplay papaq−2ℓ+p (34) For nonquadratic case we have analogously additional linea r problems for ob- jects (31). Solving these linear problems we obtain the coeffi cients of nonlinear algebraical system (16) and after that we obtain the coefficie nts of wavelet ex- pansion (19). As a result we obtained the explicit time solut ion of our problem in the base of compactly supported wavelets. We use for model ling D6, D8, D10 functions and programs RADAU and DOPRI for testing. In the following we consider extension of this approach to th e case of peri- odic boundary conditions, the case of presence of arbitrary variable coefficients and more flexible biorthogonal wavelet approach. 104 Variational Wavelet Approach for Peri- odic Trajectories We start with extension of our approach to the case of periodi c trajectories. The equations of motion corresponding to our problems may be formulated as a particular case of the general system of ordinary differe ntial equations dxi/dt=fi(xj,t), (i,j= 1,...,n), 0≤t≤1, wherefiare not more than rational functions of dynamical variables xjand have arbitrary dependence of time but with periodic boundary conditions. According to our variational approach we have the solution in the following form xi(t) =xi(0) +/summationdisplay kλk iϕk(t), x i(0) =xi(1), (35) whereλk iare again the roots of reduced algebraical systems of equati ons with the same degree of nonlinearity and ϕk(t) corresponds to useful type of wavelet bases (frames). It should be noted that coefficients of reduce d algebraical system are the solutions of additional linear problem and al so depend on particular type of wavelet construction and type of bases. This linear problem is our second reduced algebraical probl em. We need to find in general situation objects Λd1d2...dn ℓ1ℓ2...ℓn=∞/integraldisplay −∞/productdisplay ϕdi ℓi(x)dx, (36) but now in the case of periodic boundary conditions. Now we co nsider the procedure of their calculations in case of periodic boundar y conditions in the base of periodic wavelet functions on the interval [0,1] and corresponding expansion (35) inside our variational approach. Periodiza tion procedure gives us ˆϕj,k(x)≡/summationdisplay ℓ∈Zϕj,k(x−ℓ) (37) ˆψj,k(x) =/summationdisplay ℓ∈Zψj,k(x−ℓ) So, ˆϕ,ˆψare periodic functions on the interval [0,1]. Because ϕj,k=ϕj,k′if k=k′mod(2j), we may consider only 0 ≤k≤2jand as consequence our multiresolution has the form/uniondisplay j≥0ˆVj=L2[0,1] with ˆVj= span {ˆϕj,k}2j−1 k=0[15]. 11Integration by parts and periodicity gives useful relation s between objects (36) in particular quadratic case ( d=d1+d2): Λd1,d2 k1,k2= (−1)d1Λ0,d2+d1 k1,k2,Λ0,d k1,k2= Λ0,d 0,k2−k1≡Λd k2−k1(38) So, any 2-tuple can be represent by Λd k. Then our second additional linear problem is reduced to the eigenvalue problem for {Λd k}0≤k≤2jby creating a system of 2jhomogeneous relations in Λd kand inhomogeneous equations. So, if we have dilation equation in the form ϕ(x) =√ 2/summationtext k∈Zhkϕ(2x−k), then we have the following homogeneous relations Λd k= 2dN−1/summationdisplay m=0N−1/summationdisplay ℓ=0hmhℓΛd ℓ+2k−m, (39) or in such form Aλd= 2dλd, whereλd={Λd k}0≤k≤2j. Inhomogeneous equa- tions are:/summationdisplay ℓMd ℓΛd ℓ=d!2−j/2, (40) where objects Md ℓ(|ℓ| ≤N−2) can be computed by recursive procedure Md ℓ= 2−j(2d+1)/2˜Md ℓ,˜Mk ℓ=<xk,ϕ0,ℓ>=k/summationdisplay j=0/parenleftBigg k j/parenrightBigg nk−jMj 0,˜Mℓ 0= 1. (41) So, we reduced our last problem to standard linear algebraic al problem. Then we use the same methods as in part 3.3. As a result we obtained f or closed trajectories of orbital dynamics the explicit time solutio n (35) in the base of periodized wavelets (37). 5 Variational Approach in Biorthogonal Wavelet Bases Now we consider further generalization of our variational w avelet approach. Because integrand of variational functionals is represent ed by bilinear form (scalar product) it seems more reasonable to consider wavel et constructions [16] which take into account all advantages of this structur e. The action functional for loops in the phase space is F(γ) =/integraldisplay γpdq−/integraldisplay1 0H(t,γ(t))dt (42) 12The critical points of Fare those loops γ, which solve the Hamiltonian equa- tions associated with the Hamiltonian Hand hence are periodic orbits. Let us consider the loop space Ω = C∞(S1,R2n), whereS1=R/Z, of smooth loops in R2n. Let us define a function Φ : Ω →Rby setting Φ(x) =/integraldisplay1 01 2<−J˙x,x>dt −/integraldisplay1 0H(x(t))dt, x ∈Ω (43) The critical points of Φ are the periodic solutions of ˙ x=XH(x). Computing the derivative at x∈Ω in the direction of y∈Ω, we find Φ′(x)(y) =d dǫΦ(x+ǫy)|ǫ=0=/integraldisplay1 0<−J˙x− ▽H(x),y>dt (44) Consequently, Φ′(x)(y) = 0 for all y∈Ω iff the loop xsatisfies the equation −J˙x(t)− ▽H(x(t)) = 0, (45) i.e.x(t) is a solution of the Hamiltonian equations, which also sati sfiesx(0) = x(1), i.e. periodic of period 1. But now we need to take into account underlying bilinear stru cture via wavelets. We started with two hierarchical sequences of approximatio ns spaces [16]: ...V −2⊂V−1⊂V0⊂V1⊂V2..., .../tildewideV−2⊂/tildewideV−1⊂/tildewideV0⊂/tildewideV1⊂/tildewideV2..., and as usually, W0is complement to V0inV1, but now not necessarily or- thogonal complement. New orthogonality conditions have no w the following form: /tildewiderW0⊥V0, W 0⊥/tildewideV0, V j⊥/tildewiderWj,/tildewideVj⊥Wj (46) translates of ψspanW0, translates of ˜ψspan/tildewiderW0. Biorthogonality condi- tions are <ψjk,˜ψj′k′>=/integraldisplay∞ −∞ψjk(x)˜ψj′k′(x)dx=δkk′δjj′, (47) whereψjk(x) = 2j/2ψ(2jx−k). Functions ϕ(x),˜ϕ(x−k) form dual pair: <ϕ(x−k),˜ϕ(x−ℓ)>=δkl, <ϕ (x−k),˜ψ(x−ℓ)>= 0 for ∀k,∀ℓ.(48) Functionsϕ,˜ϕgenerate a multiresolution analysis. ϕ(x−k),ψ(x−k) are synthesis functions, ˜ ϕ(x−ℓ),˜ψ(x−ℓ) are analysis functions. Synthesis func- tions are biorthogonal to analysis functions. Scaling spac es are orthogonal to dual wavelet spaces. Two multiresolutions are intertwin ingVj+Wj= Vj+1,/tildewideVj+/tildewiderWj=/tildewideVj+1. These are direct sums but not orthogonal sums. 13So, our representation for solution has now the form f(t) =/summationdisplay j,k˜bjkψjk(t), (49) where synthesis wavelets are used to synthesize the functio n. But ˜bjkcome from inner products with analysis wavelets. Biorthogonali ty yields ˜bℓm=/integraldisplay f(t)˜ψℓm(t)dt. (50) So, now we can introduce this more complicated construction into our vari- ational approach. We have modification only on the level of co mputing co- efficients of reduced nonlinear algebraical system. This new construction is more flexible. Biorthogonal point of view is more stable unde r the action of large class of operators while orthogonal (one scale for m ultiresolution) is fragile, all computations are much more simpler and we accel erate the rate of convergence. In all types of (Hamiltonian) calculation, which are based on some bilinear structures (symplectic or Poissonian struct ures, bilinear form of integrand in variational integral) this framework leads to greater success. 6 Variable Coefficients In the case when we have situation when our problem is describ ed by a system of nonlinear (rational) differential equations, we need to c onsider extension of our previous approach which can take into account any type of variable coefficients (periodic, regular or singular). We can produce such approach if we add in our construction additional refinement equation , which encoded all information about variable coefficients [17]. According to our variational approach we need to compute integrals of the form /integraldisplay Dbij(t)(ϕ1)d1(2mt−k1)(ϕ2)d2(2mt−k2)dx, (51) where now bij(t) are arbitrary functions of time, where trial functions ϕ1,ϕ2 satisfy a refinement equations: ϕi(t) =/summationdisplay k∈Zaikϕi(2t−k) (52) If we consider all computations in the class of compactly sup ported wavelets then only a finite number of coefficients do not vanish. To appro ximate the 14non-constant coefficients, we need choose a different refinabl e function ϕ3 along with some local approximation scheme (Bℓf)(x) :=/summationdisplay α∈ZFℓ,k(f)ϕ3(2ℓt−k), (53) whereFℓ,kare suitable functionals supported in a small neighborhood of 2−ℓk and then replace bijin (51) by Bℓbij(t). In particular case one can take a characteristic function and can thus approximate non-smo oth coefficients locally. To guarantee sufficient accuracy of the resulting ap proximation to (51) it is important to have the flexibility of choosing ϕ3different from ϕ1,ϕ2. In the case when D is some domain, we can write bij(t)|D=/summationdisplay 0≤k≤2ℓbij(t)χD(2ℓt−k), (54) whereχDis characteristic function of D. So, if we take ϕ4=χD, which is again a refinable function, then the problem of computation o f (51) is reduced to the problem of calculation of integral H(k1,k2,k3,k4) =H(k) = (55)/integraldisplay Rsϕ4(2jt−k1)ϕ3(2ℓt−k2)ϕd1 1(2rt−k3)ϕd2 2(2st−k4)dx The key point is that these integrals also satisfy some sort o f refinement equation [17]: 2−|µ|H(k) =/summationdisplay ℓ∈Zb2k−ℓH(ℓ), µ =d1+d2. (56) This equation can be interpreted as the problem of computing an eigen- vector. Thus, we reduced the problem of extension of our meth od to the case of variable coefficients to the same standard algebraical pro blem as in the preceding sections. So, the general scheme is the same one an d we have only one more additional linear algebraic problem by which we in t he same way can parameterize the solutions of corresponding problem. 7 Numerical Calculations In this part we consider numerical illustrations of previou s analytical ap- proach. Our numerical calculations are based on compactly s upported Daubechies wavelets and related wavelet families. 15On Fig. 2 we present according to formulae (2) contributions to approxi- mation of our dynamical evolution (top row on the Fig. 3) star ting from the coarse approximation, corresponding to scale 20(bottom row) to the finest one corresponding to the scales from 21to 25or from slow to fast components (5 frequencies) as details for approximation. Then on Fig. 3 , from bottom to top, we demonstrate the summation of contributions from c orresponding levels of resolution given on Fig. 2 and as result we restore v ia 5 scales (fre- quencies) approximation our dynamical process(top row on F ig. 3 ). In this particular model case we considered for approximation simp le two frequencies harmonic process. But the same situation we have on the Fig. 5 and Fig. 6 in case when we added to previous 2-frequencies harmonic pro cess the noise as perturbation. Again, our dynamical process under invest igation (top row of Fig. 6) is recovered via 5 scales contributions (Fig. 5) to approximations (Fig. 6). The same decomposition/approximation we produce also on the level of power spectral density in the process without noise (Fig. 4) and with noise (Fig. 7). On Fig. 8 we demonstrate the family of localized con tributions to beam motion, which we also may consider for such type of appro ximation. It should be noted that complexity of such algorithms are min imal regard- ing other possible. Of course, we may use different multireso lution analysis schemes, which are based on different families of generating wavelets and apply such schemes of numerical–analytical calculations to any d ynamical process which may be described by systems of ordinary/partial differ ential equations with rational nonlinearities [13]. 160 50 100 150 200 250−0.500.5 0 50 100 150 200 250−101 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−0.200.2 Figure 2: Contributions to approximation: from scale 21to 25(without noise). 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−0.200.2 Figure 3: Approximations: from scale 21to 25(without noise). 1705101520253035404550012x 104 05101520253035404550012x 104 051015202530354045500500010000 051015202530354045500500010000 051015202530354045500510 051015202530354045500510 Figure 4: Power spectral density: from scale 21to 25(without noise) 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−0.500.5 Figure 5: Contributions to approximation: from scale 21to 25(with noise). 180 50 100 150 200 250−505 0 50 100 150 200 250−505 0 50 100 150 200 250−505 0 50 100 150 200 250−202 0 50 100 150 200 250−101 0 50 100 150 200 250−0.500.5 Figure 6: Approximations: from scale 21to 25(with noise). 051015202530354045500500010000 051015202530354045500500010000 051015202530354045500500010000 051015202530354045500500010000 051015202530354045500200400 051015202530354045500200400 Figure 7: Power spectral density: from scale 21to 25(with noise) 19Figure 8: Localized contributions to beam motion. Acknowledgments We would like to thank Professor James B. Rosenzweig and Mrs. Melinda Laraneta for nice hospitality, help, support and discussio ns before and during Workshop and all participants for interesting discussions . References [1] J.B. Rosenzweig, Fundamentals of Beam Physics, e-versi on: http://www.physics.ucla.edu/class/99F/250 Rosenzweig/notes/ [2] L. Serafini and J.B. Rosenzweig, Phys. Rev. E 55, 7565, 1997. [3] J.B. Rosenzweig, S.Anderson and L. Serafini, ‘Space Char ge Dominated Envelope Dynamics of Asymmetric Beams in RF Photoinjectors ’, Proc. PAC97 (IEEE,1998). [4] A.N. Fedorova and M.G. Zeitlin, ’Wavelets in Optimizati on and Approx- imations’, Math. and Comp. in Simulation ,46, 527, 1998. [5] A.N. Fedorova and M.G. Zeitlin, ’Wavelet Approach to Pol ynomial Me- chanical Problems’, New Applications of Nonlinear and Chaotic Dynam- ics in Mechanics , 101 (Kluwer, 1998). [6] A.N. Fedorova and M.G. Zeitlin, ’Wavelet Approach to Mec hanical Prob- lems. Symplectic Group, Symplectic Topology and Symplecti c Scales’, New Applications of Nonlinear and Chaotic Dynamics in Mecha nics, 31 (Kluwer, 1998). 20[7] A.N. Fedorova and M.G. Zeitlin, ’Nonlinear Dynamics of A ccelerator via Wavelet Approach’, CP405 , 87 (American Institute of Physics, 1997). Los Alamos preprint, physics/9710035. [8] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Wavelet Appro ach to Accel- erator Problems’, parts 1-3, Proc. PAC97 2, 1502, 1505, 1508 (IEEE, 1998). [9] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Nonlinear Effe cts in Accel- erator Physics: from Scale to Scale via Wavelets’, ’Wavelet Approach to Hamiltonian, Chaotic and Quantum Calculations in Accelera tor Physics’, Proc. EPAC98, 930, 933 (Institute of Physics, 1998). [10] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Variational A pproach in Wavelet Framework to Polynomial Approximations of Nonline ar Accel- erator Problems. CP468 , 48 ( American Institute of Physics, 1999). Los Alamos preprint, physics/990262 [11] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Symmetry, Ham iltonian Prob- lems and Wavelets in Accelerator Physics. CP468 , 69 (American Insti- tute of Physics, 1999). Los Alamos preprint, physics/990263 [12] A.N. Fedorova and M.G. Zeitlin, Nonlinear Accelerator Problems via Wavelets, parts 1-8, Proc. PAC99, 1614, 1617, 1620, 2900, 29 03, 2906, 2909, 2912 (IEEE/APS, New York, 1999). Los Alamos preprints: physics/9904039, physics/9904040, physics/- 9904041, physics/9904042, physics/9904043, physics/990 4045, physics- /9904046, physics/9904047. [13] A.N. Fedorova and M.G. Zeitlin, in press. [14] A. Latto, H.L. Resnikoff and E. Tenenbaum, Aware Technic al Report AD910708, 1991. [15] G. Schlossnagle, J.M. Restrepo and G.K. Leaf, Technica l Report ANL- 93/34. [16] A. Cohen, I. Daubechies and J.C. Feauveau, Comm. Pure. Appl. Math. , XLV, 485 (1992). [17] W. Dahmen, C. Micchelli, SIAM J. Numer. Anal. ,30, 507 (1993). 21

arXiv:physics/0003096v1 [physics.class-ph] 29 Mar 2000Classical Radiation Processes in the Weizs¨ acker-William s Approximation M.S. Zolotorev Center for Beam Physics, Lawrence Berkeley National Labora tory, Berkeley, CA 94720 K.T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (August 25, 1999) The main features of radiation by relativistic electrons are well approximated in the Weizs¨ acker-Williams method o f virtual quanta. This method is most well-known for its appli - cation to radiation during elementary particle collisions , but is equally useful in describing “classical” radiation emit ted during the interaction of a single relativistic electron wi th an extended system, such as synchrotron radiation, undulator radiation, transition radiation and ˇCerenkov radiation. I. THE WEIZS ¨ACKER-WILLIAMS APPROXIMATION Following an earlier argument of Fermi [1], Weizs¨ acker [2] and Williams [3] noted that the electromagnetic fields of an electron in uniform relativistic motion are predomi- nantly transverse, with E≈B(in Gaussian units). This is very much like the fields of a plane wave, so one is led to regard a fast electron as carrying with it a cloud of virtual photons that it can shed (radiate) if perturbed. The key feature of the frequency spectrum of the fields can be estimated as follows. To an observer at rest at distance bfrom the electron’s trajectory, the peak electric field is E=γe/b2, and the field remains above half this strength for time b/γc, so the frequency spectrum of this pulse extends up to ωmax≈γc/b. The total energy of the pulse (relevant to this observer) is U≈E2Vol≈ γ2e2/b4·b2·b/γ≈γe2/b. If the electron radiates all of this energy, the energy spectrum would be dU(ω) dω≈U ωmax≈e2 c. (1) This result does not depend on the choice of impact pa- rameter b, and is indeed of general validity (to within a factor of ln γ). The number of photons nωof frequency ωis thus dnω=dU(ω) ¯hω≈e2 ¯hcdω ω=αdω ω, (2) where α=e2/¯hc≈1/137 is the fine structure constant. The quick approximation (1)-(2) is not accurate at high frequencies. In general, additional physical arguments are needed to identify the maximum frequency of its va- lidity, called the characteristic or critical frequency ωC,or equivalently, the minimum relevant impact parameter bmin. A more detailed evaluation of the high-frequency end of the virtual photon spectrum shows it to be [1–4] dnω≈αdω ωe−2ωbmin/γc(high frequency) .(3) From this, we see the general relation between the critical frequency and the minimum impact parameter is ωC≈γc bmin, b min≈γλC. (4) The characteristic angular spread θCof the radiation pattern near the critical frequency can be estimated from (4) by noting that the radiation is much like that of a beam of light with waist bmin. Then, from the laws of diffraction we conclude that θC≈λC bmin≈1 γ. (5) This behavior is also expected in that a ray of light emit- ted in the electron’s rest frame at 90◦appears at angle 1/γto the laboratory direction of the electron. A. The Formation Length To complete an application of the Weizs¨ acker-Williams method, we must also know over what time interval the virtual photon cloud is shaken off the electron to be- come the radiation detected in the laboratory. Intense (and hence, physically interesting) radiation processes are those in which the entire cloud of virtual photons is emitted as rapidly as possible. This is usefully described by the so-called formation time t0and the corresponding formation length L0=vt0where v≈cis the velocity of the relativistic electron. The formation length (time) is the distance (time) the electron travels while a radiated wave advances one wave- length λahead of the projection of the electron’s motion onto the direction of observation. The wave takes on the character of radiation that is no longer tied to its source only after the formation time has elapsed. That is, λ=ct0−vt0cosθ≈L0(1−βcosθ)≈L0/parenleftbigg1 2γ2+θ2 2/parenrightbigg , (6) 1for radiation observed at angle θto the electron’s trajec- tory. Thus, the formation length is given by L0≈2λ θ2+ 1/γ2(7) If the frequency of the radiation is near the critical fre- quency (4), then the radiated intensity is significant only forθ<∼θC≈1/γ, and the formation length is L0≈γ2λ (λ≈λC). (8) B. Summary of the Method A relativistic electron carries with it a virtual photon spectrum of αphotons per unit frequency interval. When radiation occurs, for whatever reason, the observed fre- quency spectrum will closely follow this virtual spectrum. The spectrum of radiated photons per unit path length for intense processes is given by expressions (2)-(3) di- vided by the formation length (7): dnω dl≈α L0(ω)dω ω×/braceleftbigg1 ( ω < ω C), e−ω/ω C(ω≥ωC).(9) Synchrotron radiation, undulator radiation, transition radiation, and ˇCerenkov radiation are examples of pro- cesses which can be described within the context of clas- sical electromagnetism, but for which the Weizs¨ acker- Williams approximation is also suitable. ˇCerenkov ra- diation and transition radiation are often thought of as rather weak processes, but the Weizs¨ acker-Williams viewpoint indicates that they are actually as intense as is possible for radiation by a single charge, in the sense that the entire virtual photon cloud is liberated. II. SYNCHROTRON RADIATION Synchrotron radiation arises when a charge, usually an electron, is deflected by a magnetic field. For a large enough region of uniform magnetic field, the electron’s trajectory would be circular. However, synchrotron radi- ation as described below occurs whenever the magnetic field region is longer than a formation length. The radi- ation observed when the magnetic field extends for less than a formation length has been discussed in ref. [5]. A. The Critical Frequency An important fact about synchrotron radiation is that the frequency spectrum peaks near the critical frequency, ωC, which depends on the radius Rof curvature of the electron’s trajectory, and on the Lorentz factor γvia ωC≈γ3c R. (10)Since ω0=c/Ris the angular velocity for particles with velocity near the speed of light, synchrotron radiation occurs at very high harmonics of this fundamental fre- quency. The wavelength at the critical frequency is then λC≈R γ3. (11) For completeness, we sketch a well-known argument leading to (10). The characteristic frequency ωCis the reciprocal of the pulselength of the radiation from a single electron according to an observer at rest in the lab. In the case of motion in a circle, the electron emits a cone of radiation of angular width θ= 1/γaccording to (5) that rotates with angular velocity ω=c/R. Light within this cone reaches the fixed observer during time interval δt′=θ/ω≈R/γc. However, this time interval measures the retarded time t′at the source, not the time tat the observer. Both tandt′are measured in the lab frame, and are related by t′=t−r/cwhere ris the distance between the source and observer. When the source is heading towards the observer, we have δr=−vδt′, so δt=δt′(1−v/c)≈δt′/2γ2≈R/γ3c, from which (10) follows. B. The Formation Length The formation length L0introduced in eq. (7) applies for radiation processes during which the electron moves along a straight line, such as ˇCerenkov radiation and transition radiation. But, synchrotron radiation occurs when the electron moves in the arc of a circle of radius R. During the formation time, the electron moves by formation angle θ0=L0/Rwith respect to the center of the circle. We now reconsider the derivation of the formation time, noting that while the electron moves on the arc Rθ0=vt0of the circle, the radiation moves on the chord 2 Rsin(θ0/2)≈Rθ0−Rθ3 0/24. Hence, λ=ct0−chord≈cRθ0 v−Rθ0+Rθ3 0 24 ≈Rθ0(1−β) +Rθ3 0 24≈Rθ0 2γ2+Rθ3 0 24, (12) for radiation observed at small angles to the chord. For wavelengths longer than λC, the formation angle grows large compared to the characteristic angle θC≈ 1/γ, and the first term of (12) can be neglected compared to the second. In this case, θ0≈/parenleftbiggλ R/parenrightbigg1/3 ≈1 γ/parenleftbiggλ λC/parenrightbigg1/3 (λ≥λC),(13) and L0≈R2/3λ1/3≈γ2λC/parenleftbiggλ λC/parenrightbigg1/3 (λ≥λC),(14) 2using (11). Forλ > λ C, the formation angle θ0(λ) can also be in- terpreted as the characteristic angular width of the radi- ation pattern at this wavelength. A result not deducible from the simplified arguments given above is that for λ > λ C, the angular distribution of synchrotron radia- tion falls off exponentially: dU(λ)/dΩ∝e−θ2/2θ2 0. See, for example, sec. 14.6 of [4]. For wavelengths less than λC, the formation length is short, the formation angle is small, and the last term of (12) can be neglected. Then, we find that L0≈γ2λ (λ≤λC), (15) the same as for motion along a straight line, eq. (8). It follows that the formation angle L0/Risλ/γλ C<1/γ. However, this formation angle cannot be interpreted as the characteristic angular width of the radiation pattern, which is found to be θ0≈1 γ/radicalbigg λ λC(λ≤λC) (16) via more detailed analysis [4]. C. Transverse Coherence Length The longitudinal origin of radiation is uncertain to within one formation length L0. Over this length, the trajectory of the electron is curved, so there is an uncer- tainty in the transverse origin of the radiation as well. A measure of the transverse uncertainty is the sagitta L2 0/8R, which we label w0anticipating a useful analogy with the common notation for the waist of a focused laser beam. For λ > λ C, we have from (14), w0≈L2 0 R≈R1/3λ2/3≈γλC/parenleftbiggλ λC/parenrightbigg2/3 (λ≥λC). (17) Forλ≥λC, the characteristic angular spread (13) of the radiation obeys θ0≈λ w0, (18) consistent with the laws of diffraction. Hence, the dis- tance w0of (17) can also be called the transverse coher- ence length [6] of the source of synchrotron radiation. The analogy with laser notation is also consistent with identifying the formation length L0with the Rayleigh range z0=w0/θ0, since we see that L0≈λ θ2 0≈w0 θ0. (19) Note that the sagitta (17) is larger than the minimum transverse length (4) for which the full virtual photon cloud is shaken off.We now return to the case of short wavelengths, λ < λC. The diffraction law (18) must hold for a suitable interpretation of θ0as the characteristic angular spread, andw0as the minimum transverse extent of the radia- tion. We define θ0by eq. (16), in which case w0≈γ/radicalbigg λC λ, z0=w0 θ0≈γ2λC=L0λC λ,(λ < λ C). (20) A subtle difference between the radiation of a relativis- tic charge and a focused laser beam is that the laser beam has a Guoy phase shift [7,8] between its waist and the far field, while radiation from a charge does not. D. Frequency Spectrum The number of photons radiated per unit path length lduring synchrotron radiation is obtained from the Weizs¨ acker-Williams spectrum (9) using eqs. (14) and (15) for the formation length: dnω dl≈/braceleftbiggαdωe−ω/ω C/γ2c (λ≤λC), αω2/3 Cdω/γ2cω2/3(λ≥λC).(21) We multiply by ¯ hωto recover the energy spectrum: dU(ω) dl≈/braceleftbigge2ωdωe−ω/ω C/γ2c2(λ≤λC), e2ω2/3 Cω1/3dω/γ2c2(λ≥λC).(22) The energy spectrum varies as ω1/3at low frequencies. The total radiated power is estimated from (22) using ω≈dω≈ωC≈γ3c/R, and multiplying by v≈cto convert dltodt: dU dt≈e2γ4c R2. (23) This result is also obtained from the Larmor formula, dU/dt = 2e2a⋆2/3c2, where the rest-frame acceleration is given by a⋆=γ2a=γ2v2/R≈γ2c2/Rin terms of lab quantities. III. UNDULATOR RADIATION An undulator is a device that creates a region of trans- verse magnetic field that whose magnitude oscillates with spatial period λ0. This field is constant in time, and is usually lies in a transverse plane (although helical undu- lators have been built, and are actually somewhat easily to analyze). As an electron with velocity vtraverses the undulator, its trajectory involves transverse oscillatio ns with laboratory wavelength λ0, and laboratory frequency ω0=c/λ0. The oscillating electron then emits undulator radiation. 3This radiation is usefully described by first transform- ing to the average rest frame of the electron, which is done by a Lorentz boost of γ= 1//radicalbig 1−(v/c)2in the first approximation. The undulator wavelength in this frame is λ⋆=λ0/γ, and the frequency of the oscilla- tor is ω⋆=γω0. The electron emits dipole radiation at this frequency in its average rest frame. The laboratory radiation is the transform of this radiation. Thus, undulator radiation is readily discussed as the Lorentz transform of a Hertzian dipole oscillator, and the Weizs¨ acker-Williams approximation does not offer much practical advantage here. However, an analysis of un- dulator radiation can validate the Weizs¨ acker-Williams approximation, while also exploring the distinction be- tween undulator radiation and wiggler radiation. A. A First Estimate The characteristic angle of undulator radiation in the laboratory is θC≈1/γ, this being the transform of a ray atθ⋆= 90◦to the electron’s lab velocity. The radiation is nearly monochromatic, with frequency ωC≈2γω⋆= 2γ2ω0, (24) and wavelength λC≈λ0 2γ2. (25) The formation length, defined as the distance over which radiation pulls one wavelength ahead of the elec- tron, is L0≈γ2λ≈λ0, the undulator period. But when the electron advances one period, it continues to oscillate , and the amplitude of the radiation emitted during the second period is in phase with that of the first. Assum- ing that the radiation from successive period overlaps in space, there will be constructive interference which con- tinues over the entire length of the undulator. In this case, the radiation is not clearly distinct from the near zone of the electron until it leaves the undulator. Hence, the formation length of undulator radiation is better de- fined as L0=N0λ0, (26) where N0is the number of periods in the undulator. The frequency spread of undulator radiation narrows as the number of undulator periods increases, and ∆ω ωC≈1 N0(27) We now try to deduce the radiated photon spectrum from the Weizs¨ acker-Williams approximation (9). The constructive interference over the N0undulator periods implies that the radiated energy will be N2 0times that if there were only one period. So we multiply (9) to obtaindnω dl≈N2 0α L0dω ω≈α λ0, (28) in the narrow band (27) around the characteristic fre- quency (24). The radiated power is v¯hωC≈c¯hωCtimes (28): dU dt≈e2cγ2 λ2 0, (29) using (24). This estimate proves to be reasonable only for that part of the range of undulator parameters. To clarify this, we need to examine the electron’s trajectory through the undulator in greater detail. B. Details of the Electron’s Trajectory A magnetic field changes the direction of the electron’s velocity, but not its magnitude. As a result of the trans- verse oscillation in the undulator, the electron’s average forward velocity vwill be less than v. The boost to the average rest frame is described by γrather than γ. In the average rest frame, the electron is not at rest, but oscillates in the electric and magnetic fields ˜E≈ ˜B=γB0, where we use the symbol ˜ to indicate quan- tities in the average rest frame. The case of a helical undulator is actually simpler than that of a linear one. For a helical undulator, the average-rest-frame fields are essentially those of circularly polarized light of frequen cy ˜ω=γω0. The electron moves in a circle of radius Rat this frequency, in phase with the electric field ˜E, and with velocity ˜ vand associated Lorentz factor ˜ γ, all related by ˜γm˜v2 R= ˜γm˜v˜ω=e˜E. (30) From this we learn that ˜γ˜β=e˜E m˜ωc≈eB0 mω0c≡η, (31) and hence, ˜γ=/radicalbig 1 +η2, ˜β=η/radicalbig 1 +η2, (32) and R=˜βc ˜ω=η˜λ/radicalbig 1 +η2=ηλ0 γ/radicalbig 1 +η2(33) Thus, the dimensionless parameter ηdescribes many features of the transverse motion of an electron in an oscillatory field. It is actually a Lorentz invariant, being proportional to the magnitude of the 4-vector potential. For a linear undulator, ηis usefully defined as η=eB0,rms mω0c, (34) 4where the root-mean-square (rms) average is taken over one period. With the definition (34), the rms values of ˜β, ˜γandRfor a linear undulator of strength ηare also given by eqs. (32)-(33). We can now display a relation for γ, by noting that in the average rest frame the electron’s (average) energy is ˜γmc2=m/radicalbig 1 +η2c2, while its average momentum is zero there. Hence, on transforming back to the lab frame, we have γmc2=γ˜γmc2, and so γ=γ/radicalbig 1 +η2. (35) The transverse amplitude of the motion is obtained from (33) and (35): R=ηλ0 γ. (36) C.η >1: Wiggler Radiation The pitch angle of the helical trajectory is θ≈tanθ=R λ0=η γ. (37) Since the characteristic angle of the radiation is θC≈ 1/γ, we see that the radiation from one period of the oscillation does not overlap the radiation from the next period unless η<∼1. (38) Hence, there is no constructive interference and conse- quent sharpening of the frequency spectrum unless con- dition (38) is satisfied. Forη >1, the radiation is essentially the sum of syn- chrotron radiation from N0separate magnets each λ0 long, and this case is called wiggler radiation. The laboratory frequency of the radiation is now ωC≈2γ2ω0, (39) rather than (24). However, in the regime of undulator radiation, (38), there is little difference between the two expressions. D.η <1: Weak Undulators The estimate (29) for the power of undulator radiation holds only if essentially the whole virtual photon cloud around the electron is shaken off. This can be expected to happen only if the amplitude of the electron’s transverse motion exceeds the minimum impact parameter bmin≈ γλCintroduced in eq. (4). From eqs. (25) and (33), we see that the transverse amplitude obeysR≈ηbmin. (40) Thus, for ηless than one, the undulator radiation will be less than full strength. We readily expect that the intensity of weak radiation varies as the square of the amplitude of the motion, so the estimate (29) should be revised as dU dt≈η2e2cγ2 λ2 0,(η<∼1). (41) The radiated power can be calculated exactly via the Larmor formula, dU dt=2e2a⋆2 3c3, (42) where a⋆=eE⋆/mis the acceleration of the electron in its instantaneous rest frame. The electron is moving in a helix with its velocity perpendicular to B0, so the electric field in the instantaneous rest frame is E⋆=γβB0≈ γB0. Hence, dU dt≈2e2γ2 3c/parenleftbiggeB0 mc/parenrightbigg2 =2e2cγ2η2 3λ2 0, (43) in agreement with the revised estimate (41). In practice, η≈1 is the region of greatest interest as it provides the maximum amount of constructive undulator radiation. IV. TRANSITION RADIATION As a charged particle crosses, for example, a vac- uum/metal boundary, its interaction with charges in the material results in an acceleration and hence radiation, commonly called transition radiation. The formation length is given by (7), and a formation length extends from each boundary. Hence, the number of photons emit- ted per boundary is therefore given by (2) as αper unit frequency interval. The minimum relevant transverse scale, bmin, is the plasma wavelength λp=c/ωp, so the critical frequency isωC≈γωp, according to (4). This is well into the x-ray regime. If the boundaries are less than a formation length apart, interesting interference effects arise. While the characteristic angle of transition radiation is 1 /γ, there is only a power-law falloff at larger angles, and the optical transition radiation from an intense beam of electrons can be used to measure the spot size to accuracy of a few λ [9,10]. V.ˇCERENKOV RADIATION In the case of ˇCerenkov radiation, the charge moves with velocity v > c/n in a medium where n(λ) is the index of refraction. The radiation is emitted in a cone of 5angle θCwhere cos θC=c/nv= 1/nβ. For large θC, the formation length L0=vt0is the distance over which the electron pulls one wavelength ahead of the projection of the wave motion onto the electron’s direction: λ=vt0−c nt0cosθC=L0sin2θC. (44) Then L0=λ/sin2θC, and the photon spectrum per unit path length from (2) is dnω dl≈α L0dω ω≈αsin2θC λdω ω≈αsin2θCdω c,(45) as is well-known. If the medium extends only to a transverse radius b from the particle’s trajectory, a critical wavelength is de - fined by eq. (4) as λC≈b/γ. For wavelengths longer than this, the full virtual photon cloud is not be shaken off, and the ˇCerenkov radiation is weaker than eq. (45). Of course, ˇCerenkov radiation is limited to wavelengths for which n(λ)> c/v . [1] E. Fermi, ¨Uber die Theorie des Stoßes zwischen Atomen und elektrisch geladenen Teilchen , Z. Phys. 29, 315-327 (1924). [2] C.F. von Weizs¨ acker, Ausstrahlung bei St¨ oßen sehr schneller Elektronen , Z. Phys. 88, 612-625 (1934). [3] E.J. Williams, Correlation of Certain Collision Problems with Radiation Theory , Kgl. Danske Videnskab. Selskab Mat.-fys. Medd. 13, No. 4 (1935). [4] J.D. Jackson, Classical Electrodynamics , 3rd ed., (Wiley, New York, 1999). [5] R. Co¨ ısson, On Synchrotron Radiation in Short Magnets , Opt. Comm. 22, 135 (1977). See also V.G. Bagrov et al., Radiation of Relativistic Electrons Moving in an Arc of a Circle , Phys. Rev. D 28, 2464 (1983). [6] See, for example, chap. 4 of L. Mandel and E. Wolf, Opti- cal Coherence and Quantum Optics (Cambridge Univer- sity Press, Cambridge, 1995). [7] G. Guoy, Compt. Rendue (Paris) 110, 1251-1253 (1890); Ann. de Chim. et Phys. 24, 145-213 (1891); see also sec. 17.4 of [8]. [8] A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986). [9] X. Artru et al.,Experimental investigations on geomet- rical resolution of optical transition radiation (OTR) , Nucl. Instr. and Meth. A 410, 148-158 (1998); Resolu- tion power of optical transition radiation: Theoretical considerations ,ibid.B145, 160-168 (1998). [10] P. Catravas et al.,Beam Profile Measurement at 30 GeV Using Optical Transition Radiation Proc. PAC’99, pp. 2111-2113, http://ftp.pac99.bnl.gov/Papers/Wpac/WEA100.pdf 6

arXiv:physics/0003097v1 [physics.class-ph] 29 Mar 2000VECTOR CONSTANTS OF MOTION FOR TIME-DEPENDENT KEPLER AND ISOTROPIC HARMONIC OSCILLATOR POTENTIALS O.M. Ritter⋆ Departamento de F´ ısica Universidade Federal de Santa Catarina Trindade 88040-900 Florian´ opolis SC, Brasil F.C. Santos†and A.C. Tort‡ Instituto de F´ ısica Universidade Federal do Rio de Janeiro Cidade Universit´ aria - Ilha do Fund˜ ao - Caixa Postal 68528 21945-970 Rio de Janeiro RJ, Brasil. February 2, 2008 Abstract A method of obtaining vector constants of motion for time-in dependent as well as time-dependent central fields is discussed. Some w ell-established results are rederived in this alternative way and new ones ob tained. PACS: 45.20.Dd ⋆e-mail: fsc1omr@fisica.ufsc.br †e-mail: filadelf@if.ufrj.br ‡e-mail:tort@if.ufrj.br 11 Introduction It is well known that in classical mechanics the knowledge of all first integrals of motion of a given problem is equivalent to finding its compl ete solution. Nowadays the search for first integrals has assumed an increa sing impor- tance in the determination of the integrability of a dynamic al system. It is extremely important to know if a non-linear dynamical syste m will present chaotic behavior in some regions of the phase space. The noti on of integrabil- ity is related to the existence of first integrals of motion. S everal methods of finding first integrals are available in the literature for ex ample, Lie’s method [1], Noether’s theorem [2], or the direct method [3]. Even if not all first in- tegrals of motion associated with the problem at hand are fou nd, it may happen that the ones which are obtained contribute to the dis covery of the solution we are seeking for. Nevertheless, if we do find the so lution we are after by solving the equations of motion in a straightforwar d way, it still may be profitable to look for additional constants of motion. Suc h is the case of the Kepler problem where the knowledge of the Laplace-Runge -Lenz vector [5], [6] allows us to obtain the orbit in a simple way. Of the inexhaustible wealth of problems which we can find in cl assical mechanics one of the most aesthetically appealing and impor tant is the cen- tral field problem. Energy and angular momentum associated w ith this type of field are well known conserved quantities. However, other vector and ten- sor conserved quantities have been associated with some par ticular central fields. The Laplace-Runge-Lenz vector is a vector first integ ral of motion for the Kepler problem; the Fradkin tensor [7] is conserved for t he case of the harmonic oscillator and for any central field it is possible t o find a vector first integral of motion as was shown in [4]. In the general case the se additional integrals of motion turn out to be complicated functions of t he position rand linear momentum pof the particle probing the central field. When orbits are closed and degenerated with respect to the mechanical en ergy, however, we should expect these additional constant of motion to be si mple function ofrandp. In this article we wish to exploit further this line of reaso ning by determining the existence of such additional vector first in tegrals of motion for the time-dependent Kepler and isotropic harmonic oscil lator problems. In particular, we will show that for the time-dependent Kepl er problem the 2existence of a vector constant of motion coupled to a simple t ransformation of variables turns the problem easily integrable. The structure of this paper goes as follows: in section 2 we es tablish the conditions which guarantee the existence of a vector first in tegral for a general central force field. In section 3 we put the method to test by re deriving some well known results such as the conservation of angular momen tum in an arbitrary central field, the conservation of the Laplace-Ru nge-Lenz vector for the Kepler problem, and the conservation of the Fradkin tens or fixing en route these specific fields to which they correspond. In section 4 we consider the time-dependent case establishing generalizations of the e xamples considered before and presenting new ones. In section 5 we show that the e xistence of a vector first integral enable us to find the orbits of a test par ticle. This is accomplished for the case of harmonic oscillator, and the time-dependent Kepler problem. Also the period of the time-dependent Keple r problem is obtained. Finally, section 6 is reserved for final comments. 2 Constructing vector constants of motion The force f(r,t) acting on a test particle moving in a central but otherwise arbitrary and possible time-dependent field of force g(r,t) can be written as f(r,t) =g(r,t)r, (1) where r=r(t) is the position vector with respect to the center of force, ris its magnitude, and tis the time. To this test particle we assume that it is possible to associate a vector jwhich in principle can be written in the form j(p,r,t) =A(p,r,t)p+B(p,r,t)r, (2) where p=p(t) :=m˙ r(t) is the linear momentum, mis the reduced mass andA,Bare arbitrary scalar functions of p,randt. Taking the total time derivative of (2) and making use of Newton’s second law of mot ion we readily obtain dj dt=/parenleftBigg Ag+dB dt/parenrightBigg r+/parenleftBiggdA dt+B m/parenrightBigg p. (3) If we assume that jis a constant of motion it follows that the functions A andBmust satisfy 3Ag+dB dt= 0, (4) dA dt+B m= 0. (5) Eliminating Bbetween (4) and (5) we obtain md2A dt2−gA= 0. (6) It follows from (5) that jcan be written in the form j=Ap−mdA dtr. (7) Therefore, since (6) is equivalent to both (4) and (5) if the fi eldg(r,t) is known any solution of (6) will yield a vector constant of moti on of the form given by (7). Equation (6), however, is a differential equati on whose solution may turn out to be a hard task to accomplish. Nevertheless, we can make progress if instead of trying to tackle it directly we make pl ausible guesses concerning Athereby linking jto specific forms of the field g(r,t). This procedure is tantamount to answering the following questio n: given jwhat type of central field will admit it as a constant of motion? The answer is given in the next section. 3 Simple examples Withr,pand a unit constant vector ˆ uwe can construct the following scalars: ˆ u·r,ˆ u·pandr·p.Other possibilities will be considered later on. For the moment let us consider some simple possibilities for the scalar function A(p,r,t). Consider first A(p,r,t) =ˆ u·r.It is immediately seen that this choice for Asatisfies (6) for any function g(r,t).The constant vector jreads j=(ˆ u·r)p−(ˆ u·p)r, (8) and it can be related to the angular momentum L=r×pas follows. Firstly we recast (8) into the form j=M·ˆ u, (9) 4where M=pr−rp.Sinceˆ uis a constant vector we conclude that the constancy of jis equivalent to the constancy of Mwhose components are Mjk=pjxk−xjpk, withi,j,k = 1,2,3. The antisymmetrical tensor M is closely related to angular momentum Lof the test particle. In fact, it can be easily shown that −2Li=εijkMjk, whereLiis the i-th angular mo- mentum component and εijkis the usual permutation symbol or Levi-Civita density. Therefore, this simple choice for Aleads to conservation of angular momentum for motion under a central arbitrary field g(r,t). Consider now the choice A(p,r,t) =ˆ u·p. Then making use of Newton’s second law it follows that (7) is satisfied if we find a solution to dg(r,t) dtˆ u·r= 0. (10) For arbitrary values of ˆ u·rwe can find a solution to (10) if and only if. g(r,t)≡0,org=g0whereg0is a constant. In this case we can write j= (ˆ u·p)p−g0(ˆ u·r)r. (11) If we choose the constant to be equal to −kthen the central force field will correspond to an isotropic harmonic oscillator, f=−kr.As before, (11) can be recasted into the form j= 2mF·ˆ u, (12) where Fis given by F=pp 2m+krr 2. (13) The tensor Fis symmetrical and is known as the Fradkin tensor [7]. Finall y, considerA=r·p.For this choice of A(7) yields 1 gdg dt+3 rdr dt= 0, (14) where we have made use of (1) and also of the fact that dˆ r/dtandrare per- pendicular vectors. Equation (14) can be easily integrated ifgis considered to be a function of the radial distance ronly. If this is the case we obtain the Kepler field g(r) =−k/r3.The constant vector jis then given by j= (r·p)p−(p·p)r+mkr r, (15) 5Making use of a well known vector identity we can recast (15) i nto the form, j=L×p−mkr r. (16) Therefore jcan be equaled to minus the Laplace-Runge-Lenz vector A. From (16) and the condition j·r=−A·r= 0 the allowed orbits for the Kepler problem can be obtained in a simple way, see for example [5]. 4 Time-dependent fields Let us now consider time-dependent central force fields for w hich we can build more general vector first integrals of motion. As with the tim e-independent case there are of course several possibilities when it comes to the choice of a functionAfor a time-dependent central field. Here is one A=φ(t)r·p+ψ(t)r·r. (17) Evaluating the second derivative of (17) we obtain d2A dt2=/parenleftBiggd2φ dt2+4gφ m+4 mdψ dt/parenrightBigg r·p+/parenleftBigg 2dφ dtg+φdg dt+d2ψ dt2+2ψg m/parenrightBigg r·r +2 m2/parenleftBigg mdφ dt+ψ/parenrightBigg p·p. (18) Where we have made use of (1). If we impose the additional cond ition ψ+mdφ dt= 0, (19) we eliminate the quadratic term in p. With the condition given by (19) we can substitute for ψin (17) and (18) and take the results into (6) thus obtaining 3/parenleftBigg −md2φ dt2+gφ/parenrightBigg r·dr dt+/parenleftBigg φdg dt−md3φ dt3+gdφ dt/parenrightBigg r·r= 0.(20) Equation (20) can be rewritten as 3 2/parenleftBigg −md2φ dt2+gφ/parenrightBiggd(r2) dt+/bracketleftBiggd dt/parenleftBigg −md2φ dt2+gφ/parenrightBigg/bracketrightBigg r2= 0, (21) 6and easily integrated so as to yield g(r,t) =m φd2φ dt2+C φr3. (22) The vector first integral of motion associated with (22) is j=/parenleftBigg φr·p−mdφ dtr·r/parenrightBigg p+/parenleftBigg mdφ dtr·p−φp·p−mC r/parenrightBigg r, (23) which can be simplified and written in the form j=mφ2L×d dt/parenleftBiggr φ/parenrightBigg −mCr r. (24) where we have made used of the fact the angular momentum is con stant for any arbitrary central field whether it is time-independent o r not. If in (24) we setφ= 1 andC/negationslash= 0, then from (22) we see that g(r,t) is the Kepler field andjis minus the Laplace-Runge-Lenz vector as before; the scala r function A(r,p,t) reduces to r·pwhich we have already employed in section 3. If we set (m/φ)d2φ/dt2=−k(t),that is, ifφis an arbitrary function of the time, and alsoC= 0, we have the time-dependent isotropic harmonic oscillat or field,F(r) =−k(t)r. In this case jis equal to the first term on the R.H.S. of (24). If ( m/φ)d2φ/dt2=−kandC= 0, we have the time-independent isotopic harmonic oscillator field but this time jis not the same vector as the one we have obtained before. The reason for this is our choice (17) for the scalar function A(r,p,t) which is not reducible to the form ˆ u·pemployed previously. As a last example let us consider again the time-dependent is otropic har- monic oscillator and show how it is possible to generalize th e Fradkin tensor for this case. Let the function A(r,p,t) be written as A=φ(t)ˆ u·r+ψ(t)ˆ u·p. (25) The first and the second derivative of Aread dA dt=dφ dtˆ u·r+φ mˆ u·p+dψ dtˆ u·p+gψˆ u·r, (26) 7and d2A dt2=/parenleftBiggd2φ dt2+ 2gdψ dt+dg dtψ+gφ m/parenrightBigg ˆ u·r+/parenleftBiggd2ψ dt2+ 2gdφ dt+gψ m/parenrightBigg ˆ u·p (27) Taking (27) into (6) we obtain the condition m/parenleftBiggd2φ dt2+ 2gdψ dt+dg dtψ/parenrightBigg ˆ u·r+/parenleftBigg 2dφ dt+md2ψ dt2/parenrightBigg ˆ u·p= 0. (28) Imposing the additional condition 2dφ dt+md2ψ dt2= 0, (29) (28) becomes md3ψ dt3−4gdψ dt−2dg dtψ= 0. (30) We can solve (30) thoroughly if g(r,t) is a function of the time tonly. In this case, as before, we end up with the time-dependent isotr opic harmonic oscillator. The vector jassociated with (25) can be obtained as follows: first we integrate (29) thus obtaining φ=−m 2dψ dt+C, (31) whereCis an integration constant. Then making use of (25), (31) and (26) we arrive at j=/bracketleftBigg/parenleftBigg −m 2dψ dt+C/parenrightBigg ˆ u·r+ψˆ u·p/bracketrightBigg p (32) +/bracketleftBigg/parenleftBiggm2 2d2ψ dt2−mgψ/parenrightBigg ˆ u·r−/parenleftBiggm 2dψ dt+C/parenrightBigg ˆ u·p/bracketrightBigg r, or in terms of components ji=Fijuj, (33) whereFijis defined by 8Fij=/parenleftBigg −m 2dψ dt+C/parenrightBigg pixj+ψpipj (34) +/parenleftBiggm2 2d2ψ dt2−mgψ/parenrightBigg xixj−/parenleftBiggm 2dψ dt+C/parenrightBigg xipj. The constant Cin (34) can be made zero without loss of generality. A generalized Fradkin tensor can now be defined by Fij=/parenleftBigg −m 2dψ dt/parenrightBigg pixj+ψpipj+/parenleftBiggm2 2d2ψ dt2−mgψ/parenrightBigg xixj−m 2dψ dtxipj.(35) From (35) we can read out the diagonal components of the gener alized Frad- kin tensor Fii=−m2dψ dtxi. dxi dt+ψp2 i+/parenleftBiggm2 2d2ψ dt2−mg/parenrightBigg x2 i. (36) It is not hard to see that the trace of this generalized Fradki n tensor F(r,p,t) becomes the energy of the particle when g(r,t) is a constant field. 5 Obtaining explicit solutions: An alterna- tive way Now we wish to show how to take advantage of the vector constan tjto obtain the solution for the Kepler and the isotropic harmonic oscil lator potentials. But firstly we must establish some very general relationship s between the sought for solution r(t) andA(r,p,t) andj. First notice that (10) can be recasted into the form j=m A(r,p,t)2d dt/bracketleftBiggr A(r,p,t)/bracketrightBigg , (37) where it must be kept in mind that A(r,p,t) satisfies (6). As we have seen before in specific examples the form of the vector constant jdepends on the force acting on the particle. Integrating (37) we readily ob tain 9r(t) A(r,p,t)−r(0) A(r,p,0)=j m/integraldisplayt 0dτ A(r,p,τ)2. (38) Equation (38) can be given a simple but interesting geometri cal interpreta- tion. Assume that the initial conditions r(0) and p(0) are known and there- fore the function A(r,p,0) can be determined. The vector r(0)/A(r,p,0)is therefore a constant and completely determined vector. As t ime increases, the R.H.S. of (38) increases. The vector on the right side of ( 38) though varying in time has a fixed direction which is determined by j. Therefore, r(t)/A(r,p,t) must increase in order to close the triangle whose sides are the three vectors involved in (38). If the orbit is unlimited then it is easy to see that the following property ensues: there is an asympt ote if in the limitt→ ∞ the definite integral/integraltextt 0dt A2is constant. On the other hand, if the orbit is limited, but not necessarily closed, there wi ll be a position vector rwhose direction is parallel to that of the vector jat the instant t∗. If the length of the position vector ris finite, we can conclude that at the same instant t∗the function A(r,p,0) must be zero. Thus, we can see that the vector r(t)/A(r,p,t) must be reversed at this instant and its evolution is determined by the fact that its end is on the stra ight line that contains j. Fort=t∗+ǫ, whereǫis a positive infinitesimal number, the vector r(t∗+ǫ)/A(r,p,t∗+ǫ) changes its direction abruptly, so to speak, as shown in the figure, hence in the transition A(r,p,t∗)→A(r,p,t∗+ǫ) the scalar function must change its sign. Let us obtain the solution r(t) for the case of the isotropic harmonic oscillator. A particular solution of (17) for g=−κ,whereκis the elastic constant is given by A(t) = cos(ωt), (39) whereω=/radicalBig k mis the angular frequency. The solution given by (39) allows us to write r(t) cos(ωt)−r(0) =p(0) m/integraldisplayt 0dτ cos2(ωτ)(40) The integral can be readily performed and after some simplifi cations we fi- nally obtain r(t) = cosωtr(0) +1 mωsinωtp(0). (41) 10Therefore we have obtained the solution of the time-indepen dent harmonic oscillator in an alternative way from the knowledge of the in itial conditions as it should be. Notice that the general solution A(t) =A(0) cos (ωt+θ) would lead to the same general result. Another possible solu tion in the case of the time-independent isotropic harmonic oscillator is g iven by A(r,p,t) =ˆ u·p(t), (42) as can be shown by substituting this solution into (6). This s olution shows that the trajectories have no asymptotes. Let us now show how we can obtain the orbits in the case of the ti me- dependent Kepler problem. Let us begin by rewriting (37) in p olar coordi- nates on the plane. In these coordinates the angular momentu m conservation law is written in the form l=mr2dθ dt(43) and this allows to rewrite (37) as j=mA2d dθ/parenleftbiggr A/parenrightbiggdθ dt=l/parenleftbiggA r/parenrightbigg2d dθ/parenleftbiggr A/parenrightbigg . (44) Introducing the unitary vectors∧rand∧ θwe can write the above equation as j=l/bracketleftBigg −d dθ/parenleftbiggA r/parenrightbigg ∧r+A r∧ θ/bracketrightBigg . (45) The components of the vector jin the direction of∧rand∧ θare given by j·∧ θ=lA r, (46) and j·∧r=−ld dθ/parenleftbiggA r/parenrightbigg . (47) Equation (47) can be obtained from (46) and therefore it is re dundant. In section 4 we determined a generalized Laplace-Runge-Lenz v ector for the time-dependent Kepler problem. The scalar function A(r,p,t) associated with this vector was found to be 11A=φ(t)r·p−mdφ dtr2. (48) Making use of (43) we can rewrite the linear momentum as a func tion ofθ as follows p=mdr dtdθ dt=l r2dr dθ. (49) Taking (49) into (48) and considering Aas a function of θwe obtain A r=−ld dθ/parenleftBiggφ r/parenrightBigg . (50) Equations. (46) and (50) lead to d dθ/parenleftBiggφ r/parenrightBigg =−j·∧ θ l2=j l2sin(θ−α), (51) whereαis the angle between jand the OXaxis (see figure 2). In order to integrate (51) we assume that the initial conditions at t= 0 are known vector functions, i.e. r(0) =r0; (52) and p(0) =p0. (53) In terms of polar coordinates these initial conditions are w ritten as r(θ0) =r0, (54) and making use of (49) /parenleftBiggdr dθ/parenrightBigg θ=θ0=r2 0 lp0. (55) Upon integrating (51) we find φ r=φ0 r0+j l2[cos(θ0−α)−cos(θ−α)]. (56) 12For the usual time-independent Kepler problem, φ= 1 and in this case (56) takes the form 1 r=1 r0+j l2[cos(θ0−α)−cos(θ−α)] (57) The scalar product between jas given by (16) and r0permit us to eliminate cos(θ0−α) and leads to the usual orbit equation 1 r=−mC l2/bracketleftbigg 1 +j mCcos(θ−α)/bracketrightbigg . (58) If we define a new position vector r′according to r′:=r φ, (59) and redefine our time parameter according to dt′:=dt φ2, (60) we can recast the equation of motion for the time-dependent K epler problem, namely md2r dt2=/parenleftBiggm φd2φ dt2+C φr3/parenrightBigg r (61) into a simpler form. According to (59) and (60) the velocity a nd the accel- eration transform in the following way dr dt=r′dφ dt+1 φdr dt′′ (62) and d2r dt2=r′d2φ dt2+1 φ3d2r dt′2′ . (63) where we have taken advantage of the fact that r,r′andφcan be considered as functions of tort′. Making use of (63) the equation of motion (61) can be written as md2r dt′2′ =C (r′)3r′. (64) 13Equation (64) corresponds to the usual time-independent Ke pler problem whose solution is given by (58). Equations (59) and (60) show that the open solutions of (64) are transformed into the open solutio ns of (61) with the same angular size and that closed solutions of (64) are as sociated with spiraling solutions of (61). The period of the orbit of (64) i s related to the time interval that the spiraling particle takes to cross a fix ed straight line. Representing this time interval by T0we have T=/integraldisplayT0 0dt φ2. (65) As an application of the above remarks suppose we are looking for the form of the function φwhich yields a circular orbit with radius Ras a solution to (61)? To find this function we see from (61) that we have to so lve the following equation differential equation d2φ dt2+|C| mR3φ=|C| mR3. (66) The solution is φ(t) = 1 +φ0cos (ωt+β), (67) whereφ0is a constant and ω=/radicalBig |C| mR3andβan arbitrary phase angle . The constantφ0may be chosen so that the transformed solution will be a given ellipse as we show below. Equation (59) leads to 1 r′=R−1[1 +φ0cos (ωt+β) ]. Comparing with (58) we obtain R=l2 m|C|, (68) and φ0=−j m|C|. (69) The period of this circular orbit is given 14T0=2π ω= 2π/radicaltp/radicalvertex/radicalvertex/radicalbtmR3 |C|. (70) and using (65) we get T=/integraldisplay2π ω 0dt (1 +ecosωt)2=2π ω1 (1−e2)3 2(71) wheree=|φ0|is the eccentricity. Making use of 1 −e2=R/a, whereais the major semi-axis we finally obtain the orbital period T= 2πa3 2/radicalbigg −m C. (72) To conclude consider the total mechanical energy associate d with (64) E=p′2 2m+C r′=const. (73) Sincep′andpare related by p′=φ(t)p−m˙φ(t)r (74) andr′andrby (59) we easily obtain E=φ2p2 2m−2φdφ dtr·p+/parenleftBiggdφ dt/parenrightBigg2r2 2+Cφ r(75) which is a conserved quantity and can be interpreted as a gene ralization of the energy of the particle under the action of a time-depende nt Kepler field. 6 Laplace-Runge-Lenz type of vector constants for arbitrary central fields Equation (22) determines a time-dependent Kepler field g(r,t),where the variablesrandtare independent. If, however, we consider the orbit equa- tionr(t) we can eliminate the time variable and define the function g(r,t(r)) which can be understood as an arbitrary function of r.For the sake of sim- plicity we denote this function by g(r). The function φ(t) which transforms 15the Kepler problem when understood as a function of rtransforms the Kepler field in an arbitrary central field. Let us write Eq. (22) in the form md2φ dt2−g(r,t)φ+C r3= 0 (76) and consider the transformation d2φ dt2=d2φ dr2/parenleftBiggdr dt/parenrightBigg2 +dφ drd2r dt2(77) Energy conservation and the equation of motion allow us to wr ite /parenleftBiggdr dt/parenrightBigg2 =2 m/bracketleftBigg E−V(r)−l2 2mr2/bracketrightBigg (78) and d2r dt2=rg(r) m+l2 m2r3(79) whereEis the energy of the particle and lits angular momentum. Taking these three last equations into account Eq. (76) reads now 2/bracketleftBigg E−V(r)−l2 2mr2/bracketrightBiggd2φ dr2+/bracketleftBigg rg(r) +l2 m2r3/bracketrightBiggdφ dr−g(r)φ+C r3= 0 (80) Equation (80) permit us to determine the function φ(r) for any potential V(r). Thus, we conclude that a central field problem can be transf ormed into a time-dependent Kepler problem. When g(r) describes the Kepler field the solution of (80) is φ(r) = 1. Sometimes it is convenient to perform a second change of variables by defining the transformation φ=ψ−mC l2(81) Then Eq. (80) becomes 2/bracketleftBigg E−V(r)−l2 2mr2/bracketrightBiggd2ψ dr2+/bracketleftBiggrg(r) m+l2 m2r3/bracketrightBiggdψ dr−g(r)ψ= 0 (82) As an example consider V(r) =k/r. Then the solution of (82) is simply ψ(r) =c1/parenleftBig kmr+l2/parenrightBig +c2√ l2+ 2kmr−2mEr2 (83) 167 Conclusions In this paper we have outlined a simple and effective method fo r treating problems related with time-dependent and time independent central force fields. In particular we have dealt with the Kepler problem an d the isotropic harmonic oscillator fields. We have been able to rederive som e known results from an original point of view and generalize others. The cen tral force field has been discussed in the literature from many points of view . The difficulty in finding vector constants of motion for central fields stem f rom the fact that in general orbits for these type of problem are not closed, th erefore any new ways to attack those problems are welcome.. In our method thi s difficulty is transferred, so to speak, to the obtention for each possible central field, which can be time-dependent or not, of a certain scalar function of the position, linear momentum and time. For a given central field this scala r function is a solution of (6). In the general case, the obtention of the sc alar function is a difficult task. Judicious guesses, however, facilitate the search for solution of (6) and this is what we have done here. References [1] Olver P J 1986 Applications of Lie groups to differential equations (New York: Springer Verlag). [2] Cantrijn F and Sarlet W 1981 Siam Review 23(4) 467. [3] Whittaker E T 1937 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies 4thed. (Cambridge: Cambridge University Press). [4] C.C. Yan, J. Phys. A 244731-38. [5] Goldstein H 1980 Classical Mechanics 2nded (Reading MA: Addison- Wesley). [6] Laplace P S de 1799 Trait´ e de M´ ecanique Celeste ; Runge C 1919 Vektor- analysis V1 pg. 70 (Leipzig: S. Hirzel, Leipzig); W Pauli 1926 Z. Phyis ik 36336-63; Lenz W 1926 Z. Phyisik 24197-207; Heintz W H 1974 Am. J. Phys. 42.1078–82 See also Goldstein H 1975 Am. J. Phys. 43737-8; ibid. 1976 441123-4. 17[7] Fradkin D M1965 Am. J. Phys. 33207-11; 1967 Prog. Theor. Phys. 37 798-812. [8] Yoshida T 1987, Eur. J. Phys. 8258-59; 1989 Am. J. Phys. 57376-7. 18

arXiv:physics/0003098v1 [physics.data-an] 29 Mar 2000Tsallis’ entropy maximization procedure revisited S. Mart´ ınez1,2∗, F. Nicol´ as, F. Pennini1,2†, and A. Plastino1,2‡ 1Physics Department, National University La Plata, C.C. 727 , 1900 La Plata, Argentina 2Argentine National Research Council (CONICET) Abstract The proper way of averaging is an important question with reg ards to Tsal- lis’ Thermostatistics. Three different procedures have bee n thus far employed in the pertinent literature. The third one, i.e., the Tsalli s-Mendes-Plastino (TMP) [1] normalization procedure, exhibits clear advanta ges with respect to earlier ones. In this work, we advance a distinct (from the TM P-one) way of handling the Lagrange multipliers involved in the extremiz ation process that leads to Tsallis’ statistical operator. It is seen that the n ew approach consid- erably simplifies the pertinent analysis without losing the beautiful properties of the Tsallis-Mendes-Plastino formalism. PACS: 05.30.-d, 95.35.+d, 05.70.Ce, 75.10.-b KEYWORDS: Tsallis thermostatistics, normalization. ∗E-mail:martinez@venus.fisica.unlp.edu.ar †E-mail: pennini@venus.fisica.unlp.edu.ar ‡E-mail: plastino@venus.fisica.unlp.edu.ar 1I. INTRODUCTION Tsallis’ thermostatistics [1–6] is by now recognized as a ne w paradigm for statistical mechanical considerations. One of its crucial ingredients , Tsallis’ normalized probability distribution [1], is obtained by following the well known Ma xEnt route [7]. One maximizes Tsallis’ generalized entropy [3,8] Sq=k1−/summationtextw i=1pq i q−1, (1) (k≡k(q) tends to the Boltzmann constant kBin the limit q→1 [2]) subject to the constraints (generalized expectation values) [2] w/summationdisplay i=1pi= 1 (2) /summationtextw i=1pq iO(i) j/summationtextw i=1pq i=/an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htq, (3) where piis the probability assigned to the microscopic configuratio ni(i= 1, . . ., w ) and one sums over all possible configurations w.O(i) j(j= 1, . . ., n ) denote the nrelevant observables (the observation level [9]), whose generalized expectatio n values /an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htqare (assumedly) a priori known. The Lagrange multipliers recipe entails maximizing [1] F=Sq−α0/parenleftBiggw/summationdisplay i=1pi−1/parenrightBigg −n/summationdisplay j=1λj /summationtextw i=1pq iO(i) j/summationtextw i=1pq i− /an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htq , (4) yielding pi=f1/(1−q) i ¯Zq, (5) where fi= 1−(1−q)/summationtext jλj/parenleftBig O(i) j− /an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htq/parenrightBig k/summationtext jpq j, (6) is the so-called configurational characteristic [10] and ¯Zq=/summationdisplay if1/(1−q) i , (7) 2stands for the partition function. The above procedure, originally employed in [1], overcomes most of the problems posed by the old, unnormalized way of evaluating Tsallis’general ized mean values [1,11]. Some hardships remain, though. One of them is that numerical diffic ulties are sometimes encoun- tered, as the piexpression is explicitly self-referential . An even more serious problem is also faced: a maximum is not necessarily guaranteed. Indeed, ana lyzing the concomitant Hessian so as to ascertain just what kind of extreme we face, one encou nters the unpleasant fact that this Hessian is not diagonal. In the present effort we introduce an alternative Lagrange ro ute, that overcomes the above mentioned problems. II. THE NEW LAGRANGE MULTIPLIERS’ SET We extremize again (1) subject to the constraints (3), but, a nd herein lies the central idea,rephrase (4) by recourse to the alternative form w/summationdisplay i=1pq i/parenleftBig O(i) j− /an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htq/parenrightBig = 0, (8) withj= 1, . . ., n . We have now F=Sq−α0/parenleftBiggw/summationdisplay i=1pi−1/parenrightBigg −n/summationdisplay j=1λ′ jw/summationdisplay i=1pq i/parenleftBig O(i) j− /an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htq/parenrightBig , (9) so that, following the customary variational procedure and eliminating α0we find that the probabilities are, formally, still given by (5). However, i n terms of the new set of Lagrange multipliers, the configurational characteristics do not depend explicitly on the probabilities f′ i= 1−(1−q) k/summationdisplay jλ′ j/parenleftBig O(i) j− /an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htq/parenrightBig . (10) Comparing (10) with (6), it is clear that the Lagrange multip liersλjof the Tsallis- Mendes-Plastino formalism (TMP) [1] and λ′ j(present treatment) can be connected via λ′ j=λj/summationtext ipq i, (11) 3which leads to the nice result f′ i=fi. The probabilities that appear in (11) are those special ones that maximize the entropy , not generic ones. The ensuing, new partition function is also of the form (7), with fi>0 the well known Tsallis’ cut-off condition [3,8]. Notice tha t now the expression for the MaxEnt probabilities piis NOT explicitly self-referential. In order to ascertain the kind of extreme we are here facing we study the Hessian, that now is of diagonal form. The maximum condition simplifies then to the requireme nt ∂2F ∂p2 i<0. (12) The above derivatives are trivially performed yielding ∂2F ∂p2 i=−qpq−2 ifi, (13) which formally coincides with the maximum requirement one fi nds in the case of Tsallis’ unnormalized formalism. Since the fiare positive-definite quantities, for a maximum one should demand that q >0. Extremes found by following the celebrated Lagrange proced ure depend only on the nature of the constraints, not on the form in which they are ex pressed. Thus, the two sets of multipliers lead to the same numerical values for the micr o-state probabilities. Via (11) one is always able to establish a connection between both tre atments. The present algorithm exhibits the same nice properties of t he TMP formalism, namely: •The MaxEnt probabilities are invariant under uniform shift s of the Hamiltonian’s energy spectrum (see, for instance, the illuminating discu ssion of Di Sisto et al. [12]). Indeed, after performing the transformation ǫi→ǫi+ǫ0 (14) Uq→Uq−ǫ0, (15) on equation (5), with figiven by (10), we trivially find that the probabilities pikeep their forms invariant if the λ′ jdo not change. Due to relation (11), the λjare invariant too. 4•The mean value of unity equals unity, i.e., /an}bracketle{t/an}bracketle{t1/an}bracketri}ht/an}bracketri}htq= 1, which is not the case with the unnormalized expectation values [3,8]. •One easily finds that, for two independent subsystems A, B , energies add up: Uq(A+ B) =Uq(A) +Uq(B). III. THERMODYNAMICS We pass now to the question of writing down the basic mathemat ical relationships of Thermodynamics, as expressed with respect to the new set of L agrange multipliers λ′ j. In order to do this in the most general quantal fashion we shal l work in a basis- independent way. This requires consideration of the statis tical operator (or density operator) ˆρthat maximizes Tsallis’ entropy, subject to the foreknowle dge of Mgeneralized expectation values (corresponding to Moperators/hatwideOj). These take the form /angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q=Tr(ˆρq/hatwideOj) Tr(ˆρq), j = 1, ..., M. (16) To these we must add, of course, the normalization requireme nt Trˆρ= 1. (17) The TMP formalism, where relations are written in terms of th e “old” Lagrange multi- pliers λj, yields the usual thermodynamical relationships [1], name ly ∂S ∂/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q=kλj (18) ∂ ∂λi(lnqZq) =−/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q, (19) where lnq¯Zq=¯Z1−q q−1 1−q(20) and [1] 5lnqZq= ln q¯Zq−/summationdisplay jλj/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q, (21) so that the essential mathematical structure of Thermodyna mics is preserved. Following the standard procedure [5,13] one gets ˆρ=¯Z−1 q 1−(1−q)M/summationdisplay jλ′ j/parenleftbigg /hatwideOj−/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q/parenrightbigg 1 1−q , (22) where ¯Zqstands for the partition function ¯Zq=Tr 1−(1−q)/summationdisplay jλ′ j/parenleftbigg /hatwideOj−/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q/parenrightbigg 1 1−q . (23) Enters here Tsallis’cut-off condition [5,13]. The form (22) does not a priori guarantee that we will have a positive-definite operator. Some additio nal considerations are requested. Consider the operator /hatwideA= 1−(1−q)/summationdisplay jλ′ j/parenleftbigg /hatwideOj−/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q/parenrightbigg (24) enclosed within parentheses in (22). One must ensure its pos itive-definite character. This entails that the eigenvalues of ˆAmust be non-negative quantities. This can be achieved by recourse to an heuristic cut-off procedure. We replace (22) b y ˆρ=¯Z−1 q/bracketleftBigˆAΘ(ˆA)/bracketrightBig1/(1−q), (25) with¯Zqgiven by ¯Zq=Tr/bracketleftBigˆAΘ(ˆA)/bracketrightBig1/(1−q), (26) where Θ( x) is the Heaviside step-function. Equations (25)-(26) are t o be re-interpreted as follows. Let |i/an}bracketri}htandαi,stand, respectively, for the eigenvectors and eigenvalues of the operator (24), whose spectral decomposition is then ˆA=/summationdisplay iαi|i/an}bracketri}ht /an}bracketle{ti|. (27) In the special basis used above ˆ ρadopts the appearance 6ˆρ=¯Z−1 q/summationdisplay if(αi)|i/an}bracketri}ht /an}bracketle{ti|, (28) withf(x) defined as f(x) = 0, for x ≤0, (29) and f(x) =x1 1−q, for x > 0. (30) Notice that f(x) possesses, for 0 < q < 1,a continuous derivative for all x.Moreover, df(x) dx=/parenleftBigg1 q−1/parenrightBigg [xΘ(x)]q 1−q. (31) In terms of the statistical operator, Tsallis’ entropy Sqreads Sq=1 q−1Tr/bracketleftBig ˆρq/parenleftBig ˆρ1−q−ˆI/parenrightBig/bracketrightBig =1 q−1Tr/bracketleftBig ˆρq/parenleftBig¯Zq−1 qˆAΘ(ˆA)−ˆI/parenrightBig/bracketrightBig (32) =¯Zq−1 q q−1Tr/bracketleftBig ˆρqˆAΘ(ˆA)/bracketrightBig −Tr(ˆρq) (q−1), where ˆIis the unity operator. Obviously, ˆ ρcommutes with ˆA. The product of these two operators can be expressed in the common basis that diagonalizes them ˆρqˆAΘ(ˆA) =¯Z−q q/summationdisplay i[f(αi)]qαi|i/an}bracketri}ht/an}bracketle{ti|, (33) which entails, passing from the special basis |i/an}bracketri}htto the general situation, that ˆρqˆAΘ(ˆA) = ˆρq 1−(1−q)/summationdisplay jλ′ j/parenleftbigg /hatwideOj−/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q/parenrightbigg , (34) and, consequently Sq=¯Zq−1 q−1 q−1Tr(ˆρq) +¯Zq−1 q/summationdisplay jλ′ jTr/bracketleftbigg ˆρq/parenleftbigg /hatwideOj−/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q/parenrightbigg/bracketrightbigg . (35) Since the last term of the right-hand-side vanishes, by defin ition (8), we finally arrive at 7Sq=¯Zq−1 q−1 q−1Tr(ˆρq). (36) Now, from the very definition (in terms of ˆ ρ) of Tsallis’ entropy Sq[5,13], we find Tr(ˆρq) = 1 +(1−q) kSq, (37) so that (36) and (37) lead to Tr(ˆρq) =¯Z1−q q (38) and Sq=klnq¯Zq, (39) where ln q¯Zqhas been introduced in (20). Using (38), equation (11) can be rewritten as λ′ j=λj ¯Z1−q q. (40) Following [1] we define now lnqZ′ q= ln q¯Zq−/summationdisplay jλ′ j/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q, (41) and introduce k′=k¯Z1−q q, (42) which leads finally to (see (18) and (19)) ∂S ∂/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q=k′λ′ j (43) ∂ ∂λ′ j/parenleftBig lnqZ′ q/parenrightBig =−/angbracketleftBig/angbracketleftBig/hatwideOj/angbracketrightBig/angbracketrightBig q. (44) Equations (43) and (44) constitute the basic Information Th eory relations on which to build up, ` a laJaynes [7], Statistical Mechanics. 8Notice that k′, as defined by (42), obeys the condition k′→kBasq→1. This is a condition that this constant must necessarily fulfill (see [ 2]). Comparing (18) and (43) one arrives at the important relationship k′λ′ j=kλj, (45) which entails that the intensive variables are invariant un der the λ′→λtransformation. As a special instance of Eqs. (43) and (44) let us discuss the C anonical Ensemble, where they adopt the appearance ∂S ∂Uq=k′β′=kβ=1 T(46) ∂ ∂β′/parenleftBig lnqZ′ q/parenrightBig =−Uq, (47) where (see equation (41)) lnqZ′ q= ln q¯Zq−β′Uq. (48) ¿From Equation (46) one notices that the temperature Tis the same for both multipliers’ sets. Finally, the specific heat reads Cq=∂Uq ∂T=−k′β′2∂Uq ∂β′. (49) We conclude that the mathematical form of the thermodynamic relations is indeed pre- served by the present treatment. Both sets of Lagrange multi pliers accomplish this feat and they are connected via (11). The primed one, however, allows for a simpler treatment, as will be illustrated below. IV. SIMPLE APPLICATIONS We consider now some illustrative examples. They are chosen in such a manner that each of them discusses a different type of situation: classic al and quantal systems, the latter in the case of both finite and infinite number of levels. 9A. The classical harmonic oscillator Let us consider the classical harmonic oscillator in the can onical ensemble. We can associate with the classical oscillator a continuous energ y spectrum ǫ(n) =ǫ nwithǫ >0 andn∈ R+compatible with the cut-off condition. The ensuing MaxEnt pr obabilities adopt the appearance pq(n, t′) =[fq(n, t′)]1/(1−q) ¯Zq(t′), (50) where fq(n, t′) = 1−(1−q)(n−uq) t′, (51) and ¯Zq(t′) =/integraldisplaynmax 0[fq(n, t′)]1/(1−q)dn, (52) withuq=Uq/ǫandt′=k′T/ǫ. We have introduced also nmaxas the upper integration limit on account of Tsallis’ cut-off condition. One appreciates th e fact that nmax→ ∞ ifq >1. nmaxis, of course, the maximum n-value that keeps [ fq(n, t′)]1/(1−q)>0 forq <1. The normalization condition reads uq(t′) =/integraltextnmax 0[pq(n, t′)]qn dn/integraltextnmax 0[pq(n, t′)]qdn, (53) or, using (50), uq(t′) =/integraltextnmax 0[fq(n, t′)]q/(1−q)n dn /integraltextnmax 0[fq(n, t′)]q/(1−q)dn. (54) Due to the form of fq, equation (54) constitutes a well-defined expression. By ex plicitly performing the integrals for 1 < q < 2 (for q≥2 the integrals diverge) we obtain uq(t′) =t′2/(2−q)(1 + (1 −q)uq/t′)(2−q)/(1−q) t′(1 + (1 −q)uq/t′)1/(1−q). (55) After a little algebra, the above equation leads to the simpl e result 10uq(t′) =t′. (56) Replacing now uq=Uq/ǫandt′=k′T/ǫ, we obtain Uq=k′T, so that the specific heat reads Cq=k′. (57) It is worthwhile to remark that, in the case of this particula r example, we formally regain the usual expressions typical of the q= 1 case. Due to fact that we possess a degree of freedom in the definition of k′, we can set k′=kBand thus recover Gibbs’ Thermodynamics. Performing the pertinent integral and using (56), the parti tion function becomes ¯Zq(t′) =t′(2−q)1/(1−q). (58) According to equation (40), t′can be written in terms of tand¯Zq, allowing us to recover [1] ¯Zq(t) =t1/q(2−q)1/[q(1−q)], (59) and, consequently, uq=t1/q(2−q)1/q(60) Cq=k 2(2−q)1/qt(1−q)/q. (61) These results are identical to those of [1], but are here deri ved in a remarkably simpler fashion. B. The two-level system and the quantum harmonic oscillator Let us consider the discrete case of a single particle with an energy spectrum given by En=ǫn, where ǫ >0 and n= 0,1, ..., N . IfN= 1, we are facing the non degenerate two level system, while, if n→ ∞, the attendant problem is that of the quantum harmonic oscillator. 11The micro-state probabilities are of the form, once again pn=f1/(1−q) n ¯Zq(62) with ¯Zq=N/summationdisplay n=0fn1/(1−q). (63) The configurational characteristics take the form fn(t′) = 1−(1−q)(n−uq)/t′(64) where again (see (IVA)), t′=k′T/ǫanduq=Uq/k′. Using (62), the mean energy can be written as uq=/summationtextN n=0fq/(1−q) n n /summationtextN n=0fq/(1−q) n, (65) which, using the explicit form of fnand rearranging terms, allows one to write down the following equation N/summationdisplay n=0/bracketleftBigg 1−(1−q) t′(n−uq)/bracketrightBiggq/(1−q) (n−uq) = 0, (66) which implicitly defines uq. Notice that one does not arrive to a closed expression. Howe ver, in order to numerically solve for uq, we just face (66). This equation is easily solved by recourse to the so-called “seed” methods (cut-off always tak en care of), with quick conver- gence (seconds). This is to be compared to the TMP instance [1 ]. In their case, one faces a non-linear coupled system of equations in order to accompli sh the same task. This coupled system can be recovered from (66) and (62), writing t′in terms of t. C. Magnetic Systems Consider now a very simple magnetic model, discussed, for in stance, in [14]: a quantum system of Nspin 1/2 non-interacting atoms in the presence of a uniform, external magnetic field/vectorH=Hˆk(oriented along the unit vector ˆk). Each atom is endowed with a magnetic 12moment/hatwide/vector µ(i)=g µ0/hatwide/vectorS(i) ,where µ0=e/(2mc) is Bohr’s magneton and/hatwide/vectorS(i) = (¯h/2)/hatwide/vector σ(i), with /hatwide/vector σ(i)standing for the Pauli matrices. The concomitant interacti on energy reads ˆH=−N/summationdisplay i=1/hatwide/vector µ(i)·/vectorH=−gµ0 ¯hH/hatwideSz, (67) where/hatwide/vectorS=/summationtextN i=1/hatwide/vectorS(i) the total (collective) spin operator. The simultaneous eig envectors of /hatwide/vectorS2 and/hatwideSzconstitute a basis of the concomitant 2N-dimensional space. We have |S, M/an}bracketri}ht, with S=δ, . . . , N/ 2, M=−S, . . . , S, andδ≡N/2−[N/2] = 0 (1 /2) ifNis even (odd). The corresponding multiplicities are Y(S, M) =Y(S) =N!(2S+ 1)/[(N/2−S)!(N/2 +S+ 1)!] [14]. We recast the Hamiltonian in the simple form ˆH=−x′ β′ˆSz, (68) withx′=gµ0Hβ′/¯h. Our statistical operator can be written as ˆρ=1 ¯Zq/bracketleftbigg 1−(1−q)x′/parenleftbigg ˆSz−/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q/parenrightbigg/bracketrightbigg1/(1−q) , (69) where ¯Zq=Tr/bracketleftbigg 1−(1−q)x′/parenleftbigg ˆSz−/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q/parenrightbigg/bracketrightbigg1/(1−q) . (70) Due to the cut-off condition, 1 −(1−q)x′/parenleftbigg ˆSz−/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q/parenrightbigg >0. The mean value of the spin z-component is computed according to (16) /angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q=Tr/parenleftBig ˆρqˆSz/parenrightBig Tr(ˆρq), (71) so that, replacing (69) into (71) and rearranging then terms we arrive at Tr/braceleftBigg/bracketleftbigg 1 + (1 −q)x′/parenleftbigg ˆSz−/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q/parenrightbigg/bracketrightbiggq/(1−q)/parenleftbigg ˆSz−/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q/parenrightbigg/bracerightBigg = 0. (72) More explicitly, one has N/2/summationdisplay S=δY(S)S/summationdisplay M=−S/bracketleftbigg 1 + (1 −q)x′/parenleftbigg M−/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q/parenrightbigg/bracketrightbiggq/(1−q)/parenleftbigg M−/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q/parenrightbigg = 0, (73) 13which is the equation to be solved in order to find/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q. Notice that, once again, one faces just a single equation, that can be easily tackled. If one uses instead the TMP prescription (as discussed in [15]) one has to solve a coupled , highly non-linear system of equations. Such a system can be recover ed from (73) if one replaces x′ byx/Tr(ρq) and adds the condition Tr(ρq) from (69). As in [15], we consider now two asymptotic situations from th e present viewpoint. Forx′→0 we Taylor-expand (73) around x′= 0 and find /angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q=qx′N 4, (74) that leads to an effective particle number N0 eff=qN, (75) as in [15]. Following the same mechanism and using (74), one fi nds that Tr(ρq) = 2N(1−q). (76) Remembering that x′=x/Tr(ρq), it is possible to recover the TMP normalized solution [15] /angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q=qxN 42N(q−1), (77) and N0(3) eff=qN2N(q−1). (78) Forx′→ ∞,and for 0 < q < 1,expression (73) leads to an equation identical to that of [15] N/2/summationdisplay S=δY(S)S/summationdisplay M=−S/parenleftbigg M−/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q/parenrightbigg1/(1−q) = 0, (79) whose solution reads/angbracketleftBig/angbracketleftBigˆSz/angbracketrightBig/angbracketrightBig q=N/2. 14V. CONCLUSIONS In order to obtain the probability distribution pithat maximizes Tsallis’ entropy subject to appropriate constraints, Tsallis-Mendes-Plastino ext remize [1] F=Sq−α0/parenleftBiggw/summationdisplay i=1pi−1/parenrightBigg −n/summationdisplay j=1λj /summationtextw i=1pq iO(i) j/summationtextw i=1pq i− /an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htq , and obtain pi=f1/(1−q) i ¯Zq, where fi= 1−(1−q)/summationtext jλj/parenleftBig O(i) j− /an}bracketle{t/an}bracketle{tOj/an}bracketri}ht/an}bracketri}htq/parenrightBig k/summationtext jpq j, and¯Zqis the partition function. Two rather unpleasant facts are t hus faced, namely, •piexplicitly depends upon the probability distribution (sel f-reference). •The Hessian of Fis not diagonal. In this work we have devised a transformation from the origin al set of Lagrange multi- pliers {λj}to a new set {λ′ j}such that •Self-reference is avoided. •The Hessian of Fbecomes diagonal. As a consequence, all calculations, whether analytical or n umerical, become much simpler than in [1], as illustrated with reference to several simple examples. The primed multipliers λ′ j=λj/summationtext ipq i incorporate the pi1in their definition. Since one solves directly for the primed multipliers , such a simple step considerably simplifies the TMP treatment . Finally, we remark on the fact that the two sets of multipliers lead to thermodynamical rel ationships that involve identical intensive quantities (45). 1that maximize the entropy 15ACKNOWLEDGMENTS The financial support of the National Research Council (CONI CET) of Argentina is gratefully acknowledged. F. Pennini acknowledges financia l support from UNLP, Argentina. 16REFERENCES [1] C. Tsallis, R. S. Mendes, and A. R. Plastino, Physica A 261(1998) 534. [2] C. Tsallis, Braz. J. of Phys. 29(1999) 1, and references therein. See also http://www.sbf.if.usp.br/WWW pages/Journals/BJP/Vol129/Num1/index.htm [3] C. Tsallis, Chaos, Solitons, and Fractals 6(1995) 539, and references therein; an updated bibliography can be found in http://tsallis.cat.cbpf.br/ biblio.htm [4] C. Tsallis, Physics World 10 (July 1997) 42. [5] A.R. Plastino and A. Plastino, in Condensed Matter Theories , Volume 11, E. Lude˜ na (Ed.), Nova Science Publishers, New York, USA, p. 341 (1996) . [6] A. R. Plastino and A. Plastino, Braz. J. of Phys. 29(1999) 79. [7] E. T. Jaynes in Statistical Physics , ed. W. K. Ford (Benjamin, NY, 1963); A. Katz, Statistical Mechanics , (Freeman, San Francisco, 1967). [8] C. Tsallis, J. Stat. Phys. 52(1988) 479. [9] E. Fick and G. Sauerman, The quantum statistics of dynamic processes (Springer-Verlag, Berlin, 1990). [10] F. Pennini, A. R. Plastino and A. Plastino, Phys. Lett. A 208(1995) 309. [11] F. Pennini, A. R. Plastino and A. Plastino, Physica A 258(1998) 446. [12] R. P. Di Sisto, S. Mart´ ınez, R. B. Orellana, A. R. Plasti no, A. Plastino, Physica A 265 (1999) 590. [13] A. Plastino and A. R. Plastino, Braz. J. of Phys. 29(1999) 50. [14] M. Portesi, A. Plastino and C. Tsallis, Physical Review E 52(1995) R3317. [15] S. Mart´ ınez, F. Pennini, and A. Plastino, Physica A (2000), in press. 17

arXiv:physics/0003099v1 [physics.plasm-ph] 29 Mar 20001 RAPID ACCELERATION OF ELECTRONS IN THE MAGNETOSPHERE BY FAST-MODE MHD WAVES Danny Summers and Chun-yu Ma1 Department of Mathematics and Statistics, Memorial Univer sity of Newfoundland, St John’s, Canada Short title: ACCELERATION OF ELECTRONS BY FAST-MODE WAVES2 Abstract. During major magnetic storms enhanced fluxes of relativisti c electrons in the inner magnetosphere have been observed to correlate w ith ULF waves. The enhancements can take place over a period of several hours. I n order to account for such a rapid generation of relativistic electrons, we examine th e mechanism of transit-time acceleration of electrons by low-frequency fast-mode MHD w aves, here the assumed form of ULF waves. Transit-time damping refers to the resona nt interaction of electrons with the compressive magnetic field component of the fast-mo de waves via the zero cyclotron harmonic. In terms of quasi-linear theory, a kine tic equation for the electron distribution function is formulated incorporating a momen tum diffusion coefficient representing transit-time resonant interaction between e lectrons and a continuous broad band spectrum of oblique fast-mode waves. Pitch angle scatt ering is assumed to be sufficiently rapid to maintain an isotropic electron distrib ution function. It is further assumed that there is a substorm-produced population of ele ctrons with energies of the order of 100 keV. Calculations of the acceleration times cales in the model show that fast-mode waves in the Pc4 to Pc5 frequency range, with t ypically observed wave amplitudes (∆ B= 10–20 nT), can accelerate the seed electrons to energies of order MeV in a period of a few hours. It is therefore concluded that t he mechanism examined in this paper, namely, transit-time acceleration of electr ons by fast-mode MHD waves, may account for the rapid enhancements in relativistic elec tron fluxes in the inner magnetosphere that are associated with major storms.3 1. Introduction There is much current interest in the rapid enhancements of r elativistic ( >MeV) electrons in the Earth’s inner magnetosphere (3 ≤L≤6) taking place over tens of minutes or a few hours during major magnetic storms [e.g., Baker et al. , 1998a; Rostoker et al., 1998; Liu et al. , 1999; Hudson et al. , 1999,2000]. Part of this interest is due to the fact that relativistic electrons appearing near geostatio nary orbit ( L= 6.6) constitute a potential hazard to operational spacecraft [e.g., Baker et al., 1997]. These relativistic electrons are sometimes colloquially referred to as “kille r electrons.” Rapid energetic electron enhancements have been observed to correlate clos ely with ULF waves in the Pc4 (7–22 mHz) or Pc5 (2–7 mHz) frequency ranges. It is the refore reasonable to examine the possible role that ULF waves may have in genera ting the relativistic electron flux enhancements. Liu et al. [1999] have formulated an acceleration mechanism comprising magnetic pumping by ULF waves, while Hudson et al. [1999,2000] have proposed a drift-resonant acceleration mechanism involvi ng enhanced ULF waves, modeled by a three dimensional global MHD simulation of the J anuary 10-11, 1997, conoral-mass-ejection-driven magnetic cloud event. Notw ithstanding these studies, the acceleration mechanism of relativistic electrons in the in ner magnetosphere is not yet fully understood. It is the purpose of the present paper to ex amine the role of ULF waves in accelerating electrons in the magnetosphere from a new standpoint. Here we examine the “transit-time acceleration” of electrons by low-frequency, oblique, fast-mode (magnetosonic) MHD waves. Transit-time acceler ation in association with4 “transit-time damping” has been studied, for instance, by Stix[1962], Fisk[1976], and Achterberg [1981]. The basic physical mechanism of transit-time dampi ng, which is a resonant form of Fermi acceleration and can be regarded as th e magnetic analogue of Landau damping, is discussed in detail by Miller [1997]; the name transit-time damping arises because the gyroresonance condition defining the pro cess can be expressed as λ/bardbl/v/bardbl≈T, where v/bardblis the parallel component of particle velocity, λ/bardblis the parallel wavelength, and Tis the wave period. Thus the wave-particle interaction is st rongest when the particle transit-time across the wave compression is approximately equal to the period. It is the compressive magnetic field component of the fast-mode wave that allows for the effect of transit-time damping [ Fisk, 1976; Achterberg , 1981; Miller et al. , 1996]. While transit-time damping has been utilized as a mec hanism for accelerating energetic particles in the interplanetary medium [ Fisk, 1976], for accelerating electrons in solar flares [ Miller et al. , 1996; Miller , 1997], and for accelerating cosmic ray particles [Schlickeiser and Miller , 1998], it has not been examined as a possible acceleration mechanism of electrons in the magnetosphere. It will be show n in this paper, in fact, that transit-time damping of fast-mode MHD waves (here the a ssumed form of ULF waves) is a viable mechanism for generating the aforementio ned rapid enhancements of relativistic electrons in the inner magnetosphere. It will be assumed that electrons of energies ∼100 keV that are injected near geosynchronous orbit as a resu lt of substorm activity [e.g., Baker et al. , 1989, 1998b] form the source population for the relativist ic (>MeV) electrons that are subsequently observed. The structure of ULF waves in the Earth’s magnetosphere is co mplex. Broadly,5 MHD waves in the dipole magnetosphere are characterized by “ toroidal” and “poloidal” modes, and, in general, these modes are coupled. Toroidal mo des relate to transverse ULF waves propagating along field lines, while poloidal mode s relate to global compressional waves associated with radial oscillations o f the field lines. Field line resonance (FLR) theory describes the toroidal pulsations a s transverse Alfv´ en waves standing on dipole flux tubes with fixed ends in the ionosphere . Extensive theory has accumulated pertaining to field line resonances, global compressional modes, and associated wave excitation mechanisms [e.g., see Kivelson and Southwood , 1986, Krauss-Varban and Patel , 1988, Lee and Lysak , 1989, and references therein]. Anderson et al. [1990] give a historical review of observations of ULF waves in the magnetosphere and also present results of a statistical study of Pc3–5 puls ations during the period from August 24, 1984, to December 7, 1985. It should be emphas ized here that unlike ULF waves observed during relatively quiet magnetic condit ions, storm-associated ULF waves characteristically have large compressional compon ents. It is these components that engender the transit-time acceleration mechanism pre sented in this paper. With respect to the typical motion of energetic electrons in the inner magnetosphere, these electrons gyrate around their magnetic field line and b ounce back and forth along their field line between mirror points, while executing an ea stward drift about the Earth. Consequently, since in the interests of rendering th e theory presented herein tractable we shall assume a constant background magnetic fie ld, our formulation applies to compressional MHD waves interacting with electrons that mirror relatively close to the equator.6 We are concerned with the fast-mode MHD branch in plasma wave theory [e.g., Swanson , 1989] and, in particular, with fast-mode waves on the ω≪Ωisection of the branch, where Ω iis the proton gyrofrequency. Such waves have the dispersion relation, ω=kvA, (1) where ω, k, and vAare the wave frequency, the wave number, and the Alfv´ en spee d, respectively. In general, the condition for gyroresonant i nteraction between electrons and a wave of frequency ωis ω−k/bardblv/bardbl=n|Ωe|/γ, (2) where v/bardblis the electron parallel velocity component , k/bardbl=kcosθis the parallel wave number, θis the wave propagation angle, |Ωe|is the electron gyrofrequency, γ is the Lorentz factor, and n(= 0,±1,±2,···) denotes the cyclotron harmonic. The compressive component of the wave magnetic field can interac t with electrons through then= 0 resonance [e.g., Miller et al. , 1996], in which case (2) reduces to ω=k/bardblv/bardbl. (3) Equation (3) is the gyroresonance condition that defines tra nsit-time damping [ Stix, 1962;Fisk, 1976; Achterberg , 1981]. From (1) and (3) it follows that v/bardbl=vA/cosθ, (4) from which follows the important necessary threshold condi tion for resonance, v > v A, (5)7 where vis the particle speed. Condition (5) states that for electro ns with any pitch- angle interacting with fast-mode MHD waves propagating at a ny angle θ, resonance is only possible for electrons with speeds exceeding the Alf v´ en speed. The equivalent minimum-energy condition can be conveniently written, E > E min, Emin= (1−β2 A)−1/2−1≈β2 A/2, (6) where Eis the electron kinetic energy in units of rest-mass energy a ndβAis the Alfv´ en speed in units of the speed of light. Values for the parameter βAand the minimum energy Eminthat are representative of the inner magnetosphere are give n in Table 1. In Table 1 Table 1, we set N0= 10 cm−3as the particle number density in the inner magnetosphere outside the plasmasphere, and we use the equatorial (dipole ) magnetic field value B0= 3.12×10−5/L3T. With regard to the background electron population, in ord er for the fast-mode waves to accelerate a small fraction of the electrons in the tail of the distribution rather than to produce a bulk heating of the pop ulation, it is required that vA> vth, where vthis a characteristic thermal speed. Taking the background el ectron temperature in the magnetosphere to be Te/tildewide<1 eV, we find from Table 1 that the required condition vA> vthis satisfied. In addition, since we are assuming a substorm-p roduced source of electrons with energies ∼100 keV, Table 1 shows that the minimum-energy condition (6) is well satisfied; that is, the condition v≫vAholds. The analysis of transit-time damping of fast-mode waves tha t is carried out in this paper and presented in the following section is based on quas i-linear theory; this is8 an approximation that requires justification. In numerical simulations, Miller [1997] has found that quasi-linear theory provides an accurate des cription of transit-time acceleration even when the energy density of the fast-mode w ave turbulence is almost equal to the ambient magnetic field energy density ((∆ B/B 0)2/tildewide<1). Thus, although the analysis presented here is based formally on “s mall-amplitude” turbulence ((∆B/B 0)2≪1), the results are applicable to the large-amplitude ULF wa ves typically observed during magnetic storms. 2. Electron Momentum Diffusion Equation Consider energetic charged particles in a uniform magnetic field with superimposed small-amplitude plasma turbulence. By using the quasi-lin ear approximation [ Kennel and Engelmann , 1966; Lerche, 1968], the pitch angle averaged particle distribution function F(p, t) can be shown to satisfy the kinetic (Fokker-Planck) equati on ∂F ∂t=1 p2∂ ∂p/parenleftBigg p2D(p)∂F ∂p/parenrightBigg , (7) where D(p) =1 2/integraldisplay1 −1Dppdµ. (8) In (7) and (8), pis the relativistic momentum of the particle in units of rest -mass momentum given by p=γv/c, where vis the particle speed and γ= (1−v2/c2)−1/2= (1 +p2)1/2is the Lorentz factor, with cbeing the speed of light; tis time; µis the cosine of the pitch angle; and Dppis the momentum diffusion coefficient, which depends on the properties of the wave turbulence. In the derivation of (7), it has been assumed that9 the rate of pitch angle scattering is large enough to isotrop ize the distribution function, and the pitch angle has been eliminated from the equation by a veraging with respect to µ. The distribution function Fis normalized so that 4 πp2F(p, t)dpis the number of the particles per unit volume in the momentum interval dp. It has also been assumed in deriving (7) that there are no energy losses, that no particl es escape from the system, and that there are no additional particle sources or sinks. Associated with the (averaged) momentum diffusion coefficien tD(p) in (7) and (8) is the acceleration timescale, TA=p2/D(p). (9) In this paper we consider two forms for the transit-time damp ing diffusion coefficient, given by Miller et al. [1996] and Schlickeiser and Miller [1998], respectively. Assuming a continuous spectrum of oblique, low-frequency ( ω≪Ωi), fast-mode waves and assuming isotropic turbulence and integrating over wave pr opagation angle, Miller et al. [1996] obtain a diffusion coefficient Dppfor transit-time damping of fast-mode waves by electrons that can be expressed in the form Dpp=π 16ΩiR/parenleftBiggc∝angbracketleftk∝angbracketright Ωi/parenrightBigg γ2ββ2 A/parenleftBigg 1−β2 A β2µ2/parenrightBigg(1−µ2)2 |µ|, (10) where Ω i=eB0/(mic) is the proton gyrofrequency, with B0being the ambient magnetic field strength, mibeing the proton rest mass, and ebeing the electronic charge; R= (∆B/B 0)2is the ratio of the turbulent wave energy to magnetic field energy, with ∆ Bbeing the average fast-mode wave amplitude; c∝angbracketleftk∝angbracketright/Ωiis the mean dimensionless wave number of the wave spectrum; β=v/c; and βA=vA/cwhere10 vA=B0/(4πN0mi)1/2is the Alfv´ en speed, with N0being the particle number density. Substituting (10) into (8) and setting x=βA/β (11) yields the result D(p) =π 16ΩiR/parenleftBiggc∝angbracketleftk∝angbracketright Ωi/parenrightBigg γ2ββ2 Ag(x), (12) where g(x) = (1 + 2 x2) loge/parenleftbigg1 x/parenrightbigg +x2+x4 4−5 4, (13) forx <1. The function g(x) can be regarded as an efficiency factor [ Miller et al. , 1996], which relates to the velocity-dependent fraction of electrons that can resonate with fast-mode waves having the assumed spectrum; g(x) = 0 for x≥1, and in the limit as β→1,g(x)→loge(1/βA)−5/4, approximately, since βA≪1. Therefore, for values of βAappropriate to the Earth’s magnetosphere (see Table 1) for 3 ≤L≤6.6, withN0= 10 cm−3, the function g(x) approaches values in the range from 2.4 to 4.7 for highly relativistic electrons. Setting ∝angbracketleftk∝angbracketright=∝angbracketleftω∝angbracketright/vAin (12), where ∝angbracketleftω∝angbracketrightis the mean angular frequency (rad/s), from (9) and (12) we find that the acceleration timescale TAcan be written as TA=8 π21 ∝angbracketleftfw∝angbracketright1 R1 xg(x), (14) where ∝angbracketleftfw∝angbracketright=∝angbracketleftω∝angbracketright/2πis the mean wave frequency (in millihertz). Later in this sec tion and in the numerical results presented below, we shall find it convenient to use the previously introduced dimensionless kinetic energy E=Ek/(mec2) =γ−1, where Ek11 is the electron kinetic energy and meis the electron rest mass; we shall require the relation, β= [E(E+ 2)]1/2/(E+ 1). (15) Schlickeiser and Miller [1998] assume that the fast-mode wave turbulence is isotropic and Kolmogorov-like, with a power law spectral en ergy density distribution in wave number k. Specifically, the spectral energy density Wis assumed to take the form W(k)∝k−q, k > k min, (16) where q(>1) is the spectral index and kminis some minimum wave number. Corresponding to the Kolmogorov-like spectrum (16), the mo mentum diffusion coefficient Dpp, as given by Schlickeiser and Miller [1998], can be written Dpp=π 4(q−1)ΩiR/parenleftBiggckmin Ωi/parenrightBiggq−1/parenleftbiggme mi/parenrightbiggq−2 γ(γβ)q−1β2 Ah(µ, x), (17) where h(µ, x) =H(|µ| −x)1−µ2 |µ|/bracketleftBigg 1 +x2 µ2/bracketrightBigg/bracketleftBigg (1−µ2)/parenleftBigg 1−x2 µ2/parenrightBigg/bracketrightBiggq/2/integraldisplay∞ λJ2 1(s) s1+qds, (18) with λ=/parenleftBiggckmin Ωi/parenrightBigg/parenleftbiggme mi/parenrightbigg γβ(1−µ2)1/2/parenleftBigg 1−x2 µ2/parenrightBigg1/2 . (19) In (17)–(19), ckmin/Ωiis the minimum dimensionless wave number of the wave spectru m, His the Heaviside unit function, and J1is the Bessel function of the first kind of order unity. Substitution of (17) into (8) yields D(p) =π 4(q−1)ΩiR/parenleftBiggckmin Ωi/parenrightBiggq−1/parenleftbiggme mi/parenrightbiggq−2 γ(γβ)q−1β2 AI(x, βA, kmin), (20)12 where I(x, βA, kmin) =  c1(q) loge/parenleftBig 1 x/parenrightBig ,1< q≤2 c2(q)(γβ)2−q/parenleftBig ckmin Ωi/parenrightBig2−q/parenleftBig me mi/parenrightBig2−qloge/parenleftBig 1 x/parenrightBig , q > 2(21) with c1(q) = 21−qq 4−q2Γ(q)Γ(2−q/2) Γ3(1 +q/2),1< q < 2 c1(2) = 3 /4, (22) c2(q) =2q2−3q+ 4 4q(2q−3), q > 2 where Γ is the gamma function. In (20) we set the minimum dimensionless wave number kmin= 2πfmin/vA, where fminis the minimum wave frequency (in millihertz). From (9) and ( 20) the acceleration timescale for transit-time damping associated with the wav e spectrum (16) is found to be TA=8 q−11 (2π)q1 Ωi1 R/parenleftBiggΩi fmin/parenrightBiggq−1/parenleftbiggmi me/parenrightbiggq−21 γq−21 x3−qI, (23) where Iis given by (21) with ckmin/Ωireplaced by ( fmin/Ωi)(2π/βA). It should be noted that while the transit-time damping diffus ion coefficients (12) and (20) may appear different, they are, in fact, approximate ly equivalent. Since the coefficient (12) employs an average wave frequency, while coe fficient (20) employs a minimum frequency and a Kolmogorov spectral index, it is con venient to utilize both coefficients in order to retain some flexibility in constructi ng the acceleration timescale profiles and comparing the results with observations. Finally, it is useful to relate the mean energy change of a par ticle∝angbracketleft˙E∝angbracketrightto the13 acceleration timescale TA. Associated with the momentum diffusion process given by (7) – (8), the mean energy change [ Tsytovich , 1977; Achterberg , 1981] is given by ∝angbracketleft˙E∝angbracketright=1 p2∂ ∂p/parenleftBig βp2D(p)/parenrightBig ≈σβ pD(p), (24) and, hence, by (9) we derive the result, ∝angbracketleft˙E∝angbracketright ≈σβp TA=σE(E+ 2) (E+ 1)TA, (25) where σis a factor such that σ= 4 corresponding to the diffusion coefficient (12), and σ= 2 + qcorresponding to (20). The approximation in the second line of (24) follows from the fact that, for the electron energies considered in t his paper, the functions g andIvary only slightly with x. 3. Numerical results The acceleration timescale TAdepends on a number of parameters. Both results (14) and (23) depend on the average wave amplitude ∆ B, the electron kinetic energy E, the background plasma number density N0, and the location L. In addition, (14) depends on the mean wave frequency ∝angbracketleftfw∝angbracketright, while (23) depends on the the minimum wave frequency fminand the turbulence spectral index q. With regard to typical wave amplitudes of Pc-5 pulsations during major magnetic storms ,Barfield and McPherron [1978] and Engebretson and Cahill [1981] report ∆ B≈10 nT, while Higuchi et al. [1986] report typical values ∆ B≈70 – 90 nT corresponding to the maximum power spectral14 densities in the frequency range 5 – 12 mHz. Baker et al. [1998a] report ULF waves in the frequency range 2 –20 mHz having amplitudes ∆ B≈50 nT, rising to ∆ B≈200 nT at times. We assume a substorm-produced seed electron population wit h energies in the range 100 – 300 keV, which corresponds to the dimensionless k inetic energy Ein the approximate range 0 .2< E < 0.6. From result (25), it follows that electrons with energies in such a range accelerate to energies in the ra nge from 1 MeV to 2 MeV, approximately, over the timescale TA(E) where 0 .2< E < 0.6. In Figure 1, N0= 10 cm−3, and for the specified mean wave frequency ∝angbracketleftfw∝angbracketright= 10 mHz, curves are plotted showing TA, given by (14), as a function of energy E(eV), at the locations L= 3,4,5,6.6, for each of the wave amplitudes ∆ B= 10,20,50 nT; for reference, a (dashed) line is shown corresponding to a time of one day. In g eneral, for a fixed energy E, the timescale TAis seen to increase as the value of Ldecreases, and, as expected, TAdecreases as the wave amplitude ∆ Bincreases. Figure 1 indicates, in particular, that at L= 6.6, for the parameter values N0= 10 cm−3and∝angbracketleftfW∝angbracketright= 10 mHz, the timescales for accelerating seed electrons of energies ∼100 keV to energies ∼1 MeV are approximately 6 days, 1.5 days, and 5.8 hours correspond ing to the respective wave amplitudes ∆ B= 10,20,50 nT. The aforementioned respective times assume the approximate values 2 days, 12 hours, and 2 hours if the value o fN0= 1 cm−3is specified for the background plasma number density (N.B. It could be ar gued that N0= 10 cm−3 is too high a generic value for the background plasma number d ensity, and that N0= 1 cm−3is a more representative value).15 In Figure 2, we show the variation of TA, as given by (14), as a function of the wave amplitude ∆ B(nT), for a fixed value of particle energy, E= 1/4 (orβ= 0.6). The upper, middle, and lower panels of Figure 2 correspond re spectively to the mean wave frequencies ∝angbracketleftfw∝angbracketright= 2,10, and 22 mHz. The decrease in acceleration timescale with increase in mean frequency ∝angbracketleftfw∝angbracketright, as indicated by formula (14), is clearly shown in Figure 2. Figure 2 can be used as an illustration of the wave amplitudes ∆ B, at a given location L, and for a given mean wave frequency ∝angbracketleftfw∝angbracketright, that correspond to a particular timescale TAfor the generation of electrons of energies∼>1 MeV from seed electrons of energies∼>100 keV. In Table 1 , corresponding to (14), the required wave amplitudes ∆ Bare given that correspond to a timescale T0= 10 hours for this generation process, corresponding to the mean wave frequen cies∝angbracketleftfw∝angbracketright= 2,10,and 22 mHz at the specified locations, and with N0= 10 cm−3. In particular, we note that at L= 6.6, corresponding to the respective mean wave frequencies ∝angbracketleftfw∝angbracketright= 10,22 mHz, the required wave amplitudes are ∆ B= 39,26 nT; corresponding to N0= 1 cm−3, these respective wave amplitudes are ∆ B= 22,15 nT. In Figure 3, for N0= 10 cm−3, the acceleration timescale TA(sec) given by (23) is plotted as a function of particle energy E, for the minimum wave frequencies fmin= 2,10 mHz, for the spectral indices q= 3/2,5/3,5/2,4, at each of the locations L= 3,6.6, and for the mean wave amplitude ∆ B= 20 nT. As can be observed from Figure 3, the timescale TAdecreases as both fminandLincrease; lower values of TAare also generally favoured by lower q-values. The curves in Figure 3 show that, corresponding to ∆ B= 20 nT, the timescale TAatL= 6.6 is of the order of a few hours,16 for values of qin the range 3 /2< q < 5/3. In Table 2 , values of the wave amplitudes ∆B(nT) are given that correspond to the value of the accelerati on timescale TAgiven by (23) equal to 10 hours, for the specified values of fmin,q, and L, with E= 1/4 (or β= 0.6), and N0= 10 cm−3. Thus, Table 2 , which corresponds to (23), effectively gives the required wave amplitudes ∆ Bto generate relativistic (∼>1 MeV) electrons from seed (∼>100 keV) electrons in a timescale of 10 hours, for the specifie d values of the remaining parameters. For instance, at L= 6.6, for a minimum wave frequency fmin= 10 mHz, and with qin the range 3 /2< q < 5/3, the required wave amplitudes are in the range 5 .4 nT<∆B <7.1 nT. 4. Discussion The present paper is a new examination of ULF waves as a possib le rapid acceleration mechanism of electrons in the inner magnetosp here during storms. Specifically, we take the assumed form of ULF waves to be fast- mode (magnetosonic) MHD waves, and analyze the mechanism of transit-time accele ration of electrons under magnetic storm conditions. We assume that the seed ele ctrons in the process have energies in the range 100 – 300 keV, and are produced by su bstorm activity. In accordance with quasi-linear theory and a test particle app roach, a simple model kinetic equation (7) is formulated in which momentum diffusion is due to the gyroresonant transit-time interaction between electrons and fast-mode MHD turbulence. A continuous broad-band spectrum of oblique fast-mode waves is assumed, and it is further supposed that pitch-angle scattering is sufficiently rapid to maintai n an isotropic particle17 distribution function. The model calculations applied to t he inner magnetosphere show that the mechanism under consideration, namely transit-ti me damping of fast-mode MHD waves, can accelerate source electrons with energies 10 0 – 300 keV to relativistic electrons with energies exceeding 1 MeV, in a timescale of a f ew hours if the wave amplitudes are of the order of ∆ B= 10 – 20 nT. Since observed amplitudes of ULF waves during storm-time are in this range, it is concluded th at transit-time damping of fast-mode MHD waves, as the agent of ULF wave activity, could play an important role in generating the observed increases of relativistic elect rons during major storms. We note that the models formulated by Liu et al. [1999] and Hudson et al. [1999a, b] also show that ULF waves could be instrumental in energizin g relativistic electrons under storm conditions, though their approaches are quite d ifferent from that adopted here. Liu et al. [1999] formulate an acceleration mechanism comprising mag netic pumping with global ULF waves as the energy source and pitch- angle scattering as the catalyst, while Hudson et al. [1999a, b] propose a mechanism, further investigated byElkington et al. [1999], in which electrons are adiabatically accelerated t hrough a drift-resonance via interaction with toroidal-mode ULF wa ves. We caution that the calculations in the present paper are bas ed on an approximate, timescale analysis. A more complete investigation of elect ron acceleration by transit-time damping of fast-mode waves entails the full solution of a kin etic equation of the form (7), appropriately modified by the inclusion of terms repres enting particle and energy losses under storm conditions. Aside from the aforementioned ULF wave mechanisms, other en ergization18 mechanisms have been previously proposed to account for rel ativistic electron enhancements during storms, e.g., see Li et al. [1997] and Summers and Ma [1999] for brief summaries. Moreover, various types of storm-related energetic electron events have been observed [e.g., Baker et al. , 1997; 1998 b;Reeves , 1998; Reeves et al. , 1998]. The rapid acceleration mechanism presented in this paper ap pears well suited to major storms that produce coherent global oscillations in the mag netosphere in the Pc-4 to Pc-5 frequency range. In contrast, the gradual acceleratio n process occurring over a few days involving gyroresonant electron-whistler-mode chor us interaction [ Summers et al. , 1998, 1999; Summers and Ma , 1999; see also Ma and Summers , 1998] is expected to apply to moderate storms having long-lasting recovery phas es. Acknowledgments. This work is supported by the Natural Sciences and Engineeri ng Research Council of Canada under Grant A-0621. Additional s upport is acknowledged from the Dean of Science, Memorial University of Newfoundland.19 References Achterberg, A., On the nature of small amplitude Fermi accel eration, Astron. Astrophys., 97 , 259, 1981. Anderson, B. J., et al., A statistical study of Pc 3–5 pulsati ons observed by the AMPTE/CCE magnetic fields experiment 1. Occurrence distributions, J. Geophys. Res., 95 , 10495, 1990. Baker, D. N., et al., Relativistic electrons near geostatio nary orbit: evidence for internal magnetospheric acceleration, Geophys. Res. Lett., 16 , 559, 1989. Baker, D. N., et al., Recurrent geomagnetic storms and relat ivistic electron enhancements in the outer magnetosphere: ISTP coordinated measurements ,J. Geophys. Res., 102 , 14141, 1997. Baker, D. N., et al., A strong CME-related magnetic cloud int eraction with Earth’s magnetosphere: ISTP observations of rapid relativistic el ectron acceleration on May 15, 1997,Geophys. Res. Lett., 25 , 2975, 1998a. Baker, D. N., et al., Coronal mass ejections, magnetic cloud s, and relativistic electron events: ISTP, J. Geophys. Res., 103 , 17279, 1998b. Barfield, J. N., and R. L. Mcpherron, Storm time Pc 5 magnetic p ulsations observed at synchronous orbit and their correlation with the partial ri ng current, J. Geophys. Res., 83, 739, 1978. Elkington, S. R., et al., Acceleration of relativistic elec trons via drift-resonant interaction with toroidal-mode Pc-5 ULF oscillations, Geophys. Res. Lett., 26 , 3273, 1999.20 Engebretson, M. J., and L. J. Cahill, Jr., Pc 5 pulsations obs erved during the June 1972 geomagnetic storm, J. Geophys. Res., 86 , 5619, 1981. Fisk, L. A., On the acceleration of energetic particles in th e interplanetary medium, J. Geophys. Res., 81 , 4641, 1976. Higuchi, T., et al., Harmonic structure of compressional Pc -5 pulsations at synchronous orbit, Geophys. Res. Lett., 13 , 1101, 1986. Hudson, M. K., et al., Simulation of radiation belt dynamics driven by solar wind variations, Sun-Earth Plasma Connections ,Geophys. Monog. 109 , edited by J. L. Burch, R. L. Carovillano, S. K. Antiochos, p. 171, A.G.U., Washington, 1 999a. Hudson, M. K., et al., Increase in relativistic electron flux in the inner magnetosphere: ULF wave mode structure, Adv. Space Res. , in press, 1999b. Kennel, C. F., and F. Engelmann, Velocity space diffusion fro m weak plasma turbulence in a magnetic field, Phys. Fluids, 9 , 2377, 1966. Kivelson, M. G., and D. J. Southwood, Coupling of global magn etospheric MHD eigenmodes to field line resonances, J. Geophys. Res., 91 , 4345, 1986. Krauss-Varban, D., and V. L. Patel, Numerical analysis of th e coupled hydromagnetic wave equations in the magnetosphere, J. Geophys. Res., 93 , 9721, 1988. Lee, D. -H., and R. L. Lysak, Magnetospheric ULF wave couplin g in the dipole model: the impulsive excitation, J. Geophys. Res., 94 , 17097, 1989. Lerche, I., Quasilinear theory of resonant diffusion in a mag neto-active relativistic plasma, Phys. Fluids, 11 , 1720, 1968.21 Li, X., et al., Multi-satellite observations of the outer zo ne electron variation during the November 3 – 4, 1993, magnetic storm, J. Geophys. Res., 102 , 14123, 1997. Liu, W. W, et al., Internal acceleration of relativistic ele ctrons by large-amplitude ULF pulsations, J. Geophys. Res., 104 , 17391, 1999. Ma, C.-Y., and D. Summers, Formation of power-law energy spe ctra in space plasmas by stochastic acceleration due to whistler-mode waves, Geophys. Res. Lett., 25 , 4099, 1998. Miller, J. A., et al., Stochastic acceleration by cascading fast mode waves in impulsive solar flares, Astrophys. J., 461 , 445, 1996. Miller, J. A., Electron acceleration in solar flares by fast m ode waves: quasi-linear theory and pitch-angle scattering, Astrophys. J., 491, 939, 1997. Reeves, G. D., Relativistic electrons and magnetic storms: 1992 – 95, Geophys. Res. Lett., 25 , 1817, 1998. Reeves, G. D., et al., The relativistic electron response at geosynchronous orbit during the January 1997 magnetic storm, J. Geophys. Res., 103 , 17559, 1998. Rostoker, G., et al., On the origin of relativistic electron s in the magnetosphere associated with some geomagnetic storms, Geophys. Res. Lett., 25 , 3701, 1998. Schlickeiser, R., and J. A. Miller, Quasi-linear theory of c osmic ray transport and acceleration: the role of oblique magnetohydrodynamic waves and transit- time damping, Astrophys. J., 492 , 352, 1998. Stix, T. H., The Theory of Plasma Waves , McGraw Hill, New York, 1962. Summers, D., et al., Relativistic theory of wave-particle r esonant diffusion with application to electron acceleration in the magnetosphere, J. Geophys. Res., 103 , 20487, 1998.22 Summers, D., et al., A model for stochastic acceleration of e lectrons during geomagnetic storms, Adv. Space Res., in press, 1999. Summers, D., and C.-Y. Ma, A model for generating relativist ic electrons in the Earth’s inner magnetosphere based on gyroresonant wave-particle intera ctions, J. Geophys. Res., in press, 1999. Swanson, D. G., Plasma Waves , Academic, San Diego, Calif., 1989. Tsytovich, V. N., Theory of Turbulent Plasma , Plenum, New York, 1977. Danny Summers and Chun-yu Ma, Department of Mathematics and Statistics, Memorial Univer sity of Newfoundland, St John’s, Newfoundland, A1C 5S7, Canada (e-mail: dsummers @math.mun.ca, cyma@math.mun.ca) Received November 5, 1999; revised Febrary 4, 2000; accepte d February 23, 2000. 1On leave from Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, People’s Republic of China.23 Figure 1. Acceleration timescale TA(sec) as given by (14), as a function of the electron kinetic energy E(eV), at the indicated locations L, for the average wave amplitudes ∆B= 10,20,50 nT. The mean wave frequency ∝angbracketleftfw∝angbracketright= 10 mHz. Figure 2. Acceleration timescale TA(sec) as given by (14), as a function of the average wave amplitude ∆ B(nT), at the indicated locations L, for the mean wave frequencies ∝angbracketleftfw∝angbracketright= 2,10,22 mHz. The parameter β= 0.6. Figure 3. Acceleration timescale TA(sec) as given by (23), as a function of the electron kinetic energy E(eV), at the indicated locations L, for the given values of the spectral index q, and for the minimum wave frequencies fmin= 2,10 mHz. The average wave amplitude ∆ B= 20 nT.24 Table 1. The required average wave amplitudes ∆ B(nT), as calcu- lated from (14), that correspond to an acceleration timesca leTAof about 10 hours, at the given locations L, and for the mean wave frequencies ∝angbracketleftfw∝angbracketright= 2,10,22 mHz. The parameter β= 0.6. Also given are the ambi- ent magnetic field strength B0(10−7T), the dimensionless Alfv´ en speed βA=vA/c, and the minimum energy Emin(eV). The latter value is calcu- lated from Emin(eV) = 512 ×103Eminwhere Eminis given by (6). ∝angbracketleftfW∝angbracketright= 2∝angbracketleftfW∝angbracketright= 10 ∝angbracketleftfW∝angbracketright= 22 L B 0 βA Emin(eV) ∆ B ∆B ∆B 3 11.6 2 .67×10−2183 427 191 129 4 4.85 1 .13×10−233 222 99 67 5 2.50 5 .75×10−39 147 66 44 6.6 1.10 2 .53×10−32 87 39 2625 Table 2. The required average wave amplitudes ∆ B(nT), as calculated from (23), that correspond to an acceleration timescale TAof about 10 hours, at the given locations L, for the given values of the spectral index q, and the minimum wave frequencies fmin= 2,10 mHz. The parameter β= 0.6. fmin= 2 mHz fmin= 10 mHz L q = 1.5q= 5/3q= 2.5q= 4.0 q= 1.5q= 5/3q= 2.5q= 4.0 3 11 24 201 174 7.2 14 90 78 4 9.5 18 115 100 6.3 11 51 45 5 8.8 15 76 66 5.9 8.8 34 30 6.6 8.1 12 47 40 5.4 7.1 21 18

arXiv:physics/0003100v1 [physics.bio-ph] 30 Mar 2000Auditory sensitivity provided by self-tuned critical oscillations of hair cells S´ ebastien Camalet∗, Thomas Duke∗†‡, Frank J¨ ulicher∗and Jacques Prost∗ ∗Institut Curie, PhysicoChimie Curie, UMR CNRS/IC 168, 26 rue d’Ulm, 75248 Paris Cedex 05, France †Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Den mark ‡Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, U K We introduce the concept of self-tuned criticality as a general mechanism for signal detection in sensory systems. In the case of hearing, we argue that active amplification of faint sounds is provided by a dynam- ical system which is maintained at the threshold of an oscillatory instability. This concept can account for the exquisite sensitivity of the auditory system and its wide dynamic range, as well as its capacity to respond selectively to different frequencies. A spe- cific model of sound detection by the hair cells of the inner ear is discussed. We show that a collection of motor proteins within a hair bundle can generate os- cillations at a frequency which depends on the elastic properties of the bundle. Simple variation of bun- dle geometry gives rise to hair cells with character- istic frequencies which span the range of audibility. Tension-gated transduction channels, which primar- ily serve to detect the motion of a hair bundle, also tune each cell by admitting ions which regulate the motor protein activity. By controlling the bundle’s propensity to oscillate, this feedback automatically maintains the system in the operating regime where it is most sensitive to sinusoidal stimuli. The model explains how hair cells can detect sounds which carry less energy than the background noise. Detecting the sounds of the outside world imposes stringent demands on the design of the inner ear, where the transduction of acoustic stimuli to electrical signals takes place [1]. The hair cells within the cochlea, which act as mechanosensors, must each be responsive to a particular frequency component of the auditory input. Moreover, these sensors need the utmost sensitivity, since the weakest audible sounds impart an energy, per cycle of oscillation, which is no greater than that of thermal noise [2]. At the same time, they must operate over a wide range of volumes, responding and adapting to intensities which vary by many orders of magnitude. Clearly, some form of non-linear amplification is necessary in sound de- tection. The familiar resonant gain of a passive elastic system is far from sufficient for the required demands, be- cause of the heavy viscous damping at microscopic scales [3]. Instead, the cochlea has developed active amplifica- tory processes, whose precise nature remains to be dis- covered. There is strong evidence that the cochlea contains force-generating dynamical systems which are capable ofexecuting oscillations of a characteristic frequency [4–1 0]. In general, such a system exhibits a Hopf bifurcation [11]: as the value of a control parameter is varied, the behavior abruptly changes from a quiescent state to self-sustained oscillations. When the system is in the immediate vicin- ity of the bifurcation, it can act as a nonlinear amplifier for sinusoidal stimuli close to the characteristic frequen cy. That such a phenomenon might occur in hearing was first proposed by Gold [3], more than 50 years ago. The idea was recently revived by Choe, Magnasco and Hudspeth [12], in the context of a specific model of the hair cell. No general analysis of the amplification afforded by a Hopf bifurcation has been provided, however, and no theory has been advanced to explain how proximity to the bi- furcation point might be ensured. In this paper, we provide both a generic framework which describes the known features of acoustic detec- tion, and a detailed discussion of the specific elements which could be involved in this detection. We first derive the general resonance and amplification behavior of a dy- namical system operating close to a Hopf bifurcation and emphasize that such a system is well-suited to the ear’s needs. In order for active amplification to work reliably, tuning to the bifurcation point is crucial. We introduce the concept of a self-tuned Hopf bifurcation which per- mits the favorable amplificatory properties of a dynami- cal instability to be obtained in a robust way. Self-tuning maintains the system in the proximity of the critical point and is achieved by an appropriate feedback mechanism which couples the output signal to the control parameter that triggers the bifurcation. The concept can explain several important features of the auditory sensor such as the frequency selectivity, high sensitivity and the abilit y to respond to a wide range of amplitudes. It can also explain the intrinsic nonlinear nature of sound detection [13,14] and the occurrence of spontaneous sound emission by the inner ear [9,10]. Furthermore, self-tuned criticali ty provides a framework for understanding the role of noise in the detection mechanism. The amplificatory process, which involves a limited number of active elements, intro- duces stochastic fluctuations, which adds to those caused by Brownian motion. We show that the response to weak stimuli can take advantage of this background activity. The proposed existence of a self-tuned Hopf bifurcation raises questions about the specific mechanisms involved: What is the physical basis of the dynamical system? 1How is the self-tuning realized? It might be expected that different organisms have evolved different apparatus to implement the same general strategy. In this paper, we restrict our specific discussion to the more primitive cochleae of non-mammalian vertebrates. We propose a model of the hair cell of the inner ear which accords with data from a wide variety of physiological experiments. The model incorporates a physical mechanism which al- lows motor proteins to generate spontaneous oscillations [15]. We find that molecular motors such as dyneins in the kinocilium or myosins in the stereocilia are natural candidates for the force generators involved in the ampli- fication of hair-bundle motion. Tension-gated transduc- tion channels in the stereocilia serve primarily to detect this motion, but also have a second function: by admit- ting ions which regulate the motor protein activity, they provide the self-tuning mechanism. I. GENERIC ASPECTS Amplification and frequency filtering of a Hopf bifurcation. We discuss the behavior of a dynamical system which is controlled by a parameter C. Above a critical value, C > C c, the system is stable; for C < C cit oscillates spontaneously. At the critical point (or Hopf bi - furcation) C=Ccthe system shows remarkable response and amplification properties which do not depend on the physical mechanism at the basis of the bifurcation. These generic properties can be described as follows. Since we are interested in the response to a periodic stimu- lus with frequency ν=ω/2π, we express the hair-bundle deflection x(t) by a Fourier series x(t) =/summationtextxneinωtwith complex amplitudes xn=x∗ −n. In the vicinity of the bifurcation, the mode n=±1 is dominant and the re- sponse to an externally-applied sinusoidal stimulus force f(t) =f1eiωt+f−1e−iωtcan be expressed in terms of a systematic expansion in x1. Symmetry arguments (see Appendix A) imply that the first nonlinear term is cubic: f1=Ax1+B |x1|2x1+... (1) where A(ω, C) and B(ω, C) are two complex functions. The bifurcation point is characterized by the fact that A vanishes for the critical frequency, A(ωc, Cc) = 0. For C < C cand no external force, the system oscillates with |x1|2≃∆2(Cc−C)/Cc, where ∆ is a characteristic amplitude. For C=Ccthe response to a stimulus at the critical frequency has amplitude |x1| ≃ |B|−1/3|f1|1/3. (2) This represents an amplified response at the critical fre- quency with a gain r=|x1| |f1|∼ |f1|−2/3(3)that becomes arbitrarily large for small forces. If the stimulus frequency differs from the critical fre- quency, the linear term in Eq. (1) is non-zero and can be expressed to first order as A(ω, C c)≃A1(ω−ωc). The dramatic amplification of weak signals, implied by Eq. (3), is maintained as long as this term does not exceed the cubic term in Eq. (1). If the frequency mismatch increases such that |ω−ωc| ≫ |f1|2/3|B|1/3/|A1|, the re- sponse becomes linear |x1| ≃|f1| |(ω−ωc)A1|(4) i.e. the gain is independent of the strength of the stimu- lus. 110100 r (nm/pN) 1 5 10ν (kHz)10 10 10 1010 1-2 -1 2 31010 1010 1 -2-12 |f | (pN)|x | (nm) 11 a b FIG. 1. Response to external forces near a Hopf bifurca- tion (a) Amplitude |x1|as a function of force |f1|at various driving frequencies ν(✸2 kHz, ×5 kHz, ✷10 kHz, + 13 kHz). (b) Gain ras a function of frequency νfor different amplitudes |f1|(✸0.01 pN, △0.05 pN, + 0.1 pN, ×0.5 pN, ✷1 pN). While the form of these curves is generic, the numer- ical values of force, amplitude and frequency depend on the physical nature of the dynamical system. The values given here correspond to the specific model model of a hair cell dis- cussed in the text, with parameters chosen to give a critical frequency of approximately 10 kHz. Thus the Hopf resonance acts as a sharply tuned high- gain amplifier for weak stimuli, and as a low-gain filter for strong stimuli. This generic behavior is illustrated in Fig. 1 with data obtained by numerical simulation. Laser interferometry measurements of the motion of the basilar membrane, when the live cochlea is stimulated by pure tones, display strikingly similar features [16,17] . 2In particular, the peak response as a function of force amplitude has been demonstrated to obey a power law |x1| ∼ |f1|0.4±0.2[17]. This strongly suggests that the membrane is being driven by a dynamical system which is poised at a Hopf bifurcation. Self-tuned critical oscillations. How does the sys- tem come to be so precisely balanced at the critical point?1The control parameter must be tuned to C≃Cc, otherwise the nonlinear amplification is lost. Moreover, the value of Ccdiffers for hair cells with different char- acteristic frequency. We propose a feedback mechanism which allows the dynamical system to operate automat- ically close to the bifurcation point, whatever its char- acteristic frequency. Without loss of generality, we as- sume that the control parameter decreases as long as the system does not oscillate. After some time, critical con- ditions are reached and spontaneous oscillations ensue. The onset of oscillations triggers an increase of the con- trol parameter which tends to restore stability. Hence the system converges to an operating point close to the bifurcation point. To illustrate this general idea, we con- sider the following simple feedback which changes Cin response to deflections x: 1 C∂C ∂t=1 τ/parenleftbiggx2 δ2−1/parenrightbigg (5) where δis a typical amplitude. If no external force is ap- plied, this feedback, after a relaxation time τ, tunes the control parameter to a value Cδfor which spontaneous oscillations with |x1| ≃δoccur. If δis small compared to the characteristic amplitude ∆, this is on the oscillating side close to the bifurcation, |Cδ−Cc|/Cc≃(δ/∆)2. Two modes of signal detection are possible: (i) For transient stimuli short compared to τthe system operates at Cδ. The amplitude |x1|shows the characteristic nonlinear re- sponse discussed above. (ii) For stimuli sustained over longer times, self-tuning maintains |x1| ≃δconstant for different stimulus amplitudes. This effect of the feedback represents a perfect adaptation mechanism. However, in the presence of noise, phase-locking of the response (to be discussed later) occurs as soon as an external stimulus is applied, and this can be detected. II. MODEL Mechanosensitivity and self-tuning mechanism provided by transduction channels. We demon- strate the general principles introduced above by devising a specific model for the amplification of acoustic stim- uli by hair bundles in non-mammalian vertebrates. A 1Self-tuning to a coexistence point has been discussed pre- viously in certain dynamic first-order transitions [18].schematic representation of a hair bundle is shown in Fig. 2. It consists of several stereocilia and a single kinocil- ium [19,20]. Transduction of hair bundle deflection to a chemical signal occurs via channels located near the tip of each stereocilium. Tip links which connect neigh- boring stereocilia are believed to be the gating springs of the transduction channels. If the hair bundle is deflected, tension in the tip links triggers the opening of the chan- nels. The subsequent influx of ions (principally K+, but also Ca2+) causes a corresponding change of the mem- brane potential of the hair cell which, in turn, generates a nervous signal. The mechanosensor can be used for self-tuning, as well as signal detection. Many physiological processes are reg- ulated by ionic concentrations, so it is natural to identify the Ca2+concentration with the control parameter C. We assume that Cdecreases if the transduction chan- nels are closed, owing to the action of pumps in the cell membrane. When the hair bundle is deflected by x, the transduction channels open with probability Po(x). We therefore characterize the mechanosensor by the equation ∂C ∂t=−C τ+JoPo(x) (6) where Jois the Ca2+flux through open transduction channels. Note that this equation provides self-tuning. In this case it replaces the more simple but less realistic Eq. (5). For our numerical examples, we use a two-state model for the channels with Po(x) =1 1 +Ae−x/δ(7) where (1 + A)−1≪1 is the probability that a chan- nel is open when the hair cell is quiescent and δis the characteristic amplitude of motion to which the system is sensitive. For a sufficiently long relaxation time τ, the slow variation C0ofC≃C0(t)+C1eiωtcan be separated from the small-amplitude oscillations, giving ∂tC0≃ −C0 τ+Jo˜Po(|x1|2) (8) where ˜Po=/integraltext1/ν 0dt P o(x1eiωt+x−1e−iωt) is the averaged probability of channel opening in the presence of oscilla- tions, which increases monotonically with their ampli- tudex1. For physically relevant parameter values, the system reaches a steady state close to the bifurcation point, independent of the initial value of C. Oscillations generated by molecular motors. We still have to specify the nature and location of the oscil- lator within the hair bundle. It has been suggested that the transduction channels might be the source of the in- stability [7,12]; or, that myosin motors in the stereocilia might generate the force necessary to move the bundle [21,22]. We propose a third possibility: the kinocilium could vibrate using its internal dynein motors. Recently, 3a simple physical mechanism has been proposed which al- lows motor proteins operating in collections to generate spontaneous oscillations by traversing a Hopf bifurcation [15,23]. Typically, motors move along cytoskeletal fila- ments and elastic elements oppose this motion. In this case, two possibilities exist: The system either reaches a stable balance between opposing forces, or it oscillates around the balanced state. Three time scales characterize this behavior: The relaxation time λ/Kof passive relax- ation, where λis the total friction and Kis the elastic modulus; and the times ω−1 1andω−1 2, where ω1is the kinetic rate at which a motor detaches from a filament, andω2is the attachment rate. An explicit solution of a simple model is derived in Appendix B. We find that for an appropriate value of a control parameter C, which is related to the ratio ω1/ω2, a Hopf bifurcation occurs with critical frequency ωc≃/parenleftbiggKα λ/parenrightbigg1/2 , (9) which is the geometric mean of the passive relaxation rate K/λand the typical ATP hydrolysis rate α=ω1+ω2. The above identification of Cwith the Ca2+concentra- tion is consistent with the fact that Ca2+regulates motor protein activity [24]. The system oscillates if the elastic modulus does not exceed a maximal value Kmax≃k0N, where k0is the crossbridge elasticity of a motor and N is the total number of motors. The maximal frequency, obtained when K=Kmax, can be significantly higher than the ATP hydrolysis rate α. Ca2+ L a FIG. 2. A hair bundle consists of a single kinocilium and tens to hundreds of stereocilia. The kinocilium contains dynein motors (red). The stereocilia contain transduction channels (blue) which are gated by the tension in the tip link s (cyan); the steady-state tension is maintained constant by adaptation motors (magenta). In our model, the kinocilium is the active part of the mechanoreceptor and the stereocili a act as the detection apparatus. Feedback control of the acti ve amplificatory process involves the influx of Ca2+ions through the transduction channels; the ions diffuse into the kinocil ium and regulate the generation of force by the motor proteins.Characteristic frequency of a vibrating kinocil- ium. The kinocilium is a true cilium containing a cylin- drical arrangement of microtubule doublets and dynein motors. Since cilia have the well-established tendency to beat and vibrate with frequencies from tens of Hz up to at least 1 kHz [25,26], the kinocilium is a natu- ral candidate to be responsible for the Hopf bifurcation. A simple two-dimensional model can be used to discuss the main physical properties of a vibrating cilium near a Hopf bifurcation [27]. In this model, motors induce the bending of a pair of elastic filaments separated by a dis- tance a(corresponding to the distance between neighbor- ing microtubule doublets in the axoneme). An isolated kinocilium of length Land bending rigidity κ, fixed at the basal end but free at its tip, will vibrate in a wave-like mode with wavelength Λ ≃4L. A typical displacement zof the motors along the filaments leads to bending of the filament pair and a deflection of the tip by a dis- tance x≃zL/a. The elastic bending energy is of order (κ/L3)x2≃(κ/a2L)z2. Therefore, the total elastic mod- ulus experienced by the motors is given by K≃κ La2. (10) The viscous energy dissipation per unit time due to motion of the kinocilium is of order ηL(∂tx)2≃ (ηL3/a2)(∂tz)2. Therefore, the total friction experienced by the motors can be written as λ≃ηL3 a2+λ0ρL , (11) where λ0describes the dissipation within the kinocilium per motor and ρ=N/Ldenotes the number of dynein motors per unit length along the axoneme. If internal friction can be neglected, i.e. if L≫L0= (λ0a2ρ/η)1/2, the frequency of a vibrating cilium at the bifurcation point is given by ωc≃/parenleftbiggκα η/parenrightbigg1/21 L2. (12) Using typical values λ0≃10−9kg/s,a≃20 nm, ρ≃ 5 108m−1andη≃10−3kg/ms, we find L0≃200 nm, shorter than typical kinocilia. Using α≃103s−1,k0≃ 10−3N/m and κ≃4 10−22Nm2(the bending rigidity of 20 microtubules), the frequency range between 100 Hz and 10 kHz can naturally be spanned by changing the length of the kinocilium between 1 µm and 10 µm. The above argument neglects the contribution of the stereocilia to elasticity and motion. The elastic response of stereocilia to hair bundle deflections has been mea- sured [6,28]. It can be well described by an angular spring at the base of each stereocilium which contributes an elastic energy per stereocilium of the order of ks(x/L)2, where ksis an angular elastic modulus. The kinocilium 4length Lin a hair bundle is approximately inversely pro- portional to the number Nsof stereocilia [19], and we write Ns≃ls/L, where ls≃10−4m is the total length of stereocilia. With this assumption, we find an additional elastic modulus Ks≃lsks La2(13) contributed by the stereocilia to K. The contribution of stereocilia to the friction coefficient λcan be estimated as λs≃ηlsL2/a2. (14) The measured value of ks[28] indicates that Ksdomi- nates the contribution to Kgiven by Eq. (10). Similarly, since ls> L,λsshould dominate friction. In this case, we expect ωc≃/parenleftbiggksα ηL3/parenrightbigg1/2 (15) and the range of frequencies is somewhat reduced. If the stiffness of the ensemble of stereocilia greatly exceeds that of the kinocilium, a new situation arises. Since the kinocilium is attached to the stereocilia at its tip [20], movement of the tip is strongly reduced and the kinocilium preferentially vibrates in a mode with wave- length Λ ≃2Lfor which the relation given by Eq. (12) is again valid. Adaptation motors. Finally, we note that in order to obtain a robust self-tuning, the feedback mechanism must be sensitive only to the oscillation amplitude x1 and not to a stationary displacement x0; we assumed in Eq. (8) that ˜Pois independent of x0. The transduction mechanism of stereocilia and their tension-gated chan- nels does indeed have this property: It is well known that an ATP-dependent adaptation mechanism [29] ex- ists which removes the dependence of the channel cur- rent on a constant displacement x0. It is widely believed that this adaptation involves the motion of myosin mo- tors, which maintains constant the steady-state tension in the tip links that control the transduction channels [30]. Therefore, the stereociliar transduction mechanism has precisely the required properties to be used as a feed- back signal for self-tuning the bifurcation. III. SIMULATION The above analysis does not include the effects of noise. Brownian motion is one source of fluctuations in the movement of the hair bundle [31]. Another source of noise is caused by stochastic fluctuations in motor pro- tein force as dynein molecules bind to, and detach from the microtubules in the kinocilium. In order to investi- gate the consequences of this randomness, we performeda Monte Carlo simulation of the two-state model de- scribed in [15], using a realistic number of motor pro- teins. The motors, when attached, experienced a po- tential of amplitude Uand period l(the corresponding crossbridge elasticity is k0=U/l2). The attachment rate was constant and independent of position, ω2=α. De- tachment was localized to a region of width 0 .1l, centred on the potential minimum and the detachment rate ω1 was regulated by the Ca2+concentration C. We chose ω1/ω2= (Cq/C)3, where Cqis the steady-state Ca2+ concentration in a quiescent hair cell; the precise func- tional dependence is unimportant as long as ω1/ω2de- creases monotonically with increasing Cin a fairly sen- sitive way. We simulated systems with N= 1000-4000 motors, representing hair cells with different kinocilium length and stereociliary number. -1012345 0 1000 2000z/l01123J/JC/C STCO STCO tαo q FIG. 3. Self-tuning of a hair bundle. If the Ca2+concen- tration Cwithin the cell is artificially high, the hair bundle is initially quiescent; but as Ca2+ions are pumped from the cell, it gradually begins to oscillate with small amplitude . IfC is suddenly artificially lowered, the hair bundle becomes un - stable and executes high-amplitude spontaneous oscillati ons. Since these movements open the transduction channels, the influx Jof Ca2+ions increases; the consequent change in C regulates the motors and diminishes the amplitude of the os- cillation until it almost disappears. In the steady state, t he bundle executes self-tuned critical oscillations (STCO). For this simulation, we assumed that the probability of transdu c- tion channels opening was Po(z) = (1 + 10 e−20z/l)−1, where zis the motor displacement, and that the time constant for equilibration of the Ca2+concentration was τ= 1000 /α. Self-tuning and the characteristic frequency of spontaneous oscillations. The self-tuning of a hair cell to the vicinity of the bifurcation, where small-amplitude spontaneous oscillations occur, is demonstrated in Fig. 3. When a change of the internal Ca2+concentration is imposed, the system is transiently perturbed; but af- ter an interval of time of order τ, it returns to the same steady state. Hair cells with different numbers of motors acquire different internal concentrations of Ca2+, in or- der to adjust the motor detachment rate in such a way 5that the system approaches the critical point. The spon- taneous oscillations of three different hair bundles are shown in Fig. 4. Note that the characteristic frequency is approximately proportional to the inverse-square of the number of motors, as expected from Eq. (12), and that it can exceed the typical ATP cycle rate αwhen the total number of motors is small (short kinocilium). All three hair cells execute spontaneous oscillations with a similar amplitude, as expected from our arguments above. How- ever, the noise introduces a significant new effect: The oscillations are irregular. The incoherence of the phase of the oscillation is evident in the Fourier transform of the displacement, which exhibits a broad peak centred on the characteristic frequency. 01 0 20 40 60 80 100 tαz/l1000 2000 4000 00.05 0.1 1ν/αz( )/lν1000 2000 4000 FIG. 4. Self-tuned critical oscillations of systems compri s- ingN= 1000 ,2000 and 4000 motors. In each case, the os- cillation has small amplitude z≃0.1land is irregular, as can be seen from the broadness of the Fourier spectrum. For these simulations we chose parameters k0/λα= 4 107N−3 andK/λα = 4 1013N−4, which correspond with the scaling dependence of the spring constant and friction coefficient on kinocilium length in Eqs. (10-11), and also with the order of magnitude estimates of the physical parameters, given in th e text. Dynamic response to a tone at the characteris- tic frequency. The response of a self-tuned hair bun- dle to a sinusoidal force with a frequency approximately equal to the bundle’s characteristic frequency is illus- trated in Fig. 5. For weak stimuli,the amplitude of the oscillation does notincrease with the amplitude |f1|of the applied force; this is because the small response to the stimulus is masked by the noisy, spontaneous motion. Instead, the phase of the hair-bundle oscillation becomes more regular; as it does so, a peak emerges from the Fourier spectrum at the driving frequency. The height of the peak grows approximately as |f1|1/3for intermediate values of |f1|, and approximately linearly for very weak stimuli, and also for very strong stimuli (for which theresponse is essentially passive). Thus the Fourier compo- nent of the hair-bundle displacement at the driving fre- quency responds to the stimulus in the generic manner discussed above. 01 z/l 00.10.20.30.40.50.60.70.8 0.2 0.3 0.40 ν/α1 5 25 100 z( )/lν FIG. 5. Response of a system with N= 2000 motors to a si- nusoidal force at frequency ν= 0.3α, close to the hair bundle’s characteristic frequency. Curves are marked by the dimen- sionless amplitude of the force, |f1|/fmot, where fmot=U/l is the force produced by a motor molecule. Note that a force equal to that of a single motor is sufficient to elicit a respons e in the Fourier spectrum. Curves shaded red are the responses of an equivalent, passive hair bundle (i.e. a bundle with ide n- tical mechanical properties but no force generators). Phase-locking of the nervous signal. The flow of ions through the transduction channels depolarizes the cell membrane which, in turn, opens voltage-gated chan- nels at the base of the hair cell and generates a synaptic current [1]. This sequence of events happens fast enough for variations at the synapse faithfully to reflect the hair- bundle motion at frequencies below 1kHz. Information about the auditory stimulus is subsequently passed along the auditory nerve in the form of a spike train. Simpli- fying this transduction process, we assume that a spike is elicited whenever the transduction channel current J passes a threshold value of 0 .5Jo. The resulting nervous response is shown in Fig. 6. In the absence of a stimu- lus, the self-tuned critical oscillations of the bundle cau se the nerve to fire stochastically, at a low rate. When a weak sinusoidal stimulus is applied at the character- istic frequency, the firing rate does not increase above the spontaneous rate, but the spike train becomes de- tectably phase-locked to the stimulus. The degree of phase-locking increases rapidly as the amplitude of the stimulus increases, reflecting the growing regularity of the hair-bundle motion. It is only when the neural re- sponse is almost completely phase-locked that the firing rate begins to rise. Eventually, for strong stimuli, the spike rate saturates at the stimulus frequency. This be- 6havior is strikingly similar to that which is observed ex- perimentally. In particular, it is well known that the threshold for phase-locking in the auditory nerve fiber is 10-20 dB lower than the threshold at which the firing rate begins to rise [32,33]. 0400 φ360 01a 0400 φ360 05 0800 φ360 025 00.20.40.60.81 0.1 1 10 100vector strength firing rate / 00.3 0.2 0.1α |f |/fmot1b FIG. 6. (a) Histogram of the phase φof the driving force at the instants when nervous spikes are generated, for var- ious values of the force amplitude (marked by the value of |f1|/fmot). (b) Degree of phase-locking (blue; expressed as the vector strength/integraltext dφ p(φ)eiφ) and firing rate (red), as a function of the driving force. The frequency selectivity of the bundle can be appreciated by comparing the neural response to a stimulus with frequency close to the characteristic fre - quency, ν= 0.3α(•), with the response to other frequencies, ν= 0.15α(▽) and ν= 0.6α(△). IV. DISCUSSION The self-tuned critical oscillator which we have intro- duced as a system for signal detection has characteris- tic amplification properties which are generic and do not depend on the choice of a specific model. The oscilla- tion frequency, however, depends on the physical mech- anism involved. In our model of non-mammalian hair cells, the frequency is determined by the geometry of the hair bundle. A simple morphological gradient along the basilar membrane would endow the ear with the abil- ity to analyze a wide range of frequencies. Experimen- tally, it is well established that the height of hair bun- dles progressively increases along the cochlea, and that concurrently the characteristic frequency of the hair cell s declines [19,34]. Our proposition that the kinocilium is likely to play an active role in non-mammalian vertebrate hair cells suggests experiments which study the motility of the kinocilium and its potential for generating oscilla- tions. If the kinocilium is the only source of oscillations, its removal should suppress hair-bundle vibrations. Ifthe self-tuning mechanism is removed instead (by cut- ting the tip links, for example) our model predicts that the kinocilium should exhibit stronger spontaneous oscil- lations. Careful control of the extracellular ion concen- trations, such as Ca2+would be essential in such exper- iments. The simulations of our model reveal how a hair bundle can achieve its remarkable sensitivity to weak stimuli. By profiting from the periodicity of a sinusoidal input, and measuring phase-locking rather than the amplitude of re- sponse, the mechanosensor can detect forces considerably weaker than those exerted by a single molecular motor (if the bundle were a simple, passive structure, its response to such forces would be smaller than its Brownian mo- tion). An important implication of this detection mech- anism is that even though the hair cell selects a certain frequency, the signal must still be encoded by the interval between spikes elicited in the auditory nerve. Paradoxi- cally, the stochastic noise caused by the motor proteins serves a useful purpose. It ensures that the self-tuned critical oscillations of the hair bundle are incoherent, so that the pattern of spontaneous firing in the nerve is ir- regular. Against this background, the regular response to a periodic stimulus can easily be detected.2 Another beneficial feature of noise arises from the fact that weak stimuli do not increase the amplitude of oscil- lation above the spontaneous amplitude. Thus the Ca2+ concentration remains constant, the hair bundle stays in the critical regime, and active amplification can be sus- tained indefinitely. Stronger stimuli cause the system to drift away from the critical point, so that the degree of amplification diminishes over time. It is well known that both the perception of loudness [35] and the firing rate of the auditory nerve [36] decrease over a period of a few seconds, when a stimulus of moderate intensity is main- tained. This phenomenon, which is usually referred to as ‘adaptation’, is consistent with our self-tuning mech- anism. Following the sustained presentation of a loud stimulus, the spontaneous firing rate diminishes and the threshold to weak stimuli is augmented [36]. Such ‘fa- tigue’ is also accounted for by our self-tuning mechanism. Since self-tuning positions the system slightly on the oscillating side of the critical point, self-tuned critica lity provides a natural explanation for otoacoustic emissions [9,10]. In its normal working state, the inner ear would generate faint sounds with a broad range of frequencies. If the feedback mechanism were to fail in certain cells, the spontaneous oscillations could become large enough for distinct tones to be emitted. A self-tuned Hopf bifurcation is ideal for sound detec- tion because it provides sharp frequency selectivity and a 2The benefits of noise have been discussed in a variety of situations such as those involving stochastic resonance [3 7]. 7nonlinear gain which compresses a wide range of stimulus intensities into a narrow range of response. We therefore believe that the concept applies to all vertebrate hear- ing systems, and potentially to other mechanoreceptor systems. Kinocilia are typically absent in mammalian cochlea, and we suggest that their force-generating role has been assumed by the outer hair cells. Self-tuning of these motile cells might be realized by a mechanism sim- ilar to that presented here, using transduction channels in their hair bundles. It could, however, work very dif- ferently; for example, it might involve feedback from the inner hair cells, via the efferent nervous system. Tuning to the proximity of a critical point is likely to be a general strategy adopted by sensory systems. Simple molecular receptors [38], as well as the physiological sen- sors of higher organisms, can enhance their response to weak stimuli in this way. We propose that the physics of self-tuned criticality is the ‘central science of transduc er physiology’ spoken of by Delbr¨ uck [39]. Acknowledgements: We thank P. Martin and C. Pe- tit for useful discussions, and A.F. Huxley for referring us to the work of T. Gold. T. Duke is grateful for the hospi- tality of Institut Curie and the Niels Bohr Institute, and acknowledges the support of the Royal Society. After submission of our manuscript we learned that Eguiluz, Ospeck, Choe, Hudspeth and Magnasco have indepen- dently described the generic response of a system near a Hopf bifurcation; we thank them for communicating their unpublished results. APPENDIX A: GENERIC BEHAVIOR AT A HOPF-BIFURCATION 1. Nonlinear relation between periodic stimulus and displacements We are interested in the response x(t) of a nonlinear system to a periodic stimulus force f(t). If only one fre- quency ν=ω/2πis present we use the Fourier expansions f(t) =∞/summationdisplay n=−∞fneinωt(A1) x(t) =∞/summationdisplay n=−∞xneinωt, (A2) where the complex coefficients xnandfnobeyxn=x∗ −n andfn=f∗ −n. This representation implies that we focus on the limit cycle solution and ignore all transient relax- ation phenomena. We consider the class of systems for which the force at a given time depends in a nonlinear way on the history of the displacements x(t) alone; as we will discuss in section D more complex cases do notchange the basic properties. In this situation, the rela- tion between xandfcan be expressed as a systematic expansion of the force amplitudes fnin the amplitudes xn: fk=F(1) klxl+F(2) klmxlxm +F(3) klmnxlxmxn+O(x4), (A3) where the expansion coefficients F(n) k,k1,..,k nare symmet- ric with respect to permutations of the indices k1..kn. The limit cycle solutions are invariant with respect to translations in time t→t+ ∆t. Under these transfor- mations the amplitudes change as xn→xneinω∆tand fn→fneinω∆t. Inspection of Eq. (A3) shows that the time translation symmetry allows only for those terms F(n) k,k1,..,k nxk1..xknfor which k=k1+...+kn. For all other cases F(n) k,k1,..,k nmust vanish which significantly re- stricts the number of terms. 2. Hopf bifurcation The nonlinear system exhibits spontaneous oscillations and a Hopf-bifurcation if nontrivial solutions to Eq. (A3) withxn/negationslash= 0 exist in the case where all fk= 0, i.e. if no stimulus force is applied. Without loss of generality, we consider here an instability of the mode x1. In this case, the dominant terms allowed by symmetry read ( fk= 0) 0≃F(1) 11x1+ 2F(2) 1,2,−1x−1x2 + 6F(3) 1,1,1,−1x2 1x−1+ 6F(3) 1,1,2,−2x2x−2x1 (A4) 0≃F(2) 22x2+ 2F(2) 211x2 1. (A5) Eq. (A5) determines x2≃ −2(F(2) 211/F(2) 22)x2 1. Inserting this relation in Eq. (A4), we obtain to lowest order 0≃ Ax1+B|x1|2x1, (A6) where A ≡F(1) 11andB ≡3F(3) 1,1,1,−1−4F(2) 211F(2) 1,2,−1/F(2) 22. The coefficients A(ω, C) andB(ω, C) are complex and in general depend on frequency ωand a control parameter which we denote by C. A Hopf bifurcation occurs at a critical point C=Ccat which Avanishes for a frequency ωc, i.e. A(ωc, Cc) = 0. This can be demonstrated as follows: A spontaneously oscillating solution satisfies |x1|2=−A B(A7) Note, that such a solution can only exist if A/Bis real and negative. At the bifurcation point, A= 0 and A/B is therefore real for ω=ωc, however the corresponding amplitude |x1|2vanishes. In the vicinity of this point we expect to find solutions with finite amplitude. We use the expansion 8A(ω, C)≃(ω−ωc)A1+ (C−Cc)A2 (A8) where A1andA2are complex coefficients and we neglect higher order terms. Spontaneous oscillating solutions ex- ist only if A/Bis real. This condition is satisfied for a particular frequency ω=ωswith ωs=ωc+Im(A2/B) Im(A1/B)(Cc−C). (A9) The ratio −A/Bat this frequency ωschanges sign for C=Cc; here we assume without loss of generality that it is positive for C < C c. In this case, the system oscillates spontaneously with an amplitude which according to Eq. (A7) behaves as |x1|2= ∆2(Cc−C)/Cc, where ∆2=Cc/parenleftbigg Re(A2/B)−Re(A1/B)Im(A2/B) Im(A1/B)/parenrightbigg (A10) is a typical amplitude. We have thus demonstrated that Eq. (A6) characterizes a Hopf-bifurcation if the complex coefficient Avanishes at a critical point Ccfor a critical frequency ωc. 3. Amplified response to sinusoidal stimuli If a sinusoidal stimulus f(t) =f1eiωt+f−1e−iωt, for which all fnwithn/negationslash=±1 vanish, Eq. (A6) becomes f1≃ Ax1+B|x1|2x1. (A11) We consider a system that is tuned exactly to the bifurca- tion,C=Cc. In this situation spontaneous oscillations do not occur and A= (ω−ωc)A1. If the imposed fre- quency is equal to the critical frequency ω=ωc, the coefficient Avanishes and we can solve Eq. (A11) for |x1|to find the nonlinear response |x1| ≃ |B|−1/3|f1|1/3, (A12) as a function of the force amplitude |f1|. This behavior represents an amplified response with a gain r=|x1| |f1|∼ |f1|−2/3(A13) that becomes arbitrarily large for small forces. If the frequency ωis different from ωc, this nonlinear response still holds as long as the linear term in Eq. (A11) is small compared to the cubic term and can be neglected. This is the case if |x1|2≫ |A /B|=|ω−ωc||A1/B|. There- fore, the nonlinear regime characterized by Eq. (A12) holds for sufficiently large force amplitudes, |f1| ≫ |(ω− ωc)A1|3/2/|B|1/2, or if the frequency is sufficiently close to the critical frequency, |ω−ωc| ≪ |f1|2/3|B|1/3/|A1|. If the frequency mismatch |ω−ωc|becomes large, or if forces |f1|are small, a new regime occurs for whichthe linear term in (A11) dominates. In this regime, the response is linear, |x1| ≃|f1| |(ω−ωc)A1|, (A14) and the gain is constant. This is a passive response if the stimulus frequency is too far from the critical frequency. 4. Additional remarks The above derivation is based on an expansion (A3) in the displacements xn. This excludes some nonlinearities in the force which can lead to additional nonlinear terms in Eq. (A11). The most general form of Eq. (A11) is f1≃ Ax1+B|x1|2x1+Cx1|f1|2+Dx−1f2 1 +E|x1|2f1+Fx2 1f−1+G|f1|2f1.(A15) However, for small forces f1and small amplitudes x1, the results derived above are not affected. The regime of nonlinear reponse |f1| ∼ |x1|3, as well as the linear response regime |f1| ∼ |x1|still exist. If |f1| ∼ |x1|, the nonlinear terms in f1renormalize the third order term, which in this regime is negligable. If |f1| ∼ |x1|3, the nonlinear terms in f1are of even higher order and can be neglected. APPENDIX B: OSCILLATIONS GENERATED BY MOLECULAR MOTORS 1. Two state model The two state model describes force-generation as a re- sult of transitions between two states, a bound state and a detached state of a motor and its track filament. The interaction between a motor at position zalong the fila- ment in states 1 and 2 is characterized by two periodic potentials W1(z) =W1(z+l) and W2(z) =W2(z+l) where lis the period. We introduce the relative position ξ=zmodlwith respect to the potential period. De- tachment and attachment rates are denoted ω1(ξ) and ω2(ξ), respectively. Oscillations can occur in this model if a large number Nof motors move collectively against an external elastic element of modulus K. We introduce the probability P1(ξ) andP2(ξ) of finding a motor bound at position ξin state 1 or 2, which satisfy the normalization condition /integraldisplayl 0dξ(P1+P2) = 1 (B1) For a large number of motors collectively moving with the same velocity vthe dynamic equations read 9∂tP1+v∂ξP1=−ω1P1+ω2P2 (B2) ∂tP2+v∂ξP2=ω1P1−ω2P2 (B3) The velocity vis determined by the force-balance condi- tion f=λv+Kz+N/integraldisplayl 0dξ(P1∂ξW1+P2∂ξW2) (B4) where λis a friction coefficient describing the total fric- tion and zis the displacement of the motors, ∂tz=v. For an incommensurate arrangement of motors with respect to the track filament and a large number Nof motors, P1(ξ) +P2(ξ) = 1/land the equations of motions sim- plify: ∂tP+v∂ξP=−(ω1+ω2)P+ω2/l , (B5) where we denote for simplicity P(ξ) =P1(ξ). We discuss a simple choice for the potentials and tran- sition rates for which the Hopf bifurcation is easy to de- termine analytically. We consider the potential W1(ξ) =Ucos(2πξ/l) (B6) with amplitude U, and the potential W2to be constant. The transition rates are chosen to be periodic functions ω1(ξ) =β−βcos(2πξ/l) (B7) ω2(ξ) =α−β+βcos(2πξ/l) (B8) parameterized by two coefficients αandβ. With this choice, ω1(ξ) +ω2(ξ) =α (B9) is constant and the fact that ω1andω2are positive re- stricts βto the interval 0 ≤β≤α/2. 2. Linear response function In order to determine the linear coefficient Awhich determines the stability of the system, we look for small amplitude oscillations close to the resting state with v= 0. We write P≃p0+p1eiωt(B10) f≃f1eiωt(B11) z≃z1eiωt(B12) where p0=ω2/αl. To linear order in z1, we find from Eq. (B5) p1=−iωz1 iω+α∂xp0 (B13) The corresponding force is given byf1≃ Az1 (B14) with A=iωλ+K+χ , (B15) where the active response χof the motors is given by χ=−N/integraldisplayl 0dξiω iω+α∂ξp0∂ξW1 (B16) For the choice of Eq. (B6) and (B8) the integral can be calculated and we obtain A(C, ω) =iωλ+K−Nk0Ciω/α+ (ω/α)2 1 + (ω/α)2.(B17) Here, we have introduced the dimensionless control pa- rameter C≡2π2β/αwith 0 < C < π2and the cross- bridge elasticity k0≡U/l2of the motors. 3. Hopf bifurcation A Hopf bifurcation occurs if there is a pair of values (C, ω) for which Aas given by Eq. (B17) vanishes. Such a point indeed exists. For the critical value Cc=λα+K Nk0(B18) the bifurcation occurs for the critical frequency ωc=/parenleftbiggKα λ/parenrightbigg1/2 (B19) The critical frequency is bounded by the fact that Cc< π2. 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arXiv:physics/0003101v1 [physics.bio-ph] 30 Mar 2000Physical Aspects of Axonemal Beating and Swimming S´ ebastien Camalet and Frank J¨ ulicher PhysicoChimie Curie, UMR CNRS/IC 168, 26 rue d’Ulm, 75248 Pa ris Cedex 05, France We discuss a two-dimensional model for the dynamics of axonemal deformations driven by internally generated forc es of molecular motors. Our model consists of an elastic filamen t pair connected by active elements. We derive the dynamic equations for this system in presence of internal forces. In the limit of small deformations, a perturbative approach al - lows us to calculate filament shapes and the tension profile. We demonstrate that periodic filament motion can be gener- ated via a self-organization of elastic filaments and molecu lar motors. Oscillatory motion and the propagation of bending waves can occur for an initially non-moving state via an in- stability termed Hopf bifurcation. Close to this instabili ty, the behavior of the system is shown to be independent of microscopic details of the axoneme and the force-generatin g mechanism. The oscillation frequency however does depend on properties of the molecular motors. We calculate the osci l- lation frequency at the bifurcation point and show that a lar ge frequency range is accessible by varying the axonemal lengt h between 1 and 50 µm. We calculate the velocity of swimming of a flagellum and discuss the effects of boundary conditions and externally applied forces on the axonemal oscillations . I. INTRODUCTION Many small organisms and cells swim in a viscous en- vironment using the active motion of cilia and flagella. These are hair-like appendages of the cell which can un- dergo periodic motion and use hydrodynamic friction to induce cellular self-propulsion (Bray 1992). In this paper , we are interested in those flagella and cilia which con- tain force generating elements integrated along the whole length of the elastic filamentous structure. They repre- sent rod-like elastic structures which move and bend as a result of internal stresses. Examples for these systems are paramecium which has a large number of cilia on its sur- face; sperm, which use a single flagellum to swim; and chlamydomonas which uses two flagella to swim (Bray 1992). Cilia also occur in very different situations. An example is the kinocilium which exists in many hair bun- dles of mechanosensitive cells and has the ability to beat periodically (R¨ usch and Thurm 1990). The common structural theme of cilia and flagella is the axoneme, a characteristic structure which occurs in a large number of very different organisms and cells and which appeared early in evolution. The axoneme consists of a cylindrical arrangement of 9 doublets of parallel mi- crotubules and one pair of microtubules in the center.In addition, it contains a large number of other proteins such as nexin which provide elastic links between micro- tubule doublets, see Fig. 1. The axoneme is inherently active. A large number of dynein molecular motors are located in two rows between neighboring microtubules and can induce forces and local displacements between adjacent microtubules (Albert et al. 1994). Axonemal flagella can generate periodic waving or beating patterns of motion. In the case of sperm for ex- ample, a bending wave of the flagellum propagates from the head which contains the chromosomes towards the tail. In a viscous environment, the surrounding fluid is set in motion and hydrodynamic forces act on the fila- ment. For typical values of frequencies and length scales given by the size of a flagellum, the Reynolds number is small and inertia terms in the fluid hydrodynamics can be neglected (Taylor 1951). Therefore, only friction forces resulting from solvent viscosity can contribute to propulsion. Under such conditions self-propulsion is pos- sible if a wave propagates towards one end, a situation which breaks time-reversal invariance of the sequence of deformations of the flagellum (Purcell 1977). M D N FIG. 1. Schematic representation of the cross-section of th e axoneme. Nine doublets of microtubules (M) are arranged in a cylindrical fashion, two microtubules are located in the c en- ter. Dynein motors (D) are attached to microtubule doublets and interact with a neighboring doublet. Elastic elements ( N) such as nexin are indicated as springs. How can bending waves be generated by axone- mal dyneins which act internally within the axoneme? Dynein molecular motors induce relative forces between parallel elastic filaments, which are microtubule doublets . As a result of these forces, the filaments have the ten- dency to slide with respect to each other. If such a slid- 1ing is permitted globally, the filaments simply separate but no bending occurs. Bending results if global sliding is suppressed by rigidly connecting the filament pair in the region close to one of the two ends. In this situation, sliding is still possible locally, however only if the fila- ments undergo a bending deformation. This coupling of axonemal bending to local microtubule sliding has been demonstrated experimentally. In situations where the ax- oneme is cut at its basal end, filaments slide and in the presence of ATP separate without bending (Warner and Mitchell 1981). Small gold-beads specifically attached to microtubules in fully functioning flagella can be used to directly visualize the local relative sliding during beati ng (Brokaw 1991). The theoretical problem of oscillatory axonemal bend- ing and wave patterns have been addressed by several authors. One can distinguish two principally different mechanisms to generate oscillatory forces within the ax- oneme: (i) Deterministic forcing: a chemical oscillator could regulate the dynein motors which are activated and deactivated periodically. In this case, the regula- tory system defines a dynamical force-pattern along the axoneme and drives the system in a deterministic way (Sugino and Naitoh 1982). (ii) Self-organized beating: the axoneme oscillates spontaneously as a result of the interplay of force-generating elements and the elastic fil- aments (Machin 1963, Brokaw 1975, Brokaw 1985, Lin- demann and Kanous 1995). In particular, Brokaw has studied thoroughly self-organized patterns of beating using numerical simulations of simple models (Brokaw 1985, Brokaw 1999). Such models are based on the bend- ing elasticity of the flagellum and an assumption on the coupling of motor activity to the flagellar deformations. In the present work, we are mostly interested in self- organized beating. Our approach is conceptually differ- ent from other works as we focus on an oscillating insta- bility of the motor-filament system. In general, sponta- neous oscillations occur via a so-called Hopf-bifurcation where an initially stable quiescent state becomes unsta- ble and starts to oscillate. There is evidence for the existence of such a dynamic instability in the axoneme. Demembranated flagella show a behavior which depends on the ATP-concentration CATPin the solution. For smallCATP, the flagellum is straight and not moving. If CATPis increased, oscillatory motion sets in at a critical value of the ATP concentration and persists for larger concentrations (Gibbons 1975). This implies an instabil- ity of the initial straight state with respect to a wave-like mode. Recently, it has been demonstrated using a sim- ple model for molecular motors that a large number of motors working against an elastic element can generate oscillations via a Hopf bifurcation by a generic mecha- nism (J¨ ulicher and Prost 1995, J¨ ulicher and Prost 1997). This suggests that a system consisting only of molecu- lar motors and semiflexible filaments can in general un- dergo self-organized oscillations. This idea is supportedby the facts that flagellar dyneins are capable of gener- ating oscillatory motion (Shingyoji et al. 1998) and that experiments suggest the existence of dynamic transitions in many-motor systems (Riveline et al. 1998, Fujita and Ishiwata 1998). Patterns of motion of cilia and flag- ella can be complex and embedded in three dimensional space. Many examples of propagating bending waves, however, are planar and can thus be considered as con- fined to two-dimensions (Brokaw 1991). In the following sections, we present a systematic study of a simplified model for the axoneme which has been in- troduced recently (Camalet et al. 1999) and which cap- tures the basic physical properties that are relevant for its dynamics. In Section II, we present a thorough anal- ysis of the dynamics of flexible filaments driven by in- ternal forces. This work is inspired by recent studies of the dynamics of semiflexible filaments subject to ex- ternal forces (Goldstein and Langer 1995, Wiggins and Goldstein 1998, Wiggins et al. 1998). We show how the dynamic equations can be solved perturbatively and we calculate the shapes of bending waves, the velocity of swimming and the tension profile along the flagellum. In Section III, we briefly review a simple two-state model for a large number of coupled molecular motors. This model is well suited to represent the dynein molecular motors which act within the axoneme. Self-organized bending waves and oscillations via a Hopf bifurcation of the coupled motor-filament system are studied in section IV. We calculate the wave-patterns close to a Hopf bi- furcation and determine the frequencies selected by the system. Finally, we discuss the relevance of our simple model to real axonemal cilia and flagella and propose ex- periments which could be performed to test predictions that follow from our work. II. A SIMPLE MODEL FOR AXONEMAL DYNAMICS The cylindrical arrangement of microtubule doublets within the axoneme can be modeled effectively as an elastic rod. Deformations of this rod lead to local slid- ing displacements of neighboring microtubules. Here, we consider planar deformations. In this case, the geomet- ric coupling of bending and sliding can be captured by considering two elastic filaments (corresponding to two microtubule doublets) arranged in parallel with constant separation aalong the whole length of the rod. At one end, which corresponds to the basal end of an axoneme and which we call “head”, the two filaments are rigidly attached and not permitted to slide with respect to each other. Everywhere else, sliding is possible, see Fig. 2. The configurations of the system are described by the shape of the filament pair given by the position of the neutral line r(s) as a function of the arclength s, where 2ris a point in two dimensional space. The shapes of the two filaments are then given by r1(s) =r(s)−an(s)/2 r2(s) =r(s) +an(s)/2, (1) where nwithn2= 1 is the filament normal. The local geometry along the filament pair is characterized by the relations ˙r=t (2) ˙t=Cn (3) ˙n=−Ct, (4) where tdenotes the normalized tangent vector and C= ˙t·nis the local curvature. Throughout this paper dots denote derivatives with respect to s, i.e. ˙r≡∂sr≡ ∂r/∂s. Δ f(s) -f(s) s arrr 12 FIG. 2. Two filaments (solid lines) r1andr2at constant separation aare rigidly connected at the bottom end with s= 0, where sis the arclength of the neutral line r(dashed). Internal forces f(s) are exerted in opposite directions, tan- gential to the filaments. The sliding displacement ∆ at the tail is indicated. A. Bending and sliding of a filament pair The two filaments are assumed to be incompressible and rigidly attached to each other at one end where s= 0. Bending of the filament pair and local sliding displace- ments are then coupled by a geometric constraint. The sliding displacement at position salong the neutral line ∆(s)≡/integraldisplays 0ds′(|˙r1| − |˙r2|) (5) is defined to be the difference of the total arclengths along the two filaments up to the points r1(s) andr2(s) which face each other along the neutral line. From Eqns. (1) and (4) follows that ˙r1,2= (1±aC/2)tand thus |˙r1,2|= 1±aC/2. (6)Therefore, ∆ =a/integraldisplays 0ds′C (7) is given by the integrated curvature along the filament. B. Enthalpy functional The bending elasticity of filaments (microtubules) characterizes the energetics of the filament pair. In ad- dition to filament bending, we also have to take into ac- count the large number of passive and active elements (e.g. nexin links and dynein molecular motors) which give rise to relative forces between neighboring micro- tubule doublets. These internal forces can be charac- terized by a coarse-grained description defining the force per unit length f(s) acting at position sin opposite direc- tions on the two microtubules, see Fig. 2 (internal forces must balance). This force density is assumed to arise as the sum of active and passive forces generated by a large number of proteins. This internal force density corre- sponds to a shear stress within the flagellum which tends to slide the two filaments with respect to each other. A static configuration of a filament pair of length Lcan thus be characterized by the enthalpy functional G≡/integraldisplayL 0/bracketleftbiggκ 2C2+f∆ +Λ 2˙r2/bracketrightbigg ds . (8) Here,κdenotes the total bending rigidity of the fila- ments. The incompressibility of the system is taken into account by the Lagrange multiplier function Λ( s) which is used to enforce the constraint ˙r2= 1 which ensures that sis the arclength (Goldstein and Langer 1995). The in- ternal force density fcouples to the sliding displacement ∆ as described by the contribution/integraltext dsf∆. The varia- tion of this term under small deformations of the filament shape represents the work performed by internal stresses. Using Eq. (7) leads after a partial integration to G=/integraldisplayL 0ds/bracketleftbiggκ 2C2−aCF+Λ 2˙r2/bracketrightbigg , (9) where F(s)≡ −/integraldisplayL sds′f (10) is the force density integrated to the tail. From Eq.(9) it follows that if the internal stresses or Fare imposed, Gis minimized for a filament curvature C=C0where C0(s) =aF(s)/κis a local spontaneous curvature. The internal forces therefore induce filament bending. In or- der to derive the filament dynamics, we determine the variationδGwith respect to variations δr. Details of 3this calculation are given in Appendix A. As a result, we find δG δr=∂s[ (κ˙C−af)n−τt]. (11) Here, τ= Λ +κC2−aCF , (12) plays the role of the physical tension. This becomes ap- parent since from Eq. (11) it follows that τ(s) =t(s)·/parenleftBigg/integraldisplayL sds′δG/δr+Fext(L)/parenrightBigg (13) where Fext(L) is the external force applied at the end which satisfies Eq. (20). Therefore, τis the tangent component of the integrated forces acting on the filament. C. Dynamic equations We derive the dynamic equation with the simplifying assumption that the hydrodynamics of the surrounding fluid can be described by two local friction coefficients ξ/bardbl andξ⊥for tangential and normal motion, respectively. The equations of motion in this case are given by (Wig- gins and Goldstein 1998, Wiggins et al. 1998) ∂tr=−/parenleftbigg1 ξ⊥nn+1 ξ/bardbltt/parenrightbigg ·δG δr. (14) For the following, it is useful to introduce a coordinate system r= (X,Y) and the angle ψbetween the tangent t= (cosψ,sinψ) and the X-axis which satisfies C=˙ψ. We find with Eq. (11) ∂tr=1 ξ⊥n(−κ... ψ+a˙f+˙ψτ) +1 ξ/bardblt(κ˙ψ¨ψ−a˙ψf+ ˙τ). (15) Noting that ∂t˙r=n∂tψ, we obtain an equation of motion forψ(s) alone: ∂tψ=1 ξ⊥(−κ.... ψ+a¨f+˙ψ˙τ+τ¨ψ) +1 ξ/bardbl˙ψ(κ˙ψ¨ψ−af˙ψ+ ˙τ). (16) The tension τis determined by the constraint of in- compressibility ∂t˙r2= 2t·∂t˙r= 0. This condition and Eq. (15) leads to a differential equation for the tension profile: ¨τ−ξ/bardbl ξ⊥˙ψ2τ=a∂s(˙ψf)−κ∂s(˙ψ¨ψ) +ξ/bardbl ξ⊥˙ψ(a˙f−κ... ψ). (17)Eqns. (16) and (17) determine the filament dynamics. The filament shape follows from r(s,t) =r(0,t) +/integraldisplays 0(cosψ,sinψ)ds′, (18) wherer(0,t) can be obtained from Eq. (15) evaluated at s= 0. D. Boundary Conditions The filament dynamics depends on the imposed bound- ary conditions. The variation δGhas contributions at the boundaries which can be interpreted as externally applied forces Fextand torques Textacting at the ends, see Appendix A. At the head with s= 0, Fext= (κ˙C−af)n−τt Text=−κC−a/integraldisplayL 0ds′f . (19) Similarly, at the tail for s=L, Fext= (−κ˙C+af)n+τt Text=κC . (20) If constraints on the positions and/or angles are imposed at the ends, forces and torques have to be applied to satisfy these constraints. In the following, we will discus s different boundary conditions as specified in Table I: Case A is the situation of a filament with clamped head, i.e. both the tangent and the position at the head are fixed, the tail is free. Case B is a filament with fixed head, i.e. the tangent at the head can vary. The situation of a swimming sperm corresponds to case C where the friction force of a viscous load attached at the head is taken into account, the head is otherwise free. As an example of a situation with an external force Fext(L) applied at the tail we consider Case D. For simplicity, we assume a force parallel to the (fixed) tangent at the head. boundary heads= 0 tails=L condition A∂tr=0∂tt=0Fext=0Text= 0 B∂tr=0Text= 0Fext=0Text= 0 C Fext=−ζ∂trText= 0Fext=0Text= 0 D∂tr=0∂tt=0Fext/negationslash=0Text= 0 TABLE I. Different boundary conditions studied. (A) clamped head, free tail; (B) fixed head, free tail; (C) swim- ming flagellum with viscous load ζ; (D) clamped head, exter- nal force applied at the tail. 4E. Small deformations In the absence of internal forces f, the filament relaxes passively to a straight rod with ˙ψ= 0, i.e.ψconstant. Without loss of generality, we choose ψ= 0 in this state which implies that straight filament is parallel to the X- axis. Internal stresses f(s) induce deformations of this straight conformation. For small internal stresses, we can perform a systematic expansion of the filament dynamics in powers of the stress amplitude. We introduce a dimensionless parameter ǫwhich scales the amplitudes of the internal stresses, f(s,t) =ǫf1(s,t), wheref1is an arbitrary stress distribution. We can now solve the dynamic equations (16) and (17) perturbatively by writing ψ=ǫψ1+ǫ2ψ2+O(ǫ3) τ=τ0+ǫτ1+ǫ2τ2+O(ǫ3), (21) which allows us to determine the coefficients ψn(s,t) and τn(s,t). This procedure is described in Appendix B. We find thatψ2andτ1always vanish and τ0=σis a con- stant tension which is equal to the X-component of the external force applied at the end. Note, that for bound- ary conditions A-C, σ= 0. The filament shape is thus characterized by the behavior of ψ1(s,t) which obeys ξ⊥∂tψ1=−κ.... ψ1+σ¨ψ1+a¨f1. (22) In order to discuss the filament motion in space, we define the average velocity of swimming ¯v= lim t→∞1 t/integraldisplayt 0dt′∂tr (23) which is independent of s. This velocity is different from zero only in the case of boundary condition C. We choose theX-axis parallel to ¯vand introduce a coordinate sys- tem (x,y) which moves with the filament (x,y) = (X−¯vt,Y), (24) where ¯v=±|¯v|and the sign depends on whether motion is towards the positive or negative X-direction, see Fig. 3. It is useful to introduce the transverse and longitudinal deformations handu, respectively, which satisfy (x,y) = (s+u(s),h(s)) (25) and which vanish for ǫ= 0. The quantities h,uand ¯v can be calculated perturbatively in ǫ, see Appendix B. To second order in ǫtransverse motion satisfies ξ⊥∂th=−κ∂4 sh+σ∂2 sh+a∂sf (26) Longitudinal displacements satisfy u(s) =u(0)−1 2/integraldisplays 0(∂sh)2ds′(27)The boundary conditions for h(s) are given in table II. v_ X,xYy tn ψ FIG. 3. Resting frame ( X,Y) and frame ( x,y) moving with the filament at velocity ¯ v. The tangent tand the normal nare indicated, the angle between the X,x-axis and tis denoted ψ. F. Swimming The velocity of swimming ¯ vcan be calculated pertur- batively inǫ. We consider for simplicity periodic internal stresses with frequency ωgiven by1 f(s,t) =˜f(s)iωt+˜f∗(s)e−iωt(28) which after long time leads to periodic filament motion h(s,t) =˜h(s)eiωt+˜h∗(s)e−iωt(29) where ˜hsatisfies according to Eq. (26) κ∂4 s˜h−σ∂2 s˜h+ξ⊥iω˜h=a∂s˜f . (30) boundary heads= 0 tails=L condition F=−/integraltextL 0dsf ∂2 sh= 0 Ah= 0 ∂sh= 0κ∂3 sh=af Bh= 0 κ∂2 sh=aFκ∂3 sh=af Cζ∂th=af−κ∂3 sh κ∂2 sh=aFκ∂3 sh=af Dh= 0 ∂sh= 0κ∂3 sh −σ∂sh=af TABLE II. Boundary conditions for small amplitude mo- tion. (A) clamped head, free tail; (B) fixed head, free tail; ( C) swimming flagellum with viscous load ζ; (D) clamped head, external force applied at the tail. 1Note, that more general periodic stresses f=/summationtext n˜fneinωt can also be considered. Since the equation of motion (26) is linear, different modes superimpose linearly and we can without loss of generality restrict our discussion to a sing le moden= 1. 50.00.00.20.2 0.40.4 0.60.6 0.80.8 1.01.0 H sAB C 0.00.20.40.60.81.0sA BC -10.00.010.020.0 (a) (b) Φ FIG. 4. Fundamental modes of filament motion ˜h1(s) =He−iφclose to a Hopf bifurcation are displayed for an oscillation frequency of about 50 Hzand boundary condi- tions A-C. In case C, a viscous load ζ≃5.10−8Ns/m acts at the head. (a) Amplitude Hin arbitrary units as a function of arclength s. (b) Same plot for the gradient of the phase ˙φ. Writing ˜h=He−iφwhereHandφdenote the am- plitude and phase, respectively, filament motion can be expressed as h(s,t) =H(s)cos(ωt−φ(s)), (31) which represents bending waves for which vp=ω/˙φcan be interpreted as the local wave propagation velocity. Ex- amples of bending waves for self-organized beating dis- cussed in section IV are displayed in Figs. 4 and 5. For a freely oscillating filament with a viscous load of fric- tion coefficient ζattached (case C), we find an average propulsion velocity ¯ v=V0/(1 +ζ/ξ/bardblL) where V0=−/parenleftbiggξ⊥ ξ/bardbl−1/parenrightbiggω 2L/integraldisplayL 0dsH2˙φ , (32) is the velocity for ζ= 0. Eq. (32) which is correct to second order in ǫreveals that motion is only possible if the filament friction is anisotropic ξ/bardbl/ξ⊥/negationslash= 1 and if a wave is propagating, ˙φ/negationslash= 0. This is consistent with earlier work on swimming (Taylor 1951, Purcell 1977, Stone and Samuel 1996). For a rod-like filament with ξ/bardbl< ξ⊥, swimming motion is opposite to the direction of wave propagation.A B C FIG. 5. Filament shapes in the ( x,y) plane that correspond to the modes displayed in Fig. 4. Snapshots taken at differ- ent times to illustrate bending waves are shown for boundary conditions A-C. The arrows indicate the direction of wave propagation. III. INTERNAL FORCES GENERATED BY MOLECULAR MOTORS The filament dynamics is driven by internal stresses f(s,t) generated by a large number of dynein motors and passive elements. We describe the properties of a force- generating system using a simplified two-state model. A. Two state model for molecular motors We briefly review the two-state model for a large num- ber of motor molecules attached to a filament which slides with respect to a second filament (J¨ ulicher and Prost 1995, J¨ ulicher et al. 1997). Each motor is assumed to have two different chemical states, a strongly bound state 1 and a weakly bound state or detached state 2. The interaction between a motor and a filament in states 1 and 2 is characterized by energy landscapes W1(x) and W2(x), wherexdenotes the position of a motor along the filament. The potentials reflect the symmetry of the fila- ments: they are periodic with period l,Wi(x) =Wi(x+l) and are spatially asymmetric, Wi(x)/negationslash=Wi(−x). In the presence of ATP which is the chemical fuel that drives the system, the motors are assumed to undergo transi- tions between states. The corresponding transition rates are denoted ω1for detachments and ω2for attachments. We introduce the relative position ξof a motor with re- spect to the potential period where x=ξ+nl, 0≤ξ<l andnis an integer. The probability to find a motor in stateiat position ξat timetis denotedPi(ξ,t),P1+P2 is normalized within one period. The dynamic equations of the system are given by 6∂tP1+v∂ξP1=−ω1P1+ω2P2 ∂tP2+v∂ξP2=ω1P1−ω2P2. (33) The sliding velocity v=∂txis determined by the force- balance f=λv+ρ/integraldisplayl 0(P1∂ξW1+P2∂ξW2)dξ+Kx . (34) Here, the coefficient λdescribes the total friction per unit length in the system. The number density of motors along the filament is denoted by ρandfis the force per unit length generated by the system. The elastic modulus per unit length Koccurs in presence of elastic elements such as nexins. If motors have an incommensu- rate arrangement compared to the filament periodicity, P1+P2= 1/land the equations simplify and can be expressed by P1alone ∂tP1+v∂ξP1=−(ω1+ω2)P1+ω2/l f=λv+ρ/integraldisplayl 0P1∂ξ∆W+Kx (35) where ∆W=W1−W2. B. Molecular motors coupled to a filament pair The two state model for a large number of motors is well suited to represent the internal forces acting within the axoneme. We assume that at any position san inde- pendent two-state model described by Eq. (35) is located which generates the internal force density f(s). The fil- ament sliding displacement is identified with the motor displacement, ∆ ≡x. Note, that here we neglect for sim- plicity fluctuations which arise from the chemical activity of a finite number of force generators and we assume ho- mogeneity of all properties along the axonemal length. The energy source of the active system is the chemical activity characterized by the transition rates ω1andω2, the generated forces induce bending deformations of the filaments. The dynamic equations (35) represent a nonlinear ac- tive system which generates time-dependent forces. Since we will study periodic motion, we can express forces and displacements by a Fourier expansion f(t) =/summationdisplay nfneinωt(36) ∆(t) =/summationdisplay n∆neinωt. (37) The relation between force amplitudes fnand amplitudes of sliding displacements ∆ ncan in general be expressed in powers of the amplitudes ∆ n(J¨ ulicher and Prost 1997) fn=F(1) nk∆k+F(2) nkl∆k∆l+O(∆3 k). (38)Here, the summation over common indices is implied. The coefficients F(n)are calculated for the two-state model in Appendix C. However, Eq. (38) is more gen- eral and characterizes a whole class of active nonlinear mechanical systems. C. Symmetry considerations The interaction of the motors with the two filaments is asymmetric. Motors are rigidly connected to one of the filaments and slide along the second. In addition, the filaments are polar and filament sliding is induced by motors towards one particular end of the two filaments. As a consequence, the system has a natural tendency to create bending deformations with one particular sign of the curvature. Exchanging the role of the two filaments reverses this sign of bending deformations. This broken symmetry does however not reflect the symmetry of the axoneme which is symmetric with respect to all micro- tubule doublets. Each of the nine microtubule doublet within the cylindrical arrangement of the axoneme play identical roles and interact with rigidly attached motors as well as with motors which slide along their surfaces. If all motors generate a constant force, there is consequently no resulting bending deformation since all bending mo- ments cancel by symmetry. (a) (b) - - - - -- FIG. 6. Asymmetric (a) and symmetric (b) mo- tors/filament pair. The arrows indicate the polarity of the filaments. In case (a), spontaneous bending occurs. In case (b) both filaments play identical roles, no spontaneous bend - ing occurs. In order to introduce this axonemal symmetry in our two-dimensional model, we assume that both filaments have motors which are rigidly attached and which slide on the second filament, see Fig. 6. The consequence of this symmetry is that the filament polarity becomes unimportant. Exchanging the role of the two filaments is equivalent to replacing the sliding displacement ∆ of motors by −∆. Therefore, in the context of the two-state model this symmetrization is equivalent to the require- ment that energy landscapes Wi(x) and transition rates 7ωi(x) are symmetric functions with respect to x→ −x. As a consequence of this symmetry, there is no preferred direction of bending. In the following, we adopt this choice of a symmetric motor/filament pair. If we use the expansion given in Eq. (38), this symmetry implies that all even coefficients F(2n)= 0. D. Linear response function The perturbative treatment to study small filament de- formations introduced in section II can be naturally ex- tended to the coupled motor-filament situation using the expansion (38). Up to second order in ǫ, only the lin- ear response coefficient of the active system plays a role which for the two-state model is given by F(1) nk=χδnk with χ=K+λiω−ρ/integraldisplayl 0dξ∂ξ∆W∂ξR α+iωiω . (39) Here, we have introduced R≡ω2/αlandα=ω1+ω2. Higher order terms F(2n+1)have to be taken into account if the third or higher order in ǫis considered. The linear response function χas well as nonlinear coefficients can be calculated most easily for a simple choice of symmetric potential and transition rates ∆W(ξ) =Ucos(2πξ/l) ω1(ξ) =β−βcos(2πξ/l) ω2(ξ) =α−β+βcos(2πξ/l) (40) whereαandβareξ-independent rate constants. For this convenient choice χ(Ω,ω) =K+iλω−ρkΩiω/α+ (ω/α)2 1 + (ω/α)2,(41) wherek≡U/l2is the cross-bridge elasticity of the mo- tors. We have introduced the dimensionless parameter Ω = 2π2β/αwith 0<Ω< π2which plays the role of a control parameter of the motor-filament system, αis a characteristic ATP cycling rate. IV. SELF-ORGANIZED BEATING VIA A HOPF BIFURCATION A Hopf-bifurcation is an oscillating instability of an ini- tial non-oscillating state which occurs for a critical valu e Ωcof a control parameter. For Ω <Ωc, the system is passive and not moving, while for Ω >Ωcit exhibits spontaneous oscillations. As we demonstrate below, self- organized oscillations of the driven filament pair are a natural consequence of its physical properties. The con- trol parameter Ω is in our two-state model a ratio ofchemical rates of the ATP hydrolysis cycle. In an exper- imental situation, it could be varied e.g. by changing the ATP concentration. If the molecular motors are regu- lated by some other ion concentration such as e.g. Ca2+, this concentration could also play the role of the control parameter. A. Generic aspects For oscillations in the vicinity of a Hopf-bifurcation, filament deformations are small and we can thus use Eq. (26) to describe the filament dynamics. The force ampli- tude˜fcan be obtained from the expansion ˜f=χ(Ω,ω)˜∆ +O(˜∆3) (42) in the amplitude of sliding displacements. In addition, this amplitude is related to the filament shape: ∆(s) =a(∂sh(s)−∂sh(0)) +O(h3). (43) With Eq. (30), we find that spontaneously oscillating modes ˜h(s) at frequency ωare to linear order near the bifurcation solutions to κ∂4 s˜h−σ∂2 s˜h+ξ⊥iω˜h=a2χ∂2 s˜h (44) Note that this equation is general and its structure does not depend on the specific model chosen for the force generating elements. Boundary conditions corresponding to cases A-D follow from Table II by replacing h→˜h, ∂th→iω˜handf→˜f=aχ(∂s˜h−∂s˜h(0)). As discussed in Appendix D, Eq. (44) has the struc- ture of an eigenvalue problem, with χplaying the role of a complex eigenvalue. For every choice of parameters, there exists an infinite set of nontrivial solutions ˜hnto Eq. (44) if χis equal to the corresponding eigenvalue χn,n= 1,2,... We order these values according to their amplitude: |χn| ≤ |χn+1|. The functional dependence of the Eigenvalues χnon the model parameters is com- pletely determined by dimensionless functions ¯ χn(¯σ,¯ω) according to χn=κ L2a2¯χn(¯σ,¯ω), (45) where ¯σ≡σL2/κand ¯ω≡ωξ⊥L4/κare a dimensionless tension and frequency, respectively. The existence of discrete eigenvalues reveals that only particular values of the linear response function of the active material are consistent with being in the vicinity of a Hopf-bifurcation. If the actual value of χ(Ω,ω) of the system differs from one of the eigenvalues χn, the system is either not oscillating, or it oscillates with larg e amplitude for which higher order terms in ǫcannot be neglected. An important consequence of this observa- tion is that the modes ˜hnof filament beating close to a 8Hopf-bifurcation can be calculated without knowledge of the properties of the active elements. Not even knowl- edge of the linear response coefficient is required since it follows as an eigenvalue at the bifurcation. Therefore, filament motion close to a Hopf bifurcation has generic or universal features which do not depend on details of the problem such as the structural complexity within the axoneme. The motion is given by one of the modes ˜hn which only depend on ¯ σand ¯ω. χ_χ_ χ_χ_ χ_1 23 ω_=105 =1,7.103 =5.103 -60ω_ -60 -30-30 00 Im Reω_ FIG. 7. (a) Eigenvalues ¯ χnfor boundary conditions B. Each eigenvalue is represented by a line which traces the loc a- tions of ¯χnin the complex plane for varying reduced frequency ¯ω. The lines begin for ¯ ω= 0 on the real axis, the value of ¯ ω at the other ends are indicated. Fig. 4 displays the amplitude Hand the gradient of the phase ˙φof the fundamental mode ˜h1=He−iφ. Note, that we use arbitrary units for Hsince the amplitude of ˜h(s) is not determined by Eq. (44). Corresponding fil- ament shapes represented according to Eq. (25) in the (x,y) plane, are displayed in Fig. 5 for boundary con- ditions A-C as snapshots taken at different times. The oscillation amplitudes are chosen in such a way that the maximal value of ψ≃π/2. The direction of wave propa- gation depends on the boundary conditions. For clamped head (case A), waves propagate towards the head whereas for fixed or free head (cases B and C), they propagate towards the tail. The positions of the lowest eigenvalues ¯χnin the complex plane for boundary conditions B are displayed in Fig. 7 for ¯ σ= 0 and varying ¯ ω. B. Selection of eigenmodes and frequency In order to determine which of the modes ˜hnis selected and at what frequency ωthe system oscillates at the bi- furcation point, explicit knowledge of the linear response functionχ(Ω,ω) is necessary. As a simple example, we useχas given by Eq. (41) for a two-state model. For Ω = 0,χ=K+iλωwhich is a passive viscoelastic re- sponse. In this case, no spontaneous motion is possiblewhich can be seen by the fact that all eigenvalues ¯ χn have negative real and imaginary parts which cannot be matched by the linear response of a passive system. If Ω is increased, we are interested in a critical point Ω = Ω c for which the straight filament configuration becomes un- stable. This happens as soon as the linear response func- tionχmatches for a particular frequency ω=ωcone of the complex eigenvalues, κ a2L2¯χn(¯σ,¯ωc) =χ(Ωc,ωc), (46) where ¯ωc=ξ⊥L4ωc/κ. Sinceχis complex, both the real and the imaginary part of Eq. (46) represent indepen- dent conditions. Therefore, Eq. (46) determines both the critical point Ω cas well as the selected frequency ωc. The selected mode nis the first mode (beginning from Ω = 0) to satisfy this condition. Since |χ(Ω,ω)|typi- cally increases with increasing Ω, the instability occurs almost exclusively for n= 1 since |¯χ1|has been defined to have the smallest value. For Ω >Ωc, but close to the transition, the system oscillates spontaneously with fre- quencyωc. The shapes of filament beating are character- ized by the corresponding mode ˜hn(s) whose amplitude is determined by nonlinear terms. For the case of a con- tinuous transition, the deformation amplitude increases as|˜hn| ∼(Ω−Ωc)1/2. C. Axonemal vibrations for different lengths We choose the parameters of our model to correspond to the axonemal structure. We estimate the bending rigidityκ≃4·10−22Nm2which is the rigidity of about 20 microtubules. Furthermore, we choose a microtubule separation a≃20nm, motor density ρ≃5·108m−1, friction per unit length λ≃1kg/ms, rate constant α≃103s−1, cross-bridge elasticity k≃10−3N/m and a perpendicular friction ξ⊥≃10−3kg/ms which is of the order of the viscosity of water. A rough estimate of the elastic modulus per unit length associated with filament sliding can be obtained by comparing the num- ber of dynein heads to the number of nexin links within the axoneme. This suggests that Kis relatively small, K< ∼kρ/10. The selected frequency ωcat the bifurcation point for case B is shown in Fig. 8 as a function of the axoneme length forK= 0. For small lengths, the oscillation fre- quency is large and increases as Ldecreases. In this high-frequency regime, which occurs for L< ∼10µm, the system vibrates in a mode with no apparent wave prop- agation. For L> ∼10µm the frequency is only weakly L- dependent and the system propagates bending waves of a wave-length shorter than the filament length. The limit of small lengths corresponds to small ¯ ωand can be studied analytically. For ¯ σ= 0, the eigenvalue χ1is given to linear order in ¯ ωby ¯χ1≃ −π2/4−iγ¯ω. 9The coefficient γdepends on boundary conditions but not on any model parameters. For a clamped head (A), γ≃0.184, for a fixed head (B), γ≃0.008, see Appendix D. The criterion for a Hopf bifurcation for small Lis χ(Ωc,ωc)≃ −π2κ 4a2L2−iγξ⊥ωcL2 a2(47) Together with Eq.(41), we find the critical frequency ωc≃π 2L2/parenleftbiggκα γξ⊥/parenrightbigg1/2/bracketleftbigg1 + (L/L K)2 1 + (Lλ/L)2/bracketrightbigg1/2 ,(48) whereLK≡(π2κ/4Ka2)1/2andLλ≡(λa2/γξ⊥)1/2are two characteristic lengths. For Kandλsmall,LK≫ L≫Lλand the critical frequency behaves as ωc∼1/L2. 1 10 505110 5 0.5ω /2π (kHz)c L (µm) FIG. 8. Oscillation frequency ωc/2πat the bifurcation point as a function of the axoneme length Lfor boundary conditions B and model parameters as given in the text. The insets show characteristic patterns of motion for small and large lengths. The critical value of the control-parameter for small L is given by Ωc≃κπ2 4a2ρkL2+K+λα ρk+αγξ⊥L2 ρka2. (49) For axoneme lengths below a characteristic value, L < Lmin, the condition Ω c≤π2is violated. Therefore, os- cillations exist only for L> L min. If (K+λα)/ρk≪1, Lmin≃/radicalbig κ/4a2ρk≃1µm for our choice of the param- eters. The maximal frequency of oscillations occurs for L=Lmin. Within the range of lengths between 1 µmand 50µm, we find frequencies ωc/2πwhich vary between tens of Hz and about 20 kHz. The parameters chosen in Fig. 8 lead to frequencies above 150 Hz. Lower frequencies are obtained for larger values of the friction λor smaller chemical rate α. D. Effect of external forces applied at the tail In the presence of an external force applied at the end ¯σ/negationslash= 0 and the filament pair is under tension. The eigen-values ¯χndepend on ¯σ, therefore the tension affects shape and frequency of an unstable mode. Inspection of Eq. (44) suggests that the tension σcan be interpreted as an additive contribution to the linear response coefficient χ→χ+σ/a2. Taking into account the boundary condi- tions complicates the situation, the ¯ σ-dependence of the eigenvalues is in general nontrivial. However, for a fila- ment with clamped head and a force applied at the tail parallel to the tangent at the head (case D) the effect of tension can be easily studied. In this particular case, the tension leads to the same contribution to χboth in Eq. (44) and in the boundary conditions and the eigenvalues thus depend linearly on σ: ¯χn(¯σ,¯ω) = ¯χn(0,¯ω)−¯σ . (50) Since the tension corresponds to a change of the real part ofχ, its presence has the same effect on filament motion as an increase of the elastic modulus Kper unit length in Eq. (41) by σ/a2. Therefore, we can include the effects of tension on the critical frequency in Eq. (48) by replac- ingK→K+σ/a2which reveals that in those regimes where the frequency depends on K, it will increase for increasing tension. The critical value of the control pa- rameter Ω cis a function of the tension, with Eq. (49), we find Ωc(σ) = Ω c(0) +1 ρka2σ . (51) For our choice of parameters, ρka2≃200pN which in- dicates that forces of the order of 102pNshould have a significant effect on the bifurcation. Furthermore, Eq. (51) indicates that the tension σcan play the role of a second control parameter for the bifurcation. Consider a tensionless filament oscillating close to the bifurcation for Ω>Ωc(0). If a tension is applied, the critical value Ωc(σ) increases until the system reaches for a critical tensionσca bifurcation point with Ω c(σc) = Ω and the system stops oscillating. V. DISCUSSION In the previous sections, we have shown that many aspects of axonemal eating can be described by a simple model based on the idea of local sliding of microtubule doublets driven by molecular motors inside the axoneme. This model represents a class of physical systems termed internally driven filaments (Camalet et al. 1999) which have characteristic properties that are closely linked to the geometric constraint that couples global bending and local filament sliding. Our work shows that axonemal beating can be studied both numerically and analytically using a coarse-grained description which ignores many details of the proteins in- volved and is based on effective material properties such 10as bending rigidities, internal friction coefficient and ela s- tic modulus per unit length as well as the frequency de- pendent linear response function of the active elements. Our main results are: (i) The filament pair introduces a geometric coupling of the active elements at different po- sition along the filaments. This coupling is suited for the generation of periodic beating motion by a general self- organization mechanism of the system. (ii) The genera- tion of oscillations and propagating bending waves occurs via a Hopf bifurcation for a critical value of a control- parameter such as the ATP concentration. The patterns of motion generated close to this bifurcation can be cal- culated without any knowledge of the internal force gen- erating mechanism if the oscillation frequency is known. (iii) The frequency depends on chemical rates of the mo- tors, coefficients of internal elasticities and frictions, t he solvent viscosity and the microtubule rigidity. Our cal- culations using a simple two-state model for molecular motors suggest significant variations of the oscillation fr e- quency if the axonemal length is varied. Short axonemes are predicted to be able to oscillate at frequencies of sev- eral kHz. This fact is particularly interesting in the case of kinocilia which are located in the hair bundles of many mechanosensitive cells that are involved in the detection of sounds. If cilia are able to vibrate at high frequencies, this suggests that the kinocilium could play as active role in sound detection (Camalet et al. 2000). (iv) Forces ap- plied to the axoneme can be an important tool to study the mechanism of force generation. An external force can play the role of a second control parameter for the bifurcation which has a stabilizing effect for increasing forces. These results have been obtained using several simpli- fying assumptions. Our model represents the solvent hy- drodynamics by an anisotropic local friction acting on the rod-like filament. This approach ignores hydrodynamic interactions between different parts of the filament which lead to logarithmic corrections. These hydrodynamic ef- fects do not change the basic physics but can lead to cor- rections to the numerical results. It is straightforward to generalize our model to incorporate the effects of hydro- dynamic interaction, but this does not change our results qualitatively. A more serious simplification of our model is the restriction to a two-dimensional system and to fila- ment configurations which are planar. This choice is mo- tivated by the fact that observed bending waves of flagella are planar in many cases (Brokaw 1991). If filament con- figurations are planar then our model applies and is com- plete. However, this model cannot explain why motion is confined to a plane and it misses all non-planar modes of beating. In order to address such questions, a three dimensional generalizations have to be used. Such gen- eralizations introduce additional aspects. In particular , torsional deformations become relevant. The local sliding displacements depend in the three dimensional case both on bending and torsional deformations and the full tor-sional dynamics has to be accounted for in the dynamic equations. Finally, we have restricted most calculations to the limit of small deformations which corresponds to filament shapes that are almost straight. This regime has several important features. The filament dynamics can be studied by an analytic approach by a systematic ex- pansion in the deformation amplitude. This allows us to characterize linear and nonlinear terms both for the fila- ment dynamics as well as for the properties of the active elements. Patterns of motion close to a Hopf bifurcation are fully characterized by the linear terms of this expan- sion. These linear terms are given by the structure of the problem and their form does not depend on molecular details of the system. For example, most details of the operation of the molecular motors are unimportant for the shape of the filament oscillations at the bifurcation, only the linear response function of the active material plays a role. If filament beating with larger amplitude is of interest, nonlinearities become relevant. Nonlinearities arise due to nonlinear geometric terms in the bending energy or via non-Hookian corrections of the elasticity. Furthermore, nonlinearities in the force-generation process of molecul ar motors exist. All these nonlinearities determine the large amplitude motion and could give rise to new types of be- havior such as additional dynamic instabilities. However, in contrast to the linear terms, the form of the dominant nonlinearities does depend on structural details of the ax- oneme. Therefore, an analysis of large amplitude motion is difficult and knowledge of the nature of the dominating nonlinearities is required. However, if no new instabili- ties after the initial Hopf-bifurcation occur, the princi- pal effect of nonlinear terms is to fix the amplitude of propagating waves. We therefore expect that propagat- ing waves with larger amplitude are in many cases well approximated by our calculation to linear order. Our work shows that propagating bending waves and oscillatory motion of internally driven filaments can oc- cur naturally by a simple physical mechanism. Complex biochemical networks to control the system are not re- quired for wave propagation to occur. This suggests that the basic axonemal structure intrinsically has the ability to oscillate. Biochemical regulation systems are thus ex- pected to control this activity on a higher level but are not responsible for oscillations. Our work shows that ex- perimental studies of axonemal beating close to a Hopf bifurcation would be very valuable. Such experiments could be e.g. performed by using demembranated flag- ella and ATP concentrations not far from the level where beating sets in. The observed behavior close to the bifur- cation would give insight in the self-organization at work. Furthermore, externally applied forces and manipulation of boundary conditions could be sensitive tools to test some of the predictions of our work. 11ACKNOWLEDGMENTS We thank A. Ajdari, M. Bornens, H. Delacroix, T. Duke, R. Everaers, K. Kruse, A. Maggs, A. Parmeggiani, M. Piel, J. Prost and C. Wiggins for useful discussions. APPENDIX A: FUNCTIONAL DERIVATIVE OF THE ENTHALPY The variation δGof the enthalpy Ggiven by Eq. (8) under variations δrof a shape r(s) is δG=/integraldisplayL 0[(κC−aF)δC+ Λ˙r·δ˙r]ds . (A1) Note, that ˙r2= 1 but ( ˙r+δ˙r)2/negationslash= 1 in general. Under such a variation, δC=δ(n·¨r) =n·δ¨rsince¨r·δn=Cn·δn= 0. Therefore, δG=/integraldisplayL 0[(κC−aF)n·δ¨r+ Λt·δ˙r]ds . (A2) Two subsequent partial integrations lead to δG= [(κC−aF)n·δ˙r]L 0(A3) +/bracketleftBig/parenleftBig −(κ˙C−af)n+ (κC2−aCF+ Λ)t/parenrightBig ·δr/bracketrightBigL 0 +/integraldisplayL 0∂s/parenleftBig (κ˙C−af)n−(κC2−aCF+ Λ)t/parenrightBig ·δrds . The functional derivative δG/δrgiven by Eq. (11) can be read off from the integrand, the boundary terms provide expressions for forces and torques at the ends given by Eqs. (19) and (20). APPENDIX B: SMALL DEFORMATIONS Inserting the expansions (21) in the differential Eq. (17) for the tension profile leads at each order in ǫto a separate equation. Up to second order we find ¨τ0= 0 (B1) ¨τ1= 0 ¨τ2=∂s(˙ψ1(af1−κ¨ψ1)) +ξ/bardbl ξ⊥˙ψ1(−κ... ψ1+a˙f1+τ0˙ψ1). From the boundary conditions, it follows that τ0=σis constant and τ1= 0. Repeating the same procedure for the dynamic Eq. (16) using τ1= 0 andτ0=σ, we obtain ξ⊥∂tψ1=−κ.... ψ1+a¨f1+σ¨ψ1 ξ⊥∂tψ2=−κ.... ψ2 (B2) The Equation for ψ2is independent of f1and only de- scribes transient behavior which depends on initial con- ditions. After long times and for limit cycle motion,ψ2= 0. The transverse and longitudinal displacements h,uand the velocity ¯ vare obtained perturbatively as h=ǫh1+ǫ2h2+O(ǫ3) u=ǫu1+ǫ2u2+O(ǫ3) ¯v=ǫ¯v1+ǫ2¯v2+O(ǫ3). (B3) Using Eqns. (21) and (18) we find u1(s) =u1(0),h2(s) = h2(0) and h1(s) =h1(0) +/integraldisplays 0ψ1(s′)ds′(B4) u2(s) =u2(0)−1 2/integraldisplays 0ψ1(s′)2ds′. (B5) The dynamics of h(s) andu(s) to second order in ǫis therefore determined by ψ1(s,t) and the motion ∂trfor s= 0. For the latter, we find from Eq. (15) ¯ v1= 0, ∂tu1(0) = 0,∂th2(0) = 0 and ξ⊥∂th1(0) =−κ.... h1(0) +τ0¨h1(0) +a˙f1(0) (B6) ξ/bardbl(¯v2+∂tu2(0)) = −ξ/bardblψ1(0)∂th1(0) +κ˙ψ1(0)¨ψ1(0) −a˙ψ1(0)f1(0) + ˙τ2(0). (B7) For the lateral motion, we obtain with Eq. (B2) and (B4), ξ⊥(∂th1(s)−∂th1(0)) =/bracketleftBig −κ.... h1+σ¨h1+a˙f1/bracketrightBigs 0(B8) Using Eq. (B6), we obtain ξ⊥∂th1=−κ.... h1+σ¨h1+a˙f1, (B9) which is the result given in Eq. (26). In order to calculate τ2(s) and ¯v2we integrate Eq. (B1). Together with Eq. (B9), ˙τ2(s) = ˙τ2(0) +/bracketleftBig ˙ψ1(af1−κ¨ψ1)/bracketrightBigs 0 +ξ/bardbl/integraldisplays 0˙ψ1∂th1ds′. (B10) After elimination of ˙ τ2(0) via Eq. (B7) we find the tension profile τ2(s) =τ2(0) +sξ/bardbl(¯v2+∂tu2(0)) +/integraldisplays 0ds′/bracketleftBig ˙ψ1(af1−κ¨ψ1) +ξ/bardblψ1∂th1/bracketrightBig −ξ/bardbl/integraldisplays 0ds′/integraldisplays′ 0ds′′ψ1∂tψ1. (B11) The tension at the head τ2(0), as well as ¯ v2+∂tu2(0) are fixed using the boundary conditions. In cases A,B and D where the head is fixed ¯ v2+∂tu2(0) = 0. In this case, τ2(0) follows from Eq. (B11) at s=L. For boundary condition C, the velocity of the head is determined from 12the condition τ2(L) = 0 together with the boundary con- ditionsτ2(0) =−ψ1(0)(κ¨ψ1(0)−af1(0))+ζ(¯v2+∂tu2(0)) andκ... h1(L) =af1(L) which leads to ¯v2+∂tu2(0) =ξ⊥−ξ/bardbl ζ+ξ/bardblL/integraldisplayL 0˙h1∂th1ds +ξ/bardbl ζ+ξ/bardblL/integraldisplayL 0ds/integraldisplays 0ds′1 2∂t(˙h2 1).(B12) Averaging this equation for periodic motion h1(s,t) = H(s)cos(ωt−φ(s)) leads to Eq. (32). APPENDIX C: LINEAR AND NONLINEAR RESPONSE FUNCTIONS OF THE TWO-STATE MODEL We introduce the Fourier modes of the distribution functionP1(ξ,t) =/summationtext kP1(ξ,k)eikωt. The dynamic equa- tion (35) of the two-state model can then be expressed as P1(ξ,k) =δk,0R(ξ)−iω/summationdisplay lmlδk,l+m α+iωk∆l∂ξP1(ξ,m) (C1) Inserting the Ansatz P1(ξ,k) =Rδk,0+P(1) kl∆l+P(2) klm∆l∆m+O(∆3) (C2) into Eq. (C1), one obtains a recursion relation P(n)(ξ)k,k1,..,k n=−iω/summationdisplay llδk,kn+l α+iωk∂ξP(n−1) l,k1,..,k n−1. (C3) the knowledge of the functions P(n) k,k1,..,k nallows one to calculate the coefficients of the expansion (38). In par- ticular, with P(0)=R, one obtains the result (39) and F(n) k,k1,..,k n=ρ/integraldisplayl 0dξP(n) k,k1,..,k n(ξ)∂ξ∆W (C4) APPENDIX D: EIGENVALUES AND EIGENMODES NEAR A HOPF BIFURCATION Nontrivial solutions to Eq. (44) can be found using the Ansatz ˜h=Aeqs/Lwithqbeing solution to q4−(¯σ+ ¯χ)q2+i¯ω= 0 (D1) Here, ¯σ≡σL2/κ, ¯ω≡ωξ⊥L4/κ, and ¯χ≡χa2L2/κare a dimensionless tension, frequency and linear response coefficient, respectively. Eq. (D1) has four complex so- lutionsqi,i= 1,..,4. Therefore,˜h=4/summationdisplay i=1Aieqis/L. (D2) In order to determine the amplitudes Aithe four bound- ary conditions have to be used. Since Eq. (44) is linear and homogeneous, the equations for the coefficients Ai have the form 4/summationdisplay i=1AiMij= 0, (D3) whereMijis a 4×4 matrix which depends on ¯ χ, ¯σand ¯ω. Nontrivial solutions exist only for those values ¯ χ= ¯χn for which det Mij= 0. The corresponding eigenmode is the solution for Aiof Eq. (D3). These eigenvalues and eigenfunctions can be determined by numerically ob- taining solutions for det Mij= 0. In the limit of small ¯ω, analytic expressions can be obtained by inserting the ansatz ¯χ= ¯χ(0)+ ¯χ(1)¯ω+O(¯ω2) for the eigenvalue which leads to an expansion of det Mijin powers of ¯ ω1/2. This procedure leads for ¯ σ= 0 to ¯χ(0) n=−[(n−1)π+π/2]2 independent of boundary conditions. The linear coeffi- cient ¯χ(1) 1=−iγdepends on boundary conditions. For boundary conditions A γ=12π3−32π2+ 1728π−4608 π5+ 144π3≃0.184,(D4) while for case B, γ=192 π7/parenleftbiggπ5 32+π4 12−π3+ 2π2−2π/parenrightbigg ≃0.008.(D5) B. Alberts, D. Bray, J. Lewis, M. Raff, K. 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arXiv:physics/0003102v1 [physics.flu-dyn] 30 Mar 2000Transition from the Couette-Taylor system to the plane Coue tte system Holger Faisst and Bruno Eckhardt Fachbereich Physik, Philipps Universit¨ at Marburg, D-350 32 Marburg, Germany We discuss the flow between concentric rotating cylinders in the limit of large radii where the system approaches plane Couette flow. We discuss how in this l imit the linear instability that leads to the formation of Taylor vortices is lost and how the charac ter of the transition approaches that of planar shear flows. In particular, a parameter regime is iden tified where fractal distributions of life times and spatiotemporal intermittency occur. Experiment s in this regime should allow to study the characteristics of shear flow turbulence in a closed flow g eometry. 47.20.Lz,47.27.Cn,47.20.Ft,47.20.-k The transition to turbulence for a fluid between con- centric rotating cylinders has attracted much experimen- tal and theoretical attention. Ever since Taylor’s success [1] in predicting and observing the instabilities for the formation of vortices the system has become one of the paradigmatic examples for the transition to turbulence and a large number of bifurcations have been analyzed in considerable detail [2–5]. The limiting case of large radii and fixed gap width where the effects due to cur- vature become less important and where the system ap- proaches plane Couette flow between parallel walls has received much less attention. In this limit the character of the flow changes: plane Couette flow is linearly stable and the mechanisms that drive the transition to turbu- lence are still unclear. The question we address here is to what extend the Couette-Taylor system can be used to gain insight into the dynamics of plane Couette flow. This problem is of both experimental and theoretical interest. As mentioned, the experimental situation for Couette-Taylor flow is much better, there being numer- ous facilities and detailed studies of patterns, boundary effects and critical parameters [4–6]. The moving bound- aries in plane Couette flow reduce the experimental ac- cessibility and the possibilities of applying controlled p er- turbations. On the theoretical side it is an intriguing question how the change in stability behaviour from the Couette Taylor system to the plane Couette system oc- curs. Studies by Nagata [7] show that some states from the rotating plane Couette system survive the limiting process and appear in finite amplitude saddle node bifur- cations in the plane Couette system (see also the investi- gation of this state by Busse and Clever [8]). Unless the transition from linear instability dominated behaviour in Couette-Taylor flow to the shear flow type transition in plane Couette flow is singularly connected to the absence of any curvature it can be expected to happen at a finite radius ratio near which interesting dynamical behaviour should occur. We should mention that there are other useful embed- dings of plane Couette. Busse and Clever [8] start from a layer of fluid heated from below with cross flow and pro- ceed to study the stability and parameter dependence of the states. And Cherhabili and Ehrenstein [9] start from plane Poisseuille flow and find localized solutions, albeitat Reynolds numbers higher than the ones studied here. Our aim here is to follow some of the instabilities in the Couette-Taylor system to the limit of the plane Couette system and to identify the parameters where the change in behaviour occurs. In particular, we study the transi- tion from laminar Couette flow to Taylor vortices and the instability of vortices to the formation of wavy vortices. Note that the asymptotic situation of plane Couette flow can be characterized by a single parameter, a Reynolds number based on the velocity difference, whereas Couette Taylor flow has at least two parameters, the Reynolds numbers based on the velocities of the cylinders. This extra degree of freedom provides an additional parame- ter that can be used to modify the flow without changing the basic features. In cylindrical coordinates ( r, φ, z) the equations of mo- tion for the velocity components ( ur, uφ, uz) can be writ- ten as ∂tur+/parenleftBig u·˜∇/parenrightBig ur−ν˜∆ur+˜∇p= ν/parenleftbigg1 r∂rur−2 r2∂φuφ−1 r2ur/parenrightbigg +1 ru2 φ (1) ∂tuφ+/parenleftBig u·˜∇/parenrightBig uφ−ν˜∆uφ+˜∇p= ν/parenleftbigg1 r∂ruφ+2 r2∂φur−1 r2uφ/parenrightbigg −1 ruruφ (2) ∂tuz+/parenleftBig u·˜∇/parenrightBig uz−ν˜∆uz+˜∇p= ν1 r∂ruz (3) ˜∇ ·u=−1 rur (4) where the modified Nabla and Laplace operators are ˜∇=er∂r+eφ1 r∂φ+ez∂z, (5) ˜∆ =∂rr+1 r2∂φφ+∂zz, (6) and where eiare the unit basis vectors [10] The terms in eqs. (1)-(4) are arranged so that all the ones on the right hand side vanish when the system ap- proaches the plane Couette system, i.e. in the limit of 1large radii but finite velocities at the cylinders. The re- maining ones become the equations of motion for plane Couette flow in cartesian coordinates ( x, y, z ) if the iden- tification x=randy=φris made. However, there are other ways of taking the limit of a small gap that lead to different limiting systems. For instance, the case of almost corotating cylinders with high mean rotation rate gives rise to plane Couette flow with an additional Corio- lis term (‘rotating plane Couette flow’ [7]). Another limit corresponds to the case of counterrotating cylinders with diverging rotation rates [11]. In our numerical work we use the full equations, without any reduction in terms. This allows us to extend Nagata’s work from the rotating plane Couette flow to the full Couette-Taylor system. The velocities at the inner and outer cylinder (distin- guished by indices iando, respectively) are prescribed and define the boundary conditions uφ(r=Rx) = Ω xRx, (7) ur(r=Rx) =uz(r=Rx) = 0, x =i, o. (8) For the choice of dimensionless quantities we appeal to the plane Couette flow limit. There the relevant quantities are the velocity difference between the walls, ∆U=RiΩi−RoΩo, and the gap width d=Ro−Ri. Without loss of generality we can always assume Ω i≥0. The Reynolds number for plane Couette flow is based on half the velocity difference and half the gap width, Re=∆Ud 4ν. (9) For the Couette Taylor system there are two Reynolds numbers based on the gap width and the rotation rates of the inner and outer cylinders, Rex=RxΩxd/ν , (10) where the index xcan stands for ioro, the inner and outer cylinders. The plane Couette flow Reynolds num- ber thus is Re= (Rei−Reo)/4. The ratio of these Reynolds numbers will be called ˜µ=Reo/Rei, (11) (the tilde is used to distinguish it from µ= Ω o/Ωi, a frequently defined quantity not used here). η=Ri/Ro (12) denotes that ratio of radii. Experiments and numerical simulations show that plane Couette flow undergoes a subcritical transition to turbulence around RePCF≈320 [12–14]. The Couette- Taylor system shows a first linear instability to the for- mation of vortices (Taylor-vortex flow, TVF) at Reynolds numbers that depend on the rotation rates and the curva- ture of the cylinders. In order to see shear flow dominated dynamics the critical Reynolds number for the linear in- stability has to be above RePCF. The formation of TVFoccurs at Reynolds numbers that can be parametrized in the form Re=A(˜µ)(1−η)−1/2+B(˜µ) (13) forη<∼1 [15]. This number is larger than the tran- sitional Reynolds number for plane Couette flow if ηis sufficiently close to one. The minimal radius ratio η320 where the linear instability occurs for Re > 320 strongly depends on the ratio of the Reynolds numbers of inner and outer cylinder. A few examples for minimal radius ratios η320are summarized in Table I. Very important for the transition to turbulence in lin- early stable systems are nonlinear processes that could give rise to some finite amplitude states, perhaps station- ary or periodic, around which the turbulent state could form. One candidate that could serve as a nucleus for tur- bulence in plane Couette flow is the stationary state first calculated by Nagata [7]. He observed that the wavy vor- tices that form in a secondary instability from the TVF in the rotating plane Couette system can be followed to the limit of the plane Couette flow where they become part of a saddle node bifurcation at finite Reynolds numbers. This state was also identified and studied in a different limiting process by Busse and Clever [8]. They found that the critical axial and azimuthal wavelengths for this state are λz=πandλφ= 2π . (14) This is roughly twice the critical wave lengths that would be expected for Taylor vortices. We developed a numerical code for the solution of the full Navier-Stokes equation using Fourier modes in ax- ial and azimuthal direction and Legendre collocation in the radial direction. The pressure terms were treated by a Lagrange method. The period in zandφwas de- termined by the fundamental wave lengths (14) of wavy vortex flow. The continuation of the wavy vortex flow from the Couette-Taylor system to the plane Couette system is shown in Fig. 1 for the case of the outer cylinder at rest (˜µ= 0) and for counterrotating cylinders with ˜ µ=−1. For small ηthe wavy vortex develops from a secondary bifurcation of TVF, but for sufficiently large ηthe wavy vortex state is created first in a saddle node bifurcation. The critical Reynolds number for the formation of Tay- lor vortices diverges as ηapproaches one, but the one for the formation of wavy vortices approaches a finite value. Thus the gap in Reynolds numbers between the two transitions widens and the region where plane Cou- ette flow like behaviour can be expected increases with ηapproaching one. The radius ratios ηcand Reynolds numbers Recof the codimension two point where the in- stabilities for TVF and wavy vortex flow cross are listed in Table I. The ratio of radii ηcwhere the linear insta- bility of Couette flow and of the Taylor vortex flow cross is a non-monotonic function of the ratio ˜ µof rotation 2speed. Both the critical Reynolds numbers for the linear instability of the Couette profile and for the formation of wavy vortices increase with decreasing ˜ µ, but at different rates and with different dependencies of η. As a conse- quence there seems to be a local minimum near about 0.93 for ˜ µclose to −1. For the parameter value considered here the curvature of the cylinder walls is geometrically small (see Fig. 2). On the length of one unit cell in φ-direction the rela- tive displacement in radial direction from a planar wall is about π(1−η), i.e. only 3% for η= 0.99. The critical Reynolds number for the formation of wavy vortex flow (WVF) seems to converge to the same value for both ratios ˜ µshown in Fig. 1. The critical Reynolds number as well as the rotation speed of the wavy vortices for several different ratios ˜ µare collected in Fig. 3. The rotation speed is defined as the angu- lar phase velocity ωof WVF times the mean radius R= (Ri+Ro)/2 minus the mean azimuthal velocity v= (ωiRi+ωoRo)/2. For all ratios between the speed of inner and outer cylinder the critical Reynolds number for the formation of the wavy vortex state converges to a value of about 125 and the speed of rotation goes to zero. The limiting state that is approached is the stationary Nagata-Busse-Clever state. The velocity field of a wavy vortex solution at η= 0.993, ˜µ=−1 and Re= 124 is shown in Fig. 4; it differs little from the corresponding plane Couette state obtained by Busse and Clever [8], both in appearance and in critical Reynolds number. In the region above the wavy vortex instability but below the linear instability the dynamics of perturba- tions shows the fractal life time pictures familiar from plane Couette flow [16]. Fig. 5 shows an example at a radius ratio of η= 0.993 and a Reynolds number ratio of ˜µ=−1. The initial state was prepared by rescal- ing a WVF field obtained at very low radius ratio and Reynolds number. It is interesting to note that even with this initial condition, which is at least topologi- cally close to the Nagata-Busse-Clever state, it is not possible to realize a turbulent signal in its neighborhood: the state quickly leaves this region in phase space. One might have hoped that in spite of the linear instability of the Nagata-Busse-Clever state other states created out of secondary bifurcations could have supported some tur- bulent dynamics in its neighborhood, but the numerical experiments do not support this. The gap between the Reynolds number where the WVF state is formed and the one where typical initial conditions become turbulent is about the same as in plane Couette flow: the WVF states forms around Re= 125 and the transition to turbulence, based on the requirement that half of all perturbations induce a long living turbulent state, occurs near a value ofRetrans = 310, very much as in plane Couette flow [14]. In summary, we have identified parameter ranges in the Couette-Taylor system where some of the character- istics of the plane Couette system can be found. These parameter ranges include radius ratios that can be real-ized experimentally. Investigations in this regime should be rewarding as they open up the possibility to study the properties of the transition in a closed geometry and to switch continuously between supercritical and subcritica l transition to turbulence. The observation of a codimen- sion two point where the linear instability to TVF and the secondary instability to wavy vortex flow cross should provide a starting point for further modelling of the tran- sition in terms of amplitude equations. Acknowledgments This work was financially supported by the Deutsche Forschungsgemeinschaft. [1] G.I. Taylor. Phil. Trans. Roy. Soc. London ,223(A), 289, (1923). [2] H.A. Snyder. Int. J. Non-Linear Mechanics ,5, 659, (1970). [3] P.S. Marcus. J. Fluid Mech. ,146, 65, (1984). [4] C.D. Andereck, S.S. Liu, H.L. Swinney. J. Fluid Mech. , 164, 155, (1986). [5] R. Tagg. Nonlinear Science Today ,4(3), (1994). [6] L. Koschmieder. Cambridge University Press, (1993). [7] M. Nagata. J. Fluid Mech. ,217, 519, (1990). [8] R.M. Clever, F.H. Busse. J. Fluid Mech. ,344, 137, (1997). [9] A. Cherhabili, U. Ehrenstein. J. Fluid Mech. ,342, 159, (1997). [10] L.D. Landau, E.M. Lifshitz. Pergamon Press, (1987). [11] Y. Demay, G. Iooss, P. Laure. Eur. J. Mech., B/Fluids , 11(5), 621, (1992). [12] N. Tillmark, P.H. Alfredsson. J. Fluid Mech. ,235, 89, (1992). [13] S. Bottin, O. Dauchot, F. Daviaud. Phys. Rev. Lett. , 79(22), 4377, (1997). [14] A. Schmiegel, B. Eckhardt. Dynamics of perturbations i n plane Couette flow. submitted . [15] A. Esser, S. Grossmann. Phys. Fluids ,8(7), 1814, (1996). [16] A. Schmiegel, B. Eckhardt. Phys. Rev. Lett. ,79(26), 5250, (1997). 30.92 0.94 0.96 0.98 1.00 η0100200300400Re WVF ~µ = −1TVF ~µ = −1 WVF ~µ = 0 TVF ~µ = 0 FIG. 1. Bifurcations to Taylor vortex flow (TVF) and wavy vortex flow (WVF) in Couette-Taylor flow for the outer cylin- der at rest (˜ µ= 0) and counter-rotating cylinders (˜ µ=−1). The vertical line indicates the parameter range of the lifet ime measurements of Fig. 5 at ˜ µ=−1. η=0.9η=0.99η=0.999d2πd FIG. 2. Geometrical curvature of the cylinders in the Cou- ette-Taylor flow and the plane Couette flow limit. Shown is one fundamental azimuthal wavelength for different radiu s ratios ηas indicated.0.92 0.94 0.96 0.98 1.00 η50100150Re~µ=−∞ ~µ=−2.4142 ~µ=−1.0 ~µ=−0.4142 ~µ= 0.0 01020ωR−v FIG. 3. The convergence to the Nagata-Busse-Clever state for different rotation ratios ˜ µ. In the limit of ηgoing to one the wavy vortex states for all ˜ µapproach the same flow that moves with the mean velocity azimuthally. The top diagram shows the rotation speed and the botton one the critical Reynolds numbers. FIG. 4. The wavy vortex flow state near the Na- gata-Busse-Clever state at η= 0.993, ˜µ=−1 and Re= 124. Shown is only the disturbance, without the Couette profile. The frames from left to right show cuts through the ( r,z) plane at azimuthal wave lengths φ= 0,π/4,π/2, 3π/4 and π. The vectors indicate the randzcomponents of the ve- locity field and shading the φ-component. The inner (outer) cylinder is located at the left (right) side of each frame. 4270 290 310 330 350 Re0.010.11Amplitude 0<t<500 500<t<2000 2000<t<6000 6000<tFIG. 5. Life time distribution in Couette-Taylor flow at η= 0.993,˜µ=−1 and for the indicated range of Reynolds numbers. ˜µ A B η320 ηc Rec 0.0 10.8 0.5 0.999 0.990 109 -0.4142 16.8 0.8 0.997 0.977 110 -1.0 33.9 3.2 0.989 0.929 131 -2.4142 53.6 4.6 0.971 ≈0.94 ≈220 TABLE I. Parameters connected with the Couette-Taylor system in the limit of large radii. AandBare the coefficients in the parametrization (13) of the primary instability. η320 is the radius ratio where the primary instability lies above Re= 320; finally, ηcandRecare the parameter values for the crossing of the stability curves for Taylor vortex flow an d wavy vortex flow. 5

arXiv:physics/0003103v1 [physics.class-ph] 30 Mar 2000Acoustic coupling between two air bubbles in water Pai-Yi HSIAO Laboratoire de Physique Th´ eorique de la Mati` ere Condens´ ee Universit´ e Paris 7 – Denis Diderot case 7020, 2 place jussieu, 75251 Paris Cedex 05, FRANCE E-mail: hsiao@ccr.jussieu.fr Martin DEVAUD and Jean-Claude BACRI Laboratoire des Milieux D´ esordonn´ es et H´ et´ erog` enes Universit´ e Pierre et Marie Curie – Paris 6 case 78, 4 place jussieu, 75252 Paris Cedex 05, FRANCE E-mails: devaud @ccr.jussieu.fr ,jcbac@ccr.jussieu.fr Abstract Abstract – The acoustic coupling between two air bubbles immersed in wa ter is clearly demonstrated. The system is acoustically forced , and its response is detected. The experimental results confirm that both theore tically predicted eigenmodes, respectively symmetrical and antisymmetrica l, do exist. Their frequencies, measured as a function of the bubbles spacing, follow theoretical estimations within a 10% accuracy. Keywords: bubbles, eigenmodes, acoustics PACS : 43.20+g, 43.30Jx, 43.25Yw Typeset using REVT EX 11. INTRODUCTION Bubbles play an important role in the sound propagation in ev eryday life liquids. For example, the murmur of the brooks essentially originates, a s first suggested by Bragg [1,2], in the oscillations of air bubbles captured and dragged along b y the water. The so-called “hot chocolate effect”, namely the rising of sound pitch when one r epeatedly taps the bottom of the mug in which some instant coffee or chocolate is being diss olved, is explained by the releasing into the water of the tiny bubbles trapped in the po wder [3,4]. Bubble dynamics and acoustic properties of liquids containing a large number of bubbles have been widely studied for a long time [5–10]. Inter alia , the problem of the interaction of two neighbouring bubbles has been discussed using fluid dynamics tools [11] or the acou stic-electrestatic analogy [12]. Moreover, the free oscillations of a system of two (and even t hree) air cavities formed in a metal plate lying on a water surface have been theoretically and experimentally investigated in detail (including cubic nonlinearities) [13]. The aim of the present article is to present a simple, readily reproducible, experimental study of the forced oscillation regime of a two-air bubble system in water. We begin with a short introductory th eory in which we show that the two-bubble system is mostly equivalent to a set of two mag netically coupled electric circuits. 2. THEORETICAL MODEL An air bubble in water will be considered as a perfect sphere1of radius R(t) =R0+ξ(t), with variation ξmuch smaller than equilibrium value R0. It can be shown that ξ(t) oscillates with Minnaert’s angular frequency ω0=/radicalBig 3γP0/ρ0R2 0, where γis the specific heat ratio 1The correction to the Minnaert angular frequency due to devi ation from the spherical shape can be shown to be negligible [14–16]. 2Cp/Cvof air, and P0andρ0respectively stand for the equilibrium pressure2and mass density of water. This oscillation is damped through severa l mechanisms: of course the acoustic radiation damping (thanks to which the bubble nois e is audible), but also the viscous and thermal dampings [5,7]. We will neglect, in the f ollowing simplified theory, the last two ones. Moreover, allowing for the typical 1 KHz acoustic frequency and 1 mmbubble size we deal with in our experiment, we will neglect any sound propagation in the enclosed air. We thus deliberately restrict the present study to the ( radial) fundamental resonance of the air bubble-water system. 2.1 One-bubble free oscillation Let us consider one bubble with radius R0immersed in an infinite volume of water at equilibrium pressure P0. LetP(/vector r, t) be the actual pressure at site /vector rand instant t. The extra pressure p(/vector r, t) is defined as P(/vector r, t)−P0. According to Minnaert’s assumption, the enclosed air undergoes isentropic transformations and its (extra) p ressure p(t) is homogeneous inside the bubble. Then, neglecting air’s inertia as well as the air -water surface tension, p(t) and the radius variation ξ(t) are linked by: p(t) P0+3γξ(t) R0= 0 (1) On the other hand, it can be easily shown that (extra) pressur ep(r, t) at distance rfrom the center of the bubble follows a d’Alembert-like 1D equati on, the solution of which exactly reads, for r≥R0: 2The pressure difference accross the bubble boundary due to ai r-water surface tension is about 1%P0for a typical radius of 1 mm(see [5] eq. (65b)) and will be neglected: P0isalsothe equilibrium pressure of enclosed air. 3p(r, t) =1 rρ0R2 0/bracketleftbigg ξ′′−R0 cξ′′′+...+ (−R0 c)kξ(2+k)+.../bracketrightbigg (t−r−R0 c) (2) where cis the sound velocity in water. If the acoustic wavelength λis much larger than r(i.e., under the circumstances, if condition rω0/c≪1 is fulfilled), then p(r, t) can be approximated by: p(r, t)≃ρ0R2 0/bracketleftBiggξ′′(t) r−ξ′′′(t) c/bracketrightBigg ≃ρ0R2 0/bracketleftBiggξ′′(t) r+ω2 0 cξ′(t)/bracketrightBigg (3) Then, equalling p(t) in eq. (1) to p(R0, t) in eq. (3), one gets, all calculations carried out: ξ′′+ω2 0R0 cξ′+3γP0 ρ0R2 0ξ=ξ′′+ Γradξ′+ω2 0ξ= 0 (4) which is the well-known differential equation of a weakly3damped 1D harmonic oscillator. 2.2 Two-bubble free oscillation Let us now add a second bubble, with the same (equilibrium) ra diusR0, at a distance dapart from the first one. Let /vector ri(i= 1,2) be the (equilibrium) position of the ithbubble center, ξi(t) its radius variation, pi(t) the (inner) extra pressure of the enclosed air, and pi(/vector r, t) (resp. /vector ui(/vector r, t) the would-be (outer)extra pressure (resp. displacement w ith respect to equilibrium) at point /vector rand instant tin the water medium if bubble iwas alone. Then, allowing for the superposition principle for small displac ements, we assume that overall water extra pressure and displacement respectively read: p(/vector r, t) =p1(/vector r, t) +p2(/vector r, t) (5) /vector u(/vector r, t) =/vector u1(/vector r, t) +/vector u2(/vector r, t) (6) 3Ratio Γ rad/ω0=ω0R0/cis actually assumed to be much smaller than unity, as a conseq uence of the the validity condition of eq. (3). 4with, of course, pi(t) and ξi(t) still linked by eq.(1). On the surface of the first bubble: r1=|/vector r−/vector r1|=R0,r2=|/vector r−/vector r2| ≃d, andp(/vector r, t) =p1(t). A similar constraint is required on the surface of the second bubble, where r1≃dandr2=R0. If the bubble spacing dis much smaller than λ(i.e.ω0d/c≪1)4, then eq. (3) is available and we finally get the following pair of coupled motion equations: ξ′′ 1+αξ′′ 2+ Γrad(ξ′ 1+ξ′ 2) +ω2 0ξ1= 0 (7) αξ′′ 1+ξ′′ 2+ Γrad(ξ′ 1+ξ′ 2) +ω2 0ξ2= 0 (8) where α=R0/d(<0.5) can be regarded as a dimensionless coupling constant. Obs erve, by the way, that if double condition: R0≪d≪λis fulfilled, eqs. (7) and (8) are available (with α≃0), and dynamic variables ξiare still coupled by radiation damping, since the dissipation terms do not involve α. Defining symmetrical and antisymmetrical normal variables φs(t) and φa(t) as respec- tively the sum and the difference of ξ1(t) and ξ2(t), we get the uncoupled equations system: (1 +α)φ′′ s+ 2Γ radφ′ s+ω2 0φs= 0 (9) (1−α)φ′′ a+ω2 0φa= 0 (10) It is noteworthy that, as far as only radiation is concerned, the symmetrical mode’s damping rate is twice the single-bubble’s one, while the antisymmet rical mode is undamped.This feature is easily understood in terms of constructive (resp . destructive) interference between the acoustic waves radiated by each bubble, and parallels a w ell-known situation in the atomic physics domain (super- and sub- radiant quantum stat es of a couple of identical atoms interacting with each other through the E.M. field). Fr om eqs. (9) and (10), it is clear that the symmetrical mode has the lower angular freque ncyωs=ω0/√1 +α, and the 4In our experiments λis of order 1 m, while dranges from 1 to 5 cm. 5antisymmetrical mode the higher one ωa=ω0/√1−α. Observe that, leaving apart the calculation of radiative damping, it is very easy to derive a bove expressions of ωs,ausing the following trick. Let us consider the water as an uncompre ssible fluid (i.e. c→ ∞). The water displacement due to bubble i’s motion simply reads: /vector ui(/vector r, t) =ξi(t)R2 0 r2 i/vector eri (11) with/vector eri= (/vector r−/vector ri)/|/vector r−/vector ri|= (/vector r−/vector ri)/ri. Then, allowing for eq. (6), the overall water kinetic energy Tis: T=1 2ρ0/integraldisplay d3r(∂/vector u ∂t)2 =1 2M0/parenleftBig ξ′2 1+ξ′2 2+ 2αξ′ 1ξ′ 1/parenrightBig (12) where M0= 4πR3 0ρ0is the effective mass of either bubble. On the other hand, the t otal potential energy Vassociated with the isentropic compressibility of the encl osed air reads: V=1 2K/parenleftBig ξ2 1+ξ2 2/parenrightBig (13) where K= 12πγR 0P0is the effective stiffness of either bubble. The Lagrange equa tions derived from L=T−Vare: ξ′′ 1+αξ′′ 2+ω2 0ξ1= 0 (14) αξ′′ 1+ξ′′ 2+ω2 0ξ2= 0 (15) which is exactly the c→ ∞ limit of eqs. (7) and (8). It is worth noticing that eqs. (12) through (15) are formally equivalent to those of a system of t wo (L, C) electric circuits coupled by mutual induction with coefficient αL. In this analogy, M0andKrespectively correspond to Land 1/C, and the ξi’s to the electric charges qiof either capacitor. 62.3 Forced oscillation Let us now suppose that the above studied two-bubble system i s driven by an external acoustic source with an angular frequency ωnear Minnaert’s one, ω0. The phase difference of the driving pressures on both bubbles can therefore be neg lected, since ωd/c≪1. Let pei(t) be the external pressure undergone by bubble i. Motion eqs. (7) and (8) are then completed in: ξ′′ 1+αξ′′ 2+ Γrad(ξ′ 1+ξ′ 2) +ω2 0ξ1=−pe1(t) ρ0R0(16) αξ′′ 1+ξ′′ 2+ Γrad(ξ′ 1+ξ′ 2) +ω2 0ξ2=−pe2(t) ρ0R0(17) or equivalently: φ′′ s+2Γrad 1 +αφ′ s+ω2 sφs=Fes(t) (18) φ′′ a+ω2 aφa=Fea(t) (19) withFes(t) =−(pe1(t) +pe2(t))/ρ0R0(1 +α) and Fea(t) =−(pe1(t)−pe2(t))/ρ0R0(1−α). Solving for φsandφain above eqs. (18) and (19), one gets ξ1(t) and ξ2(t), and conse- quently (using eq. (3)) quantities p1(/vector r, t) and p2(/vector r, t) at any point /vector rof the medium. At last, comparing external (applied) pressure pe(/vector r, t) with the actual overall extra pressure p(/vector r, t) =pe(/vector r, t)+p1(/vector r, t)+p2(/vector r, t), we can experimentally measure the two-bubble system’s response as a function of ω. In this respect (and provided that the excitation-detecti on geometry allows it), resonances are expected for ω=ωsandω=ωa. 3. EXPERIMENTS Our aim is to demonstrate the existence of both above mention ed modes. From an experimental point of view, it turns out to be easier to imple ment a forced oscillation scheme than a free oscillation one. We therefore present the former hereafter. 73.1 Experimental setup A small net (see fig.1), made up with a gauze maintained with a w ire, is designed to catch up an air bubble in water and to fix it at any desired posit ion without appreciably modifying acoustic impedance and spherical symmetry. Two such devices are used for studying the two-bubble system . The external driving source is a speaker and extrapressure p(/vector r, t) is measured with a small microphone. A func- tion generator, to which the speaker is connected, produces a c.w. sinusoidal signal with a frequency slowly swept from flowtofhigh. The signal delivered by the microphone is trans- mitted to a lock-in amplifier which compares it with the refer ence one (delivered by the function generator) and decomposes it into real and imagina ry parts. Both parts can be seen on an oscilloscope and recorded with a computer (see fig. 2). In a preliminary set of experiments, without any bubble in th e aquarium, the response of the microphone is calibrated for different speaker-microph one configurations. Two kinds of configurations are presented in figure 3. In figs. 3(a) and 3(b) the configuration is deliberately asymmetrical: the microphone is mainly susceptible to bubb le 2’s motion, while the speaker selectively drives bubble 2 (fig. 3(a)) or bubble 1 (fig. 3(b)) , so that Fea(t) is nonzero: both modes can be excited and the associated motions detected. In fig. 3(c), the speaker is placed far from the bubbles; then, not only the phases, but also the a mplitudes of the external pressures p1e(t) and p2e(t) undergone on either bubble are appreciably equal. In such a symmetrical excitation configuration, Fea(t) = 0, so that the antisymmetrical mode remains unexcited. Observe, by the way, that since distances r1andr2between the bubbles and the microphone are equal, the latter would detect no contrib ution from the antisymmetrical modeeven though it was excited (see eqs.(3) and (5): r1=r2andξ1=−ξ2yields p1(r1, t)+ p(r2, t) = 0 ). 83.2 Results and discussion In figs. 4(a) and 4(b), the imaginary part Im p of the output signal from the lock- in amplifier is displayed versus the speaker frequency ffor various values of the bubbles spacing d. Figures 4(a) and 4(b) respectively correspond to configura tions 3(a) and 3(b). Two resonances can be made out in fig 4(a) and (though at a lesse r degree) in fig 4(b). Observe that the sign of the signal at the higher frequency re sonance is changed from 4(a) to 4(b), while the lower frequency one remains unchanged. Th is is consistent with the latter signal being associated with the symmetrical mode’s resona nce (ωs=ω0/√1 +α < ω 0, and Fesunchanged from configuration 3(a) to 3(b)), and the former on e with the antisymmetrical mode’s resonance ( ωa=ω0/√1−α > ω 0, and Feachanged into −Feafrom configuration 3(a) to 3(b)). It is noteworthy that both resonances have appreciably the s ame width. This is in contradiction with simplified eqs.(9) and (10) (or (18) and ( 19)), in which only the radiation damping was considered. In fact, as mentioned in introducti on, other kinds of damping (namely viscous and thermal dampings) should be taken into a ccount. Discussing this point is out of the scope of the present paper. In figure 5, we have plo tted, for both symmetrical and antisymmetrical modes, and for R0≃2mm, the inverse squared frequency f−2(multiplied by a factor of 107) versus the inverse bubble spacing d−1, in order to get a visual check of theoretical relations: 1 f2s=1 f2 0+R0 f2 0·1 d(20) 1 f2 a=1 f2 0−R0 f2 0·1 d(21) Although experimental points are appreciably aligned, the measured slopes are about 40% below theoretical prediction, suggesting that couplin g constant αhas been overesti- mated. In fact, theoretical value α=R0/dwas derived in eq.(12) when integrating the 9water kinetic energy density1 2ρ0(∂/vector u ∂t)2over the whole space5. This inertial coupling is nat- urally lowered if some obstacle lies between the bubbles and consequently screens (part of) the water flow6. Now, this is exactly what happens in configurations 3(a) and 3(b): to be able to excite the antisymmetrical mode, we are compelled to insert the speaker between the two bubbles, thus bringing about the above screening effe ct. In order to check this inter- pretation, we performed the same experiment with configurat ion 3(c), and recorded, for the symmetrical branch of the linear fitting of fig. 5, a slope of ab out 90% of the theoretically predicted value. 4. CONCLUSION As a conclusion, the acoustic inertial coupling between two air bubbles in water has been experimentally put in evidence. Theoretical analysis show s that the two-bubble system is formally equivalent to a set of two magnetically coupled ( L,C) electric circuits, with two eigenmodes, respectively symmetrical and antisymmetrica l. Experimental measurements and theoretical predictions are in 10% accuracy agreement. 5More precisely: over the whole space outside the two bubbles (the inner air’s inertia being negligible). Nevertheless, it can be shown that the coefficie nt of coupling term ξ′ 1ξ′ 2in integral (12) does not depend on the bubbles radius R0. 6The effective mass M0is modified too, but at a lesser degree. 10REFERENCES [1] M.Minnaert, Phil.Mag., 16, 235 (1933) [2] T.G.Leighton and A.J.Walton, Eur.J.Phys., 8, 98 (1987) [3] W.E.Farrell, D.P.McKenzie, and R.L.Parker, Proc.Camb .Phil.Soc., 65, 365 (1969) [4] F.S.Crawford, Am.J.Phys., 50, no.5, 398 (1982) [5] C.Devin, J.Acous.Soc.Am., 31, no.12, 1654 (1959) [6] A.Prosperetti, J.Fluid Mech., 168, 457 (1986) [7] A.Prosperetti, J.Fluid Mech., 222, 587 (1991) [8] E.L.Carstensen and L.L.Foldy, J.Acous.Soc.Am., 19, no.3, 481 (1947) [9] E.Silberman, J.Acous.Soc.Am., 29, no.8, 925 (1957) [10] L.d’Agostino and C.E.Brennen, J.Acous.Soc.Am., 84, 2126 (1988) [11] G.N.Kuznetsov and I.E.Shchekin, Akust.Zh., 18, 565 (1972) [Sov.Phys.Acoust., 18, 466 (1973)] [12] Yu.A.Kobelev and L.A.Ostrovskii, Akust.Zh., 30,715 (1984) [Sov.Phys.Acoust., 30, 427 (1984)] [13] V.V.Bredikhin, Yu.A.Kobelev, and N.I.Vasilinenko, J .Acoust.Soc.Am., 103, 1775 (1998) [14] M.Strasberg, J.Acous.Soc.Am., 25, 536 (1953) [15] M.Strasberg, J.Acous.Soc.Am., 28, 20 (1956) [16] P.H.Roberts and C.C.Wu, Phys.Fluids, 10, no.12, 3227 (1998) 11FIGURE CAPTIONS FIG.1 Simple tool for capturing the bubble FIG.2 Diagram of the experimental setup. In the experiment, we pum p air into a tube immersed in water to produce the bubbles. The radii differenc e between these bubbles is small and will be neglected. (It can be shown that a small ra dii difference yields second order correction of ωsandωa). FIG.3 Different geometrical configurations FIG.4 (a) Spectra of symmetrical and antisymmetrical modes in configu ration 3(a). (b)Change of sign of Im pfor the antisymmetrical mode when configuration 3(b) is adopted. FIG.5 Linear fitting of the plot1 f2s,a·107(Hz−2) vs.1 d(cm−1) for the two modes. Average of the resonance frequency of the two bubbles: 1499 Hz; corresponding radius: 0 .217cm; slopes for the two fitting lines: 0 .578 and −0.550 (cm·sec2); slopes of theoretical prediction: ±0.966(cm·sec2). 12FIGURES FIG. 1 13FIG. 2 14FIG. 3 15FIG. 4(a) 16FIG. 4(b) 17FIG. 5 18

arXiv:physics/0003104v1 [physics.acc-ph] 30 Mar 2000Novel approach for spin-flipping a stored polarized beam∗ Ya.S. Derbenev and V.A. Anferov Physics Department, University of Michigan, Ann Arbor, MI4 8109-1120 October 31, 2013 Abstract The traditional method of spin-flipping a stored polarized b eam is based on slowly crossing an rf induced depolarizing resonance. This paper d iscusses a novel approach where the polarization reversal is achieved by trapping the beam polarization into a stable spin-flipping motion on top of the rf induced resonance at a ha lf-revolution frequency. 1 Introduction Developing the spin-flipping technique is important for hig h energy spin experiments since frequent reversals of the beam polarization can significant ly reduce systematic errors in an experiment’s spin asymmetry measurements. To spin-flip a st ored polarized beam, one can slowly ramp the frequency of an rf magnet (either solenoid or dipole) through the rf-induced depolarizing resonance. This technique was successfully u sed to spin-flip a polarized proton beam stored in the IUCF Cooler Ring with and without Siberian Snake [1]. While slow resonance crossing rate is required to achieve good spin-fli p efficiency, it also increases the time of each spin-flip. Moreover, it makes the spin motion sensiti ve to weak synchrotron sideband or higher-order depolarizing resonances that may occur in the vicinity of the rf induced resonance. In this paper we discuss another possible way of spin-flippin g the beam polarization by rearranging the stable spin motion in such a way, that the pol arization direction alternates on every particle turn around the ring without any depolarizat ion. 2 Stable spin motion at an RF induced resonance In a circular accelerator or a storage ring with no Siberian S nakes, the spin vector of each particle precesses around the vertical magnetic field of the ring’s dipole magnets. For a particle moving along the closed orbit, the spin tune νs, which is the number of spin precessions during one turn around the ring, is proportional to the beam energy νs=Gγ, (1) whereγis the Lorentz energy factor and G= (g−2)/2 is particle’s gyromagnetic anomaly. While for protons G= 1.79285, it is much smaller for electrons ( G= 0.00116) and deuterons (G=−0.1426). This vertical spin precession can be perturbed by any horizo ntal rf magnetic field, whenever its frequency is in resonance with the spin motion fRF=fcirc(k±νs), (2) ∗Supported by a research grant from the US Department of Energ y 1wherefcircis the beam circulation frequency and kis an integer. Near the rf induced reso- nance, the spin precession becomes unstable, which could le ad to depolarization of a vertically polarized beam. However, the same rf magnetic field establis hes a new stable spin direction in the horizontal plane. To show this, let us consider the spi n motion in the presence of an rf spin perturbation ε·e−iωθ, whereεis resonance strength and we assume that the perturbation frequencyω=fRF/fcirc±kis close to the resonance condition of Eq.(2). The equation o f spin motion can be written for the spinor wave function ψin the following form [2], dψ dθ=−i 2/parenleftBigg Gγ ε ·e−iωθ ε∗·eiωθ−Gγ/parenrightBigg ·ψ, (3) whereθis the azimuthal particle coordinate in the ring. In these no tations, the diagonal terms in Eq.(3) represent spin rotation around the vertical axis, while the off diagonal terms corre- spond to horizontal spin perturbation. Transforming the ab ove equation into the resonance rotating frame, ψ=e−iωθσ3/2·ξ, the equation of spin motion becomes dξ dθ=−i 2/parenleftBigg Gγ−ω ε ε∗ω−Gγ/parenrightBigg ·ξ (4) Note that exactly on top of the rf induced resonance Gγ−ω= 0, and the spin precesses around the spin perturbing field ε, which rotates with frequency ω=fRF/fcirc±kin the laboratory frame. Thus, the direction along the rotating spin perturba tion vector becomes stable for the spin motion in the presence of an rf perturbation. Next, we co nsider how the existence of such a rotating horizontal stable spin direction can be used for s pin-flipping. 3 Spin-flipping at selected energies in the absence of Siberi an snakes As we saw in the previous section, an external rf magnetic fiel d, when at resonance with the spin precession, creates a horizontal stable spin directio n which rotates around the ring with angular frequency wθ= (fRF/fcirc±k)θ=Gγθ. (5) WhenGγis equal to a half-integer number, the horizontal spin would rotate by exactly 180 degrees in one turn around the ring. Thus, the spin would flip i ts direction after every turn. This opens a possibility to organize the spin motion in such a way that spin would arrive at the experimental section with longitudinal polarizatio n, whose sign alternates on every turn. A practical solution would require installing an rf so lenoid at the experimental straight section and matching the injected beam polarization with th e longitudinal direction at the experimental section. One can also use radial rf dipole field to create a spin-flipping motion of a horizontal spin. The stable spin direction would be radi al near the rf dipole, while in a different straight section it will be rotated towards the lon gitudinal direction by angle φ=Gγθ bend (6) whereθbendis the orbit bend between the rf dipole and the point of intere st. Note that the spin tuneGγshould be half-integer for both the rf solenoid and rf dipole induced spin-flipping. There are several effects that could potentially perturb the spin motion driven by the rf magnet. With an energy offset or spread present in the beam, pa rticles will have their spin 200.20.40.60.81 0 2 4 6 8 10Stable Polarization Resonance Strength / Spin tune shift Figure 1: The average stable polarization magnitude during the spin- flipping process is plotted against theε/∆νsratio. The solid curve is derived analytically from the spin transformation over two turns around the ring. The dashed curve is stable polarization pre dicted by Eq.(8). precession frequency slightly shifted from the rf resonanc e. An offset from the rf resonance would tilt the stable spin direction out of the horizontal pl ane by an angle β, β= tan−1/parenleftbigg∆νs ε/parenrightbigg , (7) where ∆νs=G∆γis the spin tune shift due to the energy offset. The tilt in the s table spin direction would in turn reduce the equilibrium horizontal b eam polarization Pby a factor of ∆P P= 1−cos2β≃/parenleftbigg∆νs ε/parenrightbigg2 (8) Thus, in order to maintain control over the spin motion (∆ P/P < 0.1), the rf induced resonance should dominate over the energy spread, ε≥3·∆νs. (9) For example, to overcome a spin tune shift of 0.001 the streng th of the rf induced resonance should be about 0.003. The average stable polarization magn itude during the spin-flipping process is plotted against the ε/∆νsratio in Fig. 1. Similarly, the rf induced resonance should dominate over hi gher-order horizontal spin per- turbations, which would also tend to perturb the stable spin direction. It is also important to note that, in the spin-flipping method described here, the rf solenoid would operate at exactly one half of the beam circulation frequency. Therefore, one p articular part of the beam would always pass the rf magnet when its field is very close to zero, a nd thus, would not have stability of the horizontal spin. To avoid this problem, one could crea te a gap in the stored beam while 3synchronizing the rf field with the remaining beam bunches. A nother solution could be to use a high frequency rf-magnet (or a special rf-cavity) sync hronized with every bunch in the beam. This could be done when there is an odd number of bunches in the ring; the rf magnet would then be operating at the frequency fRF=h 2fcirc, wherehis the harmonic number (odd integer) of the main rf cavities. 4 Spin-flipping in the presence of Siberian snakes In the presence of a full Siberian snake [3] in the ring, the sp in tune becomes half-integer and energy independent. With a proper choice of the betatron tun es, a set of Siberian snakes can overcome all dangerous depolarizing resonances in the ring [4]. Nevertheless, an rf magnetic field in resonance with the half-integer spin tune can depola rize the beam even in the presence of a Siberian snake. These resonances are often called ”snak e” depolarizing resonances [5]. Similarly to the case with no snake in the ring, such an rf indu ced ”snake” resonance can create a rotating stable spin direction. Since the spin tune is half-integer in the presence of a Siberian snake, this rotating stable spin direction would b e flipped after every turn around the ring. To create such a stable spin-flipping mode of the spin motion, the rf magnetic field has to be orthogonal to the unperturbed spin. With a single snake in th e ring and no spin perturbation present, the stable spin direction is horizontal and coinci des with the snake axis in the straight section opposite to the snake location. Therefore, an rf dip ole with vertical field operating at 0.5fcircwould make vertical direction stable for the spin motion, wh ile the polarization direction would flip after every turn around the ring. With an even number of snakes in the ring, the unperturbed sta ble spin direction is vertical. Ideally, one could use longitudinal rf field near the interac tion region to create the longitudinal spin stable and flipping every turn around the ring. However, solenoids become impractical at high energies since their spin rotation angle linearly decr eases with the beam momentum. An energy independent spin rotation in a dipole makes dipole ma gnets more attractive for spin manipulation at high energies. An rf dipole with radial field could create stable spin-flipping of the beam that has radial polarization near the rf dipole lo cation, while at the experimental straight section polarization would be longitudinal. The stability of the continuous spin-flipping motion could b e lost when the spin-tune moves out of the induced rf resonance. Such a spin tune shift could b e caused by a small error in the snake current or by to some high-order spin perturbation. As in the case with no Siberian snake in the ring, the strength of the rf induced resonance determi nes the tolerable spin tune shift from the half-integer value (as indicated in Eq.(8)). The effect o f the high-order spin perturbation could be reduced by a proper choice of the betatron tunes. The refore, an adequate strength of the rf magnet (to achieve ε≃10−3) and the snake current precision at the level of 10−4would provide stability of the spin-flipping motion. In practice, an rf dipole field of 0.1 T could be obtained [6] in the frequency range near 20 kHz, which corresponds to a half of the circulation frequenc y in most high energy rings. The spin perturbation strength by a 1-meter-long rf dipole of th is type would be, ε=Ge 2πmc2/integraldisplay Bdl∼0.01 (for protons) . (10) Such an rf dipole would certainly have adequate strength to c ontrol the spin motion. 4βx' x/G15πδ X0/G35/G29/G03/G4E/G4C/G46/G4E /G25/G48/G57/G44/G57/G55/G52/G51 /G50/G52/G57/G4C/G52/G51 Figure 2: Beam oscillations excited by the rf dipole are shown in the ph ase space rotating with the rf frequency. The amplitude of the excited beam oscillations, X0, is determined by the rf kick amplitude and the rf frequency separation from the betatron tune δ=νx−fRF fcirc+k. This stable mode of the excited beam oscillations is achieved when rf dipole is turn ed on adiabatically. 5 Spin manipulation of a polarized deuteron beam The technique discussed for spin-flipping seems especially attractive for polarized deuteron beams. The conventional spin manipulation methods become d ifficult due to deuteron’s small anomalous magnetic moment. A full Siberian snake would only be practical at low energies, where solenoidal magnets could provide the required spin ro tation (a 200 MeV deuteron beam would require a solenoid with 11 Tmfield integral). Similarly, the effect of the rf magnetic field on spin will be rather small, which limits applicabilit y of the conventional spin-flipping technique. In contrast, the method presented here uses the a ccelerator lattice to flip the spin while the rf field keeps the spin motion stable. This feat ure becomes an advantage for particles with small anomalous magnetic moment in the energ y region where Gγ < 10. In the case of deuteron beam, stable spin-flipping motion could be o rganized at the beam energies corresponding to half-integer Gγvalues T[GeV] = 4.7 + 13.156·n (11) wherenis an integer. A strong rf field could also create stability fo r the non-flipping longi- tudinal polarization direction when Gγis an integer. In this case, rf field should be strong enough to dominate over the imperfection fields in the accele rator. Note that there are no such first-order spin perturbations near a half-integer Gγ, and the required rf field could be much weaker. The direct effect of the rf magnetic field on the deuteron’s spi n is rather small. However, an rf-dipole also excite beam oscillations, and the spin is m ainly driven by the quasi-resonant accumulation of the spin kicks from all quadrupoles of the ac celerator lattice. To estimate the amplification of the spin perturbation due to the excited bea m oscillations, we first calculate the beam oscillation amplitude using the approach discussed in [7]. Considering the beam motion in the phase space coordinates ( x, x′βx), it is convienient to transform to the rf rotating frame 5where the beam kick by the rf dipole is constant. After the rf d ipole’s kick, the particle’s phase space vector would rotate by an angle, 2πδ= 2π/parenleftbigg νx−fRF fcirc+k/parenrightbigg , (12) whereνxis the horizontal betatron frequency and kis an integer. When the rf dipole is turned on adiabatically, the excited beam oscillations reach a sta ble mode which is shown in Fig. 2. The beam oscillation amplitude, X0, for this stable mode is given by 2πδ·X0=β0x′=β01 2/integraldisplayBRFdℓ Bρ, (13) whereβ0is the value of the horizontal beta-function near the rf dipo le. The resulting beam oscillations around the ring can be written as x(s) =X0/radicalBigg βx(s) β0cos (νRFθ), (14) whereνRF=k+fRF/fcircis the rf harmonic closest to the betatron tune (i.e. νRF∼νx). The strength of the induced spin perturbation is given by the spi n kick accumulated in the lattice quadrupoles over one turn around the ring εb=Gγ 2π/contintegraldisplay g(s)x(s)eiνsθds, (15) whereg(s) =∂By/∂x Bρis normalized strength of the quadrupoles around the ring an dνsis the spin tune. Eq.(15) should be compared with the strength of th e spin perturbation by the direct effect of the rf magnetic field, which is given by ε0=Gγ 4π/integraldisplayBRFdℓ Bρ. (16) In both cases, the effect of transverse magnetic fields is prop ortional to the anomalous magnetic moment [8]. Substituting Eq. (14) into Eq. (15) one would qui ckly obtain εb=ε0√β0 4πδ/contintegraldisplay g(s)/radicalBig βx(s)ei(νs±νRF)θds. (17) When the spin tune is close to the frequency of the induced bea m oscillations, |νs−νRF| ≪1, one can neglect the exponent in the integral. The remaining i ntegral resembles definition of the chromaticity function ξx=−1 4π/contintegraltextg(s)βx(s)≃ −νx. Therefore, the strength of the induced spin perturbation can be estimated as εb≃ε0νx δ√β0 /angbracketleft√βx/angbracketright. (18) This estimate corresponds to the maximum amplification of th e spin perturbation which occurs in the vicinity of the strong intrinsic depolarizing resona ncesνs≃νx, where the spin kicks from all lattice quadrupoles are synchronized with the spin precession. Farther away from these regions, the effect of the induced beam oscillations on spin is reduced by the exponent in the Eq. (17); in that case, smaller δwould enhance the excited beam oscillation amplitude as well as the induced spin perturbation. 6Finally, we would like to comment that acceleration of polar ized deuterons seems possible in modern high energy rings [9]. While full Siberian snakes do n ot seem practical for high energy deuterons, their depolarizing resonances are 25 times weak er and 25 times farther apart than for protons. Therefore, one could use individual resonance correction techniques developed for proton beam at the AGS ring (Brookhaven Natl. Lab.). A par tial Siberian snake could overcome all imperfection depolarizing resonances [10]. S uch a partial snake could be realized either using a solenoid magnet or a set of correction dipoles distributed around the ring to create a controlled closed orbit perturbation. Note that, i n an ideal case, reversing the axis of the partial snake could flip the longitudinal polarization o f the beam stored near an integer Gγ. However, without additional correction of the natural impe rfection resonance this spin-flipping method remains impractical due to inevitable polarization losses. The intrinsic depolarizing resonances could be handled by an rf dipole which induces coh erent beam oscillations and makes intrinsic resonances strong enough to spin-flip [11]. An rf spin perturbation could also be used as a spin rotator for polarized deuterons. For exampl e, to bring initial vertical beam polarization to the longitudinal direction, a horizontal r f dipole field could be applied for a time period which corresponds to the required π/2 spin rotation. 6 Summary In summary, we found that an external rf magnetic field can be u sed to create a stable mode of the spin motion, where the polarization direction flips af ter every turn around the ring. Such stable spin-flipping spin motion can be realized whenev er the spin tune is equal to a half integer value; this is always true in the rings equipped with Siberian snakes, while rings without Siberian snakes also reach half-integer spin tune at certai n energies. The applied rf magnetic field should be orthogonal to the unperturbed stable spin dir ection, and should operate at exactly one half of the circulation frequency. Provided tha t the rf field is synchronized with the circulating beam and is strong enough to dominate over po ssible spin tune spread, this spin-flipping motion is stable. It was also noted earlier [12 ], that the rf stabilization of the spin motion against the spin tune spread is an interesting possib ility for accelerators with Siberian snakes. References [1] D.D. Caussyn et al., Phys. Rev. Lett. 73, 2857 (1994); B.B. Blinov et al., Phys. Rev. Lett. 81, 2906 (1998). [2] B.W. Montague, Part. Accel. 11(4), 219 (1981). [3] Ya.S. Derbenev and A.M. Kondratenko, Sov. Phys. Dokl. 20, 562 (1978). [4] S.Y. Lee and E.D. Courant, Phys. Rev. D 41, 292 (1990). [5] S.Y. Lee and S. Tepikian, Phys. Rev. Lett. 56, 1635 (1986); R.A. Phelps et al., Phys. Rev. Lett. 78, 2772 (1997). [6] B. Parker et al., in Proc. of 1999 Part. Accel. Conf. (PAC-99, New York), 3336 (1999); P. Schwandt, private communications. [7] M. Bai et al., Phys. Rev. E 56, 6002 (1997). 7[8] A.M. Kondratenko, in Proc. of 9thIntl. Symposium on High Energy Spin Physics, Bonn 1990, 140 (1991). [9] E.D. Courant, in Proc. of Workshop on RHIC Spin Physics, B NL April 1998, Report BNL-65615, 275 (1998); Spin Note AGS/RHIC/SN 066 (1997). [10] V.A. Anferov et al., Phys. Rev. A46, R7383 (1992); H. Huang et al., Phys. Rev. Lett. 73, 2982 (1994). [11] M. Bai et al., Phys. Rev. Lett. 80, 4673 (1998). [12] Ya.S. Derbenev, in Proc. of DESY Workshop on Polarized P rotons at High Energies, DESY-PROC-1999-03, 225 (1999). 8

arXiv:physics/0003105v1 [physics.bio-ph] 30 Mar 2000Noise-Based Switches and Amplifiers for Gene Expression Jeff Hasty1, Joel Pradines1, Milos Dolnik1,2and J.J. Collins1 September 23, 1999 1Center for BioDynamics and Dept. of Biomedical Engineering , Boston University, 44 Cummington St., Boston, MA 02215, U.S.A. 2Dept. of Chemistry and Center for Complex Systems, Brandeis University, Waltham, MA 02454, U.S.A.ABSTRACT The regulation of cellular function is often contr olled at the level of gene transcription. Such genetic regulation usually con sists of interacting networks, whereby gene products from a single network can ac t to control their own expression or the production of protein in another netwo rk. Engineered control of cellular function through the design and manipul ation of such networks lies within the constraints of current technology. Here we d evelop a model describing the regulation of gene expression, and elucidat e the effects of noise on the formulation. We consider a single network derived fro m bacteriophage λ, and construct a two-parameter deterministic model descri bing the temporal evolution of the concentration of λrepressor protein. Bistability in the steady- state protein concentration arises naturally, and we show h ow the bistable regime is enhanced with the addition of the first operator site in the promotor region. We then show how additive and multiplicative external noise can be used to regulate expression. In the additive case, we demonstrate t he utility of such control through the construction of a protein switch, where by protein production is turned “on” and “off” using short noise pulses. In the multi plicative case, we show that small deviations in the transcription rate can lea d to large fluctuations in the production of protein, and describe how these fluctuat ions can be used to amplify protein production significantly. These novel re sults suggest that an external noise source could be used as a switch and/or ampl ifier for gene expression. Such a development could have important implic ations for gene therapy. 1Introduction Regulated gene expression is the process through which cell s control fundamental functions, such as the production of enzymatic and structural proteins , and the time sequence of this production during development [1, 2]. Many of these regulat ory processes take place at the level of gene transcription [3], and there is evidence that t he underlying reactions governing transcription can be affected by external influences from the environment [4]. As experimental techniques are increasingly capable of pro viding reliable data pertaining to gene regulation, theoretical models are becoming import ant in the understanding and manipulation of such processes. The most common theoretica l approach is to model the interactions of elements in a regulatory network as biochem ical reactions. Given such a set of chemical reactions, the individual jump processes (i .e., the creation or destruction of a given reaction species) and their associated probabili ties are considered. In its most general form, this often leads to a type of Monte Carlo simula tion of the interaction proba- bilities [5]. Although this approach suffers from a lack of an alytic tractability, its strength is its completeness – fluctuations in species’ concentratio ns are embedded in the modeling process. These internal fluctuations are important for syst ems containing modest numbers of elements, or when the volume is small. Rate equations originate as a first approximation to such a ge neral approach, whereby internal fluctuations are ignored. These deterministic diff erential equations describe the evo- lution of the mean value of some property of the set of reactio ns, typically the concentrations of the various elements involved. The existence of positive or negative feedback in a regu- latory network is thought to be common [6], and, within the re action framework, feedback leads to nonlinear rate equations [7]. Noise in the form of random fluctuations arises in these syste ms in one of two ways. As discussed above, internal noise is inherent in the biochemical reactions. Its magnitu de is proportional to the inverse of the system size, and its origi n is often thermal. On the other hand, external noise originates in the random variation of one or more of the externally set 2control parameters, such as the rate constants associated w ith a given set of reactions. If the noise source is small, its effect can often be incorporated post hoc into the rate equations. In the case of internal noise, this is done in an attempt to recap ture the lost information embod- ied in the rate equation approximation. But in the case of ext ernal noise, one often wishes to introduce some new phenomenon where the details of the effe ct are not precisely known. In either case, the governing rate equations are augmented w ith additive or multiplicative stochastic terms. These terms, viewed as a random perturbat ion to the deterministic pic- ture, can induce various effects, most notably the switching between potential attractors (i.e., fixed points, limit cycles, chaotic attractors) [8]. While impressive progress has been made in genome sequencin g and the understand- ing of certain qualitative features of gene expression, the re have been comparatively few advancements in the quantitative understanding of genetic networks. This is due to the inherent complexity of such biological systems. In this wor k, we adopt an engineering ap- proach in studying a solitary gene network. We envision that a plasmid, or genetic applet [9], containing a small, self-contained gene regulatory networ k, can be designed and studied in isolation. Such an approach has two distinct advantages. Fi rst, since the approach is inher- ently reductionist, it can make gene network problems tract able and thus more amenable to a mathematical formulation. Secondly, such an approach c ould form the basis for new techniques in the regulation of in vivo gene networks, whereby a genetic applet is designed to control cellular function. In this paper, we develop a model describing the dynamics of p rotein concentration in such a genetic applet, and demonstrate how external noise ca n be used to control the net- work. Although our results are general for networks designe d with positive autoregulation, we ground the discussion by considering an applet derived fr om the promotor region of bac- teriophage λ. Since the range of potentially interesting behavior is wid e, we focus primarily on the steady-state mean value of the concentration of the λrepressor protein. This choice is motivated by experiment; detailed dynamical informatio n is still rather difficult to obtain, 3as are statistical data concerning higher moments. We show h ow an additive noise term can be introduced to our model, and how the subsequent Langev in equation is analyzed by way of transforming to an equation describing the evolution of a probability function. We then obtain the steady-state mean repressor concentration by solving this equation in the long time limit, and discuss its relationship to the magnitu de of the external perturbation. This leads to a potentially useful application, whereby one utilizes the noise to construct a genetic switch. We then consider noise at the level of transc ription, where noise enters the formulation in a multiplicative manner. As in the additive c ase, we transform to an equation describing a probability distribution, and solve for the st eady-state mean concentration as a function of noise strength. Finally, we demonstrate how suc h a noise source can be used to amplify the repressor concentration by several orders of ma gnitude. A Model for Repressor Expression In the context of the lysis-lysogeny pathway in the λvirus, the autoregulation of λrepressor expression is well-characterized [10]. In this section, we present two models describing the regulation of such a network. We envision that our system is a plasmid consisting of the PR−PRMoperator region and components necessary for transcriptio n, translation, and degradation. Although the full promotor region in λphage contains the three operator sites known as OR1, OR2, and OR3, we first consider a mutant system whereby the operator site OR1 is absent from the region. The basic dynamical properties of this network, along with a categorization of the biochemical reactions, are as follo ws [10]. The gene cIexpresses repressor (CI), which in turn dimerizes and binds to the DNA a s a transcription factor. In the mutant system, this binding can take place at one of the tw o binding sites OR2 or OR3. (Here, we ignore nonspecific binding.) Binding at OR2 enhanc es transcription, which takes place downstream of OR3, while binding at OR3 represses tran scription, effectively turning off production. 4The chemical reactions describing the network are naturall y divided into two categories – fast and slow. The fast reactions have rate constants of ord er seconds, and are therefore assumed to be in equilibrium with respect to the slow reactio ns, which are described by rates of order minutes. If we let X,X2, and Ddenote the repressor, repressor dimer, and DNA promoter site, respectively, then we may write the equilibr ium reactions 2XK1⇀↽X2 (1) D+X2K2⇀↽DX2 D+X2K3⇀↽DX∗ 2 DX2+X2K4⇀↽DX2X2 where the DX2andDX∗ 2complexes denote binding to the OR2 or OR3 sites, respective ly, DX2X2denotes binding to both sites, and the Kiare forward equilibrium constants. We let K3=σ1K2andK4=σ2K2, so that σ1andσ2represent binding strengths relative to the dimer-OR2 strength. The slow reactions are transcription and degradation, DX2+Pkt→DX2+P+nX (2) Xkd→A where Pdenotes the concentration of RNA polymerase and nis the number of proteins per mRNA transcript. These reactions are considered irreversi ble. If we consider an in vitro system with high copy-number plasmids∗, we may define con- centrations as our dynamical variables. Letting x= [X],y= [X2],d= [D],u= [DX2], v= [DX∗ 2], and z= [DX2X2], we can write a rate equation describing the evolution of th e concentration of repressor, ˙x=−2k1x2+ 2k−1y+nktp0u−kdx+r (3) ∗This assumption is necessary since the number of relevant mo lecules per cell is small in vivo . Since there are many cells, we could alternatively use state probabilit ies as dynamical variables describing an in vivo system. 5where we assume that the concentration of RNA polymerase p0remains constant during time. The parameter ris the basal rate of production of CI, i.e., the expression ra te of the cIgene in the absence of a transcription factor. We next eliminate y,u, and dfrom Eq. (3) as follows. We utilize the fact that the reactions in Eq. (1) are fast compared to expression and degr adation, and write algebraic expressions y=K1x2(4) u=K2dy=K1K2dx2 v=σ1K2dy=σ1K1K2dx2 z=σ2K2uy=σ2(K1K2)2dx4 Further, the total concentration of DNA promoter sites dTis constant, so that dT=d+u+v+z=d(1 + (1 + σ1)K1K2x2+σ2K2 1K2 2x4) (5) Under these assumptions, Eq. (3) becomes ˙x=nktp0dtK1K2x2 1 + (1 + σ1)K1K2x2+σ2K2 1K2 2x4−kdx+r (6) Without loss of generality, we may eliminate two of the param eters in Eq. (3) by rescaling the repressor concentration xand time. To this end, we define the dimensionless variables /tildewidex=x√K1K2and/tildewidet=t(r√K1K2). Upon substitution into Eq. (3), we obtain ˙x=α x2 1 + (1 + σ1)x2+σ2x4−γ x+ 1 (7) where the time derivative is with respect to /tildewidetand we have suppressed the overbar on x. The dimensionless parameter α≡nktp0dT/ris effectively a measure of the degree in which the transcription rate is increased above the basal ra te by repressor binding, and γ≡kd/(r√K1K2) is proportional to the relative strengths of the degradati on and basal rates. 6For the mutant operator region of λphage, we have σ1∼1 and σ2∼5 [10, 11], so that the two parameters αandγin Eq. (7) determine the steady-state concentration of repr essor. For this equation, there are two types of behavior. For one se t of parameter values, we have monostability, whereby all initial concentrations evolve to the same fixed-point value. For another set, we have three fixed points, and the initial conce ntration will determine which steady state is selected. Additionally, in the multiple fixe d-point regime, stability analysis indicates that the middle fixed point xmis unstable, so that all initial values x < x mwill evolve to the lower fixed point, while those satisfying x > x mwill evolve to the upper. This bistability arises as a consequence of the competition betw een the production of xalong with dimerization and its degradation. For certain parameter va lues, the initial concentration is irrelevant, but for those that more closely balance product ion and loss, the final concentration is determined by the initial value. Graphically, we can see how bistability arises in Eq. (7) by s etting α x2/(1+2 x2+5x4) = γ x−1. In Fig. 1A we plot the functions α x2/(1 + 2 x2+ 5x4) and γ x−1 for fixed α and several values of the slope γ. We see that for γsmall (whereby degradation is minimal compared with production), there is one possible steady-st ate value of x(and therefore CI). As we increase γabove some critical value γL, we observe that three fixed-point values appear. As we increase γstill further beyond a second critical value γU, the concentration “jumps” to a lower value and the system returns to a state of mo nostability. The preceding ideas lead to a plausible method whereby the sy stem may be experimen- tally probed for bistability. We envision that αis fixed by the transcription rate and DNA binding site concentration, and that the degradation param eterγis an adjustable control. Beginning with a low initial value of γ=γ0= 5, we slowly increase the degradation rate. The effect is illustrated in Fig. 1B. We see that as γis slowly increased, the concentration of CI slowly decreases as the system tracks the fixed point. Th en, at the moment when γ is greater than γU, the concentration abruptly jumps to a lower value, followe d by a further slow increase. Now suppose we reverse course, and begin to de crease γ. Then the system 7will track along the lower fixed point until a point when γis greater than γL. At this point, the system will again jump, this time to a higher fixed-point v alue. The trademark of hys- terisis is that the two jumps, one when increasing γand the other when decreasing, occur for different values of γ. As is well-known, the full operator region of λphage contains three sites. We turn briefly to the effect of the additional site OR1 on the above network. I n order to incorporate its effect, Eq. (1) must be generalized to account for additional equilibrium reactions. This generalization amounts to the incorporation of dimer bindi ng to OR1 [10], and permutations of multiple binding possibilities at the three operator sit es. Then, using known relationships between the cooperative binding rates, the above steps can b e repeated and an equation analogous to Eq. (7) constructed. We obtain ˙x=α(2x2+ 50x4) 25 + 29 x2+ 52x4+ 4x6−γ x+ 1 (8) As can be seen, the addition of OR1 has the effect of changing th e first term on the right- hand side of the equation. While this augmentation does not a ffect the qualitative features of the above discussion, one important quantitative differe nce is depicted in Fig. 1B. In this figure, we see that the addition of OR1 has a large effect on the b istability region, increasing the overall size of the region by roughly an order of magnitud e. Additionally, the model predicts that, while the drop in the concentration of repres sor at the first bifurcation point will be approximately the same in both cases, the jump to the h igher concentration will be around five times greater in the system containing OR1. Final ly, since one effect of a larger bistable region is to make the switching mechanism more robu st to noise, these results are of notable significance in the context of the lysogeny-to-ly sis switching of λphage. Additive Noise We now focus on parameter values leading to bistability, and consider how an additive ex- ternal noise source affects the production of repressor. Phy sically, we take the dynamical 8variable xdescribed above to represent the repressor concentration w ithin a colony of cells, and consider the noise to act on many copies of this colony. In the absence of noise, each colony will evolve identically to one of the two fixed points, as discussed above. The pres- ence of a noise source will at times modify this simple behavi or, whereby colony-to-colony fluctuations can induce novel behavior. An additive noise source alters the “background” repressor production. As an example, consider the effect of a randomly varying external field on the biochemical reactions. The field could, in principle, impact the individual reaction ra tes [12, 13], and since the rate equations are probabilistic in origin, its influence enters statistically. We posit that such an effect will be small and can be treated as a random perturbatio n to our existing treatment; we envision that events induced will affect the basal product ion rate, and that this will translate to a rapidly varying background repressor produc tion. In order to introduce this effect, we generalize the aforementioned model such that ran dom fluctuations enter Eq. (8) linearly, ˙x=f(x) +ξ(t) (9) where f(x) is the right-hand side of Eq. (8), and ξ(t) is a rapidly fluctuating random term with zero mean ( < ξ(t)>= 0). In order to encapsulate the rapid random fluctuations, w e make the standard requirement that the autocorrelation be “ δ-correlated”, i.e., the statistics ofξ(t) are such that < ξ(t)ξ(t′)>=Dδ(t−t′), with Dproportional to the strength of the perturbation. Eq. (9) can be rewritten as ˙x=−∂φ(x) ∂x+ξ(t) (10) where we introduce the potential φ(x), which is simply the integral of the right-hand side of Eq. (7). φ(x) can be viewed as an “energy landscape”, whereby xis considered the position of a particle moving in the landscape. One such landscape is p lotted in Fig. 2A. Note that the stable fixed values of repressor concentration correspo nd to the minima of the potential 9φin Fig. 2A, and the effect of the additive noise term is to cause random kicks to the particle (system state point) lying in one of these minima. On occasio n, a sequence of kicks may enable the particle to escape a local minimum and reside in a n ew valley. To solve Eq. (10), we introduce the probability distributio nP(x, t), which is effectively the probability of finding the system in a state with concentr ationxat time t. Given Eq. (10), a Fokker-Planck (FP) equation for P(x, t) can be constructed [14] ∂tP(x, t) =−∂x(f(x)P(x, t)) +D 2∂x2P(x, t) (11) We focus here on the value of the steady-state mean (ssm) conc entration. To this end, we first solve for the steady-state distribution, obtaining Ps(x) =Ae−2 Dφ(x)(12) where Ais a normalization constant determined by requiring the int egral of Ps(x) over all x be unity. In Fig. 2B, we plot Ps(x), corresponding to the landscape of Fig. 2A, for two values of the noise strength D. It can be seen that for the smaller noise value the probabili ty is distributed around the lower concentration of repressor, w hile for the larger noise value the probability is split and distributed around both concentra tions. This is consistent with our conceptual picture of the landscape: low noise will enable o nly transitions from the upper state to the lower state as random kicks are not sufficient to cl imb the steep barrier from the lower state, while high noise induces transitions between b oth of the states. Additionally, the larger noise value leads to a spreading of the distributi on, as expected. Using the steady-state distribution, the steady-state mea n (ssm) value of x≡< x > ssis given by < x > ss=/integraldisplay∞ 0xAe−2 Dφ(x)dx (13) In Fig. 2C, we plot the ssm concentration as a function of D, obtained by numerically integrating Eq. (13) and transforming from the dimensionle ss variable xto repressor concen- tration. It can be seen that the ssm concentration increases withD, corresponding to the 10increasing likelihood of populating the upper state, as dis cussed previously with respect to Figs. 2A and B. Figure 2C indicates that the external noise can be used to con trol the ssm concentration. As a candidate application, consider the following protein switch. Given parameter values leading to the landscape of Fig. 2A, we begin the switch in the “off” position by tuning the noise strength to a very low value. This will cause a high popu lation in the lower state, and a correspondingly low value of the concentration. Then at so me time later, consider pulsing the system by increasing the noise to some large value for a sh ort period of time, followed by a decrease back to the original low value. The pulse will ca use the upper state to become populated, corresponding to a concentration increase and a flipping of the switch to the “on” position. As the pulse quickly subsides, the upper state rem ains populated as the noise is not of sufficient strength to drive the system across either ba rrier (on relevant time scales). To return the switch to the off position, the upper-state popu lation needs to be decreased to a low value. This can be achieved by applying a second noise pu lse of intermediate strength. This intermediate value is chosen large enough so as to enhan ce transitions to the lower state, but small enough as to remain prohibitive to upper-st ate transitions. Figure 2D depicts the time evolution of the switching proces s for noise pulses of strengths D= 1.0 and D= 0.05. Initially, the concentration begins at a level of [CI] = 1 0 nM, corresponding to a low noise value of D= 0.01. After six hours in this low state, a 30- minute noise pulse of strength D= 1.0 is used to drive the concentration to a value of [CI]∼58 nM. Following this burst, the noise is returned to its orig inal value. At 11 hours, a second 90-minute pulse of strength D= 0.05 is used to return the concentration to its original value. Multiplicative Noise We now consider the effect of a noise source which alters the tr anscription rate. Although transcription is represented by a single biochemical react ion, it is actually a complex sequence 11of reactions [16], and it is natural to assume that this part o f the gene regulatory sequence is likely to be affected by fluctuations of many internal or ext ernal parameters. We vary the transcription rate by allowing the parameter αin Eq. (8) to vary stochastically, i.e., we set α→α+ξ(t). In this manner, we obtain an equation describing the evolu tion of the protein concentration x ˙x=h(x) +ξ(t)g(x) (14) where h(x) is the right-hand side of Eq. (8), and g(x)≡2x2+ 50x4 25 + 29 x2+ 52x4+ 4x6(15) Thus, in this case, the noise is multiplicative, as opposed t o additive, as in the previous case. Qualitatively, we can use the bifurcation plot of Fig. 3A to a nticipate one effect of allowing the parameter αto fluctuate. Such a bifurcation plot is yet another way of dep icting the behavior seen in Fig. 1A; it can be seen that for certain value s ofαthere is one unique steady-state value of repressor concentration, and that fo r other values there are three. To incorporate fluctuations, if we envision αto stochastically vary in the bistable region of Fig. 3A, we notice that the steep top branch implies the cor responding fluctuations in repressor concentration will be quite large. This is contra sted with the flat lower branch, where modest fluctuations in αwill induce small variations. In order to verify this observ ation quantitatively, we simulated Eq. (14), the results of which are presented in Fig. 3B. Beginning with repressor concentration equal to its upper value of app roximately 500 nM, we notice that the immediate fluctuations are quite large even though αvaries by only a few percent (Fig. 3A). Then, at around 700 minutes, the concentration qu ickly drops to its lower value, indicating that the fluctuations envisioned in Fig. 3A were s ufficient to drive the repressor concentration to the dotted line of Fig. 3A and off the upper br anch (across the unstable fixed point). The final state is then one of very small variatio n, as anticipated. As in the previous section, the steady-state probability di stribution is obtained by trans- 12forming Eq. (14) to an equivalent Fokker-Planck equation [1 4], ∂tP(x, t) =−∂x(h(x) +D 2g(x)g′(x))P(x, t) +D 2∂2 ug2(u)P(x, t) (16) where the prime denotes the derivative of g(x) with respect to x. We again solve for the steady-state distribution, obtaining Ps(x) =Be−2 Dφm(x)(17) As before, the steady-state distribution can be used to obta in the ssm concentration. Although not originating from a deterministic equation lik e that of Eq. (7), the function φm(u) in Eq. (17) can still be viewed as a potential. We now conside r parameter values leading to one such landscape in Fig. 3C. This landscape impl ies that we will have two steady-state repressor concentrations of approximately 5 and 1200 nM. This large difference is due to the largeness of the parameter α, implying that repressor “induced” transcription amplifies the basal rate by a large amount. (Since dTenters in the numerator of the definition ofα, one could construct such a system experimentally with a hig h copy-number plasmid). This feature suggests that multiplicative noise could be us ed to amplify protein production, as described in the following example. We begin with zero pro tein concentration and very low noise strength D, leading to a highly populated lower state and low overall co ncentration. Then, at some later time, we pulse the system by increasing Dfor some short interval. This will cause the upper state to become quickly populated as it i s easy to escape the shallow valley of the landscape and move into the large basin. In Fig. 3D, we plot the temporal evolution of the mean repressor concentration obtained fro m the simulation of Eq. (14). We see that the short noise pulse at around 20 hours indeed cause s the concentration to increase abruptly by over three orders of magnitude, making this type of amplification an interesting case for experimental exploration. 13Discussion From an engineering perspective, the control of cellular fu nction through the design and manipulation of genetic regulatory networks is an intrigui ng possibility. In this paper, we have shown how external noise can be used to control the dynam ics of a regulatory network, and how such control can be practically utilized in the desig n of a genetic switch and/or amplifier. Although the main focus of this work was on a networ k derived from the promotor region of λphage, our approach is generally applicable to any autoregu latory network where a protein-multimer acts as a transcription factor. An important element of our control scheme is bistability. T his implies that a neces- sary criterion in the design of a noise-controlled applet be that the network is poised in a bistable region. This could potentially be achieved by meth ods such as the utilization of a temperature-dependent repressor protein, DNA titration, SSRA tagging, or pH control. Physically, the noise might be generated using an external fi eld. Importantly, it has been claimed that electromagnetic fields can exert biologic al effects [15]. In addition, recent theoretical [13] and experimental [12] work suggests a poss ible mechanism whereby an electric field can alter an enzyme-catalyzed reaction. These findings suggest that, although there is global charge neutrality, an external field can interact wit h local dipoles which arise through transient conformational changes or in membrane transport . Current gene therapy techniques are limited in that transfe cted genes are typically ei- ther in an “on” or “off” state. However, for the effective treat ment of many diseases, the expression of a transfected gene needs to be regulated in som e systematic fashion. Thus, the development of externally-controllable noise-based swit ches and amplifiers for gene expres- sion could have significant clinical implications. ACKNOWLEDGEMENTS. We respectfully acknowledge insightfu l discussions with Kurt 14Wiesenfeld, Farren Issacs, Tim Gardner, and Peter Jung. Thi s work was supported by the Office of Naval Research (Grant N00014-99-1-0554) and the U.S . Department of Energy. References [1] Dickson, R., Abelson, J., & Barnes, W. (1975) Science 187, 27–35. [2] Yuh, C. H., Bolouri, H., & Davidson, E. H. (1998) Science 279, 1896–1902. [3] Lewin, B. (1997) in Genes VI (Oxford University Press, Oxford). [4] Harada, Y. et. al. (1999) Biophys. J. 76, 709–715. [5] McAdams, H. H. & Arkin, A. (1997) Proc. Natl. Acad. Sci. 94, 814–819. [6] McAdams, H. H. & Shapiro, L. (1995) Science 269, 650–656. [7] Keller, A. (1995) J. Theor. Biol. 172, 169–185; Smolen, P, Baxter, D. A., & Byrne, J. H. (1998) Am. J. Physiol.–Cell Ph. 43, C531–C542; Wolf, D. M. & Eeck- man, F. H. (1998) J. Theor. Biol. 195, 167–186. [8] Horsthemke, W. & Lefever, R. (1984) in Noise-Induced Transitions (Springer-Verlag, Berlin). [9] Gardner, T. S., Cantor, C. R., & Collins, J. J. (1999) Nature , in press. [10] Ptashne, M et al. (1980) Cell19, 1–11; Johnson, A. D. et al. (1981) Nature 294, 217– 223. [11] Johnson, A. D. et al. (1980) Methods Enzymol. 65, 839–856. [12] Xie, T. D., Marszalek, P., & Chen, Y. (1994) Biophys. J. 67, 1247–1251. [13] Astumian, R. D. & Robertson, B. (1993) J. Am. Chem. Soc. 115, 11063–11068. 15[14] Van Kampen, N. G. (1992) in Stochastic Processes in Physics and Chemistry (North- Holland, Amsterdam). [15] See, for example, Berg, H. (1995) Bioelectrochem. Bioenerg. 38, 153–159; Liu, D. S. et al. (1990) J. Biol. Chem. 265, 7260–7271; Otter, M. W., McLeod, K. J., & Ru- bin, C.T. (1998) Clin. Orthop. 355, S90–S104; Asbury, C. L. & van den Engh, G. (1998) Biophys. J. 74, 1024–1130. [16] von Hippel, P. H. (1998) Science 281, 660–665. [17] Sancho, J., Miguel, M. S., & Katz, S. (1982) Phys. Rev. A 26, 1589–1593. 16Figure Captions FIG. 1. Bifurcation plots for the variable xand concentration of λrepressor. (A) Graph- ical depiction of the fixed points of Eq. (7), generated by set ting ˙x= 0 and plotting α x2/(1 + 2 x2+ 5x4) and the line γx−1. As the slope γis increased, the system traverses through a region of multistability and returns to a state of m onostability. (B) Hysterisis loops for the mutant and nonmutant systems obtained by setti ng ˙x= 0 in Eqs. (7) and (8). Beginning with concentrations of 35 nM for the mutant system and 85 nM for the nonmutant system, we steadily increase the degradation parameter γ. In both systems, the concentration of repressor slowly decreases until a bifurcation point. In the mutant (nonmutant) system, the repressor concentration abruptly drops to a lower value atγ∼16 (γ∼24). Then, upon reversing course and decreasing γ, the repressor concentration increases slowly until γ encounters a second bifurcation point at γ∼14 (γ∼6), whereby the concentration immedi- ately jumps to a value of 15 nM (mutant) or 70 nM (nonmutant). T he subsequent hysterisis loop is approximately 10 times larger in the nonmutant case. Parameter values are α= 50, K1= 0.05 nM−1, and K2= 0.026 nM−1for the mutant system, and K2= 0.033 nM−1for the nonmutant system [10]. FIG. 2. Results for additive noise with parameter values α= 10 and γ= 5.5. (A) The energy landscape. Stable equilibrium concentration values of Eq. (8) correspond to the valleys at [CI] = 10 and 200 nM, with an unstable value at [CI] = 99 nM. (B) S teady-state probability distributions for noise strengths of D= 0.04 (solid line) and D= 0.4 (dotted line). (C) The steady-state equilibrium protein concentration plotted v ersus noise strength. The concentra- tion increases as the noise causes the upper state of (A) to be come increasingly populated. (D) Simulation of Eq. (9) demonstrating the utilization of e xternal noise for protein switch- ing. Initially, the concentration begins at a level of [CI] = 10 nM corresponding to a low noise value of D= 0.01. After six hours, a large 30-minute noise pulse of strengt hD= 1.0 is used to drive the concentration to 58 nM. Following this pu lse, the noise is returned to 17its original value. At 11 hours, a smaller 90-minute noise pu lse of strength D= 0.04 is used to return the concentration to near its original value. The s imulation technique is that of Ref. [17]. FIG. 3. Results for multiplicative noise. (A) Bifurcation p lot for the repressor concentration versus the model parameter α. The steep upper branch implies that modest fluctuations in αwill cause large fluctuations around the upper fixed value of r epressor concentration, while the flat lower branch implies small fluctuations about the low er value. (B) The evolution of the repressor concentration in a single colony, obtained by simulation of Eq. (14). Relatively small random variations of the parameter α(∼6%) induce large fluctuations in the steady- state concentration until around 700 minutes and small fluct uations thereafter. (C) Energy landscape for parameter values α= 100 and γ= 8.5. (D) Large-scale amplification of the protein concentration obtained by simulation of Eq. (14). A t 20 hours, a 60-minute noise pulse of strength D= 1.0 is used to quickly increase the protein concentration by ov er three orders of magnitude. The parameter values are the same as tho se in (C). 181030507090 51015202530[CI] (nM) gMutant Nonmutant0481216 0 0.2 0.4 0.6 0.8f(x) xgg U LA B Figure 1 - Hasty et al.44.244.444.644.8 0100 200 300 400f [CI] (nM)0.020.040.060.080.100.12 0100 200 300 400P([CI]) [CI] (nM) 20406080100 00.40.81.21.62.0[CI] (nM) Noise Strength D20406080100 040080012001600[CI] (nM) Time (minutes)A B C D Figure 2 - Hasty et al. a 200400600800 0 400 800 1200 1600[CI] (nM) Time (minutes) 3.843.883.923.964.00 10 100 1000f [CI] (nM)04008001200 05001000150020002500[CI] (nM) Time (minutes)[CI] (nM) 200 150 100 50 001000200030004000A B CD Figure 3 - Hasty et al.

arXiv:physics/0003106v1 [physics.ed-ph] 30 Mar 2000Arhivele Electronice Los Alamos http://xxx.lanl.gov/physics/0003106 ELEMENTE DE MECANIC ˘A CUANTIC ˘A HARET C. ROSU e-mail: rosu@ifug3.ugto.mx fax: 0052-47187611 phone: 0052-47183089 h/2π 1Pentru tot ¸i cei atra¸ si de S ¸tiint ¸ele Fizice ¸ si se afl˘ a ˆ ın dificilii ani ai ˆ ınceputului de facultate. . Primul curs de mecanic˘ a cuantic˘ a ˆ ın romˆ ane¸ ste ˆ ın Inte rnet. Acest curs a fost scris ˆ ın Instituto de F´ ısica, Universidad de Guanajuato, Le´ on, Guanajuato, M´ exico. Copyright c∝ci∇cleco√†∇t2000 de c˘ atre autor. Orice drept comercial este rezervat. Martie 2000 2English Abstract This is the first graduate course on elementary quantum mecha nics in In- ternet written in Romanian for the benefit of Romanian speaki ng students (Romania and Moldova). It is a translation (with correction s) of the Spanish version of the course, which I did at the suggestion of Ovidiu Cioroianu, a student of physics in Bucharest. The topics included refer t o the postulates of quantum mechanics, one-dimensional barriers and wells, angular momen- tum and spin, WKB method, harmonic oscillator, hydrogen ato m, quantum scattering, and partial waves. Abstract Romˆ anesc Acesta este un curs internetizat de mecanic˘ a cuantic˘ a ele mentar˘ a pe care l-am tradus cu ˆ ımbun˘ at˘ at ¸iri din spaniol˘ a (limba ˆ ın ca re l-am predat) la rug˘ amintea studentului Ovidiu Cioroianu din Bucure¸ sti. Este destinat ˆ ın principal acelor student ¸i care se afl˘ a la primele contacte cu aceast˘ a disciplin˘ a de studiu obligatorie, de¸ si ar putea s˘ a fie de un oarecare fo los ¸ si pentru alte categorii de cititori. Sursele de ‘inspirat ¸ie’ le-am g˘ asit ˆ ın multe din excelentele manuale de mecanic˘ a cuantic˘ a care au fost pub licate de-a lungul anilor. 3CUPRINS 0. Forward - Cuvˆ ınt ˆ ınainte ... 5 1. Postulate cuantice ... 7 2. Bariere ¸ si gropi rectangulare unidimensionale ... 25 3. Moment cinetic ¸ si spin ... 47 4. Metoda WKB ... 75 5. Oscilatorul armonic ... 89 6. Atomul de hidrogen ... 111 7. Ciocniri cuantice ... 133 8. Unde part ¸iale ... 147 Include aproximativ 25 de probleme ilustrative. Unit˘ at ¸ile atomice nerelativiste de spat ¸iu ¸ si timp aH= ¯h2/mee2= 0.529·10−8cm tH= ¯h3/mee4= 0.242·10−16sec Unit˘ at ¸ile Planck relativiste de spat ¸iu ¸ si timp lP= ¯h/mPc= 1.616·10−33cm tP= ¯h/mPc2= 5.390·10−44sec 40 (E). FORWARD The energy quanta occured in 1900 in the work of Max Planck (No bel prize, 1918) on the black body electromagnetic radiation. Planck’ s “quanta of light” have been used by Einstein (Nobel prize, 1921) to expl ain the pho- toelectric effect, but the first “quantization” of a quantity having units of action (the angular momentum) belongs to Niels Bohr (Nobel P rize, 1922). This opened the road to the universalization of quanta, sinc e the action is the basic functional to describe any type of motion. However , only in the 1920’s the formalism of quantum mechanics has been develope d in a system- atic manner. The remarkable works of that decade contribute d in a decisive way to the rising of quantum mechanics at the level of fundame ntal theory of the universe, with successful technological applicatio ns. Moreover, it is quite probable that many of the cosmological misteries may b e disentan- gled by means of various quantization procedures of the grav itational field, advancing our understanding of the origins of the universe. On the other hand, in recent years, there is a strong surge of activity in t he information aspect of quantum mechanics. This aspect, which was general ly ignored in the past, aims at a very attractive “quantum computer” techn ology. At the philosophical level, the famous paradoxes of quantum mechanics, which are perfect examples of the difficulties of ‘quantum’ th inking, are actively pursued ever since they have been first posed. Perha ps the most famous of them is the EPR paradox (Einstein, Podolsky, Rosen , 1935) on the existence of elements of physical reality , or in EPR words: “If, without in any way disturbing a system, we can predict with certainty (i.e. , with probability equal to unity) the value of a physical quantity, then there e xists an element of physical reality corresponding to this physical quantit y.” Another famous paradox is that of Schr¨ odinger’s cat which is related to the fundamental quantum property of entanglement and the way we understand a nd detect it. What one should emphasize is that all these delicate poin ts are the sourse of many interesting and innovative experiments (suc h as the so-called “teleportation” of quantum states) pushing up the technolo gy. Here, I present eight elementary topics in nonrelativistic quantum me- chanics from a course in Spanish (“castellano”) on quantum m echanics that I taught in the Instituto de F´ ısica, Universidad de Guanaju ato (IFUG), Le´ on, Mexico, during the semesters of 1998. Haret C. Rosu 50 (R). CUV ˆINTˆINAINTE Cuantele de energie au ap˘ arut ˆ ın 1900 ca o consecint ¸˘ a a lu cr˘ arilor lui Max Planck (premiul Nobel 1918) asupra problemei radiat ¸iei de corp negru. “Cuantele de lumin˘ a” planckiene au fost folosite de c˘ atre Albert Einstein (premiul Nobel 1921) pentru a explica efectul fotoelectric , dar prima “cuan- tificare” a unei m˘ arimi cu unit˘ at ¸i de act ¸iune (momentul c inetic) se datoreaz˘ a lui Niels Bohr (premiul Nobel 1922). Aceasta a deschis drumu l univer- salit˘ at ¸ii cuantelor pentru c˘ a act ¸iunea este funct ¸ion ala fundamental˘ a pentru descrierea oric˘ arui tip de mi¸ scare. Chiar ¸ si ˆ ın aceste c ondit ¸ii, numai anii 1920 se consider˘ a ca adev˘ aratul ˆ ınceput pentru formalis mul cuantic, care a fost capabil s˘ a ridice mecanica cuantic˘ a la nivelul unei t eorii fundamentale a universului ¸ si s˘ a o transforme ˆ ıntr-o surs˘ a de numeroa se succese tehno- logice. Este foarte posibil ca multe dintre misteriile cosm ologice s˘ a se as- cund˘ a ˆ ın spatele diferitelor proceduri de cuantificare al e cˆ ımpului nelinear gravitat ¸ional ¸ si eventualele progrese ˆ ın aceast˘ a dire ct ¸ie ar putea contribui la o mai bun˘ a ˆ ınt ¸elegere a istoriei ¸ si evolut ¸iei univer sului. Pe de alt˘ a parte, aspectul informatic al mecanicii cuantice, care nu a fost mu lt investigat ˆ ın trecut, cunoa¸ ste ˆ ın prezent o perioad˘ a exploziv˘ a de cer cet˘ ari ˆ ın ideea con- struirii a¸ sa-numitelor “calculatoare cuantice”. ˆIn domeniul filosofic este de ment ¸ionat c˘ a ˆ ın mecanica cuan tic˘ a exist˘ a paradoxuri faimoase, care ˆ ınc˘ a se ment ¸in ˆ ın polemic˘ a ¸ si care reflect˘ a difi- cult˘ at ¸ile de logic˘ a pe care le creaz˘ a modul de “gˆ ındire cuantic˘ a” (sau prob- abilistic˘ a cuantic˘ a ). Unul dintre cele mai cunoscute par adoxuri este cel al lui Einstein (care nu a acceptat ˆ ın mod total mecanica cuant ic˘ a), Podolsky ¸ si Rosen (EPR, 1935) ˆ ın leg˘ atur˘ a cu problema dac˘ a exist ˘ a sau nu “elemente adev˘ arate de realitate fizic˘ a” ˆ ın microcosmosul studiat cu metode cuantice (dup˘ a Einstein, mecanica cuantic˘ a interzice existent ¸a independent˘ a a ac- tului de m˘ asurare de sistemele fizice m˘ asurate). Alt parad ox, de acela¸ si rang de celebritate este al “pisicii lui Schr¨ odinger”. Cee a ce trebuie sub- liniat ˆ ın leg˘ atur˘ a cu toate aceste puncte teoretice ¸ si m etateoretice delicate este c˘ a ele genereaz˘ a experimente foarte interesante (cu m ar fi, de exem- plu, cele referitoare la a¸ sa-numita “teleportare” ale st˘ arilor cuantice) care impulseaz˘ a dezvoltarea tehnologic˘ a . Ceea ce urmeaz˘ a su nt cˆ ıteva teme in- troductive ˆ ın mecanica cuantic˘ a nerelativist˘ a care au s ervit ca baz˘ a pentru cursul de graduare ˆ ın mecanica cuantic˘ a pe care l-am preda t ˆ ın Institutul de Fizic˘ a al Universitat ¸ii Statale Guanajuato din Mexic ˆ ın 1998. Haret C. Rosu 61. POSTULATE CUANTICE Urm˘ atoarele 6 postulate se pot considera ca baz˘ a pentru te orie ¸ si experiment ˆ ın mecanica cuantic˘ a ˆ ın varianta sa cea mai folosit˘ a (st andard). P1.- Fiec˘ arei m˘ arimi fizice ‘bine definit˘ a clasic’ L ˆ ıi cores punde un operator hermitic ˆL. P2.- Fiec˘ arei st˘ ari fizice stat ¸ionare ˆ ın care se poate g˘ asi un sistem fizic cuantic ˆ ıi corespunde o funct ¸ie de und˘ a normalizat˘ a ψ(∝ba∇dblψ∝ba∇dbl2 L2= 1). P3.- M˘ arimea fizic˘ a L poate s˘ a ‘ia’ experimental numai valori le proprii ale ˆL. De aceea valorile proprii trebuie s˘ a fie reale, ceea ce are l oc pentru operatori hermitici. P4.- Ceea ce se m˘ asoar˘ a este ˆ ıntotdeauna valoarea medie La m˘ arimii (op- eratorului) ˆLˆ ın stareaψn, care ˆ ın teorie este elementul de matrice diagonal <ψn|ˆL|ψn>=L. P5.- Elementele de matrice ale operatorilor coordonat˘ a ¸ si mo ment carteziene /hatwidexi¸ si/hatwiderpk, calculate intre funct ¸iile de und˘ a f ¸ si g satisfac ecuat ¸ iile de mi¸ scare Hamilton din mecanica clasica ˆ ın forma: d dt<f|/hatwidepi|g>=−<f|∂/hatwideH ∂/hatwidexi|g> d dt<f|/hatwidexi|g>=<f|∂/hatwideH ∂/hatwidepi|g> , unde/hatwideHeste operatorul hamiltonian, iar derivatele ˆ ın raport cu o per- atori se definesc ca ˆ ın punctul 3 al acestui capitol. P6.- Operatorii /hatwidepi¸ si/hatwiderxkau urm˘ atorii comutatori: [/hatwidepi,/hatwiderxk] =−i¯hδik, [/hatwidepi,/hatwiderpk] = 0, [/hatwidexi,/hatwiderxk] = 0 7¯h=h/2π= 1.0546×10−27erg.sec. 1.- Corespondent ¸a cu o m˘ arime fizic˘ a L care are analog clas icL(xi,pk) Aceasta se face substituind xi,pkcu/hatwidexi/hatwiderpk. Funct ¸ia L se presupune c˘ a se poate dezvolta ˆ ın serie de puteri. Dac˘ a funct ¸ia nu cont ¸ine produse xkpk, operatorul ˆLeste hermitic ˆ ın mod automat. Exemplu: T= (/summationtext3 ip2 i)/2m−→/hatwideT= (/summationtext3 i/hatwidep2)/2m. Dac˘ a L cont ¸ine produse mixte xipi¸ si puteri ale acestora, ˆLnu este hermitic, ˆ ın care caz L se substituie cu ˆΛ, partea hermitic˘ a a lui ˆL(ˆΛ este un operator autoadjunct). Exemplu: w(xi,pi) =/summationtext ipixi−→/hatwidew= 1/2/summationtext3 i(/hatwidepi/hatwidexi+/hatwidexi/hatwidepi). Rezult˘ a deasemenea c˘ a timpul nu este un operator fiind doar un parametru (care se poate introduce ˆ ın multe feluri). Aceasta pentru c ˘ a timpul nu depinde de variabilele canonice ci din contr˘ a . 2.- Probabilitate ˆ ın spectrul discret ¸ si continuu Dac˘ aψneste funct ¸ie proprie a operatorului ˆM, atunci: L=<n|ˆL|n>=<n|λn|n>=λn<n|n>=δnnλn=λn. Deasemenea, se poate demonstra c˘ a Lk= (λn)k. Dac˘ a funct ¸ia φnu este funct ¸ie proprie a lui ˆLse folose¸ ste dezvoltarea ˆ ın sistem complet de f.p. ale lui ˆL¸ si deci: ˆLψn=λnψn,φ=/summationtext nanψn combinˆ ınd aceste dou˘ a relat ¸ii obt ¸inem: ˆLφ=/summationtext nλnanψn. 8Putem astfel calcula elementele de matrice ale operatorulu i L: <φ|ˆL|φ>=/summationtext n,ma∗ manλn<m|n>=/summationtext m|am|2λm, ceea ce ne spune c˘ a rezultatul experimentului este λmcu o probabili- tate|am|2. Dac˘ a spectrul este discret: de acord cu P4ˆ ınseamn˘ a c˘ a|am|2, deci, coeficient ¸ii dezvolt˘ arii ˆ ıntr-un sistem complet de f.p. determin˘ a prob- abilit˘ atile de observare a valorii proprii λn. Dac˘ a spectrul este continuu: folosind urm˘ atoarea definit ¸ie φ(τ) =/integraltexta(λ)ψ(τ,λ)dλ, se calculeaz˘ a elementele de matrice pentru spectrul conti nuu <φ|ˆL|φ> =/integraltextdτ/integraltexta∗(λ)ψ∗(τ,λ)dλ/integraltextµa(µ)ψ(τ,µ)dµ =/integraltext/integraltexta∗a(µ)µ/integraltextψ∗(τ,λ)ψ(tau,µ )dλdµdτ =/integraltext/integraltexta∗(λ)a(µ)µδ(λ−µ)dλdµ =/integraltexta∗(λ)a(λ)λdλ =/integraltext|a(λ)|2λdλ. ˆIn cazul continuu se spune c˘ a |a(λ)|2este densitatea de probabilitate de a observa v.p. λdin spectrul continuu. Deasemenea, se satisface L=<φ|ˆL|φ>. Este comun s˘ a se spun˘ a c˘ a <µ|Φ>este (reprezentarea lui) |Φ>ˆ ın reprezentarea µ, unde|µ>este un vector propriu al lui ˆM. 93.- Definit ¸ia unei derivate ˆ ın raport cu un operator ∂F(ˆL) ∂ˆL= limǫ→∞F(ˆL+ǫˆI)−F(ˆL) ǫ. 4.- Operatorii de impuls cartezian Care este forma concret˘ a a lui /hatwiderp1,/hatwiderp2y/hatwiderp3, dac˘ a argumentele funct ¸iilor de und˘ a sunt coordonatele carteziene xi? Vom considera urm˘ atorul comutator: [/hatwidepi,/hatwidexi2] =/hatwidepi/hatwidexi2−/hatwidexi2/hatwidepi =/hatwidepi/hatwidexi/hatwidexi−/hatwidexi/hatwidepi/hatwidexi+/hatwidexi/hatwidepi/hatwidexi−/hatwidexi/hatwidexi/hatwidepi = (/hatwidepi/hatwidexi−/hatwidexi/hatwidepi)/hatwidexi+/hatwidexi(/hatwidepi/hatwidexi−/hatwidexi/hatwidepi) = [/hatwidepi,/hatwidexi]/hatwidexi+/hatwidexi[/hatwidepi,/hatwidexi] =−i¯h/hatwidexi−i¯h/hatwidexi=−2i¯h/hatwidexi. ˆIn general, se satisfac: /hatwidepi/hatwidexin−/hatwidexin/hatwidepi=−ni¯h/hatwidexin−1. Atunci, pentru toate funct ¸iile analitice avem: /hatwidepiψ(x)−ψ(x)/hatwidepi=−i¯h∂ψ ∂xi. Acum, fie/hatwidepiφ=f(x1,x2,x3) modul ˆ ın care act ¸ioneaz˘ a /hatwidepiasupra lui φ(x1,x2,x3) = 1. Atunci: /hatwidepiψ=−i¯h∂ψ ∂x1+f1ψ¸ si exist˘ a relat ¸ii analoage pentru x2yx3. Din comutatorul [ /hatwidepi,/hatwiderpk] = 0 se obt ¸ine∇×/vectorf= 0, ¸ si deci, fi=∇iF. Forma cea mai general˘ a a lui /hatwidepieste/hatwidepi=−i¯h∂ ∂xi+∂F ∂xi, unde F este orice funct ¸ie. Funct ¸ia F se poate elimina folosind o trans formare uni- tar˘ a/hatwideU†= exp(i ¯hF). 10/hatwidepi=/hatwideU†(−i¯h∂ ∂xi+∂F ∂xi)/hatwideU = expi ¯hF(−i¯h∂ ∂xi+∂F ∂xi)exp−i ¯hF =−i¯h∂ ∂xi rezultˆ ınd /hatwidepi=−i¯h∂ ∂xi−→/hatwidep=−i¯h∇. 5.- Calculul constantei de normalizare Orice funct ¸ie de und˘ a ψ(x)∈L2de variabil˘ a xse poate scrie ˆ ın forma: ψ(x) =/integraltextδ(x−ξ)ψ(ξ)dξ ¸ si se poate considera aceast˘ a expresie ca dezvoltare a lui ψˆ ın f.p. ale operatorului coordonat˘ a ˆ xδ(x−ξ) =ξ(x−ξ). Atunci,|ψ(x)|2 este densitatea de probabilitate a coordonatei ˆ ın starea ψ(x). De aici rezult˘ a interpretarea normei ∝ba∇dblψ(x)∝ba∇dbl2=/integraltext|ψ(x)|2dx= 1. Intuitiv, aceast˘ a relat ¸ie ne spune c˘ a sistemul descris d e c˘ atre funct ¸ia ψ(x) trebuie s˘ a se g˘ aseasc˘ a ˆ ıntr-un ‘loc’ pe axa real˘ a, chi ar dac˘ a vom ¸ sti doar aproximativ unde. Funct ¸iile proprii ale operatorului impuls sunt: −i¯h∂ψ ∂xi=piψ, integrˆ ınd se obt ¸ine ψ(xi) =Aexpi ¯hpixi,x¸ sipau spectru continuu ¸ si deci normalizarea se face cu “funct ¸ia delta. Cum se obt ¸ine constanta de normalizare ? Se poate obt ¸ine utilizˆ ınd urm˘ atoarele transform˘ ari Fo urier: f(k) =/integraltextg(x) exp−ikxdx,g(x) =1 2π/integraltextf(k)expikxdk. Deasemenea se obt ¸ine cu urm˘ atoarea procedur˘ a : Fie funct ¸ia de und˘ a nenormalizat˘ a a particulei libere φp(x) =Aexpipx ¯h¸ si formula δ(x−x′) =1 2π/integraltext∞ −∞expik(x−x′)dx 11Se vede c˘ a /integraltext∞ −∞φ∗ p′(x)φp(x)dx =/integraltext∞ −∞A∗exp−ip′ x ¯hAexpipx ¯hdx =/integraltext∞ −∞|A|2expix(p−p′ ) ¯hdx =|A|2¯h/integraltext∞ −∞expix(p−p′ ) ¯hdx ¯h = 2π¯h|A|2δ(p−p′) ¸ si deci constanta de normalizare este: A=1√ 2π¯h. Deasemenea rezult˘ a c˘ a f.p. ale operatorului impuls forme az˘ a un sistem complet (ˆ ın sensul cazului continuu) pentru funct ¸iile de clas˘ aL2. ψ(x) =1√ 2π¯h/integraltexta(p) expipx ¯hdp a(p) =1√ 2π¯h/integraltextψ(x)exp−ipx ¯hdx. Aceste formule stabilesc leg˘ atura ˆ ıntre reprezent˘ aril e x ¸ si p. 6.- Reprezentarea p (de impuls) Forma explicit˘ a a operatorilor ˆ pi¸ si ˆxkse poate obt ¸ine din relat ¸iile de comutare, dar ¸ si folosind nucleele x(p,β) =U†xU=1 2π¯h/integraltextexp−ipx ¯hxexpiβx ¯hdx 12=1 2π¯h/integraltextexp−ipx ¯h(−i¯h∂ ∂βexpiβx ¯h). Integrala are forma urm˘ atoare: M(λ,λ′) =/integraltextU†(λ,x)/hatwiderMU(λ′,x)dx, ¸ si folosind ˆxf=/integraltextx(x,ξ)f(ξ)dξ, act ¸iunea lui ˆ xasupra lui a(p)∈L2este: ˆxa(p) =/integraltextx(p,β)a(β)dβ =/integraltext(1 2π¯h/integraltextexp−ipx ¯h(−i¯h∂ ∂βexpiβx ¯h)dx)a(β)dβ =−i 2π/integraltext/integraltextexp−ipx ¯h∂ ∂βexpiβx ¯ha(β)dxdβ =−i¯h 2π/integraltext/integraltextexp−ipx ¯h∂ ∂βexpiβx ¯ha(β)dx ¯hdβ =−i¯h 2π/integraltext/integraltextexpix(β−p) ¯h∂ ∂βa(β)dx ¯hdβ =−i¯h/integraltext∂a(p) ∂βδ(β−p)dβ=−i¯h∂a(p) ∂p, unde δ(β−p) =1 2π/integraltextexpix(β−p) ¯hdx ¯h. Operatorul impuls ˆ ın reprezentarea p se caracterizeaz˘ a p rin nucleul: p(p,β) =/hatwideU†p/hatwideU =1 2π¯h/integraltextexp−ipx ¯h(−i¯h∂ ∂x)expiβx ¯hdx =1 2π¯h/integraltextexp−ipx ¯hβexpiβx ¯hdx=βλ(p−β) rezultˆ ınd ˆpa(p) =pa(p). Ceea ce se ˆ ıntˆ ımpl˘ a cu ˆ x¸ si ˆpeste c˘ a de¸ si sunt operatori hermitici pen- tru toate f(x)∈L2nu sunt hermitici exact pentru funct ¸iile lor proprii. Dac˘ a ˆpa(p) =poa(p) ¸ si ˆx= ˆx†ˆp= ˆp†, atunci 13<a|ˆpˆx|a>−<a|ˆxˆp|a>=−i¯h<a|a> po[<a|ˆx|a>−<a|ˆx|a>] =−i¯h<a|a> po[<a|ˆx|a>−<a|ˆx|a>] = 0 Partea stˆ ıng˘ a este zero, ˆ ın timp ce dreapta este indefinit ˘ a , ceea ce este o contradict ¸ie. 7.- Reprezent˘ arile Schr¨ odinger ¸ si Heisenberg Ecuat ¸iile de mi¸ scare date prin P5au diferite interpret˘ ari, datorit˘ a fap- tului c˘ a ˆ ın expresiad dt< f|ˆL|f >se poate considera dependent ¸a temporal˘ a ca apart ¸inˆ ınd fie funct ¸iilor de und˘ a fie opera torilor, fie atˆ ıt funct ¸iilor de und˘ a cˆ ıt ¸ si operatorilor. Vom considera n umai primele dou˘ a cazuri. •Pentru un operator ce depinde de timp/hatwideO=/hatwidestO(t) avem: ˆpi=−∂/hatwideH ∂ˆxi, ˆxi=∂/hatwideH ∂ˆpi [ˆp,f] = ˆpf−fˆp=−i¯h∂f ∂ˆxi [ˆx,f] = ˆxf−fˆx=−i¯h∂f ∂ˆpi ¸ si se obt ¸in ecuat ¸iile de mi¸ scare Heisenberg: ˆpi=−i ¯h[ˆp,/hatwideH], ˆxi=−i ¯h[ˆx,/hatwideH]. •Dac˘ a funct ¸iile de und˘ a depind de timp, ˆ ınc˘ a se poate fol osi ˆpi= −i ¯h[ˆpi,/hatwideH], pentru c˘ a este o consecint ¸˘ a numai a relat ¸iilor de co- mutare ¸ si deci nu depind de reprezentare d dt<f|ˆpi|g>=−i ¯h<f|[ˆp,/hatwideH]|g>. 14Dac˘ a acum ˆ pi¸ si/hatwideHnu depind de timp ¸ si se t ¸ine cont de hermitic- itate se obt ¸ine: (∂f ∂t,ˆpig) + (ˆpif,∂g ∂t) =−i ¯h(f,ˆpiˆHg) +i ¯h(f,ˆHˆpig) =−i ¯h(ˆpf,ˆHg) +i ¯h(ˆHf,ˆpig) (∂f ∂t+i ¯hˆHf,ˆpig) + (ˆpif,∂g ∂t−i ¯hˆHg) = 0 Ultima relat ¸ie seˆ ındepline¸ ste pentru orice pereche de f unct ¸iif(x) ¸ sig(x) ˆ ın momentul init ¸ial dac˘ a fiecare satisface ecuat ¸ia i¯h∂ψ ∂t=Hψ. Aceasta este ecuat ¸ia Schr¨ odinger ¸ si descrierea sistemu lui cu aju- torul operatorilor independent ¸i de timp se cunoa¸ ste ca re prezentarea Schr¨ odinger. ˆIn ambele reprezent˘ ari evolut ¸ia temporal˘ a a sistemului se caracter- izeaz˘ a prin operatorul/hatwideH, care se obt ¸ine din funct ¸ia Hamilton din mecanica clasic˘ a . Exemplu:/hatwideHpentru o particul˘ a ˆ ın potent ¸ial U(x1,x2,x3) este: /hatwideH=ˆp2 2m+U(x1,x2,x3), ¸ si ˆ ın reprezentarea x este: /hatwideH=−¯h2 2m∇2 x+U(x1,x2,x3). 8.- Leg˘ atura ˆ ıntre reprezent˘ arile S ¸ si H P5este corect ˆ ın reprezent˘ arile Schr¨ odinger ¸ si Heisenbe rg. De aceea, valoarea medie a oric˘ arei observabile coincideˆ ın cele do u˘ a reprezent˘ ari, ¸ si deci, exist˘ a o transformare unitar˘ a cu care se poate tr ece de la o reprezentare la alta. O astfel de transformare este de forma ˆs†= exp−iˆHt ¯h. Pentru a trece la reprezentarea Schr¨ odinger trebuie folo sit˘ a transformata Heisenberg ψ=ˆs†fcuf¸ siˆL, ¸ si pentru a trece la reprezentarea Heisenberg se folose¸ ste transformarea Sch r¨ odinger ˆΛ = 15ˆs†ˆLˆscuψ¸ siˆΛ. Se poate obt ¸ine ecuat ¸ia Schr¨ odinger dup˘ a cum urmeaz˘ a : cum ˆ ın transformarea ψ=ˆs†ffunct ¸iafnu depinde de timp, vom deriva transformarea ˆ ın raport cu timpul pentru a obt ¸ine: ∂ψ ∂t=∂s† ∂tf=∂ ∂t(exp−i/hatwideHt ¯h)f=−i ¯h/hatwideHexp−i/hatwideHt ¯hf=−i ¯h/hatwideHˆs†f=−i ¯h/hatwideHψ. ¸ si deci, avem: i¯h∂ψ ∂t=/hatwideHψ. ˆIn continuare obt ¸inem ecuat ¸ia Heisenberg: punˆ ınd trans formarea Schr¨ odinger ˆ ın urm˘ atoarea form˘ a ˆ sˆΛˆs†=ˆL¸ si derivˆ ındˆ ın raport cu timpul se obt ¸ine ecuat ¸ia Heisenberg ∂ˆL ∂t=∂ˆs ∂tˆΛˆs†+ ˆsˆΛ∂ˆs† ∂t=i ¯h/hatwideHexpi/hatwideHt ¯hˆΛˆs†−i ¯hˆsˆλexp−iˆHt ¯h/hatwideH =i ¯h(/hatwideHˆsˆΛˆs†−ˆsˆΛˆs†/hatwideH) =i ¯h(/hatwideHˆL−ˆL/hatwideH) =i ¯h[/hatwideH,ˆL]. Prin urmare, avem: ∂ˆL ∂t=i ¯h[/hatwideH,ˆL]. Deasemenea, ecuat ¸ia Heisenberg se poate scrie ˆ ın forma ur m˘ atoare: ∂ˆL ∂t=i ¯hˆs[/hatwideH,ˆΛ]ˆs†. ˆLse cunoa¸ ste ca integral˘ a de mi¸ scare dac˘ ad dt<ψ|ˆL|ψ >= 0 ¸ si se caracterizeaz˘ a prin urm˘ atorii comutatori: [/hatwideH,ˆL] = 0, [/hatwideH,ˆΛ] = 0. 169.- St˘ ari stat ¸ionare St˘ arile unui sistem descris prin f.p. ale lui/hatwideHse numesc st˘ ari stat ¸ionare ale sistemului, ˆ ın timp ce setul de v.p. corespunz˘ atoare s e nume¸ ste spectrul de energie (spectrul energetic) al sistemului. ˆIn astfel de cazuri, ecuat ¸ia Schr¨ odinger este : i¯h∂ψn ∂t=Enψn=/hatwideHψn. Solut ¸iile sunt de forma: ψn(x,t) = exp−iEnt ¯hφn(x). •Probabilitatea este urm˘ atoarea: δ(x) =|ψn(x,t)|2=|exp−iEnt ¯hφn(x)|2 = expiEnt ¯hexp−iEnt ¯h|φn(x)|2=|φn(x)|2. Rezult˘ a c˘ a probabilitatea este constant˘ a ˆ ın timp. •ˆIn st˘ arile stat ¸ionare, valoarea medie a oric˘ arui comuta tor de forma [/hatwideH,ˆA] este zero, unde ˆAeste un operator arbitrar: <n|/hatwideHˆA−ˆA/hatwideH|n>=<n|/hatwideHˆA|n>−<n|ˆA/hatwideH|n> =<n|EnˆA|n>−<n|ˆAEn|n> =En<n|ˆA|n>−En<n|ˆA|n>= 0. •Teorema de virial (virialului) ˆ ın mecanica cuantic˘ a - dac ˘ a/hatwideHeste un operator Hamiltonian al unei particuleˆ ın cˆ ımpul U(r), folosind ˆA= 1/2/summationtext3 i=1(ˆpiˆxi−ˆxiˆpi) se obt ¸ine: <ψ|[ˆA,/hatwideH]|ψ>= 0 =<ψ|ˆA/hatwideH−/hatwideHˆA|ψ> =/summationtext3 i=1<ψ|ˆpiˆxi/hatwideH−/hatwideHˆpiˆxi|ψ> =/summationtext3 i=1<ψ|[/hatwideH,ˆxi]ˆpi+ ˆxi[/hatwideH,ˆpi]|ψ>. 17folosind de mai multe ori comutatorii ¸ si ˆ pi=−i¯h∇i,ˆH=/hatwideT+ U(r), se poate obt ¸ine: <ψ|[ˆA,/hatwideH]|ψ>= 0 =−i¯h(2<ψ|/hatwideT|ψ>−<ψ|/vector r·∇U(r)|ψ>). Aceasta este teorema virialului. Dac˘ a potent ¸ialul este U(r) = Uorn, atunci se obt ¸ine o form˘ a a teoremei de virial caˆ ın mecani ca clasic˘ a cu unica diferent ¸˘ a c˘ a se refer˘ a la valori medii T=n 2U. •Pentru un Hamiltonian/hatwideH=−¯h2 2m∇+U(r) ¸ si [/vector r,H] =−i¯h m/vector p, cal- culˆ ınd elementele de matrice se obt ¸ine: (Ek−En)<n|/vector r|k>=i¯h m<n|ˆp|k>. 10.- Densitate de curent de probabilitate Schr¨ odinger Urm˘ atoarea integral˘ a : /integraltext|ψn(x)|2dx= 1, este normalizarea unei f.p. din spectrul discret ˆ ın reprez entarea de coordonat˘ a , ¸ si apare ca o condit ¸ie asupra mi¸ sc˘ arii mic roscopice ˆ ıntr-o regiune finit˘ a . Pentru f.p. ale spectrului continuu ψλ(x) nu se poate da ˆ ın mod direct o interpretare probabilistic˘ a . S˘ a presupunem o funct ¸ie dat˘ a φ∈L2, pe care o scriem ca o combinat ¸ie linear˘ a de f.p. ˆ ın continuu: φ=/integraltexta(λ)ψλ(x)dx. Se spune c˘ a φcorespunde unei mi¸ sc˘ ari infinite. ˆIn multe cazuri, funct ¸ia a(λ) este diferit˘ a de zero numaiˆ ıntr-o vecin˘ atate a unui punct λ=λo.ˆIn acest caz φse cunoa¸ ste ca pachet de unde. 18Vom calcula acum viteza de schimbare a probabilit˘ at ¸ii de a g˘ asi sis- temul ˆ ın volumul Ω. P=/integraltext Ω|ψ(x,t)|2dx=/integraltext Ωψ∗(x,t)ψ(x,t)dx. Derivˆ ınd integrala ˆ ın raport cu timpul g˘ asim: dP dt=/integraltext Ω(ψ∂ψ∗ ∂t+ψ∗∂ψ ∂t)dx. Utilizˆ ınd ecuat ¸ia Schr¨ odingerˆ ın integrala din partea dreapt˘ a se obt ¸ine: dP dt=i ¯h/integraltext Ω(ψˆHψ∗−ψ∗ˆHψ)dx. Folosind identitatea f∇g−g∇f=div[(f)grad(g)−(g)grad(f)] pre- cum ¸ si ecuat ¸ia Schr¨ odinger ˆ ın forma: ˆHψ=¯h2 2m∇ψ ¸ si subtituind ˆ ın integral˘ a se obt ¸ine: dP dt=i ¯h/integraltext Ω[ψ(−¯h2 2m∇ψ∗)−ψ∗(−¯h2 2m∇ψ)]dx =−/integraltext Ωi¯h 2m(ψ∇ψ∗−ψ∗∇ψ)dx =−/integraltext Ωdivi¯h 2m(ψ∇ψ∗−ψ∗∇ψ)dx. Folosind teorema divergent ¸ei pentru a transforma integra la de volum ˆ ın una de suprafat ¸˘ a obt ¸inem: dP dt=−/contintegraltexti¯h 2m(ψ∇ψ∗−ψ∗∇ψ)dx. 19M˘ arimea/vectorJ(ψ) =i¯h 2m(ψ∇ψ∗−ψ∗∇ψ) se cunoa¸ ste ca densitate de curent de probabilitate, pentru care imediat se obt ¸ine o ecuat ¸ie de continui- tate, dρ dt+div(/vectorJ) = 0. •Dac˘ aψ(x) =AR(x), unde R(x) este o funct ¸ie real˘ a , atunci: /vectorJ(ψ) = 0. •Pentru f.p. ale impulsului ψ(x) =1 (2π¯h)3/2expi/vector p/vector x ¯hse obt ¸ine: J(ψ) =i¯h 2m(1 (2π¯h)3/2expi/vector p/vector x ¯h(i/vector p ¯h(2π¯h)3/2exp−i/vector p/vector x ¯h) −(1 (2π¯h)3/2exp−i/vector p/vector x ¯hi/vector p ¯h(2π¯h)3/2expi¯h/vector p/vector x ¯h)) =i¯h 2m(−2i/vector p ¯h(2π¯h)3) =/vector p m(2π¯h)3, ceea ce ne indic˘ a c˘ a densitatea de probabilitate nu depind e de coordonat˘ a . 11.- Operator de transport spat ¸ial Dac˘ a/hatwideHeste invariant la translat ¸ii de vector arbitrar /vector a, /hatwideH(/vector r+/vector a) =/hatwideH/vector(r) , atunci exist˘ a un/hatwideT(/vector a) unitar/hatwideT†(/vector a)/hatwideH(/vector r)/hatwideT(/vector a) =/hatwideH(/vector r+/vector a). Din cauza comutativit˘ at ¸ii translat ¸iilor /hatwideT(/vector a)/hatwideT(/vectorb) =/hatwideT(/vectorb)/hatwideT(/vector a) =/hatwideT(/vector a+/vectorb), result˘ a c˘ a/hatwideTare forma/hatwideT= expiˆka, unde, ˆk=ˆp ¯h. ˆIn cazul infinitezimal: /hatwideT(δ/vector a)/hatwideH/hatwideT(δ/vector a)≈(ˆI+iˆkδ/vector a)/hatwideH(ˆI−iˆkδ/vector a), 20/hatwideH(/vector r) +i[ˆK,/hatwideH]δ/vector a=/hatwideH(/vector r) + (∇/hatwideH)δ/vector a. Deasemenea [ˆ p,/hatwideH] = 0, unde ˆ peste o integral˘ a de mi¸ scare. Sistemul are funct ¸ii de und˘ a de forma ψ(/vector p,/vector r) =1 (2π¯h)3/2expi/vector p/vector r ¯h¸ si transformarea unitar˘ a face ca expi/vector p/vector a ¯hψ(/vector r) =ψ(/vector r+/vector a). Operatorul de transport spat ¸ial /hatwideT†= exp−i/vector p/vector a ¯heste analogul lui ˆ s†= exp−iˆHt ¯h, operatorul de ‘transport’ temporal. 12.- Exemplu: Hamiltonian cristalin Dac˘ a/hatwideHeste invariant pentru o translat ¸ie discret˘ a (de exemplu ˆ ıntr-o ret ¸ea cristalin˘ a )/hatwideH(/vector r+/vector a) =/hatwideH(/vector r), unde/vector a=/summationtext i/vector aini,ni∈N¸ siai sunt vectorii barici, atunci: /hatwideH(/vector r)ψ(/vector r) =Eψ(/vector r), /hatwideH(/vector r+/vector a)ψ(/vector r+/vector a) =Eψ(/vector r+/vector a) =ˆH(/vector r)ψ(/vector r+/vector a). Rezult˘ a c˘ a ψ(/vector r) ¸ siψ(/vector r+/vector a) sunt funct ¸ii de und˘ a pentru aceea¸ si v.p. a lui/hatwideH. Relat ¸ia ˆ ıntre ψ(/vector r) ¸ siψ(/vector r+/vector a) se poate c˘ auta ˆ ın forma ψ(/vector r+/vector a) = ˆc(/vector a)ψ(/vector r) unde ˆc(/vector a) este o matrice gxg (g este gradul de degenerare al nivelului E). Dou˘ a matrici de tip coloan˘ a , ˆ c(/vector a) ˆc(/vectorb) co- mut˘ a ¸ si atunci sunt diagonalizabili simultan. ˆIn plus, pentru elementele diagonale se respect˘ a cii(/vector a)cii(/vectorb) =cii(/vector a+/vectorb), i=1,2,....,g, cu solut ¸ii de tipul cii(a) = expikia. Rezult˘ a c˘ a ψk(/vector r) = Uk(/vector r)expi/vectork/vector a, unde/vectorkeste un vector real arbitrar ¸ si funct ¸ia Uk(/vector r) este periodic˘ a de perioad˘ a /vector a,Uk(/vector r+/vector a) =Uk(/vector r). Afirmat ¸ia c˘ a funct ¸iile proprii ale unui ˆHperiodic cristalin ˆH(/vector r+/vector a) = ˆH(/vector r) se pot scrie ψk(/vector r) =Uk(/vector r)expi/vectork/vector acuUk(/vector r+/vector a) =Uk(/vector r) se cunoa¸ ste ca teorema lui Bloch. ˆIn cazul continuu, Uktrebuie s˘ a fie constant, pentru c˘ a o constant˘ a este unica funct ¸ie perio dic˘ a pentru orice/vector a. Vectorul /vector p= ¯h/vectorkse nume¸ ste cuasi-impuls (prin analogie cu cazul continuu). Vectorul /vectorknu este determinat ˆ ın mod univoc, pentru c˘ a i se poate ad˘ auga orice vector /vector gpentru care ga= 2πnunde n∈N. Vectorul/vector gse poate scrie /vector g=/summationtext3 i=1/vectorbimiundemisunt numere ˆ ıntregi ¸ sibisunt dat ¸i de 21/vectorbi= 2πˆaj×/vector ak /vector ai(/vector aj×/vector ak) pentrui∝ne}ationslash=j∝ne}ationslash=k./vectorbisunt vectorii barici ai ret ¸elei cristaline. Referint ¸e recomandate 1. E. Farhi, J. Goldstone, S. Gutmann, “How probability aris es in quantum mechanics”, Annals of Physics 192, 368-382 (1989) 2. N.K. Tyagiˆ ın Am. J. Phys. 31, 624 (1963) d˘ a o demostrat ¸ie foarte scurt˘ a pentru principiul de incertitudine Heisenberg, pe baza c˘ a ruia se afirm˘ a c˘ a m˘ asurarea simultan˘ a a doi operatori hermitici care nu com ut˘ a produce o incertitudine relat ¸ionat˘ a cu valoarea comutatorului lo r. 3. H.N. N´ u˜ nez-Y´ epez et al., “Simple quantum systems in th e momentum representation”, physics/0001030 (Europ. J. Phys., 2000) . 4. J.C. Garrison, “Quantum mechanics of periodic systems”, Am. J. Phys. 67, 196 (1999). 5. F. Gieres, “Dirac’s formalism and mathematical surprise s in quantum me- chanics”, quant-ph/9907069 (ˆ ın englez˘ a); quant-ph/990 7070 (ˆ ın francez˘ a). 1N. Note 1. Pentru “crearea mecanicii cuantice...”, Werner Heisenb erg a fost distins cu premiul Nobelˆ ın 1932 (primitˆ ın 1933). Articolul “Z¨ ur Quantenmechanik. II”, [“Asupra mecanicii cuantice.II”, Zf. f. Physik 35, 557-615 (1926) (ajuns la redact ¸ie ˆ ın 16 Noiembrie 1925) de M. Born, W. Heisenberg ¸ si P. Jordan, se cunoa¸ ste ca “lucrarea celor trei (oameni)” ¸ si este cons iderat ca cel care a deschis cu adev˘ arat vastele orizonturi ale mecanicii cuan tice. 2. Pentru “interpretarea statistic˘ a a funct ¸iei de und˘ a ” Max Born a primit premiul Nobel ˆ ın 1954. 1P. Probleme Problema 1.1 : Se consider˘ a doi operatori A ¸ si B care prin ipotez˘ a comut ˘ a. ˆIn acest caz se poate deduce relat ¸ia: eAeB=e(A+B)e(1/2[A,B]). Solut ¸ie Definim un operator F(t), ca funct ¸ie de variabil˘ a real˘ a t, prin:F(t) = e(At)e(Bt). 22Atunci:dF dt=AeAteBt+eAtBeBt= (A+eAtBe−At)F(t). Acum, aplicˆ ınd formula [ A,F(B)] = [A,B]F′(B), avem [eAt,B] =t[A.B]eAt, ¸ si deci:eAtB=BeAt+t[A,B]eAt, multiplicˆ ınd ambele p˘ art ¸i ale ecuat ¸iei ultime cu exp−At¸ si substituind ˆ ın prima ecuat ¸ie, obt ¸inem: dF dt= (A+B+t[A,B])F(t). Operatorii A , B ¸ si [A,B] comut˘ a prin ipotez˘ a . Deci, putem integra ecuat ¸ia diferent ¸ial˘ a ca ¸ si cum A+B¸ si [A,B] ar fi numere (scalare). Vom avea deci: F(t) =F(0)e(A+B)t+1/2[A,B]t2. Punˆ ındt= 0, se vede c˘ a F(0) = 1, ¸ si : F(t) =e(A+B)t+1/2[A,B]t2. Punˆ ınd acum t= 1, obt ¸inem rezultatul dorit. Problema 1.2 : S˘ a se calculeze comutatorul [ X,Dx]. Solut ¸ie Calculul se face aplicˆ ınd comutatorul unei funct ¸ii arbit rareψ(/vector r): [X,Dx]ψ(/vector r) = (x∂ ∂x−∂ ∂xx)ψ(/vector r) =x∂ ∂xψ(/vector r)−∂ ∂x[xψ(/vector r)] =x∂ ∂xψ(/vector r)−ψ(/vector r)−x∂ ∂xψ(/vector r) =−ψ(/vector r). Cum aceast˘ a relat ¸ie se satisface pentru orice ψ(/vector r), se poate deduce c˘ a : [X,Dx] =−1. Problema 1.3 : S˘ a se verifice c˘ a urma de matrice este invariant˘ a la schim b˘ ari de baze ortonormale discrete. Solut ¸ie Suma elementelor diagonale ale unei reprezent˘ ari matrice ale de un oper- ator (cuantic) A ˆ ıntr-o baz˘ a arbitrar˘ a nu depinde de baz˘ a . Se va obt ¸ine aceast˘ a propietate pentru cazul schimb˘ arii dintr-o baz˘ a ortonor- mal˘ a dicret˘ a|ui>ˆ ın alta ortonormal discret˘ a |tk>. Avem:/summationtext i<ui|A|ui>=/summationtext i<ui|(/summationtext k|tk><tk|)A|ui> 23(unde s-a folosit relat ¸ia de completitudine pentru starea tk). Partea dreapt˘ a este egal˘ a cu: /summationtext i,j<ui|tk><tk|A|ui>=/summationtext i,j<tk|A|ui><ui|tk>, (este posibil˘ a schimbarea ordinii ˆ ın produsul de dou˘ a nu mere scalare). Ast- fel, putem ˆ ınlocui/summationtext i|ui><ui|cu unu (relat ¸ia de completitudine pentru st˘ arile|ui>), pentru a obt ¸ine ˆ ın final: /summationdisplay i<ui|A|ui>=/summationdisplay k<tk|A|tk> . A¸ sadar, s-a demonstrat proprietatea cerut˘ a de invariant ¸˘ a pentru urmele matriceale. Problema 1.4 : Dac˘ a pentru operatorul hermitic Nexist˘ a operatorii her- miticiL¸ siMastfel c˘ a : [M,N] = 0, [L,N] = 0, [M,L]∝ne}ationslash= 0, atunci funct ¸iile proprii ale lui Nsunt degenerate. Solut ¸ie Fieψ(x;µ,ν) funct ¸iile proprii comune ale lui M¸ siN(fiind de comutator nul sunt observabile simultane). Fie ψ(x;λ,ν) funct ¸iile proprii comune ale luiL¸ siN(fiind de comutator nul sunt observabile simultane). Parame trii greci indic˘ a valorile proprii ale operatorilor corespunz ˘ atori. Consider˘ am, pentru a simplifica, c˘ a Nare spectru discret. Atunci: f(x) =/summationdisplay νaνψ(x;µ,ν) =/summationdisplay νbνψ(x;λ,ν). Se calculeaz˘ a acum elementul de matrice <f|ML|f >: <f|ML|f >=/integraldisplay/summationdisplay νµνaνψ∗(x;µ,ν)/summationdisplay ν′λν′bν′ψ(x;λ,ν′)dx . Dac˘ a toate funct ¸iile proprii ale lui Nsunt diferite (nedegenerate) atunci < f|ML|f >=/summationtext νµνaνλνbν. Dar acela¸ si rezultat se obt ¸ine ¸ si dac˘ a se cal- culeaz˘ a< f|LM|f >¸ si comuatorul ar fi zero. Prin urmare, unele funct ¸ii proprii ale lui Ntrebuie s˘ a fie degenerate. 242. BARIERE S ¸I GROPI RECTANGULARE Regiuni de potent ¸ial constant ˆIn cazul unui potent ¸ial cuadrat, V(x) este o funct ¸ie constant˘ a V(x) =V ˆ ıntr-o regiune oarecareˆ ın spat ¸iu. ˆIntr-o astfel de regiune, ecuat ¸ia Schr¨ odinger poate fi scris˘ a : d2 dx2ψ(x) +2m ¯h2(E−V)ψ(x) = 0 (1) Distingem mai multe cazuri: (i)E >V Introducem constanta pozitiv˘ a k, definit˘ a prin k=/radicalbig 2m(E−V) ¯h(2) Solut ¸ia ecuat ¸iei (1) se poate atunci scrie: ψ(x) =Aeikx+A′e−ikx(3) undeA¸ siA′sunt constante complexe. (ii)E <V Aceast˘ a condit ¸ie corespunde la regiuni de spat ¸iu care ar fi interzise pentru particul˘ a din punctul de vedere al mi¸ sc˘ arii mecanice cla sice.ˆIn acest caz, introducem constanta pozitiv˘ a qdefinit˘ a prin: q=/radicalbig 2m(V−E) ¯h(4) ¸ si solut ¸ia lui (1) poate fi scris˘ a : ψ(x) =Beqx+B′e−qx(5) undeB¸ siB′sunt constante complexe. (iii)E=V ˆIn acest caz special, ψ(x) este o funct ¸ie linear˘ a de x. Comportamentul lui ψ(x)la o discontinuitate a potent ¸ialului S-ar putea crede c˘ a in punctul x=x1, unde potent ¸ialul V(x) este discontinuu, funct ¸ia de und˘ a ψ(x) se comport˘ a mai ciudat, poate c˘ a ˆ ın mod discontinuu, de exemplu. Aceasta nu se ˆ ıntˆ ımpl˘ a : ψ(x) ¸ sidψ dxsunt continue, ¸ si numai a doua derivat˘ a prezint˘ a discontinuitate ˆ ın x=x1. 25Viziune general˘ a asupra calculului Procedeul pentru determinarea st˘ arii stat ¸ionare ˆ ın pot ent ¸iale rectangu- lare este deci urm˘ atorul: ˆ ın toate regiunile unde V(x) este constant, scriem ψ(x) ˆ ın oricare dintre cele dou˘ a forme (3) sau (5) ˆ ın funct ¸ie de aplicat ¸ie; ˆ ın continuare ‘lipim’ aceste funt ¸ii corespunz˘ ator cerint ¸ ei de continuitate pentru ψ(x) ¸ sidψ dxˆ ın punctele unde V(x) este discontinuu. Examinarea cˆ ıtorva cazuri simple S˘ a facem calculul cantitativ pentru st˘ arile stat ¸ionare , conform metodei descrise. Potent ¸ial treapt˘ a xV(x) V0 0III Fig. 2.1 a. CazulE >V 0;reflexie part ¸ial˘ a S˘ a punem ec. (2) ˆ ın forma: k1=√ 2mE ¯h(6) 26k2=/radicalbig2m(E−V0) ¯h(7) Solut ¸ia ec. (1) are forma din ec. (3) ˆ ın regiunile I(x<0) ¸ siII(x>0): ψI=A1eik1x+A′ 1e−ik1x ψII=A2eik2x+A′ 2e−ik2x ˆIn regiunea I ec. (1) ia forma: ψ′′(x) +2mE ¯h2ψ(x) =ψ′′(x) +k2ψ(x) = 0 iar ˆ ın regiunea II: ψ′′(x)−2m ¯h2[V0−E]φ(x) =ψ′′(x)−q2ψ(x) = 0 Dac˘ a ne limit˘ am la cazul unei particule incidente care ‘vi ne’ de lax= −∞, trebuie s˘ a alegem A′ 2= 0 ¸ si putem determina razele A′ 1/A1yA2/A1. Condit ¸iile de ‘lipire’ dau atunci: •ψI=ψII, ˆ ınx= 0 : A1+A′ 1=A2 (8) •ψ′ I=ψ′ II, ˆ ınx= 0 : A1ik1−A′ 1ik1=A2ik2 (9) Substituind A1¸ siA′ 1din (8) ˆ ın (9): A′ 1=A2(k1−k2) 2k1(10) A1=A2(k1+k2) 2k1(11) Egalarea constantei A2ˆ ın (10) ¸ si (11) implic˘ a : A′ 1 A1=k1−k2 k1+k2(12) 27¸ si din (11) obt ¸inem: A2 A1=2k1 k1+k2(13) ψ(x) este o superpozit ¸ia de dou˘ a unde. Prima (termenul ˆ ın A1) corespunde unei particule incidente, de moment p= ¯hk1, ˆ ın propagare de la stˆ ınga la dreapta. A doua (termenul ˆ ın A′ 1) corespunde unei particule reflectate, de impuls−¯hk1, ˆ ın propagare ˆ ın sens opus. Cum deja am ales A′ 2= 0,ψII(x) cont ¸ine o singur˘ a und˘ a , care este asociat˘ a cu o particul ˘ a transmis˘ a . (Se va ar˘ ata mai departe cum este posibil, folosind conceptul de c urent de prob- abilitate, s˘ a definim coeficientul de transmisie T precum ¸ s i coeficientul de reflexie R pentru potent ¸ialul treapt˘ a ). Ace¸ sti coeficien t ¸i dau probabilitatea ca o particul˘ a , sosind de la x=−∞, ar putea trece de potent ¸ialul treapt˘ a ˆ ınx= 0 sau se ˆ ıntoarce. Astfel obt ¸inem: R=|A′ 1 A1|2(14) iar pentruT: T=k2 k1|A2 A1|2. (15) T ¸inˆ ınd cont de (12) ¸ si (13), avem: R= 1−4k1k2 (k1+k2)2(16) T=4k1k2 (k1+k2)2. (17) Este u¸ sor de verificat c˘ a R+T= 1: este deci sigur c˘ a particula va fi transmis˘ a sau reflectat˘ a . Contrar predict ¸iilor mecanic ii clasice, particula incident˘ a are o probabilitate nenul˘ a de a nu se ˆ ıntoarce. Deasemenea este u¸ sor de verificat, folosind (6), (7) ¸ si (17 ), c˘ a dac˘ aE≫ V0atunciT≃1: cˆ ınd energia particulei este suficient de mareˆ ın compar at ¸ie cu ˆ ın˘ alt ¸imea treptei, pentru particul˘ a este ca ¸ si cum o bstacolul treapt˘ a nu ar exista. Considerˆ ınd solut ¸ia ˆ ın regiunea I: ψI=A1eik1x+A′ 1e−ik1x j=−i¯h 2m(φ∗▽φ−φ▽φ∗) (18) 28cuA1eik1x¸ si conjugata sa A∗ 1e−ik1x: j=−i¯h 2m[(A∗ 1e−ik1x)(A1ik1eik1x)−(A1eik1x)(−A∗ 1ik1e−ik1x)] j=¯hk1 m|A1|2 Acum cuA′ 1e−ik1x¸ si conjugata sa A∗ 1eik1xrezult˘ a : j=−¯hk1 m|A′ 1|2. Dorim ˆ ın continuare s˘ a verific˘ am proport ¸ia de curent refl ectat fat ¸˘ a de curentul incident (mai precis, dorim s˘ a verific˘ am probabi litatea ca particula s˘ a fie returnat˘ a ): R=|j(φ−)| |j(φ+)|=|−¯hk1 m|A′ 1|2| |¯hk1 m|A1|2|=|A′ 1 A1|2(19) Similar, proport ¸ia de transmisie fat ¸˘ a de incident ¸˘ a (a dic˘ a probabilitatea ca particula s˘ a fie transmis˘ a ) este, t ¸inˆ ınd acum cont de s olut ¸ia din regiunea II: T=|¯hk2 m|A2|2| |¯hk1 m|A1|2|=k2 k1|A2 A1|2(20) a. CazulE <V 0;reflexie total˘ a ˆIn acest caz avem: k1=√ 2mE ¯h(21) q2=/radicalbig 2m(V0−E) ¯h(22) ˆIn regiunea I(x<0), solut ¸ia ec. (1) [scris˘ a ψ(x)′′+k2 1ψ(x) = 0] are forma dat˘ a ˆ ın ec. (3): ψI=A1eik1x+A′ 1e−ik1x(23) iar ˆ ın regiunea II(x>0), aceea¸ si ec. (1) [acum scris˘ a ψ(x)′′−q2 2ψ(x) = 0] are forma ec. (5): ψII=B2eq2x+B′ 2e−q2x(24) 29Pentru ca solut ¸ia s˘ a fie ment ¸inut˘ a finit˘ a cˆ ınd x→+∞, este necesar ca: B2= 0 (25) Condit ¸iile de ‘lipit’ ˆ ın x= 0 dau ˆ ın acest caz: •ψI=ψII, ˆ ınx= 0 : A1+A′ 1=B′ 2 (26) •ψ′ I=ψ′ II, ˆ ınx= 0 : A1ik1−A′ 1ik1=−B′ 2q2 (27) Substituind A1¸ siA′ 1din (26) ˆ ın (27): A′ 1=B′ 2(ik1+q2) 2ik1(28) A1=B′ 2(ik1−q2) 2ik1(29) Egalarea constantei B′ 2ˆ ın (28) ¸ si (29) duce la: A′ 1 A1=ik1+q2 ik1−q2=k1−iq2 k1+iq2, (30) astfel c˘ a din (29) avem: B′ 2 A1=2ik1 ik1−q2=2k1 k1−iq2(31) Coeficientul de reflexie Reste deci: R=|A′ 1 A1|2=|k1−iq2 k1+iq2|2=k2 1+q2 2 k2 1+q2 2= 1 (32) Ca ˆ ın mecanica clasic˘ a , microparticula este ˆ ıntotdeaun a reflectat˘ a (reflexie total˘ a ). Totu¸ si, exist˘ a o diferent ¸˘ a important˘ a : dat orit˘ a existent ¸ei a¸ sa- numitei unde evanescente e−q2x, particula are o probabilitate nenul˘ a de a se g˘ asi ‘prezent˘ a ’ ˆ ın regiunea din spat ¸iu care este clas ic interzis˘ a . Aceast˘ a probabilitate descre¸ ste exponent ¸ial cu x¸ si ajunge s˘ a fie neglijabil˘ a cˆ ınd x dep˘ a¸ se¸ ste “zona” 1 /q2corespunz˘ atoare undei evanescente. S˘ a not˘ am ¸ si c˘ a A′ 1/A1este complex. O anumit˘ a diferent ¸˘ a de faz˘ a apare din cauz a reflexiei, care, fizic, se datoreaz˘ a faptului c˘ a particula este ‘ frˆ ı nat˘ a’ (ˆ ıncetinit˘ a cˆ ınd intr˘ a ˆ ın regiunea x >0. Nu exist˘ a analogie pentru aceasta ˆ ın mecanica clasic˘ a , ci doar ˆ ın optica fizic˘ a . 30Potent ¸iale tip barier˘ a 0lxV(x) V0 IIIIII Fig. 2.2 a. CazulE >V 0;rezonant ¸e S˘ a punem aici ec. (2) ˆ ın forma: k1=√ 2mE ¯h(33) k2=/radicalbig 2m(E−V0) ¯h(34) Solut ¸ia ec. (1) este ca ˆ ın ec. (3) ˆ ın regiunile I(x<0),II(0<x<a ) ¸ si III(x>a) : ψI=A1eik1x+A′ 1e−ik1x ψII=A2eik2x+A′ 2e−ik2x ψIII=A3eik1x+A′ 3e−ik1x Dac˘ a ne limit˘ am la cazul unei particule incidente care vin e de lax=−∞, trebuie s˘ a alegem A′ 3= 0. 31•ψI=ψII, ˆ ınx= 0 : A1+A′ 1=A2+A′ 2 (35) •ψ′ I=ψ′ II, ˆ ınx= 0 : A1ik1−A′ 1ik1=A2ik2−A′ 2ik2 (36) •ψII=ψIII,ˆ ınx=a: A2eik2a+A′ 2e−ik2a=A3eik1a(37) •ψ′ II=ψ′ III, ˆ ınx=a: A2ik2eik2a−A′ 2ik2e−ik2a=A3ik1eik1a(38) Condit ¸iile de continuitate ˆ ın x=adauA2¸ siA′ 2ˆ ın funct ¸ie de A3, ¸ si cele din x= 0 dauA1¸ siA′ 1ˆ ın funt ¸ie de A2¸ siA′ 2(¸ si deci ˆ ın funct ¸ie de A3). Acest procedeu este ar˘ atat ˆ ın continuare. Substituind A′ 2din ec. (37) ˆ ın (38): A2=A3eik1a(k2+k1) 2k2eik2a(39) Substituind A2din ec. (37) ˆ ın (38): A′ 2=A3eik1a(k2−k1) 2k2e−ik2a(40) Substituind A1din ec. (35) ˆ ın (36): A′ 1=A2(k2−k1)−A′ 2(k2+k1) −2k1(41) Substituind A′ 1din ec. (35) ˆ ın (36): A1=A2(k2+k1)−A′ 2(k2−k1) 2k1(42) Acum, substituind ˆ ın (41) ecuat ¸iile (39) ¸ si (40), avem: A′ 1=i(k2 2−k2 1) 2k1k2(sink2a)eik1aA3 (43) 32ˆIn final, substituind ˆ ın (42) ecuat ¸iile (39) ¸ si (40): A1= [cosk2a−ik2 1+k2 2 2k1k2sink2a]eik1aA3 (44) A′ 1/A1¸ siA3/A1[relat ¸ii care se obt ¸in egalˆ ınd ecuat ¸iile (43) ¸ si (44), ¸ si respectiv separˆ ınd ˆ ın ec. (44)] ne permit calculul coeficientului de reflexieRprecum ¸ si a celui de transmisie Tpentru acest caz simplu de barier˘ a , fiind ace¸ stia dat ¸i de: R=|A′ 1/A1|2=(k2 1−k2 2)2sin2k2a 4k2 1k2 2+ (k2 1−k2 2)2sin2k2a, (45) T=|A3/A1|2=4k2 1k2 2 4k2 1k2 2+ (k2 1−k2 2)2sin2k2a, (46) Acum este u¸ sor de verificar c˘ a R+T= 1. b. CazulE <V 0;efectul tunel Acum, fie ecuat ¸iile (2) ¸ si (4): k1=√ 2mE ¯h(47) q2=/radicalbig 2m(V0−E) ¯h(48) Solut ¸ia ec. (1) are forma ec. (3) ˆ ın regiunile I(x<0) ¸ siIII(x>a ), ˆ ın timp ce ˆ ın regiunea II(0<x<a ) are forma ec. (5): ψI=A1eik1x+A′ 1e−ik1x ψII=B2eq2x+B′ 2e−q2x ψIII=A3eik1x+A′ 3e−ik1x Condit ¸iile de ‘lipit’ ˆ ın x= 0 ¸ six=ane permit calculul coeficientului de transmisie al barierei. De fapt, nu este necesar a efectua ˆ ınc˘ a odata calculul: este suficient de a face substitut ¸ia, ˆ ın ecuat ¸i a obt ¸inut˘ a ˆ ın primul caz din aceast˘ a sect ¸iune k2cu−iq2. 33St˘ ari legate ˆ ın groap˘ a rectangular˘ a a. Groap˘ a de adˆ ıncime finit˘ a V(x) Vo x a FgiF.2.3Groapa finita ˆIn aceast˘ a parte ne limit˘ am la studiul cazului 0 <E <V 0(cazulE >V 0 este identic calculului din sect ¸iunea precedent˘ a , “bari er˘ a de potent ¸ial”. Pentru regiunile exterioare I ( x<0) ¸ si III (x>a) folosim ec. (4): q=/radicalbig 2m(V0−E) ¯h(49) Pentru regiunea central˘ a II (0 <x<a ) folosim ec. (2): k=/radicalbig2m(E) ¯h(50) Solut ¸ia ec. (1) are forma ec. (5) ˆ ın regiunile exterioare ¸ si ˆ ın forma din ec. (3) ˆ ın regiunea central˘ a : ψI=B1eqx+B′ 1e−qx ψII=A2eikx+A′ 2e−ikx 34ψIII=B3eqx+B′ 3e−qx ˆIn regiunea (0 <x<a ) ec. (1) are forma: ψ(x)′′+2mE ¯h2ψ(x) =ψ(x)′′+k2ψ(x) = 0 (51) ¸ si ˆ ın regiunile exterioare: ψ(x)′′−2m ¯h2[V0−E]φ(x) =ψ(x)′′−q2ψ(x) = 0 (52) Pentru c˘ aψtrebuie s˘ a fie finit˘ a ˆ ın regiunea I, trebuie s˘ a avem: B′ 1= 0 (53) Condit ¸iile de lipire dau: ψI=ψII, ˆ ınx= 0 : B1=A2+A′ 2 (54) ψ′ I=ψ′ II, ˆ ınx= 0 : B1q=A2ik−A′ 2ik (55) ψII=ψIII, ˆ ınx=a: A2eika+A′ 2e−ika=B3eqa+B′ 3e−qa(56) ψ′ II=ψ′ III, ˆ ınx=a: A2ikeika−A′ 2ike−ika=B3qeqa−B′ 3qe−qa(57) Substituind constantele A2¸ siA′ 2din ec. (54) ˆ ın ec. (55) obt ¸inem, re- spectiv: A′ 2=B1(q−ik) −2ik A2=B1(q+ik) 2ik(58) Substituind constanta A2¸ si constanta A′ 2din ec. (56) ˆ ın ec. (57) obt ¸inem, respectiv: B′ 3e−qa(ik+q) +B3eqa(ik−q) +A′ 2e−ika(−2ik) = 0 2ikA2eika+B′ 3e−qa(−ik+q) +B3Eqa(−ik−q) = 0 (59) 35Egalˆ ındB′ 3din ecuat ¸iile (59) ¸ si t ¸inˆ ınd cont de ecuat ¸iile (58): B3 B1=e−qa 4ikq[eika(q+ik)2−e−ika(q−ik)2] (60) ˆIns˘ aψ(x) trebuie s˘ a fie finit˘ a ¸ siˆ ın regiunea III. Prin urmare, est e necesar caB3= 0, ¸ si deci: [q−ik q+ik]2=eika e−ika=e2ika(61) Deoareceq¸ sikdepind deE, ec. (1) poate fi satisf˘ a cut˘ a pentru anumite valori ale lui E. Condit ¸ia ca ψ(x) s˘ a fie finit˘ a ˆ ın toate regiunile spat ¸iale impune cuantizarea energiei. S ¸i mai precis dou˘ a cazuri su nt posibile: (i) dac˘ a : q−ik q+ik=−eika(62) Egalˆ ınd ˆ ın ambii membri partea real˘ a ¸ si cea imaginar˘ a , respectiv, rezult˘ a : tan(ka 2) =q k(63) Punˆ ınd: k0=/radicaligg 2mV0 ¯h=/radicalig k2+q2 (64) obt ¸inem: 1 cos2(ka 2)= 1 + tan2(ka 2) =k2+q2 k2= (k0 k)2(65) Ec.(63) este astfel echivalent˘ a cu sistemul de ecuat ¸ii: |cos(ka 2)|=k k0 tan(ka 2)>0 (66) Nivelele de energie sunt determinate de c˘ atre intersect ¸i a unei linii drepte de ˆ ınclinare1 k0cu primul set de cosinusoide ˆ ıntrerupte ˆ ın figura 2.4. Astf el obt ¸inem un num˘ ar de nivele de energie, ale c˘ aror funct ¸ii de und˘ a sunt pare. Acest lucru devine mai clar dac˘ a substituim (62) ˆ ın (58) ¸ s i (60). Este u¸ sor de verificat c˘ a B′ 3=B1¸ siA2=A′ 2, astfel c˘ aψ(−x) =ψ(x). 36(ii) dac˘ a : q−ik q+ik=eika(67) Un calcul de acela¸ si tip ne duce la: |sin(ka 2)|=k k0 tan(ka 2)<0 (68) Nivelele de energie sunt ˆ ın acest caz determinate de c˘ atre intersect ¸ia aceleia¸ si linii drepte cu al doilea set de cosinusoide ˆ ınt rerupte ˆ ın figura 2.4. Nivelele astfel obt ¸inute se afl˘ a ˆ ıntre cele g˘ asite ˆ ın (i ). Se poate ar˘ ata u¸ sor c˘ a funct ¸iile de und˘ a corespunz˘ atoare sunt impare. ky 0π π π π /a23/a/a/aPIP 4I k0 Fig. 2.4 b. Groap˘ a de adˆ ıncime infinit˘ a ˆIn acest caz este convenabil s˘ a se pun˘ a V(x) zero pentru 0 <x<a ¸ si infinit ˆ ın tot restul axei. Punˆ ınd: k=/radicaligg 2mE ¯h2(69) 37ψ(x) trebuie s˘ a fie zero ˆ ın afara intervalului [0 ,a], ¸ si continu˘ a ˆ ın x= 0, cˆ ıt ¸ si ˆ ınx=a. Acum, pentru 0≤x≤a: ψ(x) =Aeikx+A′e−ikx(70) Pentru c˘ aψ(0) = 0, se poate deduce c˘ a A′=−A, ceea ce ne conduce la: ψ(x) = 2iAsin(kx) (71) ˆIn plusψ(a) = 0, astfel c˘ a : k=nπ a(72) undeneste un ˆ ıntreg pozitiv arbitrar. Dac˘ a normaliz˘ am funct ¸ ia (71), t ¸inˆ ınd cont de (72), atunci obt ¸inem funct ¸iile de und˘ a stat ¸iona re: ψn(x) =/radicalbigg2 asin(nπx a) (73) cu energiile: En=n2π2¯h2 2ma2(74) Cuantizarea nivelelor de energie este prin urmare extrem de simpl˘ a ˆ ın acest caz: energiile stat ¸ionare sunt proport ¸ionale cu p˘ atrat ele numerelor naturale. 2P. Probleme Problema 2.1: Potent ¸ialul Delta atractiv S˘ a presupunem c˘ a avem un potent ¸ial de forma: V(x) =−V0δ(x);V0>0;x∈ℜ. Funct ¸ia de und˘ a ψ(x) corespunz˘ atoare se presupune continu˘ a . a) S˘ a se obt ¸in˘ a st˘ arile legate ( E <0), dac˘ a exist˘ a , localizate ˆ ın acest tip de potent ¸ial. b) S˘ a se calculeze dispersia unei unde plane care ‘cade’ pe a cest potent ¸ial ¸ si s˘ a se obt ¸in˘ a coeficientul de reflexie R=|ψrefl|2 |ψinc|2|x=0 38undeψrefl,ψincsunt unda reflectat˘ a ¸ si respectiv cea incident˘ a . Sugestie : Pentru a evalua comportamentul lui ψ(x) ˆ ın x=0, se recomand˘ a integrarea ecuat ¸iei Schr¨ odinger ˆ ın intervalul ( −ε,+ε), dup˘ a care se aplic˘ a limitaε→0. Solut ¸ie. a) Ecuat ¸ia Schr¨ odinger este: d2 dx2ψ(x) +2m ¯h2(E+V0δ(x))ψ(x) = 0 (75) Departe de origine avem o ecuat ¸ie diferent ¸ial˘ a de forma d2 dx2ψ(x) =−2mE ¯h2ψ(x). (76) Funct ¸iile de und˘ a sunt prin urmare de forma ψ(x) =Ae−qx+Beqxpentru x>0 sau x<0,(77) cuq=/radicalig −2mE/¯h2∈ℜ.Cum|ψ|2trebuie s˘ a fie integrabil˘ a , nu putem accepta o parte care s˘ a creasc˘ a exponent ¸ial. ˆIn plus funct ¸ia de und˘ a trebuie s˘ a fie continu˘ a ˆ ın origine. Cu aceste condit ¸ii, ψ(x) =Aeqx; (x<0), ψ(x) =Ae−qx; (x>0). (78) ˆIntegrˆ ınd ecuat ¸ia Schr¨ odinger ˆ ıntre −ε¸ si +ε, obt ¸inem −¯h2 2m[ψ′(ε)−ψ′(−ε)]−V0ψ(0) =E/integraldisplay+ε −εψ(x)dx≈2εEψ(0) (79) Introducˆ ınd acum resultatul (78) ¸ si t ¸inˆ ınd cont de limi taε→0, avem −¯h2 2m(−qA−qA)−V0A= 0 (80) sauE=−m(V2 0/2¯h2) [−V2 0 4ˆ ın unit˘ at ¸i¯h2 2m].ˆIn mod clar exist˘ a o singur˘ a en- ergie discret˘ a . Constanta de normalizare se g˘ ase¸ ste c˘ a esteA=/radicalig mV0/¯h2. Funct ¸ia de und˘ a a st˘ arii legate se obt ¸ine ψo=AeV0|x|/2, cuV0ˆ ın unit˘ at ¸i¯h2 2m. 39b) Funct ¸ia de und˘ a pentru o und˘ a plan˘ a este dup˘ a cum se ¸ s tie ψ(x) =Aeikx, k2=2mE ¯h2. (81) Se mi¸ sc˘ a de la stˆ ınga la dreapta ¸ si se reflect˘ a ˆ ın potent ¸ial. Dac˘ a BsauC este amplitudinea undei reflectate sau transmise, respecti v, avem ψ(x) =Aeikx+Be−ikx; (x<0), ψ(x) =Ceikx; ( x>0). (82) Condit ¸iile de continuitate ¸ si relat ¸ia ψ′(ε)−ψ′(−ε) =−fψ(0) cuf= 2mV0/¯h2produc A+B=C B =−f f+ 2ikA, ik(C−A+B) =−fC C =2ik f+ 2ikA. (83) Coeficientul de reflexie cerut este prin urmare R=|ψrefl|2 |ψinc|2|x=0=|B|2 |A|2=m2V2 0 m2V2 0+ ¯h4k2. (84) Dac˘ a potent ¸ialul este extrem de puternic ( V0→∞) se vede c˘ a R→1, adic˘ a unda este reflectat˘ a ˆ ın totalitate. Coeficientul de transmisie , pe de alt˘ a parte, este T=|ψtrans|2 |ψinc|2|x=0=|C|2 |A|2=¯h4k2 m2V2 0+ ¯h4k2. (85) Dac˘ a potent ¸ialul este foarte puternic ( V0→∞) atunciT→0, adic˘ a , unda transmis˘ a cade rapid de cealalt˘ a parte a potent ¸ialului. Evident,R+T= 1 cum era de a¸ steptat. 40Problema 2.2: Particul˘ a ˆ ıntr-o groap˘ a de potent ¸ial fini t˘ a 1D S˘ a se rezolve ecuat ¸ia Schr¨ odinger unidimensional˘ a pen tru o groap˘ a de potent ¸ial finit˘ a descris˘ a prin condit ¸iile V(x) =/braceleftigg −V0dac˘ a|x|≤a 0 dac˘ a|x|>a. S˘ a se considere numai st˘ arile legate ( E <0). EV −V0−a +ax Fig. 2.5 Solut ¸ie. a) Funct ¸ia de und˘ a pentru |x|<a¸ si|x|>a. Ecuat ¸ia Schr¨ odinger corespunz˘ atoare este −¯h2 2mψ′′(x) +V(x)ψ(x) =Eψ(x). (86) Definim q2=−2mE ¯h2, k2=2m(E+V0) ¯h2(87) 41¸ si obt ¸inem: 1) pentru x <−a :ψ′′ 1(x)−q2ψ1= 0, ψ1=A1eqx+B1e−qx; 2) pentru−a≤x≤a :ψ′′ 2(x) +k2ψ2= 0, ψ2=A2cos(kx) +B2sin(kx); 3) pentru x >a :ψ′′ 3(x)−q2ψ3= 0, ψ3=B3eqx+B3e−qx. b) Formularea condit ¸iilor de frontier˘ a . Normalizarea st˘ arilor legate cere ca solut ¸ia s˘ a fie zero l a infinit. Aceasta ˆ ınseamn˘ a c˘ a B1=A3= 0.ˆIn plus,ψ(x) trebuie s˘ a fie continuu diferent ¸iabil˘ a . Toate solut ¸iile particulare sunt fixate ˆ ın a¸ sa fel ˆ ıncˆ ıtψprecum ¸ si prima sa derivat˘ aψ′sunt continueˆ ın acea valoare a lui x corespunzˆ ınd frontie reiˆ ıntre zona interioar˘ a ¸ si cea exterioar˘ a . A doua derivat˘ a ψ′′cont ¸ine saltul (dis- continuitatea) impus de c˘ atre potent ¸ialul particular de tip ‘cutie’ al acestei ecuat ¸ii Schr¨ odinger. Toate acestea ne conduc la ψ1(−a) =ψ2(−a), ψ 2(a) =ψ3(a), ψ′ 1(−a) =ψ′ 2(−a), ψ′ 2(a) =ψ′ 3(a). (88) c) Ecuat ¸iile de valori proprii. Din (88) obt ¸inem patru ecuat ¸ii lineare ¸ si omogene pentru coeficient ¸ii A1, A2,B2¸ siB3: A1e−qa=A2cos(ka)−B2sin(ka), qA1e−qa=A2ksin(ka) +B2kcos(ka), B3e−qa=A2cos(ka) +B2sin(ka), −qB3e−qa=−A2ksin(ka) +B2kcos(ka). (89) Adunˆ ınd ¸ si sc˘ azˆ ınd obt ¸inem un sistem de ecuat ¸ii mai u¸ sor de rezolvat: (A1+B3)e−qa= 2A2cos(ka) q(A1+B3)e−qa= 2A2ksin(ka) (A1−B3)e−qa=−2B2sin(ka) q(A1−B3)e−qa= 2B2kcos(ka). (90) Dac˘ a se presupune c˘ a A1+B3∝ne}ationslash= 0 ¸ siA2∝ne}ationslash= 0, primele dou˘ a ecuat ¸ii dau q=ktan(ka). (91) care pus ˆ ın ultimele dou˘ a d˘ a A1=B3;B2= 0. (92) 42De aici, ca rezultat, obt ¸inem o solut ¸ie simetric˘ a , ψ(x) =ψ(−x), sau de paritate pozitiv˘ a . Un calcul practic identic ne duce pentru A1−B3∝ne}ationslash= 0 ¸ si pentru B2∝ne}ationslash= 0 la q=−kcot(ka)y A 1=−B3;A2= 0. (93) Funct ¸ia de und˘ a astfel obt ¸inut˘ a este antisimetric˘ a , c orespunzˆ ınd unei parit˘ at ¸i negative . d) Solut ¸ie calitativ˘ a a problemei de valori proprii. Ecuat ¸ia care leag˘ a q¸ sik, pe care am obt ¸inut-o deja, d˘ a condit ¸ii pentru autovalorile de energie. Folosind forma scurt˘ a ξ=ka, η =qa, (94) obt ¸inem din definit ¸ia (87) ξ2+η2=2mV0a2 ¯h2=r2. (95) Pe de alt˘ a parte, folosind (91) ¸ si (93) obt ¸inem ecuat ¸iil e η=ξtan(ξ), η =−ξcot(ξ). Astfel autovalorile de energie c˘ autate pot fi obt ¸inute con struind intersect ¸ia acestor dou˘ a curbe cu cercul definit de (95), ˆ ın planul ξ-η(vezi figura 2.6). 1 32 4η ξη ξη = −ξ cot ξξ2 +η2=r2 η = ξ tan ξξ2+ η =2r2 Fig. 2.6 Cel put ¸in o solut ¸ie exist˘ a pentru valori arbitrare ale pa rametrului V0, ˆ ın cazul parit˘ at ¸ii pozitive, pentru c˘ a funct ¸ia tangent˘ a intersecteaz˘ a originea. 43Pentru paritatea negativ˘ a , raza cercului trebuie s˘ a fie ma i mare decˆ ıt o anumit˘ a valoare minim˘ a astfel c˘ a cele dou˘ a curbe s˘ a se p oat˘ a intersecta. Potent ¸ialul trebuie s˘ a aib˘ a o anumit˘ a adˆ ıncime relat ¸ ionat˘ a cu o scal˘ a spat ¸ial˘ a dat˘ aa¸ si o scal˘ a de mas˘ a dat˘ a m, pentru a permite o solut ¸ie de paritate negativ˘ a . Num˘ arul de nivele de energie cre¸ ste cu V0,a¸ si masam. Pentru cazul ˆ ın care mVa2→∞, intersect ¸iile se afl˘ a din tan(ka) =∞ −→ ka=2n−1 2π, −cot(ka) =∞ −→ ka=nπ, (96) unden= 1,2,3, ... sau combinˆ ınd: k(2a) =nπ. (97) Pentru spectrul de energie acest lucru ˆ ınseamn˘ a c˘ a En=¯h2 2m(nπ 2a)2−V0. (98) L˘ argind groapa de potent ¸ial ¸ si/sau masa particulei m, diferent ¸a ˆ ıntre dou˘ a autovalori de energie vecine va descre¸ ste. Nivelul cel mai de jos (n= 1) nu este localizat ˆ ın −V0, ci un pic mai sus. Aceast˘ a diferent ¸˘ a se nume¸ ste energia de punct zero . e) Formele funct ¸iilor de und˘ a se arat˘ a ˆ ın figura 2.7 pentr u solut ¸iile dis- cutate . 13 xψ x24ψ Fig. 2.7: Forme ale functiilor de unda , 44Problema 2.3: Particul˘ a ˆ ın groap˘ a rectangular˘ a 1D infin it˘ a S˘ a se rezolve ecuat ¸ia Schr¨ odinger unidimensional˘ a pen tru o particul˘ a care se afl˘ a intr-o groap˘ a de potent ¸ial infinit˘ a descris˘ a de: V(x) =/braceleftigg 0 pentru x′<x<x′+ 2a ∞pentrux′≥xox≥x′+ 2a. Solut ¸ia ˆ ın form˘ a general˘ a este ψ(x) =Asin(kx) +Bcos(kx) (99) unde k=/radicaligg 2mE ¯h2. (100) Cumψtrebuie s˘ a ˆ ındeplineasc˘ a ψ(x′) =ψ(x′+ 2a) = 0, se obt ¸ine: Asin(kx′) +Bcos(kx′) = 0 (101) Asin[k(x′+ 2a)] +Bcos[k(x′+ 2a)] = 0. (102) Multiplicˆ ınd (101) cu sin[ k(x′+ 2a)] ¸ si (102) cu sin( kx′) ¸ si ˆ ın continuare sc˘ azˆ ınd ultimul rezultat din primul obt ¸inem: B[ cos(kx′)sin[k(x′+ 2a)]−cos[k(x′+ 2a)]sin(kx′) ] = 0.(103) sau prin intermediul unei identit˘ at ¸i trigonometrice: Bsin(2ak) = 0 (104) Multiplicˆ ınd (101) cu cos[ k(x′+2a)] ¸ si sc˘ azˆ ınd (102) multiplicat˘ a cu cos( kx′), se obt ¸ine: A[ sin(kx′)cos[k(x′+ 2a)]−sin[k(x′+ 2a)]cos(kx′) ] = 0,(105) sau folosind aceea¸ si identitate trigonometric˘ a : Asin[k(−2ak)] =Asin[k(2ak)] = 0. (106) Cum nu se t ¸ine cont de solut ¸ia trivial˘ a ψ= 0, atunci pe baza ecuat ¸iilor (104) ¸ si (106) trebuie ca sin(2 ak) = 0, care are loc numai dac˘ a 2 ak=nπ, 45cunun ˆ ıntreg. Conform celor anterioare k=nπ/2a¸ si cumk2= 2mE/¯h2 atunci rezult˘ a c˘ a autovalorile sunt date de c˘ atre expres ia: E=¯h2π2n2 8a2m. (107) Energia este cuantizat˘ a pentru c˘ a numai pentru fiecare kn=nπ/2ale core- spunde o energie En= [n2/2m][π¯h/2a]2. Solut ¸ia ˆ ın forma general˘ a : ψn=Asin(nπx 2a) +Bcos(nπx 2a). (108) se poate normaliza: 1 =/integraldisplayx′+2a x′ψψ∗dx=a(A2+B2) (109) ceea ce ne conduce la: A=±/radicalig 1/a−B2. (110) Substituind aceast˘ a valoare a lui Aˆ ın (101) se obt ¸ine: B=∓1√asin(nπx′ 2a). (111) Substituind acum aceasta valoare a lui Bˆ ın (110) se obt ¸ine: A=±1√acos(nπx′ 2a) (112) Folosind semnele superioare pentru A¸ siB, prin substituirea valorilor aces- tora ˆ ın (108) se obt ¸ine: ψn=1√asin(nπ 2a)(x−x′). (113) Utilizˆ ınd semnele inferioare pentru A ¸ si B se obt ¸ine: ψn=−1√asin(nπ 2a)(x−x′). (114) 463. MOMENT CINETIC S ¸I SPIN Introducere DinMecanica Clasic˘ a se ¸ stie c˘ a , momentul cinetic lpentru particulele macroscopice este dat de l=r×p, (1) under¸ sipsunt respectiv vectorul de pozit ¸ie ¸ si impulsul lineal. Totu¸ si, ˆ ın Mecanica Cuantic˘ a , exist˘ a operatori de tip moment cinetic (OTMC) care nu obligatoriu se exprim˘ a (numai) prin operato rii de coor- donat˘ a ˆxj¸ si impuls ˆpkact ¸ionˆ ınd (numai) asupra funct ¸iilor proprii de coor- donate. Prin urmare, este foarte important s˘ a se stabileas c˘ a , mai ˆ ıntˆ ıi de toate, relat ¸ii de comutare generale pentru componentele O TMC. ˆInMecanica Cuantic˘ a lse reprezint˘ a prin operatorul l=−i¯hr×∇, (2) ale c˘ arui componente sunt operatori care satisfac urm˘ ato arele reguli de co- mutare [lx,ly] =ilz,[ly,lz] =ilx,[lz,lx] =ily, (3) ¸ si ˆ ın plus, fiecare dintre ele comut˘ a cu p˘ atratul momentu lui cinetic, adic˘ a l2=l2 x+l2 y+l2 z,[li,l2] = 0, i = 1,2,3. (4) Aceste relat ¸ii, ˆ ın afar˘ a de a fi corecte pentru momentul ci netic, se satisfac pentru importanta clas˘ a OTMC a operatorilor de spin, care s unt lipsit ¸i de analogi ˆ ın mecanica clasic˘ a . Aceste relat ¸ii de comutare sunt bazice pentru a obt ¸ine spe ctrul operatorilor ment ¸ionat ¸i, precum ¸ si reprezent˘ arile lor diferent ¸ia le. Momentul cinetic orbital Pentru un punct oarecare al unui spat ¸iu fix (SF), se poate int roduce o funct ¸ie ψ(x,y,z), pentru care, s˘ a consider˘ am dou˘ a sisteme carteziene Σ ¸ si Σ′, unde Σ′se obt ¸ine prin rotat ¸ia axei za lui Σ. ˆIn cazul general un SF se refer˘ a la un sistem de coordonate di ferite de Σ ¸ si Σ′. 47Acum, s˘ a compar˘ am valorile lui ψˆ ın dou˘ a puncte ale SF cu acelea¸ si coordonate (x,y,z) ˆ ın Σ ¸ si Σ′, ceea ce este echivalent cu rotat ¸ia vectorial˘ a ψ(x′,y′,z′) =Rψ(x,y,z) (5) undeReste matricea de rotat ¸ie ˆ ın R3  x′ y′ z′ = cosφ−sinφ0 sinφcosφ0 0 0 z  x y z , (6) atunci Rψ(x,y,z) =ψ(xcosφ−ysinφ,xsinφ+ycosφ,z). (7) Pe de alt˘ a parte este important de amintit c˘ a funct ¸iile de und˘ a nu depind de sistemul de coordonate ¸ si c˘ a trasformarea la rotat ¸ii ˆ ın cadrul SF se face cu ajutorul operatorilor unitari ¸ si deci pentru a stabili f orma operatorului unitarU†(φ) care trece ψˆ ınψ′, se consider˘ a o rotat ¸ie infinitezimal˘ a dφ, ment ¸inˆ ınd numai termenii lineari ˆ ın dφatunci cˆ ınd se face dezvoltarea ˆ ın serie Taylor a lui ψ′ˆ ın jurul punctului x ψ(x′,y′,z′)≈ψ(x+ydφ,xdφ +y,z), ≈ψ(x,y,z) +dφ/parenleftbigg y∂ψ ∂x−x∂ψ ∂y/parenrightbigg , ≈(1−idφlz)ψ(x,y,z), (8) unde am folosit notat ¸ia1 lz= ¯h−1(ˆxˆpy−ˆyˆpx), (9) care, dup˘ a cum se va vedea mai tˆ ırziu, corespunde operator ului de proiect ¸ie ˆ ınzal momentului cinetic de acord cu definit ¸ia din (2) ¸ si ad˘ au gˆ ınd fac- torul ¯h−1, astfel c˘ a rotat ¸iile de unghi φfinit se pot reprezenta ca exponent ¸iale de tipul: ψ(x′,y′,z) =eilzφψ(x,y,z), (10) unde ˆU†(φ) =eilzφ. (11) Pentru a reafirma conceptul de rotat ¸ie, s˘ a ˆ ıl consider˘ am ˆ ıntr-un tratament mai general cu ajutorul operatorului vectorialˆ/vectorAcare act ¸ioneaz˘ a asupra lui 1Demostrat ¸ia lui (8) se prezint˘ a ca problema 3.1 48ψ, presupunˆ ınd c˘ a ˆAx,ˆAyˆAzau aceea¸ si form˘ a ˆ ın Σ ¸ si Σ′, adic˘ a , valorile medii ale luiˆ/vectorAcalculate ˆ ın Σ ¸ si Σ′trebuie s˘ a fie egale cˆ ınd se v˘ ad din SF: /integraldisplay ψ∗(/vector r′)(ˆAxˆı′+ˆAyˆ′+ˆAzˆk′)ψ∗(/vector r′)d/vector r =/integraldisplay ψ∗(/vector r)(ˆAxˆı+ˆAyˆ+ˆAzˆk)ψ∗(/vector r)d/vector r, (12) unde ˆı′= ˆıcosφ+ ˆsinφ, ˆ′= ˆısinφ+ ˆcosφ, ˆk′=ˆk. (13) Prin urmare, dac˘ a vom combina (10), (12) ¸ si (13) obt ¸inem eilzφˆAxe−ilzφ=ˆAxcosφ−ˆAysinφ, eilzφˆAye−ilzφ=ˆAxsinφ−ˆAycosφ, eilzφˆAze−ilzφ=ˆAz. (14) Din nou considerˆ ınd rotat ¸ii infinitezimale ¸ si dezvoltˆ ı nd p˘ art ¸ile din stˆ ınga ˆ ın (14) se pot determina relat ¸iile de comutare ale lui ˆAx,ˆAy¸ siˆAzcuˆlz [lz,Ax] =iAy,[lz,Ay] =−iAx,[lz,Az] = 0, (15) ¸ si ˆ ın acela¸ si mod pentru lx¸ sily. Condit ¸iile bazice pentru a obt ¸ine aceste relat ¸ii de comu tare sunt ⋆FP se transform˘ a ca ˆ ın (7) cˆ ınd Σ →Σ′. ⋆Componentele ˆAx,ˆAy,ˆAzau aceea¸ si form˘ a ˆ ın Σ ¸ si Σ′. ⋆Vectorii valorilor medii ale lui ˆAˆ ın Σ ¸ si Σ′coincid (sunt egale) pentru un observator din SF. Deasemenea se poate folosi alt˘ a reprezentare ˆ ın care ψ(x,y,z) nu se schimb˘ a cˆ ınd Σ→Σ′¸ si operatorii vectoriali se transform˘ a ca orice vectori. Pentru a trece la o astfel de representare cˆ ınd φse rote¸ ste ˆ ın jurul lui zse folose¸ ste operatorul ˆU(φ), adic˘ a eilzφψ′(x,y,z) =ψ(x,y,z), (16) ¸ si deci e−ilzφˆ/vectorAeilzφ=ˆ/vectorA. (17) 49Utilizˆ ınd relat ¸iile date ˆ ın (14) obt ¸inem ˆA′ x=ˆAxcosφ+ˆAysinφ=e−ilzφˆAxeilzφ, ˆA′ y=−ˆAxsinφ+ˆAycosφ=e−ilzφˆAyeilzφ, ˆA′ z=e−ilzφˆAzeilzφ. (18) Pentru c˘ a transform˘ arile noii reprezent˘ ari se fac cu aju torul operatorilor unitari, relat ¸iile de comutare nu se schimb˘ a . Observat ¸ii ⋆Operatorul ˆA2este invariant la rotat ¸ii, adic˘ a e−ilzφˆA2eilzφ=ˆA′2=ˆA2. (19) ⋆Rezult˘ a c˘ a [ˆli,ˆA2] = 0. (20) ⋆Dac˘ a operatorul Hamiltonian este de forma ˆH=1 2mˆp2+U(|/vector r|), (21) atunci se ment ¸ine invariant la rotat ¸ii produseˆ ın oricar e ax˘ a care trece prin originea de coordonate [ˆli,ˆH] = 0, (22) undeˆlisunt integrale de mi¸ scare. Definit ¸ie Dac˘ a ˆAisunt componentele unui operator vectorial care act ¸ioneaz ˘ a asupra unei funct ¸ii de und˘ a dependent˘ a numai de coordonate ¸ si d ac˘ a exist˘ a opera- toriˆlicare satisfac urm˘ atoarele relat ¸ii de comutare: [ˆli,ˆAj] =iεijkˆAk,[ˆli,ˆlj] =iεijkˆlk, (23) atunci, ˆlise numesc componentele operatorului moment cinetic (orbital) ¸ si putem s˘ a deducem pe baza lui (20) ¸ si (23) c˘ a [ˆli,ˆl2] = 0. (24) 50Prin urmare, cele trei componente operatoriale asociate cu componen- tele unui moment cinetic clasic arbitrar satisfac relat ¸ii de comutare de tipul (23), (24). Mai mult, se poate ar˘ ata c˘ a aceste relat ¸ii con duc la propriet˘ at ¸i geometrice specifice ale rotat ¸iilor ˆ ıntr-un spat ¸iu eucl idean tridimensional. Aceasta are loc dac˘ a adopt˘ am un punct de vedere mai general ¸ si definim un operator moment unghiular J(nu mai punem simbolul de operator) ca orice set de trei observabile Jx,Jy¸ siJzcare ˆ ındeplinesc relat ¸iile de comutare: [Ji,Jj] =iεijkJk. (25) ˆIn plus, s˘ a introducem operatorul J2=J2 x+J2 y+J2 z, (26) p˘ atratul scalar al momentului unghiular J. Acest operator este hermitic pentru c˘ aJx,Jy¸ siJzsunt hermitici ¸ si se arat˘ a u¸ sor c˘ a J2comut˘ a cu cele trei componente ale lui J [J2,Ji] = 0. (27) Deoarece J2comut˘ a cu fiecare dintre componente rezult˘ a c˘ a exist˘ a un sistem complet de FP, respectiv J2ψγµ=f(γ2)ψγµ, Jiψγµ=g(µ)ψγµ, (28) unde, a¸ sa cum se va ar˘ ata ˆ ın continuare, FP-urile depind d e doi subindici, care se vor determina odat˘ a cu forma funct ¸iilor f(γ) ¸ sig(µ). Operatorii Ji¸ si Jk(i∝ne}ationslash=k) nu comut˘ a (ˆ ıntre ei), adic˘ a , nu au FP ˆ ın comun. Din motiv e atˆ ıt fizice cˆ ıt ¸ si matematice, suntem interesat ¸i s˘ a determin ˘ am funct ¸iile proprii comune ale lui J2¸ siJz, adic˘ a vom lua i=zˆ ın (28). ˆIn loc de a folosi componentele Jx¸ siJyale momentului unghiular J, este mai convenabil s˘ a se introduc˘ a urm˘ atoarele combinat ¸ii lineare J+=Jx+iJy, J −=Jx−iJy. (29) Ace¸ sti operatori nu sunt hermitici, a¸ sa cum sunt operator iia¸ sia†ai oscila- torului armonic (vezi capitolul 5), sunt numai adjunct ¸i. S e pot deduce u¸ sor urm˘ atoarele propriet˘ at ¸i [Jz,J±] =±J±,[J+,J−] = 2Jz, (30) [J2,J+] = [J2,J−] = [J2,Jz] = 0. (31) 51Jz(J±ψγµ) ={J±Jz+ [Jz,J±]}ψγµ= (µ±1)(J±ψγµ). (32) Prin urmare J±ψγµsunt FP ale lui Jzcorespunz˘ atoare autovalorilor µ±1, adic˘ a funct ¸iile respective sunt identice pˆ ın˘ a la fac torii constat ¸i (de determinat) αµ¸ siβµ: J+ψγµ−1=αµψγµ, J−ψγµ=βµψγµ−1. (33) Pe de alt˘ a parte α∗ µ= (J+ψγµ−1,ψγµ) = (ψγµ−1J−ψγµ) =βµ, (34) astfel c˘ a , luˆ ınd o faz˘ a de tipul eia(cuareal) pentru funct ¸ia ψγµse poate faceαµreal ¸ si egal cu βµ, ceea ce ˆ ınseamn˘ a J+ψγ,µ−1=αµψγµ,J−ψγµ=αµψγ,µ−1, (35) ¸ si deci γ= (ψγµ,[J2 x+J2 y+J2 z]ψγµ) =µ2+a+b, a= (ψγµ,J2 xψγµ) = (Jxψγµ,Jxψγµ)≥0, b= (ψγµ,J2 yψγµ) = (Jyψγµ,Jyψγµ)≥0. (36) Constanta de normalizare a oric˘ arei funct ¸ii de und˘ a nu es te negativ˘ a , ceea ce implic˘ a γ≥µ2, (37) pentruγfix, valoarea lui µare limite atˆ ıt superioar˘ a cˆ ıt ¸ si inferioar˘ a (adic˘ a ia valori ˆ ıntr-un interval finit). Fie Λ ¸ siλaceste limite (superioar˘ a¸ si inferiar˘ a de µ) pentru un γdat J+ψγΛ= 0, J −ψγλ= 0. (38) Utilizˆ ınd urm˘ atoarele identit˘ at ¸i operatoriale J−J+=J2−j2 z+Jz=J2−Jz(Jz−1), J+J−=J2−j2 z+Jz=J2−Jz(Jz+ 1), (39) act ¸ionˆ ınd asupra lui ψγΛ¸ siψγλse obt ¸ine γ−Λ2−Λ = 0, γ−λ2+λ= 0, (λ−λ+ 1)(λ+λ) = 0. (40) 52ˆIn plus Λ≥λ→Λ =−λ=J→γ=J(J+ 1). (41) Pentru unγdat (fix) proiect ¸ia momentului µia 2J+1 valori care difer˘ a printr-o unitate ˆ ıntre J¸ si−J. De aceea, diferent ¸a Λ −λ= 2Jtrebuie s˘ a fie un num˘ ar ˆ ıntreg ¸ si deci autovalorile lui Jznumerotate prin msunt ˆ ıntregi m=k, k = 0,±1,±2, ... , (42) sau semiˆ ıntregi m=k+1 2, k = 0,±1,±2, ... . (43) O stare deγ=J(J+1) dat˘ a indic˘ a o degenerare de grad g= 2J+1 fat ¸˘ a de autovalorile m(aceasta pentru c˘ a Ji, Jkcomut˘ a cu J2dar nu comut˘ a ˆ ıntre ele). Prin “stare de moment unghiular J” seˆ ınt ¸elege ˆ ın majoritatea cazurilor, o stare cuγ=J(J+ 1) care are valoarea maxim˘ a a momentului proiectat, respectivJ. Este comun s˘ a se noteze aceste st˘ ari cu ψjmsau de ket Dirac |jm∝an}b∇acket∇i}ht. S˘ a obt ¸inem acum elementele de matrice ale lui Jx, Jyˆ ın reprezentarea ˆ ın careJ2¸ siJzsunt diagonale. ˆIn acest caz, din (35) ¸ si (39) se obt ¸ine c˘ a J−J+ψjm−1=αmJ−ψjm=αmψjm−1, J(J+ 1)−(m−1)2−(m−1) =α2 m, αm=/radicalig (J+m)(J−m+ 1). (44) Combinˆ ınd (44) cu (35) se obt ¸ine J+ψjm−1=/radicalig (J+m)(J−m+ 1)ψjm, (45) rezult˘ a c˘ a elementul de matrice al lui J+este ∝an}b∇acketle{tjm|J+|jm−1∝an}b∇acket∇i}ht=/radicalig (J+m)(J−m+ 1)δnm, (46) ¸ si analog ∝an}b∇acketle{tjn|J−|jm∝an}b∇acket∇i}ht=−/radicalig (J+m)(J−m+ 1)δnm−1. (47) ˆIn sfˆ ır¸ sit, din definit ¸iile (29) pentru J+, J−se obt ¸ine u¸ sor ∝an}b∇acketle{tjm|Jx|jm−1∝an}b∇acket∇i}ht=1 2/radicalig (J+m)(J−m+ 1), ∝an}b∇acketle{tjm|Jy|jm−1∝an}b∇acket∇i}ht=−i 2/radicalig (J+m)(J−m+ 1). (48) 53Concluzii part ¸iale αPropriet˘ at ¸i ale autovalorilor lui J¸ siJz Dac˘ aj(j+ 1)¯h2¸ sim¯hsunt autovalori ale lui J¸ siJzasociate cu au- tovectorii|kjm∝an}b∇acket∇i}ht, atuncij¸ simsatisfac inegalitatea −j≤m≤j. βPropriet˘ at ¸i ale vectorului J−|kjm∝an}b∇acket∇i}ht Fie|kjm∝an}b∇acket∇i}htun eigenvector al lui J2¸ siJzcu autovalorile j(j+ 1)¯h2¸ si m¯h –(i) Dac˘ am=−jatunciJ−|kj−j∝an}b∇acket∇i}ht= 0. –(ii) Dac˘ am >−jatunciJ−|kjm∝an}b∇acket∇i}hteste un vector propriu nenul al luiJ2¸ siJzcu autovalorile j(j+ 1)¯h2¸ si (m−1)¯h. γPropriet˘ at ¸i ale vectorului J+|kjm∝an}b∇acket∇i}ht Fie|kjm∝an}b∇acket∇i}htun vector (ket) propriu al lui J2¸ siJzcu autovalorile j(j+1)¯h ¸ sim¯h ⋆Dac˘ am=jatunciJ+|kjm∝an}b∇acket∇i}ht= 0. ⋆Dac˘ am<j atunciJ+|kjm∝an}b∇acket∇i}hteste un vector nenul al lui J2¸ siJz cu autovalorile j(j+ 1)¯h2¸ si (m+ 1)¯h δConsecint ¸e ale propriet˘ at ¸ilor anterioare Jz|kjm∝an}b∇acket∇i}ht=m¯h|kjm∝an}b∇acket∇i}ht, J+|kjm∝an}b∇acket∇i}ht=m¯h/radicalig j(j+ 1)−m(m+ 1)|kjm+ 1∝an}b∇acket∇i}ht, J−|kjm∝an}b∇acket∇i}ht=m¯h/radicalig j(j+ 1)−m(m−1)|kjm+ 1∝an}b∇acket∇i}ht. Aplicat ¸ii ale momentului cinetic orbital Am considerat pˆ ın˘ a acum propriet˘ at ¸ile momentului cine tic derivate numai pe baza relat ¸iilor de comutare. S˘ a ne ˆ ıntoarcem la moment ul cinetic Lal unei particule f˘ ar˘ a rotat ¸ie intrinsec˘ a ¸ si s˘ a vedem cu m se aplic˘ a teoria din sect ¸iunea anterioar˘ a la cazul particular important [ˆli,ˆpj] =iεijkˆpk. (49) 54Mai ˆ ıntˆ ıi, ˆlz¸ si ˆpjau un sistem comun de funct ¸ii proprii. Pe de alt˘ a parte, Hamiltonianul unei particule libere ˆH=/parenleftiggˆ/vector p√ 2m/parenrightigg2 , fiind p˘ atratul unui operator vectorial are acela¸ si sistem complet de FP cu ˆL2¸ siˆlz.ˆIn plus, aceasta implic˘ a c˘ a particula liber˘ a se poate g˘ a siˆ ıntr-o stare cuE,l,mbine determinate. Valori proprii ¸ si funct ¸ii proprii ale lui L2¸ si Lz Este mai convenabil s˘ a se lucreze cu coordonate sferice (sa u polare), pentru c˘ a , a¸ sa cum vom vedea, diferit ¸i operatori de moment cinet ic act ¸ioneaz˘ a numai asupra variabilelor unghiulare θ, φ¸ si nu ¸ si asupra variabilei r.ˆIn loc de a caracteriza vectorul rprin componentele carteziene x, y, z s˘ a determin˘ am punctul corespunz˘ ator Mde raz˘ a vectoare rprin coordonatele sferice tridimensionale x=rcosφsinθ, y =rsinφsinθ, z =rcosθ, (50) cu r≥0,0≤θ≤π, 0≤φ≤2π. Fie Φ(r,θ,φ) ¸ si Φ′(r,θ,φ) funct ¸iile de und˘ a ale unei particule ˆ ın Σ ¸ si Σ′ ˆ ın care rotat ¸ia infinitezimal˘ a este dat˘ a prin δαˆ ın jurul lui z Φ′(r,θ,φ) = Φ(r,θ,φ +δα), = Φ(r,θ,φ) +δα∂Φ ∂φ, (51) sau Φ′(r,θ,φ) = (1 +iˆlzδα)Φ(r,θ,φ). (52) Rezult˘ a c˘ a∂Φ ∂φ=iˆlzΦ,ˆlz=−i∂ ∂φ. (53) Pentru o rotat ¸ie infinitezimal˘ a ˆ ın x Φ′(r,θ,φ) = Φ + δα/parenleftbigg∂Φ ∂θ∂θ ∂α+∂Φ ∂θ∂φ ∂α/parenrightbigg , = (1 +iˆlxδα)Φ(r,θ,φ), (54) 55ˆ ıns˘ a ˆ ıntr-o astfel de rotat ¸ie z′=z+yδα;z′=z+yδα;x′=x (55) ¸ si din (50) se obt ¸ine rcos(θ+dθ) =rcosθ+rsinθsinφδα, rsinφsin(θ+dθ) =rsinθsinφ+rsinθsinφ−rcosθδα, (56) adic˘ a sinθdθ= sinθsinφδα→dθ dα=−sinφ, (57) ¸ si cosθsinφdθ+ sinθcosφdφ =−cosθδα, cosφsinθdφ dα=−cosθ−cosθsinφdθ dα.(58) Substituind (57) ˆ ın (56) duce la dφ dα=−cotθcosφ . (59) Cu (56) ¸ si (58) substituite ˆ ın (51) ¸ si comparˆ ınd p˘ art ¸i le din dreapta din (51) se obt ¸ine ˆlx=i/parenleftbigg sinφ∂ ∂θ+ cotθcosφ∂ ∂φ/parenrightbigg . (60) ˆIn cazul rotat ¸iei ˆ ın y, rezultatul este similar ˆly=i/parenleftbigg −cosφ∂ ∂θ+ cotθsinφ∂ ∂φ/parenrightbigg . (61) Folosind ˆlx,ˆlyse pot obt ¸ine deasemenea ˆl±,ˆl2 ˆl±= exp/bracketleftbigg ±iφ/parenleftbigg ±∂ ∂θ+icotθ∂ ∂φ/parenrightbigg/bracketrightbigg , ˆl2=ˆl−ˆl++ˆl2+ˆlz, =−/bracketleftigg 1 sin2θ∂2 ∂φ2+1 sinθ∂ ∂θ/parenleftbigg sinθ∂ ∂θ/parenrightbigg/bracketrightigg . (62) Din (62) se vede c˘ a ˆl2este identic pˆ ın˘ a la o constant˘ a cu partea unghiular˘ a a operatorului Laplace la o raz˘ a fix˘ a ∇2f=1 r2∂ ∂r/parenleftbigg r2∂f ∂r/parenrightbigg +1 r2/bracketleftigg 1 sinθ∂ ∂θ/parenleftbigg sinθ∂f ∂θ/parenrightbigg +1 sin2θ∂2 ∂φ2/bracketrightigg .(63) 56Funct ¸ii proprii ale lui lz ˆlzΦm=mΦ =−i∂Φm ∂φ, Φm=1√ 2πeimφ. (64) Condit ¸ii de hermiticitate ale lui ˆlz /integraldisplay2π 0f∗ˆlzgdφ=/parenleftbigg/integraldisplay2π 0g∗ˆlzfdφ/parenrightbigg∗ +f∗g(2π)−f∗g(0). (65) Rezult˘ a c˘ a ˆlzeste hermitic ˆ ın clasa de funct ¸ii pentru care f∗g(2π) =f∗g(0). (66) Funct ¸iile proprii Φ male lui ˆlzapart ¸in clasei integrabile L2(0,2π) ¸ si ˆ ındeplinesc (66), cum se ˆ ıntˆ ımpl˘ a pentru orice funct ¸i e care se poate dez- volta ˆ ın Φ m(φ) F(φ) =k/summationdisplay akeikφ, k = 0,±1,±2, ... , G(φ) =k/summationdisplay bkeikφ, k =±1/2,±3/2,±5/2... , (67) cukφnumai ˆ ıntregi sau numai semiˆ ıntregi, dar nu pentru combin at ¸ii de F(φ) ¸ siG(φ). Alegerea corect˘ a a lui mse bazeaz˘ a ˆ ın FP comune ale lui ˆlz¸ siˆl2. Armonice sferice ˆIn reprezentarea{/vectorr}, funct ¸iile proprii asociate cu autovalorile l(l+ 1)¯h2, ale luiL2¸ sim¯hale luilzsunt solut ¸ii ale ecuat ¸iilor diferent ¸iale part ¸iale −/parenleftigg ∂2 ∂θ2+1 tanθ∂ ∂θ+1 sin2θ∂2 ∂φ2/parenrightigg ψ(r,θ,φ) =l(l+ 1)¯h2ψ(r,θ,φ), −i∂ ∂φψ(r,θ,φ) =m¯hψ(r,θ,φ). (68) Considerˆ ınd c˘ a rezultatele generale obt ¸inute sunt apli cabile la cazul mo- mentului cinetic, ¸ stim c˘ a leste un ˆ ıntreg sau semiˆ ıntreg ¸ si c˘ a pentru lfixat ¸i mpoate lua numai valorile −l,−l+ 1, ... ,l−1,l. 57ˆIn (68),rnu apare ˆ ın operatorul diferent ¸ial, a¸ sa c˘ a se poate cons idera ca un parametru ¸ si se poate t ¸ine cont numai de dependent ¸a ˆ ınθ, φa luiψ. Se folose¸ se notat ¸ia Ylm(θ,φ) pentru o astfel de funct ¸ie proprie comun˘ a a lui L2¸ silz, corespunz˘ atoare autovalorilor l(l+1)¯h2,m¯h¸ si se nume¸ ste armonic˘ a sferic˘ a . L2Ylm(θ,φ) =l(l+ 1)¯h2Ylm(θ,φ), lzYlm(θ,φ) =m¯hYlm(θ,φ). (69) Pentru a fi cˆ ıt mai riguro¸ si, trebuie s˘ a introducem un indi ce adit ¸ional cu scopul de a distinge ˆ ıntre diferite solut ¸ii ale lui (69), c are corespund acelea¸ si perechi de valori l, m.ˆIntr-adev˘ ar, dup˘ a cum se va vedea mai departe, aceste ecuat ¸ii au o solut ¸ie unic˘ a (pˆ ın˘ a la un factor con stant) pentru fiecare pereche de valori permise ale lui l, m; aceasta pentru c˘ a subindicii l, m sunt suficient ¸i ˆ ın contextul respectiv. Solut ¸iile Ylm(θ, φ) au fost g˘ asite prin metoda separ˘ arii variabilelor ˆ ın coordonate sferice (ve zi capitolul Atomul de hidrogen ) ψlm(r,θ,φ) =f(r)ψlm(θ,φ), (70) undef(r) este o funct ¸ie de r, care apare ca o constant˘ a de integrare din punctul de vedere al ecuat ¸iilor diferent ¸iale part ¸iale d in (68). Faptul c˘ a f(r) este arbitrar˘ a arat˘ a c˘ a L2¸ silznu formeaz˘ a un set complet de observabile2 ˆ ın spat ¸iulεr3al funct ¸iilor de /vector r(sau der,θ,φ). Cu obiectivul de a normaliza ψlm(r,θ,φ), este convenabil s˘ a se normal- izezeYlm(θ,φ) ¸ sif(r) ˆ ın mod separat: /integraldisplay2π 0dφ/integraldisplayπ 0sinθ|ψlm(θ,φ)|2dθ= 1, (71) /integraldisplay∞ 0r2|f(r)|2dr= 1. (72) Valorile perechii l, m (α):l, mtrebuie s˘ a fie ˆ ıntregi Folosindlz=¯h i∂ ∂φ, putem scrie (69) dup˘ a cum urmeaz˘ a ¯h i∂ ∂φYlm(θ,φ) =m¯hYlm(θ,φ), (73) 2Prin definit ¸ie, operatorul hermitic A este o observabil˘ a d ac˘ a acest sistem ortogonal de vectori reprezint˘ a o baz˘ a ˆ ın spat ¸iul configurat ¸iilor d e st˘ ari. 3Fiecare stare cuantic˘ a a particulei este caracterizat˘ a p rintr-o stare vectorial˘ a apart ¸inˆ ınd unui spat ¸iu vectorial abstract εr. 58care arat˘ a c˘ a Ylm(θ,φ) =Flm(θ,φ)eimφ. (74) Dac˘ a permitem ca 0 ≤φ<2π, atunci putem acoperi tot spat ¸iul pentru c˘ a funct ¸ia trebuie s˘ a fie continu˘ a ˆ ın orice zon˘ a , astfe l c˘ a Ylm(θ,φ= 0) =Ylm(θ,φ= 2π), (75) ceea ce implic˘ a eimπ= 1. (76) A¸ sa cum s-a v˘ azut, meste un ˆ ıntreg sau semiˆ ıntreg; ˆ ın cazul aplicat ¸iei la momentul cinetic orbital, mtrebuie s˘ a fie un ˆ ıntreg. ( e2imπar fi−1 dac˘ a mar fi semiˆ ıntreg). (β): Pentru o valoare dat˘ a a lui l, toate armonicele sferice Ylmcorespunz˘ atoare se pot obt ¸ine prin metode algebrice folosind l+Yll(θ,φ) = 0, (77) care combinat˘ a cu ec. (62) pentru l+duce la /parenleftbiggd dθ−lcotθ/parenrightbigg Fll(θ) = 0. (78) Aceast˘ a ecuat ¸ie poate fi integrat˘ a imediat dac˘ a not˘ am r elat ¸ia cotθdθ=d(sinθ) sinθ. (79) Solut ¸ia sa general˘ a este Fll=cl(sinθ)l, (80) undecleste o constant˘ a de normalizare. Rezult˘ a c˘ a pentru orice valoare pozitiv˘ a sau zero a lui l, exist˘ a o funct ¸ie Yll(θ,φ) care este (cu un factor constant asociat) Yll(θ,φ) =cl(sinθ)leilφ. (81) Folosind act ¸iunea lui l−ˆ ın mod repetat se poate construi tot setul de funct ¸iiYll−1(θ,φ), ... ,Yl0(θ,φ),... ,Yl−l(θ,φ).ˆIn continuare, vedem modul ˆ ın care se corespund aceste funct ¸ii cu perechea de autoval oril(l+ 1)¯h,m¯h (undeleste un ˆ ıntreg pozitiv arbitrar sau zero ¸ si meste alt ˆ ıntreg astfel c˘ a l≤m≤l). Pe baza lui (78) ajungem la concluzia c˘ a orice funct ¸ie pr oprie Ylm(θ,φ), poate fi obt ¸inut˘ a ˆ ın mod neambiguu din Yll. 59Propriet˘ at ¸i ale armonicelor sferice αRelat ¸ii de recurent ¸˘ a Urmˆ ınd rezultatele generale avem l±Ylm(θ,φ) = ¯h/radicalig l(l+ 1)−m(m±1)Ylm±1(θ,φ). (82) Folosind expresia (62) pentru l±¸ si faptul c˘ a Ylm(θ,φ) este produsul ˆ ıntre o funct ¸ie dependent˘ a numai de θ¸ sie±iφ, obt ¸inem e±iφ/parenleftbigg∂ ∂θ−mcotθ/parenrightbigg Ylm(θ,φ) =/radicalig l(l+ 1)−m(m±1)Ylm±1(θ,φ) (83) βOrtonormare ¸ si relat ¸ii de completitudine Ecuat ¸ia (68) determin˘ a numai armonicele sferice pˆ ın˘ a l a un factor constant. Acum vom alege acest factor astfel ca s˘ a avem ortonormaliza rea funct ¸iilor Ylm(θ,φ) (ca funct ¸ii de variabilele unghiulare θ, φ) /integraldisplay2π 0dφ/integraldisplayπ 0sinθdθY∗ lm(θ,φ)Ylm(θ,φ) =δl′lδm′m. (84) ˆIn plus, orice funct ¸ie continu˘ a de θ, φpot fi expresate cu ajutorul armon- icelor sferice, adic˘ a f(θ,φ) =∞/summationdisplay l=0l/summationdisplay m=−lclmYlm(θ,φ), (85) unde clm=/integraldisplay2π 0dφ/integraldisplayπ 0sinθdθY∗ lm(θ,φ)f(θ,φ). (86) Rezultatele (85), (86) sunt consecint ¸a definirii armonice le sferice ca baz˘ a ortonormal˘ a complet˘ a ˆ ın spat ¸iul εΩa funct ¸iilor de θ, φ. Relat ¸ia de com- pletitudine este ∞/summationdisplay l=0l/summationdisplay m=lYlm(θ,φ)Y∗ lm(θ′,φ) =δ(cosθ−cosθ′)δ(φ,φ), =1 sinθδ(θ−θ′)δ(φ,φ). (87) ‘Funct ¸ia’δ(cosθ−cosθ′) apare pentru c˘ a integrala peste variabila θse efectueaz˘ a folosind elementul diferent ¸ial sin θdθ=−d(cosθ). 60Operatorul de paritate Pˆ ın cazul armonicelor sferice Comportamentul lui Pˆ ın trei dimensiuni este destul de asem˘ an˘ ator celui ˆ ıntr-o singur˘ a dimensiune, respectiv, cˆ ınd se aplic˘ a u nei funct ¸iiψ(x,y,z) de coordonate carteziene ˆ ıi modific˘ a numai semnul fiec˘ areia dintre coordonate Pψ(x,y,z) =ψ(−x,−y,−z). (88) Pare propriet˘ at ¸ile unui operator hermitic ¸ si ˆ ın plus est e un operator unitar precum ¸ si proiector deoarece P2este un operator identitate ∝an}b∇acketle{tr|P|r′∝an}b∇acket∇i}ht=∝an}b∇acketle{tr|−r′∝an}b∇acket∇i}ht=δ(r+−r′), P∈|r∝an}b∇acket∇i}ht=P(P|r∝an}b∇acket∇i}ht=P|−r∝an}b∇acket∇i}ht=|r∝an}b∇acket∇i}ht, (89) ¸ si deci P2=ˆ1, (90) pentru care valorile proprii sunt P=±1. FP-urile se numesc pare dac˘ a P=∞¸ si impare dac˘ aP=−∞.ˆIn mecanica cuantic˘ a nerelativist˘ a , operatorul ˆHpentru un sistem conservativ este invariant la transform˘ a ri unitare discrete PˆHP=P−1ˆHP=ˆH. (91) Astfel ˆHcomut˘ a cuP¸ si deci paritatea st˘ arii este constant˘ a de mi¸ scare. DeasemeneaPcomut˘ a cu operatorii ˆl [P,ˆli] = 0,[P,ˆl±] = 0. (92) Dac˘ a ˆHeste par ¸ si consider˘ am o FP |Φn∝an}b∇acket∇i}htcare are paritate bine definit˘ a (P|Φn∝an}b∇acket∇i}ht), necolinear˘ a cu |ψn∝an}b∇acket∇i}ht, din faptul c˘ aPcomut˘ a cu ˆHdeducem c˘ a (P|Φn∝an}b∇acket∇i}ht) este un vector propriu al lui ˆHcu aceea¸ si valoare proprie ca ¸ si |Φn∝an}b∇acket∇i}ht). Dac˘ aψeste FP a tripletei P,ˆl¸ siˆlz, din (92) rezult˘ a c˘ a parit˘ at ¸ile st˘ arilor care difer˘ a numai ˆ ın ˆlzcoincid. ˆIn felul acesta se identific˘ a paritatea unei particule de moment unghiular ˆl. ˆIn coordonate sferice, pentru acest operator vom considera urm˘ atoarea substitut ¸ie r→r, θ→π−θ φ→π+φ. (93) Prin urmare, dac˘ a folosim o baz˘ a standard ˆ ın spat ¸iul fun ct ¸iilor de und˘ a a unei particule f˘ ar˘ a ‘rotat ¸ie proprie’, partea radial˘ a a funct ¸iilor ψklm(/vector r) din baz˘ a nu este modificat˘ a de c˘ atre operatorul paritate. Doa r armonicele sferice se schimb˘ a . Transform˘ arile (93) se traduc trigonometric ˆ ın: sin(π−θ)→sinθ, cos(π−θ)→−cosθ eim(π+φ→(−1)meimφ(94) 61¸ si ˆ ın aceste condit ¸ii, funct ¸ia Yll(θ,φ) se transform˘ a ˆ ın Yll(φ−θ,π+φ) = (−1)lYll(θ,φ). (95) Din (95) rezult˘ a c˘ a paritatea lui Ylleste (−1)l. Pe de alt˘ a parte, l−(ca ¸ si l+este invariant la transform˘ arile: ∂ ∂(π−θ)→−∂ ∂θ,∂ ∂(π+φ)→∂ ∂φei(π+φ)→−eiφcot(π−θ)→−cotθ. (96) Cu alte cuvinte l±sunt pari. Prin urmare, deducem c˘ a paritatea oric˘ arei armonice sferice este ( −1)l, adic˘ a este invariant˘ a la schimb˘ ari azimutale: Ylm(φ−θ,π+φ) = (−1)lYlm(θ,φ). (97) A¸ sadar, armonicele sferice sunt funct ¸ii de paritate bine definit˘ a , indepen- dent˘ a dem, par˘ a dac˘ a leste par ¸ si impar˘ a dac˘ a leste impar. Operatorul de spin Unele particule, posed˘ a nu numai moment cineticˆ ın raport cu axe exterioare ci ¸ si un moment propriu , care se cunoa¸ ste ca spin¸ si pe care ˆ ıl vom nota cu ˆS. Acest operator nu este relat ¸ionat cu rotat ¸ii normale fat ¸˘ a de axe ‘reale’ ˆ ın spat ¸iu, dar respect˘ a relat ¸ii de comutare de acela¸ si tip ca ale momentului cinetic orbital, respectiv [ˆSi,ˆSj] =iεijkˆSk, (98) odat˘ a cu urm˘ atoarele propriet˘ at ¸i: (1). Pentru spin, se satisfac toate formulele de la (23) pˆ ın ˘ a la (48) ale momentului cinetic. (2). Spectrul proiect ¸iilor de spin, este o secvent ¸˘ a de nu mere ˆ ıntregi sau semiˆ ıntregi care difer˘ a printr-o unitate. (3). Valorile proprii ale lui ˆS2sunt ˆS2ψs=S(S+ 1)ψs. (99) (4). Pentru un Sdat, componentele Szpot s˘ a ia numai 2 S+ 1 valori, de la−Sla +S. 62(5). FP-urile particulelor cu spin, pe lˆ ıng˘ a dependent ¸a de/vector r¸ si/sau/vector p, de- pind de o variabil˘ a discret˘ a (proprie spinului) σ, care denot˘ a proiect ¸ia spinului pe axa z. (6). FP-urile ψ(/vector r,σ) ale unei particule cu spin se pot dezvolta ˆ ın FP-uri cu proiect ¸ii date ale spinului Sz, adic˘ a ψ(/vector r,σ) =S/summationdisplay σ=−Sψσ(/vector r)χ(σ), (100) undeψσ(/vector r) este partea orbital˘ a ¸ si χ(σ) este partea spinorial˘ a . (7). Funct ¸iile de spin (spinorii) χ(σi) sunt ortogonale pentru orice pereche σi∝ne}ationslash=σk. Funct ¸iile ψσ(/vector r)χ(σ) din suma (100) sunt componentele unei FP a unei particule cu spin. (8). Funct ¸ia ψσ(/vector r) se nume¸ ste parte orbital˘ a a spinorului, sau mai scurt orbital. (9) Normalizarea spinorilor se face ˆ ın felul urm˘ ator S/summationdisplay σ=−S||ψσ(/vector r)||= 1. (101) Relat ¸iile de comutare permit s˘ a se stabileasc˘ a forma con cret˘ a a operato- rilor (matricelor) de spin care act ¸ioneaz˘ a ˆ ın spat ¸iul F P-urilor operatorului proiect ¸iei spinului. Multe particule ‘elementare’, cum ar fi electronul, neutron ul, protonul, etc. au spin 1 /2 ¸ si de aceea proiect ¸ia spinului lor ia numai dou˘ a valori, respectivSz=±1/2 (ˆ ın unit˘ at ¸i ¯ h). Fac parte din clasa fermionilor, datorit˘ a statisticii pe care o prezint˘ a cˆ ınd formeaz˘ a sisteme de m ulte corpuri. Pe de alt˘ a parte, matricele Sx, Sy, Szˆ ın spat ¸iul FP-urilor lui ˆS2,ˆSz sunt Sx=1 2/parenleftigg 0 1 1 0/parenrightigg , S y=1 2/parenleftigg 0−i i0/parenrightigg , Sz=1 2/parenleftigg 1 0 0−1/parenrightigg , S2=3 4/parenleftigg 1 0 0 1/parenrightigg . (102) 63Definit ¸ia matricelor Pauli Matricele σi= 2Si (103) se numesc matricele (lui) Pauli. Sunt matrici hermitice ¸ si au aceea¸ si ecuat ¸ie caracteristic˘ a λ2−1 = 0, (104) ¸ si prin urmare, autovalorile lui σx, σy¸ siσzsunt λ=±1. (105) Algebra acestor matrici este urm˘ atoarea: σ2 i=ˆI, σkσj=−σjσk=iσz, σjσk=i/summationdisplay lεjklσl.+δjkI .(106) ˆIn cazul ˆ ın care sistemul cu spin are simetrie sferic˘ a ψ1(r,+1 2), ψ 1(r,−1 2), (107) sunt solut ¸ii diferite datorit˘ a proiect ¸iilor Szdiferite. Valoarea probabilit˘ at ¸ii uneia sau alteia dintre proiect ¸ii este determinat˘ a de mod ulul p˘ atrat||ψ1||2 sau||ψ2||2ˆ ın a¸ sa fel c˘ a ||ψ1||2+||ψ2||2= 1. (108) Cum FP ale lui Szare dou˘ a componente, atunci χ1=/parenleftigg 1 0/parenrightigg , χ 2=/parenleftigg 0 1/parenrightigg , (109) astfel c˘ a FP a unei particule de spin 1 /2 se poate scrie ca o matrice de o column˘ a ψ=ψ1χ1+ψ2χ2=/parenleftigg ψ1 ψ2/parenrightigg . (110) ˆIn continuare, orbitalii vor fi substituit ¸i prin numere dat orit˘ a faptului c˘ a ne intereseaz˘ a numai partea de spin. 64Transform˘ arile la rotat ¸ii ale spinorilor Fieψfunct ¸ia de und˘ a a unui sistem cu spin ˆ ın Σ. S˘ a determin˘ am proba- bilitatea proiect ¸iei spinuluiˆ ıntr-o direct ¸ie arbitra r˘ a ˆ ın spat ¸iul tridimensional (3D) care se poate alege ca ax˘ a z′a lui Σ′. Cum am v˘ azut deja pentru cazul momentului cinetic, exist˘ a dou˘ a metode de abordare ¸ si so lut ¸ionare a acestei probleme: (α)ψnu se schimb˘ a cˆ ınd Σ →Σ′¸ si operatorul ˆΛ se transform˘ a ca un vector. Trebuie s˘ a g˘ asim FP-urile proiect ¸iilor S′ z¸ si s˘ a dezvolt˘ am ψˆ ın aceste FP-uri. P˘ atratele modulelor coeficient ¸ilor dau re zultatul ˆS′ x=ˆSxcosφ+ˆSysinφ=e−ilφˆSxeilφ, ˆS′ y=−ˆSxsinφ+ˆSycosφ=e−ilφˆSyeilφ, ˆS′ z=−ˆSz=eilφˆSz, (111) cu rotat ¸ii infinitezimale ¸ si din relat ¸iile de comutare se poate g˘ asi ˆL=ˆSz, (112) unde ˆLeste generatorul infinitezimal. (β) A doua reprezentare este: ˆSnu se schimb˘ a cˆ ınd Σ →Σ′¸ si componentele lui ψse schimb˘ a . Trans- formarea la aceast˘ a reprezentare se face cu o transformare unitar˘ a de forma ˆV†ˆS′ˆV=ˆΛ,/parenleftigg ψ′ 1 ψ′ 2/parenrightigg =ˆV†/parenleftigg ψ1 ψ2/parenrightigg . (113) Pe baza lui (111) ¸ si (113) rezult˘ a c˘ a ˆV†e−iˆSzφˆSeiˆSzφˆV=ˆS, ˆV†=eiˆSzφ, (114) ¸ si din (114) se obt ¸ine /parenleftigg ψ′ 1 ψ′ 2/parenrightigg =eiˆSzφ/parenleftigg ψ1 ψ2/parenrightigg . (115) 65Folosind forma concret˘ a a lui ˆSz¸ si propriet˘ at ¸ile matricelor Pauli se obt ¸ine forma concret˘ a ˆV† z, astfel c˘ a ˆV† z(φ) =/parenleftigg ei 2φ0 0e−i 2φ/parenrightigg . (116) Un rezultat al lui Euler Se poate ajunge la orice sistem de referint ¸˘ a Σ′de orientare arbitrar˘ a fat ¸˘ a de Σ prin numai trei rotat ¸ii, prima de unghi φˆ ın jurul axei z, urm˘ atoarea rotat ¸ie de unghi θˆ ın jurul noii axe de coordonate x′¸ si ultima de unghi ψa ˆ ın jurul lui z′. Acest rezultat important apart ¸ine lui Euler. Parametrii ( ϕ,θ,ψa) se numesc unghiurile (lui) Euler ˆV†(ϕ,θ,ψa) =ˆV† z′(ψa)ˆV† x′(θ)ˆV† z(ϕ). (117) Matricele ˆV† zsunt de forma (116), ˆ ın timp ce ˆV† xeste de forma ˆV† x(ϕ) =/parenleftigg cosθ 2isinθ 2 isinθ 2cosθ 2/parenrightigg , (118) astfel c˘ a ˆV†(ϕ,θ,ψa) =/parenleftigg eiϕ+ψa 2cosθ 2ieiψa−ϕ 2sinθ 2 ieiϕ−ψa 2sinθ 2e−iϕ+ψa 2cosθ 2/parenrightigg . (119) Rezult˘ a deci c˘ a prin rotat ¸ia lui Σ, componentele funct ¸i ei spinoriale se transform˘ a dup˘ a cum urmeaz˘ a ψ′ 1=ψ1eiϕ+ψa 2cosθ 2+iψ2eiψa−ϕ 2sinθ 2, ψ′ 2=iψ1eiϕ−ψa 2sinθ 2+ψ2e−iϕ+ψa 2cosθ 2. (120) Din (120) se poate vedea c˘ a unei rotat ¸ii ˆ ın E3ˆ ıi corespunde o transformare linear˘ aˆ ınE2, spat ¸iul euclidean bidimensional, relat ¸ionat˘ a cu cele dou˘ a com- ponente ale funct ¸iei spinoriale. Rotat ¸ia ˆ ın E3nu implic˘ a o rotat ¸ie ˆ ın E2, ceea ce ˆ ınseamn˘ a ∝an}b∇acketle{tΦ′|ψ′∝an}b∇acket∇i}ht=∝an}b∇acketle{tΦ|ψ∝an}b∇acket∇i}ht= Φ∗ 1ψ1+ Φ∗ 2ψ2. (121) 66Din (119) se obt ¸ine c˘ a (121) nu se satisface, totu¸ si exist ˘ a o invariant ¸˘ a ˆ ın transform˘ arile (119) ˆ ın spat ¸iul E2al funct ¸iilor spinoriale, {Φ|ψ}=ψ1Φ2−ψ2Φ1. (122) Transform˘ arile lineare care ment ¸in invariante astfel de forme bilineare se numesc binare. O m˘ arime fizic˘ a cu dou˘ a componente pentru care o rotat ¸ie a sistemului de coordonate este o transformare binar˘ a se nume¸ ste spin de ordinul ˆ ıntˆ ıi sau pe scurt spin. Spinorii unui sistem de doi fermioni Funct ¸iile proprii ale lui iˆs2iˆsz, cui= 1,2 au urm˘ atoarea form˘ a i|+∝an}b∇acket∇i}ht=/parenleftigg 1 0/parenrightigg i, i|−∝an}b∇acket∇i}ht=/parenleftigg 0 1/parenrightigg i. (123) Un operator foarte folosit ˆ ıntr-un sistem de doi fermioni e ste spinul total ˆS=1ˆS+2ˆS (124) Spinorii lui ˆ s2ˆszsunt ket-uri|ˆS,σ∝an}b∇acket∇i}ht, care sunt combinat ¸ii lineare ale spinorilor iˆs2iˆsz |+ +∝an}b∇acket∇i}ht=/parenleftigg 1 0/parenrightigg 1/parenleftigg 1 0/parenrightigg 1,|+−∝an}b∇acket∇i}ht=/parenleftigg 1 0/parenrightigg 1/parenleftigg 0 1/parenrightigg 2, |−+∝an}b∇acket∇i}ht=/parenleftigg 0 1/parenrightigg 2/parenleftigg 1 0/parenrightigg 1,|−−∝an}b∇acket∇i}ht =/parenleftigg 0 1/parenrightigg 2/parenleftigg 0 1/parenrightigg 2.(125) Funct ¸iile spinoriale din (125) se consider˘ a ortonormali zate.ˆInEnket-ul |+ +∝an}b∇acket∇i}hteste deSz= 1 ¸ si ˆ ın acela¸ si timp este funct ¸ie proprie a operatorulu i ˆS=1ˆs2+ 2(1ˆs)(2ˆs) +2ˆs2. (126) Dup˘ a cum se poate vedea din ˆS2=|+ +∝an}b∇acket∇i}ht=3 2|+ +∝an}b∇acket∇i}ht+ 2(1ˆsx·2ˆsx+1ˆsy·2ˆsy+1ˆsz·2ˆsz)|+ +∝an}b∇acket∇i}ht,(127) ˆS2=|+ +∝an}b∇acket∇i}ht= 2|+ +∝an}b∇acket∇i}ht= 1(1 + 1)|+ +∝an}b∇acket∇i}ht. (128) 67Dac˘ a se introduce operatorul ˆS−=1ˆs−+2ˆs−, (129) se obt ¸ine c˘ a [ˆS−,ˆS2] = 0. (130) Atunci ( ˆS−)k|1,1∝an}b∇acket∇i}htse poate scrie ˆ ın funct ¸ie de FP-urile operatorului ˆS2, respectiv ˆS−|1,1∝an}b∇acket∇i}ht=ˆS−|+ +∝an}b∇acket∇i}ht=√ 2|+−∝an}b∇acket∇i}ht+√ 2|−+∝an}b∇acket∇i}ht. (131) Rezult˘ a c˘ a Sz= 0 ˆ ın starea ˆS−|1,1∝an}b∇acket∇i}ht. Pe de alt˘ a parte, din condit ¸ia de normalizare avem |1,0∝an}b∇acket∇i}ht=1√ 2(|+−∝an}b∇acket∇i}ht+|−+∝an}b∇acket∇i}ht) (132) ˆS−|1,0∝an}b∇acket∇i}ht=|−−∝an}b∇acket∇i}ht +|−−∝an}b∇acket∇i}ht =α|1,−1∝an}b∇acket∇i}ht. (133) Din condit ¸ia de normalizare |1,−1∝an}b∇acket∇i}ht=|−,−∝an}b∇acket∇i}ht. (134) Exist˘ a ˆ ınc˘ a o singur˘ a combinat ¸ie linear independent˘ a de funct ¸ii de tip (125) diferit˘ a de|1,1∝an}b∇acket∇i}ht,|1,0∝an}b∇acket∇i}hty|1,−1∝an}b∇acket∇i}ht, respectiv ψ4=1√ 2(|+−∝an}b∇acket∇i}ht−|− +∝an}b∇acket∇i}ht), (135) ˆSzψ4= 0, ˆS2ψ4. (136) Prin urmare ψ4=|0,0∝an}b∇acket∇i}ht. (137) ψ4descrie starea unui sistem de doi fermioni cu spinul total eg al zero. Acest tip de stare se nume¸ ste singlet . Pe de alt˘ a parte, starea a doi fermioni de spin total egal unu se poate numi triplet avˆ ınd un grad de degenerare g= 3. Moment unghiular total Momentul unghiular total este un operator care se introduce ca suma mo- mentului unghiular orbital ¸ si de spin, respectiv ˆJ=ˆl+ˆS, (138) 68undeˆl¸ siˆS, a¸ sa cum am v˘ azut, act ¸ioneaz˘ a ˆ ın spat ¸ii diferite, dar p˘ atratele luiˆl¸ siˆScomut˘ a cu ˆJ, adic˘ a [ˆJi,ˆJj] =iεijkˆJk,[ˆJi,ˆl2] = 0,[ˆJi,ˆS2] = 0, (139) Din (139) rezult˘ a c˘ a ˆl2¸ siˆS2au un sistem comun de FP-uri cu ˆJ2¸ siˆJz. S˘ a determin˘ am spectrul proiect ¸iilor lui ˆJzpentru un fermion. Starea de proiect ¸ie de ˆJzmaxim se poate scrie ¯ψ=ψll/parenleftigg 1 0/parenrightigg =|l,l,+∝an}b∇acket∇i}ht (140) ˆzψ= (l+1 2)¯ψ,→j=l+1 2. (141) Introducem operatorul ˆJ−definit prin ˆJ−=ˆl−+ˆS−=ˆl−+/parenleftigg 0 0 1 0/parenrightigg . (142) Pe baza normaliz˘ arii α=/radicalbig (J+M)(J−M+ 1) se obt ¸ine ˆJ−ψll/parenleftigg 1 0/parenrightigg =√ 2l|l,l−1,+∝an}b∇acket∇i}ht+|l,l−1,−∝an}b∇acket∇i}ht, (143) astfel c˘ a valoarea proiect ¸iei lui ˆj−ˆ ınˆj−¯ψva fi ˆz= (l−1) +1 2=l−1 2. (144) Rezult˘ a c˘ a ˆ −mic¸ soreaz˘ a cu o unitate act ¸iunea lui ˆJz. ˆIn cazul general avem ˆk −=ˆlk −+kˆlk−1 −ˆS−. (145) Se observ˘ a c˘ a (145) se obt ¸ine din dezvoltarea binomial˘ a considerˆ ınd ˆ ın plus c˘ a ˆs2 −¸ si toate puterile superioare ale lui ˆ ssunt zero. ˆk −|l,l,+∝an}b∇acket∇i}ht=ˆlk −|l,l,+∝an}b∇acket∇i}ht+kˆlk−1 −|l,l,−∝an}b∇acket∇i}ht. (146) S ¸tim c˘ a (ˆl−)kψl,l=/radicalbigg k!(2l)! (2l−k)!ψl,l−k 69¸ si folosind-o se obt ¸ine ˆk −|l,l,+∝an}b∇acket∇i}ht=/radicalbigg k!(2l)! (2l−k)!|l,l−k,+∝an}b∇acket∇i}ht+/radicalbigg (k+1)!(2l)! (2l−k+1)!k|l,l−k+ 1,−∝an}b∇acket∇i}ht.(147) Not˘ am acum m=l−k ˆl−m −|l,l,+∝an}b∇acket∇i}ht=/radicalbigg (l−m)!(2l)! (l+m)!|l,m,+∝an}b∇acket∇i}ht+/radicalbigg (l−m−1)!(2l)! (2l+m+1)!(l−m)|l,m+1,−∝an}b∇acket∇i}ht.(148) Valorile proprii ale proiect ¸iei momentului unghiular tot al sunt date de secvent ¸a de numere care difer˘ a printr-o unitate de la j=l+1 2pˆ ın˘ a laj=l−1 2. Toate aceste st˘ ari apart ¸in acelea¸ si funct ¸ii proprii a l uiˆJca ¸ si|l,l,+∝an}b∇acket∇i}htpentru c˘ a [ˆJ−,ˆJ2] = 0 ˆJ2|l,l,+∝an}b∇acket∇i}ht= (ˆl2+ 2ˆlˆS+ˆS2)|l,l,+∝an}b∇acket∇i}ht, = [l(l+ 1) + 2l1 2+3 4]|l,l,+∝an}b∇acket∇i}ht (149) undej(j+ 1) = (l+1 2)(l+3 2). ˆIn partea dreapt˘ a a lui (149) o contribut ¸ie diferit˘ a de ze ro d˘ a numai j=ˆlzˆSz. Atunci FP-urile obt ¸inute corespund perechii j=l+1 2,mj=m+1 2 ¸ si sunt de forma |l+1 2,m+1 2∝an}b∇acket∇i}ht=/radicaligg l+m+ 1 2l+ 1|l,m,+∝an}b∇acket∇i}ht+/radicaligg l−m 2l+ 1|l,m+ 1,−∝an}b∇acket∇i}ht.(150) Num˘ arul total de st˘ ari linear independente este N= (2l+ 1)(2s+ 1) = 4l+ 2, (151) din careˆ ın (150) s-au construit (2j+1)=2l+3. Restul de 2 l−1 funct ¸ii proprii se pot obt ¸ine din condit ¸ia de ortonormalizare: |l−1 2,m−1 2∝an}b∇acket∇i}ht=/radicalig l−m 2l+1|l,m,+∝an}b∇acket∇i}ht−/radicalig l+m+1 2l+1|l,m+ 1,−∝an}b∇acket∇i}ht. (152) Dac˘ a dou˘ a subsisteme suntˆ ın interact ¸iuneˆ ın a¸ sa felˆ ıncˆ ıt fiecare moment unghiular ˆjise conserv˘ a , atunci FP-urile operatorului moment unghiul ar total ˆJ= ˆ1+ ˆ2, (153) se pot obt ¸ine printr-o procedur˘ a asem˘ an˘ atoare celei an terioare. Pentru valori proprii fixe ale lui ˆ 1¸ si ˆ2exist˘ a (2j1+ 1)(2j2+ 1) FP-uri ortonormalizate 70ale proiect ¸iei momentului unghiular total ˆJz, iar cea care corespunde valorii maxime a proiect ¸iei ˆJz, adic˘ aMJ=j1+j2, se poate construi ˆ ın mod unic ¸ si prin urmare J=j1+j2este valoarea maxim˘ a a momentului unghiular total al sistemului. Aplicˆ ınd operatorul ˆJ= ˆ1+ ˆ2ˆ ın mod repetat funct ¸iei |j1+j2,j1+j2,j1+j2∝an}b∇acket∇i}ht=|j1,j1∝an}b∇acket∇i}ht·|j2,j2∝an}b∇acket∇i}ht, (154) se pot obt ¸ine toate cele 2( j1+j2) + 1 FP ale lui ˆJ=j1+j2cu diferit ¸iM: −(j1+j2)≤M≤(j1+j2). De exemplu, FP pentru M=j1+j2−1 este: |j1+j2,j1+j2−1,j1,j2∝an}b∇acket∇i}ht=/radicaligg j1 j1+j2|j1,j1−1,j2,j2∝an}b∇acket∇i}ht+/radicaligg j2 j1+j2|j1,j1,j2,j2−1∝an}b∇acket∇i}ht. (155) Aplicˆ ınd ˆ ın continuare de mai multe ori operatorul ˆJ−se pot obt ¸ine cele 2(j1+j2−1)−1 funct ¸ii ale lui J=j1+j2−1. Se poate demonstra c˘ a |j1−j2|≤J≤j1+j2 astfel c˘ a maxJ/summationdisplay minJ(2J+ 1) = (2J1+ 1)(2J2+ 1) (156) ¸ si deci |J,M,j 1,j2∝an}b∇acket∇i}ht=/summationdisplay m1+m2=M(j1m1j2m2|JM)|j1,m1,j2,m2∝an}b∇acket∇i}ht, (157) unde coeficient ¸ii ( j1m1j2m2|JM) determin˘ a contribut ¸ia diferitelor funct ¸ii |j1,m1,j2,m2∝an}b∇acket∇i}htˆ ın funct ¸iile proprii ale luiˆJ2,ˆJzde valori proprii J(J+ 1), M¸ si sunt numit ¸i coeficient ¸ii Clebsch-Gordan. Referint ¸e : 1. H.A. Buchdahl, “Remark concerning the eigenvalues of orb ital angular momentum”, Am. J. Phys. 30, 829-831 (1962) 713N. Not˘ a : 1. Operatorul corespunz˘ ator vectorului Runge-Lenz din p rob- lema Kepler clasic˘ a se scrie ˆ/vectorA=ˆ r r+1 2/bracketleftigg (ˆl׈p)−(ˆp׈l)/bracketrightigg . unde s-au folosit unit˘ at ¸i atomice ¸ si s-a considerat Z= 1 (atomul de hidro- gen). Acest operator comut˘ a cu Hamiltonianul atomului de h idrogen ˆH= ˆp2 2−1 r, adic˘ a este integral˘ a de mi¸ scare cuantic˘ a . Componente le sale au comu- tatori de tipul [ Ai,Aj] =−2iǫijklk·H, iar comutatorii componentelor Runge- Lenz cu componentele momentului cinetic sunt de tipul [ li,Aj] =iǫijkAk, adic˘ a respect˘ a condit ¸iile (23). De demonstrat toate ace ste relat ¸ii poate fi un exercit ¸iu util. 3P. Probleme Problema 3.1 S˘ a se arate c˘ a orice operator de translat ¸ie, pentru care ψ(y+a) =Taψ(y), se poate scrie ca un operator exponent ¸ial ¸ si s˘ a se aplice ace st rezultat pentru y=/vector r¸ si pentru rotat ¸ia finit˘ a αˆ ın jurul axei z. Solut ¸ie Demonstrat ¸ia se obt ¸ine dezvoltˆ ınd ψ(y+a)) ˆ ın serie Taylor ˆ ın vecin˘ atatea infinitezimal˘ a a punctului x, adic˘ a ˆ ın puteri ale lui a ψ(y+a) =∞/summationdisplay n=0an n!dn dxnψ(x) Observ˘ am c˘ a∞/summationdisplay n=0andn dxn n!=ead dx ¸ si deciTa=ead dxˆ ın cazul 1D. ˆIn 3D,y=/vector r¸ sia→/vector a. Rezultatul este T/vector a=e/vector a·/vector∇. Pentru rotat ¸ia finit˘ a αˆ ın jurul lui zavemy=φ¸ sia=α. Rezult˘ a Tα=Rα=eαd dφ. O alt˘ a form˘ a exponent ¸ial˘ a a rotat ¸ieiˆ ın zeste ceaˆ ın funct ¸ie de operatorul moment cinetic a¸ sa cum s-a comentat ˆ ın acest capitol. Fie x′=x+dx, ¸ si 72considerˆ ınd numai primul ordin al seriei Taylor ψ(x′,y′,z′) =ψ(x,y,z) + (x′−x)∂ ∂x′ψ(x′,y′,z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectorr′=/vector r +(y′−y)∂ ∂y′ψ(x′,y′,z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectorr′=/vector r +(z′−z)∂ ∂z′ψ(x′,y′,z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectorr′=/vector r. T ¸inˆ ınd cont de faptul c˘ a ∂ ∂x′ iψ(/vector r′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle /vector r′=∂ ∂xiψ(/vector r), x′=x−ydφ, y′=y+xdφ, z′=z, se poate reduce seria din trei dimensiuni la numai dou˘ a ψ(/vector r′) =ψ(/vector r) + (x−ydφ−x)∂ψ(/vector r) ∂x+ (y+xdφ−y)∂ψ(/vector r) ∂y′, =ψ(/vector r)−ydφ∂ψ(/vector r) ∂x+xdφx∂ψ(/vector r) ∂y, =/bracketleftbigg 1−dφ/parenleftbigg −x∂ ∂y+y∂ ∂x/parenrightbigg/bracketrightbigg ψ(/vector r). Cumiˆlz=/parenleftig x∂ ∂y−y∂ ∂x/parenrightig rezult˘ a c˘ aR=/bracketleftig 1−dφ/parenleftig x∂ ∂y−y∂ ∂x/parenrightig/bracketrightig .Continuˆ ınd ˆ ın ordinul doi se poate ar˘ ata c˘ a se obt ¸ine1 2!(iˆlzdφ)2¸ si a¸ sa mai departe. Prin urmare,Rpoate fi scris ca exponent ¸ial˘ a R=eiˆlzdφ. Problema 3.2 S˘ a se arate c˘ a pe baza expresiilor date ˆ ın (14) se poate aju nge la (15). Solut ¸ie S˘ a consider˘ am numai termenii lineari ˆ ın dezvoltarea ˆ ın serie Taylor (rotat ¸ii infinitezimale) eiˆlzdφ= 1 +iˆlzdφ+1 2!(iˆlzdφ)2+... , 73¸ si deci (1 +iˆlzdφ)ˆAx(1−iˆlzdφ) = ˆAx−ˆAxdφ, (ˆAx+iˆlzdφˆAx)(1−iˆlzdφ) = ˆAx−ˆAxdφ, ˆAx−ˆAxiˆlzdφ+iˆlzdφˆAx+ˆlzdφˆAxˆlzdφ=ˆAx−ˆAxdφ, i(ˆlzˆAx−ˆAxˆlz)dφ=−ˆAydφ. Ajungem u¸ sor la concluzia c˘ a [ˆlz,ˆAx] =iˆAy. Deasemenea, [ ˆlz,ˆAy] =iˆAxse obt ¸ine din: (1 +iˆlzdφ)ˆAy(1−iˆlzdφ) = ˆAxdφ−ˆAy, (ˆAy+iˆlzdφˆAy)(1−iˆlzdφ) = ˆAxdφ−ˆAy, ˆAy−ˆAyiˆlzdφ+iˆlzdφˆAy+ˆlzdφˆAyˆlzdφ=ˆAxdφ−ˆAy, i(ˆlzˆAy−ˆAyˆlz)dφ=−ˆAxdφ. Problema 3.3 S˘ a se determine operatoruldˆσx dtpe baza Hamiltonianului unui electron cu spin aflat ˆ ıntr-un cˆ ımp magnetic de induct ¸ie /vectorB. Solut ¸ie Hamiltonianul ˆ ın acest caz este ˆH(ˆp,ˆr,ˆσ) =ˆH(ˆp,ˆr) + ˆσ·/vectorB, unde ultimul termen este Hamiltonianul Zeeman pt. electron. Cum ˆ σxcomut˘ a cu im- pulsurile ¸ si coordonatele, aplicarea ecuat ¸iei de mi¸ sca re Heisenberg conduce la: dˆσx dt=i ¯h[ˆH,ˆσx] =−i ¯he¯h 2me((ˆσyBy+ ˆσzBz)ˆσx−ˆσx(ˆσyBy+ ˆσzBz)) Folosind [σx,σy] =iσz, rezult˘ a : dˆσx dt=e me(ˆσyBz−ˆσzBy) =e me(/vector σ×/vectorB)x. 744. METODA WKB Pentru a studia potent ¸iale mai realiste decˆ ıt cele de bariere ¸ si gropi de potent ¸ial , este necesar s˘ a se foloseasc˘ a metode care s˘ a permit˘ a re zolvarea ecuat ¸iei Schr¨ odinger pentru clase mai generale de potent ¸iale ¸ si care s˘ a fie o bun˘ a aproximare a solut ¸iilor exacte. Scopul diferitelor metode de aproximare este s˘ a ofere solu t ¸ii suficient de bune ¸ si simple, care s˘ a permit˘ a ˆ ın acest fel ˆ ınt ¸eleger ea comportamentului sistemului ˆ ın form˘ a cuasianalitic˘ a . ˆIn cadrul mecanicii cuantice, una dintre cele mai vechi ¸ si e ficiente metode a fost dezvoltat˘ a ˆ ın mod aproape simultan de c˘ atre G. Wentzel, H. A. Kramers ¸ si L. Brillouin ˆ ın 1926, de la al c˘ aror nume deriv˘ a acronimul WKB sub care este cunoscut˘ a aceast˘ a metod˘ a (corect este JWKB , vezi nota 4N). Este important de ment ¸ionat c˘ a metoda WKB se aplic˘ a mai al es ecuat ¸iilor Schr¨ odinger 1D ¸ si exist˘ a dificult˘ at ¸i serioase ˆ ın gene raliz˘ arile la dimensiuni superioare. Pentru a rezolva ecuat ¸ia Schr¨ odinger −¯h2 2md2ψ dy2+u(y)ψ=Eψ (1) presupunem c˘ a potent ¸ialul are forma: u(y) =u0f/parenleftigy a/parenrightig (2) ¸ si facem schimbul de variabil˘ a : ξ2=¯h2 2mu0a2(3) η=E u0(4) x=y a. (5) Din ecuat ¸ia (5) obt ¸inem: d dx=dy dxd dy=ad dy(6) d2 dx2=d dx/parenleftig ad dy/parenrightig =/parenleftig ad dx/parenrightig/parenleftig ad dx/parenrightig =a2d2 dy2(7) 75¸ si ec. Schr¨ odinger se scrie: −ξ2d2ψ dx2+f(x)ψ=ηψ . (8) Multiplicˆ ınd cu−1/ξ2¸ si definind r(x) =η−f(x), este posibil s˘ a o scriem ˆ ın forma: d2ψ dx2+1 ξ2r(x)ψ= 0. (9) Pentru a rezolva (9) se propune urm˘ atoarea solut ¸ie: ψ(x) = exp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg . (10) A¸ sadar: d2ψ dx2=d dx/parenleftbiggdψ x/parenrightbigg =d dx/braceleftigg i ξq(x)exp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg/bracerightigg =⇒d2ψ dx2=i ξ/braceleftigg i ξq2(x)exp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg +∂q(x) ∂xexp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg/bracerightigg . Factorizˆ ınd ψavem: d2ψ x2=/bracketleftigg −1 ξ2q2(x) +i ξdq(x) dx/bracketrightigg ψ . (11) Neglijˆ ınd pe moment dependent ¸a ˆ ın x, ecuat ¸ia Schr¨ odinger se poate scrie : /bracketleftigg −1 ξ2q2+i ξ∂q ∂x+1 ξ2r/bracketrightigg ψ= 0 (12) ¸ si cum ˆ ın general ψ∝ne}ationslash= 0, avem: iξdq dx+r−q2= 0, (13) care este o ecuat ¸ie diferent ¸ial˘ a linear˘ a de tip Riccati , a c˘ arei solut ¸ii se caut˘ a ˆ ın forma unei serii de puteri ale lui ξcu presupunerea c˘ a ξeste foarte mic. Mai exact seria se propune de forma: q(x) =∞/summationdisplay n=0(−iξ)nqn(x). (14) 76Substituind-o ˆ ın Riccati obt ¸inem: iξ∞/summationdisplay n=0(−iξ)ndqn dx+r(x)−∞/summationdisplay µ=0(−iξ)µqµ∞/summationdisplay ν=0(−iξ)νqν= 0. (15) Printr-o rearanjare a termenilor obt ¸inem: ∞/summationdisplay n=0(−1)n(iξ)n+1dqn dx+r(x)−∞/summationdisplay µ=0∞/summationdisplay ν=0(−iξ)µ+νqµqν= 0. (16) Seriile duble au urm˘ atoarea proprietate: ∞/summationdisplay µ=0∞/summationdisplay ν=0aµν=∞/summationdisplay n=0n/summationdisplay m=0am,n−m, unde:µ=n−m ,ν =m. ˆIn acest fel: ∞/summationdisplay n=0(−1)n(iξ)n+1dqn dx+r(x)−∞/summationdisplay n=0n/summationdisplay m=0(−iξ)n−m+mqmqn−m= 0.(17) S˘ a vedem explicit cˆ ıt ¸iva termeni ˆ ın fiecare din seriile d in ecuat ¸ia (17): ∞/summationdisplay n=0(−1)n(iξ)n+1dqn dx=iξdq0 dx+ξ2dq1 dx−iξ3dq2 dx+... (18) ∞/summationdisplay n=0n/summationdisplay m=0(−iξ)nqmqn−m=q2 0−i2ξq0q1+... (19) Pentru ca ambele serii s˘ a cont ¸in˘ a pe iξˆ ın primul termen trebuie s˘ a le scriem: ∞/summationdisplay n=1(−1)n−1(iξ)ndqn−1 dx+r(x)−q2 0−∞/summationdisplay n=1n/summationdisplay m=0(−iξ)nqmqn−m= 0, ceea ce ne conduce la: ∞/summationdisplay n=1/bracketleftigg −(−iξ)ndqn−1 dx−n/summationdisplay m=0(−iξ)nqmqn−m/bracketrightigg +/bracketleftigg r(x)−q2 0/bracketrightigg = 0.(20) Pentru ca aceast˘ a ecuat ¸ie s˘ a se satisfac˘ a avem condit ¸i ile: r(x)−q2 0= 0⇒q0=±/radicalig r(x) (21) 77−(−iξ)ndqn−1 dx−n/summationdisplay m=0(−iξ)nqmqn−m= 0 ⇒dqn−1 dx=−n/summationdisplay m=0qmqn−mn≥1. (22) Aceasta ultima este o relat ¸ie de recurent ¸˘ a care apare ˆ ın metoda WKB. Este momentul s˘ a amintim c˘ a am definit r(x) =η−f(x), η=E u0&f(x) =u u0 ¸ si cu ajutorul ec. (21) obt ¸inem: q0=±/radicalig η−f(x) =±/radicaligg E u0−u u0=±/radicaligg 2m(E−u) 2mu0, (23) care ne indic˘ a natura clasic˘ a a impulsului WKB a particule i de energie Eˆ ın potent ¸ialul u¸ si ˆ ın unit˘ at ¸i√2mu0. Astfel: q0=p(x) =/radicalig η−f(x) nu este un operator . Dac˘ a aproxim˘ am pˆ ın˘ a la ordinul doi, obt ¸inem: q(x) =q0−iξq1−ξ2q2 ¸ si folosind relat ¸ia de recurent ¸˘ a WKB (22) calcul˘ am q1¸ siq2: dq0 dx=−2q0q1⇒q1=−1 2dq0 dx q0=−1 2d dx(ln|q0|) ⇒q1=−1 2d dx(ln|p(x)|) (24) dq1 dx=−2q0q2−q2 1⇒q2=−dq1 dx−q2 1 2q0. (25) Din ecuat ¸ia (24), ne d˘ am seama c˘ a m˘ arimea q1este panta cu semnul schimbat a lui ln |q0|; cˆ ındq0este foarte mic, q1≪0⇒ −ξq1≫0 ¸ si prin urmare seria diverge. Pentru a evita acest lucru se impu ne urm˘ atoarea condit ¸ie WKB : |q0|≫|−ξq1|=ξ|q1|. Este important de observat c˘ a aceast˘ a condit ¸ie WKB nu se s atisface pentru acele puncte xkunde: q0(xk) =p(xk) = 0. 78Darq0=p=/radicalig 2m(E−u) 2mu0¸ si deci ecuat ¸ia precedent˘ a ne conduce la: E=u(xk). (26) ˆIn mecanica clasic˘ a punctele xkcare satisfac (26) se numesc puncte de ˆ ıntoarcere pentru c˘ a ˆ ın ele are loc schimbarea sensului de mi¸ scare a particulei macroscopice. ˆIn baza acestor argumente, putem s˘ a spunem despre q0c˘ a este o solut ¸ie clasic˘ a a problemei examinate ¸ si c˘ a m˘ arimile q1&q2sunt respectiv prima ¸ si a doua corect ¸ie cuantic˘ a ˆ ın problema WKB. Pentru a obt ¸ine funct ¸iile de und˘ a vom considera numai sol ut ¸ia clasic˘ a ¸ si prima corect ¸ie cuantic˘ a a problemei pe care le substituim ˆ ın forma WKB a luiψ: ψ= exp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg = exp/bracketleftigg i ξ/integraldisplayx a(q0−iξq1)dx/bracketrightigg ⇒ψ= exp/parenleftigg i ξ/integraldisplayx aq0dx/parenrightigg ·exp/parenleftigg/integraldisplayx aq1dx/parenrightigg . Pentru al doilea factor avem: exp/parenleftigg/integraldisplayx aq1dx/parenrightigg = exp/bracketleftigg −1 2/integraldisplayx ad dx(ln|p(x)|)dx/bracketrightigg = = exp/bracketleftigg −1 2(ln|p(x)|)/vextendsingle/vextendsingle/vextendsinglex a/bracketrightigg =A/radicalbigp(x), cuAo constant˘ a , ˆ ın timp ce pentru primul factor se obt ¸ine: exp/parenleftigg i ξ/integraldisplayx aq0dx/parenrightigg = exp/bracketleftigg ±i ξ/integraldisplayx ap(x)dx/bracketrightigg . Astfel putem scrie ψˆ ın urm˘ atoarea form˘ a : ψ±=1/radicalbig p(x)exp/bracketleftigg ±i ξ/integraldisplayx ap(x)dx/bracketrightigg , (27) care sunt cunoscute ca solut ¸ii WKB ale ecuat ¸iei Schr¨ odinger unidi- mensionale . Solut ¸ia general˘ a WKB ˆ ın regiunea ˆ ın care condit ¸ia WKB se satisface se scrie: ψ=a+ψ++a−ψ−. (28) 79A¸ sa cum deja s-a ment ¸ionat, nu exist˘ a solut ¸ie WKBˆ ın pun ctele deˆ ıntoarcere, ceea ce ridic˘ a problema modului ˆ ın care se face trecerea de laψ(x<xk) la ψ(x>xk). Rezolvarea acestei dificult˘ at ¸i se face prin introducer ea formulelor de conexiune WKB. Formulele de Conexiune Deja s-a v˘ azut c˘ a solut ¸iile WKB sunt singulare ˆ ın puncte le de ˆ ıntoarcere clasic˘ a ; totu¸ si aceste solut ¸ii sunt corecte la stˆ ınga ¸ si la dreapta acestor punctexk. Ne ˆ ıntreb˘ am deci cum schimb˘ am ψ(x < xk) ˆ ınψ(x > xk) ˆ ın aceste puncte. R˘ aspunul este dat de a¸ sa numitele formule d e conexiune. Din teoria ecuat ¸iilor diferent ¸iale ordinare ¸ si pe baza a nalizei de funct ¸ii de variabil˘ a complex˘ a se poate demonstra c˘ a formulele de conexiune exist˘ a ¸ si c˘ a sunt urm˘ atoarele: ψ1(x) =1 [−r(x)]1 4exp/parenleftbigg −/integraldisplayxk x/radicalig −r(x)dx/parenrightbigg → →2 [r(x)]1 4cos/parenleftbigg/integraldisplayx xk/radicalig r(x)dx−π 4/parenrightbigg , (29) undeψ1(x) are numai comportament atenuant exponent ¸ial pentru x<xk. Prima formul˘ a de conexiune arat˘ a c˘ a funct ¸ia ψ(x), care la stˆ ınga punctu- lui de ˆ ıntoarcere se comport˘ a atenuant exponent ¸ial, tre ce ˆ ın dreapta acelui punct ˆ ıntr-o cosinusoid˘ a de faz˘ a φ=π 4¸ si de amplitudine dubl˘ a fat ¸˘ a de amplitudinea exponent ¸ialei. ˆIn cazul unei funct ¸ii ψ(x) mai generale, respectiv o funct ¸ie care s˘ a aib˘ a un comportament exponent ¸ial cresc˘ ator ¸ si atenuant, for mula de conexiune corespunz˘ atoare este: sin/parenleftbigg φ+π 4/parenrightbigg1 [−r(x)]1 4exp/parenleftbigg/integraldisplayxk x/radicalig −r(x)dx/parenrightbigg ← ←1 [r(x)]1 4cos/parenleftbigg/integraldisplayx xk/radicalig r(x)dx+φ/parenrightbigg , (30) cu condit ¸ia ca φs˘ a nu ia o valoare prea apropiat˘ a de −π 4. Motivul este c˘ a dac˘ aφ=−π 4, funct ¸ia sinus se anuleaz˘ a . Aceast˘ a a doua formul˘ a de co nex- iune signific˘ a c˘ a o funct ¸ie care se comport˘ a ca o cosinuso id˘ a la dreapta unui 80punct de ˆ ıntoarcere trece ˆ ın partea sa stˆ ınga ca o exponen t ¸ial˘ a cresc˘ atoare cu amplitudinea modulat˘ a de c˘ atre o sinusoid˘ a . Pentru a studia detaliile procedurii de obt ¸inere a formule lor de conexiune se poate consulta expunerea din cartea Mathematical Methods of Physics de J. Mathews & R.L. Walker. Estimarea erorii introduse de aproximat ¸ia WKB Am g˘ asit solut ¸ia ecuat ¸iei Schr¨ odinger ˆ ın orice regiun e unde se satis- face condit ¸ia WKB. Totu¸ si, solut ¸iile WKB diverg ˆ ın punc tele de ˆ ıntoarcere a¸ sa cum am semnalat. Vom analiza, de¸ si superficial, aceast ˘ a problematic˘ a cu scopul de a propune formulele de conexiune ˆ ıntr-o vecin˘ atate redus˘ a a punctelor de ˆ ıntoarcere. S˘ a presupunem c˘ a x=xkeste un punct de ˆ ıntoarcere, unde avem: q0(xk) =p(xk) = 0⇒E=u(xk). La stˆ ınga lui xk, adic˘ aˆ ın semidreapta x < xk, vom presupune c˘ a E < u (x), astfel c˘ a ˆ ın aceast˘ a regiune solut ¸ia WKB este: ψ(x) =a /bracketleftigu(x)−E u0/bracketrightig1 4exp −1 ξ/integraldisplayxk x/radicaligg u(x)−E u0dx + +b /bracketleftigu(x)−E u0/bracketrightig1 4exp 1 ξ/integraldisplayxk x/radicaligg u(x)−E u0dx . (31) ˆIn acela¸ si mod, la dreapta lui xk( ˆ ın semidreapta x > xk) presupunem E >u (x).ˆIn consecint ¸˘ a solut ¸ia WKB ˆ ın aceast˘ a zon˘ a este: ψ(x) =c /bracketleftigE−u(x) u0/bracketrightig1 4exp i ξ/integraldisplayx xk/radicaligg E−u(x) u0dx + +d /bracketleftigE−u(x) u0/bracketrightig1 4exp −i ξ/integraldisplayx xk/radicaligg E−u(x) u0dx . (32) Dac˘ aψ(x) este o funct ¸ie real˘ a , va avea aceast˘ a proprietate atˆ ıt la dreapta cˆ ıt ¸ si la stˆ ınga lui xk. Vom numi acest fapt “condit ¸ia de realitate” , care ˆ ınseamn˘ a c˘ a dac˘ a a,b∈ℜ, atuncic=d∗. 81Problema noastr˘ a este de a conecta aproximat ¸iile din cele dou˘ a laturi ale luixkpentru ca ele s˘ a se refere la aceea¸ si solut ¸ie. Aceasta ˆ ın seamn˘ a a g˘ asi c¸ siddac˘ a se cunosc a¸ sib, precum ¸ si viceversa. Pentru a efectua aceast˘ a conexiune, trebuie s˘ a utiliz˘ am o solut ¸ie aproximat˘ a , c are s˘ a fie corect˘ a de-a lungul unui drum care leag˘ a regiunile din cele dou˘ a laturi ale luixk, unde solut ¸iile WKB s˘ a fie deasemenea corecte. Cel mai comun este s˘ a se recurg˘ a la o metod˘ a propus˘ a de c˘ a treZwann ¸ siKemble care const˘ a ˆ ın a ie¸ si de pe axa real˘ a ˆ ın vecin˘ atatea lui xk, pe un contur ˆ ın jurul lui xkˆ ın planul complex. Pe acest contur se consider˘ a c˘ a solut ¸iile WKB continu˘ a s˘ a fie corecte. ˆIn aceast˘ a prezentare vom folosi aceast˘ a metod˘ a , dar numai cu scopul de a obt ¸ine un mijloc d e a estima erorile ˆ ın aproximat ¸ia WKB. Estimarea erorilor este ˆ ıntotdeauna important˘ a pentru s olut ¸iile aproxi- mate prin diferite metode ¸ si ˆ ın plus ˆ ın cazul WKB aproxima t ¸ia se face pe intervale mari ale axei reale, ceea ce poate duce la acumular ea erorilor ¸ si la eventuale artefacte datorate ¸ sifturilor de faz˘ a astfel i ntroduse. S˘ a definim funct ¸iile WKB asociate ˆ ın felul urm˘ ator: W±=1 /bracketleftigE−u(x) u0/bracketrightig1 4exp ±i ξ/integraldisplayx xk/radicaligg E−u(x) u0dx , (33) pe care le consider˘ am ca funct ¸ii de variabil˘ a complex˘ a . Vom folosi t˘ aieturi pentru a evita discontinuit˘ at ¸ile din zerourile lui r(x) =E−u(x) u0. Aceste funct ¸ii satisfac o ecuat ¸ie diferent ¸ial˘ a care se poate o bt ¸ine diferent ¸iindu-le ˆ ın raport cux, conducˆ ınd la: W′ ±=/parenleftbigg ±i ξ√r−1 4r′ r/parenrightbigg W± W′′ ±+/bracketleftigg r ξ2+1 4r′′ r−5 16/parenleftbiggr′ r/parenrightbigg2/bracketrightigg W±= 0. (34) S˘ a not˘ am: s(x) =1 4r′′ r−5 16/parenleftbiggr′ r/parenrightbigg2 , (35) atunciW±sunt solut ¸ii exacte ale ecuat ¸iei: W′′ ±+/bracketleftbigg1 ξ2r(x) +s(x)/bracketrightbigg W±= 0, (36) 82dar satisfac numai aproximativ ecuat ¸ia Schr¨ odinger, car e este regular˘ a ˆ ın x=xkˆ ın timp ce ecuat ¸ia pentru funct ¸iile WKB asociate este sin gular˘ a ˆ ın acel punct. Vom defini funct ¸iile α±(x) care s˘ a satisfac˘ a urm˘ atoarele dou˘ a relat ¸ii: ψ(x) =α+(x)W+(x) +α−(x)W−(x) (37) ψ′(x) =α+(x)W′ +(x) +α−(x)W′ −(x), (38) undeψ(x) este solut ¸ie a ecuat ¸iei Schr¨ odinger. Rezolvˆ ınd ecuat ¸iile anterioare pentruα±avem: α+=ψW′ −−ψ′W− W+W′−−W′+W−α−=−ψW′ +−ψ′W+ W+W′−−W′+W−, unde num˘ ar˘ atorul este exact Wronskianul luiW+¸ siW−. Nu este dificil de demonstrat c˘ a acesta ia valoarea −2 ξi, astfel c˘ aα±se simplific˘ a la forma: α+=ξ 2i/parenleftbigψW′ −−ψ′W−/parenrightbig(39) α−=−ξ 2i/parenleftbigψW′ +−ψ′W+/parenrightbig. (40) Efectuˆ ınd derivata ˆ ın xˆ ın ecuat ¸iile (39) ¸ si (40), avem: dα± dx=ξ 2i/parenleftbigψ′W′ ∓+ψW′′ ∓−ψ′′W∓−ψ′W′ ∓/parenrightbig. (41) ˆIn paranteze, primul ¸ si al patrulea termen se anuleaz˘ a ; am intim c˘ a : ψ′′+1 ξ2r(x)ψ= 0 &W′′ ±+/bracketleftbigg1 ξ2r(x) +s(x)/bracketrightbigg W±= 0 ¸ si deci putem scrie ecuat ¸ia (41) ˆ ın forma: dα± dx=ξ 2i/bracketleftbigg −ψ/parenleftbiggr ξ2+s/parenrightbigg W∓+r ξ2ψW∓/bracketrightbigg dα± dx=∓ξ 2is(x)ψ(x)W∓(x), (42) care ˆ ın baza ecuat ¸iilor (33) ¸ si (37) devine: dα± dx=∓ξ 2is(x) [r(x)]1 2/bracketleftbigg α±+α∓exp/parenleftbigg ∓2 ξi/integraldisplayx xk/radicalig r(x)dx/parenrightbigg/bracketrightbigg . (43) 83Ecuat ¸iile (42) ¸ si (43) se folosesc pentru a estima eroarea care se comite ˆ ın aproximat ¸ia WKB ˆ ın cazul unidimensional. Motivul pentru caredα± dxse poate considera ca m˘ asur˘ a a erorii WKB este c˘ a ˆ ın ecuat ¸iile (31) ¸ si (32) constantele a,b¸ sic,d, respectiv, ne dau numai solut ¸iiψaproximative, ˆ ın timp ce funct ¸iile α±introduse ˆ ın ecuat ¸iile (37) ¸ si (38) produc solut ¸ii ψexacte. Din punct de vedere geometric derivata d˘ a panta dreptei tangente la aceste funct ¸ii ¸ si indic˘ a m˘ asu ra ˆ ın careα±deviaz˘ a fat ¸˘ a de constantele a,b,c¸ sid. 4N. Not˘ a : Articolele (J)WKB originale sunt urm˘ atoarele: G. Wentzel, “Eine Verallgemeinerung der Wellenmechanik”, [“O generalizare a mecanicii ondulatorii”], Zeitschrift f¨ ur Physik 38, 518-529 (1926) [primit 18 June 1926] L. Brillouin, “La m´ ecanique ondulatoire de Schr¨ odinger: une m´ ethode g´ en´ erale de resolution par approximations successives”, [“Mecanic a ondulatorie a lui Schr¨ odinger: o metod˘ a general˘ a de rezolvare prin aproxi m˘ ari succesive”], Comptes Rendus Acad. Sci. Paris 183, 24-26 (1926) [primit 5 July 1926] H.A. Kramers, “Wellenmechanik und halbzahlige Quantisier ung”, [“Mecanica ondulatorie ¸ si cuantizarea semiˆ ıntreag˘ a ”], Zf. Physik 39, 828-840 (1926) [primit 9 Sept. 1926] H. Jeffreys, “On certain approx. solutions of linear diff. eqs . of the second order”, [“Asupra unor solut ¸ii aproximative a ecuat ¸iilor diferent ¸iale lineare de ordinul doi”], Proc. Lond. Math. Soc. 23, 428-436 (1925) 4P. Probleme Problema 4.1 S˘ a se foloseasc˘ a metoda WKB pentru o particul˘ a de energie Ecare se mi¸ sc˘ a ˆ ıntr-un potent ¸ial u(x) de forma ar˘ atat˘ a ˆ ın figura 4.1. 84Eu(x) x x x 1 2 Fig. 4.1 Solut ¸ie Ecuat ¸ia Schr¨ odinger corespunz˘ atoare este: d2ψ dx2+2m ¯h2[E−u(x)]ψ= 0. (44) Dup˘ a cum putem vedea: r(x) =2m ¯h2[E−u(x)]/braceleftigg este pozitiv˘ a pentru a<x<b este negativ˘ a pentru x<a,x>b . Dac˘ aψ(x) corespunde zonei ˆ ın care x < a , la trecerea ˆ ın intervalul a<x<b , formula de conexiune este dat˘ a de ecuat ¸ia (29) ¸ si ne spun e c˘ a : ψ(x)≈A [E−u]1 4cos /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4  (45) undeAeste o constant˘ a arbitrar˘ a . Cˆ ındψ(x) corespunde zonei x > b , la trecerea ˆ ın intervalul a < x < b avem ˆ ın mod similar: ψ(x)≈−B [E−u]1 4cos /integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4 , (46) 85undeBeste o constant˘ a arbitrar˘ a . Motivul pentru care formula d e conex- iune este din nou ecuat ¸ia (29), se ˆ ınt ¸elege examinˆ ınd ce se ˆ ıntˆ ımpl˘ a cˆ ınd particula ajunge la al doilea punct clasic de ˆ ıntoarcere x=b. Acesta pro- duce inversia direct ¸iei de mi¸ scare ¸ si atunci particula a pare ca venind de la dreapta spre stˆ ınga. Cu alte cuvinte, ne g˘ asimˆ ın prima si tuat ¸ie (de la stˆ ınga la dreapta) numai c˘ a v˘ azut˘ a ˆ ıntr-o oglind˘ a ˆ ın punctul x=a. Aceste dou˘ a expresii trebuie s˘ a fie acelea¸ si independent de constantele A¸ siB, astfel c˘ a : cos /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4 =−cos /integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4  ⇒cos /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4 + cos /integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4 = 0. (47) Amintind c˘ a : cosA+ cosB= 2cos/parenleftbiggA+B 2/parenrightbigg cos/parenleftbiggA−B 2/parenrightbigg , ecuat ¸ia (47) se scrie: 2cos 1 2 /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4+/integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4  · ·cos 1 2 /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4−/integraldisplayb x/radicaligg 2m ¯h2(E−u)dx+π 4  = 0,(48) ceea ce implic˘ a pentru argumentele acestor cosinusoide c˘ a sunt multipli ˆ ıntregi deπ 2; argumentul celui de-a doua cosinusoide nu ne duce la nici un rezultat netrivial, a¸ sa c˘ a ne fix˘ am atent ¸ia numai asupra argumentului primei cosinusoide, care este esent ¸ial pentru obt ¸inerea unui re zultat important: 1 2 /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4+/integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4 =n 2πpt. n impar ⇒/integraldisplayb a/radicaligg 2m ¯h2(E−u)dx−π 2=nπ 86⇒/integraldisplayb a/radicaligg 2m ¯h2(E−u)dx= (n+1 2)π ⇒/integraldisplayb a/radicalig 2m(E−u)dx= (n+1 2)π¯h . (49) Acest rezultat este foarte similar regulilor de cuantizare Bohr - Sommerfeld . Amintim c˘ a postulatul lui Bohr stabile¸ ste c˘ a momentul un ghiular al unui electron care se mi¸ sc˘ a pe o ‘orbit˘ a permis˘ a ’ ˆ ın jurul nu cleului atomic este cuantizat˘ a ¸ si valoarea sa este: L=n¯h,n= 1,2,3,.... Amintim deasemenea c˘ a regulile de cuantizare Wilson - Sommerfeld stabilesc c˘ a orice coordonat˘ a a unui sistem fizic care variaz˘ a periodic ˆ ın timp trebuie s˘ a satisfac˘ a condit ¸ia cuantic˘ a :/contintegraltextpqdq=nqh; undeqeste o coordonat˘ a periodic˘ a , pqeste impul- sul asociat acesteia, nqeste un num˘ ar ˆ ıntreg ¸ si heste constanta lui Planck. Se vede c˘ a rezultatul obt ¸inut ˆ ın aproximat ¸ia WKB este cu adev˘ arat foarte asem˘ an˘ ator. Problema 4.2 S˘ a se estimeze eroarea care se comite ˆ ın solut ¸ia WKB ˆ ıntr -un punct x1∝ne}ationslash=xk, cuxkun punct clasic de ˆ ıntoarcere, pentru ecuat ¸ia diferent ¸i al˘ a y′′+xy= 0.Solut ¸ia acestei probleme este important˘ a ˆ ın studiul cˆ ı mpurilor uniforme, a¸ sa cum sunt cele gravitat ¸ionale sau cele elect rice produse de pl˘ aci plane ˆ ınc˘ arcate. Solut ¸ie: Pentru aceast˘ a ecuat ¸ie diferent ¸ial˘ a avem: ξ= 1, r (x) =x &s(x) =−5 16x−2. r(x) =xare un singur zero ˆ ın xk= 0, astfel c˘ a pentru x≫0: W±=x−1 4exp/parenleftbigg ±i/integraldisplayx 0√xdx/parenrightbigg =x−1 4exp/parenleftbigg ±2 3ix3 2/parenrightbigg . (50) Derivˆ ındW±pˆ ın˘ a la a doua derivat˘ a ˆ ın x, ne d˘ am seama c˘ a se satisface urm˘ atoarea ecuat ¸ie diferent ¸ial˘ a : W′′ ±+ (x−5 16x−2)W±= 0. (51) 87Solut ¸ia exact˘ a y(x) a acestei ecuat ¸ii diferent ¸iale o scriem ca o combinat ¸ie linear˘ a de W±, a¸ sa cum s-a indicat ˆ ın sect ¸iunea corespondent˘ a estim˘ arii erorii ˆ ın aproximat ¸ia WKB; amintim c˘ a s-a propus o combin at ¸ie linear˘ a de forma: y(x) =α+(x)W+(x) +α−(x)W−(x) Pentruxmari, o solut ¸ie general˘ a a ecuat ¸iei noastre diferent ¸ia le se poate scrie ˆ ın aproximat ¸ia WKB ˆ ın forma: y(x) =Ax−1 4cos/parenleftbigg2 3x3 2+δ/parenrightbigg cˆ ındx→∞, (52) astfel c˘ aα+→A 2eiδ¸ siα−→A 2e−iδpentrux→∞. Vrem s˘ a calcul˘ am eroarea datorat˘ a acestei solut ¸ii WKB. O m˘ asur˘ a a aceste i erori este devierea luiα+¸ si a luiα−fat ¸˘ a de constantele A. Pentru aceasta folosim ecuat ¸ia: dα± dx=∓ξ 2is(x)/radicalbig r(x)/bracketleftbigg α±+α∓exp/parenleftbigg ∓2i/integraldisplayx xk/radicalig r(x)dx/parenrightbigg/bracketrightbigg ¸ si efectuˆ ınd substitut ¸iile corespunz˘ atoare avem: dα± dx=∓i 2/parenleftbigg −5 16x−2/parenrightbigg x−1 2/bracketleftbiggA 2e±iδ+A 2e∓iδexp/parenleftbigg ∓2i2 3x3 2/parenrightbigg/bracketrightbigg .(53) S ¸tim c˘ a ∆α±reprezint˘ a schimb˘ arile pe care le prezint˘ a α±cˆ ındxvariaz˘ a ˆ ıntrex1¸ si∞, ceea ce ne permite calculul prin intermediul lui: ∆α± A/2=2 A/integraldisplay∞ x1dα± dxdx= =±i5 32e±iδ/bracketleftbigg2 3x−3 2 1+e∓2iδ/integraldisplay∞ x1x−5 2exp/parenleftbigg ∓i4 3x3 2/parenrightbigg dx/bracketrightbigg . (54) Al doilea termen din paranteze este mai put ¸in important dec ˆ ıt primul pentru c˘ a exponent ¸iala complex˘ a oscileaz˘ a ˆ ıntre 1 ¸ si −1 ¸ si decix−5 2< x−3 2. Prin urmare:∆α± A/2≈±5 48ie±iδx−3 2 1 (55) ¸ si cum putem vedea eroarea care se introduce este ˆ ıntr-ade v˘ ar mic˘ a dac˘ a tinem cont ¸ si de faptul c˘ a exponent ¸iala complex˘ a oscile az˘ a ˆ ıntre−1 ¸ si 1, iar x−3 2 1este deasemenea mic. 885. OSCILATORUL ARMONIC (OA) Solut ¸ia ecuat ¸iei Schr¨ odinger pentru OA Oscilatorul armonic (OA) poate fi considerat ca o paradigm˘ a a Fizicii. Util- itatea sa apare ˆ ın marea majoritate a domeniilor, de la fizic a clasic˘ a pˆ ın˘ a la electrodinamica cuantic˘ a ¸ si teorii ale obiectelor colap sate gravitat ¸ional. Din mecanica clasic˘ a ¸ stim c˘ a multe potent ¸iale complica te pot fi aproximate ˆ ın vecin˘ atatea pozit ¸iilor de echilibru printr-un poten t ¸ial OA V(x)∼1 2V′′(a)(x−a)2. (1) Acesta este un caz unidimensional. Pentru acest caz, funct ¸ ia Hamil- tonian˘ a clasic˘ a a unei particule de mas˘ a m, oscilˆ ınd cu frecvent ¸a ωare urm˘ atoarea form˘ a : H=p2 2m+1 2mω2x2(2) ¸ si Hamiltonianul cuantic corespunz˘ ator ˆ ın spat ¸iul de c onfigurat ¸ii este : ˆH=1 2m(−i¯hd dx)2+1 2mω2x2(3) ˆH=−¯h2 2md2 dx2+1 2mω2x2. (4) Dat faptul c˘ a potent ¸ialul este independent de timp, FP Ψ n¸ si autovalorile Ense determin˘ a cu ajutorul ecuat ¸iei Schr¨ odinger independ ent˘ a de timp : ˆHΨn=EnΨn. (5) Considerˆ ınd Hamiltonianul pentru OA , ecuat ¸ia Schr¨ odin ger pentru acest caz este : d2Ψ dx2+/bracketleftigg 2mE ¯h2−m2ω2 ¯h2x2/bracketrightigg Ψ = 0. (6) Am suprimat subindicii lui E¸ si Ψ pentru c˘ a nu au nici o important ¸˘ a aici. Definind: k2=2mE ¯h2(7) λ=mω ¯h, (8) 89ecuat ¸ia Schr¨ odinger devine: d2Ψ dx2+ [k2−λ2x2]Ψ = 0, (9) cunoscut˘ a ca ecuat ¸ia diferent ¸ial˘ a Weber ˆ ın matematic ˘ a . Vom face ˆ ın continuare transformarea: y=λx2. (10) ˆIn general, cu schimbul de variabil˘ a de la xlay, operatorii diferent ¸iali iau forma:d dx=dy dxd dy(11) d2 dx2=d dx(dy dxd dy) =d2y dx2d dy+ (dy dx)2d2 dy2. (12) Aplicˆ ınd aceast˘ a regul˘ a evident˘ a transform˘ arii prop use obt ¸inem urm˘ atoarea ecuat ¸ie diferent ¸ial˘ a ˆ ın variabila y: yd2Ψ dy2+1 2dΨ dy+ [k2 4λ−1 4y]Ψ = 0, (13) sau, definind : κ=k2 2λ=¯k2 2mω=E ¯hω, (14) obt ¸inem: yd2Ψ dy2+1 2dΨ dy+ [κ 2−1 4y]Ψ = 0. (15) S˘ a trecem la rezolvarea acestei ecuat ¸ii, efectuˆ ınd mai ˆ ıntˆ ıi analiza sa asimptotic˘ a ˆ ın limita y→∞. Pentru aceasta. se rescrie ecuat ¸ia anterioar˘ a ˆ ın forma : d2Ψ dy2+1 2ydΨ dy+ [κ 2y−1 4]Ψ = 0. (16) Observ˘ am c˘ a ˆ ın limita y→∞ ecuat ¸ia se comport˘ a astfel: d2Ψ∞ dy2−1 4Ψ∞= 0. (17) Aceast˘ a ecuat ¸ie are ca solut ¸ie: Ψ∞(y) =Aexpy 2+Bexp−y 2. (18) 90Elimin˘ am expy 2luˆ ındA= 0 pentru c˘ a diverge ˆ ın limita y→ ∞ ¸ si r˘ amˆ ınem cu exponent ¸iala atenuat˘ a . Putem sugera acum c˘ a Ψ are urm˘ atoarea forma: Ψ(y) = exp−y 2ψ(y). (19) Substituind-o ˆ ın ecuat ¸ia diferent ¸ial˘ a pentru y( ec. 15) se obt ¸ine: yd2ψ dy2+ (1 2−y)dψ dy+ (κ 2−1 4)ψ= 0. (20) Ceea ce am obt ¸inut este ecuat ¸ia hipergeometric˘ a confluen t˘ a4: zd2y dz2+ (c−z)dy dz−ay= 0. (21) Solut ¸ia general˘ a a acestei ecuat ¸ii este : y(z) =A1F1(a;c,z) +B z1−c1F1(a−c+ 1;2−c,z), (22) unde funct ¸ia hipergeometric˘ a confluent˘ a este definit˘ a p rin : 1F1(a;c,z) =∞/summationdisplay n=0(a)nxn (c)nn!. (23) Comparˆ ınd acum ecuat ¸ia noastr˘ a, cu ecuat ¸ia hipergeome tric˘ a confluent˘ a, se observ˘ a c˘ a solut ¸ia general˘ a a primei este : ψ(y) =A1F1(a;1 2,y) +B y1 21F1(a+1 2;3 2,y) (24) unde a=−(κ 2−1 4). (25) Dac˘ a ment ¸inem aceste solut ¸ii ˆ ın forma ˆ ın care se prezin t˘ a , condit ¸ia de normalizare pentru funct ¸ia de und˘ a nu se satisface, pen tru c˘ a din com- portamentul asimptotic al funct ¸iei hipergeometrice confl uente5rezult˘ a ( considerˆ ınd numai comportamentul dominant exponent ¸ial ) : Ψ(y) =e−y 2ψ(y)→const. ey 2ya−1 2. (26) 4Deasemenea cunoscut˘ a ca ecuat ¸ia diferent ¸ial˘ a Kummer. 5Comportamentul asimptotic pentru |x|→ ∞ este: 1F1(a;c, z)→Γ(c) Γ(c−a)e−iaπx−a+Γ(c) Γ(a)exxa−c. 91Aceast˘ a ultim˘ a aproximat ¸ie ne duce la o divergent ¸˘ a ˆ ın integrala de nor- malizare, care fizic este inacceptabil˘ a . Ceea ce se face ˆ ın acest caz, este s˘ a se impun˘ a condit ¸ia de terminare a seriei6, adic˘ a , seria are numai un num˘ ar finit de termeni fiind deci un polinom de grad n. Observ˘ am astfel c˘ a faptul de a cere ca integrala de normali zare s˘ a fie finit˘ a (dup˘ a cum ¸ stim condit ¸ie obligatorie pentru semnificat ¸i a fizic˘ a ˆ ın termeni de probabilit˘ at ¸i), ne conduce la truncarea seriei, fapt car e ˆ ın acela¸ si timp pro- duce cuantizarea energiei. Consider˘ am ˆ ın continuare cele dou˘ a cazuri posibile : 1)a=−n¸ siB= 0 κ 2−1 4=n . (27) FP-urile sunt date de: Ψn(x) =Dnexp−λx2 21F1(−n;1 2,λx2) (28) ¸ si energia este: En= ¯hω(2n+1 2). (29) 2)a+1 2=−n¸ siA= 0 κ 2−1 4=n+1 2. (30) FP-urile sunt acum: Ψn(x) =Dnexp−λx2 2x1F1(−n;3 2,λx2), (31) iar energiile stat ¸ionare sunt: En= ¯hω[(2n+ 1) +1 2]. (32) Polinoamele obt ¸inute ˆ ın urma acestei trunc˘ ari a seriei h ipergeometrice se numesc polinoame Hermite ¸ si se pot scrie ca urm˘ atoarele funct ¸ii hiperge- ometrice : H2n(η) = (−1)n(2n)! n!1F1(−n;1 2,η2) (33) 6Condit ¸ia de truncare a seriei pentru funct ¸ia hipergeomet ric˘ a confluent˘ a 1F1(a;c, z) estea=−n, cunun ˆ ıntreg nenegativ ( adic˘ a , include zero ). 92H2n−1(η) = (−1)n2(2n+ 1)! n!η1F1(−n;3 2,η2). (34) Putem acum combina rezultatele obt ¸inute ( pentru c˘ a unele ne dau val- orile pare ¸ si altele pe cele impare ) ˆ ıntr-o singur˘ a expre sie pentru autovalori ¸ si funct ¸ii proprii : Ψn(x) =Dnexp−λx2 2Hn(√ λx) (35) En= (n+1 2)¯hω n = 0,1,2... (36) Spectrul de energie al OA este echidistant, adic˘ a , exist˘ a aceea¸ si diferent ¸˘ a ¯hωˆ ıntre oricare dou˘ a nivele. Alt˘ a observat ¸ie pe care o put em face, este ˆ ın leg˘ atur˘ a cu valoarea minim˘ a de energie pe care o are oscil atorul; poate ˆ ın mod surprinz˘ ator este diferit˘ a de zero; acesta se conside r˘ a un rezultat pur cuantic, pentru c˘ a dispare dac˘ a ¯ h→0. Se cunoa¸ ste ca energia de punct zero ¸ si faptul c˘ a este diferit˘ a de zero , este o caracteristic˘ a a tuturor potent ¸ialelor confinante . Constanta de normalizare poate fi calculat˘ a u¸ sor ¸ si are va loarea: Dn=/bracketleftigg/radicaligg λ π1 2nn!/bracketrightigg1 2 . (37) Prin urmare se obt ¸in funct ¸iile proprii normalizate ale OA unidimensional : Ψn(x) =/bracketleftigg/radicaligg λ π1 2nn!/bracketrightigg1 2 exp(−λx2 2)Hn(√ λx). (38) Operatori de creare ˆa†¸ si anihilare ˆa Exist˘ a o alt˘ a form˘ a de a trata oscilatorul armonic fat ¸˘ a de cea convent ¸ional˘ a de a rezolva ecuat ¸ia Schr¨ odinger. Este vorba de metoda alg ebric˘ a sau metoda operatorilor de scar˘ a , o metod˘ a foarte eficient˘ a c are se poate aplica cu mult succes a multe probleme de mecanic˘ a cuantic˘ a de spe ctru discret. Definim doi operatori nehermitici a¸ sia†: a=/radicalbiggmω 2¯h(x+ip mω) (39) 93a†=/radicalbiggmω 2¯h(x−ip mω). (40) Ace¸ sti operatori sunt cunoscut ¸i ca operator de anihilare ¸ sioperator de creare , respectiv (justificarea acestor denumiri se va vedea mai de parte, de¸ si se poate spune c˘ a vine din teoria cuantic˘ a a cˆ ımpuri lor ). S˘ a calcul˘ am acum comutatorul acestor doi operatori: [a,a†] =mω 2¯h[x+ip mω,x−ip mω] =1 2¯h(−i[x,p] +i[p,x]) = 1, (41) unde am folosit comutatorul: [x,p] =i¯h . (42) Prin urmare operatorii de creare ¸ si anihilare nu comut˘ a , s atisf˘ acˆ ınd relat ¸iile de comutare : [a,a†] = 1. (43) S˘ a definim deasemenea importantul operator de num˘ ar ˆN: ˆN=a†a . (44) Acest operator este hermitic dup˘ a cum se poate demonstra u¸ sor folosind (AB)†=B†A†: ˆN†= (a†a)†=a†(a†)†=a†a=ˆN . (45) Considerˆ ınd acum c˘ a : a†a=mω 2¯h(x2+p2 m2ω2) +i 2¯h[x,p] =ˆH ¯hω−1 2(46) observ˘ am c˘ a Hamiltonianul se scrie ˆ ıntr-o form˘ a simpl˘ a ˆ ın funct ¸ie de oper- atorul de num˘ ar : ˆH= ¯hω(ˆN+1 2). (47) Operatorul de num˘ ar are acest nume datorit˘ a faptului c˘ a a utovalorile sale sunt exact subindicii funct ¸iei de und˘ a asupra c˘ arei a act ¸ioneaz˘ a : ˆN|n>=n|n> , (48) unde am folosit notat ¸ia: |Ψn>=|n> . (49) 94Aplicˆ ınd acest fapt lui (47) avem : ˆH|n>= ¯hω(n+1 2)|n> . (50) Dar ¸ stim din ecuat ¸ia Schr¨ odinger c˘ a ˆH|n >=E|n >pe baza c˘ areia rezult˘ a c˘ a autovalorile energetice sunt date de : En= ¯hω(n+1 2). (51) Acest rezultat este identic (cum ¸ si trebuia s˘ a fie ) cu rezul tatul (36). ˆIn continuare s˘ a ar˘ at˘ am de ce operatorii a¸ sia†au numele pe care le au. Pentru aceasta s˘ a calcul˘ am comutatorii: [ˆN,a] = [a†a,a] =a†[a,a] + [a†,a]a=−a , (52) rezultate care se obt ¸in din [ a,a] = 0 ¸ si (43). Similar, s˘ a calcul˘ am: [ˆN,a†] = [a†a,a†] =a†[a,a†] + [a†,a†]a=a†. (53) Cu ace¸ sti doi comutatori putem s˘ a scriem: ˆN(a†|n>) = ([ ˆN,a†] +a†ˆN)|n> = (a†+a†ˆN)|n> (54) =a†(1 +n)|n>= (n+ 1)a†|n> . Cu un procedeu similar se obt ¸ine deasemenea: ˆN(a|n>) = ([ ˆN,a] +aˆN)|n>= (n−1)a|n> . (55) Expresia (54) implic˘ a c˘ a se poate considera ket-ul a†|n >ca eigenket al operatorului de num˘ ar , unde autovaloarea increment˘ a c u unu, adic˘ a , a fost produs˘ a o cuant˘ a de energie prin act ¸iunea lui a†asupra ket-ului. Aceasta explic˘ a numele de operator de creare (creat ¸ie). C omentarii urmˆ ınd aceea¸ si linie de rat ¸ionament acela¸ si tip de conc luzie rezult˘ a pentru operatorul a, ceea ce ˆ ıi d˘ a numele de operator de anihilare ( o cuant˘ a de energie este eliminat˘ a cˆ ınd act ¸ioneaz˘ a acest operator ). Ecuat ¸ia (54) deasemenea implic˘ a proport ¸ionalitatea ke t-urilora†|n >¸ si |n+ 1>: a†|n>=c|n+ 1> , (56) 95undeceste o constant˘ a care trebuie determinat˘ a . Considerˆ ınd ˆ ın plus c˘ a : (a†|n>)†=<n|a=c∗<n+ 1|, (57) putem realiza urm˘ atorul calcul: <n|a(a†|n>) =c∗<n+ 1|(c|n+ 1>) (58) <n|aa†|n>=c∗c<n + 1|n+ 1> (59) <n|aa†|n>=|c|2. (60) Dar din relat ¸ia de comutare pentru operatorii a¸ sia†: [a,a†] =aa†−a†a=aa†−ˆN= 1, (61) avem c˘ a : aa†=ˆN+ 1 (62) Substituind ˆ ın (60): <n|ˆN+ 1|n>=<n|n>+<n|ˆN|n>=n+ 1 =|c|2.(63) Cerˆ ındcs˘ a fie real ¸ si pozitiv ( prin convent ¸ie ), obt ¸inem urm˘ ato area valoare: c=√ n+ 1. (64) Cu aceasta avem relat ¸ia: a†|n>=√ n+ 1|n+ 1> . (65) Urmˆ ınd acela¸ si procedeu se poate ajunge la o relat ¸ie pent ru operatorul de anihilare : a|n>=√n|n−1> . (66) S˘ a ar˘ at˘ am acum c˘ a valorile lui ntrebuie s˘ a fie ˆ ıntregi nenegativi. Pentru aceasta, recurgem la cerint ¸a de pozitivitate a normei, apl icˆ ınd-o ˆ ın special vectorului de stare a|n>. Aceast˘ a condit ¸ie ne spune c˘ a produsul interior (intern) al acestui vector cu adjunctul s˘ au (( a|n>)†=<n|a†) trebuie s˘ a fie mai mare sau egal˘ a cu zero : (<n|a†)·(a|n>)≥0. (67) Dar aceast˘ a relat ¸ie nu este decˆ ıt : <n|a†a|n>=<n|ˆN|n>=n≥0. (68) 96Prin urmare nnu poate fi negativ ¸ si trebuie s˘ a fie ˆ ıntreg pentru c˘ a dac˘ a nu ar fi prin aplicarea consecutiv˘ a a operatorului de anihil are ne-ar duce la valori negative ale lui n, ceea ce este ˆ ın contradict ¸ie cu ce s-a spus anterior. Este posibil s˘ a se exprime starea n(|n>) direct ˆ ın funct ¸ie de starea baz˘ a (|0>) folosind operatorul de creare. S˘ a vedem cum se face aceast ˘ a iterat ¸ie important˘ a : |1>=a†|0> (69) |2>= [a† √ 2]|1>= [(a†)2 √ 2!]|0> (70) |3>= [a† √ 3]|2>= [(a†)3 √ 3!]|0> (71) ... |n>= [(a†)n √ n!]|0> (72) Putem deasemenea aplica aceast˘ a metod˘ a pentru a obt ¸ine F P-urile ˆ ın spat ¸iul configurat ¸iilor. Pentru a realiza acest lucru, vo m pleca din starea baz˘ a : a|0>= 0. (73) ˆIn reprezentarea xavem: ˆaΨ0(x) =/radicalbiggmω 2¯h(x+ip mω)Ψ0(x) = 0. (74) Amintindu-ne forma pe care o ia operatorul impuls ˆ ın reprez entareax, putem ajunge la o ecuat ¸ie diferent ¸ial˘ a pentru funct ¸ia d e und˘ a a st˘ arii fun- damentale; s˘ a introducem deasemenea urm˘ atoarea definit ¸ iex0=/radicalig ¯h mω, cu care avem : (x+x2 0d dx)Ψ0= 0 (75) Aceast˘ a ecuat ¸ie se poate rezolva u¸ sor, ¸ si prin normaliz are ( integrala sa de la−∞la∞trebuie s˘ a fie pus˘ a egal˘ a cu unu ), ajungem la funct ¸ia de un d˘ a a st˘ arii fundamentale : Ψ0(x) = (1/radicalig√πx0)e−1 2(x x0)2 (76) 97Restul de FP, care descriu st˘ arile excitate ale OA , se pot ob t ¸ine folosind operatorul de creat ¸ie. Procedeul este urm˘ atorul: Ψ1=a†Ψ0= (1√ 2x0)(x−x2 0d dx)Ψ0 (77) Ψ2=1√ 2(a†)2Ψ0=1√ 2!(1√ 2x0)2(x−x2 0d dx)2Ψ0. (78) Continuˆ ınd, se poate ar˘ ata prin induct ¸ie c˘ a : Ψn=1/radicalig√π2nn!1 xn+1 2 0(x−x2 0d dx)ne−1 2(x x0)2 . (79) Evolut ¸ia temporal˘ a a oscilatorului ˆIn aceast˘ a sect ¸iune vom ilustra prin intermediul OA modul ˆ ın care se lu- creaz˘ a cu reprezentarea Heisenberg ˆ ın care st˘ arile sunt fixate ˆ ın timp ¸ si se permite evolut ¸ia temporal˘ a a operatorilor. Vom consid era operatorii ca funct ¸ii de timp ¸ si vom obt ¸ine ˆ ın mod concret cum evolut ¸i oneaz˘ a operatorii de pozit ¸ie , impuls, a¸ sia†ˆ ın timp pentru cazul OA. Ecuat ¸iile de mi¸ scare Heisenberg pentru p¸ sixsunt : dˆp dt=−∂ ∂ˆxV(ˆ x) (80) dˆx dt=ˆp m. (81) De aici rezult˘ a c˘ a ecuat ¸iile de mi¸ scare pentru xypˆ ın cazul OA sunt: dˆp dt=−mω2ˆx (82) dˆx dt=ˆp m. (83) Prin urmare, dispunem de o pereche de ecuat ¸ii cuplate , care sunt echiva- lente unei perechi de ecuat ¸ii pentru operatorii de creat ¸i e ¸ si anihilare, care ˆ ıns˘ a nu sunt cuplate. ˆIn mod explicit : da dt=/radicalbiggmω 2¯hd dt(ˆx+iˆp mω) (84) da dt=/radicalbiggmω 2¯h(dˆx dt+i mωdˆp dt). (85) 98Substituind (82) ¸ si (83) ˆ ın (85) : da dt=/radicalbiggmω 2¯h(ˆp m−iωˆx) =−iωa . (86) Similar, se poate obt ¸ine o ecuat ¸ie diferent ¸ial˘ a pentru operatorul de creat ¸ie : da† dt=iωa†(87) Ecuat ¸iile diferent ¸iale pe care le-am obt ¸inut pentru evo lut ¸ia temporal˘ a a op- eratorilor de creat ¸ie ¸ si anihilare , pot fi integrate imedi at, dˆ ınd evolut ¸ia explicit˘ a a acestor operatori: a(t) =a(0)e−iωt(88) a†(t) =a†(0)eiωt. (89) Se poate remarca din aceste rezultate ¸ si din ecuat ¸iile (44 ) ¸ si (47) c˘ a atˆ ıt Hamiltonianul ca ¸ si operatorul de num˘ ar , nu depind de timp , a¸ sa cum era de a¸ steptat. Cu cele dou˘ a rezultate anterioare , putem s˘ a obt ¸inem oper atorii de pozit ¸ie ¸ si impuls ca funct ¸ii de timp, pentru c˘ a sunt dat ¸i ˆ ın func t ¸ie de operatorii de creat ¸ie ¸ si anihilare: ˆx=/radicaligg ¯h 2mω(a+a†) (90) ˆp=i/radicaligg m¯hω 2(a†−a). (91) Substituind-i se obt ¸ine: ˆx(t) = ˆx(0)cosωt+ˆp(0) mωsinωt (92) ˆp(t) =−mωˆx(0)sinωt+ ˆp(0)cosωt . (93) Evolut ¸ia temporal˘ a a acestor operatori este aceea¸ si ca ˆ ın cazul ecuat ¸iilor clasice de mi¸ scare. Astfel, am ar˘ atat forma explicit˘ a de evolut ¸ie a patru ope ratori baziciˆ ın cazul OA , ar˘ atˆ ınd modul ˆ ın care se lucreaz˘ a ˆ ın reprezentarea Heisenberg. 99OA tridimensional La ˆ ınceputul analizei noastre a OA cuantic am f˘ acut coment arii ˆ ın leg˘ atur˘ a cu important ¸a pentru fizic˘ a a OA . Dac˘ a vom considera un ana log tridi- mensional, ar trebui s˘ a consider˘ am o dezvoltare Taylor ˆ ı n trei variabile7 ret ¸inˆ ınd termeni numai pˆ ın˘ a ˆ ın ordinul doi inclusiv, c eea ce obt ¸inem este o form˘ a cuadratic˘ a (ˆ ın cazul cel mai general). Problema de rezolvatˆ ın aceast˘ a aproximat ¸ie nu este chiar atˆ ıt de simpl˘ a cum ar p˘ area din tr-o prim˘ a exam- inare a potent ¸ialului corespunz˘ ator : V(x,y,z) =ax2+by2+cz2+dxy+exz+fyz . (94) Exist˘ a ˆ ıns˘ a multe sisteme care posed˘ a simetrie sferic˘ a sau pentru care aproximat ¸ia acestei simetrii este satisf˘ ac˘ atoare. ˆIn acest caz: V(x,y,z) =K(x2+y2+z2), (95) ceea ce este echivalent cu a spune c˘ a derivatele part ¸iale s ecunde ( nemixte ) iau toate aceea¸ si valoare ( ˆ ın cazul anterior reprezenta te prinK). Putem ad˘ auga c˘ a aceast˘ a este o bun˘ a aproximat ¸ieˆ ın cazulˆ ın care valorile derivatelor part ¸iale secunde mixte sunt mici ˆ ın comparat ¸ie cu cele ne mixte. Cˆ ınd se satisfac aceste condit ¸ii ¸ si potent ¸ialul este da t de (95) spunem c˘ a sistemul este un OA tridimensional sferic simetric . Hamiltonianul pentru acest caz este de forma: ˆH=−¯h2 2m▽2+mω2 2r2, (96) unde Laplaceanul este dat ˆ ın coordonate sferice ¸ si reste variabila sferic˘ a radial˘ a . Fiind vorba de un potent ¸ial independent de timp, energia se conserv˘ a ; ˆ ın plus dat˘ a simetria sferic˘ a , momentul cinetic deasemenea se conserv˘ a . Avem deci dou˘ a m˘ arimi conservate, ceea ce ne permite s˘ a spunem c˘ a fiec˘ areia ˆ ıi corespunde un num˘ ar cuantic. Putem s˘ a presupunem c˘ a func t ¸iile de und˘ a depind de dou˘ a numere cuantice (de¸ si ˆ ın acest caz vom vede a c˘ a apare ˆ ınc˘ a unul ). Cu aceste comentarii, ecuat ¸ia de interes este : 7Este posibil s˘ a se exprime dezvoltarea Taylorˆ ın jurul lui r0ca un operator exponent ¸ial : e[(x−xo)+(y−yo)+(z−zo)](∂ ∂x+∂ ∂y+∂ ∂z)f(ro). 100ˆHΨnl=EnlΨnl. (97) Laplaceanul ˆ ın coordonate sferice este : ▽2=∂2 ∂r2+2 r∂ ∂r−ˆL2 ¯h2r2(98) ¸ si rezult˘ a din faptul cunoscut : ˆL2=−¯h2[1 sinθ∂ ∂θ(sinθ∂ ∂θ) +1 sinθ2∂2 ∂ϕ2]. (99) Funct ¸iile proprii ale lui ˆL2sunt armonicele sferice , respectiv: ˆL2Ylml(θ,ϕ) =−¯h2l(l+ 1)Ylml(θ,ϕ) (100) Faptul c˘ a armonicele sferice poart˘ a num˘ arul cuantic mlface ca acesta s˘ a fie introdus ˆ ın funct ¸ia de und˘ a Ψ nlml. Pentru a realiza separarea variabilelor ¸ si funct ¸iilor se propune substitut ¸ia: Ψnlml(r,θ,ϕ) =Rnl(r) rYlml(θ,ϕ). (101) Odat˘ a introdus˘ a ˆ ın ecuat ¸ia Schr¨ odinger va separa part ea spat ¸ial˘ a de cea unghiular˘ a ; ultima se identific˘ a cu un operator proport ¸i onal cu operatorul moment cinetic p˘ atrat, pentru care funct ¸iile proprii sun t armonicele sferice, ˆ ın timp ce ˆ ın partea spat ¸ial˘ a obt ¸inem ecuat ¸ia : R′′ nl+ (2mEnl ¯h2−m2ω2 ¯h2r2−l(l+ 1) r2)Rnl(r) = 0. (102) Folosind definit ¸iile (7) ¸ si (8) , ecuat ¸ia anterioar˘ a ia e xact forma lui (9), cu except ¸ia termenului de moment unghiular, care ˆ ın mod comu n se cunoa¸ ste cabarier˘ a de moment unghiular . R′′ nl+ (k2−λ2r2−l(l+ 1) r2)Rnl= 0. (103) Pentru a rezolva aceast˘ a ecuat ¸ie , vom pleca dela analiza s a asimptotic˘ a . Dac˘ a vom considera mai ˆ ıntˆ ıi r→∞, observ˘ am c˘ a termenul de moment unghiular este neglijabil, astfel c˘ a ˆ ın aceast˘ a limit˘ a comportamentul asimp- totic este identic aceluia a lui (9), ceea ce ne conduce la: 101Rnl(r)∼exp−λr2 2ˆ ın limr→∞. (104) Dac˘ a studiem acum comportamentulˆ ın jurul lui zero, vedem c˘ a termenul dominant este cel de moment unghiular, adic˘ a , ecuat ¸ia dif erent ¸ial˘ a (102) se converte ˆ ın aceast˘ a limit˘ a ˆ ın : R′′ nl−l(l+ 1) r2Rnl= 0. (105) Aceasta este o ecuat ¸ie diferent ¸ial˘ a de tip Euler8, a c˘ arei rezolvare duce la dou˘ a solut ¸ii independente: Rnl(r)∼rl+1saur−lˆ ın limr→0. (106) Argumentele anterioare ne conduc la a propune substitut ¸ia : Rnl(r) =rl+1exp−λr2 2φ(r). (107) S-ar putea deasemenea face ¸ si urm˘ atoarea substitut ¸ie: Rnl(r) =r−lexp−λr2 2v(r), (108) care ˆ ıns˘ a ne conduce la acelea¸ si solut ¸ii ca ¸ si (107) ( de ar˘ atat acest lucru este un bun exercit ¸iu). Substituind (107) ˆ ın (103) , se obt ¸ine urm˘ atoarea ecuat ¸ie diferent ¸ial˘ a pentru φ: φ′′+ 2(l+ 1 r−λr)φ′−[λ(2l+ 3)−k2]φ= 0. (109) Cu schimbarea de variabil˘ a w=λr2, obt ¸inem: wφ′′+ (l+3 2−w)φ′−[1 2(l+3 2)−κ 2]φ= 0, (110) unde am introdus κ=k2 2λ=E ¯hω. Am ajuns din nou la o ecuat ¸ie diferent ¸ial˘ a de tip hipergeometric confluent˘ a cu solut ¸iile ( a se vedea ( 21) ¸ si (22)): φ(r) =A1F1[1 2(l+3 2−κ);l+3 2,λr2]+B r−(2l+1)1F1[1 2(−l+1 2−κ);−l+1 2,λr2]. (111) 8O ecuat ¸ie de tip Euler este de forma : xny(n)(x) +xn−1y(n−1)(x) +···+xy′(x) +y(x) = 0 . Solut ¸iile ei sunt de tipul xα, care se substituie ¸ si se g˘ ase¸ ste un polinom ˆ ın α. 102A doua solut ¸ie particular˘ a nu poate fi normalizat˘ a , pentr u c˘ a diverge puternic ˆ ın zero, ceea ce oblig˘ a a lua B= 0, deci : φ(r) =A1F1[1 2(l+3 2−κ);l+3 2,λr2]. (112) Folosind acelea¸ si argumente ca ˆ ın cazul OA unidimensiona l, respectiv, a impune ca solut ¸iile s˘ a fie regulare ˆ ın infinit, ˆ ınseamn˘ a condit ¸ia de truncare a seriei, ceea ce implic˘ a din nou cuantizarea energiei . Trun carea este ˆ ın acest caz:1 2(l+3 2−κ) =−n , (113) unde introducˆ ınd explicit κ, obt ¸inem spectrul de energie : Enl= ¯hω(2n+l+3 2). (114) Putem observa c˘ a pentru OA tridimensional sferic simetric exist˘ a o en- ergie de punct zero3 2¯hω. Funct ¸iile proprii nenormalizate sunt: Ψnlm(r,θ,ϕ) =rle−λr2 21F1(−n;l+3 2,λr2)Ylm(θ,ϕ). (115) 5P. Probleme Problema 5.1 S˘ a se determine autovalorile ¸ si funct ¸iile proprii ale OA ˆ ın spat ¸iul impulsurilor . Hamiltonianul cuantic de OA este: ˆH=ˆp2 2m+1 2mω2ˆx2. ˆIn spat ¸iul impulsurilor, operatorii ˆ x¸ si ˆpau urm˘ atoarea form˘ a : ˆp→p ˆx→i¯h∂ ∂p. Prin urmare Hamiltonianul cuantic OA ˆ ın reprezentarea de i mpuls este : ˆH=p2 2m−1 2mω2¯h2d2 dp2. 103Avem de rezolvat problema de autovalori ( ceea ce ˆ ınseamn˘ a de obt ¸inut funct ¸iile proprii ¸ si autovalorile) dat˘ a prin (5) , care c u Hamiltonianul ante- rior, este urm˘ atoarea ecuat ¸ie diferent ¸ial˘ a : d2Ψ(p) dp2+ (2E m¯h2ω2−p2 m2¯h2ω2)Ψ(p) = 0. (116) Se poate observa c˘ a aceast˘ a ecuat ¸ie diferent ¸ial˘ a , est e identic˘ a , pˆ ın˘ a la constante, cu ecuat ¸ia diferent ¸ial˘ a din spat ¸iul configu rat ¸iilor ( ec. (6) ). Pentru a exemplifica o alt˘ a form˘ a de a o rezolva, nu vom urma e xact acela¸ si drum. Definim doi parametri, analogi celor din (7) ¸ si (8): k2=2E m¯h2ω2λ=1 m¯hω. (117) Cu aceste definit ¸ii, ajungem la ecuat ¸ia diferent ¸ial˘ a (9 )¸ si deci solut ¸ia c˘ autat˘ a ( ˆ ın urma efectu˘ arii analizei asimptotice ) este de forma: Ψ(y) =e−1 2yφ(y), (118) undeyeste dat de y=λp2¸ siλeste definit ˆ ın (117). Substituind (118) ˆ ın (116) , avˆ ınd grij˘ a s˘ a punem (118) ˆ ın variabila p. Se obt ¸ine astfel o ecuat ¸ie diferent ¸ial˘ a pentru φ: d2φ(p) dp2−2λpdφ(p) dp+ (k2−λ)φ(p) = 0. (119) Vom face acum schimbul de variabil˘ a u=√ λp, care ne conduce la ecuat ¸ia Hermite : d2φ(u) du2−2udφ(u) du+ 2nφ(u) = 0, (120) cunun ˆ ıntreg nenegativ , ¸ si unde am pus : k2 λ−1 = 2n . De aici ¸ si din definit ¸iile dateˆ ın (117) rezult˘ a c˘ a autov alorile de energie sunt date de : En= ¯hω(n+1 2). Solut ¸iile pentru (120) sunt polinoamele Hermite φ(u) =Hn(u) ¸ si funct ¸iile proprii nenormalizate sunt : Ψ(p) =Ae−λ 2p2Hn(√ λp). 104Problema 5.2 S˘ a se demonstreze c˘ a polinoamele Hermite pot fi expresate ˆ ın urm˘ atoarea reprezentare integral˘ a : Hn(x) =2n √π/integraldisplay∞ −∞(x+iy)ne−y2dy . (121) Aceast˘ a reprezentare a polinoamelor Hermite nu este foart e uzual˘ a , de¸ si se poate dovedi util˘ a ˆ ın unele cazuri. Ceea ce vom face pent ru a realiza demonstrat ¸ia , este s˘ a dezvolt˘ am integrala ¸ si s˘ a demon str˘ am c˘ a ceea ce se obt ¸ine este identic cu reprezentarea ˆ ın serie a polinoame lor Hermite pentru care avem : [n 2]/summationdisplay k=0(−1)kn! (n−2k)!k!(2x)n−2k, (122) unde simbolul [ c] unde se termin˘ a seria signific˘ a cel mai mare ˆ ıntreg mai mic sau egal cu c. Primul lucru pe care ˆ ıl vom face este s˘ a dezvolt˘ am binomul din integral˘ a folosind binecunoscuta teorem˘ a a binomului: (x+y)n=n/summationdisplay m=0n! (n−m)!m!xn−mym. Astfel: (x+iy)n=n/summationdisplay m=0n! (n−m)!m!imxn−mym, (123) care substituit ˆ ın integral˘ a duce la: 2n √πn/summationdisplay m=0n! (n−m)!m!imxn−m/integraldisplay∞ −∞yme−y2dy . (124) Din forma expresiei din integral˘ a putem s˘ a vedem c˘ a este d iferit˘ a de zero cˆ ındmeste par , ˆ ın cazul impar integrala se anuleaz˘ a din motive e vidente. Folosind notat ¸ia par˘ a m= 2kavem: 2n √π[n 2]/summationdisplay k=0n! (n−2k)!(2k)!i2kxn−2k2/integraldisplay∞ 0y2ke−y2dy . (125) 105Cu schimbul de variabil˘ a u=y2, integrala devine o funct ¸ie gamma : 2n √π[n 2]/summationdisplay k=0n! (n−2k)!(2k)!i2kxn−2k/integraldisplay∞ 0uk−1 2e−udu , (126) respectiv Γ( k+1 2) , care ˆ ın plus se poate exprima prin factoriali ( desigur pentrukˆ ıntreg ) : Γ(k+1 2) =(2k)! 22kk!√π . Substituind aceast˘ a expresie ˆ ın sum˘ a ¸ si folosind faptu l c˘ ai2k= (−1)kse obt ¸ine [n 2]/summationdisplay k=0(−1)kn! (n−2k)!k!(2x)n−2k, (127) care este identic cu (122) , ceea ce completeaz˘ a demonstrat ¸ia. Problema 5.3 S˘ a se arate c˘ a relat ¸ia de incertitudine Heisenberg se sat isface efectuˆ ınd calculul cu funct ¸iile proprii ale OA . Trebuie s˘ a ar˘ at˘ am c˘ a pentru oricare Ψ nse satisface: <(∆p)2(∆x)2>≥¯h2 4, (128) unde notat ¸ia <>ˆ ınseamn˘ a valoare medie. Vom calcula ˆ ın mod separat <(∆p)2>¸ si<(∆x)2>, unde fiecare din aceste expresii este : <(∆p)2>=<(p−<p> )2>=<p2−2p<p> +<p>2>=<p2>−<p>2, <(∆x)2>=<(x−<x> )2>=<x2−2x<x> +<x>2>=<x2>−<x>2. Mai ˆ ıntˆ ıi vom ar˘ ata c˘ a atˆ ıt media lui xcˆ ıt ¸ si a lui psunt zero. Pentru media luixavem: <x> =/integraldisplay∞ −∞x[Ψn(x)]2dx . 106Aceast˘ a integral˘ a se anuleaz˘ a datorit˘ a imparit˘ at ¸ti i expresiei de integrat, care este manifest˘ a . Rezult˘ a deci c˘ a : <x> = 0. (129) Acelea¸ si argumente sunt corecte pentru media lui p, dac˘ a efectu˘ am calculul ˆ ın spat ¸iul impulsurilor, respectiv cu ajutorul funct ¸iilor obt ¸inute ˆ ın problema 1 . Este suficient s˘ a vedem c˘ a forma funt ¸ional˘ a e ste acea¸ si (se schimb˘ a doar simbolul). Deci: <p> = 0. (130) S˘ a calcul˘ am acum media lui x2. Vom folosi teorema virialului9. Ob- serv˘ am mai ˆ ıntˆ ıi c˘ a : <V > =1 2mω2<x2> . Prin urmare este posibil˘ a relat ¸ionarea mediei lui x2direct cu media potent ¸ialului ˆ ın acest caz (¸ si deci folosirea teoremei virialului). <x2>=2 mω2<V > . (131) Avem nevoie deasemenea de media energiei totale : <H > =<T > +<V > , pentru care din nou se poate folosi teorema virialului ( pent run= 2 ) : <H > = 2<V > . (132) Astfel, se obt ¸ine: <x2>=<H > mω2=¯hω(n+1 2) mω2(133) 9Amintim c˘ a teorema virialului ˆ ın mecanica cuantic˘ a afirm ˘ a c˘ a : 2< T > =<r· ▽V(r)> . Pentru un potent ¸ial de forma V=λxnse satisface: 2< T > =n < V > , unde Treprezint˘ a energia cinetic˘ a ¸ si V este energia potent ¸ia l˘ a . 107<x2>=¯h mω(n+1 2). (134) Similar, media lui p2se poate calcula explicit: <p2>= 2m<p2 2m>= 2m<T > =m<H > =m¯hω(n+1 2).(135) Cu (133) ¸ si (135) avem: <(∆p)2(∆x)2>= (n+1 2)2¯h2. (136) Pe baza acestui rezultat ajungem la concluzia c˘ a ˆ ın st˘ ari le stat ¸ionare ale OA, care practic nu au fost folosite ˆ ın mod direct, relat ¸ia de incertitudine Heisenberg se satisface ¸ si are valoarea minim˘ a pentru sta rea fundamental˘ a n= 0. Problema 5.4 S˘ a se obt ¸in˘ a elementele de matrice ale operatorilor a,a†,ˆx¸ siˆp. S˘ a g˘ asim mai ˆ ıntˆ ıi elementele de matrice pentru operato rii de creat ¸ie ¸ si anihilare, care sunt de mult ajutor pentru a obt ¸ine elem ente de matrice pentru restul operatorilor. Vom folosi relat ¸iile (65) ¸ si (66), care duc la: <m|a|n>=√n<m|n−1>=√nδm,n−1. (137) ˆIn mod similar pentru operatorul de creat ¸ie avem rezultatu l: <m|a†|n>=√ n+ 1<m|n+ 1>=√ n+ 1δm,n+1. (138) S˘ a trecem acum la calculul elementelor de matrice ale opera torului de pozit ¸ie. Pentru al efectua , s˘ a exprim˘ am operatorul de po sit ¸ie ˆ ın funct ¸ie de operatorii de creat ¸ie ¸ si anihilare. Folosind definit ¸i ile (39) ¸ si (40) , se demonstreaz˘ a imediat c˘ a operatorul de posit ¸ie este dat d e : ˆx=/radicaligg ¯h 2mω(a+a†). (139) 108Folosind acest rezultat, elementele de matrice ale operato rului ˆxpot fi calculate ˆ ın manier˘ a imediat˘ a : <m|ˆx|n> =<m|/radicaligg ¯h 2mω(a+a†)|n> =/radicaligg ¯h 2mω[√nδm,n−1+√ n+ 1δm,n+1].(140) Urmˆ ınd acela¸ si procedeu putem calcula elementele de matr ice ale opera- torului impuls, considerˆ ınd c˘ a ˆ peste dat ˆ ın funct ¸ie de operatorii de creat ¸ie ¸ si anihilare ˆ ın forma : ˆp=i/radicaligg m¯hω 2(a†−a), (141) ceea ce ne conduce la: <m|ˆp|n>=i/radicaligg m¯hω 2[√ n+ 1δm,n+1−√nδm,n−1]. (142) Se poate vedea u¸ surint ¸a cu care se pot face calculele dac˘ a se folosesc elementele de matrice ale operatorilor de creat ¸ie ¸ si anih ilare. Finaliz˘ am cu o observat ¸ie ˆ ın leg˘ atur˘ a cu nediagonalitatea elementel or de matrice obt ¸inute. Aceasta este de a¸ steptat datorit˘ a faptului c˘ a reprezent area folosit˘ a este cea a operatorului de num˘ ar ¸ si nici unul dintre cei patru opera tori nu comut˘ a cu el. Problema 5.5 S˘ a se g˘ aseasc˘ a valorile medii ale lui ˆx2¸ siˆp2pentru OA unidimen- sional ¸ si s˘ a se foloseasc˘ a acestea pentru calculul valor ilor medii (de a¸ steptare) ale energiei cinetice ¸ si celei potent ¸iale. S ˘ a se compare acest ultim rezultat cu teorema virialului. Mai ˆ ıntˆ ıi s˘ a obt ¸inem valoarea medie a lui ˆ x2. Pentru aceasta recurgem la expresia (139), care ne conduce la : ˆx2=¯h 2mω(a2+ (a†)2+a†a+aa†). (143) Se aminte¸ ste c˘ a operatorii de creat ¸ie ¸ si anihilare nu co mut˘ a ˆ ıntre ei . Avˆ ınd (143) putem calcula valoarea medie a lui ˆ x2: <ˆx2>=<n|ˆx2|n> 109=¯h 2mω[/radicalig n(n−1)δn,n−2+/radicalig (n+ 1)(n+ 2)δn,n+2 +nδn,n+ (n+ 1)δn,n], (144) ceea ce arat˘ a c˘ a : <ˆx2>=<n|ˆx2|n>=¯h 2mω(2n+ 1). (145) Pentru calcularea valorii medii a lui ˆ p2folosim (141) pentru a exprima acest operator ˆ ın funct ¸ie de operatorii de creat ¸ie ¸ si anihila re: ˆp2=−m¯hω 2(a2+ (a†)2−aa†−a†a), (146) ceea ce ne conduce la: <ˆp2>=<n|ˆp2|n>=m¯hω 2(2n+ 1). (147) Ultimul rezultat ne d˘ a practic media energiei cinetice : <ˆT >=<ˆp2 2m>=1 2m<ˆp2>=¯hω 4(2n+ 1). (148) Valoarea medie a energiei potent ¸iale este: <ˆV >=<1 2mω2ˆx2>=1 2mω2<ˆx2>=¯hω 4(2n+ 1), (149) unde s-a folosit (145). Observ˘ am c˘ a aceste valori medii coincid pentru orice n, ceea ce este ˆ ın conformitate cu teorema virialului, care ne spune c˘ a pentr u un potent ¸ial cuadratic ca cel de OA, valorile medii ale energiei cinetice ¸ si potent ¸iale tre- buie s˘ a coincid˘ a ¸ si deci s˘ a fie jum˘ atate din valoarea med ie a energiei totale. 1106. ATOMUL DE HIDROGEN Introducere ˆIn acest capitol vom studia atomul de hidrogen, rezolvˆ ınd e cuat ¸ia Schr¨ odinger independent˘ a de timp cu un potent ¸ial produs de dou˘ a parti cule ˆ ınc˘ arcate electric cum este cazul electronului ¸ si protonului, cu Lap laceanul ˆ ın coordo- nate sferice. Din punct de vedere matematic, se va folosi met oda separ˘ arii de variabile, dˆ ınd o interpretare fizic˘ a funct ¸iei de und˘ a ca solut ¸ie a ecuat ¸iei Schr¨ odinger pentru acest caz important, odat˘ a cu interpr etarea numerelor cuantice ¸ si a densit˘ at ¸ilor de probabilitate. Scala spat ¸ial˘ a foarte mic˘ a a atomului de hidrogen intr˘ a ˆ ın domeniul de apli- cabilitate al mecanicii cuantice, pentru care fenomenele a tomice au fost o arie de verificare ¸ si interpretare a rezultatelor ˆ ınc˘ a d e la bun ˆ ınceput. Cum mecanica cuantic˘ a d˘ a , ˆ ıntre altele, relat ¸ii ˆ ıntre m˘ arimile observabile ¸ si cum principiul de incertitudine modific˘ a radical definit ¸i a teoretic˘ a a unei “observabile” este important s˘ a ˆ ınt ¸elegem ˆ ın mod cˆ ıt m ai clar not ¸iunea cuantic˘ a de observabil˘ a ˆ ın cˆ ımpul atomic. De acord cu pr incipiul de in- certitudine, pozit ¸ia ¸ si impulsul unei particule nu se pot m˘ asura simultan sub o anumit˘ a precizie impus˘ a de comutatorii cuantici. De fapt, m˘ arimile asupra c˘ arora mecanica cuantic˘ a d˘ a rezultate ¸ si pe care le relat ¸ioneaz˘ a sunt ˆ ıntotdeauna probabilit˘ at ¸i. ˆIn loc de a afirma, de exemplu, c˘ a raza orbitei electronului ˆ ıntr-o stare fundamental˘ a a atomul ui de hidrogen este ˆ ıntotdeauna 5 .3×10−11m, mecanica cuantic˘ a afirm˘ a c˘ a aceasta este doar raza medie; dac˘ a efectu˘ am un experiment adecuat, vom obt ¸ ine exact ca ˆ ın cazul experimentelor cu detectori macroscopici pe prob e macroscopice diferite valori aleatorii dar a c˘ aror medie va fi 5 .3×10−11m. A¸ sadar, din punctul de vedere al erorilor experimentale nu exist˘ a nici o diferent ¸˘ a fat ¸˘ a de fizica clasic˘ a . Dup˘ a cum se ¸ stie, pentru calculul valorilor medii ˆ ın meca nica cuantic˘ a este necesar˘ a o funct ¸ie de und˘ a corespunz˘ atoare Ψ. De¸ s i Ψ nu are o inter- pretare fizic˘ a direct˘ a , modulul p˘ atrat |Ψ|2calculat ˆ ıntr-un punct arbitrar din spat ¸iu ¸ si la un moment dat este proport ¸ional cu probab ilitatea de a g˘ asi particula ˆ ıntr-o vecin˘ atate infinitezimal˘ a a acel ui punct acel loc ¸ si la momentul dat. Scopul mecanicii cuantice este determinarea lui Ψ pentru o microparticul˘ a ˆ ın diferite condit ¸ii experimentale. ˆInainte de a trece la calculul efectiv al lui Ψ pentru cazul el ectronului hidrogenic, trebuie s˘ a stabilim unele rechizite generale (care trebuie s˘ a se 111respecte ˆ ın orice situat ¸ie). ˆIn primul rˆ ınd, pentru c˘ a |Ψ|2este proport ¸ional cu probabilitatea P de a g˘ asi particula descris˘ a prin Ψ, in tegrala|Ψ|2pe tot spat ¸iul trebuie s˘ a fie finit˘ a , pentru ca ˆ ıntr-adev˘ ar particula s˘ a poat˘ a fi localizat˘ a . Deasemenea, dac˘ a /integraldisplay∞ −∞|Ψ|2dV= 0 (1) particula nu exist˘ a , iar dac˘ a integrala este ∞nu putem avea semnificat ¸ie fizic˘ a ;|Ψ|2nu poate fi negativ˘ a sau complex˘ a din simple motive matemat - ice, astfel c˘ a unica posibilitate r˘ amˆ ıne ca integrala s˘ a fie finit˘ a pentru a avea o descriere acceptabil˘ a a unei particule reale. ˆIn general, este convenabil de a identifica|Ψ|2cu probabilitatea P de a g˘ asi particula descris˘ a de c˘ atre Ψ ¸ si nu doar simpla proport ¸ionalitate cu P. Pentru ca |Ψ|2s˘ a fie egal˘ a cu P se impune/integraldisplay∞ −∞|Ψ|2dV= 1, (2) pentru c˘ a/integraldisplay∞ −∞PdV= 1 (3) este afirmat ¸ia matematic˘ a a faptului c˘ a particula exist˘ aˆ ıntr-un loc din spat ¸iu la orice moment. O funct ¸ie care respect˘ a ec. 2 se spune c˘ a e ste normalizat˘ a . Pe lˆ ıng˘ a aceast˘ a condit ¸ie fundamental˘ a , Ψ trebuie s˘ a aib˘ a o valoare unic˘ a , pentru c˘ a P are o singur˘ a valoare ˆ ıntr-un loc ¸ si la un mome nt determinat. O alt˘ a condit ¸ie pe care Ψ trebuie s˘ a o satisfac˘ a este c˘ a at ˆ ıt ea cˆ ıt ¸ si derivatele sale part ¸iale∂Ψ ∂x,∂Ψ ∂y,∂Ψ ∂ztrebuie s˘ a fie continue ˆ ın orice punct arbitrar. Ecuat ¸ia Schr¨ odinger este considerat˘ a ecuat ¸ia fundame ntal˘ a a mecanicii cuantice ˆ ın acela¸ si sens ˆ ın care legea fort ¸ei este ecuat ¸ia fundamental˘ a a mecanicii newtoniene cu deosebirea important˘ a c˘ a este o e cuat ¸ie de und˘ a pentru Ψ. Odat˘ a ce energia potent ¸ial˘ a este cunoscut˘ a , se poate re zolva ecuat ¸ia Schr¨ odinger pentru funct ¸ia de und˘ a Ψ a particulei, a c˘ ar ei densitate de probabilitate|Ψ|2se poate determina pentru x,y,z,t .ˆIn multe situat ¸ii, energia potent ¸ial˘ a a unei particule nu depinde explicit d e timp; fort ¸ele care act ¸ioneaz˘ a asupra ei se schimb˘ a ˆ ın funct ¸ie numai de pos it ¸ia particulei. ˆIn aceste condit ¸ii, ecuat ¸ia Schr¨ odinger se poate simplific a eliminˆ ınd tot ce se refer˘ a lat. S˘ a not˘ am c˘ a se poate scrie funct ¸ia de und˘ a unidimensio nal˘ a a unei particule libere ˆ ın forma Ψ(x,t) =Ae(−i/¯h)(Et−px) 112=Ae−(iE/¯h)te(ip/¯h)x =ψ(x)e−(iE/¯h)t. (4) Ψ(x,t) este produsul ˆ ıntre o funct ¸ie dependent˘ a de timp e−(iE/¯h)t¸ si una stat ¸ionar˘ a , dependent˘ a numai de pozit ¸ie ψ(x). ˆIn cazul general ˆ ıns˘ a , ecuat ¸ia Schr¨ odinger pentru o sta re stat ¸ionar˘ a se poate rezolva numai pentru anumite valori ale energiei E. Nu este vorba de dificult˘ at ¸i matematice, ci de un aspect fundamental. “A re zolva” ecuat ¸ia Schr¨ odinger pentru un sistem dat ˆ ınseamn˘ a a obt ¸ine o fun ct ¸ie de und˘ a ψ care nu numai c˘ a satisface ecuat ¸ia ¸ si condit ¸iile de fron tier˘ a impuse, ci este o funct ¸ie de und˘ a acceptabil˘ a , respectiv, funct ¸ia ¸ si de rivata sa s˘ a fie continue, finite ¸ si univoce. Astfel, cuantizarea energiei apare ˆ ın m ecanica ondulatorie ca un element teoretic natural, iar ˆ ın practic˘ a ca un fenom en universal, caracteristic tuturor sistemelor microscopice stabile . Ecuat ¸ia Schr¨ odinger pentru atomul de hidrogen ˆIn continuare, vom aplica ecuat ¸ia Schr¨ odinger atomului d e hidrogen, despre care se ¸ stia pe baza experimentelor Rutherford c˘ a este for mat dintr-un pro- ton, particul˘ a cu sarcina electric˘ a + e¸ si un electron de sarcin˘ a - e¸ si care fiind de 1836 de ori mai u¸ sor decˆ ıt protonul este cu mult mai mobil . Dac˘ a interact ¸iuneaˆ ıntre dou˘ a particule este de tipul u(r) =u(|/vector r1−/vector r2|), problema de mi¸ scare se reduce atˆ ıt clasic cˆ ıt ¸ si cuantic la mi¸ scarea unei singure particule ˆ ın cˆ ımpul de simetrie sferic˘ a . ˆIntr-adev˘ ar Lagrangeanul: L=1 2m1˙/vector r2 1+1 2m2˙/vector r2 2−u(|/vector r1−/vector r2|) (5) se transform˘ a folosind: /vector r=/vector r1−/vector r2 (6) ¸ si /vectorR=m1/vector r1+m2/vector r2 m1+m2, (7) ˆ ın Lagrangeanul: L=1 2M˙/vectorR2+1 2µ˙/vector r2−u(r), (8) unde M=m1+m2 (9) 113¸ si µ=m1m2 m1+m2. (10) Pe de alt˘ a parte, introducerea impulsului se face cu formul ele Lagrange /vectorP=∂L ∂˙/vectorR=M˙/vectorR (11) ¸ si /vector p=∂L ∂˙/vector r=m˙/vector r , (12) ceea ce permite scrierea funct ¸iei Hamilton clasice H=P2 2M+p2 2m+u(r). (13) Astfel, se poate obt ¸ine operatorul hamiltonian pentru pro blema core- spunz˘ atoare cuantic˘ a cu comutatori de tipul [Pi,Pk] =−i¯hδik (14) ¸ si [pi,pk] =−i¯hδik. (15) Ace¸ sti comutatori implic˘ a un operator Hamiltonian de for ma ˆH=−¯h2 2M∇2 R−¯h2 2m∇2 r+u(r), (16) care este fundamental pentru studiul atomului de hidrogen c u ajutorul ecuat ¸iei Schr¨ odinger ˆ ın forma stat ¸ionar˘ a ˆHψ=Eψ , (17) ceea ce presupune c˘ a nu se iau ˆ ın considerare efecte relati viste (viteze apro- priate de cele ale luminii ˆ ın vid). Energia potent ¸ial˘ a u(r) este cea electrostatic˘ a u=−e2 4πǫ0r(18) Exist˘ a dou˘ a posibilit˘ at ¸i: prima, de a exprima uˆ ın funct ¸ie de coordo- natele carteziene x,y,z substituind rprin/radicalbig x2+y2+z2, a doua, de a ex- prima ecuat ¸ia Schr¨ odinger ˆ ın funct ¸ie de coordonatele p olare sferice r,θ,φ. 114ˆIn virtutea simetriei sferice a situat ¸iei fizice, vom trata ultimul caz pentru c˘ a problema matematic˘ a se simplific˘ a considerabil. Prin urmare, ˆ ın coordonate polare sferice, ecuat ¸ia Schr¨ odinger este 1 r2∂ ∂r/parenleftbigg r2∂ψ ∂r/parenrightbigg +1 r2sinθ∂ ∂θ/parenleftbigg sinθ∂ψ ∂θ/parenrightbigg +1 r2sin2θ∂2ψ ∂φ2+2m ¯h2(E−u)ψ= 0 (19) Substituind (18) ¸ si multiplicˆ ınd toat˘ a ecuat ¸ia cu r2sin2θ, se obt ¸ine sin2θ∂ ∂r/parenleftbigg r2∂ψ ∂r/parenrightbigg +sinθ∂ ∂θ/parenleftbigg sinθ∂ψ ∂θ/parenrightbigg +∂2ψ ∂φ2+2mr2sin2θ ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg ψ= 0. (20) Aceast˘ a ecuat ¸ie este ecuat ¸ia diferent ¸ial˘ a cu derivat e part ¸iale pentru funct ¸ia de und˘ aψ(r,θ,φ) a electronuluiˆ ın atomul de hidrogen. ˆImpreun˘ a cu diferitele condit ¸ii pe care ψ(r,θ,φ) trebuie s˘ a le ˆ ındeplineasc˘ a [de exemplu, ψ(r,θ,φ) trebuie s˘ a aib˘ a o valoare unic˘ a pentru fiecare punct spat ¸ ial (r,θ,φ)], aceast˘ a ecuat ¸ie specific˘ a de manier˘ a complet˘ a comportamentul e lectronului hidro- genic. Pentru a vedea care este acest comportament, vom rezo lva ec. 20 pentruψ(r,θ,φ) ¸ si vom interpreta rezultatele obt ¸inute. Separarea de variabile ˆ ın coordonate sferice Ceea ce este cu adev˘ arat util ˆ ın scrierea ecuat ¸iei Schr¨ o dinger ˆ ın coordonate sferice pentru atomul de hidrogen const˘ a ˆ ın faptul c˘ a ast fel se poate re- aliza u¸ sor separarea ˆ ın trei ecuat ¸ii independente, fieca re unidimensional˘ a . Procedeul de separare este de a c˘ auta solut ¸iile pentru car e funct ¸ia de und˘ a ψ(r,θ,φ) are forma unui produs de trei funct ¸ii, fiecare ˆ ıntr-una di n cele trei variabile sferice: R(r), care depinde numai de r; Θ(θ) care depinde numai de θ; ¸ si Φ(φ) care depinde numai de φ¸ si este practic analog separ˘ arii ecuat ¸iei Laplace. Deci ψ(r,θ,φ) =R(r)Θ(θ)Φ(φ). (21) Funct ¸iaR(r) descrie variat ¸ia funct ¸iei de und˘ a ψa electronului de-a lungul razei vectoare dinspre nucleu, cu θ¸ siφconstante. Variat ¸ia lui ψcu unghiul zenitalθde-a lungul unui meridian al unei sfere centrat˘ a ˆ ın nucleu este descris˘ a numai de c˘ atre funct ¸ia Θ( θ) pentrur¸ siφconstante. ˆIn sfˆ ır¸ sit, funct ¸ia Φ(φ) descrie cum variaz˘ a ψcu unghiul azimutal φde-a lungul unei paralele a unei sfere centrat˘ a ˆ ın nucleu, ˆ ın condit ¸iile ˆ ın carer¸ siθsunt ment ¸inute constante. 115Folosindψ=RΘΦ, vedem c˘ a ∂ψ ∂r= ΘΦdR dr, (22) ∂ψ ∂θ=RΦdΘ dθ, (23) ∂ψ ∂φ=RΘdΦ dφ. (24) Evident, acela¸ si tip de formule se ment ¸ine pentru derivat ele de ordin superior nemixte. Subtituindu-le ˆ ın ec. 20, dup˘ a ˆ ımp˘ art ¸irea cu RΘΦ se obt ¸ine sin2θ Rd dr/parenleftbigg r2dR dr/parenrightbigg +sinθ Θd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg +1 Φd2Φ dφ2+2mr2sin2θ ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg = 0. (25) Al treilea termen al acestei ecuat ¸ii este funct ¸ie numai de unghiulφ, ˆ ın timp ce ceilalt ¸i doi sunt funct ¸ii de r¸ siθ. Rescriem ecuat ¸ia anterioar˘ a ˆ ın forma sin2θ R∂ ∂r/parenleftbigg r2∂R ∂r/parenrightbigg +sinθ Θ∂ ∂θ/parenleftbigg sinθ∂Θ ∂θ/parenrightbigg +2mr2sin2θ ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg =−1 Φ∂2Φ ∂φ2. (26) Aceast˘ a ecuat ¸ie poate fi corect˘ a numai dac˘ a cei doi membr i sunt egali cu aceea¸ si constant˘ a , pentru c˘ a sunt funct ¸ii de variabile diferite. Este conven- abil s˘ a not˘ am aceast˘ a constant˘ a cu m2 l. Ecuat ¸ia diferent ¸ial˘ a pentru funct ¸ia Φ este −1 Φ∂2Φ ∂φ2=m2 l. (27) Dac˘ a se subtituie m2 lˆ ın partea dreapt˘ a a ec. 26 ¸ si se divide ecuat ¸ia rezultant ˘ a cu sin2θ, dup˘ a o regrupare a termenilor, se obt ¸ine 1 Rd dr/parenleftbigg r2dR dr/parenrightbigg +2mr2 ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg =m2 l sin2θ−1 Θ sinθd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg . (28) ˆInc˘ a odat˘ a se prezint˘ a o ecuat ¸ie ˆ ın care apar variabile diferite ˆ ın fiecare membru, ceea ce oblig˘ a la egalarea ambilor cu aceea¸ si cons tant˘ a . Din motive care se vor vedea mai tˆ ırziu, vom nota aceast˘ a const ant˘ a prinl(l+1). Ecuat ¸iile pentru funct ¸iile Θ( θ) ¸ siR(r) sunt m2 l sin2θ−1 Θ sinθd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg =l(l+ 1) (29) 116¸ si 1 Rd dr/parenleftbigg r2dR dr/parenrightbigg +2mr2 ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg =l(l+ 1). (30) Ecuat ¸iile 27, 29 ¸ si 30 se scriu ˆ ın mod normal ˆ ın forma d2Φ dφ2+m2 lΦ = 0, (31) 1 sinθd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg +/bracketleftigg l(l+ 1)−m2 l sin2θ/bracketrightigg Θ = 0, (32) 1 r2d dr/parenleftbigg r2dR dr/parenrightbigg +/bracketleftigg 2m ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg −l(l+ 1) r2/bracketrightigg R= 0. (33) Fiecare dintre aceste ecuat ¸ii este o ecuat ¸ie diferent ¸ia l˘ a ordinar˘ a pentru o funct ¸ie de o singur˘ a variabil˘ a . ˆIn felul acesta s-a reu¸ sit simplificarea ecuat ¸iei Schr¨ odinger pentru atomul de hidrogen care, ini t ¸ial, era o ecuat ¸ie diferent ¸ial˘ a part ¸ial˘ a pentru o funct ¸ie ψde trei variabile. Interpretarea constantelor de separare: numere cuan- tice Solut ¸ia pentru partea azimutal˘ a Ec. 31 se rezolv˘ a u¸ sor pentru a g˘ asi urm˘ atoarea solut ¸ie Φ(φ) =Aφeimlφ, (34) undeAφeste constanta de integrare. Una dintre condit ¸iile stabil ite mai ˆ ınainte pe care trebuie s˘ a le ˆ ındeplineasc˘ a o funct ¸ie d e und˘ a (¸ si prin urmare deasemenea Φ, care este o component˘ a a funct ¸iei complete ψ) este s˘ a aib˘ a o valoare unic˘ a pentru fiecare punct din spat ¸iu f˘ ar˘ a exce pt ¸ie. De exemplu, se observ˘ a c˘ a φ¸ siφ+ 2πse identific˘ a ˆ ın acela¸ si plan meridian. De aceea, trebuie ca Φ( φ) = Φ(φ+ 2π), adic˘ aAeimlφ=Aeiml(φ+2π). Aceasta se poate ˆ ındeplini numai cˆ ınd mleste 0 sau un num˘ ar ˆ ıntreg pozitiv sau negativ (±1,±2,±3,...). Acest num˘ ar mlse cunoa¸ ste ca num˘ arul cuantic magnetic al atomului de hidrogen ¸ si este relat ¸ionat cu direct ¸ia mo mentului cinetic Lpentru c˘ a s-a putut fi asociat cu efectele cˆ ımpurilor magne tice axiale asupra electronului. Num˘ arul cuantic magnetic mleste determinat de c˘ atre 117num˘ arul cuantic orbital l, care la rˆ ındul s˘ au determin˘ a modulul momentului cinetic al electronului. Interpretarea num˘ arului cuantic orbital lnu este nici ea f˘ ar˘ a unele prob- leme. S˘ a examin˘ am ec. 33, care corespunde p˘ art ¸ii radial eR(r) a funct ¸iei de und˘ aψ. Aceast˘ a ecuat ¸ie este relat ¸ionat˘ a numai cu aspectul ra dial al mi¸ sc˘ arii electronilor, adic˘ a , cu apropierea ¸ si dep˘ ar tarea de nucleu (pentru elipse); totu¸ si, este prezent˘ a ¸ si energia total˘ a a elec tronuluiE. Aceast˘ a en- ergie include energia cinetic˘ a a electronului ˆ ın mi¸ scar e orbital˘ a care nu are nimic de-a face cu mi¸ scarea radial˘ a . Aceast˘ a contradict ¸ie se poate elimina cu urm˘ atorul rat ¸ionament: energia cinetic˘ a Ta electronului are dou˘ a p˘ art ¸i: Tradial datorat˘ a mi¸ sc˘ arii de apropiere ¸ si dep˘ artare de nucleu , ¸ siTorbital da- torat˘ a mi¸ sc˘ arii ˆ ın jurul nucleului. Energia potent ¸ia l˘ aVa electronului este energia electrostatic˘ a . Prin urmare, energia sa total˘ a e ste E=Tradial+Torbital−e2 4πǫ0r. (35) Substituind aceast˘ a expresie a lui Eˆ ın ec. 33 obt ¸inem, dup˘ a o regrupare a termenilor, 1 r2d dr/parenleftbigg r2dR dr/parenrightbigg +2m ¯h2/bracketleftigg Tradial+Torbital−¯h2l(l+ 1) 2mr2/bracketrightigg R= 0. (36) Dac˘ a ultimii doi termeni din paranteze se anuleaz˘ aˆ ıntre ei, obt ¸inem o ecuat ¸ie diferent ¸ial˘ a pentru mi¸ scarea pur radial˘ a . Impunem dec i condit ¸ia Torbital =¯h2l(l+ 1) 2mr2. (37) Energia cinetic˘ a orbital˘ a a electronului este ˆ ıns˘ a Torbital =1 2mv2 orbital (38) ¸ si cum momentul cinetic Lal electronului este L=mvorbitalr , (39) putem exprima energia cinetic˘ a orbital˘ a ˆ ın forma Torbital =L2 2mr2(40) 118De aceea avem L2 2mr2=¯h2l(l+ 1) 2mr2(41) ¸ si deci L=/radicalig l(l+ 1)¯h . (42) Interpretarea acestui rezultat este c˘ a , ˆ ıntrucˆ ıt num˘ a rul cuantic orbital l este limitat la valorile l= 0,1,2,...,(n−1), electronul poate avea numai momentele cinetice Lcare se specific˘ a prin intermediul ec. 42. Ca ¸ siˆ ın cazul energiei totale E, momentul cinetic se conserv˘ a ¸ si este cuantizat, iar unit atea sa natural˘ a de m˘ asur˘ a ˆ ın mecanica cuantic˘ a este ¯ h=h/2π= 1.054×10−34 J.s. ˆIn mi¸ scarea planetar˘ a macroscopic˘ a , num˘ arul cuantic c are descrie mo- mentul unghiular este atˆ ıt de mare c˘ a separarea ˆ ın st˘ ari discrete ale momen- tului cinetic nu se poate observa experimental. De exemplu, un electron al c˘ arui num˘ ar cuantic orbital este 2, are un moment cinetic L= 2.6×10−34 J.s., ˆ ın timp ce momentul cinetic al planetei noastre este 2 .7×1040J.s.! Se obi¸ snuie¸ ste s˘ a se noteze st˘ arile de moment cinetic cu literaspentru l= 0, cuppentrul= 1,dpentrul= 2, etc. Acest cod alfabetic provine din clasificarea empiric˘ a a spectrelor ˆ ın a¸ sa numitele serii principal˘ a , difuz˘ a ¸ si fundamental˘ a , care este anterioar˘ a mecanicii cuantice. Combinarea num˘ arului cuantic total cu litera corespunz˘ a toare momen- tului cinetic este o alt˘ a notat ¸ie frecvent folosit˘ a pent ru st˘ arile atomice. De exemplu, o stare ˆ ın care n= 2,l= 0 este o stare 2 s, iar una ˆ ın care n= 4, l= 2 este o stare 4 d. Pe de alt˘ a parte, pentru interpretarea num˘ arului cuantic magnetic, vom t ¸ine cont c˘ a la fel ca pentru impulsul lineal, momentul cin etic este un vector ¸ si deci pentru al descrie se necesit˘ a specificarea direct ¸ iei, sensului ¸ si modu- lului s˘ au. Vectorul Leste perpendicular planului ˆ ın care are loc mi¸ scarea de rotat ¸ie ¸ si direct ¸ia ¸ si sensul s˘ au sunt date de regula mˆ ıinii drepte (de pro- dus vectorial): degetul mare are direct ¸ia ¸ si sensul lui Lcˆ ınd celelalte patru degete sunt ˆ ın direct ¸ia de rotat ¸ie. Dar ce semnificat ¸ie se poate da unei direct ¸ii ¸ si sens ˆ ın sp at ¸iul limitat al unui atom de hidrogen ? R˘ aspunsul este simplu dac˘ a ne gˆ ı ndim c˘ a un electron care gireaz˘ aˆ ın jurul unui nucleu reprezint˘ a un circuit minuscul, care ca dipol magnetic prezint˘ a un cˆ ımp magnetic corespunz˘ at or.ˆIn consecint ¸˘ a , un electron atomic cu moment cinetic interact ¸ioneaz˘ a cu u n cˆ ımp magnetic externB. Num˘ arul cuantic magnetic mlspecific˘ a direct ¸ia lui L, determinat˘ a 119de componenta lui Lˆ ın direct ¸ia cˆ ımpului. Acest fenomen se cunoa¸ steˆ ın mod comun drept cuantizare spat ¸ial˘ a . Dac˘ a alegem direct ¸ia cˆ ımpului magnetic ca ax˘ a z, componenta lui Lˆ ın aceast˘ a direct ¸ie este Lz=ml¯h . (43) Valorile posibile ale lui mlpentru o valoare dat˘ a a lui l, merg de la + lpˆ ın˘ a la −l, trecˆ ınd prin 0, astfel c˘ a orient˘ arile posibile ale vect orului moment cinetic Lˆ ıntr-un cˆ ımp magnetic sunt 2 l+ 1. Cˆ ınd l= 0,Lzpoate avea numai valoarea zero; cˆ ınd l= 1,Lzpoate fi ¯h, 0, sau−¯h; cˆ ındl= 2,Lzia numai una dintre valorile 2¯ h, ¯h, 0,−¯h, sau−2¯h, ¸ si a¸ sa mai departe. Ment ¸ion˘ am c˘ a Lnu poate fi exact alineat (paralel sau antiparalel) cu B, pentru c˘ a Lzeste ˆ ıntotdeauna mai mic decˆ ıt modulul/radicalbig l(l+ 1)¯hmomentului unghiular total. Cuantizarea spat ¸ial˘ a a momentului cinetic orbital al ato mului de hidro- gen se arat˘ a ˆ ın fig. 6.1. Fig. 6.1: Cuantizarea spat ¸ial˘ a a momentului cinetic pent ru st˘ aril= 2,L=√ 6¯h. Trebuie s˘ a consider˘ am electronul caracterizat de c˘ atre un anumitmlca avˆ ınd o orientare determinat˘ a a momentului s˘ au cinetic Lfat ¸˘ a de un cˆ ımp magnetic extern ˆ ın cazul ˆ ın care acesta se aplic˘ a . ˆIn absent ¸a cˆ ımpului magnetic extern, direct ¸ia axei zeste complet arbi- trar˘ a . De aceea, componenta lui Lˆ ın orice direct ¸ie pe care o alegem este 120ml¯h; cˆ ımpul magnetic extern ofer˘ a o direct ¸ie de referint ¸˘ a privilegiat˘ a din punct de vedere experimental. Dece este cuantizat˘ a numai componenta lui L? R˘ aspunsul se relat ¸ioneaz˘ a cu faptul c˘ a Lnu poate fi direct ¸ionat de manier˘ a arbitrar˘ a ; ˆ ıntotdeau na descrie un con centrat pe axa de cuantizare ˆ ın a¸ sa fel ˆ ıncˆ ıt proiect ¸ia sa Lz esteml¯h. Motivul pentru care se produce acest fenomen este principi ul de incertitudine: dac˘ a Lar fi fix ˆ ın spat ¸iu, ˆ ın a¸ sa fel ˆ ıncˆ ıt Lx,Ly¸ siLzar avea valori bine definite, electronul ar fi confinat ˆ ıntr-un plan b ine definit. De ex- emplu, dac˘ a Lar fi fixat de-a lungul direct ¸iei z, electronul ar avea tendint ¸a de a se ment ¸ine ˆ ın planul xy(fig. 6.2a). Fig. 6.2: Principiul de incertitudine interzice o direct ¸i e fix˘ a ˆ ın spat ¸iu a momentului cinetic. Acest lucru poate s˘ a aib˘ a loc numai ˆ ın situat ¸ia ˆ ın care c omponenta pza impulsului electronului ˆ ın direct ¸ia zeste infinit de incert˘ a , ceea ce desigur este imposibil dac˘ a face parte din atomul de hidrogen. Totu ¸ si, cum ˆ ın real- itate numai o component˘ a Lza luiLˆ ımpreun˘ a cu L2au valori definite ¸ si |L|>|Lz|, electronul nu este limitat la un plan unic (fig. 6.2b), iar da c˘ a ar fi a¸ sa, ar exista o incertitudine ˆ ın coordonata za electronului. Direct ¸ia luiLse schimb˘ a ˆ ın mod constant (fig. 6.3), astfel c˘ a valorile m edii ale lui Lx¸ siLysunt 0, de¸ si Lzare ˆ ıntotdeauna valoarea ml¯h. 121Fig. 6.3: Vectorul moment cinetic prezint˘ a o precesie cons tant˘ a ˆ ın jurul axei z. Solut ¸ia pentru Φ trebuie s˘ a satisfac˘ a deasemenea condit ¸ia de normalizare, care este dat˘ a de c˘ atre ec. 2. Deci pentru Φ avem /integraldisplay2π 0|Φ|2dφ= 1 (44) ¸ si substituind Φ se obt ¸ine /integraldisplay2π 0A2 φdφ= 1. (45) AstfelAφ= 1/√ 2π¸ si deci Φ normalizat˘ a este dat˘ a de Φ(φ) =1√ 2πeimlφ. (46) Solut ¸ia pentru partea polar˘ a Ecuat ¸ia diferent ¸ial˘ a pentru partea polar˘ a Θ( θ) are o solut ¸ie mai complicat˘ a fiind dat˘ a de polinoamele Legendre asociate Pml l(x) = (−1)ml(1−x2)ml/2dml dxmlPl(x) = (−1)ml(1−x2)ml/2 2ll!dml+l dxml+l(x2−1)l. (47) 122Aceste funct ¸ii satisfac urm˘ atoarea relat ¸ie de ortogona litate /integraldisplay1 −1[Pml l(cosθ)]2dcosθ =2 2l+ 1(l+ml)! (l−ml)!. (48) ˆIn cazul mecanicii cuantice, Θ( θ) este dat˘ a de polinoamele Legendre nor- malizate, respectiv, dac˘ a Θ(θ) =AθPml l(cosθ), (49) atunci condit ¸ia de normalizare este dat˘ a de /integraldisplay1 −1A2 θ[Pml l(cosθ)]2dcosθ = 1. (50) Prin urmare constanta de normalizare pentru partea polar˘ a este Aθ=/radicaligg 2l+ 1 2(l−ml)! (l+ml)!(51) ¸ si prin urmare, funct ¸ia Θ( θ) deja normalizat˘ a este Θ(θ) =/radicaligg 2l+ 1 2(l−ml)! (l+ml)!Pml l(cosθ) (52) Pentru obiectivele noastre, cea mai important˘ a proprieta te a acestor funct ¸ii este c˘ a , a¸ sa cum s-a ment ¸ionat deja, exist˘ a num ai cˆ ınd constanta l este un num˘ ar ˆ ıntreg egal sau mai mare decˆ ıt |ml|, care este valoarea ab- solut˘ a a lui ml. Aceast˘ a condit ¸ie se poate scrie sub forma setului de valo ri disponibile pentru ml ml= 0,±1,±2,...,±l . (53) Unificarea p˘ art ¸ilor azimutal˘ a ¸ si polar˘ a : armonicele s ferice Solut ¸iile pentru p˘ art ¸ile azimutal˘ a ¸ si polar˘ a se pot u ni pentru a forma armon- icele sferice, care depind de φ¸ siθ¸ si contribuie la simplificarea manipul˘ arilor algebrice ale funct ¸iei de und˘ a complet˘ a ψ(r,θ,φ). Armonicele sferice se in- troduc ˆ ın felul urm˘ ator: Yml l(θ,φ) = (−1)ml/radicaligg 2l+ 1 4π(l−ml)! (l+ml)!Pml l(cosθ)eimlφ. (54) Factorul suplimentar ( −1)mlnu produce nici o problem˘ a pentru c˘ a ec. Schr¨ odinger este linear˘ a ¸ si homogen˘ a ¸ si este convenab il pentru studiul del momentului cinetic. Se cunoa¸ ste ca factorul de faz˘ a Condo n-Shortley, efectul s˘ au fiind de a introduce o alternant ¸˘ a a semnelor ±. 123Solut ¸ia pentru partea radial˘ a Solut ¸ia pentru partea radial˘ a R(r) a funct ¸iei de und˘ a ψa atomului de hidro- gen este ceva mai complicat˘ a ¸ si aici este unde apar diferen t ¸e mai mari fat ¸˘ a de ecuat ¸ia Laplace ˆ ın electrostatic˘ a . Rezultatul final s e exprim˘ a ˆ ın funct ¸ie de polinoamele asociate Laguerre (Schr¨ odinger 1926). Ecu at ¸ia radial˘ a se poate rezolva ˆ ın form˘ a analitic˘ a exact˘ a numai cˆ ınd E es te pozitiv sau pen- tru una din urm˘ atoarele valori negative En(ˆ ın care caz electronul este legat atomului) En=−me4 32π2ǫ2 0¯h2/parenleftbigg1 n2/parenrightbigg , (55) undeneste un num˘ ar ˆ ıntreg numit num˘ arul cuantic principal ¸ si descrie cuantizarea energiei electronului ˆ ın atomul de hidrogen. Aceast spectru discret a fost obt ¸inut pentru prima dat˘ a de c˘ atre Bohr cu m etode empirice de cuantizare ˆ ın 1913 ¸ si apoi de c˘ atre Pauli ¸ si respectiv Schr¨ odinger ˆ ın 1926. O alt˘ a condit ¸ie care trebuie s˘ a fie satisf˘ acut˘ a pentru a rezolva ecuat ¸ia radial˘ a este ca ns˘ a fie ˆ ıntotdeauna mai mare decˆ ıt l. Valoarea sa minim˘ a estel+ 1. Invers, condit ¸ia asupra lui leste l= 0,1,2,...,(n−1) (56) Ecuat ¸ia radial˘ a se poate scrie ˆ ın forma r2d2R dr2+ 2rdR dr+/bracketleftigg 2mE ¯h2r2+2me2 4πǫ0¯h2r−l(l+ 1)/bracketrightigg R= 0, (57) Dup˘ a ˆ ımp˘ art ¸irea cu r2, se folose¸ ste substitut ¸ia χ(r) =rRpentru a elim- ina termenul ˆ ındR dr¸ si a obt ¸ine forma standard a ec. Schr¨ odinger radiale cu potent ¸ial efectiv U(r) =−const/r+l(l+ 1)/r2(potent ¸ial electrostatic plus barier˘ a centrifugal˘ a ). Aceast˘ a procedur˘ a se aplic˘ a n umai pentru a discuta o nou˘ a condit ¸ie obligatorie de frontier˘ a , obti ¸nerea sp ectrului fiind prin in- termediul ecuat ¸iei pentru R. Diferent ¸a ˆ ıntre o ec. Schr¨ odinger radial˘ a ¸ si una ˆ ın toat˘ a linia real˘ a este c˘ a o condit ¸ie de frontier˘ a suplimentar˘ a trebuie impus˘ a ˆ ın origine ( r= 0). Potent ¸ialul coulombian apart ¸ine unei clase de potent ¸iale care se numesc slab singulare, pentru care lim r→0=U(r)r2= 0. Se ˆ ıncearc˘ a solut ¸ii de tipul χ∝rν, ceea ce implic˘ a ν(ν−1) =l(l+ 1) cu solut ¸iileν1=l+ 1 ¸ siν2=−l, exact ca ˆ ın cazul electrostaticii. Solut ¸ia negativ˘ a se elimin˘ a ˆ ın cazul l∝ne}ationslash= 0 pentru c˘ a duce la divergent ¸a integralei de normalizare ¸ si deasemenea nu respect˘ a normalizarea la fu nct ¸ia deltaˆ ın cazul 124spectrului continuu, iar cazul ν2= 0 se elimin˘ a din condit ¸ia de finitudine a energiei cinetice medii.Concluzia final˘ a este c˘ a χ(0) = 0 pentru orice l. Revenind la analiza ecuat ¸iei pentru funct ¸ia radial˘ a R, se pune mai ˆ ıntˆ ıi problema adimensionaliz˘ arii ecuat ¸iei. Aceasta se face o bservˆ ınd c˘ a se poate forma o singur˘ a scal˘ a de spat ¸iu ¸ si timp din combinat ¸ii a le celor trei constante fizice care intr˘ a ˆ ın aceast˘ a problem˘ a respectiv e2,m¸ si ¯h. Acestea sunt raza Bohra0= ¯h2/me2= 0.529·10−8cm. ¸ sit0= ¯h3/me4= 0.24210−16sec., care se numesc unit˘ at ¸i atomice. Folosind aceste unit˘ at ¸ i obt ¸inem d2R dr2+2 rdR dr+/bracketleftbigg 2E+2 r−l(l+ 1) r2/bracketrightbigg R= 0, (58) unde ne intereseaz˘ a spectrul discret ( E <0). Cu notat ¸iile n= 1/√ −E¸ si ρ= 2r/nse ajunge la: d2R dρ2+2 ρdR dρ+/bracketleftbiggn ρ−1 4−l(l+ 1) ρ2/bracketrightbigg R= 0. (59) Pentruρ→∞, ecuat ¸ia se reduce lad2R dρ2=R 4cu solut ¸iiR∝e±ρ/2. Se accept˘ a pe baza condit ¸iei de normalizare numai exponent ¸ iala atenuat˘ a . Pe de alt˘ a parte asimptotica de zero, a¸ sa cum am comentat deja , esteR∝ρl. Prin urmare, putem substitui Rprintr-un produs de trei funct ¸ii radiale R=ρle−ρ/2F(ρ), dintre care primele dou˘ a sunt p˘ art ¸ile asimptotice, ia r a treia este funct ¸ia radial˘ a ˆ ın regiunea intermediar˘ a , c are ne intereseaz˘ a cel mai mult pentru c˘ a ne d˘ a spectrul energetic. Ecuat ¸ia pent ruFeste ρd2F dρ2+ (2l+ 2−ρ)dF dρ+ (n−l−1)F= 0. (60) care este un caz particular de ecuat ¸ie hipergeometric˘ a co nfluent˘ a ˆ ın care cei doi parametri ‘hiper’geometrici depind de n,l¸ si care se poate identifica cu ecuat ¸ia pentru polinoamele Laguerre asociate L2l+1 n+l(ρ) ˆ ın fizica matematic˘ a . Astfel, forma normalizat˘ a a lui Reste: Rnl(r) =−2 n2/radicaligg (n−l−1)! 2n[(n+l)!]3e−ρ/2ρlL2l+1 n+l(ρ), (61) unde s-a folosit condit ¸ia de normalizare a polinoamelor La guerre: /integraldisplay∞ 0e−ρρ2l[L2l+1 n+l(ρ)]2ρ2dρ=2n[(n+l)!]3 (n−l−1)!. (62) 125Avem deci solut ¸iile fiec˘ areia dintre ecuat ¸iile care depi nd numai de o singur˘ a variabil˘ a ¸ si prin urmare putem construi funct ¸i a de und˘ a pentru fiecare stare electronic˘ a ˆ ın atomul de hidrogen, respecti v dac˘ aψ(r,θ,φ) = R(r)Θ(θ)Φ(φ), atunci funct ¸ia de und˘ a complet˘ a este ψ(r,θ,φ) =NH(αr)le−αr/2L2l+1 n+l(αr)Pml l(cosθ)eimlφ, (63) undeNH=−2 n2/radicalbigg 2l+1 4π(l−ml)! (l+ml)!(n−l−1)! [(n+l)!]3¸ siα= 2/na0. Utilizˆ ınd armonicele sferice, solut ¸ia se scrie ˆ ın felul urm˘ ator ψ(r,θ,φ) =−2 n2/radicaligg (n−l−1)! [(n+l)!]3(αr)le−αr/2L2l+1 n+l(αr)Yml l(θ,φ).(64) Aceast˘ a formul˘ a se poate considera rezultatul matematic final pentru solut ¸ia ec. Schr¨ odinger ˆ ın cazul atomului de hidrogen pe ntru oricare stare stat ¸ionar˘ a a electronului s˘ au. ˆIntr-adev˘ ar, se pot vedea ˆ ın mod explicit atˆ ıt dependent ¸a asimptotic˘ a cˆ ıt ¸ si cele dou˘ a seturi ortogo nale complete, poli- noamele Laguerre asociate ¸ si respectiv armonicele sferic e, corespunz˘ atoare acestei ecuat ¸ii lineare cu derivate part ¸iale de ordinul d oi. Coordonatele parabolice [ ξ=r(1−cosθ),η=r(1+ cosθ),φ=φ], sunt un alt set de vari- abile ˆ ın care ec. Schr¨ odinger pentru atomul de hidrogen es te u¸ sor de separat (E. Schr¨ odinger, Ann. Physik 80, 437, 1926; P.S. Epstein, Phys. Rev. 28, 695, 1926; I. Waller, Zf. Physik 38, 635, 1926). Solut ¸ia final˘ a se exprim˘ a ca produsul unor factori de natur˘ a asimptotic˘ a, armonice azimutale ¸ si dou˘ a seturi de polinoame Laguerre asociate ˆ ın ξ, respectiv η. Spectrul energetic (−1/n2) ¸ si degenerarea ( n2) evident nu se modific˘ a. Densitatea de probabilitate electronic˘ a ˆIn modelul lui Bohr al atomului de hidrogen, electronul se ro te¸ ste ˆ ın jurul nucleului pe traiectorii circulare sau eliptice. Dac˘ a se r ealizeaz˘ a un exper- iment adecuat, s-ar putea vedea c˘ a electronul ar fi ˆ ıntotde auna situat ˆ ın limitele expeimentale la o distant ¸˘ a fat ¸˘ a de nucleu r=n2a0(undeneste num˘ arul cuantic care numeroteaz˘ a orbita ¸ si a0= 0.53˚Aeste raza orbitei celei mai apropiate de nucleu, cunoscut˘ a ca raza Bohr) ¸ siˆ ın planul ecuatorial θ= 90o, ˆ ın timp ce unghiul azimutal φpoate varia ˆ ın timp. Teoria cuantic˘ a a atomului de hidrogen modific˘ a concluzii le modelului lui Bohr ˆ ın dou˘ a aspecte importante. ˆIn primul rˆ ınd, nu se pot da valori 126exacte pentru r,θ,φ, ci numai probabilit˘ at ¸i relative de a g˘ asi electronul ˆ ı ntr- o zon˘ a infinitezimal˘ a dat˘ a a spat ¸iului. Aceast˘ a imprec izie este, desigur, o consecint ¸˘ a a naturii ondulatorii a electronului. ˆIn al doilea rˆ ınd, nu se poate gˆ ındi c˘ a electronul se mi¸ sc˘ a ˆ ın jurul nucleului ˆ ın sen sul convent ¸ional clasic, pentru c˘ a densitatea de probabilitate |ψ|2nu depinde de timp ¸ si poate varia considerabil ˆ ın funct ¸ie de zona infinitezimal˘ a und e se calculeaz˘ a . Funct ¸ia de und˘ a ψa electronului ˆ ın atomul de hidrogen este ψ=RΘΦ undeR=Rnl(r) descrie cum se schimb˘ a ψcurcˆ ınd numerele cuantice orbital ¸ si total au valorile n¸ sil; Θ = Θlml(θ) descrie la rˆ ındul lui variat ¸ia luiψcuθcˆ ınd numerele cuantice magnetic ¸ si orbital au valorile l¸ siml; ˆ ın sfˆ ır¸ sit, Φ = Φ ml(φ) d˘ a schimbarea lui ψcuφcˆ ınd num˘ arul cuantic magnetic are valoarea ml. Densitatea de probabilitate |ψ|2se poate scrie |ψ|2=|R|2|Θ|2|Φ|2. (65) Densitatea de probabilitate |Φ|2, care m˘ asoar˘ a posibilitatea de a g˘ asi elec- tronul la un unghi azimutal φdat, este o constant˘ a care nu depinde de φ. Prin urmare, densitatea de probabilitate electronic˘ a est e simetric˘ a fat ¸˘ a de axaz, independent de starea cuantic˘ a “magnetic˘ a ” (atˆ ıta tim p cˆ ıt nu se aplic˘ a un cˆ ımp magnetic extern), ceea ce face ca electronu l s˘ a aib˘ a aceea¸ si probabilitate de a se g˘ asi ˆ ın orice direct ¸ie azimutal˘ a . Partea radial˘ a Ra funct ¸iei de und˘ a , spre deosebire de Φ, nu numai c˘ a variaz˘ a cur, ci ¸ si o face ˆ ın mod diferit pentru fiecare combinat ¸ie de numere cuantic en¸ sil. Fig. 6.4 arat˘ a grafice ale lui Rˆ ın funct ¸ie de rpentru st˘ arile 1 s, 2s, ¸ si 2pale atomului de hidrogen. Reste maxim ˆ ın centrul nucleului ( r= 0) pentru toate st˘ arile s, ˆ ın timp ce este zero ˆ ın r= 0 pentru toate st˘ arile care au moment cinetic. 2p1s 2s51015r(a )0Rnl(r) Fig. 6.4: Grafice aproximative ale funct ¸iilor radiale R1s,R2s,R2p; (a0= 0.53˚A). 1271s 2s 2p 10155 r(a )0dP/dr=4 πr R2 2 Fig. 6.5: Densitatea de probabilitate de a g˘ asi electronul atomului de hidrogen ˆ ıntrer¸ sir+drfat ¸˘ a de nucleu pentru st˘ arile 1 s, 2s, 2p. Densitatea de probabilitate electronic˘ aˆ ın punctul r,θ,φ este proport ¸ional˘ a cu|ψ|2, dar probabilitatea real˘ a ˆ ın elementul de volum infinitez imaldV este|ψ|2dV.ˆIn coordonate polare sferice dV=r2sinθdrdθdφ , (66) ¸ si cum Θ ¸ si Φ sunt funct ¸ii normalizate, probabilitatea nu meric˘ a real˘ a P(r)dr de a g˘ asi electronul la o distant ¸˘ a fat ¸˘ a de nucleu cuprin s˘ a ˆ ıntrer¸ sir+dr este P(r)dr=r2|R|2dr/integraldisplayπ 0|Θ|2sinθdθ/integraldisplay2π 0|Φ|2dφ =r2|R|2dr (67) P(r) este reprezentat˘ aˆ ın fig. 6.5 pentru acelea¸ si st˘ ari ale c˘ aror funct ¸ii radiale Rapar ˆ ın fig. 6.4; ˆ ın principiu, curbele sunt foarte diferit e. Observ˘ am imediat c˘ aP(r) nu este maxim˘ a ˆ ın nucleu pentru st˘ arile s, a¸ sa cum este R, avˆ ınd maximul la o distant ¸˘ a finit˘ a de acesta. Valoarea ce a mai probabil˘ a a lui rpentru un electron 1 seste exacta0, care este raza Bohr. Totu¸ si, valoarea 128medie a lui rpentru un electron 1 seste 1.5a0, ceea ce pare ciudat la prima vedere, pentru c˘ a nivelele de energie sunt acele¸ si ˆ ın mec anica cuantic˘ a ¸ si ˆ ın modelul lui Bohr. Aceast˘ a aparent˘ a discrepant ¸˘ a se e limin˘ a dac˘ a se t ¸ine cont de faptul c˘ a energia electronului depinde de 1 /r¸ si nu direct de r, iar valoarea medie a lui 1 /rpentru un electron 1 seste exact 1 /a0. Funct ¸ia Θ variaz˘ a cu unghiul polar θpentru toate numerele cuantice l¸ si ml, except ¸ie f˘ acˆ ınd l=ml= 0, care sunt st˘ ari s. Densitatea de probabilitate |Θ|2pentru o stare seste o constant˘ a (1/2), ceea ce ˆ ınseamn˘ a c˘ a , ˆ ıntrucˆ ıt |Φ|2este deasemenea constant˘ a , densitatea de probabilitate e lectronic˘ a |ψ|2are aceea¸ si valoare pentru o valoare a lui rdat˘ a , ˆ ın toate direct ¸iile. ˆIn alte st˘ ari, electronii au un comportament unghiular care u neori ajunge s˘ a fie foarte complicat. Aceata se poate vedea ˆ ın fig.6.5, unde se a rat˘ a densit˘ at ¸ile de probabilitate electronic˘ a pentru diferite st˘ ari atom ice ˆ ın funct ¸ie de r¸ siθ. (Termenul care se reprezint˘ a este |ψ|2¸ si nu|ψ|2dV). Deoarece|ψ|2este independent de φ, o reprezentare tridimensional˘ a a lui |ψ|2se obt ¸ine prin rotat ¸ia unei reprezent˘ ari particulareˆ ın jurul unei axe verticale, ceea ce poate ar˘ ata c˘ a densit˘ at ¸ile de probabilitate pentru st˘ arile sau simetrie sferic˘ a , ˆ ın timp ce toate celelalte nu o posed˘ a . Se obt ¸in ˆ ın acest fel l oburi mai mult sau mai put ¸in pronunt ¸ate, care au forme caracteristice pe ntru fiecare stare ˆ ın parte ¸ si care ˆ ın chimie joac˘ a un rol important ˆ ın dete rminarea modului ˆ ın care interact ¸ioneaz˘ a atomii ˆ ın interiorul molecule lor. 6N. Not˘ a : 1. E. Schr¨ odinger a obt ¸inut premiul Nobelˆ ın 1933 (ˆ ımpre un˘ a cu Dirac) pen- tru “descoperirea de noi forme productive ale teoriei atomi ce”. Schr¨ odinger a scris o remarcabil˘ a serie de patru articole intitulat˘ a “ Quantisierung als Eigenwertproblem” [“Cuantizarea ca problem˘ a de autovalo ri”] (I-IV, prim- ite la redact ¸ia revistei Annalen der Physik ˆ ın 27 Ianuarie , 23 Februarie, 10 Mai ¸ si 21 Iunie 1926). 6P. Probleme Problema 6.1 - S˘ a se obt ¸in˘ a formulele pentru orbitele stabile ¸ si pent ru nivelele de energie ale electronului ˆ ın atomul de hidrogen folosind numai argumente bazate pe lungimea de und˘ a de Broglie asociat˘ ae lectronului ¸ si valoarea ‘empiric˘ a ’ 5 .3·10−11m pentru raza Bohr. Solut ¸ie : Lungimea de und˘ a a electronului este dat˘ a de λ=h mvˆ ın timp ce dac˘ a egal˘ am fort ¸a electric˘ a cu fort ¸a centripet˘ a , r espectivmv2 r=1 4πǫ0e2 r2 129obt ¸inem c˘ a viteza electronului este dat˘ a de v=e√4πǫ0mr.ˆIn aceste condit ¸ii, lungimea de und˘ a a electronului este λ=h e/radicalig 4πǫ0r m. Acum, dac˘ a folosim valoarea 5.3×10−11m pentru raza ra orbitei electronice, vedem c˘ a lungimea de und˘ a a electronului este λ= 33×10−11m. Aceast˘ a lungime de und˘ a are exact aceea¸ si valoare ca circumferint ¸a orbitei electron ului, 2πr= 33×10−11 m. Dup˘ a cum se poate vedea, orbita electronului ˆ ın atomul d e hidrogen corespunde astfel unei unde “ˆ ınchis˘ a ˆ ın ea ˆ ıns˘ a¸ si” (a dic˘ a de tip stat ¸ionar). Acest fapt se poate compara cu vibrat ¸iile unui inel de alam˘ a . Dac˘ a lungimile de und˘ a sunt un submultiplu al circumferint ¸ei sale, inelu l ar putea continua starea sa vibratorie pentru foarte mult timp cu disipare red us˘ a (st˘ ari ‘proprii’ de vibrat ¸ie sau unde stat ¸ionare). Dac˘ a ˆ ıns˘ a num˘ arul d e lungimi de und˘ a nu este ˆ ıntreg se va produce o interferent ¸˘ a negativ˘ a pe m˘ a sur˘ a ce undele se propag˘ a de-a lungul inelului ¸ si vibrat ¸iile vor disp˘ are a foarte repede. Astfel, se poate afirma c˘ a un electron se poate roti indefinitˆ ın juru l nucleului f˘ ar˘ a a radia energia de care dispune atˆ ıta timp cˆ ıt orbita cont ¸i ne un num˘ ar ˆ ıntreg de lungimi de und˘ a de Broglie. Cu acestea, avem condit ¸ia de stabilitate nλ= 2πrn, underneste raza orbitei care cont ¸ine nlungimi de und˘ a . Substituind λ, avem nh e/radicalbigg4πǫ0rn m= 2πrn, ¸ si deci orbitele stabile ale electronului sunt rn=n2¯h2ǫ0 πme2. Pentru nivelele de energie, avem E=T+V¸ si prin substituirea energiilor potent ¸ial˘ a ¸ si cinetic˘ a obt ¸inem E=1 2mv2−e2 4πǫ0r, sau echivalent En=−e2 8πǫ0rn. Substituind valoarea lui rnˆ ın ultima ecuat ¸ie obt ¸inem En=−me4 8ǫ2 0¯h2/parenleftbigg1 n2/parenrightbigg . 130Problema 6.2 - Teorema lui Uns¨ old spune c˘ a , pentru orice valoare a num˘ arului cuantic orbital l, densit˘ at ¸ile de probabilitate, sumate peste toate subst˘ arile posibile, de la ml=−lpˆ ın˘ a laml= +ldau o constant˘ a independent˘ a de unghiurile θsauφ, adic˘ a +l/summationdisplay ml=−l|Θlml|2|Φml|2=ct. Aceast˘ a teorem˘ a arat˘ a c˘ a orice atom sau ion cu subst˘ ari ˆ ınchise prezint˘ a o distribut ¸ie sferic simetric˘ a de sarcin˘ a electric˘ a . S ˘ a se verifice teorema Uns¨ old pentru l= 0,l= 1 ¸ sil= 2. Solut ¸ie : Avm pentru l= 0, Θ 00= 1/√ 2 ¸ si Φ 0= 1/√ 2π, deci vedem c˘ a |Θ0,0|2|Φ0|2=1 4π. Pentrul= 1 avem +1/summationdisplay ml=−1|Θlml|2|Φml|2=|Θ1,−1|2|Φ−1|2+|Θ1,0|2|Φ0|2+|Θ1,1|2|Φ1|2. Pe de alt˘ a parte funct ¸iile de und˘ a sunt: Θ 1,−1= (√ 3/2)sinθ, Φ−1= (1/√ 2π)e−iφ, Θ1,0= (√ 6/2)cosθ, Φ0= 1/√ 2π, Θ1,1= (√ 3/2)sinθ, Φ1= (1/√ 2π)eiφ, care substituite ˆ ın ecuat ¸ia anterioar˘ a conduc la +1/summationdisplay ml=−1|Θlml|2|Φml|2=3 8πsen2θ+3 4πcos2θ+3 8πsen2θ=3 4π ¸ si din nou obt ¸inem o constant˘ a . Pentrul= 2 avem +2/summationdisplay ml=−2|Θlml|2|Φml|2= |Θ2,−2|2|Φ−2|2|Θ2,−1|2|Φ−1|2+|Θ2,0|2|Φ0|2+|Θ2,1|2|Φ1|2+|Θ2,2|2|Φ2|2 ¸ si funct ¸iile de und˘ a sunt: Θ 2,−2= (√ 15/4)sin2θ, Φ−2= (1/√ 2π)e−2iφ, Θ2,−1= (√ 15/2)sinθcosθ , Φ−1= (1/√ 2π)e−iφ, Θ2,0= (√ 10/4)(3cos2θ− 1311), Φ 0= 1/√ 2π, Θ2,1= (√ 15/2)sinθcosθ , Φ1= (1/√ 2π)eiφ, Θ2,2= (√ 15/4)sin2θ, Φ2= (1/√ 2π)e2iφ, care substituite ˆ ın ecuat ¸ia anterioar˘ a dau +2/summationdisplay ml=−2|Θlml|2|Φml|2=5 4π, ceea ce din nou verific˘ a teorema Uns¨ old. Problema 6.3 - Probabilitatea de a g˘ asi un electron atomic a c˘ arui funct ¸ie de und˘ a radial˘ a este cea de stare fundamental˘ a R10(r) ˆ ın afara unei sfere de raz˘ a Bohr a0centrat˘ a ˆ ın nucleu este /integraldisplay∞ a0|R10(r)|2r2dr . S˘ a se calculeze probabilitatea de a g˘ asi electronul ˆ ın st area fundamental˘ a atomic˘ a la o distant ¸˘ a de nucleu mai mare de a0. Solut ¸ie : Funct ¸ia de und˘ a radial˘ a care corespunde st˘ arii fundam entale este R10(r) =2 a3/2 0e−r/a0. Substituind-o ˆ ın integral˘ a obt ¸inem /integraldisplay∞ a0|R(r)|2r2dr=4 a3 0/integraldisplay∞ a0r2e−2r/a0dr , sau /integraldisplay∞ a0|R(r)|2r2dr=4 a3 0/bracketleftigg −a0 2r2e−2r/a0−a2 0 2re−2r/a0−a3 0 4e−2r/a0/bracketrightigg∞ a0. Aceasta ne conduce la: /integraldisplay∞ a0|R(r)|2r2dr=5 e2≈68% !!, care este probabilitatea cerut˘ a ˆ ın aceast˘ a problem˘ a . 1327. CIOCNIRI CUANTICE Introducere Pentru init ¸iere ˆ ın teoria cuantic˘ a de ˆ ımpr˘ a¸ stiere ne vom servi de rezultate deja cunoscute de la ˆ ımpr˘ a¸ stierea clasic˘ a ˆ ın cˆ ımpuri centrale ¸ si vom pre- supune anumite situat ¸ii care vor simplifica calculele f˘ ar ˘ a ˆ ıns˘ a a ne ˆ ındep˘ arta prea mult de problema “real˘ a”. S ¸tim c˘ aˆ ın studiul experi mental al unei cioc- niri putem obt ¸ine date care ne pot ajuta s˘ a ˆ ınt ¸elegem dis tribut ¸ia materiei “t ¸int˘ a”, sau mai bine spus interact ¸iunea ˆ ıntre fascicu lul incident ¸ si “t ¸int˘ a”. Ipotezele pe care le vom presupune corecte sunt: i) Particulele nu au spin, ceea ce nuˆ ınseamn˘ a c˘ a acesta nu este important ˆ ın ciocniri. ii) Ne vom ocupa numai de dispersia elastic˘ a pentru care pos ibila struc- tur˘ a intern˘ a a particulelor nu se ia ˆ ın considerare. iii) T ¸inta este suficient de subt ¸ire pentru a putea neglija ˆ ımpr˘ a¸ stierile multiple. iv) Interact ¸iunile sunt descrise printr-un potent ¸ial ca re depinde numai de pozit ¸ia relativ˘ a a particulelor. Aceste ipoteze elimin˘ a o serie de efecte cuantice ¸ si m˘ a re sc corectitudinea unor rezultate bazice din teoria ciocnirilor clasice. Astf el definim: dσ dΩ=I(θ,ϕ) I0, (1) undedσeste elementul de unghi solid, I0este num˘ arul de particule incidente pe unitate de arie ¸ si IdΩ este num˘ arul de particule dispersate ˆ ın elementul de unghi solid. Cu aceste concepte binecunoscute, ajungem la: dσ dΩ=ρ sinθ|dρ dθ|. (2) Dac˘ a dorim s˘ a cunoa¸ stem ˆ ın termeni cuantici fenomenolo gia de cioc- nire, trebuie s˘ a studiem evolut ¸ia ˆ ın timp a unui pachet de unde. Fie Fi fluxul de particule al fascicolului incident, adic˘ a , num˘ a rul de particule pe unitate de timp care intersecteaz˘ a o suprafat ¸˘ a unitar˘ a transversal˘ a axei de propagare. Vom pozit ¸iona un detector departe de zona de act ¸iune efectiv˘ a a potent ¸ialului, care subˆ ıntinde un unghi solid dΩ; cu acesta putem ˆ ınregistra 133num˘ arul de particule dn/dt dispersate ˆ ın unitatea de timp ˆ ın dΩ ˆ ın direct ¸ia (θ,ϕ). z V(r)0Detector D (dn/dt ~ F d Ω θd Ωi) Flux inc. F iFig. 7.1 dn/dt este proport ¸ional cu dΩ ¸ siFi. S˘ a numim σ(θ,ϕ) coeficientul de proport ¸ionalitate ˆ ıntre dn¸ siFidΩ: dn=σ(θ,ϕ)FidΩ, (3) care este prin definit ¸ie sect ¸iunea diferent ¸ial˘ a transv ersal˘ a . Num˘ arul de particule pe unitatea de timp care ajung la detec tor este egal cu num˘ arul de particule care intersecteaz˘ a o suprafat ¸˘ a σ(θ,ϕ)dΩ situat˘ a perpendicular pe axa fasciculului. Sect ¸iunea total˘ a de d ispersie este prin definit ¸ie: σ=/integraldisplay σ(θ,ϕ)dΩ. (4) Cum putem orienta axele de coordonate conform alegerii dori te, o vom face ˆ ın a¸ sa fel ca axa fasciculului incident de particule s ˘ a coincid˘ a cu axa z (aceasta pentru simplificarea calculelor, unde vom folosi c oordonatele sferice). ˆIn regiunea negativ˘ a a axei, pentru tnegativ mare, particula va fi practic liber˘ a : nu este afectat˘ a de V(r) ¸ si starea sa se poate reprezenta prin unde plane. Prin urmare funct ¸ia de und˘ a trebuie s˘ a cont ¸in˘ a t ermeni de forma 134eikz, undekeste constanta care apare ˆ ın ecuat ¸ia Helmholtz. Prin anal ogie cu optica, forma undei dispersate este: f(r) =eikr r. (5) ˆIntr-adev˘ ar: (∇2+k2)eikr∝ne}ationslash= 0 (6) ¸ si (∇2+k2)eikr r= 0 (7) pentrur>r 0, under0este orice num˘ ar pozitiv. Presupunem c˘ a mi¸ scarea particulei este descris˘ a de Hami ltonianul: H=p2 2µ+V=H0+V . (8) V este diferit de zero numaiˆ ıntr-o mic˘ a vecin˘ atate ˆ ın ju rul originii. S ¸tim c˘ a un pachet de unde ˆ ın t= 0 se poate scrie: ψ(r,0) =1 (2π)3 2/integraldisplay ϕ(k)exp[ik·(r−r0)]d3k, (9) undeψeste o funct ¸ie semnificativ nenul˘ aˆ ın segmentul (l˘ argim ea) ∆kcentrat ˆ ın jurul lui k0. Presupunem deasemena c˘ a k0este paralel la r0, dar de sens opus. Pentru a vedea ˆ ın mod cantitativ ce se ˆ ıntˆ ımpl˘ a cu p achetul de unde cˆ ınd la un moment ulterior ciocne¸ se t ¸inta ¸ si este disper sat de aceasta ne putem folosi de dezvoltarea lui ψ(r,0) ˆ ın funct ¸iile proprii ψn(r) ale luiH, respectivψ(r,0) =/summationtext ncnψn(r). Astfel, pachetul de unde la timpul teste: ψ(r,t) =/summationdisplay ncnϕn(r)exp(−i ¯hEnt). (10) Aceasta este o funct ¸ie proprie a operatorului H0¸ si nu a luiH, dar putem substitui aceste funct ¸ii proprii cu funct ¸ii proprii part iculare ale lui H, pe care le vom nota cu ψ(+) k(r). Forma asimptotic˘ a a acestora din urm˘ a este de tipul: ψ(+) k(r)≃eik·r+f(r)eikr r, (11) unde, cum este uzual, p= ¯hk¸ siE=¯h2k2 2m. 135Aceasta corespunde unei unde plane ca fascicul incident ¸ si o und˘ a sferic˘ a divergent˘ a, despre care se poate spune c˘ a este rezultatul interact ¸iunii ˆ ıntre fascicul ¸ si t ¸int˘ a. Aceste solut ¸ii ale ec. Schr¨ odinger exist˘ a ˆ ın realitate, ¸ si putem dezvolta ψ(r,0) ˆ ın unde plane ¸ si ψk(r): ψ(r,0) =/integraldisplay ϕ(k)exp(−ik·r0)ψk(r)d3k , (12) unde ¯hω=¯h2k2 2m. Se poate spune deci c˘ a unda sferic˘ a divergent˘ a nu are nic i o contribut ¸ie la pachetul de unde init ¸ial. ˆImpr˘ a¸ stierea unui pachet de unde Orice und˘ a sufer˘ a ˆ ın cursul propag˘ arii o dispersie. De a ceea nu se poate ignora efectul undei divergente din acest punct de vedere. S e poate folosi urm˘ atorul truc: ω=¯h 2mk2=¯h 2m[k0+ (k−k0)]2=¯h 2m[2k0·k−k2 0+ (k−k0)2],(13) pentru a neglija ultimul termen ˆ ın paranteze. Substituind ωˆ ınψ, cerem ca: ¯h 2m(k−k0)2T≪1, undeT≃2mr0 ¯hk0¸ si deci: (∆k)2r0 k0≪1. (14) Aceast˘ a condit ¸ie ne spune c˘ a pachetul de unde nu se disper seaz˘ a ˆ ın mod apreciabil chiar ¸ si atunci cˆ ınd se deplaseaz˘ a pe o distan t ¸˘ a macroscopic˘ a r0. Alegˆ ınd direct ¸ia vectorului kal undei incidente de-a lungul uneia dintre cele trei direct ¸ii carteziene, putem scrie ˆ ın coordonate sferice ψk(r,θ,ϕ)≃eikz+f(k,θ,ϕ)eikr r. ˆIntrucˆ ıtH, operatorul Hamiltonian (que hemos considerat pˆ ın˘ a acum nu ca operator pentru c˘ a rezultatele sunt acelea¸ si) este inv ariant la rotat ¸iile ˆ ın axa z, putem alege condit ¸iile de frontier˘ a deasemenea inv ariante, astfel c˘ a : ψk(r,θ,ϕ)≃eikz+f(θ)eikr r. Acest tip de funct ¸ii se cunosc ca unde de ˆ ımpr˘ a¸ stiere. Co eficientulf(θ) al undei divergente se cunoa¸ ste ca amplitudine de ˆ ımpr˘ a¸ st iere. 136Amplitudinea de probabilitate Ecuat ¸ia Schr¨ odinger de rezolvat este: i¯h∂ψ ∂t=−¯h2 2m∇2ψ+V(r,t)ψ . (15) Expresia P(r,t) =ψ∗(r,t)ψ(r,t) =|ψ(r,t)|2(16) se interpreteaz˘ a , cf. lui Max Born, ca o densitate de probab ilitate dac˘ a funct ¸ia de und˘ a se normalizeaz˘ a astfel ca: /integraldisplay |ψ(r,t)|2d3r= 1. (17) Desigur integrala de normalizare a lui ψtrebuie s˘ a fie independent˘ a de timp. Acest lucru se poate nota ˆ ın felul urm˘ ator: I=∂ ∂t/integraldisplay ΩP(r,t)d3r=/integraldisplay Ω(ψ∗∂ψ ∂t+∂ψ∗ ∂tψ)d3r (18) ¸ si din ec. Schr¨ odinger: ∂ψ ∂t=i¯h 2m∇2ψ−i ¯hV(r,t)ψ (19) rezult˘ a : I=i¯h 2m/integraldisplay Ω[ψ∗∇2−(∇2ψ∗)ψ]d3r=i¯h 2m/integraldisplay Ω∇·[ψ∗∇ψ−(∇ψ∗)ψ]d3r= =i¯h 2m/integraldisplay A[ψ∗∇ψ−(∇ψ∗)ψ]ndA , (20) unde s-a folosit teorema Green pentru evaluarea integralei de volum.dAeste elementul de suprafat ¸˘ a pe frontiera care delimiteaz˘ a re giunea de integrare ¸ si [ ]ndenot˘ a componenta ˆ ın direct ¸ia normal˘ a la elementul de s uprafat ¸˘ adA. Definind: S(r,t) =¯h 2im[ψ∗∇ψ−(∇ψ∗)ψ], (21) obt ¸inem: I=∂ ∂t/integraldisplay ΩP(r,t)d3r=−/integraldisplay Ω∇·Sd3r=−/integraldisplay ASndA , (22) 137pentru pachete de und˘ a ˆ ın care ψse pune zero la distant ¸e mari ¸ si integrala de normalizare converge, integrala de suprafat ¸˘ a este zer o cˆ ınd Ω este tot spat ¸iul. Se poate demonstra (se poate consulta P. Dennery & A. Krzywicki, Mathematical methods for physicists ) c˘ a integrala de suprafat ¸˘ a este zero, astfel c˘ a integrala de normalizare este constant˘ a ˆ ın tim p ¸ si deci se satisface cerint ¸a init ¸ial˘ a . Din aceea¸ si ecuat ¸ie pentru Sobt ¸inem: ∂P(r,t) ∂t+∇·S(r,t) = 0, (23) care este o ecuat ¸ie de continuitate cu fluxul de densitate P¸ si curent de den- sitateS, f˘ ar˘ a nici un fel de surse (pozitive sau negative). Dac˘ a i nterpret˘ am ¯h im∇ca un fel de ‘operator’ vitez˘ a (ca ¸ si ˆ ın cazul timpului nu s e poate vorbi de un operator vitez˘ a ˆ ın sens riguros), atunci: S(r,t) =Re(ψ∗¯h im∇ψ). (24) Funct ¸ia Green ˆ ın teoria de ˆ ımpr˘ a¸ stiere O alt˘ a form˘ a de a scrie ecuat ¸ia Schr¨ odinger de rezolvat e ste (−¯h2 2m∇2+V)ψ= Eψsau (∇2+k2)ψ=Uψunde:k2=2mE ¯h2, ¸ siU=2mV ¯h2. Rezult˘ a mai convenabil de transformat aceast˘ a ecuat ¸ie l a o form˘ a inte- gral˘ a . Aceasta se poate face dac˘ a vom considera Uψdin partea dreapt˘ a a ecuat ¸iei ca o inomogeneitate, ceea ce ne permite s˘ a constr uim solut ¸ia ecuat ¸iei cu ajutorul funct ¸iei Green (nucleu integral), care prin de finit ¸ie este solut ¸ia lui: (∇2+k2)G(r,r′) =δ(r−r′). (25) Solut ¸ia ecuat ¸iei Schr¨ odinger se d˘ a ca suma solut ¸iei ec uat ¸iei omogene ¸ si a solut ¸iei inomogene de tip Green: ψ(r) =λ(r)−/integraldisplay G(r,r′)U(r′)ψ(r′)d3r′. (26) C˘ aut˘ am acum o funct ¸ie Gcare s˘ a fie un produs de funt ¸ii linear indepen- dente, cum sunt de exemplu undele plane: G(r,r′=/integraldisplay A(q)eiq·(r−r′)dq . (27) Folosind ecuat ¸ia 25, avem: /integraldisplay A(q)(k2−q2)eiq·(r−r′)dq=δ(r−r′), (28) 138care se transform˘ a ˆ ıntr-o identitate dac˘ a : A(q) = (2π)−3(k2−q2)−1. (29) De aici rezult˘ a : G(r,r′) =1 (2π)3/integraldisplayeiqR k2−q2d3q , (30) cuR=|r−r′|. Dup˘ a un calcul folosind metode de variabil˘ a complex˘ a10, ajungem la: G(r) =−1 4πeikr r. (31) Aceast˘ a funct ¸ie nu este determinat˘ a ˆ ın mod univoc; func t ¸ia Green poate fi oricare solut ¸ie a ecuat ¸iei 25; Alegerea uneia particula re se face prin im- punerea condit ¸iilor de frontier˘ a asupra funct ¸iilor pro priiψk(r). Funct ¸ia Green obt ¸inut˘ a ˆ ın aceste condit ¸ii este: G(r,r′) =−/parenleftigg eik|r−r′| 4π|r−r′|/parenrightigg . (32) ˆIn acest fel, ajungem la ecuat ¸ia integral˘ a pentru funct ¸i a de und˘ a de ciocnire: ψ(k,r) =ϕ(k,r)−m 2π¯h2/integraldisplayeik|r−r′| r−r′U(r′)ψ(k,r)dr, (33) undeϕeste o solut ¸ie a ecuat ¸iei Helmholtz. Notˆ ınd |r−r′|=R: (∇2+k2)ψ= (∇2+k2)[ϕ+/integraldisplay G(r,r′)U(r′)ψ(r′)d3r′] (34) ¸ si presupunˆ ınd c˘ a putem schimba ordinea ¸ si pune operato rul∇ˆ ın interiorul integralei: (∇2+k2)ψ=/integraldisplay (∇2+k2)G(r,r′)U(r′)ψ(r′)d3r′=U(r)ψ(r),(35) ceea ce ne arat˘ a c˘ a se verific˘ a faptul c˘ a G(R) =1 4πeikR Reste solut ¸ie. 10Se poate vedea problema 7.1. 139Teorema optic˘ a Sect ¸iunea diferent ¸ial˘ a total˘ a este dat˘ a de: σtot(k) =/integraldisplaydσ dΩdΩ. (36) S˘ a exprim˘ am acum f(θ) ca funct ¸ie de ¸ siftul de faz˘ a Sl(k) =e2iδl(k)ˆ ın forma: f(θ) =1 k∞/summationdisplay l=0(2l+ 1)eiδi(k)sinδl(k)Pl(cosθ) (37) atunci σtot=/integraldisplay [1 k∞/summationdisplay l=0(2l+ 1)eiδl(k)sinδl(k)Pl(cosθ)] [/integraldisplay [1 k∞/summationdisplay l′=0(2l′+ 1)eiδl′(k)sinδl′(k)Pl′(cosθ)]. (38) Folosind acum/integraltextPl(cosθ)Pl′(cosθ) =4π 2l+1δll′obt ¸inem σtot=4π k2∞/summationdisplay l=0(2l+ 1)sinδl(k)2. (39) Ceea ce ne intereseaz˘ a este c˘ a : Imf(0) =1 k∞/summationdisplay l=0(2l+ 1)Im[eiδl(k)sinδl(k)]Pl(1) =1 k∞/summationdisplay l=0(2l+ 1)sinδl(k)2= k 4πσtot. (40) Aceast˘ a relat ¸ie este cunoscut˘ a ca teorema optic˘ a . Semnificat ¸ia sa fizic˘ a este c˘ a interferent ¸a undei incidente cu unda dispersat˘ a la unghi zero pro- duce “ie¸ sirea” particulei din unda incident˘ a , ceea ce per mite conservarea probabilit˘ at ¸ii. Aproximat ¸ia Born S˘ a consider˘ am situat ¸ia din Fig. 7.2: 140LO Pur |r − r |’M Fig. 7.2 Punctul de observare M este departe de P, care se afl˘ a ˆ ın regi unea de influent ¸˘ a a potent ¸ialului U, cur≫L,r′≪l. Segmentul MP, care core- spunde la|r−r′|, este ˆ ın aceste condit ¸ii geometrice aproximativ egal cu proiect ¸ia lui MP pe MO: |r−r′|≃r−u·r′, (41) undeueste vectorul unitar (versor) ˆ ın direct ¸ia r. Atunci, pentru rmare: G=−1 4πeik|r−r′| |r−r′|≃r→∞−1 4πeikr re−iku·r. (42) Substituim Gˆ ın expresia integral˘ a pentru funct ¸ia de und˘ a de ciocnir e pentru a obt ¸ine: ψ(r) =eikz−1 4πeikr r/integraldisplay e−iku·rU(r′)ψ(r′)d3r′. (43) Aceasta deja nu mai este o funct ¸ie de distant ¸a r=OM, ci numai de θ¸ siψ; atunci: f(θ,ψ) =−1 4π/integraldisplay e−iku·rU(r′)ψ(r′)d3r′. (44) Definim acum vectorul de und˘ a incident kica un vector de modul kdirijat de-a lungul axei polare a fasciculului astfel c˘ a : eikz=eiki·r; similar, kd, de modulk¸ si cu direct ¸ia fixat˘ a prin θ¸ siϕ, se nume¸ ste vector de und˘ a ‘deplasat’ ˆ ın direct ¸ia ( θ,ϕ):kd=ku Vectorul de und˘ a transferat ˆ ın direct ¸ia ( θ,ϕ) se introduce prin: K= kd−ki. 141kd kiθ Fig. 7.3K Cu aceasta, putem scrie ecuat ¸ia integral˘ a de dispersie ˆ ı n forma: ψ(r) =eiki·r+/integraldisplay G(r,r′)U(r′)ψ(r′)d3r′(45) Acum putem ˆ ıncerca rezolvarea acestei ecuat ¸ii ˆ ın mod ite rativ. Punˆ ınd r→r′;r′→r′′, putem scrie: ψ(r′) =eiki·r′+/integraldisplay G(r′,r′′)U(r′′)ψ(r′′)d3r′′. (46) Substituind ˆ ın 45 obt ¸inem: ψ(r) =eiki·r+/integraldisplay G(r,r′)U(r′)eiki·r′d3r′+ /integraldisplay /integraldisplay G(r,r′)U(r′)G(r′,r′′)U(r′′)ψ(r′′)d3r′′d3r′. (47) Primii doi termeni din partea dreapt˘ a sunt cunoscut ¸i ¸ si n umai al treilea cont ¸ine funct ¸ia necunoscut˘ a ψ(r). Putem repeta acest procedeu: ˆ ınlocuind rcur′′¸ sir′cur′′′obt ¸inemψ(r′′) , pe care putem s˘ a o reintroducem ˆ ın ec. 47: ψ(r) =eiki·r+/integraldisplay G(r,r′)U(r′)eiki·r′+ /integraldisplay /integraldisplay G(r,r′)U(r′)G(r′,r′′)U(r′′)eiki·r′′d3r′d3r′′+ /integraldisplay /integraldisplay /integraldisplay G(r,r′)U(r′)G(r′,r′′)U(r′′)eiki·r′′G(r′′,r′′′)U(r′′′)ψ(r′′′).(48) Primii trei termeni sunt cunoscut ¸i; funct ¸ia necunoscut˘ aψ(r) se afl˘ a ˆ ın al patrulea termen. ˆIn acest fel, prin iterat ¸ii construim funct ¸ia de und˘ a de d is- persie stat ¸ionar˘ a . Not˘ am c˘ a fiecare termenˆ ın dezvolta reaˆ ın serie prezint˘ a o putere superioar˘ aˆ ın potent ¸ial fat ¸˘ a de cel precedent. Putem continuaˆ ın acest fel pˆ ın˘ a cˆ ınd obt ¸inem o expresie neglijabil˘ a ˆ ın parte a dreapt˘ a , ¸ si obt ¸inem ψ(r) ˆ ın funct ¸ie numai de m˘ arimi cunoscute. 142Substituind expresia lui ψ(r)ˆ ınf(θ,ϕ) obt ¸inem dezvoltareaˆ ın serie Born a amplitudinii de ˆ ımpr˘ a¸ stiere. Limitˆ ındu-ne la primul ordin ˆ ınU, trebuie s˘ a se fac˘ a doar substituirea lui ψ(r′) cueiki·r′ˆ ın partea dreapt˘ a a ecuat ¸iei pentru a obt ¸ine: f(B)(θ,ϕ) =−1 4π/integraldisplay eiki·r′U(r′)e−iku·r′d3r′=−1 4π/integraldisplay e−i(kd−ki)·r′U(r′)d3r′= −1 4π/integraldisplay e−iK·r′U(r′)d3r′(49) Keste vectorul de und˘ a dispersat˘ a definit maiˆ ınaite. Sect ¸iunea de dispersie se relat ¸ioneaz˘ a astfelˆ ın mod simplu cu transformata Fou rier a potent ¸ialului, dac˘ a t ¸inem cont de V(r) =¯h2 2mU(r) ¸ siσ(θ,ϕ) =|f(θ,ϕ)|2. Rezultatul este: σ(B)(θ,ϕ) =m2 4π2¯h4|/integraldisplay e−iK·rV(r)d3r|2(50) Direct ¸ia ¸ si modulul vectorului undei dispersate Kdepinde de modulul k al luiki¸ sikdprecum ¸ si de direct ¸ia de ˆ ımpr˘ a¸ stiere ( θ,ϕ). Pentruθ¸ siϕdat ¸i, sect ¸iunea eficace este o funct ¸ie de k, energia fasciculului incident. Analog, pentru o energie dat˘ a , σ(B)este o funct ¸ie de θ¸ siϕ. Aproximat ¸ia Born per- mite ca studiind variat ¸ia sect ¸iunii eficace diferent ¸ial eˆ ın funct ¸ie de direct ¸ia de ˆ ımpr˘ a¸ stiere ¸ si energia incident˘ a s˘ a obt ¸inem inform at ¸ii asupra potent ¸ialului V(r). 7N. Not˘ a : Unul dintre primele articole de ˆ ımpr˘ a¸ stiere cuantic˘ a este: M. Born, “Quantenmechanik der Stossvorg¨ ange” [“Mecanica cuantic˘ a a pro- ceselor de ciocnire”], Zf. f. Physik 37, 863-867 (1926) 7P. Probleme Problema 7.1 Calculul de variabil˘ a complex˘ a a funct ¸iei Green Reamintim c˘ a am obt ¸inut deja rezultatul: G(r,r′) =1 (2π)3/integraltexteiqR k2−q2d3q ,cuR=|r−r′|. Cumd3q=q2sinθdqdθdφ , ajungem, dup˘ a ce integr˘ am ˆ ın variabilele unghiulare , la : G(r,r′) =i 4π2R/integraltext∞ −∞(e−iqR−eiqR) k2−q2qdq . S˘ a punem: C=i 4π2R; ¸ si s˘ a separ˘ am integrala ˆ ın dou˘ a p˘ art ¸i: 143C(/integraltext∞ −∞e−iqR k2−q2qdq−/integraltext∞ −∞eiqR k2−q2qdq). S˘ a facem acum q→−qˆ ın prima integral˘ a :/integraltext∞ −∞e−i(−q)R k2−(−q)2(−q)d(−q) =/integraltext−∞ ∞eiqR k2−q2qdq=−/integraltext∞ −∞eiqR k2−q2qdq astfel c˘ a : G(r,r′) =−2C(/integraltext∞ −∞qeiqR k2−q2dq). Substituind C, obt ¸inem: G(r,r′) =−i 2π2R/integraltext∞ −∞qeiqR k2−q2dq ˆIn aceast˘ a form˘ a integrala se poate evalua cu ajutorul rez iduurilor polilor pe careˆ ıi posed˘ a , folosind metodele de variabil˘ a comple x˘ a . Not˘ am c˘ a exist˘ a poli simpli ˆ ın pozit ¸iile q=+ −k. Fig. 7.4: Reguli de contur ˆ ın jurul polilor pentru G+¸ siG− Folosim conturul din figura 7.4, care ˆ ınconjoar˘ a polii ˆ ın modul ar˘ atat, pentru c˘ a acesta d˘ a efectul fizic corect, pentru c˘ a de acor d cu teorema rezidu- urilor, G(r) =−1 4πeikr r(Imk>0) , G(r) =−1 4πe−ikr r(Imk<0) Solut ¸ia care ne intereseaz˘ a este prima, pentru c˘ a d˘ a und e dispersate divergente , ˆ ın timp ce a doua solut ¸ie reprezint˘ a unde dispersate con vergente. Mai mult, combinat ¸ia linear˘ a 1 2limǫ→0[Gk+iǫ+Gk−iǫ] =−1 4πcoskr r corespunde undelor stat ¸ionare. 144Evaluarea formal˘ a a integralei se poate face luˆ ınd k2−q2→k2+iǫ−q2, astfel c˘ a :/integraltext∞ −∞qeiqR k2−q2dq→/integraltext∞ −∞qeiqR (k2+iǫ)−q2dq . Aceasta este posibil pentru R >0, de aceea conturul pentru calcul va fi situat ˆ ın semiplanul complex superior. Astfel, polii inte grantului se afl˘ a ˆ ın: q=+ −√ k2+iǫ≃+ −(k+iǫ 2k). Procedeul de luare a limitei cˆ ınd ǫ→0 trebuie efectuat dup˘ a evaluarea integralei. Problema 7.2 Forma asimptotic˘ a a funct ¸iei radiale Cum s-a v˘ azut deja ˆ ın capitolul Atomul de hidrogen partea radial˘ a a ec. Schr¨ odinger se poate scrie: (d2 dr2+2 rd dr)Rnlm(r)−2m ¯h2[V(r) +l(l+1)¯h2 2mr2]Rnlm(r) +2mE ¯h2Rnlm(r) = 0. n,l,m sunt numerele cuantice sferice. De acum ˆ ınainte nu se vor ma i scrie din motive de comoditate. Reste funct ¸ia de und˘ a radial˘ a (depinde numai der). Vom presupune c˘ a potent ¸ialele cad la zero mai repede dec ˆ ıt 1/r, ¸ si ˆ ın plus c˘ a lim r→0r2V(r) = 0. Folosim acum u(r) =rR, ¸ si cum: (d2 dr2+2 rd dr)u r=1 rd2 dr2u, avem d2 dr2u+2m ¯h2[E−V(r)−l(l+1)¯h2 2mr2]u= 0. Not˘ am c˘ a potent ¸ialul prezint˘ a un termen suplimentar: V(r)→V(r) +l(l+1)¯h2 2mr2, care corespunde unei bariere centrifugale repulsive. Pent ru o particul˘ a liber˘ a V(r) = 0 ¸ si ecuat ¸ia devine [d2 dr2+2 rd dr)−l(l+1) r2]R+k2R= 0. Introducˆ ınd variabila ρ=kr, obt ¸inem d2R dρ2+2 ρdR dρ−l(l+1) ρ2R+R= 0. Solut ¸iile acestei ecuat ¸ii sunt a¸ sa numitele funct ¸ii Bessel sferice . Solut ¸ia regular˘ a este: jl(ρ) = (−ρ)l(1 ρd dρ)l(sinρ ρ), iar cea iregular˘ a : nl(ρ) =−(−ρ)l(1 ρd dρ)l(cosρ ρ). Pentruρmare, funct ¸iile de interes sunt funct ¸iile Hankel sferice : h(1) l(ρ) =jl(ρ) +inl(ρ) ¸ sih(2) l(ρ) = [h(1) l(ρ)]∗. De interes deosebit este comportamentul pentru ρ≫l: jl(ρ)≃1 ρsin(ρ−lπ 2) (51) nl(ρ)≃−1 ρcos(ρ−lπ 2). (52) 145¸ si atunci h(1) l≃−i ρei(ρ−lπ/2). Solut ¸ia regular˘ a ˆ ın origine este: Rl(r) =jl(kr) Forma asimtotic˘ a este (folosind ec. 51) Rl(r)≃1 2ikr[e−ikr−lπ/2−eikr−lπ/2]. Problema 7.3 Aproximat ¸ia Born pentru potent ¸iale Yukawa S˘ a consider˘ am un potent ¸ial de forma: V(r) =V0e−αr r, (53) cuV0¸ siαconstante reale ¸ si αpozitiv˘ a . Potent ¸ialul este atractiv sau re- pulsiv ˆ ın funct ¸ie de semnul lui V0; cu cˆ ıt este mai mare |V0|, cu atˆ ıt este mai intens potent ¸ialul. Presupunem c˘ a |V0|este suficient de mic pentru ca aproximat ¸ia Born s˘ a funct ¸ioneze. Conform formulei obt ¸ inute anterior, am- plitudinea de dispersie este dat˘ a de: f(B)(θ,ϕ) =−1 4π2mV0 ¯h2/integraltexte−iK·re−αr rd3r . Cum acest potent ¸ial depinde numai de r, integr˘ arile unghiulare se pot face u¸ sor, ajungˆ ınd astfel la forma: f(B)(θ,ϕ) =1 4π2mV0 ¯h24π |K|/integraltext∞ 0sin|K|re−αr rrdr . A¸ sadar, obt ¸inem: f(B)(θ,ϕ) =−2mV0 ¯h21 α2+|K|2. Din figur˘ a se observ˘ a c˘ a : |K|= 2ksinθ 2; prin urmare: σ(B)(θ) =4m2V2 0 ¯h41 [α2+4k2sinθ 22]2. Sect ¸iunea de dispersie total˘ a se obt ¸ine prin integrare: σ(B)=/integraltextσ(B)(θ)dΩ =4m2V2 0 ¯h44π α2(α2+4k2). 1468. UNDE PART ¸IALE Introducere Metoda undelor part ¸iale se refer˘ a la particule care inter act ¸ioneaz˘ a ˆ ın regiuni foarte mici de spat ¸iu cu o alta, care prin caracteristicile sale este cunoscut˘ a ca centru de ˆ ımpr˘ a¸ stiere (de exemplu faptul c˘ a se poate c onsidera fix˘ a ). ˆIn afara acestei regiuni, interact ¸ia ˆ ıntre cele dou˘ a parti cule se poate considera neglijabil˘ a . ˆIn acest fel este posibil s˘ a se descrie particula ˆ ımpr˘ a¸ s tiat˘ a cu urm˘ atorul Hamiltonian: H=H0+V , (1) undeH0corespunde Hamiltonianului de particul˘ a liber˘ a . Deci pr oblema noastr˘ a este de a rezolva urm˘ atoarea ecuat ¸ie: (H0+V)|ψ∝an}b∇acket∇i}ht=E|ψ∝an}b∇acket∇i}ht. (2) Este evident c˘ a spectrul va fi continuu (studiem cazul ˆ ımpr ˘ a¸ stierii elas- tice). Solut ¸ia ecuat ¸iei precedente este dat˘ a de: |ψ∝an}b∇acket∇i}ht=1 E−H0V|ψ∝an}b∇acket∇i}ht+|φ∝an}b∇acket∇i}ht. (3) Cu o analiz˘ a u¸ soar˘ a putem s˘ a vedem c˘ a pentru V= 0 obt ¸inem solut ¸ia |φ∝an}b∇acket∇i}ht, adic˘ a , solut ¸ia corespunz˘ atoare particulei libere. Tr ebuie notat c˘ a operatorul1 E−H0ˆ ıntr-un anumit sens este anomal, pentru c˘ a are un continuu de poli pe axa real˘ a care coincid cu valorile proprii ale lui H0. Pentru a ‘sc˘ apa’ de aceast˘ a problem˘ a s˘ a producem o mic˘ a deplas are ˆ ın direct ¸ia imaginar˘ a (±iǫ) a t˘ aieturii de pe axa real˘ a : |ψ±∝an}b∇acket∇i}ht=1 E−H0±iεV|ψ±∝an}b∇acket∇i}ht+|φ∝an}b∇acket∇i}ht (4) Aceast˘ a ecuat ¸ie este cunoscut˘ a ca ecuat ¸ia Lippmann-Sc hwinger. ˆIn final deplasarea polilor va fi ˆ ın sens pozitiv de la axa imaginar˘ a (pentru ca prin- cipiul de cauzalitate s˘ a nu fie violat [cf. Feynman]). S˘ a lu ˘ am reprezentarea x: ∝an}b∇acketle{tx|ψ±∝an}b∇acket∇i}ht=∝an}b∇acketle{tx|φ∝an}b∇acket∇i}ht+/integraldisplay d3x′/angbracketleftbigg x|1 E−H0±iε|x′/angbracketrightbigg ∝an}b∇acketle{tx′|V|ψ±∝an}b∇acket∇i}ht.(5) Primul termen din partea dreapt˘ a corespunde unei particul e libere ˆ ın timp ce al doilea termen se interpreteaz˘ a ca o und˘ a sferic˘ a care ‘iese’ din 147centrul de ˆ ımpr˘ a¸ stiere. Nucleul integralei anterioare se poate asocia cu o funct ¸ie Green (sau propagator) ¸ si este foarte simplu s˘ a s e calculeze: G±(x,x′) =¯h2 2m/angbracketleftbigg x|1 E−H0±iε|x′/angbracketrightbigg =−1 4πe±ik|x−x′| |x−x′|, (6) undeE= ¯h2k2/2m. A¸ sa cum am v˘ azut mai ˆ ınainte funct ¸ia de und˘ a se poate scrie ca o und˘ a plan˘ a plus una sferic˘ a care iese di n centrul de ˆ ımpr˘ a¸ stiere (pˆ ın˘ a la un factor constant): ∝an}b∇acketle{tx|ψ+∝an}b∇acket∇i}ht=ek·x+eikr rf(k,k′). (7) M˘ arimeaf(k,k′) care apare ˆ ın ec. 7 se cunoa¸ ste ca amplitudine de dispersie ¸ si se poate scrie explicit ˆ ın forma: f(k,k′) =−1 4π(2π)32m ¯h2∝an}b∇acketle{tk′|V|ψ+∝an}b∇acket∇i}ht. (8) S˘ a definim acum un operator T astfel c˘ a : T|φ∝an}b∇acket∇i}ht=V|ψ+∝an}b∇acket∇i}ht (9) Dac˘ a multiplic˘ am ecuat ¸ia Lippmann-Schwinger cu V ¸ si fo losim definit ¸ia anterioar˘ a obt ¸inem: T|φ∝an}b∇acket∇i}ht=V|φ∝an}b∇acket∇i}ht+V1 E−H0+iεT|φ∝an}b∇acket∇i}ht. (10) Iterˆ ınd ecuat ¸ia anterioar˘ a (caˆ ın teoria de perturbat ¸ ii) putem obt ¸ine aproximat ¸ia Born ¸ si corect ¸iile sale de ordin superior. Metoda undelor part ¸iale S˘ a consider˘ am acum cazul unui potent ¸ial central nenul. ˆIn acest caz, pe baza definit ¸iei (9) se deduce c˘ a operatorul Tcomut˘ a cu /vectorL2¸ si/vectorL; de aici se spune c˘ aTeste un operator scalar. ˆIn acest fel pentru a u¸ sura calculele este convenabil s˘ a se foloseasc˘ a coordonate sferice, pen tru c˘ a dat˘ a simetria problemei, operatorul Tva fi diagonal. Acum, s˘ a vedem ce form˘ a ia expresia (8) pentru amplitudinea de dispersie: f(k,k′) = const./summationdisplay lml′m′/integraldisplay dE/integraldisplay dE′∝an}b∇acketle{tk′|E′l′m′∝an}b∇acket∇i}ht∝an}b∇acketle{tE′l′m′|T|Elm∝an}b∇acket∇i}ht∝an}b∇acketle{tElm|k∝an}b∇acket∇i}ht, (11) 148unde const.=−1 4π2m ¯h2(2π)3. Dup˘ a cˆ ıteva calcule se obt ¸ine: f(k,k′) =−4π2 k/summationdisplay l/summationdisplay mTl(E)Ym l(k′)Ym∗ l(k). (12) Alegˆ ınd sistemul de coordonate astfel ca vectorul ks˘ a aib˘ a aceea¸ si direct ¸ie cu axa orientat˘ a z, se ajunge la concluzia c˘ a la amplitudin ea de dispersie vor contribui numai armonicele sferice cu m egal cu zero; dac ˘ a definimθca unghiul ˆ ıntre k¸ sik′vom avea: Y0 l(k′) =/radicaligg 2l+ 1 4πPl(cosθ). (13) Cu urm˘ atoarea definit ¸ie: fl(k)≡−πTl(E) k, (14) ec. (12) se poate scrie ˆ ın forma urm˘ atoare: f(k,k′) =f(θ) =∞/summationdisplay l=0(2l+ 1)fl(k)Pl(cosθ). (15) Pentrufl(k) se poate da o interpretare simpl˘ a pe baza dezvolt˘ arii une i unde plane ˆ ın unde sferice. Astfel putem scrie funct ¸ia ∝an}b∇acketle{tx|ψ+∝an}b∇acket∇i}htpentru valori mari ale lui rˆ ın forma ∝an}b∇acketle{tx|ψ+∝an}b∇acket∇i}ht=1 (2π)3/2/bracketleftigg eikz+f(θ)eikr r/bracketrightigg = 1 (2π)3/2/bracketleftigg/summationdisplay l(2l+ 1)Pl(cosθ)/parenleftigg eikr−ei(kr−lπ) 2ikr/parenrightigg +/summationdisplay l(2l+ 1)fl(k)Pl(cosθ)eikr r/bracketrightigg =1 (2π)3/2/summationdisplay l(2l+ 1)Pl(cosθ) 2ik/bracketleftigg [1 + 2ikfl(k)]eikr r−ei(kr−lπ) r/bracketrightigg .(16) Aceast˘ a expresie se poate interpreta dup˘ a cum urmeaz˘ a . C ei doi termeni exponent ¸iali corespund unor unde sferice, primul unei und e emergente ¸ si al doilea uneia convergente; ˆ ın plus efectul deˆ ımpr˘ a¸ stie re se vede convenabilˆ ın coeficientul undei emergente, care este egal cu unu cˆ ınd nu e xist˘ a dispersor. 149Deplas˘ ari (¸ sifturi) de faz˘ a S˘ a ne imagin˘ am acum o suprafat ¸˘ a ˆ ınchis˘ a centrat˘ a ˆ ın dispersor. Dac˘ a pre- supunem c˘ a nu exist˘ a creare ¸ si nici anihilare de particul e se verific˘ a : /integraldisplay j·dS= 0, (17) unde regiunea de integrare este suprafat ¸ definit˘ a mai ˆ ına inte ¸ si jeste densi- tatea de curent de probabilitate. ˆIn plus, datorit˘ a conserv˘ arii momentului cinetic ecuat ¸ia anterioar˘ a trebuie s˘ a se verifice pentru fiecare und˘ a part ¸ial˘ a (cu alte cuvinte, toate undele part ¸iale au diferite valori ale proiect ¸iilor mo- mentului cinetic, ceea ce le face diferite. Formularea teor etic˘ a ar fi echiva- lent˘ a dac˘ a se consider˘ a pachetul de unde ca un flux de parti cule care nu interact ¸ioneaz˘ a ˆ ıntre ele; mai mult, pentru c˘ a potent ¸ ialul problemei noastre este central, momentul cinetic al fiec˘ arei “particule” se v a conserva ceea ce ne permite s˘ a spunem c˘ a particulele continu˘ a s˘ a fie acele a¸ si). Cu aceste considerat ¸ii, putem s˘ a afirm˘ am c˘ a atˆ ıt unda divergent˘ a cˆ ıt ¸ si cea emergent˘ a difer˘ a cel mult printr-un factor de faz˘ a . Deci, dac˘ a defin im: Sl(k)≡1 + 2ikfl(k) (18) trebuie s˘ a avem |Sl(k)|= 1. (19) Rezultatele anterioare se pot interpreta cu ajutorul conse rv˘ arii proba- bilit˘ at ¸ilor ¸ si erau de ‘a¸ steptat’ pentru c˘ a nu am presu pus c˘ a exist˘ a creare ¸ si anihilare de particule, astfel c˘ a influent ¸a centrului dis persor consist˘ a pur ¸ si simplu ˆ ın a ad˘ auga un factor de faz˘ a ˆ ın componentele unde lor emergente ¸ si ˆ ın virtutea unitarit˘ at ¸ii factorului de faz˘ a ˆ ıl putem s crie ˆ ın forma: Sl=e2iδl, (20) undeδleste real ¸ si este funct ¸ie de k. Pe baza definit ¸iei (18) pute m s˘ a scriem: fl=e2iδl−1 2ik=eiδlsin(δl) k=1 kcot(δl)−ik. (21) Sect ¸iunea total de ˆ ımpr˘ a¸ stiere ia forma urm˘ atoare: σtotal=/integraldisplay |f(θ)|2dΩ = 1501 k2/integraldisplay2π 0dφ/integraldisplay1 −1d(cos(θ))/summationdisplay l/summationdisplay l′(2l+ 1)(2l′+ 1)eiδlsin(δl)eiδl′sin(δl′)PlPl′ =4π k2/summationdisplay l(2l+ 1)sin2(δl′). (22) Determinarea ¸ sifturilor de faz˘ a S˘ a consider˘ am acum un potent ¸ial V astfel c˘ a se anul˘ a pen trur > R , unde parametrul R se cunoa¸ ste ca “raz˘ a de act ¸iune a potent ¸ial ului”, astfel c˘ a regiunear>R evident trebuie s˘ a corespund˘ a unei unde sferice liber˘ a ( neper- turbat˘ a ). Pe de alt˘ a parte, forma cea mai general˘ a de dezv oltare a unei unde plane ˆ ın unde sferice este: ∝an}b∇acketle{tx|ψ+∝an}b∇acket∇i}ht=1 (2π)3/2/summationdisplay lil(2l+ 1)Al(r)Pl(cosθ) (r>R ), (23) unde coeficientul Aleste prin definit ¸ie: Al=c(1) lh(1) l(kr) +c(2) lh(2) l(kr), (24) ¸ si undeh(1) l¸ sih(2) lsunt funct ¸iile Hankel sferice ale c˘ aror forme asimpto- tice sunt: h(1) l∼ei(kr−lπ/2) ikr h(2) l∼−e−i(kr−lπ/2) ikr. Examinˆ ınd forma asimptotic˘ a a expresiei (23) care este: 1 (2π)3/2/summationdisplay l(2l+ 1)Pl/bracketleftigg eikr 2ikr−e−i(kr−lπ) 2ikr/bracketrightigg , (25) se poate vedea c˘ a : c(1) l=1 2e2iδlc(2) l=1 2. (26) Aceasta permite scrierea funct ¸iei de und˘ a radial˘ a pentr ur>R ˆ ın forma: Al=e2iδl[cosδljl(kr)−sinδlnl(kr)]. (27) 151Folosind ecuat ¸ia anterioar˘ a putem evalua evalua derivat a sa logaritmic˘ a ˆ ın r=R, i.e., exact la frontiera zonei de act ¸iune a potent ¸ ialului: βl≡/parenleftbiggr AldAl dr/parenrightbigg r=R=kR/bracketleftigg j′ lcosδl−n′ l(kR)sinδl jlcosδl−nl(kR)sinδl/bracketrightigg . (28) j′ leste derivata lui jlˆ ın raport cu kr¸ si evaluat˘ a ˆ ın r=R. Alt rezultat important pe care ˆ ıl putem obt ¸ine cunoscˆ ınd resultatul a nterior este ¸ siftul de faz˘ a : tanδl=kRj′ l(kR)−βljl(kR) kRn′ l(kR)−βlnl(kR). (29) Pentru a obt ¸ine solut ¸ia completa a problemei ˆ ın acest caz este necesar s˘ a se fac˘ a calculele pentru r < R , adic˘ a , ˆ ın interiorul razei de act ¸iune al potent ¸ialului. Pentru cazul unui potent ¸ial central, ecu at ¸ia Schr¨ odinger ˆ ın trei dimensiuni este: d2ul dr2+/parenleftbigg k2−2m ¯h2V−l(l+ 1) r2/parenrightbigg ul= 0, (30) undeul=rAl(r) este supus˘ a condit ¸iei de frontier˘ a ul|r=0= 0. Astfel, putem calcula derivata logaritmic˘ a, care ˆ ın virtutea pro priet˘ at ¸ii de continu- itate a derivatei logaritmice (care este echivalent˘ a cu co ntinuitatea derivatei ˆ ıntr-un punct de discontinuitate) ne conduce la: βl|interior =βl|exterior. (31) Un exemplu: ˆ ımpr˘ a¸ stierea pe o sfer˘ a solid˘ a S˘ a trat˘ am acum un caz specific. Fie un potent ¸ial definit pri n: V=/braceleftigg ∞r<R 0r>R .(32) Se ¸ stie c˘ a o particul˘ a nu poate penetra ˆ ıntr-o regiune un de potent ¸ialul este infinit, astfel c˘ a funct ¸ia de und˘ a trebuie s˘ a se anul eze ˆ ınr=R; din faptul c˘ a sfera este impenetrabil˘ a rezult˘ a deasemenea c ˘ a : Al(r)|r=R= 0. (33) Astfel, din ec. (27) avem: tanδl=jl(kR) nl(kR). (34) 152Se vede c˘ a se poate calcula u¸ sor ¸ siftul de faz˘ a pentru ori cel. S˘ a con- sider˘ am acum cazul l= 0 (ˆ ımpr˘ a¸ stiere de und˘ a s) pentru care avem: δl=−kR ¸ si din ec. (27): Al=0(r)∼sinkr krcosδ0+coskr krsinδ0=1 krsin(kr+δ0). (35) Vedem c˘ a fat ¸˘ a de mi¸ scarea liber˘ a exist˘ a o contribut ¸i e adit ¸ional˘ a de o faz˘ a . Este clar c˘ aˆ ıntr-un caz mai general diferitele unde vor av ea diferite ¸ sifturi de faz˘ a ceea ce provoac˘ a o distorsie tranzitorie ˆ ın pachetu l de unde dispersat. S˘ a studiem acum cazul energiilor mici, i.e., kR<< 1.ˆIn acest caz, expresiile pentru funct ¸iile Bessel (folosite pentru a descrie funct ¸ iile Hankel sferice) sunt urm˘ atoarele: jl(kr)∼(kr)l (2l+ 1)!!(36) nl(kr)∼−(2l−1)!! (kr)l+1(37) care ne conduc la: tanδl=−(kR)2l+1 (2l+ 1)[(2l−1)!!]2. (38) Din aceast˘ a formul˘ a putem s˘ a vedem c˘ a o contribut ¸ie apr eciabil˘ a la ¸ siftul de faz˘ a este dat de undele cu l= 0 ¸ si cum δ0=−kRobt ¸inem pentru sect ¸iunea eficace: σtotal=/integraldisplaydσ dΩdΩ = 4πR2. (39) De aici se ajunge la concluzia c˘ a sect ¸iunea eficace de ˆ ımpr ˘ a¸ siere cuantic˘ a este de patru ori mai mare decˆ ıt sect ¸iunea eficace clasic˘ a ¸ si coincide cu aria total˘ a a sferei dure. Pentru valori mari ale energiei p achetului incident se poate lucra cu ipoteza c˘ a toate valorile lui lpˆ ın˘ a la o valoare maxim˘ a lmax∼kRcontribuie la sect ¸iunea eficace total˘ a : σtotal=4π k2l∼kR/summationdisplay l=0(2l+ 1)sin2δl. (40) ˆIn acest fel, pe baza ec. (34) avem: sin2δl=tan2δl 1 + tan2δl=[jl(kR)]2 [jl(kR)]2+ [nl(kR)]2∼sin2/parenleftbigg kR−lπ 2/parenrightbigg ,(41) 153unde am folosit expresiile: jl(kr)∼1 krsin/parenleftbigg kr−lπ 2/parenrightbigg nl(kr)∼−1 krcos/parenleftbigg kr−lπ 2/parenrightbigg . Vedem c˘ aδldescre¸ ste cuπ 2de fiecare dat˘ a c˘ a lse increment˘ a cu o unitate, ¸ si deci este evident c˘ a seˆ ındepline¸ ste sin2δl+sin2δl+1= 1. Aproximˆ ınd sin2δl cu valoarea sa medie1 2, este simplu de obt ¸inut rezultatul pe baza sumei de numere impare: σtotal=4π k2(kR)21 2= 2πR2. (42) Odat˘ a ˆ ın plus rezultatul calculului bazat pe metodele de m ecanic˘ a cuan- tic˘ a , de¸ si asem˘ an˘ ator, difer˘ a totu¸ si de rezultatul c lasic. S˘ a vedem care este originea factorului 2; mai ˆ ıntˆ ıi vom separa ec. (15) ˆ ın do u˘ a p˘ art ¸i: f(θ) =1 2ikl=kR/summationdisplay l=0(2l+1)e2iδlPlcos(θ)+i 2kl=kR/summationdisplay l=0(2l+1)Plcos(θ) =frefl+fumbr˘ a. (43) Evaluˆ ınd/integraltext|frefl|2dΩ: /integraldisplay |frefl|2dΩ =2π 4k2lmax/summationdisplay l=0/integraldisplay −11 (2l+ 1)2[Plcos(θ)]2d(cosθ) =πlmax2 k2=πR2. (44) Analizˆ ınd acum fumbr˘ apentru unghiuri mici avem: fumbr˘ a∼i 2k/summationdisplay (2l+ 1)J0(lθ)∼ik/integraldisplayR 0bJ0(kbθ)db=iRJ1(kRθ) θ.(45) Aceast˘ a formul˘ a este destul de cunoscut˘ aˆ ın optic˘ a , fii nd formula pentru difract ¸ia Fraunhofer; cu ajutorul schimbului de variabil ˘ az=kRθputem s˘ a evalu˘ am integrala/integraltext|fumbr˘ a|2dΩ: /integraldisplay |fumbr˘ a|2dΩ∼2πR2/integraldisplay∞ 0[J1(z)]2 zdz∼πR2. (46) ˆIn sfˆ ır¸ sit, neglijˆ ınd interferent ¸a ˆ ıntre frefl¸ sifumbr˘ a(pentru c˘ a faza oscileaz˘ aˆ ıntre 2 δl+1= 2δl−π). Se obt ¸ine astfel rezultatul (42). Am etichetat unul dintre termeni cu titlul de umbr˘ a , pentru c˘ a originea sa se explic˘ a u¸ sor 154dac˘ a se apeleaz˘ a la comportamentul ondulatoriu al partic ulei dispersate (din punct de vedere ‘fizic’ nu exist˘ a nici o diferent ¸˘ a ˆ ıntre u n pachet de und˘ a ¸ si o particul˘ a ˆ ın acest caz). Originea sa const˘ a ˆ ın compone ntele pachetului de unde ˆ ımpr˘ a¸ stiate ˆ ınapoi ceea ce produce o diferent ¸˘ a d e faz˘ a fat ¸˘ a de undele incidente ducˆ ınd la o interferent ¸˘ a distructiv˘ a . ˆImpr˘ a¸ stiere ˆ ın cˆ ımp coulombian S˘ a consider˘ am acum un exemplu clasic ¸ si ceva mai complica t: ˆ ımpr˘ a¸ stierea de particule ˆ ıntr-un cˆ ımp coulombian. Pentru acest caz ec uat ¸ia Schr¨ odinger este: /parenleftigg −¯h2 2m∇2−Z1Z2e2 r/parenrightigg ψ(r) =Eψ(r), E > 0, (47) undemeste masa redus˘ a a sistemuluiˆ ın interact ¸ie ¸ si evident E >0 deoarece trat˘ am cazul dispersiei f˘ ar˘ a producere de nici un fel de s t˘ ari legate. Ecuat ¸ia anterioar˘ a este echivalent˘ a urm˘ atoarei expresii (pent ru valori adecuate ale constantelor k¸ siγ) : /parenleftbigg ∇2+k2+2γk r/parenrightbigg ψ(r) = 0. (48) Dac˘ a nu consider˘ am bariera centrifugal˘ a a potent ¸ialul ui efectiv (unde s) ne g˘ asim ˆ ın condit ¸iile unei interact ¸iuni coulombiene p ure ¸ si putem propune o solut ¸ie de forma: ψ(r) =eik·rχ(u), (49) cu u=ikr(1−cosθ) =ik(r−z) =ikw , k·r=kz . ψ(r) este solut ¸ia complet˘ a a ecuat ¸iei Schr¨ odinger ¸ si se po ate a¸ stepta un comportament asimptotic format din dou˘ a p˘ art ¸i, respect iv de und˘ a plan˘ a eik·r¸ si und˘ a sferic˘ a r−1eikr. Definind noi variabile: z=z w =r−z λ =φ , cu ajutorul relat ¸iilor anterioare, ec. (48) ia forma: /bracketleftigg ud2 du2+ (1−u)d du−iγ/bracketrightigg χ(u) = 0. (50) 155Pentru a rezolva aceast˘ a ecuat ¸ie, trebuie studiat mai ˆ ın tˆ ıi comportamen- tul s˘ au asimptotic, dar cum acesta a fost deja prezentat, fu nct ¸ia de und˘ a asimptotic˘ a normalizat˘ a care se obt ¸ine ˆ ın final ca rezul tat al tuturor cal- culelor anterioare este: ψk(r) =1 (2π)3/2/parenleftigg ei[k·r−γln(kr−k·r)]+fc(k,θ)ei[kr+γln2kr] r/parenrightigg . (51) Dup˘ a cum vedem, funct ¸ia de und˘ a anterioar˘ a prezint˘ a te rmeni care o fac s˘ a difere apreciabil de ec. (7). Acest lucru se datoreaz ˘ a faptului c˘ a fort ¸a coulombian˘ a este de raz˘ a infinit˘ a de act ¸iune. Efe ctuarea calculului exact pentru amplitudinea de ˆ ımpr˘ at ¸iere coulombian˘ a e ste destul de dificil de realizat. Aici vom da numai rezultatul final pentru funct ¸ ia de und˘ a normalizat˘ a : ψk(r) =1 (2π)3/2/parenleftigg ei[k·r−γln(kr−k·r)]+g∗ 1(γ) g1(γ)γ 2ksin(θ/2)2ei[kr+γln2kr] r/parenrightigg , (52) undeg1(γ) =1 Γ(1−iγ). ˆIn ceea ce prive¸ ste analiza de unde part ¸iale o vom reduce la prezentarea rezultatelor deja discutate ˆ ıntr-un mod cˆ ıt mai clar posi bil. Mai ˆ ıntˆ ıi scriem funct ¸ia de und˘ a (49) ψ(r) ˆ ın urm˘ atoarea form˘ a : ψ(r) =eik·rχ(u) =Aeik·r/integraldisplay Ceuttiγ−1(1−t)−iγdt , (53) undeAeste o constant˘ a de normalizare ¸ si toat˘ a partea integral ˘ a este trans- formata Laplace invers˘ a a transformatei directe a ecuat ¸i ei (50). O form˘ a convenabil˘ a a ecuat ¸iei anterioare este: ψ(r) =A/integraldisplay Ceik·r(1−t)eikrt(1−t)d(t,γ)dt (54) cu d(t,γ) =tiγ−1(1−t)−iγ−1. (55) ˆIn cadrul analizei de unde part ¸iale proced˘ am la a scrie: ψ(r) =∞/summationdisplay l=0(2l+ 1)ilPl(cosθ)Al(kr), (56) unde Al(kr) =A/integraldisplay Ceikrtjl[kr(1−t)](1−t)d(t,γ). (57) 156Aplicˆ ınd relat ¸iile ˆ ıntre funct ¸iile Bessel sferice ¸ si funct ¸iile Hankel sferice avem: Al(kr) =A(1) l(kr) +A(2) l(kr). (58) Evaluarea acestor coeficient ¸i nu o vom prezenta aici (fiind d estul de com- plicat˘ a ). Rezult˘ a c˘ a : A(1) l(kr) = 0 (59) A(2) l(kr)∼−Aeπγ/2 2ikr[2πig1(γ)]/parenleftig e−i[kr−(lπ/2)+γln 2kr]−e2iηl(k)ei[kr−(lπ/2)+γln2kr]/parenrightig (60) unde e2iηl(k)=Γ(1 +l−iγ) Γ(1 +l+iγ). (61) Calculul amplitudinii de ˆ ımpr˘ a¸ stiere coulombian˘ a Dac˘ a efectu˘ am transformata Laplace a ec. (50) obt ¸inem: χ(u) =A/integraldisplay Ceuttiγ−1(1−t)−iγdt . (62) ConturulCmerge de la−∞la∞¸ si se ˆ ınchide pe deasupra axei reale. ˆIn aceste condit ¸ii, vedem c˘ a exist˘ a doi poli: cˆ ınd t= 0 ¸ sit= 1. Cu schimbul de variabil˘ a s=utobt ¸inem: χ(u) =A/integraldisplay C1essiγ−1(u−s)−iγ. (63) χ(u) trebuie s˘ a fie regular˘ a ˆ ın zero ¸ si ˆ ıntr-adev˘ ar: χ(0) = (−1)−iγA/integraldisplay C1es sds .= (−1)−iγA2πi (64) Luˆ ınd acum limita u→∞, s˘ a facem o deplasare infinitezimal˘ a (pentru a elimina faptul c˘ a polii sunt pe contur) ¸ si cu un schimb de v ariabil˘ a astfel c˘ as u=−(s0±iε) iκ, vedem c˘ a aceast˘ a expresie tinde la zero cˆ ınd u→ ∞ . Deci, putem s˘ a dezvolt˘ am ( u−s) ˆ ın serie de puteri des upentru polul cu s= 0. Dar aceast˘ a dezvoltare nu este bun˘ a ˆ ın s= 1, pentru c˘ a ˆ ın acest cazs=−s0+i(κ±ε) ¸ si de aici rezult˘ a c˘ as u= 1−(s0±iε) κtinde la 1 cˆ ınd 157κ→∞; dar dac˘ a facem schimbul de variabil˘ a s′=s−uaceast˘ a problem˘ a se elimin˘ a : χ(u) =A/integraldisplay C2/parenleftbigg [essiγ−1(u−s)−iγ]ds+ [es′+u(−s′)iγ(u+s′)iγ−1]ds′/parenrightbigg . (65) Dezvoltˆ ınd seriile de puteri este u¸ sor de calculat integr alele precedente, dar ˆ ın rezultat trebuie s˘ a se ia limitas u→0 pentru a obt ¸ine formele asimp- totice corecte pentru ˆ ımpr˘ a¸ stierea coulombian˘ a : χ(u)∼2πiA/bracketleftig u−iγg1(γ)−(−u)iγ−1eug2(γ)/bracketrightig 2πg1(γ) =i/integraldisplay C2essiγ−1ds 2πg2(γ) =i/integraldisplay C2ess−iγds . (66) Dup˘ a acest ¸ sir de schimb˘ ari de variabile, ne ˆ ıntoarcem l as-ul original pentru a obt ¸ine: (u∗)iγ= (−i)iγ[k(r−z)]iγ=eγπ/2eiγlnk(r−z) (u)−iγ= (i)−iγ[k(r−z)]−iγ=eγπ/2e−iγlnk(r−z). (67) Calculul lui χodat˘ a efectuat, este echivalent cu a avea ψk(r) pornind de la (49). Aproximat ¸ia eikonal˘ a Vom face o scurt˘ a expozit ¸ie a aproximat ¸iei eikonale a c˘ a rei filosofie este aceea¸ si cu cea care se face cˆ ınd se trece de la optica ondula torie la optica geometric˘ a ¸ si de aceea este corect˘ a cˆ ınd potent ¸ialul v ariaz˘ a put ¸in pe distant ¸e comparabile cu lungimea de und˘ a a pachetului de unde disper sat, adic˘ a , pentru cazul E >>|V|. Astfel aceast˘ a aproximat ¸ie poate fi considerat˘ a ca o aproximat ¸ie cuasiclasic˘ a . Maiˆ ıntˆ ıi propunem c˘ a func t ¸ia de und˘ a cuasiclasic˘ a are forma binecunoscut˘ a : ψ∼eiS(r)/¯h, (68) unde S satisface ecuat ¸ia Hamilton-Jacobi, cu solut ¸ia: S ¯h=/integraldisplayz −∞/bracketleftbigg k2−2m ¯h2V/parenleftig/radicalbig b2+z′2/parenrightig/bracketrightbigg1/2 dz′+ constant˘ a . (69) 158Constanta aditiv˘ a se alege ˆ ın a¸ sa fel ˆ ıncˆ ıt: S ¯h→kz pentru V→0. (70) Termenul care multiplic˘ a potent ¸ialul se poate interpret a ca un schimb de faz˘ a al pachetului de unde, avˆ ınd urm˘ atoarea form˘ a ex plicit˘ a ∆(b)≡−m 2k¯h2/integraldisplay∞ −∞V/parenleftig/radicalbig b2+z2/parenrightig dz . (71) ˆIn cadrul metodei de unde part ¸iale aceast˘ a aproximat ¸ie a re urm˘ atoarea aplicat ¸ie. S ¸tim c˘ a aproximat ¸ia eikonal˘ a este corect˘ a la energii ˆ ınalte, unde exist˘ a multe unde part ¸iale care contribuie la dispersie. Astfel putem con- sideralca o variabil˘ a continu˘ a ¸ si prin analogie cu mecanica clas ic˘ a punem l=bk.ˆIn plus, cum deja am ment ¸ionat lmax=kR, care substituit ˆ ın expresia (15) conduce la: f(θ) =−ik/integraldisplay bJ0(kbθ)[e2i∆(b)−1]db . (72) 8P. Probleme Problema 8.1 S˘ a se obt ¸in˘ a deplasarea de faz˘ a (¸ siftul) ¸ si sect ¸iune a diferent ¸ial˘ a deˆ ımpr˘ a¸ stiere la unghiuri mici pentru un centru de ˆ ımpr˘ a¸ stiere de poten t ¸ialU(r) =α r2. S˘ a se t ¸in˘ a cont de faptul c˘ a ˆ ın ˆ ımpr˘ a¸ stierile de ungh iuri mici principala contribut ¸ie o dau undele part ¸iale cu lmari. Solut ¸ie : Rezolvˆ ınd ecuat ¸ia R′′ l+/bracketleftigg k2−l(l+ 1) r2−2mα ¯h2r2/bracketrightigg = 0 cu condit ¸iile de frontier˘ a Rl(0) = 0,Rl(∞) =N, undeNeste un num˘ ar finit, obt ¸inem Rl(r) =A√rIλ(kr), 159undeλ=/bracketleftigg (l+1 2)2+2mα ¯h2/bracketrightigg1/2 ¸ siIeste prima funct ¸ia Bessel modificat˘ a . Pentru determinarea lui δlse folose¸ ste expresia asimptotic˘ a a lui Iλ: Iλ(kr)∝/parenleftbigg2 πkr/parenrightbigg1/2 sin(kr−λπ 2+π 4). Prin urmare δl=−π 2/parenleftbigg λ−l−1 2/parenrightbigg =−π 2 /bracketleftigg (l+1 2)2+2mα ¯h2/bracketrightigg1/2 −/parenleftbigg l+1 2/parenrightbigg . Condit ¸ialmari de care se vorbe¸ ste ˆ ın problem˘ a ne conduce la: δl=−πmα (2l+ 1)¯h2, de unde se vede c˘ a |δl|≪1 pentrulmari. Din expresia general˘ a a amplitudinii de ˆ ımpr˘ a¸ stiere f(θ) =1 2ik∞/summationdisplay l=0(2l+ 1)Pl(cosθ)(e2iδl−1), la unghiuri mici e2iδl≈1 + 2iδl¸ si ∞/summationdisplay l=0Pl(cosθ) =1 2sinθ 2. Astfel: f(θ) =−παm k¯h21 2sinθ 2. Prin urmare: dσ dΩ=π3α2m 2¯h2Ectgθ 2. 160

arXiv:physics/0003107v1 [physics.gen-ph] 31 Mar 2000Hierarchic Theory of Condensed Matter: Long relaxation, macroscopic oscillations and the effects of magnetic field Alex Kaivarainen JBL, University of Turku, FIN-20520, Turku, Finland http://www.karelia.ru/˜alexk H2o@karelia.ru Materials, presented in this original article are based on following publications: [1]. A. Kaivarainen. Book: Hierarchic Concept of Mat- ter and Field. Water, biosystems and elementary particles. New York, NY, 1995, ISBN 0-9642557-0-7 [2]. A. Kaivarainen. New Hierarchic Theory of Matter General for Liquids and Solids: dynamics, thermodynam- ics and mesoscopic structure of water and ice (see URL: http://www.karelia.ru/˜alexk [see New articles]). [3]. Hierarchic Concept of Condensed Matter and its Interaction with Light: New Theories of Light Refraction, Brillouin Scattering and M¨ ossbauer effect (http://www.ka relia.ru/˜alexk [see New articles]). [4]. A. Kaivarainen. Hierarchic Concept of Condensed Matter : Interrelation between mesoscopic and macroscopic properties (see URL: http://www.karelia.ru/˜alexk [see New articles]). See also set of previous articles at Los-Alamos archives: http://arXiv.org/find/physics/1/au:+Kaivarainen A/0/1/0/all/0/1 Contents: Summary of new Hierarchic theory, general for liquids and solids. 1. Theoretical background for macroscopic oscillations in condensed matter 2. The hypothesis of [entropy - mass - time] interrelation 3. The entropy - information content of matter as a hierarchic system 4. Experimentally revealed macroscopic oscillations 5. Phenomena in water and aqueous systems, induced by magnetic field Coherent radio-frequency oscillations in water, revealed by C. Smith 16 Influence of weak magnetic field on the properties of solid bodies 7 Possible mechanism of perturbations of nonmagnetic materials under magnetic treatment Summary of new Hierarchic theory, general for liquids and solids A basically new hierarchic quantitative theory, general fo r solids and liquids, has been developed. It is assumed, that unharmonic oscillations of particles in any con- densed matter lead to emergence of three-dimensional (3D) s uperpo- sition of standing de Broglie waves of molecules, electroma gnetic and acoustic waves. Consequently, any condensed matter could b e con- sidered as a gas of 3D standing waves of corresponding nature . Our approach unifies and develops strongly the Einstein’s and De bye’s models. Collective excitations, like 3D standing de Broglie waves o f molecules, representing at certain conditions the mesoscopic molecul ar Bose con- densate, were analyzed, as a background of hierarchic model of con- densed matter. The most probable de Broglie wave (wave B) length is deter- mined by the ratio of Plank constant to the most probable impu lse of molecules, or by ratio of its most probable phase velocity to fre- quency. The waves B are related to molecular translations (t r) and librations (lb). As the quantum dynamics of condensed matter does not follow i n general case the classical Maxwell-Boltzmann distribution, the re al most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular Bose condensat ion in solids and liquids at temperatures, below boiling point. It is one o f the most important results of new theory, which we have confirmed by computer sim ulations on examples of water and ice. Four strongly interrelated new types of quasiparticles (collective excita- tions) were introduced in our hierarchic model: 1.Effectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the effectons (flickering clusters); 3.Transitons are the intermediate [ a⇋b] transition states of the trandlb effectons; 24.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons andconvertons. Primary effectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating ions, atoms or molecules. The volume of effectons (tr and lb) may contain fro m less than one, to tens and even thousands of molecules. The first condition m eans validity ofclassical approximation in description of the subsystems of the effect ons. The second one points to quantum properties of coherent clusters due to mesoscopic molecular Bose condensation . The liquids are semiclassical systems because their primar y (tr) effectons contain less than one molecule and primary (lb) effectons - mo re than one molecule. The solids are quantum systems totally because both kind of t heir primary effectons (tr and lb) are molecular Bose condensates .These conse- quences of our theory are confirmed by computer calculations . The 1st order [ gas→liquid ] transition is accompanied by strong decreasing of rotational (librational) degrees of freedom due to emerg ence of primary (lb) effectons and [ liquid →solid] transition - by decreasing of translational degrees of freedom due to Bose-condensation of primary (tr) effecton s. In the general case the effecton can be approximated by par- allelepiped with edges corresponding to de Broglie waves le ngth in three selected directions (1, 2, 3), related to the symmetry of the molecular dynamics. In the case of isotropic molecular moti on the effectons’ shape may be approximated by cube. The edge-length of primary effectons (tr and lb) can be consid ered as the ”parameter of order”. The in-phase oscillations of molecules in the effectons corr espond to the effecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the effectons differ in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state of the effectons has a common feature with Fr¨ olich’s polar mode. The(a→b)or(b→a)transition states of the primary effectons (tr and lb), defined as primary transitons, are accompanied b y a change in molecule polarizability and dipole moment withou t density fluctuations. At this case they lead to absorption or radiati on of IR photons, respectively. Superposition (interception) of three internal standing I R pho- tons of different directions (1,2,3) - forms primary electro magnetic deformons (tr and lb). On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density fluctuations, leading to absorption or radiation of phonons . Superposition resulting from interception of standing phonons in three direc- tions (1,2,3), forms secondary acoustic deformons (tr and lb). 3Correlated collective excitations of primary and secondary effectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromag- netic deformons ,lead to origination of macroeffectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeffec tonsin the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereffectons. In turn, the coherent excitation of both: tr andlb macrodeformons and macroconvertons in the same volume means creation of superdeformons. Su- perdeformons are the biggest (cavitational) fluctuations, leading to microbub- bles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, reflects all of possible combinations of the four basic ones [ 1-4], intro- duced above. This set of collective excitations in the form o f ”gas” of 3D standing waves of three types: de Broglie, acoustic and el ectro- magnetic - is shown to be able to explain virtually all the pro perties of all condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need the semi-empiric intermolecular potentials f or calculations, which are unavoidable in existing theories of many body systems. T he potential energy of intermolecular interaction is involved indirectly in di mensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into ac- count their different properties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra; [2]- Sound velocity; [3]- Density; [4]- Refraction index (extrapolated to the infinitive wave leng th of photon ). The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, to ev aluate more than 300 important characteristics of any condensed matter. Amo ng them are such as: total internal energy, kinetic and potential energies, heat-capacity and ther- mal conductivity, surface tension, vapor pressure, viscos ity, coefficient of self- diffusion, osmotic pressure, solvent activity, etc. Most of calculated parameters are hidden, i.e. inaccessible to direct experimental measu rement. The new interpretation and evaluation of Brillouin light sc attering and M¨ ossbauer effect parameters may also be done on the basis of h ierarchic the- ory. Mesoscopic scenarios of turbulence, superconductivi ty and superfluity are elaborated. Some original aspects of water in organization and large-sc ale dynamics of biosystems - such as proteins, DNA, microtubules, membrane s and regulative role of water in cytoplasm, cancer development, quantum neu rodynamics, etc. have been analyzed in the framework of Hierarchic theory. 4Computerized verification of our Hierarchic theory of matte r on examples of water and ice is performed, using special comput er pro- gram: Comprehensive Analyzer of Matter Properties (CAMP, c opy- right, 1997, Kaivarainen). The new optoacoustic device (CA MP), based on this program, with possibilities much wider, than t hat of IR, Raman and Brillouin spectrometers, has been proposed (see U RL: http://www.karelia.ru/˜alexk). This is the first theory able to predict all known experimenta l temperature anomalies for water and ice. The conformity bet ween theory and experiment is very good even without any adjustab le pa- rameters. The hierarchic concept creates a bridge between micro- and m acro- phenomena, dynamics and thermodynamics, liquids and solid s in terms of quantum physics. =================================================== ========== 1. Theoretical background for macroscopic oscillations in condensed matter One of the consequences of our concept is of special interest . It is the possibility for oscillation processes in solids and liq uids. The law of energy conservation is not violated thereupon because th e energies of two quasiparticle subsystems related to effectons and def ormons, can change in opposite phases. The total internal energy of m atter keeps almost constant. The equilibrium shift between subsystems of condensed matt er can be in- duced by any external factor, i.e. pressure or field. The rela xation time, neces- sary for system to restore its equilibrium, corresponding t o minimum of potential or free energy after switching off external factor can be term ed ”memory” of system. The energy redistribution between primary and secondary eff ec- ton and deformon subsystems may have a periodical character , cou- pled with the oscillation of the (a⇔b)equilibrium constant of pri- mary effectons (Ka⇔b)and correlated oscillations of primary electro- magnetic deformons concentration if dissipation processe s are weak or reversible. According to our model (Table 1 of [1, 2]) the (a→b) transition of primary effecton is related to photon absorpti on, i.e. a decrease in primary electromagnetic deformon concentrati on, while the(b→a)transition on the contrary, radiate photons. If, there- fore, the [a⇔b]and/bracketleftbig ¯a⇔¯b/bracketrightbig equilibriums are shifted right ward, and equilibrium constants Ka⇔band¯Ka⇔bdecreases, then concentrations of primary and secondary deformons (ndand¯nd)also decreases. If Ka⇔bgrows up, i.e. the concentration of primary effectons in a-st ates increases, then ndincreases. We remind that (a)and(b)states of 5the primary effectons correspond to the more and less stable m olec- ular clusters (see Introduction). In accordance with our mo del, the strong interrelation exists between dynamic equilibrium o f primary and secondary effectons. Equilibrium of primary effectons is more sensitive to any perturbations. However, the equilibrium s hift of sec- ondary effectons affect the total internal energy, the entrop y change and possible mass defect (see below) stronger than that of pr imary effectons. As we have shown (Fig. 28a,b of [1] and [3]), the scattering ab ility of A-states is more than two times as high as that of B-states. Their polarizability, refraction index and dielectric permeabi lity are also higher. It makes possible to register the oscillations in th e condensed matter in different ways. In accordance with our theory the oscillation of refraction index must induce the corresponding changes of viscosity and self -diffusion in condensed matter (see Chapter 11 of [1] and [4]). The diffus ion variations are possible, for example, in solutions of macro molecules or other Brownian particles. In such a way self-organizatio n in space and time gradually may originate in appropriate solvents, s olutions, colloid systems and even in solid bodies. The period and amplitude of these oscillations depend on the times of relaxation processes which are related to the activation energy of equilibrium shifts in the effectons, polyeffectons or cohere nt super- clusters of primary effectons subsystems. The reorganizations in the subsystems of translational and libra- tional effectons, macro- and supereffectons, as well as chain -like poly- effectons, whose stabilities and sizes differ from each other , must go on at different rates. It should, therefore, be expected that in the exper- iment the presence of several oscillation processes would b e revealed. These processes are interrelated but going with different pe riods and amplitudes. Concomitant oscillations of self-diffusion ra te also must be taken into account. In such a way Prigogine’s dissipative struc- tures could be developed (Prigogine, 1984). Instability in the degree of ordering in time and space is accompanied by the slow oscil lation of entropy of the whole macroscopic system. The coherent extraterrestrial cosmic factors and gravitat ional in- stabilities can induce long relaxation and oscillation pro cesses in wa- ter and other kind of condensed matter (Udaltsova, et. al., 1 987). 2. The hypothesis of [entropy - mass - time] interrelation The second law of thermodynamics for the closed system havin g the perma- nent number of particles and the constant internal energy is : dQi T=dSi≥0 (1) 6where: dQiappears to be due to the irreversible processes within the sy stem and is referred to as uncompensated heat (Prigogine, 1980, 1984, Babloyantz, 1986). This law means the only possibility of the processes, accomp anied by the increase of entropy (S) (related neither to chemical or nucl ear reactions). The statistical interpretation of entropy is expressed wit h the Boltzmann formula: S= k·lnP≃k·lnW (2) where: kis the Boltzmann constant, P is the statistical weight, whic h is proportional to the number of realizations (number of micro states) for the given state of the macroscopic system (W). Correspondingly the probability of one of this microstates is:pi= 1/W. It follows from (2) that the second law of thermodynamics exp resses the fact that the system tends to the most probable state. dS=k dlnW ≥0 (3) or dQi=TdS i=kT dlnW i (4) Boltzmann has put forward the hypothesis that the irreversi bility (i.e. asymme- try) of time is determined with the irreversibility of proce sses according to the second law of thermodynamics. Prigogine has modified and dev eloped this idea (Prigogine, 1984), introducing the notions of the internal time and micro- scopic operator of entropy . The ”ARROW OF TIME” and irreversibility is a result of asymmetry of physically accessible states after Prigogin. The micro- scopic mechanism of irreversibility is discussed also in Ch apter 8 of Part II of book [1]. It follows from the second law and formulae (1.12) from Part I I of this book, that the mass of a system of N similar particles with the mass ( M≃Nm) either does not change in the course of time, or decreases: dt t=−1 2dM M≥0, (5) where: dM is the change in the mass of a macroscopic system rel ated to electromagnetic and acoustic radiation, and also to interm olecular interaction induced small mass defect. It follows from (5) that the positive pace of time in the syste m is related to the decrease in its mass, while the negative pace of time - t o the increase in its mass. Hence, the coherent oscillation processes in th e system (which 7are not related to chemical reactions) are related to oscill ation of mass [M]. It is known that phonons and photons do not have the rest masses. Therefore, the coherent periodical energy exchange between subsystem s of effectons and deformons (primary and secondary) must be accompanied by ch anges in the mass of the effecton subsystem and total system. The macroscopic fluctuations of mass values for solid bodies were registered experimentally indeed (Kozyrev, 1958). The oscillations of temperature and thermal conductivity a lso must accom- pany the oscillation of secondary acoustic deformons conce ntration. The total internal energy of the closed system can be represe nted as a sum of contributions from the primary ( UaandUb), secondary ( ¯Uaand¯Ub), macro (UA, UB) and super ( UA∗, UB∗) effectons, convertons and contributions of cor- responding deformons and transitons (see formula 4.3). AtUtot=const the exchange of energies between the subsystems of effectons and deformons can be approximately represented as: −d(mc2 x)N≈d(h¯νd)N (6) where: Nis the number of particles in the system;/bracketleftbig cx= (vgrvph)1/2=const/bracketrightbig is the characteristic wave B velocity in the sys- tem, which is equal to the product of the generalized group ( vgr) and phase ( vph) velocities of wave B of particles; [ d(h¯νd)N] is the change of energy contribution of secondary deformons, which is much bigger than that of pri mary deformons due to low concentration of latter; [ dm] is the defect of mass per particle in the system as a result of energy exchange between subsystems of t he effectons and deformons. Because phonons are carriers of the heat energy, then the unc ompensated heat ( dQi) in eq.(1) could be easily related to the irreversible incre ment of secondary acoustic deformons and convertons energy or to th e uncertainty in this energy: dQi= TdS = Nd(hνd)tr,lb (7) or dQi= TdS = −d(mc2 x)N=−d(Mc2 x)tr,lb (8) where: ¯νd= [(¯νd)tr,lb+ (¯νd)ca,cb,cMd] + (νd)tr,lb (¯νd)tr,lbis the resulting frequency of secondary deformons (tr and lb ); (¯νd)ca,cb,cMdis the resulting frequency of convertons and related deform ons; M=Nmis the total mass of particles in the system. 8(νd)tr,lbare a frequencies of tr and lb IR photons. Let us enter the notion of absolute entropy (S) as a measure of the uncompensated heat change (∆Qi/T), which could occur at (¯a⇔¯b) transitions of mean effectons and convertons subsystems at t he given temperature: S=/integraldisplaydQi T(9) Putting (7) and (8) into (9) and multiplying numerator and de nominator, to Boltzmann constant (k) keeping in mind (5), we derive a new ap proximate formula for absolute entropy: S=∆Q T=N∆(3h¯νd) T≈3Nk/parenleftbiggh¯νa kT/parenrightbigg ≈ −∆Mc2 x T=2M T∆t ttr,lb(10) where: ¯ νd=|¯νa−¯νb| ∼¯νais a frequency of secondary acoustic deformon; ¯ νa is calculated from formula (2.54). Formula (10) relates the positive entropy value not only to h eat radiation from the system or the increase of the deformons contributio n into the total internal energy (∆ Qi>0), determined the corresponding decrease of its mass (∆M <0), but also to positive time course in this system (∆ t >0). The interesting experiments done by Kosyrev (1958) confirm o ne of the consequences of our theory, that the shift of equilibrium be tween subsystems of the effectons and acoustic deformons to the latter ones sho uld decrease the mass of body. In his experiments with special balance it was s hown that the activation of phonons with sufficiently high energy by means o f sound generator - decreases the mass of solid body: (100 −800)g. The value of decreasing was proportional to the total mass of body. The relative mass decreasing was estimated as: ∆M/M=(2.3−3.4)·10−5 In liquids and solids, the concentration of secondary defor mons can be lower than the concentration of molecules. Therefore, in a genera l case N in the formula (10) must be substituted by the number of secondary d eformons in the system: (Nd)tr,lb=V(nd)tr,lb, (11) where V is the volume of the system; (¯nd)tr,lb=8 9π/parenleftbigg¯νd vres/parenrightbigg3 tr,lb 9is the concentration of secondary translational and librat ional effectons; νres ph=/parenleftbig ¯ν1 ph¯ν2 ph¯ν3 ph/parenrightbig1/3(12) is the resulting frequency of secondary deformons; ¯ ν1,2,3 phare calculated from (3.15);vresis the resulting thermal phonons velocity, which is equal to isotropic hypersonic velocity in liquids and to transversal one ( v) in solids. The total entropy of a condensed substance ( Stot) is approximately a sum of secondary effecton contribution ( tr and lb ) and contribution ofconvertons. Using (10) and (11), we obtain: Stot≈Str+Slb+Scon=V0h T/bracketleftbig¯ntr d/parenleftbig 3¯νres d/parenrightbig tr+ ¯nlb d/parenleftbig 3¯νres d/parenrightbig lb/bracketrightbig + +V0h Tncon3/bracketleftbig(νa ef)tr−(νa ef)lb/bracketrightbig con(13) where: ¯ntr dand ¯nlb dcorrespond to (11); nconis a concentration of convertons, equal to that of primary librational effectons ( nlb); (νd)res tr,lbare the resulting frequencies of secondary deformons (tr an d lb). Knowing the positions of translational and librational ban ds in the oscilla- tory spectra and sound velocity, one can estimate the entrop y of matter at each temperature using (13). At constant pressure and volume the change of enthalpy (H) is equal to the change of internal energy (U). Therefore, the formulae that we have obtained for (U) and (S) allow to calculate also changes in free energy : ∆GP,V= ∆U−T∆S (14) The qualitative correctness of the formula obtained for ent ropy is obvious from (10) in the form: Str,lb≈Nk(3hνa/kT)tr,lb+Nk(3h∆νa/kT)acon≃Nk(3hνa/kT)tr,lb (15) It follows from (2.54) and (15) that: a) at T→0 :h¯νa/kT→0 and S→0; b) at decreasing νpin the melting point at the growth of Tboth h¯νa/kT andSgrow up; c) if the mixing of liquids and gases leads to the weakening of pair interaction in effectons or clusters, then νp= (Eb−Ea)/hdecreases and Sincreases. Equalizing (2) and right part of (15) we have in the framework of our ap- proximations: 10S=klnP≈k/parenleftbigg N0hνa kT/parenrightbigg =Rhνa kT(16) from (16) we can derive an approximate formula for statistic al weight: P≈exp/parenleftbigg N0h¯νa kT/parenrightbigg tr,lb(17) According to (16) the law of entropy growth means the striving of the real quantum properties of the substance to ideal properties. It means the tending of primary effectons energy in the (a) state ( hνa) to the thermal equilibrium value (kT): ∆S >0, if [ hνa→kT] (18) Competition between the discrete quantum energy distribut ion and its tendency to kT may be a reason for instability of different parame- ters of condensed matter. This may lead to origination of mac roscopic oscillations in the system interrelated with entropy, temp erature and mass (eq.10). Such oscillations could be considered as a kin d of self- organization process due to feedback links in a hierarchic s ystem of interrelated quasiparticles of matter. 3. The entropy - information content of matter as a hierarchi c system The statistical weigh for macrosystem (P), equal to number o f microstates (W), corresponding to given macrostate, neces sary for entropy calculation using (2) could be presented as: W=N! N1!·N2!·. . .·Nq!(19) where: N=N1+N2+...Nq (20) is the total number of molecules in macrosystem; Niis the number of molecules in the i-th state; qis the number of independent states of all quasiparticles in macrosys- tem. We can subdivide macroscopic volume of 1 cm3into 24 types of quasiparticles in accordance with our hierarchic model (see Table 1 of [1, 2] ). 11In turn, each type of the effectons (primary, secondary, macr o- and super- effectons) is subdivided on two states: ground (a,A) and exci ted (b,B) states. Taking into account two ways of the effectons origination - du e to thermal trans- lations (tr) and librations (lb), excitations, related to [ lb/tr] convertons, macro- and super deformons, the total number of independent states is 24 also. It is equal to number of independent relative probabilities of ex citations, composing partition function Z (see eq.4.2 of [1, 2]). Consequently, i n eqs.(4.19 and 4.20) we have: q= 24 Thenumber of molecules, in the unit of volume of condensed matter (1cm3), participating in each of 24 excitation states (i) can be calc ulated as: Ni=(v)i V0/N0·ni·Pi Z=N0 V0Pi Z(21) where: ( v)i= 1/niis the volume of (i) quasiparticle, equal to reciprocal valu e of its concentration ( ni);N0andV0are Avogadro number and molar volume, correspondingly; Z is partition function and Piare relative probabilities of in- dependent excitations in composition of Z(eq.4.2). The total number of molecules of (i)-type of excitation in an y big volume of matter ( VMac) is equal to Ni Mac=NiVMac=VMacN0 V0Pi Z(21a) Putting (20) into (18) and (19), we can calculate the statist ical weight and entropy from eq.(2). For large values of N iit is convenient to use a Stirling formula: Ni= (2πN)1/2(N/e)N·exp(Θ /12N)∼(2πN)1/2(N/Θ)N(21b) Using this formula and (20), one can obtain the following exp ression for entropy: S=k·lnW=−k·q/summationdisplay i(Ni+1 2)lnNi+ const = S1+S2+...Si (22) From this eq. we can see that the temperature increasing or [s olid→liquid] phase transition will lead to the entropy elevation: ∆S=SL−SS=k·ln(WL/WS)>0 (23) 12It follows from (22, 20) and (19) that under conditions when ( Pi) and Niun- dergoes oscillations it can lead to oscillations of contrib utions of different types of quasiparticles to the entropy of system and even to oscill ations of total en- tropy of system as an additive parameter. The coherent oscil lations of Piand Nican be induced by different external fields: acoustic, electr omagnetic and gravitational. Macroscopic autooscillations may arise sp ontaneously also in the sensitive and highly cooperative systems. Experimental evidence for such phenomena will be discussed in the next section. The notions of probability of given microstate ( pi= 1/W), entropy ( Si) and information ( Ii) are strongly interrelated. The smaller the probability th e greater is information (Nicolis 1986): Ii= lg21 pi=−lg2pi= lg2Wi (24) where piis defined from the Boltzmann distribution as: pi=exp(−Ei/kT)/summationtext∞ m=0exp(−nmhνi/kT)(25) where n mis quantum number; h is the Plank constant; Ei=hνiis the energy of (i)-state. There is strict relation between the entropy and informatio n, leading from comparison of (24) and (2): Si= (kBln 2)Ii= 2.3·10−24Ii (26) The information entropy is given as expectation of the infor mation in the system (Nicolis,1986; Haken, 1988). < I > = ΣPilg2(1/pi) =−Σpilg2(pi) (27) From (26) and (22) we can see that variation of probability piand/or Niin (20) will lead to changes of entropy and information, characteri zing the matter as a hierarchical system. Thereduced information (entropy), characterizing its quality , related to selected collective excitation of any type of condensed m atter, we introduce here as a product of corresponding component of information [Ii] to the number of molecules (atoms) with similar dynamic properties in com position of this excitation: qi= (vi/vm) =N0/(V0ni) (27a) 13where: vi= 1/niis the volume of quasiparticle, reversible to its concentra tion (ni);vm=V0/N0is the volume, occupied by one molecule. The product of (27) and (27a), i.e. the reduced information gives the quantitative characteristic not only about quantity but al so about the quality of the information: (Iq)i=pilg2(1/pi)·N0/(V0ni) (27b) This new formula could be considered as a useful modification of known Shennon equation. 4. Experimentally revealed macroscopic oscillations A series of experiments was conducted in our laboratory to st udy oscil- lations in the buffer (pH 7.3) containing 0.15 M NaCl as a contr ol system and immunoglobulin G solutions in this buffer at the followin g concentrations: 3·10−3; 6·10−3; 1.2·10−2and 2.4·10−2mg/ml . The turbidity ( D∗) of water and the solutions were measured every 10 sec- onds with the spectrophotometer at λ= 350 nm. Data were obtained automat- ically with the time constant 5 s during 40 minutes. The numbe r ofD∗values in every series was usually equal to 256. The total number of t he fulfilled series was more than 30. The time series of D∗were processed by the software for time series analysis. The time trend was thus subtracted and the autocovariance fu nction and the spectral density were calculated. The empty quartz cuvette with the optical path about 1 cm were used as a basic control. Only the optical density of water and water dissolved substa nces, which really exceeded background optical density in the control s eries were taken into account. It is shown that the noise of the photoelectronic mu ltiplier does not contribute markedly to dispersion of D∗. The measurements were made at temperatures of 17 ,28,32, and 370. The period of the trustworthily registered oscillation proces ses related to changes inD∗, had 2 to 4 discrete values over the range of (30 −600)sunder our conditions. It does not exclude the fact that the autooscill ations of longer or shorter periods exist. For example, in distilled water at 320Cthe oscillations of the scattering ability are characterized by periods of 30, 1 20 and 600 s and the spectral density amplitudes 14, 38 and 78 (in relative units ), respectively. With an increase in the oscillation period their amplitude also i ncreases. At 280C the periods of the values 30, 41 and 92s see have the correspon ding normalized amplitudes 14.7, 10.6 and 12.0. Autooscillations in the buffer solution at 280Cin a 1 cm wide cuvette with the optical way length 1 cm (i.e. square section) are charact erized with periods: 34,52,110 and 240 s and the amplitudes: 24 ,33,27 and 33 relative units. In the cuvette with a smaller (0.5 cm) or larger (5 cm) optical wa velength at the 14same width (1 cm) the periods of oscillations in the buffer cha nge insignificantly. However, amplitudes decreased by 50% in the 5 cm cuvette and b y 10-20% in the 0.5 cm-cuvette. This points to the role of geometry of space w here oscillations occur, and to the existence of the finite correlation radius o f the synchronous processes in the volume. But this radius is macroscopic and c omparable with the size of the cuvette. The dependence of the autooscillations amplitude on the con centration of the protein - immunoglobulin G has a sharp maximum at the conc entration of 1.2·10−2mg/ml . There is a background for considering it to be a manifesta- tion of the hydrodynamic Bjorkness forces between the pulsi ng macromolecules (K¨ aiv¨ ar¨ ainen, 1987). Oscillations in water and water solutions with nearly the sa me periods have been registered by the light-scattering method by Cherniko v (1985). Chernikov (1990d) has studied the dependence of light scatt ering fluctua- tions on temperature , mechanic perturbation and magnetic fi eld in water and water hemoglobin and DNA solution. It has been shown that an i ncrease in temperature results in the decline of long-term oscillatio n amplitude and in the increase of short-time fluctuation amplitude. Mechanical m ixing removes long- term fluctuations and over 10 hours are spent for their recove ry. Regular fluc- tuations (oscillations) appear when the constant magnetic field above 240 A/m is applied; the fluctuations are retained for many hours afte r removing the field. The period of long-term oscillations has the order of 10 minu tes. It has been assumed that the maintenance of long-range correlation of m olecular rotation- translation fluctuation underlies the mechanism of long-te rm light scattering fluctuations. It has been shown (Chernikov, 1990b) that a pulsed magnetic fi eld (MF), like constant MF, gives rise to light scattering oscillations in water and other liquids containing H atoms: glycerin, xylol, ethanol, a mixture of u nsaturated lipids. All this liquids also have a distinct response to the constan t MF. ”Spontaneous” and MF-induced fluctuations are shown to be associated with t he isotropic com- ponent of scattering. These phenomena do not occur in the non proton liquid (carbon tetrachloride) and are present to a certain extent i n chloroform (con- taining one hydrogen atom in its molecule). The facts obtain ed indicate an important role of hydrogen atoms and cooperative system of h ydrogen bonds in ”spontaneous” and induced by external perturbations macro scopic oscillations. The understanding of such phenomena can provide a physical b asis for of self-organization (Prigogine, 1980, 1984, Babloyantz, 19 86), the biological sys- tem evolution (Shnol, 1979, Udaltsova et al., 1987), and che mical processes oscillations (Field and Burger, 1988). It is quite probable that macroscopic oscillation processe s in biological liq- uids, e.g. blood and liquor, caused by the properties of wate r are involved in animal and human physiological processes. We have registered the oscillations of water activity in the protein-cell system by means of light microscopy using the apparatus ”Morphoqua nt”, through the change of the erythrocyte sizes, the erythrocytes being ATP -exhausted and fulfilling a role of the passive osmotic units. The revealed o scillations have a 15few minute-order periods. Preliminary data obtained from the analysis of oscillation processes in the human cerebrospinal liquor indicate their dependence on so me pathology. Per- haps, the autooscillations spectrum of the liquor can serve as a sensitive test for the physiological status of the organism. The liquor is a n electrolyte and its autooscillations can be modulated with the electromagn etic activity of the brain. We suggested that the activity of the central nervous system and the biological rhythms of the organism are dependent with th e os- cillation processes in the liquor. If it is the case, then the directed influence on these autooscillation processes, for example, by means of magnetic field makes it possible to regulate the state of the o rganism and its separate organs. Some of reflexotherapeutic effects c an be caused by correction of biorhythms. During my stay in laboratory of Dr. G.Salvetty in the Institu te of Atomic and Molecular Physics in Pisa (Italy) in 1992, the oscillati ons of heat capac- ity [Cp] in 0.1 M phosphate buffer (pH7) and in 1% solution of lysozyme in the same buffer at 200Cwere revealed. The sensitive adiabatic differential mi- crocalorimeter was used for this aim. The biggest relative a mplitude changing: [∆Cp]/[Cp]∼(0.5±0.02)% occurs with period of about 24 hours, i.e. corre- sponds to circadian rhythm. Such oscillations could be stimulated by the variation of ma gnetic and gravitational conditions of the Earth during this perio d. 5. Phenomena in water and aqueous systems, induced by magnet ic field In the works of (Semikhina and Kiselev, 1988, Kiselev et al., 1988, Berezin et al., 1988) the influence of the weak magnetic field was revea led on the di- electric losses, the changes of dissociation constant, den sity, refraction index, light scattering and electroconductivity, the coefficient o f heat transition, the depth of super-cooling for distilled water and for ice also. This field used as a modulator a geomagnetic action. The absorption and the fluorescence of the dye (rhodamine 6G) and protein in solutions also changed under the action of weak fields on wa ter. The latter circumstance reflects feedback links in the guest-host, or s olute -solvent system. The influence of constant and variable magnetic fields on wate r and ice in the frequency range 104−108Hzwas studied. The maximum sensitivity to field action was observed at the frequency νmax= 105Hz. In accordance with our calculations, this frequency corresponds to frequency of superdeformons excitations in water (see Fig. 48d of [1] and article [2]). A few of physical parameters changed after the long (nearly 6 hour) influence of the variable fields ( ˜H), modulating the geomagnetic field of the tension [ H= Hgeo] with the frequency ( f) in the range of (1 −10)·102Hz(Semikhina and Kiselev, 1988, Kiselev et al., 1988): 16H=Hcos2πft (28) In the range of modulating magnetic field (H) tension from 0 .08A/mto 212 A/m theeight maxima of dielectric losses tangent in the above mentioned ( f) range were observed. Dissociation constant decreases more than other param- eters (by 6 times) after the incubation of ice and water in mag netic field. The relaxation time (”memory”) of the changes, induced in water by fields was in the interval from 0.5 to 8 hours. The authors interpret the experimental data obtained as the influence of magnetic field on the probability of proton transfer along th e net of hydrogen bonds in water and ice, which lead to the deformation of this n et. Theequilibrium constant for the reaction of dissociation: H2O⇔OH−+H+ in ice is less by almost six orders ( ≃106) than that for water. On the other hand the values of the field-induced effects in ice are several times more than in water, and the time for reaching them in ice is less. So, the above in terpretation is doubtful. In the framework of our concept all the aforementioned pheno m- ena could be explained by the shift of the (a⇔b)equilibrium of primary translational and librational effectons to the left .In turn, this shift stimulates polyeffectons or coherent superclusters g rowth, under the in- fluence of magnetic fields. Therefore, parameters such as the refraction index, dielectric permeability and light scattering have to enhan ce symbatically, while theH2Odissociation constant depending on the probability of supe rdeformons must decrease. The latter correlate with declined electric conductance. As far, the magnetic moments of molecules within the coheren t superclusters or polyeffectons formed by primary libration al effectons are additive, then the values of changes induced by magnetic field must be proportional to polyeffecton sizes. These sizes are m arkedly higher in ice than in water and decrease with increasing temp erature. Inasmuch the effectons and polyeffectons interact with each o ther by means of phonons (i.e. the subsystem of secondary deformons), and the velocity of phonons is higher in ice than in water, then the saturation of all concomitant effects and achievement of new equilibrium state in ice is fas ter than in water. The frequencies of geomagnetic field modulation, at which ch anges in the properties of water and ice have maxima can correspond to the eigen-frequencies of the [ a⇔b] equilibrium constant of primary effectons oscillations, d etermined by [assembly ⇔disassembly] equilibrium oscillations for coherent super clusters or polyeffectons. The presence of dissolved molecules (ions, proteins) in wat er or ice can influ- ence on the initial [ a⇔b] equilibrium dimensions of polyeffectons and,consequentl y the interaction of solution with outer field. 17Narrowing of1H-NMR lines in a salt-containing water and calcium bicar- bonate solution was observed after magnetic field action. Th is indicates that the degree of ion hydration is decreased by magnetic treatment. On the other hand, the width of the resonance line in distilled water remains unchanged after 30 minute treatment in the field (135 kA/m ) at water flow rate of 60 cm/s (Klassen, 1982). The hydration of diamagnetic ions ( Li+, Mg2+, Ca2+) decreases, while the hydration of paramagnetic ions ( Fe3+, Ni2+, Cu2+) increases. It leads from corresponding changes in ultrasound velocity in ion soluti ons (Duhanin and Kluchnikov, 1975). There are numerous data which pointing to an increase the coa gulation of different particles and their sedimentation velocity after magnetic field treat- ment. These phenomena provide a reducing the scale formatio n in heating sys- tems, widely used in practice. Crystallization and polymer ization also increase in magnetic field. It points to decrease of water activity. Increasing of refraction index (n) and dielectric permeabi lity(ǫ≃ n2)and symbatic enhancement of water viscosity (Minenko, 1981 ) are in total accordance with our viscosity theory (eqs. 11.44 an d 11.45 of [1] and article [4]). It follows from our mesoscopic model that the increase of (n) is related to the increase of molecular polarizability ( α) due to the shift of ( a⇔b)tr,lb equilibrium of primary effectons leftward under the action o f magnetic field. On the other hand, distant Van der Waals interactions and conse quently dimensions of primary effectons depend on α. This explains the elevation of surface tension of liquids after magnetic treatment (see Chapter 11 of [1] or [4]). The leftward shift of ( a⇔b)tr,lbequilibrium of primary effectons must lead to decreasing of water activity due to (n2) increasing and structural fac- tor (T/U tot) decreasing its structure ordering. Corresponding change s in the vapor pressure, freezing, and boiling points, coagulation , polymerization and crystallization are the consequences of this shift and wate r activity decreasing. It follows from mesoscopic theory that any changes in conden sed matter properties must be accompanied by change of such para meters as: 1) density; 2) sound velocity; 3) positions of translational and librational bands in osci llatory spectra; 4) refraction index. Using our equations and computer simulations by means of ela bo- rated software (CAMP: Comprehensive Analyzer of Matter Pro per- ties), it is possible to obtain from these changes very detai led infor- mation (more than 200 parameters) about even small perturba tions of matter on meso- and macroscopic levels. 18Available experimental data indicate that all of above ment ioned 4 experi- mental parameters of water have been changed indeed after ma gnetic treatment. Minenko (1981) has shown that bidistilled water density increases by about 0 .02% after magnetic treatment (540 kA/m , flow rate 80 cm/s). Sound velocity in distilled water increases to 0.1% after treatment under conditions: 160 kA/m and flow rate 60 cm/s. Thepositions of the translational and librational bands of water were also changed after magnetic treatment in 415 kA/m (Klassen, 1982). Coherent radio-frequency oscillations in water, revealed by C. Smith It was shown experimentally by C. Smith (1994) that the water display a coherent properties. He shows that water is capabl e of retaining the frequency of an alternating magnetic field. Fo r a tube of water placed inside a solenoid coil, the threshold for the alternating magnetic field, potentising electromagnetic frequencies i nto water, is 7.6µT(rms). He comes to conclusion that the frequency informatio n is carried on the magnetic vector potential. He revealed also that in a course of yeast cells culture synch ronously dividing, the radio-frequency emission around 1 MHz (1061/s), 7- 9 MHz (7-9 ×1061/s)and 50-80 MHz (5-9 ×1071/s)with very narrow bandwidth (˜50 Hz) might be observed for a few minutes. These frequencies could correspond to frequencies of differ ent wa- ter collective excitations, introduced in our Hierarchic t heory, like [lb/tr] macroconvertons, the [ a⇋b]lbtransitons, etc. (see Fig. 48 of [1] and [2]), taking into account the deviation of water prop erties in the colloid and biological systems as respect to pure one. Cyril Smith has proposed that the increasing of coherence ra dius in water could be a consequence of coherent water clusters asso ciation due to Josephson effect (Josephson, 1965): tunneling of mole cules between clusters. As far primary librational effectons are r esulted from partial Bose-condensation of molecules, this idea loo ks quite acceptable in the framework of our Hierarchic theory. The coherent oscillations in tube with water, revealed by C. Smith could be induced by coherent electromagnetic radiation of m icro- tubules of cells, produced by correlated intra-MTs water ex citations (see Section 17.5 and Fig. 48 of [1]). The biological effects of magnetically treated water are ver y im- portant practically. For example, hemolysis of erythrocyt es is more vigorous in magnetically pretreated physiological soluti ons (Trincher, 1967). Microwave radiation induces the same effect (Il’ina e t al., 1979). But after boiling such effects in the treated solution s have been disappeared. It is shown that magnetic treatment of water st rongly stimulates the growth of corn and plants (Klassen, 1982). 19Now it is obvious that a systematic research program is neede d to understand the physical background of multilateral effects of magne- tized water. 6. Influence of weak magnetic field on the properties of solid b odies It has been established that as a result of magnetic field acti on on solids with interaction energy ( µBH) much less than kT, many properties of matter such as hardness, parameters of crystal cells and others cha nge significantly. The short-time action of magnetic field on silicon semicondu ctors is followed by a very long (many days) relaxation process. The action of m agnetic field was in the form of about 10 impulses with a length of 0.2 ms and an am plitude of about 105A/m. The most interesting fact was that this relaxation had an o scil- latory character with periods of about several days (Maslov sky and Postnikov, 1989). Such a type of long period oscillation effects has been found i n magnetic and nonmagnetic materials. This points to the general nature of the macroscopic oscilla tion phenomena in solids and liquids. The period of oscillations in solids is much longer than in li quids. This may be due to stronger deviations of the energy of ( a) and ( b) states of primary effectons and polyeffectons from thermal equilibrium and muc h lesser probabil- ities of transiton and deformon excitation. Consequently, the relaxation time of (a⇔b)tr,lbequilibrium shift in solids is much longer than in liquids. T he oscillations originate due to instability of dynamic equil ibrium between the sub- systems of effectons and deformons. 7. Possible mechanism of perturbations of nonmagnetic mate rials under magnetic treatment We shall try to discuss the interaction of magnetic field with dia- magnetic matter like water as an example. The magnetic susce pti- bility ( χ) of water is a sum of two opposite contributions (Eisenberg and Kauzmann, 1969): 1) average negative diamagnetic part, induced by external m ag- netic field: ¯χd=1 2(χxx+χyy+χzz)∼=−14.6(±1.9)·10−6 2) positive paramagnetism related to the polarization of wa ter molecule due to asymmetry of electron density distribution , existing without external magnetic field. Paramagnetic susceptibil ity(χp)of H2Ois a tensor with the following components: 20χp xx= 2.46·10−6;χp yy= 0.77·10−6;χp zz= 1.42·10−6(29) The resulting susceptibility: χH2= ¯χd+ ¯χp∼=−13·10−6 The second contribution in the magnetic susceptibility of w ater is about 10 times lesser than the first one. But the first contribu tion to the magnetic moment of water depends on external magnetic field and must disappear when it is switched out in contrast to seco nd one. The coherent primary librational effectons of water even in l iquid state con- tain about 100 molecules/bracketleftBig (nef M)lb≃100/bracketrightBig at room temperature (Fig. 7a of [1] or Fig.4a of [2]). In ice ( nef M)lb≥104. In (a)-state the vibrations of all these molecules are synchronized in the same phase, and in ( b)-state - in counterphase. Correlation of H2Oforming effectons means that the energies of interaction of water molecules with external magnetic field are additive: ǫef=nef M·µpH (30) In such a case this total energy of effecton interaction with fi eld may exceed thermal energy: ǫef> kT (31) In the case of polyeffectons formation this inequality becom es much stronger. It follows from our model that interaction of magnetic field w ith (a)-state of the effectons must be stronger than that with ( b)-state due to the additivity of the magnetic moments of coherent molecules: ǫef a> ǫef b(32) Consequently, magnetic field shifts ( a⇔b)tr,lbequilibrium of the effectons leftward. At the same time it minimizes the potential energy of matter, because potential energy of ( a)-state ( Va) is lesser than ( Vb): Va< Vband Ea< E b, (33) where Ea=Va+Ta kin;Eb=Vb+Tb kinare total energies of the effectons. We keep in mind that the kinetic energies of ( a) and ( b)-states are equal: Ta kin=Tb kin=p2/2m. 21These energies decreases with increasing of the effectons di mensions, deter- mined by the most probable impulses in selected directions: λ1,2,3=h/p1,2,3 The energy of interaction of magnetic field with deformons as a transition state of effectons must be even less than ǫef bdue to lesser order of molecules in this state and reciprocal compensation of their magnetic moment s: ǫd< ǫef b≤ǫef a (34) This important inequality means that as a result of external magnetic field action the shift of ( a⇔b)tr,lbleftward is reinforced by leftward shift of equilib- rium [effectons ⇋deformons] subsystems of matter. If water is flowing in a tube it increases the relative orienta tions of all ef- fectons in volume and stimulate the coherent superclusters formation. All the above discussed effects must increase. Similar ordering phe nomena happen in a rotating tube with liquid. After switching off the external magnetic field the relaxatio n ofinduced ferro- magnetism in water begins. It may be accompanied by the oscillatory beh avior of (a⇔b)tr,lbequilibrium. All the experimental effects discussed above c an be explained as a consequence of orchestrated in volume ( a⇔b) equilibrium oscillations. Remnant ferromagnetism in water was experimentally establ ished using a SQUID superconducting magnetometer by Kaivarainen et al. in 1992 (unpublished data). Water was treated in constant ma gnetic field 50Gfor two hours. Then it was frozen and after switching off external magnetic field the remnant ferromagnetism was regi stered at helium temperature. Even at this low temperature a slow re lax- ation time- dependent decrease of ferromagnetic signal was revealed. These results point to the correctness of the proposed mecha nism of magnetic field - water interaction. The attempt to make a theory of magnetic field influence on wate r, based on other model were made earlier (Yashkichev, 1980). H ow- ever, this theory does not take into account the quantum prop erties of water and cannot be considered as satisfactory one. The comprehensive material obtained by Udaltsova, Kolombe t and Shnol (1987) when studying various macroscopic oscillatio ns reveals their fundamental character and their dependence on gravit ation fac- tor. The correlated changes of time, entropy and mass of any condensed matter follows from our theory. *************************************************** *********************** 22REFERENCES Babloyantz A. Molecules, Dynamics and Life. An intro- duction to self-organization of matter. John Wiley & Sons, Inc. New York, 1986. Berezin M.V., Lyapin R.R., Saletsky A.N. Effects of weak magnetic fields on water solutions light scattering. Preprint of Physical Department of Moscow University, No.21, 1988. 4 p. (in Russian). Chernikov F.R. Lightscattering intensity oscillations in water- protein solutions. Biofizika (USSR )1985,31,596. Chernikov F.R. Effect of some physical factors on light scattering fluctuations in water and water biopolymer so- lutions. Biofizika (USSR ) 1990 a,35,711. Chernikov F.R. Superslow light scattering oscillations in liquids of different types. Biofizika (USSR )1990b ,35,717. Duhanin V.S., Kluchnikov N.G. The problems of theory and practice of magnetic treatment of water. Novocherkassk , 1975, p.70-73 (in Russian). Einstein A. Collection of works. Nauka, Moscow, 1965. Eisenberg D., Kauzmann W. The structure and proper- ties of water. Oxford University Press, Oxford, 1969. Egelstaff P. A. Static and dynamic structure of liquids and glasses. J.Non-Crystalline solids.1993, 156, 1-8. Fild R., Burger M. (Eds.). Oscillations and progressive waves in chemical systems. Mir, Moscow, 1988. K¨ aiv¨ ar¨ ain en A.I. Solvent-dependent flexibility of proteins and princip les of their function. D.Reidel Publ.Co., Dordrecht, Boston, Lancaster, 1985,pp.290. K¨ aiv¨ ar¨ ainen A.I. The noncontact interaction between macromolecules revealed by modified spin-label method. Biofizika (USSR ) 1987 ,32,536. K¨ aiv¨ ar¨ ainen A.I. Thermodynamic analysis of the sys- tem: water-ions-macromolecules. Biofizika (USSR ),1988,33,549. K¨ aiv¨ ar¨ ainen A.I. Theory of condensed state as a hier- archical system of quasiparticles formed by phonons and three-dimensional de Broglie waves of molecules. Applica- tion of theory to thermodynamics of water and ice. J.Mol.Liq . 1989a,41,53−60. K¨ aiv¨ ar¨ ainen A.I. Mesoscopic theory of matter and its interaction with light. Principles of selforganization in ice, water and biosystems. University of Turku, Finland 1992, pp.275. K¨ aiv¨ ar¨ ainen A., Fradkova L., Korpela T. Separate con- tributions of large- and small-scale dynamics to the heat ca - 23pacity of proteins. A new viscosity approach. Acta Chem.Sca nd. 1993,47,456−460. Kampen N.G., van. Stochastic process in physics and chemistry. North-Holland, Amsterdam, 1981. Kiselev V.F., Saletsky A.N., Semikhina L.P. Theor. experim .khimya (USSR ),1988,2,252−257. Klassen V.I. Magnetization of the aqueous systems. Khimiya , Moscow, 1982 (in Russian). Kozyrev N.A. Causal or nonsymmetrical mechanics in a linear approximation. Pulkovo. Academy of Science of the USSR. 1958. Maslovski V.M., Postnikov S.N. In: The treatment by means of the impulse magnetic field. Proceedings of the IV seminar on nontraditional technology in mechanical engi- neering. Sofia-Gorky, 1989. Minenko V.I. Electromagnetic treatment of water in ther- moenergetics. Harkov, 1981 (in Russian). Nicolis J.C. Dynamics of hierarchical systems. Springer, Berlin, 1986. Nicolis J.C., Prigogine I. Self-organization in nonequili b- rium systems. From dissipative structures to order through fluctuations. Wiley and Sons, N.Y., 1977. Prigogine I. From Being to Becoming: time and com- plexity in physical sciences. W.H.Freeman and Company, San Francisco, 1980. Prigogine I., Strengers I. Order out of chaos. Haine- mann, London, 1984. Semikhina L.P., Kiselev V.F. Izvestiya VUZov. Fizika (USSR), 1988, 5, 13 (in Russian). Semikhina L.P. Kolloidny jurnal (USSR), 1981, 43, 401. Shih Y., Alley C.O. Phys Rev.Lett. 1988, 61,2921. Shnol S.E. Physico-chemical factors of evolution. Nauka, Moscow, 1979 (in Russian). Udaltsova N.B., Kolombet B.A., Shnol S.E. Possible cos- mophysical effects in the processes of different nature. Push chino, 1987 (in Russian). Yashkichev V.I. J.Inorganic Chem.(USSR ),1980, 25, 327. See also set of previous articles at Los-Alamos archives: http://arXiv.org/find/physics/1/au:+Kaivarainen A/0/1/0/all/0/1 24

arXiv:physics/0003108v1 [physics.gen-ph] 31 Mar 2000Hierarchic Models of Turbulence, Superfluidityand Superconductivity Alex Kaivarainen JBL, University of Turku, FIN-20520, Turku, Finland http://www.karelia.ru/˜alexk H2o@karelia.ru Materials, presented in this original article are based on following publications: [1]. A. Kaivarainen. Book: Hierarchic Concept of Mat- ter and Field. Water, biosystems and elementary particles. New York, NY, 1995, ISBN 0-9642557-0-7. [2]. A. Kaivarainen. New Hierarchic Theory of Matter General for Liquids and Solids: dynamics, thermodynamics and mesoscopic structure of water and ice ( http://www.kare lia.ru/˜alexk [see New articles]). [3]. A. Kaivarainen. Hierarchic Theory of Condensed Matter: Interrelation between mesoscopic and macroscopic properties (see URL: http://www.karelia.ru/˜alexk [see New articles]). Also see archives of Los-Alamos: http://arXiv.org/find/ph ysics/1/au:+kaivarainen/0/1/0//0/1 Contents of Article Introduction to new Hierarchic theory of condensed matter 1 Turbulence. General description 2 Mesoscopic mechanism of turbulence 3 Superfluidity. General description 4 Mesoscopic scenario of fluidity 5 superfluidity as a hierarchic self-organization process 6 Superfluidity in3He 7 Superconductivity General properties of metals and semiconductors Plasma oscillations Cyclotron resonance Electroconductivity 8. Microscopic theory of superconductivity (BCS) 9. Mesoscopic scenario of superconductivity Interpretation of experimental data in the framework of mes o- scopic model of superconductivity 1=================================================== =========== Introduction to new Hierarchic Theory of Condensed Matter A basically new hierarchic quantitative theory, general fo r solids and liquids, has been developed. It is assumed, that unharmonic oscillations of particles in any con- densed matter lead to emergence of three-dimensional (3D) s uperpo- sition of standing de Broglie waves of molecules, electroma gnetic and acoustic waves. Consequently, any condensed matter could b e con- sidered as a gas of 3D standing waves of corresponding nature . Our approach unifies and develops strongly the Einstein’s and De bye’s models. Collective excitations, like 3D standing de Broglie waves o f molecules, representing at certain conditions the mesoscopic molecul ar Bose con- densate, were analyzed, as a background of hierarchic model of con- densed matter. The most probable de Broglie wave (wave B) length is deter- mined by the ratio of Plank constant to the most probable impu lse of molecules, or by ratio of its most probable phase velocity to fre- quency. The waves B are related to molecular translations (t r) and librations (lb). As the quantum dynamics of condensed matter does not follow i n general case the classical Maxwell-Boltzmann distribution, the re al most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular Bose condensat ion in solids and liquids at temperatures, below boiling point. It is one o f the most important results of new theory, which we have confirmed by computer sim ulations on examples of water and ice. Four strongly interrelated new types of quasiparticles (collective excita- tions) were introduced in our hierarchic model: 1.Effectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the effectons (flickering clusters); 3.Transitons are the intermediate [ a⇋b] transition states of the trandlb effectons; 4.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons andconvertons. Primary effectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating ions, atoms or 2molecules. The volume of effectons (tr and lb) may contain fro m less than one, to tens and even thousands of molecules. The first condition m eans validity ofclassical approximation in description of the subsystems of the effect ons. The second one points to quantum properties of coherent clusters due to molecular Bose condensation on mesoscopic spatial scale . The liquids are semiclassical systems because their primar y (tr) effectons contain less than one molecule and primary (lb) effectons - mo re than one molecule. The solids are quantum systems totally because both kind of t heir primary effectons (tr and lb) are molecular Bose condensates .These conse- quences of our theory are confirmed by computer calculations . The 1st order [ gas→liquid ] transition is accompanied by strong decreasing of rotational (librational) degrees of freedom due to emerg ence of primary (lb) effectons and [ liquid→solid] transition - by decreasing of translational degrees of freedom due to Bose-condensation of primary (tr) effecton s. In the general case the effecton can be approximated by par- allelepiped with edges corresponding to de Broglie waves le ngth in three selected directions (1, 2, 3), related to the symmetry of the molecular dynamics. In the case of isotropic molecular moti on the effectons’ shape may be approximated by cube. The edge-length of primary effectons (tr and lb) can be consid ered as the ”parameter of order”. The in-phase oscillations of molecules in the effectons corr espond to the effecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the effectons differ in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state of the effectons has a common feature with Fr¨ olich’s polar mode. The(a→b)or(b→a)transition states of the primary effectons (tr and lb), defined as primary transitons, are accompanied b y a change in molecule polarizability and dipole moment withou t density fluctuations. At this case they lead to absorption or radiati on of IR photons, respectively. Superposition (interception) of three internal standing I R pho- tons of different directions (1,2,3) - forms primary electro magnetic deformons (tr and lb). On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density fluctuations, leading to absorption or radiation of phonons . Superposition resulting from interception of standing phonons in three direc- tions (1,2,3), forms secondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondary effectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromag- netic deformons ,lead to origination of macroeffectons, macrotransitons andmacrodeformons (tr and lb respectively) . 3Correlated simultaneous excitations of tr and lb macroeffec tonsin the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereffectons. In turn, the coherent excitation of both: tr andlb macrodeformons and macroconvertons in the same volume means creation of superdeformons. Su- perdeformons are the biggest (cavitational) fluctuations, leading to microbub- bles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, reflects all of possible combinations of the four basic ones [ 1-4], intro- duced above. This set of collective excitations in the form o f ”gas” of 3D standing waves of three types: de Broglie, acoustic and el ectro- magnetic - is shown to be able to explain virtually all the pro perties of all condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need the semi-empiric intermolecular potentials f or calculations, which are unavoidable in existing theories of many body systems. T he potential energy of intermolecular interaction is involved indirectly in di mensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into ac- count their different properties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra; [2]- Sound velocity; [3]- Density; [4]- Refraction index (extrapolated to the infinitive wave leng th of photon ). The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, to ev aluate more than 300 important characteristics of any condensed matter. Amo ng them are such as: total internal energy, kinetic and potential energies, heat-capacity and ther- mal conductivity, surface tension, vapor pressure, viscos ity, coefficient of self- diffusion, osmotic pressure, solvent activity, etc. Most of calculated parameters are hidden, i.e. inaccessible to direct experimental measu rement. The new interpretation and evaluation of Brillouin light sc attering and M¨ ossbauer effect parameters may also be done on the basis of h ierarchic the- ory. Mesoscopic scenarios of turbulence, superconductivi ty and superfluidityare elaborated. Some original aspects of water in organization and large-sc ale dynamics of biosystems - such as proteins, DNA, microtubules, membrane s and regulative role of water in cytoplasm, cancer development, quantum neu rodynamics, etc. have been analyzed in the framework of Hierarchic theory. Computerized verification of our Hierarchic concept of matt er on examples of water and ice is performed, using special comput er pro- gram: Comprehensive Analyzer of Matter Properties (CAMP, c opy- right, 1997, Kaivarainen). The new optoacoustic device (CA MP), 4based on this program, with possibilities much wider, than t hat of IR, Raman and Brillouin spectrometers, has been proposed (see U RL: http://www.karelia.ru/˜alexk). This is the first theory able to predict all known experimenta l temperature anomalies for water and ice. The conformity bet ween theory and experiment is very good even without any adjustab le pa- rameters. The hierarchic concept creates a bridge between m icro- and macro- phenomena, dynamics and thermodynamics, liquid s and solids in terms of quantum physics. =================================================== =========== 1. Turbulence. General description The type of flow when particles move along the straight trajec tory without mixing with adjoining layers, is termed laminar flow . If the layers of the liquid of the laminar flow are moving relat ive to each other at different velocities, then the forces of intern al friction (Ffr)or viscosity forces originate between them: Ffr=η/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆v ∆d/vextendsingle/vextendsingle/vextendsingle/vextendsingleS, (1) where: ∆vis relative liquid layer velocity; Sis the contact surface; ηis dynamic viscosity;/vextendsingle/vextendsingle∆v ∆d/vextendsingle/vextendsingleis the module of velocity gradient directed to the surface of layers. Near the walls of a straight tube the velocity of laminar flow i s equal to zero and in the center of the tube it is maximum. The relation between the layer velocity and its distance fro m cen- tral axes of the tube (r) is parabolic: v(r) =v0/parenleftbigg 1−r2 a2 t/parenrightbigg (2) where: a tis tube radius; v0is the velocity of the liquid on the central axis, depending on the difference of pressure at the ends of th e tube: ∆P=P1−P2 (3) as follows: v0=P1−P2 4ηla2 t (4) 5where: lis tube length and ηis dynamic viscosity. Theflux of liquid , i.e.the volume of liquid flowing over the cross-section of the tube during a time unit is determined by the Poiseuille fo rmula: Q=(P1−P2)πa2 t 8ηl(5) This formula has been used frequently for estimation of dyna mic viscosity η. The corresponding mass of flowing liquid is equal: m=ρQ (6) and corresponding kinetic energy: Tk=ρ 4Qv2 0 (7) where:ρis density of liquid. The work of internal friction force is: A=−4ηv0lQ/ρR2 In the case of the laminar movement of a spherical body relati ve to liquid the force of internal friction (viscosity force) is determined by the Stokes law : Ffr= 6πavη, (8) where: (a) is the radius of sphere and vis its relative velocity. As a result of liquid velocity ( v) and/or the characteristic dimension (a) increasing, the laminar type of liquid flow could change to the turbulent one. This begins at certain values of the dimensionless Reynolds number: R=ρvca/η=vca/ν, (9) where:ρis liquid density; v cis characteristic (average) flow velocity; ν=η/ρ iskinematic viscosity of liquid. For a round tube with radius (a) the critical value of R is abou t 1000. A turbulent type of flow is accompanied by rapid irregular puls ations of liquid velocity and pressure, representing a kind of self-organiz ation. In the case of nonstationary movement, the flow can be charact er- ized by two additional dimensionless parameters like: Strouhal number: 6S=vcτ/a (10) where:τis the characteristic time of velocity ( vc) pulsations; andMarch number : M=vc/vs (11) where:vsis sound velocity in liquid. 2. Mesoscopic mechanism of turbulence The physical scenario of transition from a laminar flow to a tu rbulent one is still unclear. It is possible to propose the mechanism of t his transition based on mesoscopic concept of matter. Let us start from the assumption that in the case of laminar flo w, the thick- ness of parallel layers is determined by the thickness of pri mary electromagnetic deformons (translational ∼5·105˚Aand librational ∼105˚A), equal to linear dimensions of corresponding macrodeformons. The total internal energy and the internal pressure of neigh boring layers are not equal. The surface between such layers can be characterized by corr esponding sur- face tension [ σ(r)]. Surface tension prevents mixing between layers with diff erent laminar flow velocities. According to our model the thickness of the two outer borders of each layer (skin-surface) is determined by the effective linear dimens ions of primary (tr and lib) effectons [ ltr,lb∼(3−15)˚A] related to the corresponding most probable wave B length ( λtr,lb) of molecules in liquid (see eq. 11.30): ltr,lb= (Vef/V2/3 ef) (12) Decreasing of ltr,lb,depending on most probable impulse of liquid molecules, as a result of increased flow velocity [ v1(r)] see eq. (13) and/or temperature elevation in accordance with our theory of surface tension ( see eq. 11.33 of [1] or [3]), leads to reducing of σ(r). In turn, this effect strongly decreases the work of cavitational fluctuations, i.e. the bubbles formation (2 7) and increases their concentration (29). These bubbles lead to mixing of laminar layers, the instability of laminar flow and its changing to a turbulent ty pe. The critical flow velocity: vc=v1(r) is determined by the critical librational wave B length: λ1,2,3 lb=h/m·/bracketleftbig/parenleftbigv1,2,3 gr/parenrightbig lb+v1(r)/bracketrightbig (13) 7corresponding to the condition (6.6) of liquid-gas first ord er phase transition: [(Vef)lib/(V0/N0)] =/bracketleftbigg9 4π(λ(1)·λ(2)·λ(3))lib/(V0/N0)/bracketrightbigg ≤1 (14) where:mis molecular mass; v1,2,3 gris the most probable librational group velocity of molecules of liquid in selected directions (1 ,2,3);v1(r) is the flow velocity of a liquid layer in the tube at the distance (r) from the central axes of the tube (2). Increasing of v1(r) atr→0 decreases λ1 lband (Vef)lbin accordance with (13) and (14). The value of λ1 lbis also related to phase velocity ( va ph) and frequency (νa 1) of the primary librational effecton in (a) state (see 2.60): λ1 lib=h/m/bracketleftbig(v1 gr)lb+v1(r)/bracketrightbig =/parenleftbiggva ph νa 1/parenrightbigg lb= = (va ph/ν1 p)/bracketleftbig exp/parenleftbigh(ν1 p)lb/kT)−1/parenrightbig /bracketrightbig (15) where: (ν1 p)lb= (νb 1−νa 1)lb=c(ν1 p)lb (16) is the frequency of ( a⇔b)lbtransitions of the primary librational effecton of aflowing liquid determined by librational band wave number (˜ ν1 p)lbin the oscillatory spectra. It was calculated earlier for water under stationary condit ions, that the ele- vation of temperature from 00to 1000Ctill the phase transition condition (14), is accompanied by the increase in ( vgr)lbfrom (1.1 to 4.6)·103cm/s (Fig. 12b of [1]). This means that at 300C, when (vgr)lb≃2·103cm/s, the critical flow velocity v1(r), necessary for mechanical boiling of water (condition 14) should be about 2.5m/s. The reduced number of primary librational effectons ( Nef) in the volume (VM D) of primary electromagnetic deformons (tr and lib) also inc reases with temperature and/or flow velocity: /parenleftbigNef/parenrightbigD tr,lb=/bracketleftbiggPa+Pb Znef·VM D/bracketrightbigg (17) The reduced number of primary transitons ( Nt) has a similar dependence on T andv1(r), due to increasing of nef, andVM Das far: /parenleftbig Nt/parenrightbigD tr,lb=/parenleftbig Nef/parenrightbigD tr,lb(18) 8The analysis of eq.(15) predicts that at T = const an increase in v1(r) must be accompanied by the low-frequency shift of the librational b and: ˜ν(1) lib≃700cm−1 and/or by the decrease in sound velocity ( va ph)lb(eq.2.74of[1]) in the direction of flow. It follows also from our mesoscopic theory that these change s should be accompanied by a rise in dynamic viscosity (11.45 of [1] or [3 ]) due to increased structural factor ( Tkin/Utot). Turbulent pulsations of flow velocity (∆ v) originate under developed turbu- lence conditions: ∆v=vtur=v−v (19) where: ¯ v is average flow velocity and ( v) is instant flow velocity. The frequencies of large-scale pulsations have the order of : ν=v/λ, (20) where:λis the main scale of pulsations. λcan correlate with the dimensions of electromagnetic defor mons and can be determined by the transverse convection rate, depending on the bubbles dimensions. The pulsations of flow velocity (∆v)can result from the a) mixing of parallel layers with different flow velocity and b) fluctuation of viscosity force (eq. 8) due to fluctuations i n bub- bles radius and concentration, as well as density, viscosit y and thermal conductivity; c) movements and emergence of the bubbles as a result of the Archimedes force. The bubbles have two opposite types of influence on instant velocity. The layer mixing effect induced by them can increase flow velocity . On the other hand, the bubbles can simultaneously decrease fl ow velocity due to enhanced internal friction. In the case of developed turbulence with different scales of p ulsations it is reasonable to introduce the characteristic Reynolds numbe r: Rλ=vturλ/νtur (21) where:λis a scale of pulsations; vturis the velocity of pulsation and νtur= (η/ρ)turcharacteristic kinematic viscosity. The ratio between turbulent kinematic viscosity ( νtur) and a laminar one ( ν) is related to the corresponding Reynolds numbers like (Land au, Lifshits, 1988): 9νtur ν∼R Rtur(22) One can see from (21) that it means: vtur·λ∼const (23) Based on dimension relations turbulent kinematic viscosit y can be expressed as follows: νtur∼∆v·l∼vλ·λ (24) and energy dissipation as: ǫdis∼νtur(vtur/λ)2∼v3 tur λ(25) This expression leads to the Kholmogorov-Obuchov law: vtur∼(ǫdisλ)1/3(26) Large-scale pulsations correspond to high λvalues and low νturvalues. i.e. high characteristic turbulence Reynolds numbers (see 21). According to our model the maximum energy dissipation occur s in the vol- ume of superdeformons (or supertransitons). Mechanically induced boiling under conditions of turbulence (eqs. 13 and 14) is accompanied by the emergence of gas bubbles relate d to the increased superdeformons probability and decreased surface tension between layers. Critical bubble creation work ( W) is strongly dependent on inter- layer sur- face tension ( σ).A general classical theory (Nesis, 1973) gives: W=4 3πa2σ=16πσ3 3(P−Pext)2(27) where: P=Pext+2σ a=Pa=∞·exp/parenleftbigg −2σVe akT/parenrightbigg (28) P is the internal gas pressure in a bubble with radius ( a);Veis the volume of liquid occupied by one molecule. The bubbles quantity ( Nb) has an exponential dependence on W: 10Nb= exp/parenleftbigg −W kT/parenrightbigg (29) One can see from our mesoscopic theory of surface tension (eq s. 11.31 - 11.33 of [1]; [3] and eq.12) that under mechanical boiling conditions the skin-surface thickness (12) : l→(V0/N0)1/3andqs→1, the interlayer surface tension ( σ) tends to zero, Wdecreases and Nbincreases. We can conclude that the mesoscopic scenario of mechanical b oil- ing presented here can provide a background for elaboration of a quantitative physical theory of turbulence and other hydro dynamic instabilities like Taylor’s and Benar’s ones. 3. Superfluidity. General description superfluidityhas been revealed for two liquids only: helium iso- topes:4He with boson’s properties (S= 0)and3He with fermion properties (S= 1/2). The interactions between the atoms of these liquids is very weak. It will be shown below that the values of nor- mal sound velocity at temperatures higher than those of seco nd order phase transition (λ-point) are lower than the most probable thermal velocities of the atoms of these liquids. The first theories of superfluidity were proposed by Landau (1 941) and Feynman (1953). First order phase transition [gas →liquid] occurs at 4.22K. Second order phase transition, when superfluidity originat es,4He →He II takes place at Tλ= 2.17K(Pext= 1atm.). This transition is accompanied by: a) heat capacity jump to higher values; b) abruptly increased thermal conductivity; c) markedly decreased cavitational fluctuations and bubble s in liq- uid helium. For explanation of experimental data Landau supposed that a tT <T λthe He II consists of two components: - thesuperfluidity component with relative fraction of density ρS/ρ, increas- ing from zero at T=Tλto 1 at T = 0 K. The properties of this component are close to those of an ideal liquid with a potential type of flow. The entropy of this component is zero and it does not manifest the viscous fr iction on flowing through narrow capillaries; - thenormal component with density ρn=ρ−ρs (30) 11decreasing from 1 at T=Tλto zero at T = 0 K. This component behaves as a usual viscous liquid which exhibits dumping of the oscil lating disk in He II. Landau considered this component to be a gas of two types o f excitations: phonons androtons . The hydrodynamics of normal and superfluid components of He I I are char- acterized by two velocities : normal (vn) and superfluid one: vsf= (/planckover2pi1/m)∇ϕ (31) where∇ϕ∼ksf= 1/Lsfis a phase of Bose-condensate wave function - see eq. 36. As a result of two types of hydrodynamic velocities and densi ties, the corre- sponding 2 types of sound waves propagate in the volume of He II. Thefirst sound (U1) is determined by the usual formula valid for normal condensed matter: U2 1= (∂P/∂ρ )S (32) In this case density oscillations spread in the form of phono ns. Thesecond sound (U2) is related to oscillations of temperature and entropy (S): U2 2=ρSTS2/cρn (33) In normal condensed media the temperature oscillation fade at the distance of the order of wave length. Landau considered the second sound as density waves in the ga s of quasi- particles: rotons and phonons. The third sound (U3)propagates i n the thin surface films of He II in the form of ”ripplons”, i.e. quantum capillary waves relate d to the isothermal oscillations of the superfluid component. U3= (ρS/ρS)·d·∂E ∂d(1 +TS/L ), (34) where: (¯ρS/ρS) is the relative density of superfluid component averaged in the thickness of the film (d); E is the potential of Van- der-Waals interactions of 4He atoms with the bottom surface; L is evaporation heat. Thefourth sound (U4) propagates in He II, located in very narrow capillaries, when the length of quasiparticles (phonons and rotons) free run is compatible or bigger than the diameter of these capillaries or pores. The hydrodynamic velocity ( vn) of the normal component under such con- ditions is zero and ρn/ρ≪ρsf/ρ: 12U2 4= (ρS/ρ)U2 1+ (ρn/ρ)U2 2≃(ρS/ρ)U2 1 (35) In accordance with Bose-Einstein statistics, a decrease in temperature, when T→Tλ, leads to condensation of bosons in a minimum energy state. This process results in the origination of a superfluid compo nent of He II with the coherent thermal and hydrodynamic movement of atom s. Coherence means that this movement can be described by the si ngle wave function: ψ=ρ1/2 S·eiϕ(36) The movement of the superfluid component is potential as far its velocity ( /vector vsf) is determined by eq.31 and: rotvsf= 0 (37) Vortex filaments in He II When the rotation velocity of a cylindrical vessel containi ng He II is high enough, then the emergency of so-called vortex fila ments becomes thermodynamically favorable. The filament is forme d by the superfluid component of He II in such a way that their impul se of movement decreases the total energy of He II in a rotating v essel. The shape of filaments in this case is like a straight rod and th eirthickness is of the order of atom’s dimensions, increasing with loweri ng the temperature atT <T λ. Vortex filaments are continuous. They are closed or limited w ithin the boundaries of a liquid. For each surface surrounding a vorte x filament the condition (37) is valid. The values of velocity of circulation around the axis of filam ents are deter- mined (Landau, 1941) as follows: /contintegraldisplay vsfdl= 2πrvsf= 2πκ (38) and vsf=κ/r (39) 13Increasing the radius of circulation (r) leads to decreased circulation velocity (vsf). Substituting vsfin eq.31, we obtain: /contintegraldisplay vsfdl=/planckover2pi1 m∆Φ, (40) where: ∆Φ = n2πis a phase change as a result of circulation, n= 1,2,3...is the integer number. Comparing (40) and (38) gives: κ=n/planckover2pi1 m(41) It has been shown that only curls with n= 1 are thermodynamically stable. Taking this into account, we have from (39) and (41): r=n/planckover2pi1 mvsf(42) An increase in the angle frequency of rotation of the cylinde r con- taining He II results in the increased density distribution of vortex filaments on the cross-section of the cylinder. As a result of interaction between the filament and the normal component of He II, the filaments move in the rotating cylinde r with normal liquid. The flow of He II through the capillaries can be accompanied by emergence of vortex filaments. In ring-shaped vessels the circulation of closed vortex fila ments is stable. Stability is related to the quantum pattern of cir culation change (eqs. 38 and 41). Let us consider now the phenomena of superfluidityin He II in t he framework of our mesoscopic concept. 4. Mesoscopic scenario of superfluidity It will be shown below how our mesoscopic model (Table 1) can be used to explain He II properties, its excitation spectrum (Fig. 1), increased heat capacity at λ-point and the vortex filaments formation. We assume here, that the formulae obtained earlier for inter nal energy (Utot−eq.4.3), viscosity (eqs. 11.48, 11.49 and 11.55 of [1] or see [3]), thermal conductivity (eq. 11.37), vapor pressure (eq. 11.26) remain valid for both components of He II. The theory proposed by Landau (Lifshits, Pitaevsky, 1978) q uali- tatively explains only the lower branch (a) in the spectrum ( Fig. 1) as a result of phonons and rotons excitation. 14But the upper branch (b) points that the real process is more complicated and needs introduction of other quasiparticle s and ex- cited states for its explanation. Our mesoscopic model of superfluidity interrelates the lowe r branch with the ground acoustic (a) state of primary effectons in liq uid4He and the upper branch with their excited optical (b) state. In ac- cordance with our model, the dissipation and viscosity fric tion (see section 11.6) arise in the normal component of He II due to the r- mal phonons radiated and absorbed in the course of the ¯b→¯aand ¯a→¯btransitions of secondary effectons correspondingly, symba tic to macrodeformons excitation. Fig. 1. Excitation spectrum of liquid4He from neutron scat- tering measurements (March and Parrinello, 1982). Spectru m is characterized by two branches, corresponding to (a) and (b) states of the primary effectons according to the mesoscopic model. Landau described the minimum in the region of λ-point using the expression: E= ∆0+(P−P0)2 2m∗, (43) where ∆ 0and P 0are the energy and impulse of liquid4He atλ-point (Fig. 1) andm∗= 0.16mis the effective mass of the4He atom (mHe= 4·1.44·10−24g= 5.76·10−24g). The effective mass m∗can be determined experimentally. Feynman (1953) explained the same part of the excitation spe ctra by the nonmonotonic behavior of the structure factor S(k) and the f ormula: 15E=/planckover2pi1ω=/planckover2pi12k2 2mS=/planckover2pi12 2mL2S(44) where: k= 1/L= 2π/λ (45) is the wave number of neutron interacting with liquid4He. Our mesoscopic model allows to unify Landau’s and Feynman’s approaches. The total energy of de Broglie wave either free or as part of co ndensed matter can be expressed through its amplitude squared ( A2), or effective mass ( m∗) in the following manner (see 2.45 and 2.46): Etot=Tk+V=mvgrvph=/planckover2pi12 2mA2=/planckover2pi12 2m∗L2(46) wherevgrandvphare the most probable group and phase velocities. In accordance with our model (eq. 2.46a), the structural fac tor S(k) is equal to the kinetic ( Tk) to total (Etot) energy ratio of wave B: S=Tk/Etot=A2/L2=m∗/m (47) where: Tk=P2/2m=/planckover2pi1 2mL(48) Combining (46), (47) and (48), we obtain the following set of equation for the energy of4He at transition λ-point: ∆0=E0=/planckover2pi12 2mA2 0=/planckover2pi12 2m∗L2 0 ∆0=/planckover2pi12 2mL2 0S=T0 k S  (49) These approximate formulae for the total energy of liquid4He made it possible to estimate the most probable wave B length, forming the prim ary librational (or rotational effectons) at λ-point: λ0=h mv0gr= 2πL0= 2πA0/parenleftbigm/m∗/parenrightbig1/2, (50) where the critical amplitude of wave B: 16A0=/planckover2pi1/parenleftbigg1 2mE0/parenrightbigg1/2 (51) can be calculated from the experimental E0values (Fig.1). Putting in (51) and (50) the available data: ∆0=E0=kB·8.7K= 1.2·10−15erg; the mass of atom: m(4He) = 5.76·10−24gand (m∗/m) = 0.16, we obtain: λ0∼=14·10−8cm= 14˚A (52) the corresponding most probable group velocity of4Heatoms is:v0 gr= 8.16· 103cm/s. It is known from the experiment that the volume occupied by one atom of liquid4Heis equal:v(4He)= 46˚A3/atom. The edge length of the corresponding cubic volume is: l=/parenleftbigv4He/parenrightbig1/3= 3.58˚A (53) From (52) and (53) we can calculate the number of4He atoms in the volume of primary librational (rotational) effecton at λ-point: n0 V=Vef v4He=(9/4π)λ0 l3= 43 atoms (54) One edge of such an effecton contains (43)1/3∼=3.5 atoms of liquid4He. We must take into account, that these parameters can be lower than the realones as in above simple calculations we did not consider the c ontribution of secondary effectons, transitons and deformons to total inte rnal energy (see eq. 4.3). On the other hand, in accordance with the mesoscopic model, t he conditions of the maximum stability of primary effectons correspond to t heinteger number of particles in the edge of these effectons (see Chapter 6 and F ig. 7a of [1] or Fig.4a of [2]). Consequently, we have to assume that the true number of4Heatoms forming a primary effecton at λ-point is equal to n0 V= 64. It means that the edge of cube as the effecton shape approximation contains q0= 4 atoms of4He: n0 e= (n0 V)1/3= 641/3= 4 (56) 17The primary librational effectons of such a type may correspo nd to rotons in- troduced by Landau to explain the high heat capacity of He II. The thermal impulses of4He atoms in these coherent clusters can totally compensate each other and the resulting impulse of primary e ffectons is equal to zero. Further decline in temperature gives rise to dimens ions of primary effectons. The most stable of them contain in their ribs the in teger number of helium atoms: q=q0+n (56a) where:n= 1,2,3... λ0, n0 Vandn0 ecan be calculated more accurately using eqs. (2.60) and (3.5) of [1], if the required experimental data on oscillato ry spectroscopy and sound velocimetry are available. 5. Superfluidity as a hierarchic self-organization process Let us consider now the consequence of the phenomena observe d in4He in the course of temperature decline to explain Fig. 1 in th e framework of mesoscopic model: 1. In accordance with our model lowering the temperature til l the 4.2 K and gas-liquid first order phase transition occurs u nder condition (6.6). This condition means that the most probabl e wave B length of atoms related to their rotations or librations st arts to exceed the average distance between4He atoms in a liquid phase: λ=h/mv gr≥3.58˚A (57) The corresponding value of the most probable group velocity is vgr≤3.2·104cm/s. The translational thermal impulses of particles are usuall y bigger and waves B length smaller than those related to librations. In a ccordance with our model of first order phase transitions (Section 6.2 o f [1]), this fact determines the difference in the temperatures of [gas →liquid] and [liquid →solid] transitions. The freezing of liquid4He occurs at a sufficiently high pressure of∼25atm only and means the emergency of primary translational effectons. A pressure increase as well as the drop in temperat ure declines the impulses between particles and stimulates dis tant Van der Waals interaction between them, responsible for cohere nt clusters formation. 18In normal component of liquid4He II like in a usual liquid at T >0K,the existence of primary and secondary effectons, converton s, transitons and deformons is possible. The contributions of each of these quasiparticles determine total internal energy (eq. 4.3 of [1, 2]), kinetic and potential energies (eqs. 4.33 and 4.36 of [1, 2]) , viscosity (11.45), thermal conductivity (11.35), vapor pressure (11 .26) of [1], paper [3] and many other parameters. We assume that the lower branch in the excitation spectrum of Fig. 1 reflects the (a) state and the upper branch the (b) state of primary (lb and tr) effectons. 2. Decreasing the temperature to λ-point:Tλ= 2.17Kis accom- panied by the condition (55), which stimulates Bose-conden sation of atoms, increasing the dimensions of primary effectons as w ell as Bose-condensation of secondary effectons with nonzero resu lting im- pulse. This leads to emergency of primary polyeffectons supe rfluid subsystem due to distant Van der Waals interactions and Jose ph- son junctions between neighboring effectons. It is accompan ied by the (a)-states probability jump-way increasing (Pa→1)and that of (b)-states decreasing (Pb→0). Probability of primary and secondary deformons (Pd=Pa·Pb;¯Pd=¯Pa·¯Pb)decreases correspondingly. In the excitation spectrum (Fig.1) these processes are displa yed as the tendency of (b)-branch to (a)-branch due to degeneration of b-branch at very law temperature. Like in the theory of 2nd order phase transitions proposed by Lan- dau (Landau and Lifshits, 1976), we can introduce here the pa rameter of order as: η= 1−κ= 1−Pa−Pb Pa+Pb(58) where:κ=Pa−Pb Pa+Pbis an equilibrium parameter. One can see that at Pa=Pb, the equilibrium parameter κ= 0and η= 1 (the system is far from 2nd order phase transition). On the other hand, at conditions of phase transition: T→Tλwhen Pb→0, κ→1and parameter of order (η)tends to zero. According to Landau’s theory, the equality of his specific pa ram- eter of order to zero is a criterion of 2nd order phase transit ion. As usual, this transition is followed by a decrease in structur al symmetry with a decline in temperature. The important point of our scenario of superfluidityis the st ate- ment that the leftward shift of (a⇔b)equilibrium of the primary effectons (tr and lb) becomes stable starting from Tλdue to their polymerization ”side by side”. This process of Bose-conden sation, including conversion of secondary effectons to primary ones , differs from condensation of an ideal Bose-gas described by eq. (1.2 6). Such 19kind of Bose-condensation means the enhancement of the conc entra- tion of (a)-state of primary effectons with lower energy, rel ated to de- generation of the all others. The polymerization of primary effectons in He II gives rise to macroscopically long filament-like (or chain-like) polyeffectons. This process can be considered as self-organ ization on macroscopic scale. These filament-like polyeffectons, repr esenting su- perfluid component, can form closed circles or three-dimens ional (3D) isotropic networks in a vessel with He II. The remnant fraction of liquid represent normal fraction of He II. Polyeffectons are characterized by the dynamic equilibrium :[assembly ⇔deassembly ]. Temperature decrease and pressure increase have to shift th is equi- librium to the left, increasing the surface of the primary eff ectons side-by-side interaction and number of Josephson junction s. The probability of tunneling between coherent clusters inc reases also correspondingly. The relative movement (sliding) of flexible ”snake-like” po lyeffec- tons occurs without phonons excitation in the volumes of IR d efor- mons, equal to that of macrodeformons. Just macrodeformons ex- citation is responsible for dissipation and viscosity in no rmal liquids (see section 11.6). The absence of macrodeformons excitation, related to polye ffec- tons emergency, explains the superfluidity phenomenon acco rding to our model. Breaking of symmetry in a three-dimensional polyeffecton ne twork and its violation can be induced by external fields, like the gravita tional gradient, me- chanical perturbation and surface effects. It is possible as far the coherent polyeffecton system is highly cooperative and interaction b etween individual effectons as 3D standing waves is small. In rotating cylindrical vessel, filament-like polyeffecton s originate from 3D isotropic net and they tend to be oriented along the cylinder with their own ro- tation round their own axis in the direction opposite to that of cylinder rotation. In accordance with our model, this phenomenon represents th e vortex filaments in He II discussed above. The radius of the filaments (42) is de termined by the group velocity of the coherent4He atoms, which form part of the primary effectons(vgr=vsf). The numerical value of vgrmust be equal to or less than 6·103cm/s, this corresponding to conditions (55 and 56). At T→0,vgrde- creases and the filament radius (42) increases to reach the va lues corresponding tovmin gr=v0determined by the zero-point oscillations of4He atoms. Under these conditions the aggregation or polymerization of tran slational primary ef- fectons in (a)-state can occur, leading to liquid-solid pha se transition in4He. The self-organization of highly cooperative coherent poly effectons in λ-point and (a ⇋b) equilibrium leftward shift should be accompanied by a heat capacity jump. The mechanism, leading to stabilization of (a)- state of pri mary effectons as the first stage of their polymerization, is a formation of coherent super- 20clusters from primary effectons without direct contacts. Stabilizat ion of (a) states in superclusters could be resulted from macroscopic self-organization of matter in the volume of electromagnetic IR deformon due to distant Van der Waals interaction and Vibro-gravitational interaction, introduced in our theory between primary effectons (see section 10.4). These interac tions between acous- tic (a) states of the effectons are more effective, than betwee n optic (b) states. They minimize the potential energy of the system and increas e the probability of macroscopic Bose-condensation. The successive mechanisms of super-clusterization and pol ymer- ization of primary effectons could be responsible for second order phase transitions, leading to emergency of superfluidity an d super- conductivity. The second sound in such a model can be attributed to phase velocity in a system of polyeffectons or superclusters. The propagation o f the second sound through chain polyeffectons or superclusters should be acco mpanied by their elastic deformation and [assembly ⇔disassembly] equilibrium oscillations. The third sound can be also related to the elastic deformation of polyeffec- tons and equilibrium constant oscillations of supercluste rs, but in the surface layer with properties different from those in bulk volume. In accordance with the mesoscopic theory, such a difference in surface and volum e parameters is responsible (see eq.11.31-11.33) of [1] and [3] for surface tension (σ) in He II and its jump at λ-point. this increase in σexplain also the disappearance of cavitational bubbles at T <T λ. The fourth sound is the consequence of the increase in primary effecton dimensions and the change in their phase velocity as a result of He II interaction with narrow capillary’s walls and thermal movement immobil ization. The normal component of He II is related to the fraction of He II atoms not involved in polyeffectons formation. This fract ion com- poses individual primary and secondary effectons, maintain ing the ability for (a⇔b)and/parenleftbig ¯a⇔¯b/parenrightbig transitions. In accordance with our mesoscopic model, these transitions in composition of macr oeffectons and macrodeformons are accompanied by the emission and abso rption of heat phonons. The manifestation of viscous properties in normal liquid an d nor- mal component of He II is related to fluctuations of concentra tion in the volume of macrodeformons (VM D)- see eqs. 11.48, 11.49 and 11.55 of [1]. On the other hand, macro- and superdeformons are absent in th e superfluid component, as far in primary polyeffectons at T <T λ: the probability of B-state of macroeffectons: PB=Pb·¯Pb→0; the proba- bility of A-state of the effectons: PA=Pa·¯Pa→1and, consequently, the probability of macrodeformons tends to zero: PM D=PB·PA→0. Decreasing the probability of superdeformons PS D= (PM D)tr·(PM D)lb→0 means the decreased concentration of cavitational bubbles and vapor 21pressure. 3. We can explain the decrease in E(k) in Fig. 1 around T=Tλ by reducing the contributions related to (b) of the primary e ffectons, degeneration of secondary effectons due to their Bose-conde nsation and concomitant elimination of the contribution of seconda ry acoustic deformons (i.e. phonons) to the total energy of liquid4He. One can see from eqs. (11.45 - 11.53 of [1] and paper [3]) that u nder conditions of superfluidity at the absence of secondary effec tons, when the life-time of secondary effectons and cycle-period of mac roeffectons (τM)tr,lbtends to zero, the viscosity also tends to zero: η→0. In accordance with the mesoscopic theory of thermal conduct ivity (see eqs. 11.35−11.37), the elimination of secondary acoustic defor- mons atT≤Tλmust lead also to enhanced thermal conductivity. This effect was registered experimentally, indeed. 4. The increase in E(k)in Fig. 1 at T <T λcan be induced by the enhanced contribution of primary polyeffectons to the total energy of He II and the factor: Utot/Tk=S−1in the state equation (11.8 of [1]) and eq.(44). The activity of the normal component of He II as a solvent for polyeffectons reduces and tends to zero at T→0. Under such condi- tion(T= 0)superpolymerization and total Bose-condensation occur in4He. The maximum in Fig. 1 at 0<T <T λis the result of competition of two opposite factors: a rise in the total energy of He II due to progress of primary effectons polymerization and its reduct ion due to the decline in the most probable group velocity (vgr),accompa- nied by secondary effectons and deformons degeneration (see eq. 46). The latter process predominates at T→0. The development of a polyeffectons superfluid subsystem is accompanied by a corre spond- ing diminution of the normal component in He II (ρS→1andρ→0). The normal component has a bigger internal energy than super fluid one. The own dimensions of primary translational and librationa l effec- tons in composition of polyeffectons increases at T→0. Verification of the inaccessibility of the b-state of primar y effectons atT≤Tλ Let us analyze our formula (2.74 of [1], se also [2]) for the ph ase velocity of primary effectons in the (a)-state for the condition T≤Tλ, when filament- like polyeffectons originate: 22va ph=vS1−fd fa 1 +Pb Pa/parenleftBig νbres νbres/parenrightBig (59) where:vSis the sound velocity; PbandPaare the thermoaccessibility of the (b) and (a) states of primary effectons; fdandfaare the probabilities of primary deformons and primary effectons in (a) state excitations (se e eq. 2.66 of [1]). One can see from (59), that if: Pb→0, then P d=Pb·Pa→0and f d→0at T≤Tλ then phase velocity of the effecton in (a) state tends to sound velocity: va ph→vS (60) For theseλ−point conditions, the total energy of4He atoms forming poly- effectons due to Bose-condensation of secondary effectons (s ee 46) can be pre- sented as: Etot∼Ea=mvgrva ph→mvgrvS (61) where the empirical sound velocity in He II is vS= 2.4·104cm/s . The kinetic energy of wave B at the same conditions is Tk=mv2 gr/2. Di- vidingEtotbyTkwe have, using (47): vS vgr=Etot 2Tk=1 2S=1 2(m∗/m)(62) and v0 gr=vs·2S0= 2.4·104·0.32 = 7.6·103cm/s. (63) m∗= 0.16mis the semiempirical effective mass at T=Tλ. The most probable wave B length corresponding to (63) at λ-point: λ0=h/mv0 gr= 15.1˚A (64) The number of4He atoms in the volume of the same effecton calculated in accordance with (54) is equal: q0= (n0 v)1/3= 3.8. This result is even closer to one predicted by the mesoscopic model (see 55) than (53). It confirms that at T≤Tλthe probability of b-state Pb→0 and conditions (60) and (61) take place indeed. In such a way our mesoscopic model of superfluidity explains t he available experimental data on liquid4He in a noncontradiction manner as a limit case of our mesoscopic viscosity theory for normal liquids. 236. Superfluidity in3He The scenario of superfluity, described above for Bose-liqui d of4He (S= 0) in principle is valid for Fermi-liquid of3He (S=±1/2) as well. A basic differ- ence is determined by an additional preliminary stage relat ed to the formation of Cooper pairs of3He atoms with total spins, equal to 1, i.e. with boson’s properties. The bosons only can form effectons as coherent cl usters containing particles with equal energies. We assume in our model that Cooper’s pairs can be formed be- tween neighboring3He atoms. It means that the minimum number of3He atoms forming part of the primary effecton’s edge at λ-point must be 8, i.e. two times more than that in4He (condition 55). Cor- respondingly, the number of3He atoms in the volume of an effecton is(n0 V)3He= 83= 312. These conditions explains the fact that super- fluidityin3He arises at temperature T= 2.6·10−3K, i.e. lower than that in4He. The formation of flexible filament-like polyeffectons, re p- resenting macroscopic Bose-condensate in liquid3He responsible for superfluidity, is a process, similar to that in4He described above. 7. Superconductivity General properties of metals and semiconductors The dynamics of conductance electrons in metals and semicon duc- tors is determined by three main factors (Kittel, 1978, Ashk roft and Mermin, 1976, Blakemore, 1985): 1. The electric field influencing the energy of electrons. 2. The magnetic fields changing the direction of electrons mo tions. 3. Scattering on the other electrons, ions, phonons, defect s. The latter factor determines the values of the electron cond uctance and resistance. In spite of the small mean distances between electrons in met als (2-3) ˚A their mean free run length at room temperatures exceeds 104˚Aand grows by several orders at T→0. It is related to the fact that only electrons having energy higher than Fermi energy (ǫF)may be involved in collisions. The fraction of these electrons in the total number of electr ons is very small and decreases on lowering the temperature as ( kT/ǫ F)2. At room temperatures the scattering of electrons in metals occurs mainly on phonons. The mean free run length of electrons in indium at 2K is about 3 0 cm. The analysis of electric and magnetic fields influence on an el ectron needs the notion of its effective mass ( m∗). It is introduced as a proportionality coefficient between the force acting on the electron and the acceleratio n (a) in the electric field (E): 24F=−eE=m∗a;a=dvgr/dt (65) In a simple case of an isotropic solid body the effective mass o f an electron is a scalar (Kittel, 1978): m∗=h2 d2ǫ/dk2(66) whereǫis the kinetic energy of an electron, having a quadratic depe ndence on the wave number ( k= 1/LB): ǫ=/planckover2pi12k2 m∗=/planckover2pi12 2mL2(67) In a general case, for electrons in solid bodies with a comple x periodic structure, the effective mass is a tensor: [m∗ ij] =/planckover2pi12/[∂2ǫ/∂k i∂kj] (68) The effective mass tensor can have positive components for so me directions and negative for others. Plasma oscillations At every displacement of the electron gas relative to the sub system of ions in a solid body, a returning electric field appears. As a conse quence of that, the subsystem of electrons will oscillate relative to the subsy stem of ions with the characteristic plasma frequency (Ashkroft and Mermin, 197 6): ωpl= 2πνpl=/parenleftbigg4πne2 m∗/parenrightbigg1/2 (69) where: (n) is the number of electrons in 1 cm3,(e) is the charge and ( m∗) is the effective mass of an electron. The quantified collective oscillations of electron plasma a re termed plasmons . With decreasing (n) from 1022to 1010cm−3the frequencies ωpldecrease from 6·1015s−1to 6·103s−1. For metals ωplcorresponds to an ultraviolet frequency range, and for semiconductors - to an IR frequency range. For longitudinal plasma oscillations at small wave vectors the dependence of frequency on the wave number ( k= 1/L= 2π/λ) can be approximately represented as (Kittel, 1978): 25ω≈ωpl·/parenleftBigg 1 +3k2v2 F 10ω2 pl+.../parenrightBigg (70) where: v Fis the Fermi velocity of an electron (see eq.77). The screening length (l), characterizing the electron-ele ctron interaction in plasmon when Fermi-gas is degenerated is equal to: l=vF/νp∼1˚Ain metals (71) For the cases of non-degenerated Fermi-gas, when the concen trations of free electrons are sufficiently low (in semiconductors) or at high temperatures T∼ 104K, the screening length ( ld) is dependent on thermal electron velocity: vth=/parenleftbig3kBT/m∗/parenrightbig1/2(72) and lD=vth/νp∼=/parenleftbiggǫkbT 4πne2/parenrightbigg1/2 (73) where:νplcorresponds to (69), ǫis the dielectric constant. For example, if in a semiconductor n= 5·1017cm−3andǫ= 12,thenlD= 60˚A(March, Parrinello, 1982). Fermi energy The notion of Fermi energy ( ǫF) can be derived from the Pauli principle forbidding the fermions to be in the same energeti c states. The formula for Fermi energy for the case of ideal electron ga s includes the electron mass (m), the Plank constant (h= 2π/planckover2pi1)and the concentration of free electrons (ne=Ne/V): ǫF=h2 2m/parenleftbigg3 8πme/parenrightbigg3/2 =2π2/planckover2pi12 m/parenleftbigg3 8πme/parenrightbigg3/2 (73a) whereNeis the number of free electrons in selected volume ( V). For a real electron gas, mmust be substituted by its effective mass: m→m∗. The formula (73a) can also be derived using the idea of standi ng waves B of the unbind electrons of matter. The condition unde r which 26the concentration of twice polarized standing waves B of ele ctrons is equal to the concentration of electrons themselves: nF B=Ne V=8π 3(λF B)3(74) The wave B length of an electron corresponding to this condit ion is: λF B=/parenleftbigg8πV 3Ne/parenrightbigg1/3 =h mvf gr(75) The kinetic energy of the unbind electrons waves B ( Tk) could be expressed through their length and mass. It appears that the kinetic en ergy of the electrons standing waves B, limited by their concentration is equal to Fermi energy: TF k=h2 2mλ2 F=h2 2m/parenleftbigg3ne 8π/parenrightbigg2/3 =P2 F 2m=ǫF, (76) where Fermi impulse: PF=mvF=h/parenleftbigg3ne 8π/parenrightbigg1/3 =/planckover2pi1(3π2ne)1/3(77) The Fermi energy corresponds to Fermi temperature ( TF): ǫF=kTF=hνF (78) AtT < T Felectron gas is in a strongly ”compressed” state. The more the relation (T/T F) =kT/ǫ F, the more the probability of the appearance of ”free volume” in a dense electron gas. On lower ing the temperature, when the impulse of electrons decreases and th e heat wave B length increases, the ”effective pressure” of the elec tron gas grows, leading to its Bose-condensation. Cyclotronic resonance The magnetic field B zin the direction (z) influencing the electron by the Lorentz force, changes the direction of its motion wit hout changing the energy. If an electron’s energy does not dissip ate, then the electrons rotate in the plane xy, around z-axis. Such an e lectron with the effective mass m∗has a circulation orbit of the radius r, with rotation frequency ωc. From the condition of equality between the 27Lorentz force (rωceBz)and the centrifugal force (m∗ω2 cr)the formula is derived for angular cyclotron frequency (Kittel, 1978): ωc=eBz/m∗(79) The kinetic energy, corresponding to the rotation is equal t o: Tk=1 2m∗(ωc)2r2(80) In the range of radio-frequencies ( ω) such a value of the magnetic induction Bz can be selected that at this value the resonance energy absor ption occurs, when ω=ωc. Such experiments on the cyclotron resonance can be done to de termine m∗ in selected directions. In a simple case, an electron revolves around the Fermi spher e with the zero impulse component in z-direction. The radius of this sphere is determined by the Fermi impulse P F (see eq. 77). In the real space: rF∼/planckover2pi1/PF (81) The energy of free particles near the Fermi surface: ǫ(PF) =vF(P−PF) (82) where: v FandPF=m∗vFare the Fermi velocity and impulse: P >P Fis the impulse of thermal electron at T >0 near the Fermi surface. The solution of the Schr¨ odinger equation, modified by Landa u for electrons in a magnetic field in real space leads to the following total e nergy eigenvalues (Blakemore, 1985): ǫ=/planckover2pi12k2 z m∗+/parenleftbigg l+1 2/parenrightbigg /planckover2pi1ωc, (83) where the first term of the right part represent the energy of translational motion of electrons, which does not depend on magnetic field magnitu de;kz= 1/Lz is the wave number of this motion; the second term is responsi ble for rotational energy,l= 0,1,2...is the integer quantum number for rotational motion in magnetic field Bz. Every value of lmeans a corresponding Landau level. Thus, free electrons in a magnetic field move along the helica l trajectory of the radius: rl= [(2l+ 1)/planckover2pi1/m∗ωc]1/2(84) 28At the transition from real space to the wave number space, th e radius of the orbit (kp)and its area is quantified as: Sl=πk2 p=2πeB /planckover2pi1c/parenleftbigg l+1 2/parenrightbigg (85) This formula is valid not only for the free electron model, bu t also for real metals. The magnitude 2π(/planckover2pi1c/e)termed a flux quantum. In a strong magnetic field the quantization of electrons ener gy leads to the periodic dependence of the metal magnetic momen t on the magnetic field (B): the de Haaz - van Alfen effect (Kittel, 1 978, Ashkroft and Mermin, 1976). Electroconductivity According to the Sommerfeld theory (Blakemore, 1985), elec troconductivity (σ) depends on the free run time of an electron ( τ) between collisions: σ=ne2τ/m, (86) where:n is the concentration of electrons, (e) and (m) are el ectron charge and mass. The free run time is equal to the ratio of the average free run d istance (λ) of electrons to the Fermi speed ( vF): τ=λ/vF (87) The free run distance is determined by scattering at defects (λD) and scattering at phonons ( λph): 1/λ= 1/λD+ 1/λph (88) The resistance ( R= 1/σ) could be expressed as: R= 1/σD+ 1/σph=RD+Rph (89) the contribution R Ddepends mainly on the concentration of the conductors defects, and the phonon contribution Rphdepends on temperature. Formula (89) expresses the Mattisen rule. A transition to a superconducting state means that the free r un time and distance tend to infinity: τ→ ∞;λ≃λef=h/Pef→ ∞, while the resulting group velocity of the electrons ( vres gr) and impulses tends to zero: 29Pef=m·vres gr→0 The emergency of macroscopic Bose-condensation of seconda ry ionic effectons and Cooper pairs corresponds to this condition. We assume in our hierarchic model, that the absence of the non - elastic scattering and dissipation of electrons energy is o bserved as superconductivity, when the probability of secondary ioni c effectons and deformons tends to zero, leading to emergency of primary elec- tronic polyeffectons. Let us consider first a conventional microscopic approach to the problem of superconductivity. 8. Microscopic theory of superconductivity (BCS) This theory (BCS) was created by Bardin, Cooper and Schriffer in 1957. The basic, experimentally proven assumption of thi s theory, is that electrons at sufficiently low temperatures are groupe d into Cooper pairs with oppositely directed spins - Bose-particl es with a zero spin. The charge of the pair is equal to e∗= 2eand massm∗= 2me. Such electron pairs obey the Bose-Einstein statistic. The B ose- condensation of this system at the temperature below the Bos e-gas condensation temperature (T < T k)leads to the superfluidityof the electron liquid. This superfluidity(analogous to the super fluidityof liquid helium) is manifested as superconductivity. According to BCS’s theory, the Cooper electron pair formati on mechanism is the consequence of virtual phonon exchange thr ough the lattice. The energy of binding between the electrons in a pair is very l ow: 2∆∼3kTc. It determines a minimum energetic gap (∆)separating a state of superconductivity from a state of usual conductivi ty. Notwithstanding that the kinetic energy of electrons in a su per- conducting state is greater than ǫF,, the contribution of the potential energy of attraction between electron pairs is such that the total en- ergy of the superconducting state (Ee a)is smaller than the Fermi en- ergy (ǫF) (Kittel, 1978). The presence of the energetic gap ∆makes a superconducting state stable after switching-off externa l voltage. The middle of the gap coincides with the Fermi level. The rupture of a pair can happen due to photon absorption by superconductor with the energy: ∆ =hνp≈3kTc. Superconductivity usually disappears in the frequency range 109<νp<1014s−1. In the BCS theory, the magnitude ∆is proportional to the number of Cooper pairs and grows on lowering the temperature. The excitation energy of quasiparticles in a superconducti ng state, which is characterized by the wave number (k), is: 30Ek= (ǫ2 k+ ∆2)1/2, (90) where ǫk=/planckover2pi12 m(k2−k2 F)≈/planckover2pi12 mkF(k−kF) (91) and δkF= (k−kF)≪kF= 1/LF The critical speed of the electron gas ( vc), for exciting a transition from a superconducting state to a normal one is determined from the condition: Ek=/planckover2pi1kvcandvc=Ek /planckover2pi1k(92) The wave function Φ(r), which describes the properties of el ectron pairs in the BCS theory, is the superposition of one-electron functions with energies in a range of about 2∆ near ǫF. Therefore, the dispersion of impulses for one- electron levels involved in the formation of pairs is expres sed as: ∆ =δǫF=δ/parenleftbiggP2 F 2m/parenrightbigg =/parenleftbiggPF m/parenrightbigg δPF≈vFδPF (93) The characteristic coherence length ( ξc) of the pair function Φ( r) has the value (Ashkroft and Mermin, 1976, Lifshits and Pitaevsky, 1978): ξc∼/planckover2pi1/δPF≃/planckover2pi1vF ∆≃1 kFǫF ∆(94) The magnitude ( ǫF/∆) is usually 103−104, andkF= 1/LF∼108cm−1. Thus, from (94): ξc∼(103−104)˚A (95) Inside the region of coherence length ( ξc) there are millions of pairs. The im- pulses of pairs in such regions are correlated in such a way th at their resulting impulse is equal to zero. At T>0 some of the pairs turn to a dissociated state and the concent ration of superconducting electrons ( ns) decreases. The coherence length ( ξc) also tends to zero with increase in temperature. 31The important parameter, characterizing the properties of a superconductor is the value of the critical magnetic field ( Hc), above which the superconductor switches to a normal state. With a rise in the temperature of the superconductor when T→Tc, the critical field tends at zero: Hc→0. And vice versa, at lowering of temperature, when T <T c,theHcgrows up as: Hc=H0[1−(T/T c)2] whereH0corresponds to T= 0. The Meisner effect - the ”forcing” of the outer magnetic field o ut of the superconductor is also an important feature of supercon ductivity. The depth of magnetic field penetration into the superconduc tor (λ) (Kittel, 1978, Ashkroft and Mermin, 1976) is: λ= (mǫ0c2/e2ns)1/2≃(10−6−10−5)cm, wherensthe density of electrons in a superfluid state; ǫ0- dielectric constant. On temperature raising from 0 K to Tc,theλgrows as: λ=λ0/bracketleftbigg 1−/parenleftBig T Tc/parenrightBig4/bracketrightbigg1/2, (96) where:λ0corresponds to λatT= 0. The superconductors with magnetic field penetration depth ( λ) less than coherence length ξ:: λ≪ξ (97) are termed first order superconductors and those with λ≫ξ (98) are second order ones. Nowadays, in connection with the discovery of high temperat ure superconductivity (Bednorz, Muller, 1986, Nelson, 1987) t he mecha- nism of stabilizing electron pairs by means of virtual phono ns in the BCS theory evokes doubts. 329. Mesoscopic scenario of superconductivity We propose a new mechanism of electron pair formation and the ir subsequent Bose-condensation, without virtual phonons as mediators. Such a process is analogous to the formation of primary polye ffectons in liquid helium related to superfluidityphenomena (see Sec tion 5). Two basic questions must be answered in relation to the emerg ence of superconductivity: I. Why does energy dissipation in the system [conductivity e lec- trons + lattice] disappear at T≤Tc? II. How does the coherence in this system, related to electro n pair formation, originate ? It will be shown below how these problems can be solved in the framework of our Hierarchic (mesoscopic) theory. The following factors can affect electron’s dynamics and sca ttering near Fermi energy: 1. Interaction of electrons with primary and secondary ioni c ef- fectons in acoustic (a) and (¯ a) states, stimulating Cooper pairs for- mation; 2. Interaction with primary and secondary effectons of latti ce in acoustic (b) and ( ¯b) states, leading to origination of polarons and Cooper pairs dissociation; 3. Interaction with transitons in the course of (a⇔b)and (¯a⇔¯b)transitions of primary and secondary effectons; 4. Interaction with [tr/lb]convertons (interconversions between primary translational and librational effectons; 5. Interaction with primary electromagnetic deformons; 6. Interaction with secondary acoustic deformons (possibi lity of polaron formation); 7. Interaction with macroeffectons in A- and B-states; 8. Interaction with macro- and superdeformons (possible em er- gence of defectons). It follows from our model that the oscillations of all types o f quasiparticles in conductors and semiconductors are accompanied not only by e lectron-phonon scattering, but also by electromagnetic interaction of pri mary deformons with unbind electrons. AtT > T cthe fluctuations of unbind electrons with energy higher than Fermi one under the influence of factors (1 - 8) are random (noi se-like) and no selected order of fluctuations in normal conductors exists. It means that ideal Fermi-gas approximation for such electrons is sufficiently g ood. In this case, the effective electron mass can be close to that of a free electron (m∗≃m). 33Electric current in normal conductors at external voltage s hould dissipate due to fluctuations and energy exchange of the elec trons with lattice determined by factors (1 - 8). Coherent in-phase acoustic oscillation of the ionic primar y effec- tons in (a)-state is the ”ordering factor” simulating elect ron gas co- herence due to electromagnetic interactions. But its contr ibution in normal conductors at T >T cis very small. For an ideal electron gas, the total energy (Etot)of each electron as wave B is equal to its kinetic energy (Tk), as far potential energy (V= 0): Etot=/planckover2pi1ω=/planckover2pi12 2mA2=Tk+V=/planckover2pi12 2mL2(99) where: Tk=/planckover2pi12 2mL2(100) and L=/planckover2pi1 mvgr=1 k(101) One can see from (100) that for an ideal gas, when m=m∗, the most probable amplitude (A) and wave B length (L) are equal: A=L, ifEtot=TkandV= 0 (102) Like in case of liquid helium at conditions of superfluity, Bo se- con- densation in metals and semiconductors is related to an incr ease in the concentration of the (a)-state of primary effectons with the low- est energy and a corresponding decrease in the concentratio ns of all other excitations. The Bose-condensation and degeneratio n of sec- ondary ionic effectons and deformons, followed by formation of elec- tronic polyeffectons from Cooper pairs is responsible for se cond order phase transition like superconductivity. The cooperative character of 2nd order phase transition [co nductor →super- conductor] is determined by a feedback reaction between the lattice and electron subsystems. It means that the collective Bose-condensatio n in both subsystems is starting at the same temperature: T=Tc. Under such conditions the probabilities of the (a)-states o f ionic (Pi a) and electronic ( Pe a) effectons tend to 1 at T≤Tc: Pi a→1;Pe a→1 Pi b→0;Pe b→0/bracerightbigg (103) 34The equilibrium parameter for both subsystems: κi,e=/parenleftbiggPa−Pb Pa+Pb/parenrightbiggi,e →1 (104) and the order parameter : ηi,e= (1−κi,e)→0 (105) like in a 2ndorder phase transition for liquid helium (see eq. 58). In our model the coherent Cooper pairs are formed as Bose part i- cles with resulting spin equal to 0 and 1 from neighboring ele ctrons, in contrast to the BCS theory, which assumes phonons - mediat ed interaction between distant electrons. Such pairs can compose primary electron’s effectons (e-effec tons) as a co- herent cluster with a resulting impulse equal to zero . Formation of secondary e-effectons with nonzero resulting impulse is possible also . Just the interaction of this secondary e-effectons with lattice is responsible fo r electric resistance in normal conductors. Degeneration of such type of excitation s in the process of their Bose-condensation and their conversion to primary e- effectons means the emergency of superconductivity. The in-phase coherent oscillations of the integer number of elec- tron pairs forming primary e-effectons correspond to its aco ustic (a)- state, and the counterphase oscillations to its optic (b) st ate, like in ionic or molecular effectons. We assume that superconductivity can originate only when th e fraction of unbind coherent electrons forming primary e-eff ectons in certain regions of conductor strongly prevails over the f raction of noncoherent secondary e-effectons. Due to feedback reactio n between subsystems of lattice and electrons this fraction should be equal to ratio of wave length of primary and secondary ionic effectons . This condition can be introduced as: (II) :/parenleftbiggλe a ¯λea/parenrightbigg Tc=/parenleftbiggvs/λa vs/¯λa/parenrightbigg Tc≥10 (106) where:λe a/¯λe ais the ratio of wave lengths of primary and sec- ondary e- effectons, equal to that of primary and secondary eff ectons of lattice; sound velocity is equal to lattice primary effect ons phase velocity in (a) state: vs≃va phunder conditions corresponding to (60). λe a=λi a≃10·¯λe a≃20·n1/3 c (107) 35are the periods of electron’s oscillations modulated by the oscilla- tions of primary and secondary ionic effecton’s in the acoust ic states. As far the oscillations of e-pairs in the Bose-condensate (a -state of e- effectons) are modulated by the electromagnetic field, rad iated by oscillating ions in the (a)-state of primary ionic effectons they must have the same frequency. Condition (106) means that the number of electrons in the vol ume (Ve∼λ3)of primary e-effectons is about 103times more than that in secondary e-effectons. The lattice and electronic effectons subsystems are spatially compatible. As far the effective mass (m∗)of the electrons in the coherent macroscopic Bose-condensate organized by e-polyeffectons at condi- tions of superconductivity tends to infinity: T <T c m∗/mapsto−→ ∞ T→0(108) the plasma frequency (eq. 69) tends to zero: 2πνpl=ωplT <T c /mapsto−→0 at T→0 and, consequently, the screening length (eq.71)tends to infinity: l→ ∞. This condition also corresponds to that of macroscopic Bos e- condensation emergency Under these conditions an impulse (see eq.93) originates in addi- tion to Fermi’s one: δPF=/planckover2pi1δkF>0 (109) but a decrease in the potential energy of both electron’s and ion’s subsystems due to leftward (a⇔b)equilibrium shift leads to the emergence of the gap near the Fermi surface (2 ∆) depending on the difference of energy between (a) and (b) states of primary effe ctons. The linear dimension of coherent primary electronic effecto ns (e- effectons), which is equal to coherence length in the BCS theo ry (see eq. 94) is determined by additional impulse δPF: ξ=λe a=vs νa=h/∆pF=h/m e∆vF (110) In turn, the primary e-effecton in (a)-state can form e- polye ffec- tons as a result of their polymerization. The starting point of this 36collective process represents macroscopic Bose - condensa tion and second order phase transition in accordance with our model. The energy gap between normal and superconductive states ca n be calculated directly from our mesoscopic theory, as the diffe rence be- tween the total energy of matter before/bracketleftbig UT>T c tot/bracketrightbig and after/bracketleftbig UT<T c tot/bracketrightbig the second order phase transition: 2∆ =UT>T c−UT<T c(111) However, such experimental parameters as sound velocity, d ensity and the positions of bands in a far IR region must be available around transition temperature (Tc)for calculation of (111). This gap must be close to the energy of (a→b)itransitions of ionic primary effectons, related to the energy of (a→b)etransitions of the electronic e-effectons (see 103 - 105). This statement of our superconductivity model coincides we ll with the experimental destruction of superconductivity state b y IR-radiation with minimum frequency (νg), corresponding to the energy gap (2∆) at given temperature: hνg= 2∆∼(Eb−Ea) (112) Another general feature of superconductivity for low- and h igh- temperature superconductors is the almost constant ratio: 2∆0 kBTc≃3.5 (113) where the gap: ∆ = ∆ 0atT=Tcand ∆ = 0 at T >T c. It will be shown below that the experimental result (113) is r elated to condition (106) of our mesoscopic model of superconducti vity. Considering (112), (113) and (2.27), the frequency of a prim ary ionic effectons in a-state near transition temperature is: νi a=ν0 g exp/parenleftBig 2∆ kTc/parenrightBig −1≃ν0 g/32.1 = 0.03(2∆/h) (114) consequently: hνi a= 0.03·2∆ (115) 37The frequency of secondary lattice effectons in ( a)−state in accor- dance with (2.54) is: νia=νi a exp/parenleftBig hνia kTc/parenrightBig −1 as far: hνa kTc= 0.032∆ kTc= 0.1≪1 (116) we have:/bracketleftbigg exp/parenleftbigghνi a kTc/parenrightbigg −1/bracketrightbigg−1 ∼kTc hνa∼10 consequently: νa≃kTc/h, (117) Now, using (17), (116) and (113) we confirm the correctness of condition (106): λe a/λea= (¯νa/νa)i=/bracketleftbigg exp/parenleftbigghνi a kTc/parenrightbigg −1/bracketrightbigg−1 ∼10 (118) where: λe a=h/2me(va gr)e= (vs/νa)i(119) is the most probable wave B length of coherent electron pairs compos- ing a primary e-effecton; (va gr)eis a group velocity of electron pairs in a-state of primary e-effectons, stimulated by ionic lattice oscillations: λea=h/2m∗vagr= (vs/νa)i(120) is the mean wave B length of electron pair forming the effectiv e sec- ondary e-effecton. Our theory predicts also the another condition of coherency be- tween ionic and electronic subsystems, leading to supercon ductivity, when the linear dimension of primary translational ionic eff ectons grows up to the value of coherence length (see eq.94): (λi a)tr=vs (νia)tr≥ Tcζ=/planckover2pi1vF ∆(121) 38In simple metals a relation between sound (vs)and Fermi velocities (vF)is determined by the electron to ion mass ratio (me/M)1/2(March and Parinello, 1982): vs=/parenleftBigzme 2M/parenrightBig1/2 ·vF, (122) where: z is the valence of ions in a metal. Putting (νi a)trfrom eq.(115) and eq.(122) in condition (121), and introducing instead electron mass its effective mass (m∗)in composi- tion of e-effecton ,we obtain at T=Tc: m∗Tc≃2·10−4·M z(123) As far we assume here that at transition temperature (T c)the volumes of primary lattice (ionic) effectons and primary e-e ffectons coincide, then the number of electrons in the volume of prima ry ef- fectons is: Ne=ne·Vi ef=ne·9 4π/parenleftbiggvs νia/parenrightbigg3 Tc(124) where:neis a concentration of the electrons; Vi ef=Ve efis the volume of primary translational ionic effectons. Using the relation between primary and secondary ionic wave s B (106) at T casλa= 10λa, taking into account (117) and (121) we got: /parenleftbiggzm∗ e 2/parenrightbigg1/2 ·1 TcM1/2=k 10·2π∆(125) or: Tc(M/z)1/2=10π∆ k·(2m∗ e)1/2(126) If for the different isotopes the energetic gap is constant (2 ∆≃ const), then the left part of (126) is constant also. Such an importa nt correlation between transition temperature (Tc)and isotope mass (M) is experimentally confirmed for many metals: Tc≃/parenleftbiggh kB/parenrightbigg ·vs 2n1/3 e(127) This result as well as (117) can be considered as evidence in p roof for our model of superconductivity. 39It leads from (127) that the more rigid is lattice and the bigg er is sound velocity (vs),the higher is transition temperature. Anisotropy ofvsmeans the anisotropy of superconductor properties and can b e affected by external factors such as pressure. It was shown in this article, that all most important phenom- ena, related to turbulence, superfluidityand superconduct ivity can be explained in the framework of our Hierarchic theory. =================================================== =========== REFERENCES Ashkroft N., Mermin N. Solid state physics. N.Y.:Helt, Rinehart and Winston, 1976. Bardeen J., Cooper L.N., Schrieffer J.R. Phys. Rev., 108, 1175, 1957 Bardeen J., Schrieffer J.R. Progr. Low. 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arXiv:physics/0003109v1 [physics.atom-ph] 31 Mar 2000The ZEKE-effect in cold Rydberg gases S. K. Dutta, D. Feldbaum, G. Raithel University of Michigan, Physics Department, Ann Arbor, MI 4 8109-1120 (September 25, 2013) Cold, dense Rydberg gases produced in a cold-atom trap are investigated using spectroscopic methods and time - resolved electron counting. On the discrete Rydberg reso- nances we observe large trap losses and long lasting elec- tron emission from the Rydberg gas ( >30ms). Our observa- tions are explained by quasi-elastic l-mixing collisions between Rydberg atoms and slow electrons that lead to the population of long-lived high-angular-momentum Rydberg states. Thes e atoms thermally ionize slowly and with large probabilities , leading to the observed effects. 32.80.Pj, 52.25.Ya, 34.60.+z Laser-cooled atoms can be used to study highly ex- cited Rydberg atoms [1] at both large densities and low atomic velocities. Due to the low velocity of the Rydberg atoms, ionizing Rydberg-Rydberg collisions that domi- nate the behavior of hot Rydberg gases [2–4] are largely suppressed. Therefore, the interactions between the Ryd- berg atoms, free electrons and ions result in a variety of novel phenomena that are specific to translationally cold Rydberg gases. Density-dependent effects have been ob- served in the resonant excitation transfer between cold Rydberg atoms [5,6]. A minute increase of the frequency of the Rydberg excitation laser leads to the production of metastable cold plasmas rather than cold Rydberg gases [7]. At a critical density corresponding to about one atom per atomic volume, Rydberg gases might un- dergo a Mott transition that would lead to a new kind of metastable matter [8–10]. The formation of metastable Rydberg matter might proceed through an intermediate phase, in which the Rydberg population accumulates in long-lived high-angular-momentum (high- l) states [10]. In the present paper, we show that in cold Rydberg gases high-lRydberg states are efficiently produced by a ro- bust mechanism, which is, in a similar form, also at work in ZEKE (ZEro Kinetic Energy) electron - spectroscopy, a powerful technique that has revolutionized molecular spectroscopy [11]. In our experimental cycle,87Rb atoms are collected and cooled in a magneto-optic trap (MOT [12], see Fig. 1) for about 950ms. 1ms after the shutdown of the MOT a 5µslong diode laser pulse ( λ= 780nm) resonant with the 5S1/2, F= 2→5P3/2, F= 3 transition is applied. While the 780nm pulse is on, a blue dye laser pulse ( λ≈480nm, 10ns width, bandwidth ≈15GHz, repetition rate 10Hz) excites ns- and nd-Rydberg states from the intermediate 5P3/2-level. The maximum photon fluence of one bluepulse easily exceeds the saturation fluence at the pho- toionization threshold (7 ×1016cm−2atλion= 479 .1nm [13]). If the dye laser operates with the oscillator only, the broad-band ASE (amplified spontaneous emission) contained in the blue pulse is <1% of the pulse energy (≈50µJ). With the dye amplifiers on, the ASE contains ≈10% of the pulse energy, is about 5nm wide and cen- tered at λ= 478nm, which is above the ionization thresh- old. The pulsed dye laser is pumped by the 3rd harmonic of a Nd-YAG laser (355nm), a small fraction of which (<1mJ) can be diverted to partially ionize the atomic cloud before the blue laser pulse arrives. The Rydberg excitation causes a reduction of the ground-state pop- ulation, which reduces the area density of ground-state atoms that we measure with a low-intensity probe laser pulse resonant on the 5 S1/2, F= 2→5P3/2, F= 3 tran- sition. A microchannel-plate (MCP) detector located about 10cm from the atomic cloud is used to detect elec- trons emitted from the Rydberg gas. Atomic CloudRb87ns,□ndIonization Limit 5P3/2 5S1/2Diode LaserBeamsplittersTo□photon counterTripled Nd:YAG LaserDye (”Blue”) Laser Spectrometer Tripled Nd:YAG Laser 355□nm Diode Laser 780□nmMCPDye□Laser ~480□nm Vacuum Chamber FIG. 1. Level scheme (left) and outline of the experimental setup (right). Fig. 2a) shows a trap-loss spectrum taken with both the 5S1/2→5P3/2and the Rydberg transition saturated. In the continuum as well as on the discrete Rydberg res- onances up to about 70% of the atoms are removed. The trap-loss spectra are largely independent of the time at which the probe pulse is applied. Thus, the observed trap loss is mostly due to the permanent loss of atoms from the trap upon excitation. The observed trap loss is a cu- mulative effect, as each trapped atom experiences many blue laser pulses during the MOT loading time (2 .5s). We have used a simple model of the MOT loading dynamics to estimate the single-pulse loss fraction X. In Fig. 2b), a floor of X≈1% is observed between the discrete Ryd- berg lines, which is due to direct photoionization by the above discussed 10% ASE of the blue pulse. The discrete Rydberg lines peak at an Xof up to 8% above the floor, a value that corresponds to one-third loss of the excited 1Rydberg atom population. Further, there is no signifi- cant change in Xat the continuum threshold. Our ob- servations show that there is a process by which initially bound Rydberg atoms permanently leave the trap with an efficiency that rivals direct optical photoionization. 00.020.040.060.080.1 479479.5480480.5481481.5482482.5483Single-pulse trap loss fraction27d wavelength (nm)b) λion2 1083 1084 1085 1086 1087 1088 108 479479.5480480.5481481.5482482.5483Area density ( cm-2 ) wavelength (nm)27dno blue laser a) λion FIG. 2. Rydberg excitation spectrum of atoms in a mag- neto-optic trap. Panel a): central area density of trapped atoms vs. the wavelength of the blue laser. Panel b): Corre- sponding estimated single-pulse loss fraction X. Two-body ionizing Rydberg-Rydberg collisions could, in principle, cause a trap loss. Based on an ionization cross section given in [4], a Rydberg atom velocity of 0.1m/s, a lifetime of 100 µs (n≈40), and a Rydberg atom density of 5 ×109cm−3we estimate a collisional ionization probability of ≈0.5% of the excited Rydberg population, corresponding to an X≈0.13%. This figure is too small for two-body ionizing Rydberg-Rydberg collisions to be the dominant source of trap loss. To estimate the importance of thermally induced mi- crowave ionization of the excited nsandndRydberg states, we have performed rate equation simulations of the population flow among the bound atomic states and the continuum. We use a basis ( n, l) of discrete levels up ton= 100 with all allowed values of land the proper quantum defects, and a grid of 10100 continuum states (ǫ, l) with energies up to ǫ= 130meV and l= 0,1, ..,101. Due to isotropy, we can assume uniform distributions over the magnetic substates and use m-averaged transi- tion rates [1]. The obtained thermal ionization probabil- ities for an ideal 300K blackbody spectrum are displayed in Fig. 3. While direct thermal ionization of the initially excited states does not cause enough ionization to explain the observed trap loss - see Fig. 3 -, it produces ions mov- ing at about 1m/s and electrons with about 8meV av- erage kinetic energy; the latter figure results from the rate-equation calculations. Based on Fig. 3 and on the number of excited Rydberg atoms, we estimate that upto∼105electron-ion pairs are created. If the blue laser is used with its amplifiers active, the ionizing ASE of the blue laser pulse adds a significant amount of additional electron-ion pairs (electron energy ≈10meV). As a re- sult, conditions are such that a metastable cold plasma is formed [7]: A fraction of the electrons quickly evapo- rates, leaving behind a plasma with a net positive charge that acts as an electron trap. If, under our conditions, the initial number of electrons exceeds about 1000, the trap becomes deep enough to retain a fraction of the elec- trons [7]. The net positive charge causes a slow Coulomb expansion, due to which all trapped electrons eventually evaporate. The electron storage time is thereby limited to of order 100 µs, which is long enough for the retained electrons to frequently collide with the main product of the laser excitation - the bound Rydberg atoms floating in the plasma. The collisions initiate a sequence of events we refer to as the ZEKE-effect. 0102030405060 102030405060708090Ionization Probability neffFull: l-mixing up to specified time Open: No l-mixing level required for 50% trap loss at a 2.5s MOT loading time500µs 25µs d-states s-states FIG. 3. Thermal ionization probabilities of Rydberg atoms vs. the effective principal quantum number obtained from rate-equation calculations. Open circles: nd-states, open di- amonds: ns-states, full circles: l-mixing for t <500µs, full boxes: l-mixing for t <25µs. The curves are explained in the text. In step Aof the ZEKE-effect (see Fig. 4), the atoms are promoted from their initial sord-states into a high- l- state by the electric-field sweep produced by a bypassing electron. The field sweep brings the initial Rydberg state in contact with a hydrogenic manifold of high- lstates, and state-mixing causes a quasi-elastic transition of the atom into a superposition of high- l-states. To model these collisions, we have numerically solved the time- dependent Schr¨ odinger equation. For a given initial state |n0, l0, m0/angbracketright, electron velocity vand collision parameter b the calculation yields a final probability P(n0, l0, v, b) of finding the atom in the hydrogenic manifold, i.e. in a state with l≥4. For l0/negationslash= 0, we run the calculation for the allowed values of m0and average the resultant prob- abilities over m0. The cross section σfor a transition into l≥4 is then defined as σ(n0, l0, v) =/integraldisplayb=∞ b=0P(n0, l0, v, b)2πbdb (1) 2Forl0= 0, 1 or 2 it is easy to determine an upper cutoff value for bwhere P→0. The range b < a 0n2, where the neglected ionizing processes should become dominating [4], does not significantly contribute to σ. -0.00135-0.0013-0.00125 0 5 10-81 10-71.5 10-7Energy (at. un.) Electric Field (at.un.)23s 21d 22pABC FIG. 4. The ZEKE effect visualized using the Stark map of Rb in the vicinity of n= 20. The three steps A,Band C, explained in the text, lead to the production of long-lived Rydberg atoms and time-delayed thermal ionization. The stepAis induced by quasi-elastic collisions between electrons and Rydberg atoms. 0.010.11101001000 20 30 40 50 60708090 100σ(d) σ(s)σ (µm2) neff70 90 0.1110 0.1 1101001000σ (µm2) Electron energy (meV)33s31d63s61d FIG. 5. Left: Calculated l-mixing cross sections σfor col- lisions of Rydberg atoms in the indicated initial states wit h electrons versus the electron energy. Right: l-mixing cross sections σwithn5 eff-fits (dotted) for s- and dinitial states at an electron energy of 4.5meV versus the effective quantum number. Fig. 5 shows the results of our cross section calcula- tions for the sandd-states of two values of n. The drop ofσat higher electron energy reflects the fact that the passage behavior of the Rydberg atoms in the Stark map (Fig. 4) turns diabatic. The s-states generally have cross sections about 2.5 times smaller than the cross sections of the nearest d-state. This reflects the diabatic crossing behavior of the s-state atoms with the lowest few states of the nearest hydrogenic manifold - note the narrow an- ticrossings in Fig. 4. For fixed electron energy, the cross sections approximately scale as n5 eff. The average values oflof the atoms that made a transition into the hy- drogenic manifold are of order n/2 (not shown), i.e. the l-distribution becomes pretty well randomized within the hydrogenic manifold.The plasma volume ( ≈1mm−3), the electron num- ber (≥1000), the electron velocity ( ≈50000 m/s), the electron storage time ( ≈100µs), and the cross sections depicted in Fig. 5 lead to the conclusion that the step A in Fig. 4 happens with certainty for nlarger than about 20. The presence of the plasma that temporarily traps the electrons is crucial, as it keeps the electrons from leaving and causes them to frequently collide with the abundant, bound Rydberg atoms floating in the plasma. Collisions between Rydberg atoms and ions are ineffec- tive, because the ions are very slow ( ≈1m/s), as are the Rydberg atoms themselves. After step Ain Fig. 4, the weak but rapidly varying microfields generated by more distant electrons will be sufficient to further randomize the Rydberg population among the quantum-defect-free hydrogenic states (step B). Step Bis more probable than step A, but it is not required for the subsequent step C. Once all plasma elec- trons have evaporated, the only electric fields the Ryd- berg atoms are still exposed to are the fields generated by the very slowly moving ions. Therefore, we expect that the plasma dynamics and the internal dynamics of the Rydberg atoms decouple at about 100 µs after the excitation. 101001000 0 5 10 15 20measured rates simulation (25µs l-mixing)Count Rate (ms-1) Time (ms)n=50n=20 FIG. 6. Electron count rates for the indicated Rydberg d-states vs. the delay time between the Rydberg excitation and the counting gate. The signal lasts at least 100 times as long as the natural lifetime of the initially excited states , and shares the qualitative features of simulated results (dott ed). Subsequently, while the cloud of high- lRydberg atoms produced by the steps AandBslowly expands, the atoms decay or thermally ionize on a slow time scale (step Cin Fig. 4). To model the overall dynamics of the Rydberg population, we have included an initial ”plasma phase” in our rate-equation simulations. During that phase, the populations in the n-manifolds and the nearby non- hydrogenic states are periodically averaged over the al- lowed l-values (with weights ∼(2l+1)); after the “plasma phase” the averaging ceases. The obtained ionization probabilities are displayed in Fig. 3 for 25 µs and 500 µs long plasma phases. The ionization probabilities are sta- ble against variations of the duration of the plasma phase and are large enough to explain the experimentally ob- served trap losses. The discussed model is supported by the results de- 3scribed in the following. Using the MCP detector lo- cated near the atom trap, we have measured the thermal ionization current vs. time (Fig. 6). When we excite discrete Rydberg levels, we find long-lived electron sig- nals that extend beyond 30ms and that only occur if the frequency of the blue laser is resonant with a discrete Rydberg line. The delayed electron signal, which is char- acteristic for the ZEKE-effect [11], is due to thermal ion- ization of high- lRydberg states. Fig. 7 shows the electron current in a time-delayed counting window vs. the wavelength of the blue laser for conditions well below the saturation of the Rydberg transition. The clarity of the ionization threshold in the long-lived electron signal, which is typical for the ZEKE- effect [11], shows that the delayed electrons are linked to the initial optical excitation of bound Rydberg atoms. In Fig. 7a) the blue pulse is generated with the dye laser oscillator only, which has practically no ASE. Therefore, all electron-ion pairs that are created within about 100 µs from the Rydberg excitation originate in thermal ioniza- tion of the excited ns- ornd-Rydberg atoms. Since the 5P→ns-photoexcitation cross sections are about six times smaller than the ones of the neigboring d-states, and since the thermal ionization probability of the s- states is only half that of the neighboring d-states (see Fig. 3), the thermal electron yield on the s-lines is less than one-tenth of the yield on the neighboring d-lines. Recalling that of order 1000 slow electrons are needed to form the essential cold-plasma electron trap - with some electrons left in it -, it follows that there is a wide range of parameters where the ZEKE-signal should occur on thed-lines, but not on the neighboring s-lines. Fig. 7a), where the s-lines barely appear, shows one such case. We have used two methods to force a ZEKE-signal on the s-lines in cases where it would normally not ap- pear. In Fig. 7b) we have used the same parameters as in a), except that the blue laser pulse has been generated with a dye-laser amplifier being active. The pulse has then been attenuated to the same pulse energy as in a). The ASE produced by the amplifier creates enough slow electron-ion pairs to form the cold-plasma electron trap, independent of whether the coherent part of the laser ex- citesdorsRydberg atoms. As a result, we observe both sanddlines in the ZEKE-signal. In Fig. 7c), the blue laser pulse is the same as in a), i.e. it has no ASE, but we ionize a small fraction of the atoms by a UV laser pulse that hits the cloud a few ns before the blue laser pulse (see Fig. 1). The electrons produced by the UV pulse have an energy of about 1eV and therefore all leave. The remaining electron-free potential well captures any slow electrons that are subsequently produced. The compara- tively few thermal electrons produced on the s-lines now don’t escape, as in the case of Fig. 7 a), but are trapped and trigger the ZEKE-effect (note the s-lines in Fig. 7c)).479 480 481 482 48302550 λion25d c) Excitation Wavelength (nm)Electron count rate (ms-1) 479 480 481 482 4830100200300 λion27sb) 479 480 481 482 4830100200300 λion a) FIG. 7. Electron current emitted by the Rydberg gas in a time window from 5ms to 6ms after the excitation as a function of the wavelength of the blue laser. a) <1% ASE in the laser spectrum. b) ≈10% ASE in the laser spectrum. c) as a), but a weak UV pulse is used to ionize about 0 .1% of the atoms a few ns before the blue laser pulse. In this paper we have shown that dense, cold Rydberg gases in a room-temperature thermal radiation field de- cay via the ZEKE-effect, which involves the quasi-elastic collisional production of high- lRydberg states and un- usually slow thermal ionization. The effect hinges on the temporary existence of a cold plasma, which acts as a transient electron trap. The stability of the observed phenomenon makes it likely that it represents the generic decay pattern of cold, dense Rydberg gases. It appears likely that the spontaneous and efficient production of high-lRydberg states could aid the formation of con- densed Rydberg matter [9,10]. Since it has become ap- parent that the radiation temperature is one of the most important parameters of the system, we intend to per- form future studies in a cryogenic enclosure with variable wall temperature. We thank Prof. P. Bucksbaum for inspiring discussions and generous loaning of equipment. Support by NSF and DoE is acknowledged. [1] T. F. Gallagher, Rydberg Atoms , Cambridge University Press, Cambridge 1994. [2] M. Ciocca et al., Phys. Rev. Lett. 56, 704 (1986). [3] M. W. McGeoch, R. E. Schlier, G. K. Chawla, Phys. Rev. Lett.61, 2088 (1988). [4] G. Vitrant, J. M. Raimond, M. Gross, S. Haroche, J. Phys.B 15, L49 (1982). [5] W. R. Anderson, J. R. Veale, T. F. Gallagher, Phys. Rev. Lett.80, 249 (1998). 4[6] I. Mourachko et al. Phys. Rev. Lett. 80, 253 (1998). [7] T. C. Killian et. al., Phys. Rev. Lett. 83, 4776 (1999) [8] N. F. Mott, Proc. R. Soc. London A382 , 1 (1982) [9] E. A. Manykin, M. I. Ozhovan, P. P. Poluektov, Sov. Phys. JETP 57, 256 (1983), L. Homlid, E. A. Manykin, Zh. Eksp. Theor. Fiz. 111, 1601 (1997). [10] R. Svensson, L. Holmlid, L. Lundgren, J. Appl. Phys. 70, 1489 (1991), E. R. Olsson, R. Svensson, J. Davidsson, J. Phys.D 28, 479 (1995), R. Svensson, L. Holmlid, Phys. Rev. Lett. 83, 1739 (1999). [11] the field is reviewed by E. W. Schlag, R. D. Levine, Comm. At. Mol. Phys. 33, 159 (1997). The importance of collisions has been stressed by W. A. Chupka, J. Chem. Phys. 99, 4580 (1993) and P. Bellomo, D. Farelly, T. Uzer, J. Chem. Phys. 108, 5259 (1998). [12] E. L. Raab, M. G. Prentiss, A. E. Cable, A. Clairon, S. Chu, D. E. Pritchard, Phys. Rev. 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arXiv:physics/0004001v1 [physics.gen-ph] 1 Apr 2000The Fractal Universe B.G. Sidharth Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Hyderabad 500 063 (India) Abstract In this talk, we touch upon the chaotic and fractal aspects of the Universe. 1 Introduction It was thought that a subject like Celestial Mechanics belon ged to the domain of deterministic Mechanics[1]. However studies by Laskar[ 2, 3] subsequently confirmed, that the inner planetary system is chaotic with a s mall inverse Lyapunov exponent sheds new light on the topic. This need not be surpris- ing because the solar system is really a many body system. Fur ther, it is possible to treat a system with time varying constant of grav itation as a dynamical system[4]. Indeed in certain cosmological schem es the universal constant of gravitation Gvaries, with time[5, 6, 7]. On the other hand there has been an ongoing debate whether the Universe is homogeneous at the largest scales or if space time is indeed a differentiable manifold at the smallest scales[8]. It must be observed that from one point of view, the universe i s not a ”con- tinuum” but rather, displays a fractal character. Thus with in an atom, the nucleus occupies a very tiny fraction of the volume, roughly ∼10−15. Then there is a wide gap till we reach the orbiting electrons. Simi larly there are in- termolecular distances, interplanetary, interstellar, i ntergalactic.... distances 0E-mail:birlasc@hd1.vsnl.net.in 1which provide relatively huge gaps. This is not in the spirit of a uniform continuum. We will now argue that this is because, for example the nucleo ns are bound together, so also the electrons and the nucleons are bound to gether, the atoms in the molecules are bound together... and so on with su bsequent gaps, which leads to some interesting scale dependent conse quences, all this in the context of a Brownian underpinning. 2 The Fractal Universe In [8] it was argued that we could introduce a ”Scaled” Planck Constant given by h1∼1093(1) for super clusters; h2∼1074(2) for galaxies and h3∼1054(3) for stars. This was directly related to the fact that we have the followi ng Random Walk relations: R≈l1/radicalBig N1 (4) R≈l2/radicalBig N2 (5) l2≈l3/radicalBig N3 (6) R∼l√ N (7) where N1∼106is the number of superclusters in the universe, l1∼1025cms is a typical supercluster size N2∼1011is the number of galaxies in the uni- verse and l2∼1023cmsis the typical size of a galaxy, l3∼1 light year is a typical distance between stars and N3∼1011is the number of stars in a galaxy, Rbeing the radius of the universe ∼1028cms, N ∼1080is the num- ber of elementary particles, typically pions in the univers e and lis the pion Compton wavelength. The relation (7) was observed nearly a century ago by Weyl and Heddington. 2It was shown (Cf.ref.[7]) that far from being empirical this relation can be deduced on the basis of the fluctuational creation of particl es from a back- ground Zero Point Field or Quantum Vacuum, in a scheme which l eads to a cosmology consistent also with Dirac’s large number coinci dences[9] and in which the gravitational constant Gvaries with time, G∝T−1. From this point of view the Random Walk character of equation (7) is not accidental, and this reasoning could be extended to equatio ns (4),(5) and (6), in the light of equations (1), (2) and (3). This is agains t the spirit of deterministic mechanics and it may be mentioned that it lead s to a fractal character[10]. It may be observed that in all these cases we have a length, the Compton wavelength or its analogue which defines regions of matter se parating rela- tively empty spaces. Infact it was also argued in[8] that these scaled ”Compton wa velengths” and scaled ”Planck Constants” arise due to the well known equati on of gravita- tional orbits, GM L∼v2(8) On the other hand equation (8) can be viewed as resulting from the Virial Theorem[11], where the velocity is replaced by the velocity dispersion. This velocity vwould be different at different scales. For example for a black hole it would be the velocity of light, while for galaxies it i s∼107cmsper second[12]. It is this circumstance that produces the above scales leading to fractality. We could go one step further, because we expect that the same e ffect would apply to solar type systems: The planets and other objects ar e bound quite close to the sun compared to the interstellar distances. Inf act we can verify that this is so for Kuiper Belt objects which have been studie d in the recent past[13]. In this case a typical size is ∼5km, the distances are ∼1015cm, masses are ∼1019gmstheir number is ∼1010while an application of equation (8) shows that the velocities are ∼105cmsper second. It can now be shown quite easily that this defines a scaled Planck Constant h4∼1034. Incidentally from (8) we could easily deduce that the angula r momentum J is given by J∝M2(9) 3It is quite remarkable that the equation (9) also applies to e lementary par- ticles and Regge trajectories[14]. This is a further substa ntiation of the rationale for the fractal structure given above in the light of bound systems separated by relatively large and relatively empty spaces a nd applies right upto the level of galaxies[15]. References [1] Prigogine, I., and Nicolis, G., ”Exploring Complexity” , (1989), W.H. Freeman, San Francisco. [2] Laskar, J., (1989) Nature 338 , p.237. [3] Nottale, L., Chaos, Solitons & Fractals, (1994) 4, 3, 361 -388 and refer- ences therein. [4] Melnikov, V.N., and Romero, C., (1990) ”Gravitation and Cosmology”, 14p.277. [5] Barrow, J.D., (1992) ”Theories of Everything”, Vintage , London. [6] Narlikar, J.V., (1983) Foundations of Physics, Vol.13. No.3. [7] Sidharth, B.G., Int.J.Mod.Phys.A, (1998) 13 (15), p.25 99ff. [8] Sidharth, B.G., ”The Scaled Universe” to appear in Chaos Solitons and Fractals and other references therein. [9] B.G. Sidharth, Int.J.Th.Phys., (1998), 37 (4), p.1307ff . [10] Mandelbrot, B.B., ”The Fractal Geometry of Nature”, (1 982) W.H. Freeman, New York, pg.2,18,27. [11] Nottale, L., ”Fractal Space-Time and Microphysics: To wards a Theory of Scale Relativity”, (1993), World Scientific, Singapore, p.312. [12] Narlikar, J.V., (1993) ”Introduction to Cosmology”, F oundation Books, New Delhi. [13] David Jewitt, Physics World, July 1999, p.37ff. 4[14] Tassie, L.J., (1973) ”The Physics of Elementary Partic les”, Longman Group Limited, London. [15] Carneiro, S., Found.Phys.Letts. (1998) 11 (1), p.95ff. 5

arXiv:physics/0004002v1 [physics.gen-ph] 1 Apr 2000A BRIEF NOTE ON FLUCTUATIONS AND INTERACTIONS B.G. Sidharth∗ B.M. Birla Science Centre, Hyderabad 500 063 (India) Abstract In this brief note we re-emphasize the fact that an underpinn ing of fluctuations characterizes the fundamental interaction s in the light of El Naschie’s recent work. In a recent paper El Naschie has introduced the concept of a flu ction [1], a result of geometric fluctuation which could lead towards a u nification of fundamental forces. It is pointed out in this brief note that recent work by the author does indeed emphasize the underpinning of fluctua tions for fun- damental interactions. In this recent work[2, 3, 4, 5, 6, 7], it was pointed out firstly that the fluctu- ation of the electromagnetic field (or the Zero Point Field) l eads to[8], ∆B∼√ ¯hc/L2, (1) where Lis the spatial extent. It was pointed out that if L∼Compton wavelength of a typical elementary particle then from (1) we recover the mass and energy of this particle. In other words at the Compto n wavelength the elementary particle ”condenses” out of the background Z ero Point Field. Similarly a fluctuation in the metric leads to (Cf.refs.[1, 2 ]), ∆Γ∼∆g L∼lP/L2(2) 0∗E-mail:birlasc@hd1.vsnl.net.in 1where lP∼10−33cms∼Planck scale. Unlike in equation (1), if Lin (2) is taken to be ∼lPthen from (2) we get the gravitational interaction. That fluctuations tie up equations (1) and (2) can be seen expl icitly as fol- lows. As is known, given N∼1080elementary particles in the universe, the fluctuation in the particle number is ∼√ Nwhich leads to a fluctuational electromagnetic energy which in the above scheme is the ener gy of the typical elementary particle, so that we have (Cf. also[9]) e2√ N R=mc2(3) Using in (3) the fact that[3, 10], R=GNm c2 we get the well known relation e2∼Gm2·√ N=Gm2·1040(4) Equation (4) is usually interpreted as an adhoc or empirical relation compar- ing the strengths of gravitational and electromagnetic for ces. But once the fluctuational underpinning has been taken into account, we h ave deduced (4) and can now see the connection between electromagnetic and g ravitational interactions. Indeed from (4) one can deduce that[11] at the Planck scale the electromagnetic and gravitational forces become equal, or alternatively the Planck scale of mass ∼10−5gmsis a Schwarszchild black hole. Indeed in the model referred to earlier, elementary particl es like electrons are Kerr-Newman type black holes giving at once both the elec tromagnetic and gravitational fields including the Quantum Mechanical a nomalous gyro magnetic ratio g= 2[12]. From this point, it was shown that the strong interactions fo llow at the Comp- ton wavelength scale itself, where the dimensionality is lo w (Cf.ref.[6, 4, 5]). Infact within the same scheme, it was shown that the very puzz ling character- istics of quarks namely their fractional charge, handednes s and confinement besides the order of their massses can be deduced. It is by the same argument of the fluctuation of the number of pa rticles that it was shown that the weak interactions can also be explained [7, 13]. In- deed similar arguments in a different context were put forwar d years ago by 2Hayakawa[14]. Briefly if the weak force is mediated by a particle of mass Mand Compton wavlength Lwe get from the fluctuation of particle number, this time g2√ NL2≈Mc2∼10−14, whence the weak interaction can be characterised. The conclusion is that the spirit of El Naschie’s fluction is v indicated (Cf.also ref.[15]). References [1] M.S. El Naschie, CSF, 11 (2000) 1459-1469. [2] B.G. Sidharth, Ind. J. of Pure and Applied Physics, 35, p. 456ff (1997). [3] B.G. Sidharth, Int.J.Mod.Phys.A, 13 (15), p.2599ff (199 8). [4] B.G. Sidharth, Mod. Phys. Lett. A., 14 (5), pg.387ff (1999 ). [5] B.G. Sidharth, ”Universe of Chaos and Quanta”, in Chaos, Solitons and Fractals, in press. xxx.lanl.gov.quant-ph: 9902028. [6] B.G. Sidharth, in Instantaneous Action at a Distance in M odern Physics: ”Pro and Contra” , Eds., A.E. Chubykalo et. al., Nova Science Publish- ing, New York, 1999. [7] B.G. Sidharth, ”From the Neutrino to the Edge of the Unive rse”, to appear in Chaos Solitons and Fractals. [8] C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitation ”, W.H. Free- man, San Francisco (1973). [9] S. Hayakawa, Suppl. of PTP, 1965, pp532-541. [10] B.G. Sidharth, Int.J.Th.Phys., 37 (4) p.1307ff (1998). [11] B.G. Sidharth, ”The Emergence of the Planck Scale”, to a ppear in Chaos Solitons and Fractals”. 3[12] B.G. Sidharth, Gravitation & Cosmology, 4 (2) (14), 199 8, 158ff. [13] B.G. Sidharth, ”Quantum Mechanical Black Holes: Issue s and Ramifi- cations”, Proceedings of ”Frontiers of Fundamental Physic s”, in Press. [14] S. Hayakawa, PTP, (Letters to the Editor), 1965, pp.538 -539. [15] M.S. El Naschie, ”Towards the Unification of Fundamenta l Interac- tions....”, to appear in Chaos Solitons and Fractals. 4

arXiv:physics/0004003v1 [physics.gen-ph] 1 Apr 2000THE SCALED UNIVERSE II B.G. Sidharth∗ B.M. Birla Science Centre, Hyderabad 500 063 (India) Abstract In an earlier paper we had pointed out that Quantum Mechanica l type effects are seen at different scales in the macro universe also. In this paper we obtain a rationale for this, which lies in the pi cture of bound material systems, spanning a Compton wavelength type extent, separated by much larger and relatively much less dense dist ances. 1 Introduction In a previous communication[1] it was shown that the mysteri ous quantum prescription of microphysics has analogues at the much larg er scale of stars, galaxies and superclusters. The common feature in all these cases is a Brow- nian type fractality. We now examine this circumstance in gr eater detail to obtain a rationale for this ”Scaled Quantum Mechanical” beh aviour. At the outset it must be observed that in a sense, the universe is not a contin- uum. Thus within an atom, the nucleus occupies a very tiny fra ction of the volume, roughly ∼10−15. Then there is a wide gap till we reach the orbiting electrons. Similarly there are intermolecular distances, interplanetary, inter- stellar, intergalactic.... distances which provide relat ively huge gaps. This is not in the spirit of a uniform continuum. We will now argue that this is because, for example the nucleo ns are bound together, so also the electrons and the nucleons are bound to gether, the atoms in the molecules are bound together... and so on with su bsequent gaps, which leads to some interesting scale dependent conse quences, all this in the context of a Brownian underpinning. 0∗E-mail:birlasc@hd1.vsnl.net.in 12 A Rationale for Scaled Effects The starting point in[1] was the fact that we have the followi ng Random Walk type Relations: R≈l1/radicalBig N1 (1) R≈l2/radicalBig N2 (2) l2≈l3/radicalBig N3 (3) R∼l√ N (4) where N1∼106is the number of superclusters in the universe, l1∼1025cms is a typical supercluster size N2∼1011is the number of galaxies in the uni- verse and l2∼1023cmsis the typical size of a galaxy, l3∼1 light year is a typical distance between stars and N3∼1011is the number of stars in a galaxy, Rbeing the radius of the universe ∼1028cms, N ∼1080is the num- ber of elementary particles, typically pions in the univers e and lis the pion Compton wavelength. On this basis it was argued that we could introduce a ”Scaled” Planck Con- stant given by h1∼1093(5) for super clusters; h2∼1074(6) for galaxies and h3∼1054(7) for stars. The relation (4) was observed nearly a century ago by Weyl and Eddington. It was shown (Cf.ref.[2]) that far from being empirical this relation can be deduced on the basis of the fluctuational creation of particl es from a back- ground Zero Point Field or Quantum Vacuum, in a scheme which l eads to a cosmology consistent also with Dirac’s large number coinci dences[3]. From this point of view the Random Walk character of equation (4) i s not acci- dental, and this reasoning could be extended to equations (1 ),(2) and (3), in the light of equations (5), (6) and (7). It may be observed that in all these cases we have a length, the Compton 2wavelength or its analogue which defines regions of matter se parating rela- tively empty spaces. Infact it was also argued in[1] that these scaled ”Compton wa velengths” and scaled ”Planck Constants” arise due to the well known equati on of gravita- tional orbits, GM L∼v2(8) On the other hand equation (8) can be viewed as resulting from the Virial Theorem[4], where the velocity is replaced by the velocity d ispersion. This velocity vwould be different at different scales. For example for a black hole it would be the velocity of light, which would then give t he Schwarzchild radius from (8). For galaxies it is ∼107cmsper second[5]. It is this circum- stance that produces the above scales leading to fractality . We could go one step further, because we expect that the same e ffect would apply to solar type systems: The planets and other objects ar e bound quite close to the sun compared to the interstellar distances. Inf act we can verify that this is so for Kuiper Belt objects which have been studie d in the recent past[6]. In this case a typical size is ∼5km, the distances are ∼1015cm, masses are ∼1019gmstheir number is ∼1010while an application of equa- tion (8) shows that the velocities are ∼105cmsper second which is indeed so. It can now be shown quite easily that this defines a scaled Plan ck Constant h4∼1034 In other words equation (8) characterizes ”Black Holes” wit h different max- imal velocities. Incidentally from (8) we could easily deduce that the angula r momentum J is given by J∝M2(9) It is quite remarkable that the equation (9) also applies to e lementary parti- cles and Regge trajectories[7]. Indeed this is no coincidence as can be seen as follows: It was shown that a pion can be considered to be a bound state of an electron and a positron within the context of Quantum Mechanical Black Holes[8]. In this case we have instead of equation (8) c2∼e2 mer(10) 3where meis the electron mass. Whence we have h≈mπcr∼e2×102 c∼10−27(11) It is remarkable that (11) gives us the value of Planck’s cons tant, given the pion and electron masses, and at the same time we can deduce an equation like (9) at the micro level also. Equation (11) is on the same f ooting as equations (5), (6) and (7). In this case it is the electromagn etic interaction which replaces the gravitational interaction in the equati on (8). Thus the rationale for the fractal structure seen above is in bound systems separated by relatively large and relatively empty spaces. References [1] Sidharth, B.G., ”The Scaled Universe”, to appear in Chao s Solitons and Fractals. [2] Sidharth, B.G., Int.J.Mod.Phys.A, (1998) 13 (15), p.25 99ff. [3] Sidharth, B.G., Int.J.Th.Phys., (1998), 37 (4), p.1307 ff. [4] Nottale, L., ”Fractal Space-Time and Microphysics: Tow ards a Theory of Scale Relativity”, (1993), World Scientific, Singapore, p.312. [5] Narlikar, J.V., (1993) ”Introduction to Cosmology”, Fo undation Books, New Delhi. [6] Jewitt, D., Physics World, July 1999, p.37ff. [7] Tassie, L.J., (1973) ”The Physics of Elementary Particl es”, Longman Group Limited, London. [8] Sidharth, B.G., Ind. J. of Pure and Applied Physics, (199 7) 35, p.456ff. 4

arXiv:physics/0004004v1 [physics.gen-ph] 1 Apr 2000SCALE DEPENDENT DIMENSIONALITY B.G. Sidharth∗ B.M. Birla Science Centre, Hyderabad 500 063 (India) Abstract We argue that dimensionality is not absolute, but that it dep ends on the scale of resolution, from the Planck to the macro scale . 1 Introduction Is dimensionality dependent on the scale of resolution, or i s it independent of this scale. This question becomes relevant in the light of some recent work (for example Cf.[1]). It has ofcourse been pointed out that t he spin half character of a collection of Fermions leads to the usual thre e dimensionality of our space[2, 3], while the spin half itself is associated w ith the Comp- ton wavelength as discussed in recent papers (Cf. for exampl e[4]). Further, it was argued that as we approach the Compton scale, we encoun ter lower dimensionality[5, 6]. In this paper we point out that indeed the dimension- ality is scale dependent. 2 Scale Dependence We first notice that at Planck scale lP, we have N3/4lP∼R (1) where N∼1080is the number of elementary particles and R∼1028cmthe radius of the universe. This is not an empirical relation but rather can be 0∗E-mail:birlasc@hd1.vsnl.net.in 1deduced on the basis of a fluctuational creation of particles scheme recently discussed (Cf. for example[7, 8]). In this scheme,√ Nparticles are fluc- tuationally created and this happens in the Compton time τof a typical elemental particle, a pion. Further this corresponds to a fluctuational creation of N1/4Planck particles as recently argued[9], in the Planck time τP. Indeed, we have ˙N∼N1/4/τP∼√ N/τ (2) Equation (2) leads to (1). (1) shows that at the Planck length, the fluctuational dimens ionality is 4 /3. Interestingly this is the dimension of a Koch curve and a coas tline [10]. With this dimensionality we should have M∝R4/3, which indeed is true[11]. At the Compton scale of resolution, we have[7], as indeed can be deduced from (2) the well known Eddington formula, R∼√ Nl (3) (3) shows the two dimensional character at the Compton lengt h. Indeed as noted in the introduction three dimensionality is at scal es much greater than the Compton wavelength - as we approach the Compton wave length we encounter two dimensionality as can be seen from (3) - indeed this was the key to explain puzzling characteristics of quarks includin g their fractional charge and handedness[6]. Finally at scales L∼10cm, we have N1/3L∼R (4) (4) shows up the usual three dimensionality. Interestingly, if we take the typical elementary particle t he pion, and con- sider it successively as a 4 /3 dimensional object at the Planck scale, a two dimensional object at the Compton scale and three dimension al at our macro scale, and consider successive densities ρP∼m/(lP)4/3, ρπ∼m/l2and ρ∼m/L3, we have, M∼ρPR4/3∼ρπR2∼ρR3, as required. 2References [1] M.S. El Naschie, ”Towards Unification of Fundamental Int eractions...” to appear in Chaos Solitons and Fractals. [2] C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitation ”, W.H. Free- man, San Francisco (1973). [3] R. Penrose, ”Angular Momentum: An approach to combinati onal space- time” in, ”Quantum Theory and Beyond”, Ed., Bastin, T., Camb ridge University press, Cambridge (1971). [4] B.G. Sidharth, Ind. J. of Pure and Applied Physics, 35, p. 456ff (1997). [5] B.G. Sidharth, ”Universe of Chaos and Quanta”, in Chaos, Solitons and Fractals, in press. xxx.lanl.gov.quant-ph: 9902028. [6] B.G. Sidharth, Mod. Phys. Lett. A., 14 (5), pg.387ff (1999 ). [7] B.G. Sidharth, Int.J.Mod.Phys.A, 13 (15), p.2599ff (199 8). [8] B.G. Sidharth, Int.J.Th.Phys., 37 (4) p.1307ff (1998). [9] B.G. Sidharth, ”The Emergence of the Planck Scale”, to ap pear in Chaos Solitons and Fractals”. [10] B.B. Mandelbrot, ”The Fractal Geometry of Nature”, W.H . Freeman, New York, pg.2,18,27 (1982). [11] Sidharth, B.G., and Popova, A.D., (1996), Differential Equations and Dynamical Systems, 4 (3/4), 431-440. 3

arXiv:physics/0004005v1 [physics.class-ph] 3 Apr 2000Lorentz-Covariant Hamiltonian Formalism. A. B´ erarda, H. Mohrbachaand P. Gosselinb. a)L.P.L.I. Institut de Physique, 1 blvd.F.Arago, 57070, Metz, France. b)Universit´ e Grenoble I , Institut Fourier, UMR 5582 CNRS, UFR de Math´ ematique, BP 74, 38402 Saint Martin d’H` eres, Cedex, France. Abstract The dynamic of a classical system can be expressed by means of Poisson brackets. In this paper we generalize the relation between t he usual non covariant Hamiltonian and the Poisson brackets to a covaria nt Hamiltonian and new brackets in the frame of the Minkowski space. These br ackets can be related to those used by Feynman in his derivation of Maxwe ll’s equations. The case of curved space is also considered with the introduc tion of Christoffel symbols, covariant derivatives, and curvature tensors. I. INTRODUCTION A remarkable formulation of classical dynamics is provided by Hamiltonian mechanics. This is an old subject. However, new discoveries are still be en made; we quote two ex- amples among several: the Arnold duality transformations, which generalize the canonical transformations [1,2], and the extensions of the Poisson br ackets to differential forms and multi-vector fields by A.Cabras and M.Vinogradov [3]. In thi s context the transition from classical to relativistic mechanics raises the question of Hamiltonian covariance, the physical significance of which is discussed by Goldstein [4]. In the fir st part of this paper we briefly 1recall the Poisson brackets approach and the covariant Hami ltonian formalism. Then we in- troduce new brackets to study the dynamics associated to thi s covariant Hamiltonian, which define an algebraic structure between position and velocity , and does not have an explicit formulation. We examine the close link between these bracke ts and those used by Feynman for his derivation of the Maxwell equations [5–8]. A very int eresting way to arrive at the same sort of result was found by Souriau in the frame of his sym plectic classical mechanics [9]. In the final part of this work we consider the dynamics in c urved space, using Christoffel symbols, covariant derivatives, and curvature tensors exp ressed in terms of these brackets. II. BRIEF REVIEW OF ANALYTIC MECHANICS A. Poisson brackets The dynamics of a classical particle in a 3-dimensional flat s pace with vector position qi and vector momentum pi(i= 1,2,3) is defined by the Hamilton equations:   .qi=dqi dt=∂H ∂pi .pi=dpi dt=−∂H ∂qi(1) where the Hamiltonian H(qi, pi) is a form on the phase space ( the cotangent fiber space). They can be also expressed in a symmetric manner by means of Po isson brackets:   .qi={qi, H} .pi={pi, H}(2) These brackets are naturally defined as skew symmetric bilin ear maps on the space of func- tions on the phase space in the following form: {f, g}=∂f ∂qi∂g ∂pi−∂g ∂qi∂f ∂pi(3) 2B. Covariant Hamiltonian Except in the electromagnetic situation, the Hamiltonian i s not the total energy when it is time-dependent, and its generalization to relativist ic problems with the M4Minkowski space is not trivial because it is not Lorentz covariant. In the electromagnetic case the answer to this question is gi ven by the introduction of the following covariant expression [4] : H=uµpµ−L=uµ(muµ+q cAµ) (4) where Lis the usual invariant electromagnetic Lagrangian : L=1 2m uµuµ+q cuµAµ (5) anduµthe quadri-velocity defined by means of the proper time tp, here used as an invariant parameter: uµ=dxµ dtp(6) Finally we have the covariant Hamiltonian H=1 2m uµuµ (7) with the corresponding eight Hamilton equations:   ∂H ∂pµ=dxµ dtp=uµ ∂H ∂xµ=−dpµ dtp(8) It is interesting to recall that this structure is only possi ble in the situation where the potential can be put in a covariant manner as in the electroma gnetism theory. III. LORENTZ COVARIANT HAMILTONIAN AND BRACKETS FORMALISM Now we want to generalize the relation between the usual non c ovariant relativistic Hamiltonian and the Poisson brackets to a covariant Hamilto nianHand new formal brackets 3introduced in the frame of the Minkowski space. It is importa nt to remark that, in a different manner, P.Bracken also studied the relation between this Fe ynman problem and the Poisson brackets [10]. In this context a ”dynamic evolution law” is given by means of a one real parameter group of diffeomorphic transformations : g(IR×M4)−→M4:g(τ, x) =gτx=x(τ) The ”velocity vector” associated to the particle is natural ly introduced by: .xµ=d dτgτxµ(9) where the ”time” τis not identified with the proper time as we shall see later. Th e derivative with respect to this parameter of an arbitrary function defin ed on the tangent bundle space can be written, by means of the covariant Hamiltonian, as: df(x,.x, τ) dτ= [H, f(x,.x, τ)] +∂f(x,.x, τ) ∂τ(10) where for Hwe take the following definition: H=1 2mdxµ dτdxµ dτ=1 2m.xµ.xµ (11) Equation (10) giving the dynamic of the system, is the definit ion of our new brackets struc- ture, and is the fundamental equation of the paper. We require for these new brackets the usual first Leibnitz law : [A, BC ] = [A, B]C+ [A, C]B (12) and the skew symmetry: [A, B] =−[B, A] (13) where the quantities A, B andCdepend of xµand.xµ. In the case of the vector position xµ(τ) we have from (10): 4.xµ= [H, xµ] =m/bracketleftBig.xν, xµ/bracketrightBig.xν (14) and we easily deduce that: m/bracketleftBig.xν, xµ/bracketrightBig =gµν(15) where gµνis the metric tensor of the Minkowski space. As in the Feynman approach the time parameter is not the prope r time. To see this we borrow Tanimura’s argument [6]. Consider the relation gµνdxµ dtpdxν dtp= 1 (16) which implies /bracketleftBigg .xλ, gµνdxµ dtpdxν dtp/bracketrightBigg = 0. (17) and is in contradiction with: /bracketleftBigg .xλ, gµνdxµ dτdxν dτ/bracketrightBigg =−2 m.xλ. (18) But differently from Feynman, the fact that gµνis the metric is a consequence of the for- malism and is not imposed by hand. In addition, contrary to Fe ynman, we do not need to impose the Leibnitz condition: d dτ[A, B] =/bracketleftBiggdA dτ, B/bracketrightBigg +/bracketleftBigg A,dB dτ/bracketrightBigg (19) (AandBbeing position- and velocity-dependent functions) becaus e the time derivative is given by the fundamental equation (10). We impose the usual locality property: [xµ, xν] = 0 (20) which directly gives for an expandable function of the posit ion or the velocity the following useful relations: 5  [xµ, f(.x)] =−1 m∂f(.x) ∂.xµ /bracketleftBig.xµ, f(x)/bracketrightBig =1 m∂f(x) ∂xµ(21) which reduce in the particular cases of the position and velo city to:   /bracketleftBig xµ,.xν/bracketrightBig =−1 mgµρ∂.xν ∂.xρ=−1 m∂.xν ∂.xµ=−gµν m /bracketleftBig.xµ, xν/bracketrightBig =1 mgµρ∂xν ∂xρ=1 m∂xν ∂xµ=gµν m(22) To compute the bracket between two components of the velocit y we require in addition the Jacobi identity: /bracketleftBig/bracketleftBig.xµ,.xν/bracketrightBig , xρ/bracketrightBig +/bracketleftBig/bracketleftBig xρ,.xµ/bracketrightBig ,.xν/bracketrightBig +/bracketleftBig/bracketleftBig.xν, xρ/bracketrightBig.xµ/bracketrightBig = 0 (23) which by using (15) gives: /bracketleftBig.xµ,.xν/bracketrightBig =−Nµν(x) m(24) where Nµν(x) is a skew symmetric tensor. The second derivative of the position vector is: ..xµ=d.xµ dτ=/bracketleftBig H,.xµ/bracketrightBig =Nµν.xν (25) and we write: Fµν=m qNµν(26) in order to recover the Lorentz equation of motion. remark 1. We can easily calculate: [H, H] =1 4m2/bracketleftBig.xµ.xµ,.xν.xν/bracketrightBig =−q.xµ.xνFµν= 0 (27) and then deduced: dH dτ=∂H ∂τ(28) 6which is the expected result. In the same manner, we get for the 4-orbital momentum: dLµν dτ=md dτ/parenleftBig xµ.xν−.xµxν/parenrightBig =m/parenleftBig xµ..xν−..xµxν/parenrightBig =q(xµFνρ.xρ−xνFµρ.xρ) = [H, Lµν] (29) as expected. IV. MAXWELL EQUATIONS Our formal construction will give the Maxwell equations bec ause it leads to the fun- damental result (15) which is the starting point of Feynman’ s proof of the first group of Maxwell equations. The difference is that our main property i s equation (10) and not the Leibnitz rule (19). So our derivation will be obtained differ ently and will give in addition the two groups of Maxwell equations. •To be general, we choose like in [8], the following definition for the gauge curvature: /bracketleftBig. xµ,. xν/bracketrightBig =−1 m2(qFµν+g∗Fµν) (30) where g will be interpreted as the magnetic charge of the Dira c monopole, the *-operation being the Hodge duality. •A simple derivative gives: d(qFµν(x) +g∗Fµν(x)) dτ=q∂ρFµν(x).xρ+g∂ρ∗Fµν(x).xρ (31) and by means of the fundamental relation (10) we obtain: d(qFµν(x) +g∗Fµν(x)) dτ= [H, qFµν(x) +g∗Fµν(x)] =−m3 q/bracketleftBig.xρ,/bracketleftBig.xµ,.xν/bracketrightBig/bracketrightBig.xρ (32) Now using the Jacobi identity we rewrite this expression as: 7d(qFµν(x) +g∗Fµν(x)) dτ=m3 q/parenleftBig/bracketleftBig.xµ,/bracketleftBig.xν,.xρ/bracketrightBig/bracketrightBig.xρ+/bracketleftBig.xν,/bracketleftBig.xρ,.xµ/bracketrightBig/bracketrightBig.xρ/parenrightBig.xρ =−q(∂µFνρ+∂νFρµx).xρ−g(∂µ∗Fνρ+∂ν∗Fρµx).xρ (33) By comparing equations (31) and (33) we deduce the following field equation: q(∂µFνρ+∂νFρµ+∂ρFµν) +g(∂µ∗Fνρ+∂ν∗Fρµ+∂ρ∗Fµν) = 0 (34) that is:   ∂µFνρ+∂νFρµ+∂ρFµν=gNµνρ ∂µ∗Fνρ+∂ν∗Fρµ+∂ρ∗Fµν=−qNµνρ(35) where Nµνρis a tensor to be interpreted. Using the differential forms language defined on the Minkowsk i space ( M4) we write the preceding equations in a compact form:   dF=gN d∗F=−qN(36) where Fand∗F∈ ∧2(M4) and N∈ ∧3(M4) . If we put:   gN=−∗k qN=∗j(37) where jandk∈ ∧1(M4), we deduce:   δF=j dF=−∗k(38) 8δis the usual codifferential δ:∧k(M4)→ ∧k−1(M4) defined here as: δ= (−)k(4−k+1)+1(∗d∗) Interpreting the 1-forms jandkas the electric and magnetic four dimensional current densities, we obtained the two groups of Maxwell equations i n the presence of a magnetic monopole. The situation without monopole is evidently obta ined by putting the 1-form k equal to zero. We easily see by means of the Poincar´ e theorem that: δ2F=δj= 0 (39) which is nothing else that the current density continuity eq uation: ∂µjµ=m[.xµ, jµ] = 0, (40) From the skew property of the brackets, we can choose: jµ=ρ.xµ, (41) ρis the charge density whose dynamic evolution is given by: dρ dτ= [H, ρ] =m/bracketleftBig.xµ, ρ/bracketrightBig.xµ= (∂µρ).xµ=∂µjµ= 0 (42) We see that Hautomatically takes into account the gauge curvature. It pl ays the role of a Hamiltonian not with the usual Poisson brackets, but with ne w four-dimensional brackets which can be related to for example, those used by Feynman in h is derivation of Maxwell equations as published by Dyson [5]. 9V. APPLICATION TO A CURVED SPACE In this section we extend the previous analysis to the case of a general space time metric gµν(x). In this case we define the covariant Hamiltonian from the usua l fundamental quadratic formds2in the following manner: H=1 2m/parenleftBiggds dτ/parenrightBigg2 =1 2mgµν(x).xµ.xν In the same manner asin section 3, we can prove the relation be tween the metric tensor and the brackets structure: m/bracketleftBig.xν, xµ/bracketrightBig =gµν(x) The law of motion is: ..xµ=/bracketleftBig H,.xµ/bracketrightBig =1 2m/bracketleftBig gνρ,.xµ/bracketrightBig.xν.xρ+m/bracketleftBig.xν,.xµ/bracketrightBig.xν =−1 2∂µgνρ.xν.xρ−Nνµ.xν (43) where we define Nµν(x,.x) as: /bracketleftBig.xµ,.xν/bracketrightBig =−Nµν(x,.x) m(44) Note that this tensor is now velocity-dependent, in contras t to the Minkowski case. By means of equation (23) and (43), we deduce the relation: ∂Nµν ∂.xρ=∂νgρµ−∂µgρν(45) then: Nµν(x,.x) =−(∂µgρν−∂νgρµ).xρ+nµν(x) (46) where the tensor nµν(x) is only position dependent. If we introduce this equation i n (43), we find: 10..xµ=−1 2∂µgνρ.xν.xρ−(∂µgρν−∂νgρµ).xν.xρ+nµν(x)xν =1 2∂µgνρ.xν.xρ−(∂µgρν−1 2∂νgρµ−1 2∂ρgνµ).xν.xρ+nµν(x).xν =−Γνρµ.xν.xρ+nµν(x).xν (47) where we have defined the Christoffel symbols by: Γνρµ=1 2/parenleftBig/bracketleftBig.xρ,/bracketleftBig.xν, xµ/bracketrightBig/bracketrightBig −/bracketleftBig.xν,/bracketleftBig.xρ, xµ/bracketrightBig/bracketrightBig −/bracketleftBig.xµ,/bracketleftBig.xρ, xν/bracketrightBig/bracketrightBig/parenrightBig (48) =1 2(∂ρgνµ−∂νgρµ−∂µgρν) (49) Comparing with the usual law of motion of a particle in an elec tromagnetic field, as in the situation of a flat space, we can put: Fµν(x) =m qnµν(x) (50) and get the equation of motion of a particle in a curved space: md.xµ dτ=−mΓµ νρ.xν.xρ−q Fνµ.xυ (51) so that: /bracketleftBig H,.xµ/bracketrightBig =−Γµ νρ.xν.xρ−q mFνµ.xυ (52) Note the difference between the two tensor NµνandFµνwhose definitions are:   /bracketleftBig.xµ,.xν/bracketrightBig =−Nµν m=−gµρgνσNρσ m [.xµ,.xν] =−Fµν m=−gµρgνσFρσ m(53) and more generally:   /bracketleftBig.xµ, f(.x, τ)/bracketrightBig =Nµν m∂f(.x,τ) ∂.xν [.xµ, f(.x, τ)] =Fµν m∂f(.x,τ) ∂.xν(54) As in the case of flat Minkowski space, it is not difficult to reco ver the two groups of Maxwell equations with or without monopoles. In this last ca se we must take the following definition for the dual field: 11∗Fµν=1 2√−gεµνρσFρσ. (55) Now we will show that the covariant derivative and the curvat ure tensor can be naturally introduced with our formalism. A. Covariant derivative As in the flat-space case, the equation of motion can be rewrit ten in the two following manners: md.xµ dτ=−mΓµ νρ.xν.xρ−q Fνµ.xυ (56) and: md.xµ dτ=m∂.xµ ∂xν.xν(57) we then put: ∂.xµ ∂xν=−Γµ νρ.xρ+q mFµ ν=/bracketleftBig H′,.xµ/bracketrightBig (58) From equation (58), a covariant derivative can be defined by m eans of the brackets. For an arbitrary vector we put: m[.xν, Vµ(x)] =∂Vµ(x) ∂xν(59) We then define as the usual covariant derivative: [Dν, Vµ] =∂Vµ ∂xν+ Γµ νρVρ(60) and for an arbitrary mixed tensor: [Dν, Tµ σ] =∂Tµσ ∂xν+ Γµ νρTρ σ−Γρ νσTµ ρ (61) For the particular case of the velocity we get: /bracketleftBig Dν,.xµ/bracketrightBig =∂.xµ ∂xν+ Γµ νρ.xρ=q mFµ ν (62) and in addition we recover the standard result: [Dν, gµν] = 0 12B. Curvature tensor From this definition of the covariant derivative we can natur ally express a curvature tensor by means of the brackets. Let’s compute the following expressions: [Dµ,[Dν, Vρ]] =/bracketleftBig.xµ, ∂νVρ+ Γνρ σVσ/bracketrightBig + Γµν α(∂αVρ+ Γαρ σVσ) + Γµρ α(∂νVα+ Γαν σVσ) =∂µ∂νVρ+∂µ(Γνρ σ)Vσ+ Γνρ σ(∂µVσ) + Γµν α(∂αVρ+ Γαρ σVσ) +Γµρ α(∂νVα+ Γαν σVσ) (63) and therefore: [Dµ,[Dν, Vρ]]−[Dν,[Dµ, Vρ]] =∂µ(Γνρ σ)Vσ−∂ν(Γµρ σ)Vσ+ Γµρ αΓαν σVσ−Γνρ αΓαµ σVσ +Γνµ α(∂αVρ+ Γαρ σVσ)−Γµν α(∂αVρ+ Γαρ σVσ) =Rµνρ σVσ+ Ωµν αDαVρ(64) where we have introduced the torsion tensor Ωµνα= Γνµ α−Γµν α= 0,and the curvature tensor Rµνρσ. Due to symmetric property of the Christoffel symbols, the cu rvature tensor is reduced to: Rµνρ σVσ=∂µ(Γνρ σ)Vσ−∂ν(Γµρ σ)Vσ+ Γµρ αΓαν σVσ−Γνρ αΓαµ σVσ(65) The Jacobi identity gives: [Dµ,[Dν, Vρ]] + [Dν,[Vρ, Dµ]] + [Vρ,[Dµ, Dν]] = 0 (66) that is: [Dµ,[Dν, Vρ]]−[Dν,[Dµ, Vρ]] = [[Dµ, Dν], Vρ] = 0 (67) and finally: [[Dµ, Dν], Vρ] =Rµνρ σVσ(68) remark 2. We can also define the Ricci and the electromagnetic energy-i mpulsion tensors, but we were unable to deduce the Einstein equation from this f ormalism. Naturally, we can write this equation with our brackets as a constraint equati on. 13remark 3. We can generalize the covariant derivative in including the skew-symmetric tensor Fµνin the definition. For this we take into account the gauge curv ature for the determination of the new covariant derivative. For a vector ial function of the velocity we write: [∆ν, fµ(.x)] =∂fµ(.x) ∂xν+ Γµ νρfρ(.x)−q mFρν∂fµ(.x) ∂.xρ(69) and then for the velocity: /bracketleftBig ∆ν,.xµ/bracketrightBig =∂.xµ ∂xν+ Γµ νρ.xρ−q mFµ ν= 0 (70) The covariant derivatives, are then simultaneously covari ant under both local internal and external gauges. If we want to keep a synthetic form for th e formulas using the curvature and torsion tensors, we must suppose for an arbitrary vector the relation: [∆ν, Vµ] =∂Vµ ∂xν+ Γµ νρVρ−AνVµ(71) where the vector Aνis defined by the following equation: Fµν=m/parenleftBig/bracketleftBig.xµ, Aν/bracketrightBig −/bracketleftBig.xν, A/bracketrightBigµ/parenrightBig (72) therefore we have: [∆µ,[∆ν, Vρ]]−[∆ν,[∆µ, Vρ]] = [[∆µ,∆ν], Vρ] =Rµνρ σVσ+ Ωµν α∆αVρ+FµνVρ(73) We define a new ”generalized” curvature tensor which matches the local electromagnetism internal symmetry with the local external symmetry: Rµνρ σVσ=Rµνρ σVσ+FµνVρ(74) then: [[∆µ,∆ν], Vρ] =Rµνρ σVσ(75) 14VI. CONCLUSION The goal of this work was to study the dynamic associated with the Lorentz-covariant Hamiltonian well known in analytic mechanic. For this, we in troduced a four dimensional bracket structure which gives an algebraic structure betwe en the position and velocity and generalizes the Poisson brackets. This leads us to introduc e a new time parameter which is not the proper time, but is the conjugate coordinate of thi s covariant Hamiltonian. This formal construction allows to recover the two groups of Maxw ell equations in flat space. This approach is close to the one used by Feynman in his own der ivation of the first group of Maxwell equations. The principal interest of this method, besides the phase spa ce formalism, is in the study of theories with gauges symmetries because it avoids the int roduction of the non-gauge invariant momentum. Our formalism can be directly extrapolated to the curved spa ce, where the principal notions are introduced in a natural manner. A five-dimension al structure can also be studied by considering the τparameter as a fifth coordinate. In such a case equations take a simpler form, particularly the group of Maxwell equations, but the m eaning of this new coordinate is still difficult to interpret, and could be perhaps understo od in the context of Kaluza-Klein compactification. Just after finishing this work we received a paper referring t o the covariant Hamiltonian in the context of Feynman’s proof of the Maxwell equations [1 1] . Acknowledgment: We would like to thank Y.Grandati for helpful discussions. 15REFERENCES [1] V.Arnold, ”Huyghens and Barrow, Newton and Hooke ”Birkh auser-Verlag, Bassel (1990). [2] D.Stump , Jour.Math.Phys. 39, 7 (1998) 3661-3669 . [3] A.Cabras and A.Vinogradov, Jour.Geom.Phys. 9 (1992) 75 -100. [4] H.Goldstein, ”Classical Mechanics”, Addison-Wesley, Reading MA (1950). [5] F.Dyson, Am.J.Phys.583 (1990) 209. [6] S.Tanimura, Ann.Phys.220 (1992) 229. [7] A.B´ erard, Y.Grandati and H.Mohrbach, Phys.Lett . A 254 (1999) 133-136. [8] A.B´ erard, Y.Grandati and H.Mohrbach, Jour.Math.Phys . 40 (1999) 3732-3737. [9] J.M.Souriau, ”Structure des Syst` emes Dynamiques.” Du nod, Paris (1970). [10] P.Bracken, Int.Jour.Theor.Phys. 35 (1996) 2125-2138 . [11] M.Montesinos and A.P´ erez-Lorenzana, Int.Jour.Theo r.Phys. 38 (1999) 901-910. 16

arXiv:physics/0004006v1 [physics.ins-det] 3 Apr 2000Measurement of Carbon Disulfide Anion Diffusion in a TPC Tohru Ohnukia,1, Daniel P. Snowden-IfftaC. J. Martoffb aDepartment of Physics, Occidental College, 1600 Campus Roa d, Los Angeles 90041-3314, USA bDepartment of Physics, Temple University, 1900 N. 13-th St. , Philadelphia, PA 19122-6082, USA Abstract A Negative Ion Time Projection Chamber was used to measure th e field dependence of lateral and longitudinal diffusion for CS 2anions drifting in mixtures of CS 2and Ar at 40 Torr. Ion drift velocities and limits on the capture d istance for electrons as a function of field and gas mixture are also reported. Key words: carbon disulfide, CS 2, diffusion, mobility, TPC PACS Classification: 29.40.Cs, 29.40.Gx, 51.50.+v 1 Introduction The Directional Recoil Identification From Tracks (DRIFT) d etector has recently been proposed [1] to search for dark matter. This detector is unique in that it drifts negative ions, instead of electrons, in a time projection ch amber (TPC). A detailed de- scription of the operation, motivation, and other uses of a n egative ion TPC (NITPC) can be found in [2]. Briefly, an electronegative component in the gas captures ionized electrons forming negative ions. These anions then drift to the anode wires where, provided the electronegative component allows it, the anio ns are ionized and normal electron avalanche occurs. The motivation for such an arran gement is that it allows transport of charge to the anode with the minimum possible di ffusion. Using a pro- portional chamber, Crane showed that CS 2has the desired electronegative properties [3]. Using a single wire drift chamber of unusual design, Mar toff et al. [2] measured the drift velocity and lateral diffusion of CS 2ions in two different gas mixtures. In 1Email: ohnukit@oxy.edu Preprint submitted to Elsevier Preprint 2 February 2008this paper we used a NITPC to measure the drift velocity, late ral diffusion and lon- gitudinal diffusion in a variety of CS 2-Ar gas mixtures. These measurements allowed limits to be set on the electron capture distance in these mix tures. 2 Theory The diffusion of charged particles being drifted through a ga s has been parameterized by an expression of the form [4]: σ2=4εkL 3eE(1) where Lis the drift distance, Eis the drift field and εkis the characteristic (average) energy of the electron or ion. For electrons, εkvaries from thermal ( ∼kBT) at low drift fields to several eV at higher E/p. The nonlinear variation of εkfor electrons arises from the mass mismatch between electrons and gas atom s which prevents elas- tic electron–atom collisions from efficiently thermalizing the electron energy gained between collisions [5]. Ions, on the other hand, have masses comparable to the gas atoms and hence would be expected to remain well thermalized even if the energy gained between collisions became comparable with ∼kBT. Thus εkfor ions should remain constant and one should be able to reduce the diffusion of ions (as 1/√ E) to much lower values than for electrons. The NITPC concept re lies critically on this hypothesis. The purpose of the present work was to test it in d etail. 3 Experimental Method 3.1 Apparatus The NITPC used in this experiment consisted of a drift region attached to a multi- wire proportional chamber (MWPC) (Figure 1). The drift regi on was composed of a solid copper cathode and field cage. The field cage consisted o f 4 loops of 500 µm wire supported by acrylic posts and connected by a ladder of preci sion 10 MΩ resistors. The lateral dimensions of the field cage were 15 cm by 20 cm. The MWPC was made of three wire planes: two grid planes sandwiching an anode pl ane. The grid and anode wires were 100 µm and 20 µm gold coated tungsten respectively. Anode wire spacing was 2 mm, grid wire spacing was also 2 mm but at 45 degrees to the anodes and the gap was 1.1 cm. The drift distance, from the solid copper cath ode to the nearest grid plane, was 10 cm. A single negative high voltage supply connected at the catho de was used to set the 2Cathode Field CageGridAnodesVacuum Chamber Wall Caliper Mounted Pinhole Xenon StrobelampGrid -HV Fig. 1. TPC in Situ (not to scale) drift and MWPC potentials via the field cage resistance and a v ariable resistance between anode and grid. During operation, the MWPC voltage w as kept as high as possible to yield the greatest gas gain and to insure 100% tra nsparency [4]. The detector was mounted with wire planes vertical in an evac uable chamber. Gas mixtures were prepared in the chamber by introducing gases o ne by one through a manifold. CS 2vapor was evolved from a liquid source by the low pressure in t he cham- ber. All experiments were done at a pressure of 40.0 Torr. The chamber pressure was measured with a capacitive manometer allowing accurate (0. 1 Torr) determination of pressure and mixture. Before each run, the chamber was evacu ated to ∼50 mTorr measured using a Convectron gauge, then backfilled with the d esired gas mixture and pressure. A gas filling would be used for several hours, durin g which time the pressure rise was about a percent. The diffusion measurements were made using photoelectrons g enerated from the solid copper cathode in the following way. Near the NITPC was a port containing a fused silica window. A UV flashlamp projected light through a movab le 200 µm aperture, through the silica window, through the grid and anode wire pl anes, and onto the cathode. The projected spot on the cathode was ∼1.3 mm by ∼2 mm measured using photographic film. To maintain the photoelectron yiel d, the copper cathode was cleaned with abrasive every three to five runs. A photodiode v iewing the flashlamp directly provided a start signal for drift time measurement s. The anode wires were connected to Amptek A-250 charge sensit ive pre-amplifiers which were housed inside the chamber to reduce noise pickup. The outputs of the Ampteks were amplified and shaped by Ortec 855 spectroscopy a mplifiers. These signals were digitized and stored on a 20 MHz digital oscillo scope. The scope used the photodiode signal as a trigger. Waveform averaging was u sed to improve signal to noise. 33.2 Measurement To measure the lateral diffusion, the maximum pulse heights o n two adjacent wires were measured as the aperture was moved laterally through 40 , 125µm steps. Pulse height maxima were measured on waveforms averaged over eith er 128 or 256 individual flashlamp shots. Space charge should not affect the results of this experiment. The number of photoelectrons produced was always less than ∼50 per flash. The lateral diffusion of the anions was always greater than 0.7 mm. Since t he gain was low, ∼ 3000, the size of the avalanche was ∼0.05 mm [4]. In those cases where two avalanches overlapped, the reduction in gain for the second avalanche w as less than 1% [4]. In addition to these theoretical arguments, the gain linearit y was tested by confirming that a ×10 reduction in flash energy resulted in an order of magnitude reduction in pulse height. Figure 2 shows a set of measurements, with aperture position on the horizontal axis and pulse height of the averaged waveforms on the vertical ax is. The data for each peak was fit to a function that allowed deconvolution of the de posited charge from a known electronic artifact. The distance between the two pea ks represented the 2 mm wire spacing and was used to normalize the lateral measureme nt. The +’s and ×’s in Figure 2 are the data for each wire, the line is the fitted funct ion. The longitudinal diffusion was determined by measuring the p ulse width in time on a single wire. To convert this measurement to distance, the d rift speed as a function of drift field was determined for each gas mixture. This was do ne by measuring the delay time between the light pulse (photoelectron generati on) and the peak of the anode signal. Though the majority of the delay was due to the t ime ions spent in the drift field, a correction was made for the delay in the MWPC. 0.3 0.4 0.50246 Position (inches)Signal (Volts)Signal vs Position 8/4/98 run 1 40Torr CS2 6.5kV app Vg=3.66kV Ed=229 V/cm 200µ hole 2.25 dist ratio wiresep 2mm 855 gain 500 digitized average 128 waveforms strobe 0.1J ~4Hz2-Gauss Fit x0=0.4232 a0=5.434 w0=0.0322 x1=0.4106 a1=0.5084 w1=0.1508 Fig. 2. Two–Gaussian Fit to Typical Data Set 4In order to minimize variations due to changing chamber cond itions, a set of runs at different drift fields would be done with a given gas mixture. A dditionally, all of the above measurements (lateral and longitudinal diffusion and drift velocity) were done concurrently at a given drift field. The delay time and pulse t ime-width were both measured at the aperture position corresponding to maximum pulse height for the wire under test. To reduce the effects of gas aging, a gas mixtu re was kept for only one set of runs, and was flushed at the end of the day. 4 Results and Discussion There are a number of contributions to the measured σ. The main ones are the finite spot size, geometry of the chamber, capture distance, and th e diffusion itself(given by Equation 1). Since the gap distance in the MWPC was so small and the field so large the diffusion there was assumed to be minimal. The vario us contributions can be assumed to be uncorrelated and hence to add in quadrature: σ2 measured =σ2 spot+σ2 geometry +σ2 capture +2kBTeffL e1 E(2) where εkin Equation 1 has been characterized in terms of an effective t emperature Teff. The NITPC hypothesis is that Teffwill remain constant and close to room temperature up to very high drift fields. For the lateral measurements, the σspotwas significant while for the longitudinal measurements it was not due to the short duration of the pulse . In the lateral mea- surements σgeometry arises from the 2 mm wire spacing and is significant. For the longitudinal measurements σgeometry arises from the different pathlengths ions can take to get to the anode wires and is also significant. None of t hese is a function of the field. Sinceσcapture is expected to increase with increasing field, a linear relat ionship between σ2 measured and 1/Esupports the assumption that σcapture is slowly varying or constant. In that case, Teffcan be deduced from the slope. The capture distance can be ded uced from the intercept if the geometry and spot size contributio ns are known or estimated, since according to Equation 2 σ2 capture =σ2 intercept −σ2 geometry −σ2 spot (3) 4.1 Lateral Diffusion The main lateral diffusion results are summarized in Figure 3 and Table 1. The plot shows the lateral diffusion squared ( σ2) versus the inverse of the drift field. On the 50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70 1 2 3 4 5 1/Reduced Field (cm/VTorr)σ2 (mm2) 100% CS2 90%CS 2−10%Ar 50%CS 2−50%Ar 25%CS 2−75%Ar 50%propane −50%Ar Fig. 3. Lateral Diffusion for Various Gas Mixtures Table 1 Lateral Diffusion Results Gas Mixture Slope Temperature Y-Intercept σcapture 100%CS 2 0.16±0.02 Vmm/Torr 360 ±40 K 0.50 ±0.05 mm2<0.4 mm 90%CS 2-10%Ar 0.13 ±0.03 Vmm/Torr 300 ±80 K 0.52 ±0.07 mm2<0.4 mm 50%CS 2-50%Ar 0.11 ±0.02 Vmm/Torr 260 ±40 K 0.56 ±0.05 mm2<0.5 mm 25%CS 2-75%Ar 0.10 ±0.02 Vmm/Torr 240 ±50 K 0.74 ±0.06 mm2<0.6 mm graph, the marks represent the data taken for different gas mi xtures at a variety of drift fields. The triangles are electrons drifted in 50% prop ane-50%argon for reference. The points very close to the origin are from experiments wher e the drift region was removed and the MWPC used alone with a copper cathode in place of one of the grid planes. The copper cathode was in approximately the same pos ition relative to the light source for both the MWPC and NITPC measurements so that the spot size was constant. The reason that several diffusion values appear fo r each drift field is that two wires were used during the measurement. Since the uncert ainty in the Gaussian fits for the points is much smaller than the marks, the separat ion between the points at identical fields represents a systematic error. This is be lieved to be due to the modulation of UV light intensity by the grid and anode wires. Clearly, the electron data (open triangles in Figure 3) do no t obey Equation (2), but all of the ion data do. The ion data were fit to lines from which Teffand the intercept were extracted. As can be seen in Table 1, all of the temperatu res are consistent with 60 2 4 6 8 100 10 20 30 40 50 60 Reduced Field (V/(cm Torr))Drift Velocity (m/s) 100% CS2 90%CS 2−10%Ar 50%CS 2−50%Ar 25%CS 2−75%Ar Fig. 4. Drift Velocity vs. Field thermal diffusion with perhaps a hint that the lateral diffusi on is decreasing with increasing concentration of Ar. The fact that the high field points lie on the general trend of t he data is important in that it shows that electrons are captured even in very large E /p. For the 100% CS 2 data the high field was 83 V/cmTorr while for the 25% CS 2- 75%Ar data it was 45 V/cmTorr. The intercept data shown in Table 1 and Equation 3 can now be used to figure out σcapture. Given the crude nature of our photographic measurement σspot could not be estimated with any reliability. However, σgeometry = 2mm/√ 12 which allows to us to place the upper limits on σcapture shown in Table 1. 4.2 Mobility Figure 4 shows the drift velocity data for the gas mixtures pl otted against the reduced field. The data from each of the gas mixtures was fitted to v=m(E/p) +n(E/p)2. Table 2 shows the drift velocity coefficients for the different gas mixtures. The drift velocity increases with increasing concentration of argon . This is not unexpected as an Ar atom is much smaller than a CS 2molecule and therefore the mean free path of the CS 2ions is much larger, the mean time to collision is increased, thus yielding a higher drift velocity. In addition the larger mass of the CS 2molecule means that it will preferentially scatter off of the Ar atoms in the forward direction. 7Table 2 Mobility Coefficients Gas Mixture m n 100%CS 2 5.2±0.2 -0.20 ±0.03 90%CS 2-10%Ar 4.6 ±0.03 -0.057 ±0.004 50%CS 2-50%Ar 6.77 ±0.03 -0.090 ±0.005 25%CS 2-75%Ar 9.4 ±0.1 -0.10 ±0.02 4.3 Longitudinal Diffusion The longitudinal diffusion results are shown in Figure 5 and T able 3. As in the lateral case, the rms diffusion squared is plotted against the invers e of the reduced drift field. As before, the marks represent the data for the various gas mi xtures. All of the data look linear with the obvious exception of the 25%CS 2-75%Ar mixture. The linear datasets were fit to straight lines, giving the Teffvalues and intercepts shown in Table 3. Notice that in this case the trend is for the temperature to increase with increasing concentration of Ar. Unlike the lateral case, σspotis small since photoelectrons are created at the same time. Also unlike the lateral case, σgeometry is difficult to calculate since it arises from the different path lengths traveled by the ions. It is probabl y significant on the scale 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0 0.5 1.0 1.5 2.0 2.5 1/Reduced Field (cm Torr/V)σ2 (mm2) 100% CS2 90%CS 2−10%Ar 50%CS 2−50%Ar 25%CS 2−75%Ar Fig. 5. Longitudinal Diffusion 8Table 3 Longitudinal Diffusion Results Gas Mixture Slope Temperature Y-Intercept σcapture 100%CS 2 0.10±0.01 Vmm/Torr 230 ±20 K 0.17 ±0.02 mm2<0.4 mm 90%CS 2-10%Ar 0.11 ±0.02 Vmm/Torr 260 ±40 K 0.14 ±0.04 mm2<0.4 mm 50%CS 2-50%Ar 0.13 ±0.01 Vmm/Torr 300 ±20 K 0.14 ±0.04 mm2<0.4 mm of these measurements (i.e. sub-mm). Therefore we can again only infer an upper limit on the capture distance, shown in Table 3 for the three m easurements which indicated a constant capture distance. Given the discussion accompanying Equation 2 the interpret ation of the 25%CS 2- 75%Ar data is that σcapture is a stronger function of the field for this mixture. This interpretation is bolstered by a measurement of a 10%CS 2-90%Ar mixture in which some of the electrons drifted directly to the anode wires wit hout being captured by the CS 2. This was seen as a “direct” pulse, arriving within several m icroseconds after the light pulse instead of the several milliseconds that the ions take. 4.4 Trends The lateral and longitudinal ion diffusion data are generall y consistent with diffusion at thermal ion energies, with the exceptions noted above (25 %CS 2-75%Ar mixtures). If a further trend exists it is for the lateral diffusion to dec rease with increasing Ar concentration while the opposite occurs for the longitud inal diffusion. Removing the systematic error discussed above may help to resolve the se trends. Both of these deviations from Equation 1 can be understood from the fact th at CS 2is much heavier than Ar and therefore tends to preserve its direction of moti on after a collision with Ar, violating one of the assumptions underlying Equation 1. Theσcapture limits are harder to understand. For gas mixtures 50% Ar or le ss the lateral and longitudinal data both indicate a constant valu e with respect to field and gas mixture. One would expect each of these to increase wi th increasing Ar concentration but this increase may be masked by the constan t terms in Equation 3. At 75% Ar the lateral and longitudinal data are very different . The lateral data seem to indicate an elevated though field independent σcapture while the longitudinal data clearly indicate a field dependent σcapture. At 90% Ar the prompt arrival of electrons is evidence for a σcapture larger than 10 cm. 95 Conclusion The lateral and longitudinal diffusion of CS 2anions were systematically studied as a function of field and gas mixture. While mostly consistent w ith a thermal model certain trends were identified which need to be explored in mo re detail. The drift velocity of CS 2anions and limits on the capture distance for producing thos e anions were also measured. These measurements are critical for the operation of the DRIFT detector and for future uses for NITPCs. The results show tha t the NITPC concept will permit detectors such as DRIFT[1] to achieve submillim eter track diffusion for drift distances of the order of a meter, over a useful range of gas mixtures and drift fields. 6 Acknowledgments Funding for this project was provided by: Academic Student Project Award of Occidental College (spri ng 98, fall 98) Ford-Anderson Summer Research Grant (summer 98) Research Corporation Grant No. CC–4512 (summer 98) 7 References References [1] Daniel P. Snowden-Ifft, C. Jeff Martoff, and Juan M. Burwell . Low pressure negative ion drift chamber for dark matter search. Accepted for publication in Physical Review D , 2000. [2] C. Jeff Martoff, Daniel P. Snowden-Ifft, Tohru Ohnuki, Neil J. C. Spooner, and Matthew Lehner. Supressing drift chamber diffusion without magneti c field. Nucl. Instr. & Meth. in Phys. Res. A , 440:335, 2000. [3] H. R. Crane. CO 2-CS2geiger counter. Rev. Sci. Inst. , 32:953, 1961. [4] Luigi Rolandi and Walter Blum. Particle Detection with Drift Chambers . Springer- Verlag, Berlin, Germany, 1994. [5] F. Sauli. Experimental Techniques in High Energy Physics , page 81. Addison-Wesley Publishing, 1987. 10

arXiv:physics/0004007v1 [physics.acc-ph] 3 Apr 2000A microscopic derivation of Special Relativity: simple harmonic oscillations of a moving space-time lattic e Richard Lieu1 1Department of Physics, University of Alabama, Huntsville, AL 35899, U.S.A. Received ; accepted– 2 – ABSTRACT The starting point of the theory of Special Relativity1is the Lorentz transfor- mation, which in essence describes the lack of absolute meas urements of space and time. These effects came about when one applies the Second Rel ativity Postulate (which states that the speed of light is a universal constant ) to inertial observers. Here I demonstrate that there is a very elegant way of explain ing how exactly nature enforces Special Relativity, which compels us to con clude that Einstein’s theory necessitated quantization of space and time. The mod el proposes that microscopically the structure of space-time is analogous t o a crystal which con- sists of lattice points or ‘tickmarks’ (for measurements) c onnected by identical ‘elastic springs’. When at rest the ‘springs’ are at their na tural states. When set in motion, however, the lattice vibrates in a manner desc ribed by Einstein’s theory of the heat capacity of solids, with consequent widen ing of the ‘tickmarks’ because the root-mean-square separation now increases. I a ssociate a vibration temperature Twith the speed of motion vvia the fundamental postulate of this theory, viz. the relationv2 c2=e−ǫ kTwhere ǫis a quantum of energy of the lattice harmonic oscillator. A moving observer who measures distan ces and time inter- vals with such a vibrating lattice obtains results which are precisely those given by the Lorentz transformation. Apart from its obvious beaut y, this approach provides many new prospects in understanding space and time . For example, an important consequence of the model is the equation ǫ=κx2 o, where xois the basic ‘quantum of space’ and κis the spring constant which holds together the lattice. Thus space-time, like mass, has an equivalence wit h energy.– 3 – In Special Relativity theory the distance between two point s is not absolute. If an inertial observer Σ′measures simultaneously the positions of the ends of a uniformly moving rod to obtain the rod length, and if the rod is oriented parall el to its velocity v, the result will be smaller than that of another inertial observer Σ who m easures the length from a frame of reference with respect to which the rod is at rest. Th e difference between the two data values is the Lorentz factor γ= (1−v2/c2)−1 2, which means the length obtained by Σ′ is much smaller in the limit v→c. The same conclusion applies to the measurement of time. Thus if Σ′witnesses a time difference between the decay of two moving el ementary particles, both having the same velocity vand disintegrated in the same spatial position, the result is smaller than that of Σ, who measures decay times from a reference frame which does not move with respect to either particles. The above manner of formulating the two paramount results of Special Relativity considers (a) the effects of motion on position measurements at the same time, and (b) time measurements at the same position. Once each phenomena is understood without interference from the other, changes in space and time can be superposed to form a general Lorentz transformation. Einstein originally arrived at th is transformation by invoking the Second Relativity Postulate. I propose in this Letter a new more fruitful approach which demonstrates how Lorentz transformation is realized by a dy namic space-time lattice. We begin with a treatment of the relativity of length, i.e. situ ation (a). The treatment of time then follows immediately by considering the time lattice as being the same as that of space, except re-scaled by the speed of light cto form a different dimension. The measurement ruler of Σ is calibrated according to the top most part of Figure 1, and is used to determine the length of the rod drawn immediate ly beneath. A minimum unit of length is assumed to exist (i.e. space is quantized), which we designate as xo, and is also the distance between any pair of black dots (hereafte r referred to as lattice points)– 4 – on the ruler. The length of the rod is then expressed in units o fxo(such a regular lattice is not a necessary part of the theory, but for simplicity of ar gument we shall assume it). If the ruler of a moving observer measures a smaller length for t he same rod, this is equivalent to having a larger quantum of distance for Σ′, i.e. the lattice points of Σ′are more widely spaced, as shown in the lower half of Figure 1. Specifically th e length contraction effect will be achieved if, for Σ′: xo→xo/parenleftBigg 1 1−v2 c2/parenrightBigg1 2 (1) where v, the velocity of Σ′relative to Σ, is parallel to the length of the rod. For the lattice of Σ′every lattice point moves with velocity vlike all others, so why would this lead to a widening of the lattice ? The question ari ses only because we consider motion of the lattice as a rigid body. It is entirely possible that microscopically the space lattice points are connected by springs, and a moving lattic e necessarily vibrates: the amplitude of oscillation increasing with v. Henceforth I shall address the previous sentence (to be made even more precise in the enusing discussions) as t hePostulate of Lattice Space-Time . Now there is an effective increase in the lattice spacing xbecause, while the mean separation < x > remains at xo, theroot mean square xrms=√ < x2>is larger. Indeed, an interesting aspect of Equ (1) is that the right sid e has the form xo√ Nwhere N≥1, which reminds one of random walk. Moreover, Nhas a denominator reminiscent of quantum statistical mechanics. To explain the Lorentz transformation, I would then begin wi th a harmonic oscillator. We write x=xo+x1, with x1representing pure sinusoidal motion, implying < x1>= 0, and the lattice spacing may be written as: xrms=xo/parenleftbigg 1 +< x2 1> x2o/parenrightbigg1 2 (2) Next, we assume a classical oscillator and, to quantify the f orementioned Postulate of Lattice Space-Time, we make the reasonable conjecture that/radicalbig < x2 1>∝v=αv(meaning– 5 – of course that/radicalbig <˙x2 1>is also∝v; such a relation applies, e.g. to a uniformly moving lattice which started from rest as a result of an external imp ulse). Equ (2) now reads: xrms=xo/parenleftbigg 1 +α2v2 x2 o/parenrightbigg1 2 (3) We can therefore derive the length contraction of Equ (1) in t he non-relativistic limit of v≪cif in Equ (3) we set x2 o/α2=c2. However, apart from the obvious lack of elegance and completeness, the model is at best heuristic. An exact tr eatment requires quantum mechanics, when the energy of a harmonic oscillator becomes E= (n+ 1/2)ǫwhere ǫis the basic quantum of the system. Once the complete theory is in pl ace, we will see that the case of v≪cis not even classical. The proposed model works successfully if it allows the latti ce oscillators to have varying energies E, but the mean energy and degree of variation is governed by a t emperature T which replaces Eas the monitoring parameter of the overall vibration level ( i.e.Tincreases withv). The model used for such a thermal quantum lattice is Einste in’s theory for the heat capacity of solids. Here the reader is reminded of the Pa rtition function and mean energy per oscillator. Ignoring the zero point energy they a re, respectively: Z=1 1−e−ǫ kT(4) and¯E=ǫe−ǫ kT×Z. Thus if we define κas the lattice spring constant, and refer the average < x2 1>to its quantum expectation value, we may write < x2 1>=¯E/κ=ǫZ κe−ǫ kT (5) When Equ (4) and (5) are substituted into Equ (2), the result i s in close resemblance with that of length contraction (Equ (1)). In fact, the analogy is striking, and provides clear guideli nes on how precisely Tis related to v. Thus, to quantitatively complete the Postulate of Lattice Space-Time, I– 6 – propose this relation as: v2 c2=e−ǫ kT (6) Use of Equ (4), (5), and (6) gives: < x2 1>=ǫ κv2 c2 1−v2 c2(7) Equ (2) and (7) now combine to read: xrms=xo/parenleftBigg 1 +ǫ κx2 ov2 c2 1−v2 c2/parenrightBigg1 2 (8) The coincidence in form between Equ (8) and Equ (1), with the f ormer being based on a totally natural model which involves no manipulation of te rms, renders it extremely difficult to draw conclusions other than the one which says tha t we are indeed confronted with a microscopic realization of Relativity. In fact, Equ ( 8) agrees perfectly with Equ (1) when the following simple relationship between the three fu ndamental parameters of the space-time lattice holds: ǫ=κx2 o (9) A similar equation which gives no extra information applies to the time lattice, and is responsible for the relativity of time described earlier . This is because in Lorentz transformation time behaves exactly like space, with the su bstitution x=ct. Thus time and space are controlled by the same underlying lattice. In conclusion , Special Relativity already revealed that ev en the arrangement of the smallest quanta of our space-time fabric is highly ordered. However, one must also consider Equ (9), which implies that the space-time ‘crystal’ has an e nergy equivalence. A more immediate consequence is the quantization of the Lorentz fa ctorγat low v. Could this be responsible for the discreteness in the mass of elementary p articles ? The reported findings may also be of relevance to understanding astrophysical pro cesses, including (and especially)– 7 – those of the early universe. Finally, it opens the question a s to how the notion of space-time microstates presented here could facilitate further under standing and development of the theory of General Relativity. I am indebted to Dr Massimilano Bonamente for suggesting a ha rmonic oscillator model for the space-time lattice of a moving observer. Reference [1] Einstein, A., 1905, Annalen der Physik ,18, 891.– 8 – Figure Caption The space-time lattice of stationary (Σ) and moving (Σ′) observers are illustrated here for the case of distance measurements. The ‘tickmarks’ of the ruler of Σ are marked as the topmost set of black dots. The rod to be measured is the s hort bar immediately beneath, and is at rest with respect to Σ. Observer Σ′measures the length of this rod while in motion, by simultaneously acquiring data on the position s of the front and rear end of the rod. It is postulated that effectively Σ′is using a moving set of ‘tickmarks’, and if microscopically these are connected by ‘elastic springs’ w hich can extend while in motion, the ‘tickmarks’ widen as depicted in the lower half of the dia gram. Consequently Σ′obtains a smaller value for the length of the rod.

arXiv:physics/0004008v1 [physics.class-ph] 3 Apr 2000Dirac monopole with Feynman brackets Alain. B´ erard LPLI-Institut de Physique, 1 blvd D.Arago, F-57070 Metz, Fr ance Y. Grandati LPLI-Institut de Physique, 1 blvd D.Arago, F-57070 Metz, Fr ance Herv´ e Mohrbach M.I.T, Center for Theoretical Physics, 77 Massachusetts Av enue, Cambridge, MA 02139-4307 USA and LPLI-Institut de Physique, 1 blvd D.Arago, F-57070 Metz, Fr ance Abstract We introduce the magnetic angular momentum as a consequence of the struc- ture of the sO(3) Lie algebra defined by the Feynman brackets. The Poincar´ e momentum and Dirac magnetic monopole appears as a direct res ult of this framework. I. INTRODUCTION In 1990, Dyson [1] published a proof due to Feynman of the Maxw ell equations, assuming only commutation relations between position and velocity. In this article we don’t use the commutation relations explicitly. In fact what we call a com mutation law is a structure of algebra between position and velocity called in this letter Feynman’s brackets. With this minimal assumption Feynman never supposed the existence of an Hamiltonian or Lagrangian 1formalism and didn’t need the not gauge invariant momentum. Tanimura [2] extended Feynman’s derivation to the case of the relativistic partic le. In this letter one concentrates only on the following point: the study of a nonrelativistic particle using Feynman brackets. We show that Poincare’s ma gnetic angular momentum is the consequence of the structure of the sO(3) Lie algebra defi ned by Feynman’s brackets. II. FEYNMAN BRACKETS Assume a particle of mass m moving in a three dimensional Eucl idean space with position: xi(t) (i= 1,2,3) depending on time. As Feynman we consider a non associativ e internal structure (Feynman brackets) between the position and the v elocity. The starting point is the bracket between the various components of the coordinat e: [xi, xj] = 0 (1) We suppose that the brackets have the same properties than in Tanimura’s article [2], that is: [A, B] =−[A, B] (2) [A, BC ] = [A, B]C+ [A, C]B (3) d dt[A, B] = [. A, B] + [A,. B] (4) where the arguments A,BandCare the positions or the velocities. The following Jacobi identity between positions is also tri vially satisfied: [xi,[xj, xk]] + [xj,[xk, xi]] + [xk,[xi, xj]] = 0 (5) In addition we will need also a “Jacobi identity” mixing posi tion and velocity such that: [.xi,[.xj, xk]] + [.xj,[xk,.xi]] + [xk,[.xi,.xj]] = 0 (6) 2Deriving (1) gives: [.xi, xj] + [xi,.xj] = 0 (7) This implies: [xi,.xj] =gij(xk), (8) where gij(xk) is a symmetric tensor. We consider here only the case where: gij=δij m(9) this gives the following relations: [xi, f(xj)] = 0 (10) [xi, f(xj,.xj)] =1 m∂f(.xj) ∂.xi(11) [.xi, f(xj)] =−1 m∂f(xj) ∂xi(12) III. ANGULAR MOMENTUM Suppose first the following relation: [.xi,.xj] = 0 (13) which permits to say that the force law is velocity independe nt: ..xi=..xi(xj) (14) By definition the orbital angular momentum is: Li=mεijkxj.xk (15) which satisfies the standard sO(3) Lie algebra for Feynman’s brackets: 3[Li,Lj] =εijkLk (16) The transformation law of the position and velocity under th is symmetry is: [xi,Lj] =εijkxk (17) [.xi, Lj] =εijk.xk (18) We consider as Feynman [1], the case with a ”gauge curvature” : [.xi,.xj] =α m2Fij (19) where Fmust be an antisymmetric tensor (electromagnetic tensor fo r our example) and α a constant. The goal of our work is to see what happens if we kee p the structure of the Lie algebra of the angular momentum and the transformation law o f the position and velocity. Using (6) we get the relations: α∂Fjk ∂.xi=−m2[xi,[.xj., xk]] (20) =−m2[.xj,[xi,.xk]] + [.xk,[.xj,xi]] = 0 then the electromagnetic tensor is independent of the veloc ity: Fjk=Fjk(xi) (21) By deriving (8) we have: [xi,..xj] =−[.xi,.xj] =−αFij m2(22) then: m∂..xj ∂.xi=αFji(xk) (23) or: m..xi=α(Ei(xk) +Fij(xk).xj) (24) 4We get the ” Lorentz force’s law”, where the electric field app ears as a constant of integration (this is not the case for the relativistic problem, see [2]). Now the force law is velocity dependent: ..xi=..xi(xj,.xj) (25) For the case (19), the equations (16), (17)and (18) become : [xi,Lj] =εijkxk (26) [.xi,Lj] =εijk.xk+αεjklxkFil m(27) [Li,Lj] =εijkLk+αεiklεjmsxkxmFls (28) Introducing the magnetic field we write Fin the following form: Fij=εijkBk, (29) We get then the new relations: [.xi,Lj] =εijk.xk+α m{xiB−δij(→r.→ B)} (30) [Li,Lj] =εijk{Lk+αxk(→r.→ B)} (31) To keep the standard relations we introduce a generalized an gular momentum: L/angbracketright=L/angbracketright+M/angbracketright (32) We call Mithe magnetic angular momentum because it depends on the field→ B. It has no connection with the spin of the particle, which can be intr oduced by looking at the spinorial representations of the sO(3) algebra. Now we impo se for the {αj}’s the following commutation relations: [.xi,L|] =ε/angbracketright|/bardbl§/bardbl (33) 5[.xi,L|] =ε/angbracketright|/bardbl. §/bardbl (34) [L/angbracketright,L|] =ε/angbracketright|/bardblL/bardbl (35) This first relation gives: Mi=Mi(xj) (36) and the second: [.xi,Mj] =α m[δij(→r.→ B)−xiBj] (37) If we replace it in (35) we deduce: Mi=−α(→r.→ B)xi (38) Putting this result in (34) gives the following equation of c onstraint for the field− →B: xiBj+xjBi=−xjxk∂Bk ∂xi(39) One solution has the form of a radial vector field centered at t he origin: → B=β→r r3(40) The generalized angular momentum then becomes: → L=m(→r∧.→r)−α(→r.→ B)→r (41) We can check the conservation of the total angular momentum: d→ L dt=m(→r∧..→r)−α{→r∧(.− →r∧− →B)}= 0 (42) because the particle satisfies the usual equation of motion: md2..→r dt2=α(.→r∧→ B) (43) 6If we choose: α=qandβ=g, where qandgare the electric and magnetic charges, we obtain as a the special case the Poincar´ e [3] magnetic angul ar momentum: → M=−qg 4π→r r(44) and the Dirac [4] magnetic monopole: → B=g 4π→r r3(45) In addition we find that for the Dirac monopole the source of th e field is localized at the origin: div− →B= [.xi,[.xj,.xk]] + [.xj,[.xk,.xi]] + [.xk,[.xi,.xj]] =g 4π[.xi,xi r3] =gδ(→r) (46) We see that in the construction of the Feynman’s brackets alg ebra the fact that we didn’t im- pose the Jacobi identity between the velocities is a necessa ry condition to obtain a monopole solution. In summary, we used the Feynman’s algebra between position a nd velocity to compute the algebra of the angular momentum of a non relativistic par ticle in a electromagnetic field. The Dirac monopole and magnetic angular momentum is a d irect consequence of the conservation of the form of the standard sO(3) Lie algebra. IV. CASIMIR OPERATOR In the same spirit, it is interesting to introduce L2,the Casimir operator of sO(3) Lie alge- bra. Again we want to keep the same commutation relations in t he two cases corresponding to zero and non zero curvature. In the first case, we easily see that: [xi,L2] = 2(→ L∧→r)i (47) [.xi,L2] = 2(→ L∧.→r)i (48) 7[Li,L2] = 0 (49) and in presence of a curvature: [xi,L2] = 2(→ L∧→r)i (50) [.xi,L2] = 2[(→ L∧.→r)i+α(→ L∧→r)lFil] (51) [Li,L2] = 2α(→ L∧→r)i(→r.→ B) (52) then we want: [xi,L∈] =∈(→ L∧→ ∇)/angbracketright (53) [.xi,L∈] =∈(→ L∧.→ ∇)/angbracketright (54) [L/angbracketright,L∈] =′ (55) and we can deduce: [xi,M2] = 2(→ M∧→r)i (56) [.xi,M2] = 2[(→ M∧→r)i−α(→ L∧→r)lFil (57) 2α(→ L∧→r)i(→r.→ B) + [Li, M2] + [Mi,L2] = 0 (58) The last equation becomes after a straightforward computat ion: (→ M∧→r)(→ L∧.→r)−(→ L∧→r)(→ M∧.→r)−(→ M∧.→r)(→ L∧→r) + (→ M∧→r)(→ L∧.→r) = 0 (59) We can check that this equation of constraint is in particula r satisfied for the Poincar´ e angular momentum. 8V. CONCLUSION We find that the structure of Feynman’s brackets (without an H amiltonian or La- grangian), illuminates the connections between the spaces with gauge curvature, the sO(3) Lie algebra and the existence of the Poincar´ e magnetic angu lar momentum. It seems that more than the phase space formalism, the Feynman’s one is a go od approach of the me- chanics in a space with gauge symmetry, because it avoids the introduction of the not gauge invariant momentum. Further, other applications of this me thod, for example, the case of the Minkowski space with Lorentz Lie algebra, will be consid er in the future. 9REFERENCES [1] F.Dyson,Am.J.Phys.58,209(1990). [2] S.Tanimura,Ann.Phys.220,229(1992). [3] H.Poincar´ e,C.R.Acad.Sci.Paris,123,530(1896). [4] P.A.M.Dirac,Proc.R.Soc.A 113,60(1931). 10

arXiv:physics/0004009v1 [physics.bio-ph] 4 Apr 2000Coupled Two-Way Clustering Analysis of Gene Microarray Dat a G. Getz, E. Levine and E. Domany Department of Physics of Complex Systems, Weizmann Inst. of Science, Rehovot 76100, Israel February 2, 2008 Abstract We present a novel coupled two-way clustering approach to gene microarray data analys is. The main idea is to identify subsets of the genes and samples, such that when o ne of these is used to cluster the other, stable and significant partitions emerge. The search for such subse ts is a computationally complex task: we present an algorithm, based on iterative clustering, which performs s uch a search. This analysis is especially suitable for gene microarray data, where the contributions of a variety of bio logical mechanisms to the gene expression levels are entangled in a large body of experimental data. The method wa s applied to two gene microarray data sets, on colon cancer and leukemia. By identifying relevant subsets of the data and focusing on them we were able to discover partitions and correlations that were masked and hidden whe n the full dataset was used in the analysis. Some of these partitions have clear biological interpretation; others can serve to identify possible directions for future research. Introduction In a typical DNA microarray experiment expression levels of thousands of genes are recorded over a few tens of dif- ferent samples1[1, 3, 4]. Hence this new technology gave rise to a new computational challenge: to make sense of such massive expression data [5, 6, 7]. The sizes of the datasets and their complexity call for multi-variant clus- tering techniques [8, 9], which are essential for extractin g correlated patterns and the natural classes present in a set ofNdata points, or objects , represented as points in the multidimensional space defined by Dmeasured features . Gene microarray data are fairly special in that it makes good sense to perform clustering analysis in two ways [1, 2]. The first views the nssamples as the N=ns objects to be clustered, with the nggenes’ levels of ex- pression in a particular sample playing the role of the fea- tures, representing that sample as a point in a D=ng dimensional space. The different phases of a cellular pro- cess emerge from grouping together samples with similar or related expression profiles. The other, not less natural way, looks for clusters of genes that act correlatively on the different samples. This view considers the N=ng genes as the objects to be clustered, each represented by its expression profile, as measured over all the samples, as a point in a D=nsdimensional space. Whereas in previous work [1, 2, 10] the samples and genes were clustered completely independently, we in- troduce and perform here a coupled two-way clustering (CTWC) analysis. Our philosophy is to narrow down both the featuresthat we use and the data points that are clustered. We believe that only a small subset of the genes participate in any cellular process of interest, which takes place only in a subset of the samples; by focusing on small subsets, we lower the noise induced by the other samples and genes. We look for pairs of a relatively small subset Fof features (either genes or samples) and of objects O, (samples or genes), such that when the set Ois clustered using the features F, stable and significant partitions are obtained. Finding such pairs of subsets is a rather complex mathe- matical problem; the CTWC method produces such pairs in an iterative clustering process. CTWC can be performed with any clustering algo- rithm. We tested it in conjunction with several clustering methods, but present here only results that were obtained using the super-paramagnetic clustering algorithm (SPC) [16, 11, 12], which is especially suitable for gene microar- ray data analysis due to its robustness against noise and its “natural” ability to identify stable clusters. The CTWC clustering scheme was applied to two gene microarray data sets, one from a colon cancer experiment [1] and the other from a leukemia experiment [3]. From both datasets we were able to “mine” new partitions and correlations that have notbeen obtained in an unsuper- vised fashion by previously used methods. Some of these new partitions have clear, well understood biological inte r- pretation. We do notreport here discoveries of biologically relevant, previously unknown results. The main point of our message is twofold: (a) we wereable to identify bio- logically relevant partitions in an unsupervised way and (b) other, not less natural new partitions were also found, 1By “sample” we refer to any kind of living matter that is being tested, e.g. different tissues[1] cell populations collect ed at different times[2] etc. 1which maycontain new, important information and for which one should seek biological interpretation. Coupled Two Way Clustering Motivation and Algorithm The results of every gene microarray experiment can be summarized as a set of numbers, which we organize in anexpression level matrix A. A row of this matrix cor- responds to a single gene, while each column represents a particular sample. Our normalization is described in detail later. In a typical experiment simultaneous expression lev- els of thousands of genes are measured. Gene expression is influenced by the cell type, cell phase, external signals and more [13]. The expression level matrix is therefore the result of all these processes mixed together. Our goal is to separate and identify these processes and to extract as much information as possible about them. The main point is that each biological process on which we wish to focus may involve a relatively small subset of the genes that are present on a microarray; the large majority of the genes constitute a noisy background which may mask the effect of the small subset. The same may happen with respect to samples. The CTWC procedure which we now describe is de- signed to identify subsets of genes and samples, such that a single process is the main contributor to the expression of the gene subset over the sample subset. We start with clustering the samples and the genes of the full data set and identify all stable clusters of either samples or genes. We scan these clusters one by one. The expression levels of the genes of each cluster are used as the feature set Fto represent object sets. The different object sets Ocontain either all the samples or any sample cluster. Similarly, we scan all stable clusters of samples and use them as the feature set Fto identify stable clusters of genes. We keep track of all the stable clusters that are generated, of both genes, denoted as vg, and samples vs. The gene clusters are accumulated in a list Vgand the sample clusters in Vs. Furthermore, we keep all the chain of clustering analyses that has been performed (which subset was used as ob- jects, which subset was used as features, and which were the stable clusters that have been identified). When new clusters are found, we use them in the next iteration. At each iteration step we cluster a subset of the objects (either samples or genes) using a subset of the features (genes or samples). The procedure stops when no new relevant information is generated. The outcome of the CTWC algorithm are the final sets VgandVsand the pointers that identify how all stable clusters of genes and samples were generated. A precise, step by step definition of the algorithm is given in Fig. 1.Analyzing the clusters obtained by CTWC The output of CTWC has two important components. First, it provides a broad list of gene and sample clus- ters. Second, for each cluster (of samples, say) we know which subset (of samples) was clustered to find it, and which were the features (genes) used to represent it. We also know for every cluster C, which other clusters can be identified by using Cas the feature set. We present here a brief selection of the possible ways one can utilize this kind of information. Implementations of the particular uses listed here are described in the Applications section. Identifying genes that partition the samples ac- cording to a known classification. This particular application is supervised . Denote by Ca known classi- fication of the samples, say into two classes, c1andc2. CTWC provides an easy way to rank the clusters of genes inVgby their ability to separate the samples according toC. It should be noted that CTWC not only provides a list of candidate gene clusters one should check, but also a unique method of testing them. First we evaluate for each cluster of samples vsinVs two scores, purity andefficiency , which reflect the extent to which assignment of the samples to vscorresponds to the classification C. These figures of merit are defined (for c1, say) as purity( vs|c1) =|vs∩c1| |vs|; efficiency( vs|c1) =|vs∩c1| |c1|. Once a cluster vswith high purity and efficiency has been found, we can use the saved pointers to read off the clus- ter (or clusters) of genes that were used as the feature set to yield vsin our clustering procedure. Clustering, as op- posed to classification, discovers only those partitions of the data which are, in some sense, “natural”. Hence by this method we identify the most natural group of genes that can be used to induce a desired classification. Needless to say, one can also test a gene cluster vg that was provided by CTWC using more standard statis- tics, such as the t-test[14] or the Jensen-Shannon distance [15]. Both compare the expression levels of the genes of vgon the two groups of samples, c1, c2, partitioned ac- cording to C. Alternatively, one can also use the genes of vgto train a classifier to separate the samples according toC[3], and use the success of the classifier to measure whether the expression levels of the genes in vgdo or do not correspond to the classification. Discovering new partitions. Every cluster vsofVs is a subset of all the samples, the members of which have been linked to each other and separated from the other samples on the basis of the expression levels of some co- expressed subset of genes. It is reasonable therefore to argue that the cluster vshas been formed for some biolog- ical or experimental reason. As a first step to understand the reason for the for- mation of a robust cluster vs, one should try to relate it to some previously known classification (for example, in terms of purity and efficiency). Clusters which cannot be 2Step 1. Initialization 1a. Letvg 0be the cluster of all genes, and vs 0be the cluster of all samples. 1b. Initialize sets of gene clusters, Vg, and sample clusters, Vs, such that Vg={vg 0}andVs={vs 0}. 1c. Add each known class of genes as a member of Vg, and each known class of samples as a member of Vs. 1d. Define a new set W=∅. This set is needed to keep track of clustering analyses that have already been performed. Step 2. For each pair ( vg, vs)∈(Vg×Vs)\W: 2a. Apply the clustering algorithm on the genes of vg using the samples of vsas its features and vice versa. 2b. Add all the robust gene clusters generated by Step 2a toVg, and all the robust sample clus- ters to Vs. 2c. Add ( vg, vs) toW. Step 3. For each new robust cluster uin either VgorVs define and store a pair of labels Pu= (uo, uf). Of these, uois the cluster of objects which were clus- tered to find u, and ufis the cluster of features used in that clustering. Step 4. Repeat Step 2 until no new clusters are added to either VgorVs. Figure 1: CTWC algorithm. The input of the algorithm is the full expres sion matrix. The output is a set Vgof stable gene clusters and a set Vsof stable sample clusters. For each stable cluster u, found in a clustering operation, the clusters which provided the objects and those that served as the features fo r this operation are stored as a label Pu. associated with any known classification, have to be in- spected more carefully. Useful hints for the meaning of such a cluster of samples may come from the identity of the cluster of genes which was used to find it. Clearly, the CTWC clusters can be used in the same way to inter- pret clusters of genes which were not previously known to belong to the same process. CTWC is a sensitive tool to identify sub- partitions. Some of the sample clusters in Vsmay have have emerged from clustering a subset of the samples, say vs 0. These clusters reflect a sub-partition of the samples which belong to vs 0. When trying to cluster the full sample set, this sub-partition may be missed, since other samples, unrelated to vs 0, are masking it. CTWC reveals conditional correlations among genes. The CTWC method collects stable gene clusters inVg. In many cases the same groups of genes may be added to Vgmore than once. This is caused by the fact that some genes are co-regulated in all cells, and there- fore are clustered together, no matter which subset of the samples is used as the feature set. For example, ribosomal proteins are expected to be clustered together for any set of samples which is not unreasonably small. Some gene clusters, however, are different; they are co- regulated only in a specific subset of samples. We call this situation conditional correlation. The identity of the sam -ple cluster which reveals the conditionally correlated gen e cluster is clearly important to understand the biological process which makes these genes correlated. Clustering method and similarity measures Any reasonable choice of clustering method and defini- tion of stable clusters can be used within the framework of CTWC. We describe here the benefits of the particu- lar clustering algorithm and similarity measure we used, which we found to be particularly suitable to handle the special properties of gene microarray data. SPC provides clear identification of stable clusters in a robust manner. Super-paramagnetic clustering (SPC) is a hierarchical clus- tering method recently introduced by Blatt et al[16]. The intuition that led to it is based on an analogy to the physics of inhomogeneous ferromagnets. Full details of the algorithm and the underlying philosophy are given else- where [11, 17]. As for many hierarchical clustering algorithms, the in- put for SPC is a distance or similarity matrix dijbetween 3the objects O, calculated according to the feature set F. A tunable parameter T(’temperature’) controls the reso- lution of the performed clustering. One starts at T= 0, with a single cluster that contains all the objects. As T increases, phase transitions take place, and this cluster breaks into several sub-clusters which reflect the structur e of the data. Clusters keep breaking up as Tis further in- creased, until at high enough values of Teach object forms its own cluster. Blatt et al showed that the SPC algorithm is ro- bust against variation of its parameters, initialization a nd against noise in the data . The following advantages of SPC makes it especially suitable for gene microarray data analysis: ( i) No prior knowledge of the structure of the data is assumed; ( ii) SPC provides information about the different self organizing regimes of the data; ( iii) The num- ber of “macroscopic” clusters is an output of the algorithm; and (iv) Hierarchical organization of the data is reflected in the manner clusters merge or split when the control parameter (the ’temperature’ T) is varied. Moreover, the control parameter can be used to provide a natural measure for the stability of any particular cluste r by the range of temperatures ∆ Tat which the cluster re- mains unchanged. A stable cluster is expected to ’survive’ throughout a large ∆ T, one which constitutes a significant fraction of the range it takes the data to break into single point clusters. Inspection of the gene dendrograms of Fig. 4 reveals stable clusters and stable branches. Normalization of the gene expression array The Pearson correlation is commonly used as the similar- ity measure between genes or samples [18, 2, 1]. This mea- sure conforms with the intuitive biological notion of what it means for two genes to be co-expressed; this statistic captures similarity of the “shapes” of two expression pro- files, and ignores differences between the magnitudes of the two series of measurements [2]. The correlation co- efficient is high between two genes that are affected by the same process, even if each has a different gain due to the process, over different background expression lev- els (caused by other processes). One problem of using the correlation coefficient is that its reliability depends on the absolute expression level of the compared genes; a positive correlation between two highly expressed genes is much more significant than the same value between two poorly expressed genes. This information is ignored in the clustering process. However, we find that correlations do not always cap- ture similarity between samples. For example, consider two samples taken at different stages of some process, with the expression levels of a family of genes much below av- erage in one sample and much higher in the other. Even if the expression levels of the two samples over these genes are correlated, one would like to assign them into differ- ent clusters. Furthermore, the distance between the two samples should be affected by the statistical significance of their expression differences.We therefore used the following normalization scheme. Denote by Dthe matrix of the raw data. Dis ang×ns matrix, where ngis the number of genes and nsthe num- ber of samples. We normalize our expression level matrix in two steps. First, divide each column by its mean: D′ij=Dij/¯Dj; ¯Dj=1 ng/summationtextng i=1Dij. We then normalize each row, such that its mean vanishes and its norm is one: Aij=D′ij−¯D′i /bardblD′i/bardbl, where ¯D′i=1 ns/summationtextns j=1D′ijand /bardblD′i/bardbl2= /summationtextns j=1/parenleftbig D′ij−¯D′i/parenrightbig2. For genes and samples we use the Euclidean distance as the dissimilarity measure. For two genes (rows of A) the Euclidean distance is closely related to the Pearson correlation between them. Applications In order to show the strength of the CTWC algorithm, we apply it to two gene microarray experiment data sets. Here we report only the results which were obtained by CTWC, and could not be found using a straightforward clustering analysis. We highlight a small subset of the partitions that our method was able to extract from the data; these are the results for which we were able to find satisfactory biological explanation. We do notreport here new discoveries of biologically relevant, previously unknown results. Rather, we claim to have discovered a method that is capable to minesuch information out of the available data. New, relevant information maybe con- tained in the new partitions which were found, to which we were not yet able to assign biological meaning. These new, uninterpreted results are reviewed briefly below; full lists of the clusters associated with these results, as well as their constituent samples or genes can be found at http://www.weizmann.ac.il/physics/complex/compphys. Analysis of Leukemia samples We analyzed data obtained by Golub et al[3] from 72 sam- ples collected from acute leukemia patients at the time of diagnosis. 47 cases were diagnosed as ALL (acute lym- phoblastic leukemia) and the other 25 as AML (acute myeloid leukemia). RNA prepared from the bone marrow mononuclear cells was hibridized to high-density oligonu- cleotide micorarrays, produced by Affymetrix, containing 6817 human genes. After rescaling the data in the manner described by Golub et al, we selected only those genes whose minimal expression over all samples is greater than 20. As a result of this thresholding operation 1753 genes were left. The resulting array was then normalized as described previ- ously, to give the 1753 ×72 expression level matrix A(see Fig. 2). 4We found that two iterations of the CTWC algorithm sufficed to converge to 49 stable gene clusters (LG1-49) and 35 stable sample clusters (LS1-35). We highlight here four of our findings, which demonstrate the power of the method to solve problems listed above. Identifying genes that partition the samples ac- cording to a known classification. First we use the known ALL/AML classification of the samples to de- termine which gene clusters can distinguish between the two classes. We found only a single gene cluster (LG1) which enables stable separation into AML/ALL clusters2. This well demonstrates the strength of CTWC, since it turned out that SPC was not able to clearly identify the AML/ALL separation using the full set of genes. Discovering new partitions. Next, we search the stable sample clusters for unknown partitions of the sam- ples. We focus our attention on sample clusters which were repeatedly found to be stable. One such cluster, de- noted LS1, may be of interest; it includes 37 samples and was found to be stable when either a cluster of 27 genes (LG2) or another unrelated cluster of 36 genes (LG3) was used to provide the features. LG3 includes many genes that participate in the glycolysis pathway. Due to lack of additional information about the patients we cannot deter- mine the biological origin of the formation of this sample cluster. Identifying sub-partitions Using a 28 gene cluster (LG4) as features, we tried to cluster only the samples that were identified as AML patients (leaving out ALL samples). A stable cluster, LS2, of 16 samples was found (see Fig. 2(B)); it contains most of the samples (14/15) that were taken from patients that underwent treatment and whose treatment results were known (either success or failure). For none of the other AML patients was any in- formation about treatment available in the data. Some of the 16 genes of this cluster, LG4, are ribosomal proteins and some others are related to cell growth. Apparently these genes can partition the AML patients according to whether they did or did not undergo treatment. This result demonstrates a possible diagnostic use of the CTWC approach; one can identify different responses to treatment, and the groups of genes to be used as the appropriate probe. We repeated the same procedure, but discarding AML and keeping only the ALL samples. We discovered that when any one of 5 different gene clusters (LG4-8) are used to provide the features, the ALL samples break into two stable clusters; LS5, which consists mostly of T-Cell ALL patients and LS4, that contains mostly B-Cell ALL pa- tients (see Fig. 2(A)). When all the genes were used to cluster all samples, no such clear separation into T-ALL vs B-ALL was observed. One of the gene clusters used, LG5, with T/B separating ability, contains 29 genes, many of which are T-cell related. Another gene cluster, LG6, which also gave rise to T/B differentiation contains many HLA histocompatability genes.These results demonstrate how CTWC can be used to characterize different types of cancer. Imagine that the na- ture of the sub-classification of ALL had not been known. On the basis of our results we could predict that there are two distinct sub-classes of ALL; moreover, by the fact that many genes which induce separation into these sub-classes are either T-Cell related or HLA genes, one could suspect that these sub-classes were immunology related. As a different possible use of our results, note that some of the genes in the T-Cell related gene cluster LG5 have no determined function, and may be candidates for new T-Cell genes. This assumption is supported both by the fact that these genes were found to be correlated with other T-Cell genes, and by the fact that they support the differentiation between T-ALL and B-ALL. Analysis of Colon cancer data The data set we consider next contains 40 colon tumor samples and 22 normal colon samples, analyzed with an Affymetrix oligonucleotide array complementary to more than 6500 human genes and ESTs. Following Alon et al [1], we chose to work only with the 2000 genes of greatest minimal expression over the samples. We normalized the data to get a 2000 ×62 expression level matrix A. The CTWC algorithm was applied to this data set. 97 stable gene clusters (CG1-97) and 76 stable sample clus- ters (CS1-76) were obtained in two iterations. Identifying genes that partition the samples ac- cording to a known classification. Again we search first for gene clusters which differentiate the samples ac- cording to the known normal/tumor classification. We found 4 gene clusters (CG1-4) that partition the samples this way. The genes of these clusters can be used if one wishes to construct a classifier for diagnosis purposes (see Fig. 3(A)). Discovering new partitions. Five clusters of genes (CG2,CG4-CG7) generated very stable clusters of sam- ples. Two of the five (CG2,CG4) differentiated tumor and normal; two other were less interesting since the clusters they generated contained most of the samples. The gene cluster CG5, however, gave rise to a clear partition of the samples into two clusters, of 39 and 23 tissues (see Fig. 3(B)). Checking with the experimentalists3We discov- ered that this separation coincides almost precisely with a change of the experimental protocol; 22 RNA samples were extracted using a poly-A detector (’protocol-A’), and the other 40 samples were prepared by extracting total RNA from the cells (’protocol-B’). Cursory examination did not yield any obvious common features among the 29 genes of the cluster CG5 that gave rise to this separation of the tissues. Identifying conditionally correlated genes and sub-partitions Finally, we turn to identify conditionally correlated genes by comparing stable gene clusters formed when using different sample sets as features. We found 2A cluster is identified with a certain class if both its purity and efficiency exceeds 3/4. 3U.Alon, K.Gish, D.Mack & A.Levine, Private communication. 5genes samplesBBAA 204060200 400 600 800 1000 1200 1400 1600 0 5 10 15 20 2510152025303540T samples(B)0510152025303540450 10203040T (A) Figure 2: The expression level matrix of the leukemia experiment is sh own on the left. Rows correspond to different genes, ordered by clustering them using all the samples. The two box es contain expression data from (A) ALL patients, measured on one gene cluster and (B) AML patients, on another gene clus ter. Clustering the ALL samples, using the data in box (A), yields good separation between T-ALL (black) and B-ALL (whi te). Clustering of AML samples, using the data in box (B) yields a stable cluster, which contains all patients who wer e treated, with results known to be either success (black) or failure (gray). The vertical axis is the “temperature” parameter Tand on the horizontal axis the samples are ordered according to the dendrogram. that most gene clusters form irrespectively of the samples that are used. We did find, however, 4 special groups of genes (CG8-11) that formed clear and stable clusters when using only the tumor samples as features, but were relatively uncorrelated, i.e. spread across the dendrogra m of genes, when clustering was performed based on all the samples or only the normal ones. One of these 4 clusters, (CG9), breaks up, at a higher resolution, into two sub-clusters, as shown in Fig. 4(B). One of these sub-clusters, (CG12), consists of 51 genes, all of which are related to cell growth (ribosomal proteins and elongation factors). The other sub-cluster, (CG13), contains 17 genes, many of which are related to intestinal epithelial cells ( e.g.mucin, cathespin proteases). Interest- ingly, when clustering the genes on the basis of the either all samples or only the normal ones, both clusters (CG12 and CG13) appear as two uncorrelated distinct clusters, and their positions in the dendrogram are quite far from each other (Fig. 4). The high correlation between growth genes and epithe- lial genes, observed in tumor tissue, suggests that it is the epithelial cells that are rapidly growing. In the normal samples there is smaller correlation, indicating that the expression of growth genes is not especially high in the normal epithelial cells. These results are consistent with the epithelial origin of colon tumor.Two other groups of genes formed clusters only over the tumor cells. One (CG11, of 34 genes) is related to the immune system (HLA genes and immunoglobulin re- ceptors). The second (CG10, of 62 genes) seems to be a concatenation of genes related to epithelial cells (endoth e- lial growth factor and retinoic acid), and of muscle and nerve related genes. We could not find any common func- tion for the genes in the fourth cluster (CG8). Clustering the genes on the basis of their expression over only the normal samples revealed three gene clusters (CG14-16) which did not form when either the entire set of samples or the tumor tissues were used. Again, we could not find a clear common function for these genes. Each cluster contains genes that apparently take part in some process that takes place in normal cells, but is suppressed in tumor tissues. Summary and discussion We proposed a new method for analysis of gene microarray data. The main underlying idea of our method is to zero in on small subsets of the massive expression patterns ob- tained from thousands of genes for a large number of sam- ples. A cellular process of interest may involve a relativel y small subset of the genes in the dataset, and the process 6A BA Bgenes samples204060200 400 600 800 1000 1200 1400 1600 1800 2000 0 10 20 30 40 50 600 1020304050T samples(B)0 10 20 30 40 50 600 1020304050T (A) Figure 3: The expression level matrix of the colon experiment is shown on the left. Rows correspond to different genes, ordered by clustering them using all the samples. The two boxes conta in expression data of all samples for two gene clusters. (A) U sing the genes of the first cluster, clear separation between tumo r samples (white) and normal ones (black) is obtained. (B) An other separation of the samples is obtained using the second gene c luster. This separation is consistent with two distinct exp erimental protocols, denoted by short and long bars. The vertical axis is the “temperature” parameter Tand on the horizontal axis the samples are ordered according to the dendrogram. may take place only in a small number of samples. Hence when the full data set is analyzed, the “signal” of this process may be completely overwhelmed by the “noise” generated by the vast majority of unrelated data. We are looking for a relatively small group of genes, which can be used as the features used to cluster a subset of the samples. Alternatively, we try to identify a subset of the samples that can be used in a similar way to identify genes with correlated expression levels. Identifying pair s of subsets of genes and samples, which produce significant stable clusters in this way, is a computationally complex task. We demonstrated that the Coupled Two-Way Clus- tering technique provides an efficient method to produce such subgroups. The CTWC algorithm provides a broad list of stable gene and sample clusters, together with various connec- tions among them. This information can be used to per- form the most important tasks in microarray data anal- ysis, such as identification of cellular processes and the conditions for their activation; establishing connection be- tween gene groups and biological processes; and finding partitions of known classes of samples into sub-groups. We reemphasize that CTWC is applicable with any reasonable choice of clustering algorithm, as long as it is capable of identifying stable clusters. In this work we re- ported results obtained using the super-paramagnetic clus- tering algorithm (SPC), which is especially suitable forgene microarray data analysis due to its robustness against noise which is inherent in such experiments. The power of the CTWC method was demonstrated on data obtained in two gene microarray experiments. In the first experiment the gene expression profile in bone mar- row and peripheral blood cells of 72 leukemia patients was measured using gene microarray technology. Our main re- sults for this data were the following: ( i) The connection between T-Cell related genes and the sub-classification of the ALL samples, into T and B-ALL, was revealed in an unsupervised fashion. ( ii) We found a stable partition of the AML patients into two groups: those who were treated (with known results), and all others. This partition was revealed by a cluster of cell growth related genes. This observation may serve as a clue for a possible use of the CTWC method in understanding the effects of treatment. The second experiment used gene microarray technol- ogy to probe the gene expression profile of 40 colon tumor samples and 22 normal colon tissues. Using CTWC we find a different, less obvious stable partition of the sam- ples into two clusters. To find this partition, we had to use a subset of the genes. The new partition turned out to reflect two different experimental protocols. We de- duce that the genes which gave rise to this partition of the samples are the ones which were sensitive to the change of protocol. Another result that was obtained in an unsupervised 7200 400 600 800 1000 1200 1400 160020304050607080T 21 genes200 400 600 800 1000 1200 1400 160020304050607080T 12 genes Figure 4: Clustering genes of the colon cancer experiment, (A) using a ll samples and (B) using only tumor samples as the feature sets. Each node of this dendrogram represents a clus ter; only clusters of size larger than 9 genes are shown. The l ast such clusters of each branch, as well as non-terminal cluste rs that were selected for presentation and analysis are show n as boxes. In each dendrogram the genes are ordered according to the corresponding cluster analysis. The two circled cluste rs of the first dendrogram are reproduced also in the second, but th ere the two share a common ’parent’ in the tree. Note that the stability of a cluster is easily read off a dendrogram produce d by the SPC algorithm. manner using CTWC, is the connection between epithe- lial cells and the growth of cancer. When we looked at the expression profiles over only the tumor tissues, a clus- ter of cell growth genes was found to be highly correlated with epithelial genes. This correlation was absent when the normal tissues were used. These novel features, discovered in data sets which were previously investigated by conventional clustering analysis, demonstrate the strength of CTWC. We find CTWC to be especially useful for gene microarray data analysis, but it may be a useful tool for investigating other kinds of data as well. Acknowledgments We thank N. Barkai for helpful discussions. 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arXiv:physics/0004010v1 [physics.chem-ph] 4 Apr 2000Exact Topological Twistons in Crystalline Polyethylene E. Ventura and A. M. Simas Departamento de Qu´ ımica Fundamental, Universidade Feder al de Pernambuco 50670-901, Recife, Pernambuco, Brazil and D. Bazeia† Departamento de F´ ısica, Universidade Federal da Para´ ıba Caixa Postal 5008, 58051-970, Jo˜ ao Pessoa, Para´ ıba, Braz il February 2, 2008 Abstract We investigate the presence of topological twistons in crys talline polyethylene. We describe crystalline polyethylene with a model that couples the tors ional and longitudinal degrees of free- dom of the polymeric chain by means of a system of two real scal ar fields. This model supports topological twistons, which are described by exact and stab le topological solutions that appear when the interaction between torsional and longitudinal fie lds is polynomial, containing up to the sixth power in the fields. We calculate the energy of the topol ogical twiston, and the result is in very good agreement with the value obtained via molecular si mulation. †Corresponding author. Fax: +55 83 216 7542; E-mail: bazeia@ fisica.ufpb.br 1The existence of twistons in crystalline polyethylene (PE) was postulated [1] two decades ago, and refers to a twist of 1800that extends smoothly over several CH2groups in crystalline PE, in the plane orthogonal to the chain direction, with the corres ponding CH2unit length contraction along the polymeric chain. These twiston configurations app ear in crystalline PE as a result of its large torsional flexibility, and may contribute to eluci date some of its properties, in particular the dielectric αrelaxation [2, 3, 4, 5, 6, 7]. There are some interesting models of twistons in crystallin e PE [8, 9, 10, 11, 12]. The works [8, 9] are almost simultaneous to the work [13], which introd uces solitons to describe conductivity in polyacetylene via distortions of the single-double bond alternations. In the PE chain, however, the bounds are always single bounds, which require at least o ne bosonic degrees of freedom to describe the torsional flexibility of this unsaturated po lymer. Despite these two decades of investigations, we believe that the issue of topological tw istons playing some role in explaining properties of the crystalline PE chain is still incomplete, requiring further investigations both in the theoretical and experimental grounds. This is the mai n motivation of the present work, where we follow an alternate route to topological twistons i n PE to bring new facts to the former theoretical investigations. This new route was introduced in Ref. [14], and here we complete the investigation, including the calculation of the energies o f the exact topological twistons. We start our investigations by first reviewing the basic feat ures of the several distinct mechan- ical models used to describe twistons in crystalline PE. The most important ones are described in Refs. [8, 9, 10, 11, 12]. In the pioneer work [8] the author c onsiders a system which couples the torsional and translational degrees of freedom. In Ref. [9] the authors consider a simpler system, describing only the torsional motion along the crystalline chain, and this is also considered in the subsequent work [10]. It is only more recently [11, 12] th at one includes interactions between radial, torsional and longitudinal degrees of freedom. In t his case one uses cilindrical coordinates to describe a generic CH2unit via ( rn, θn, zn), which correspond to the three degrees of freedom of the rigid molecular group. A simplification can be introdu ced, and concerns freezing the rn’s, so that the radial motion is neglected. In [11] one further ig nores the translational degree of free- dom, the zncoordinates, to get to a simple model described via the torsi onal variable that in the continuum limit can be taken as θn(t)→θ(z, t). The model reproduces the double sine-Gordon model, according to the assumptions there considered to des cribe the intermolecular interaction. The other more recent work [12] on twiston in crystalline PE g ives another step toward a more realistic model. This is the first time the radial, torsional and longitudinal degrees of freedom are simultaneously considered to model twiston in crystall ine PE. The model is very interesting, although it is hard to find exact solutions and investigate th e corresponding issues of stability. The problem engenders several intrinsic difficulties, which have inspired us to search for an alter- nate model, in the form of two coupled fields belonging to the c lass of systems investigated in the recent works [15, 16, 17, 18]. The basic assumptions introduced in the former models for tw istons in crystalline PE may be described considering cilindrical coordinates. The Lagra ngian presents the usual form L=T−U, 2where T=1 2m/summationdisplay n(˙r2 n+r2 n˙θ2 n+ ˙z2 n) (1) U=Uintra+Uinter (2) Heremis the mass associated to the molecular group CH2, and UintraandUinterare potentials used to model the intramolecular and intermolecular intera ctions in the crystalline environment, respectively. The intramolecular potential can be conside red as Uintra=1 2/summationdisplay nK1(θn+1−θn)2+1 2/summationdisplay nK2(zn+1−zn)2+· · · (3) where K1andK2are coefficients related to the harmonic approximation for to rsional and longitu- dinal motions, respectively. The intramolecular potentia l may contain derivative coupling between the torsional and longitudinal motions. In this case we shou ld add to Uintrathe contribution [11] 1 2/summationdisplay nK3(θn+1−θn)2(zn+1−zn) (4) However, instead of the above coupling we can consider deriv ative coupling between the radial and longitudinal motions. In this other situation we add to Uintrathe contribution [12] 1 2/summationdisplay nK4(rn−r0)2+1 2/summationdisplay nK5(rn+1−rn) (zn+1−zn) (5) The above terms (4) and (5) are two among several other possib ilities of introducing derivative coupling between the torsional, longitudinal and radial de grees of freedom. We shall not consider such possibilities in the present work, although in [18] one shows a route for taking derivative coupling into account. As we are going to show below, we consi der the standard harmonic approximation for Uintrain order to follow the basic steps of the first works [8, 9, 10] o n twistons in crystalline PE. The second potential in (2) is Uinter. It responds for the intermolecular interactions and is usually given in the form Uinter=/summationtext n[U0(θn) +U1(θn)Ul(zn)]. Here U0(θn) and U1(θn) are used to model torsional mobility and Ul(zn) to describe the longitudinal motion along the chain. In the works [8, 9, 10, 11], after freezing radial and translati onal motion, the above intermolecular potential is described by the U0(θn) contributions. We can get to models for the torsional motio n alone, and in the continuum limit they may be described via th e sine-Gordon potential [8, 9, 10] A1[1−cos(2θ)], or the polynomial potential [9] A2θ2+B1θ4, or yet the double sine-Gordon potential [11] A3[1−cos(2θ)] +B2[1−cos(4θ)]. Here AiandBiare real constants, used to parametrize the corresponding interactions. Evidently, t he above potentials lead to different models for the torsional field, and are introduced to account for the specific motivations presented in the respective works [8, 9, 10, 11]. In the more recent work [12], one considers coupling 3between the radial, torsional and longitudinal degrees of f reedom, but the analytical solutions there obtained are found under assumptions that ultimately decouple the system. The above models show that the basic idea introduced in Ref. [ 1] has survived along the years, although there have been interesting quantitative c ontributions to investigate the presence of twistons in crystalline PE. In particular, in Ref. [12] on e includes the most relevant degrees of freedom when one considers the CH2group in the form of rigid molecular group along the crystalline chain in crystalline PE. However, in the model c onsidered in [12] we could not fully understand the reason for not considering harmonic interac tions between neighbor radial coordi- nates, while taking into account interactions between radi al and longitudinal degrees of freedom in the intramolecular potential. For this reason, we think w e can introduce another mechanical model for the polymeric chain, where we modify some assumpti ons presented in the former works [11, 12]. The difficulties inherent to the problem of describi ng topological twistons in crystalline PE bring motivations for simplifying former assumptions, w ith the aim of offering an alternate model that presents exact solutions for twistons in crystal line PE. Toward this goal, let us use cilindrical coordinates to describe the molecular groups u nder the assumption of rigidity. We start with the kinetic energy (1), rewriting it in the form T=1 2m r2 0/summationdisplay n/parenleftBigg ˙φ2 n+/parenleftbiggc r0/parenrightbigg2 ˙χ2 n+ ˙ρ2 n/parenrightBigg (6) Here we have set φn=θn−[1−(−1)n](π/2),χn= (zn−nc)/candρn= (rn−r0)/r0, where r0is the equilibrium position of the radial coordinate and cis the longitudinal distance between consecutive molecular groups. Now φn,χnandρnare all dimensionless variables, and in the continuum limit they can be seen as real fields φ(z, t),χ(z, t) and ρ(z, t). Before going to the continuum version of the PE chain, however, let us reconside r the intramolecular potential given by Eq. (3). We use the harmonic approximation to write Uintra=1 2/summationdisplay nkt(φn+1−φn)2+1 2/summationdisplay nkl(χn+1−χn)2+1 2/summationdisplay nkr(ρn+1−ρn)2(7) where kt,klandkrare spring-like constants, related to the torsional, longi tudinal and radial degrees of freedom, respectively. The harmonic interactions present in the intramolecular te rm (7) makes the dynamics to appear as the dynamics of relativistic fields, in the same way it happens with the standard harmonic chain. We use (6) and (7) to write the following Lagr angian density for the continuum version of the mechanical model for crystalline PE Lm=1 2m cr2 0/parenleftBigg∂φ ∂t/parenrightBigg2 −1 2ktc/parenleftBigg∂φ ∂z/parenrightBigg2 +1 2m cc2/parenleftBigg∂χ ∂t/parenrightBigg2 − 1 2klc/parenleftBigg∂χ ∂z/parenrightBigg2 +1 2m cr2 0/parenleftBigg∂ρ ∂t/parenrightBigg2 −1 2krc/parenleftBigg∂ρ ∂z/parenrightBigg2 −Vinter(φ, χ, ρ ) (8) The quantity m/cidentifies the mass density along the chain, and ktc=κt, hlc=κl, krc=κr are Young parameters related to the torsional, longitudina l and radial motion, respectively. 4The above mechanical model is still incomplete, but it conta ins the basic assumption that we are dealing with an harmonic chain, and deviation from the ha rmonic behavior is to be included inVinter. Although in this case we can not introduce any other derivat ive coupling, we still have the freedom to specify Vinterand so introduce nonlinearity via the presence of the surrou nding environment in the crystalline material. This is the model w e keep in mind to introduce the following field theoretic considerations. We follow the lines of the former mechanical model, which lea d us to introduce a field theoretic model that contains three real scalar fields. The Lagrangian density describing the fields φ= φ(x, t),χ=χ(x, t) and ρ=ρ(x, t) was introduced in Ref. [14]. The model is defined by the potential, which is supposed to have the form V(φ, χ, ρ ) =1 2H2 φ+1 2H2 χ+1 2H2 ρ (9) HereHφ=∂H/∂φ and so forth. H=H(φ, χ, ρ ) is a smooth but otherwise arbitrary function of the fields. This restriction is introduced along the lines of the former investigations [15, 16, 17], and leads to interesting properties, such as the ones explor ed below. We focus attention on the crystalline PE chain. In this case i t is a good approximation [8, 9, 10, 11, 12] to descard radial motion in the PE chain. Thi s simplification leads to a system of two fields, describing torsional and longitudinal motion s simultaneously. However, we first consider the simpler system, described by the torsional fiel d alone. In this case, in accordance with the Refs. [11, 12], investigations on molecular simula tion allows introducing the following torsional potential V1(φ) =1 2λ2φ2(φ2−π2)2(10) Fortunately, this potential is generated by the function H1(φ) = (1 /2)λ φ2(φ2/2−π2). Also, it has three degenerate minima, one at φ= 0 and the other two at φ2=π2. We use the potential V1(φ) to get the masses of the elementary excitations around φ= 0 and φ=±πin the form mφ(0) = |λ|π2andmφ(±π) = 2 |λ|π2. These results identify an asymmetry in the spectra of excitations of the torsional motion around the minima φ= 0 and φ2=π2. This asymmetry appears in consequence of the polynomial potenti al (10), and is small for small λ. It is related to the asymmetry between the well at φ= 0, and the well at φ2=π2. Since the mass of the field corresponds to the minimum energy necessary to ex citate elementary mesons into the system, we realize that the value |λ|π2, the difference 2 |λ|π2− |λ|π2may be seen as the energy for the field φgo from φ= 0 to φ=±π, that is the energy to overcome the torsional barrier in this simplified model. To get to a more realistic model we couple the torsional field t o the longitudinal motion along the chain. We model the presence of interactions by extendin g the former function H1(φ) to H2(φ, χ) given by H2(φ, χ) =1 2λ φ2(1 2φ2−π2) +1 2µφ2χ2(11) 5This gives the system L2=1 2/parenleftBigg∂φ ∂t/parenrightBigg2 −1 2/parenleftBigg∂φ ∂z/parenrightBigg2 +1 2/parenleftBigg∂χ ∂t/parenrightBigg2 −1 2/parenleftBigg∂χ ∂z/parenrightBigg2 −V2(φ, χ) (12) where V2(φ, χ) =1 2λ2φ2(φ2−π2)2+λµφ2(φ2−π2)χ2+1 2µ2φ2χ4+1 2µ2φ4χ2(13) We are using natural units, as in Ref. [14]. The above potenti al presents interesting features. For instance, V2(φ,0) =V1(φ), which reproduces the torsional model V1(φ) when one freezes the longitudinal motion. Also, V2(0, χ) = 0 and V2(±π, χ) =1 2µ2π4χ2+1 2µ2π2χ4(14) We can evaluate the quantity ∂2V/∂φ∂χ to see that it contributes with vanishing values at the minima (0 ,0) and ( ±π,0). This shows that the spectra of excitations of the torsion al motion around the ground states are unaffected by the presence of the longitudinal motion. Thus, we can use V1(φ) to investigate the behavior of the torsional motion around the equilibrium config- urations. The masses of the φfield are now given by mφ(0,0) =|λ|π2andmφ(±π,0) = 2 |λ|π2, around the minima (0 ,0) and ( ±π,0), respectively. Accordingly, for the χfield we see that it is massless at (φ= 0, χ= 0), and at ( ±π,0) the mass is mχ(±π,0) = |µ|π2. These results identify an asymmetry in the spectra of excitations of both the torsiona l and longitudinal motion around the minima (0 ,0) and ( π2,0). This asymmetry appears in consequence of the polynomial potential (13), and is small for small parameters λandµ. These results allow introducing the ratio mχ/mφ between the masses of the φandχfields – see Ref. [14]. The topological solutions connect distinct, adjacent mini ma of the potential. The energy corresponding to the classical configurations can be writte n in the general form [17] Eij=|H(¯φi,¯χi,¯ρi)−H(¯φj,¯χj,¯ρj)| (15) where ( ¯φi,¯χi,¯ρi) and ( ¯φj,¯χj,¯ρj) stand for two vacuum states, that is, two adjacent points iand jin the field space ( φ, χ, ρ ) that minimize the potential. Let us first consider the case of a single field, the φfield that describes torsional motion along the polymeric chain. We use former results to write the equat ion of motion for static configuration in the form d2φ dz2=λ2φ(φ2−π2)(3φ2−π2) (16) This equation is solved by solutions of the first-order equat ion dφ dz=λφ(φ2−π2) (17) 6There are topological twistons, given by [15] φ(±) (t)(z) =±π/radicalBig (1/2)[1−tanh(λπ2z)] (18) Here we are taking z= 0 as the center of the soliton, but this is unimportant becau se the continuum, infinity chain presents translational invarian ce. The sign of λidentifies kink and antikink solutions, connecting the minima 0 and πor 0 and −π. These solutions are stable and can be boosted to their time-dependent form by just changing ztoξ= (z−vt)/(1−v2)1/2. This model can be seem as an alternate model to the ones introduced in the former works [8, 9, 10, 11]. The amplitude of the torsional field φisπ, which is the angle the chain rotates to form the twiston. The width of the twiston, L(t), which is the length along the chain where the angular position of CH2groups appreciately deviates from the crystalographic pos itions, is inversely proportional to the quantity |λ|π2. We can also get the energy corresponding to the static twiston. We use Eq. (15) to get the value E(t)=1 4|λ|π4(19) We now consider the model that describes interactions betwe en the torsional and longitudinal fields. The equations of motion for static fields φ=φ(z) and χ=χ(z) are given by d2φ dz2=λ2φ(φ2−π2)(3φ2−π2) + 2λµφ(2φ2−π2)χ2+µ2φ(χ2+ 1)χ2(20) d2χ dz2= 2λµφ2(φ2−π2)χ+ 2µ2φ2χ3+µ2φ2χ (21) Although there is no general way of solving these equations, we recognize that they follow from the potential in Eq. (13), defined via the function introduce d in Eq. (11), and so they are solved by dφ dz=λφ(φ2−π2) +µφχ2(22) dχ dz=µφ2χ (23) which are first-order differential equations, easier to inve stigate. To find explicit solutions we use the trial orbit method intro duced in Ref. [19]. We consider the orbit λ(φ2−π2) +µχ2=µ(φ2−π2) (24) We note that this orbit is compatible with the first-order Eqs . (22) and (23). Also, from Eq. (22) we get φ(±) (t,l)(z) =±π/radicalBig (1/2)[1−tanh(µπ2z)] (25) 7This result and the orbit (24) are now used to obtain, χ(±) (t,l)(z) =±π/radicalBigg λ µ−1/radicalBig (1/2)[1 + tanh( µπ2z)] (26) These solutions are valid for λ/µ > 1 and are similar to the solutions found in Ref. [12] to describe the torsional and longitudinal degrees of freedom that describe topological twistons in the crystalline PE chain. The amplitude of the twiston is still π, while the amplitude of the longitudinal motion is given byπ[(λ/µ)−1]1/2. This result requires that λ/µ > 1, which is compatible with the investigation of Ref. [14]. In this more sofisticated model the width L(t,l)of the topological twiston is proportional to 1/(|µ|π2). It depends inversely on µ. We compare L(t)andL(t,l)to see that L(t,l)> L(t)since λ/µ > 1 for the topological twiston of the model of two coupled field s. This result is new, and shows that the presence of the longitudinal motions cont ributes to enlarge the width of the topological twiston. Another result follows after calculating the energy of thes e solutions. We use Eq. (15) to get E(t,l)=E(t)= (1/4)|λ|π4, which equals the value of the energy of the simpler model, wh ere one discards the motion of the longitudinal field. This result sh ows that although the more general model changes some of the features of the simpler model, whic h describes only the twiston field, it does not change the energy of the twiston. We understand th is result as follows: the first-order equations (22) and (23) also present the pair of solutions ¯φ(±) (t,l)(z) =±π/radicalBig (1/2)[1−tanh(λπ2z)] ¯ χ(t,l)(z) = 0 (27) This pair of solutions and the former one, given by Eqs. (25) a nd (26), are at the same topological sector and present the very same energy, given in Eq. ( ??). However, when one sets χ→0 in the coupled model, the system changes to the simpler model, a nd so the energy of the pair (27) is necessarily equal to the energy of the twiston in the singl e field system. This fact explains our results, and shows that the torsional energy is the main quan tity to calculate the energy of the topological twiston. We use this point of view to rewrite the energy as E(t,l)= (1/4) (|λ|π2)π2. We have already identified |λ|π2and 2 |λ|π2as the masses of the twiston field, which show that when the φfield varies from 0 to ±π, that is when a twiston is formed, one changes from the energy |λ|π2to the energy 2 |λ|π2, and this requires the value |λ|π2. We then identify this value with the energy for twiston formation along the polymeric ch ain. According to Ref. [1], the energy contribution of the localized twisted region to the criatio n of the twist defect is 7.3 Kcal/mol. In fact, in Fig. [6] and Table II of Ref. [1] we see that U0= 9.8−2.5 = 7.3 Kcal/mol, which is to be regarded as the contribution of the localized twisted reg ion to the criation of the twist defect [20]. We then change |λ|π2→7.3 in the energy to obtain E(t,l)= 17.99 Kcal /mol (28) This is the energy of the topological twiston, and is in good a greement with the energy values of 18.01 Kcal/mol [1], 18-19 Kcal/mol [10], and 17.2 Kcal/mo l [12], obtained using different numerical simulations and models. 8We conclude this letter recalling that we have investigated a system of two coupled real scalar fields to model topological twistons in crystalline PE. This model describes no radial motion, but it couples the torsional and longitudinal degrees of freedo m in a very interesting way. We have found exact solutions, which engender several features, an d here we offer the following remarks. The limit µ→λtransforms the solutions (25) and (26) into the solutions (1 8) of the former case, that describes the torsional motion alone. The solutions of the model of a single field present width proportional to 1 /|λ|, and for the two field model it is proportional to 1 /|µ|. The width of the solutions of the two fields is exactly the same, in agreeme nt with the topological features of the solutions, and with the orbit (24), used to solve the coup led equations (22) and (23). This result is intuitive, since one expects that when the torsion al motion completes the 180orotation and returns to its crystalographic position the longitudin al motion should simultaneously return to its crystal register. The amplitude of the torsional field is given by the solution (25) and isπ, in agreement with the model we use for the twiston configurat ion. The amplitude of the longitudinal motion is given by the solution (26) and is π[(λ/µ)−1]1/2. In the PE chain we have to set this to unit, to make it compatible with c, the full longitudinal motion. This picture follows in accordance with the fact that crystalline PE presents deg enerate ground states, obtained from each other by a rotation of 1800or by a translation of calong the polymer chain. We have also obtained the energy of the twistons. It is E= (1/4)|λ|π4. We have used |λ|π2 to identify the mass difference for the torsional field in the m inima φ=πandφ= 0. This and results of Ref. [1] allow getting |λ|π2= 7.3 Kcal/mol, which gives the energy of the topological twiston as 17 .99 Kcal/mol, in good agreement with values known in the liter ature. The results presented in this work completes the former inve stigation [14]. They show that the approach of using systems of coupled fields and the corres ponding field theoretic analysis to describe topologically non-trivial excitations in contin uum versions of polymeric chains seems to work correctly. The procedure describes interesting aspec ts of the problem, and allows obtaining the energy of the topological excitation in a direct way. We b elieve that similar polymeric chains can also be investigated by similar systems, and this makes u s to think on modelling topological twistons for instance in the family of systems where one chan ges some CH2groups by oxygens periodically, to make chains with the basic units CH2−O,CH2−CH2−O,CH2−CH2−CH2−O, etc. Despite the presence of oxygen the bounds are still sigm a bounds, and the torsional motion seems to be similar to the PE chain. Thus, we may use a twiston m odel to explore properties of the family ( CH2)n−O, in particular in the case of CH2−CH2−O, the Poly(oxyethylene), POE. This and other related investigations are presently un der consideration. D.B. and E.V. would like to thank Roman Jackiw and Robert Jaffe for hospitality at the Center for Theoretical Physics, MIT, where this work has beg un. We would like to thank R. H. Boyd for the exchange of informations related to Ref.[1]. We also thank the brazilian agencies CAPES, CNPq, and PRONEX for partial support. 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1 Mechanism of Discordant Alternans in Spatially Homogeneous Tissue. short title: theory of discordant alternans Mari A. Watanabe, MD, PhD; Flavio H Fenton*, PhD; Steven J Evans, MD**; Harold M Hastings, PhD*; Alain Karma, PhD Physics Department Dana Building Northeastern University Boston, MA 02115 Telephone: 617-373-2929 Fax: 617-373-2943 karma@presto.physics.neu.edu * Mathematics Department, Hofstra University, Hempstead, NY ** Cardiology Department, Beth Israel Hospital, New York, NY subject code: 5, 106, 1322 abstract Discordant alternans, the phenomenon of separate cardiac tissue locations exhibiting action potential duration (APD) alternans of opposite phase, appears to be a potential mechanism for electrocardiographic T wave alternans, but its initiation mechanism is unknown. We studied behavior of one- and two- dimensional cardiac tissue spatially homogeneous in all respects, including APD restitution and conduction velocity restitution, using the Beeler Reuter ion channel model. We found that discordant alternans were initiated when spatial gradients of APD arose dynamically, such as from fixed rate pacing of a cable end (sinus node scenario), or from fixed rate pacing at one site preceded by a single excitation wavefront from another site (ectopic focus scenario). In the sinus node scenario, conduction velocity restitution was necessary to initiate discordant alternans. Alternating regimes of concordant and discordant alternans arose along the length of the cable, with regimes delimited by nodes of fixed APD. The number of observable nodes depended upon pacing rate and tissue length. Differences in beat to beat conduction velocity values at steady state were small. In the ectopic focus scenario, variable conduction velocity was not required for induction of discordant alternans. In both scenarios, stability of node position was dependent upon electrotonic coupling. Other mathematical models produced qualitatively similar results. We conclude that spatial inhomogeneities of electrical restitution are not required to produce discordant alternans; rather, discordant alternans can arise dynamically from interaction of APD and conduction velocity restitution with single site pacing, or from APD restitution alone in two site pacing. key words: discordant alternans, T wave alternans, restitution3 Introduction Occurrences of electrocardiographic T wave alternans (TWA) were noted soon after the first electrocardiograms were published 90 years ago, but such overt TWA was rare enough to elicit case reports [1]. Overt TWA was frequently associated with profibrillatory conditions such as angina, acute myocardial ischemia [reviewed in 2] and long QT syndrome [3]. Recent advances in signal processing techniques have led to the ability to detect microscopic (microvolt level) TWA from the body surface [4] and to the discovery that microscopic TWA is a fairly common phenomenon at higher heart rates. Clinical studies show a high correlation between large amplitude microscopic TWA and sudden cardiac arrest [5, 6]. Commercial instruments can now quantify microscopic TWA in cardiology offices, and one such test was cleared by the FDA last year as the first non-invasive test to identify patients at risk for sudden cardiac death [7]. Many experimental studies have been conducted to elucidate the mechanism of electrocardiographic alternans. Some have focused on alternans of action potential duration (APD) and mechanical contraction [reviewed in 8] that occurs at fast heart rates. Others have studied the correlation between alternans and ischemia [reviewed in 2], or long QT syndrome [3, 9, 10]. A recent optical mapping study by Pastore et al strongly suggests that TWA is related to discordant alternans (two spatially discrete sites exhibiting APD alternans of opposite phases) [11], a phenomenon described previously between epicardial and endocardial electrocardiograms [12]. The purpose of our study was to describe the origins of discordant alternans in spatially homogeneous cardiac tissue, using mathematical modeling and non-linear dynamics theory. Previous theoretical work has already suggested that sustained APD alternans [13] arises when the slope of APD restitution (the dependence of APD on duration of electrical diastole, i.e., diastolic interval or DI) is greater than 1.4 Methods A discrete model was used to illustrate initiation mechanisms of discordant alternans. It was composed of a 100 cell cable with cell-cell distance of 0.0375 mm, and no electrotonic coupling. Each cell adhered to the APD restitution equation APD [ms] = 300-250*exp(-DI/100) and to the intercellular conduction velocity restitution equation CV [mm/ms] = 0.025-0.02*exp(-DI/40) or = 0.025. To simulate long term behavior of discordant alternans, we used the Beeler Reuter (BR) ion channel model [14] in a cable or a sheet of cardiac cells. We used the forward Euler method to solve the partial differential equations, with step-sizes of 0.25 mm and 0.02 ms. Restitution of APD and conduction velocity (dependence of conduction velocity on DI) are both monotonically increasing curves in the BR model [15]. To confirm validity of results from the BR model, we also analyzed discordant alternans in the Noble ion channel model [16] and in 2- and 3- variable models in which restitution parameters could be varied [15, 17]. Figures 1 and 2 are from the discrete model, all others are from the BR model. Results Basic mechanisms of discordant alternans initiation Initiation of discordant alternans requires two conditions to be met, an APD restitution slope greater than 1 at the stimulus site, which allows sustained APD alternans at that site, and an initial spatial variation of DI. Initial variation of DI can be produced even in absence of spatial heterogeneity of APD and conduction velocity restitution curves. To illustrate, we will use a standard geometrical technique called cobwebbing ( figure 1 ). Drawing a vertical line from the x axis to the restitution curve APD = f(DI) gives an APD value for a given DI. If time between stimuli is S1S1, then new DI = S1S1 - old APD. Therefore, new DI can also be obtained graphically by drawing a horizontal line from the coordinates (old DI, old APD) on the restitution curve until it intersects the auxiliary line DI + APD= S1S1 at5 (new DI, old APD). This process of drawing a vertical line to the restitution curve and a horizontal line to the auxiliary line, can be repeated indefinitely to produce sequential DI, APD values. The intersection of the restitution curve and auxiliary line is called the fixed point (DI*, APD*), because the DI and APD are fixed in time at that point. The cobwebbing technique shows that if initial DI < DI*, then the subsequent DI and APD are always longer than the current values (a short-long sequence), and conversely, if initial DI > DI*, then the next DI and APD are always shorter (a long-short sequence). I.e., any spatial gradient of DI in which DI values flank DI* must produce discordant alternans. Two scenarios of discordant alternans initiation The laddergrams in figure 2 show two scenarios for producing initial DI gradients. In the ectopic focus scenario ( figure 2A ), we assumed that the tissue was excited uniformly before the first S1 stimulus (S1 1), and that depolarization wavefronts traveled with fixed velocity, making time from depolarization to depolarization 290 ms everywhere along the cable. At the stimulus site, the first DI was 62 ms (white space). APD (grey space) was then 165 ms from f(62)=165. The second DI was 125 from 290-165=125. The second APD was then f(125)=228. The third DI was 290-228=62, identical to the initial DI, proving the sequence would repeat itself stably. Farther from the stimulus site, the fact that the wavefront could not travel instantaneously to other sites gave rise to longer DI with distance traversed. Increasing DI produced increasing APD. This in turn led to a reversal in spatial distribution of DI when the wavefront from S1 2 travelled down the cable. Therefore, at the stimulus site, APD alternated with a short-long-short-long sequence, while at the bottom end of the cable, the same stimuli produced a long- short-long-short sequence, or discordant alternans. Cell 19, where DI equaled DI*, separated the concordant and discordant alternans regimes. Below the bottom edge of the laddergram, conduction block occurred when APD grew to equal S1S1.6 With sufficient time, APD values came to alternate between 165 and 228 ms for cells 0-18, and cells 20-100 with different phases, while cell 19 remained at APD*. This unphysiological end state of a sharp 63 ms APD difference between two cable segments was due to absence of electrotonic coupling in this discrete model. In the sinus node scenario, similar to the stimulation protocol of ref [11], stimuli were introduced following long quiescence ( figure 2B). There was only one pacemaker site, but conduction velocity was allowed to vary with DI. Short DI (2 ms) produced by S1 2 at cell 1 produced slow conduction. Slow conduction gave rise to longer DI as the wavefront traveled down the cable, which in turn sped up conduction and shortened DI, thereby producing sinusoidal DI gradients. The conduction velocity restitution itself produced the initial variation of DI, which spanned the DI* of 91 ms. APD node position where the APD was identical for two consecutive beats gradually moved towards the stimulus end. For example, the first node formed just beyond cell 100 and node APD value was approximately 167 ms. The node was next seen at cell 49, with node APD value of 207. Subsequent nodes were seen at cells 35 and 28 respectively. The remainder of the results section describes the dynamics of discordant alternans in this sinus node initiation scenario, using the BR model. Spatial gradients of APD, cycle length and conduction velocity APD gradients evolve in time. How the gradients grow in magnitude until the even and odd beat gradients intersect is shown in figure 3 for a cable length of 80mm, basic cycle length (BCL) of 310 ms. The intersection (node) asymptotically approaches a location near the pacing site. The final state is shown in figure 5A . Figure 4 shows the steady state distribution of voltage over the length of a 1-D cable for a full alternans cycle, at a BCL of 310 ms, for two cable lengths. The vertical axis is time, so rotating these figures 90 degrees clockwise produces the laddergram orientation. The voltage tracings in the left panel exhibit several features. The two7 depolarization wavefronts look roughly linear, similar to the laddergram, indicating that fluctuation of conduction velocity is relatively small. APD and DI can be visualized from high voltage and low voltage segments along the vertical axis separated by densely apposed voltage traces. The upstroke of the tenth voltage trace from the bottom shows the approximate location where APD is fixed in time. To the left of this position, APD alternates concordantly with the stimulus end, and to the right, discordantly. It is important to note that some of the voltage distributions exhibit a minimun and maximum. The significance of this phenomenon is discussed later in relation to electrotonic interactions. The right panel shows voltage distribution when 3 nodes exist at steady state. Figures 5 A-C show even and odd steady state APD gradients for 3 different tissue lengths at a BCL of 310 ms. There are 1, 2 and 3 nodes for cables of length 80, 117.5 and 135 mm, respectively. Multiple nodes imply alternating regimes of concordant and discordant alternans. If a node exists for a particular tissue length, it occurs at the same position regardless of cable length. However, the cable must be considerably longer than where a node is expected, for the node to exist at all. E.g., there is only one node for the 80 mm cable, although judging from the 117.5 mm cable, two nodes might be expected. This is because the node first arises where the spatial APD gradients cross, moves in the direction of the stimulus site, and becomes fixed in position. Even though a node can exist or "fit" in a particular length of tissue, the tissue has to be long enough for the node to be born. Where the node is born depends on the conduction velocity restitution curve. E.g., increased conduction velocity moves the initial APD node position farther from the stimulus site, and slope must not be zero for the gradients to cross. Figures 5 C-E show the relationship between steady state APD, cycle length, and conduction velocity gradients for a cable length of 135 mm. At the APD nodes, cycle length alternates by approximately 6 ms. At the cycle length nodes where cycle8 length is fixed, APD alternates, replicating the condition at the stimulated end of the cable. Conduction velocity nodes are closely but not exactly aligned with the APD nodes, and alternate by approximately 50 mm/s. Relationship between BCL and node position and number In experiments [11], shortening BCL causes concordant tissue to become discordant. The division of length, BCL parameter space into regions of 0 (concordant) through 4 node alternans in figure 6A illustrates a mechanism for this phenomenon. Above a BCL of 320 ms, there is no permanent alternans of APD at the stimulus site. Below a BCL of 280 ms, stimulus rate is high and produces refractory block of every other stimulus. BCL between these two threshold values produces stable alternans of APD at the stimulus site, and concordant or multi- nodal discordant alternans in the tissue as a whole. As tissue length increases, more nodes can exist for a given BCL. For a fixed tissue length, the number of nodes can increase by one as BCL is shortened. E.g., at a tissue of length 30 mm, alternans is concordant at or above a BCL of 290 ms. Shortening BCL to 285 ms or lower produces one node, i.e., one region of discordant alternans, similar to experiments. Electrocardiograms computed from an electrode placed above the center of a 30 mm cable for BCL of 290 and 285 ms are shown at the bottom of the figure. Both electrocardiograms show some alternans of QRS. The lower electrocardiogram exhibits T wave alternans. Figure 6B shows steady state APD node position versus BCL for a cable length of 140 mm. Nodes move closer to the paced end of the cable and internodal distances shrink as cycle length is reduced. Restitution curve during alternans The outer graph in Figure 7 shows the relationship between the standard restitution curve of the BR model (dotted line) and the two DI, APD relationship curves that arise dynamically during discordant alternans. Both of the dynamic9 curves are lower than the standard restitution curve for most DI, and higher for a small range of short DI. A particular DI value can produce two APD values, as shown by the curling ends of the dynamic restitution curves. The split of the dynamic curves shows why DI can alternate, yet still produce the same APD. The inset similarly shows standard and dynamic conduction velocity restitution curves. It is important to note that the different dynamic restitution curves do not signify an a priori spatially heterogenous distribution of restitution curves, but a dynamic modulation of inherent restitution properties by electrotonic coupling. 2-D simulation and results from other models Figure 8 shows temporal evolution of APD gradient for alternate beats in a 2- dimensional sheet in the sinus node scenario. The APD node line, where APD was identical on consecutive beats, moved towards and stabilized near the pacing site. APD gradient on the diagonal line of the 2-D sheet is shown at the bottom. Simulations of discordant alternans in other models [15-17] produced results qualitatively similar to simulations from the BR model, demonstrating the robustness of mechanisms of initiation and evolution across different models. One quantitative difference was noted. In the BR model, QRS alternans and T wave alternans arose simultaneously, due to similar DI over which APD and conduction velocity restitution had large slopes, while in the other models, T wave alternans could precede QRS alternans with rate increase. Discussion The first condition that must be satisfied to initiate discordant alternans is to have non-transient alternans at the pacing site. In theory, a restitution curve with a maximum slope greater than 1 will produce stable alternation of APD for some range of BCL [13,18,19], although there are experiments showing some discrepancy from theory [20], and some experimental [21,22] and theoretical [23] results ascribe a more important role to slope of dynamic rather than standard restitution curve in10 alternans production. The APD restitution curve of the BR model has a maximum slope that is greater than 1, and stable alternans was produced at the pacing site in accordance with theory. Steep (>1) restitution curve slope has also been shown to produce spiral wave break-up [15,24] in mathematical models, a phenomenon believed to be an analog for ventricular fibrillation. The slope of restitution curve is thus an indirect link between ventricular fibrillation and alternans, and may be related to the close relationship between TWA and ventricular fibrillation. The second condition necessary to produce discordant alternans was a non- uniform initial distribution of DI in space. We found two ways to accomplish this in electrically homogeneous tissue. In the ectopic focus scenario, conduction velocity could be fixed, but required two stimulus sites ( figure 2A). That situation is analogous in a 2 dimensional system to a straight edge depolarization wavefront coming from one direction, followed by straight edge depolarization wavefronts coming from a second direction, such as would happen when a sinus beat is followed by beats from a regularly firing ectopic focus, or the case where a ventricular premature contraction is followed by sinus beats. In the sinus node scenario, conduction velocity restitution produced spatial gradients of DI with only one stimulus site. The boundary between discordant and concordant alternans was represented by nodes where APD was identical on two consecutive beats. In the sinus node scenario, nodes formed, gradually moved towards the stimulus site, and stopped at a distance determined by BCL and cable length. Multiple nodes indicated multiple regimes in space where the phase of APD alternans was alternately concordant or discordant with the phase at the stimulated end. The number of nodes seen depended on tissue length. Although only single nodes have been observed experimentally [11], the simulation predicts that more may be found if larger tissue sizes are studied. Simulations replicated the experimental observation of faster11 pacing converting concordant to discordant alternans. During discordant alternans, there was alternans not only of APD, but of DI, cycle length, and conduction velocity. The relative importance of APD alternans and conduction velocity alternans has been debated, with some giving alternans of conduction velocity the primary role [25] and others, alternans of APD [26-28]. Two recent studies support primacy of APD alternans; activation times were identical from beat to beat [29, 30]. The results of our simulations show that both APD and conduction velocity alternans are important. It is possible that the magnitude of conduction velocity alternans is too small (50 mm/s compared to an overall velocity of 450 mm/s) to produce measurable differences of activation time in experiments. The difference of the dynamic restitution curves from the standard restitution curve was found to be due to electrotonic coupling of cardiac tissue. This is best explained by studying Figure 4 . When DI is very short, a minimum appears in the spatial distribution of voltage with the next depolarization. APD at the voltage minimum is longer than expected from the restitution curve, because electrotonic current flows towards the minimum, keeping voltage elevated. Conversely, when DI is long, a maximum appears in the spatial distribution of voltage with the next depolarization. APD at the voltage maximum is shorter than expected from the restitution curve, because electrotonic current flows away from the maximum. The split between the two dynamic curves themselves was also found to be due to electrotonic coupling. There existed an asymmetry of electrotonic coupling with respect to propagation direction for a given APD gradient. For example, consider three adjacent segments in a cable, a, b, and c, with DI-dependent APD values APD a, APD b, and APD c, in the uncoupled state. If the excitation sequence is a -> b -> c, segment a supplies electrotonic current during its phases 1-3 to segment b, while segment c supplies electrotonic current during its phases 0-3 to segment b. If12 the excitation sequence were reversed, segment b would receive phase 1-3 current from segment c and phase 0-3 current from segment a. The sum of electrotonic currents passing from segments a and c to segment b during APD b is therefore different depending on depolarization direction, even if the uncoupled scalar values APD a and APD c remain the same. This asymmetry can be quantified rigorously, and is part of a separate manuscript in progress. The split of dynamic restitution curves due to electrotonic coupling is directly responsible for stability of APD node position, and therefore of discordant alternans. Based on our results, we expect that spatial heterogeneity in the form of smooth gradients of electrical restitution would not change initiation and evolution characteristics of discordant alternans. However, effects of discontinuous tissue heterogeneities such as non-conducting scar tissue require further study. In summary, discordant alternans arises from the simple cardiac tissue characteristics of APD restitution and conduction velocity restitution. It does not require spatially heterogeneous tissue properties. Previous theory predicts that concordant alternans could be prevented by reducing heart rate, or alternatively by reducing APD restitution curve slope, such as by calcium channel antagonists [21,30 and references therein]. Results from the present study suggest that discordant alternans, and by inference, TWA, might be prevented by pharmacological agents that reduce conduction velocity restitution slope, even in presence of alternans. Acknowledgments This study was supported by American Heart Association Grant in Aid 96009660 and NIH SCOR in Sudden Cardiac Death 1PL50HL-52319.13 References 1 Kimura E, Yoshida K. A case showing electrical alternans of the T wave without change in the QRS complex. Am Heart J. 1963;65:391-395 2 Verrier RL, Nearing BD Electrophysiologic basis for T wave alternans as an index of vulnerability to ventricular fibrillation. J Cardiovasc Electrophysiol 1994;5:445-461 3 Schwartz PJ, Malliani A. Electrical alternans of the T-wave: Clinical and experimental evidence of its relationship with the sympathetic nervous system and with the long Q-T syndrome. Am Heart J 1975;89:45-50 4 Smith JM, Clancy EA, Valeri R, Ruskin JN, Cohen RJ. Electrical alternans and cardiac electrical instability. Circulation 1988;77:110-21. 5 Rosenbaum DS, Jackson LE, Smith JM, Garan H, Ruskin JN, Cohen RJ. Electrical alternans and vulnerability to ventricular arrhythmias. N Engl J Med 1994;330:235- 41. 6 Estes NAM, Michaud G, Zipes DP, El-Sherif N, Venditti FJ, Rosenbaum DS, Albrecht P, Wang PJ, Cohen RJ Electrical alternans during rest and exercise as predictors of vulnerability to ventricular arrhythmias. Am J Cardiol 1997;80:1314- 18. 7 Cambridge Heart CH 2000 Cardiac Diagnostic System. Cambridge Heart, Inc. Bedford, MA 8 Surawicz B, Fisch C. Cardiac alternans: Diverse mechanisms and clinical manifestations. J Am Coll Cardiol 1992;20:483-499 9 Hiejima K, Sano T. Electrical alternans of the TU wave in Romano-Ward syndrome. Br Heart J 1976;38:767 10 Rosenbaum MB, Acunzo RS. Pseudo 2:1 atrioventricular block and T wave alternans in the long QT syndromes. J Am Coll Cardiol 1991;18:1363-1366 11 Pastore JM, Girouard SD, Laurita KR, Akar FG, Rosenbaum DS. Mechanism linking T-wave alternans to the genesis of cardiac fibrillation. Circulation14 1999:99:1385-1394. 12 Hellerstein HK, Liebow JM. Electrical alternation in experimental coronary artery occlusion. Am J Physiol 1950;160:366-74 13 Nolasco JB, Dahlen RW. A graphic method for the study of alternation in cardiac action potentials. J Appl Physiol. 1968;25:191-196 14 Beeler GW, Reuter HO Reconstruction of the action potential of ventricular myocardial fibres. J Physiol 1977;268:177-21 15 Karma A. Electrical alternans and spiral wave breakup in cardiac tissue. Chaos 1994;4:461-472 16 Noble D. A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker poentials. J Physiol 1962;160:317-352 17 Fenton F, Karma A. Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: filament instability and fibrillation. Chaos 1998;8:20-47 18 Watanabe M, Otani NF, Gilmour RF Jr. Biphasic restitution of action potential duration and complex dynamics in ventricular myocardium. Circ Res. 1995;76:915- 921 19 Guevara MR, Glass L, Electrical alternans and period doubling bifurcations Computers in Cardiology 1984;pp167-170 20 Hall GM, Bahar S, Gauthier DJ. Prevalence of rate-dependent behaviors in cardiac muscle Phys Rev Lett. 1999;82:2995-2998 21 Koller ML, Riccio ML, Gilmour RF Jr. Dynamic restituion of action potential duration during electrical alternans and ventricular fibrillation Am J Physiol. 1998;275:H1635-1642 22 Riccio ML, Koller ML, Gilmour RF Jr. Electrical restitution and spatiotemporal organization during ventricular fibrillation Circ Res. 1999;84:955-963 23 Fenton FH, Evans SJ, Hastings HM Memory in an excitable medium: A15 mechanism for spiral wave breakup in the low-excitability limit. Phys Rev Lett 1999;83:3964-3967 24 Xie F, Qu Z, Garfinkel A. Dynamics of reentry around a circular obstacle in cardiac tissue. Phys Rev E 1998;58:6355 25 Downar E, Manse M, Durrer D. The effect of acute coronary artery occlusion on subepicardial transmembrane potentials in the intact heart. Circulation 1977;56:217- 224 26 Hoffman BF, Suckling EE. Effect of heart rate on cardiac membrane potentials and unipolar electrogram Am J Physiol 1954;179:123-130 27 Kleinfeld M, Stein E. Electrical alternans of components of the action potential. Am Heart J 1968;75:528-530 28 Dilly SG, Lab MJ. Electrophysiological alternans and restitution during acute regional ischemia in myocardium of anesthetized pig. J Physiol (Lond) 1988:402:315- 333 29 Shimizu W, Antzelevitch C. Cellular and ionic basis for T-wave alternans under long-QT conditions. Circulation 1999:99:1499-1507 30 Hirayama Y, Saitoh H, Atarashi H, Hayakawa H. Electrical and mechanical alternans in canine myocardium in vivo. Circulation 1993;88:2894-290216 Figure legends Figure 1 Cobweb diagram showing mechanism of discordant alternans initiation. The curve represents APD restitution equation APD = 300-250*exp(-DI/100), the -45 ° line represents equation DI+APD=290. The vertical line shows DI*, the fixed point value of DI. The L-shaped arrows illustrate how small value DI and APD to the left of DI* are followed by large value DI and APD to the right of DI*, and vice versa. The square represents the stable alternation between DI, APD pairs (62, 165) and (125, 228). Figure 2. Laddergrams showing two scenarios of discordant alternans initiation. The vertical axis represents 100 cells of a cable, and the horizontal axis represents time. Stimuli were applied at the top end of the cable. Bands of white and grey depict electrical diastole and action potential respectively. Boxed numerical values indicate DI and APD. Bold lines depict depolarization wavefronts. Computation of DI and APD was independent of values at adjacent points, i.e., lacked electrotonic effects. A: Ectopic focus scenario. The entire cable had depolarized and repolarized before S1 1. Conduction velocity was fixed, but increasing DI with distance from stimulus end provided the DI gradient necessary for discordant alternans initiation. B: Sinus node scenario. The cable was in quiescence before S1 1. Conduction velocity restitution provided the DI gradient necessary for discordant alternans initiation. The cell number at which APD values (in boxes) were the same for two consecutive beats gradually decreased during initiation of discordant alternans. Figure 3 Evolution of APD gradients in sinus node scenario, cable length 80 mm, BCL 310 ms. The gradient for even (hashed) and odd (solid line) beats are shown for stimulus numbers 2,3 ( A), 30, 31 ( B), 50, 51 ( C) and 150, 151 ( figure 5A ). The APD node (gradient intersection) moved leftwards with increasing stimulus number. Figure 4 Steady state distribution of voltage over a cable for a full alternans cycle at a BCL of 310 ms, for cable lengths of 80 ( left) and 135 mm ( right ). Voltage traces are17 plotted every 6 ms with upward shift for clarity. Arrows indicate time of stimulation. Figure 5 Steady state APD, cycle length, and conduction velocity gradients over a cable for even (hashed) and odd (solid line) beats. BCL was 310 ms. A, B, C : APD node position and number of APD nodes at 3 cable lengths 80, 117.5 and 135 mm. C, D, E : The relationship between APD gradient, cycle length gradient, and conduction velocity gradients for a cable length of 135 mm. Figure 6 Relationship between APD node position, number of nodes, BCL, and cable length at steady state. A. The parameter space is divided into zero (concordant), 1, 2, 3 and 4 node regimes. As cycle length is shortened for a given cable length, the number of nodes can increase. Electrocardiograms computed for BCL of 290 and 285 ms at tissue length of 30 mm are also shown. B. Node position as a function of cycle length for cable length of 140 mm. As cycle length was reduced, internodal distances decreased. Figure 7 The relationship between the standard (dotted line) and dynamic restitution curves during discordant alternans, (hashed line - even, solid line - odd beats), in a cable 62.5 mm long, for BCL of 305 ms. Large graph: APD restitution curves. Inset: Magnified view of conduction velocity restitution curves. Figure 8 APD gradients in an 80 mm x 80 mm 2-dimensional sheet produced by a point stimulus applied at the bottom left corner every 310 ms. Left to right - Top row: APD gradient for stimulus number 2, 20 and 52. Center row : APD gradient for stimulus numbers 3, 21 and 53. Bottom row : APD gradients on the diagonal of the 2-D sheets. The black lines in the 2-d sheets indicate APD node line position where APD was fixed for consecutive beats. When stimulation was continued, the APD node line became a quarter circle (a line equidistant from stimulus site).050100150200250300 0 50 100 150 200 250 300 DI (ms) Figure 1DI*APD*18 (DI , APD )(DI , APD ) s sl l(DI , APD ) (DI , APD )S SL LS1 S1 S1 S1 S1 S1 290 290 290 290 290 290 1 2 3 4 5 6 0 10 20 30 40 50 60 70 80 90 10062 165 125 228 62 165 125 228 62 165 125 228 91 199 91 199 91 199 91 199 91 199 91 199 125 228 62 165 125 228 62 165 125 228 62 165 0 10 20 30 40 50 60 70 80 90 100S1 S1 S1 S1 S1 S1 1 2 3 4 5 6 290 290 290 290 290 290 A B 192 192204 204 167 17019 Figure 2300 288 2 56 234 276 14 83 207 269 22 98 207 2070100200300APD (ms)0100200300APD (ms)0100200300APD (ms) 80 mmA B C20 Figure 380 mm21 Figure 4135 mm0100200300APD (ms) 0100200300APD (ms) 0100200300APD (ms) 300320CL (ms) 380480CV (mm/s)80 mm 117.5 mm 135 mm22 A Figure 5B C D E135 mm 135 mm2 4 6 8 10 12 14 L (cm)275285295305315CL (ms) 280 290 300 310 320 CL (ms)02468101214L (cm)Node 1 Node 2 Node 3 Node 4 012 3 4 ECG1 ECG2 Figure 6 23 A B0 100 200 3000100200300 50 100 150 200 250390410430450CV (mm/s)APD (ms) DI (ms) Figure 7 24 DI (ms)

arXiv:physics/0004012v1 [physics.pop-ph] 8 Apr 2000TWO NOVEL SPECIAL RELATIVISTIC EFFECTS: SPACE DILATATION AND TIME CONTRACTION J.H.Field D´ epartement de Physique Nucl´ eaire et Corpusculaire Univ ersit´ e de Gen` eve . 24, quai Ernest-Ansermet CH-1211 Gen` eve 4. Abstract The conventional discussion of the observed distortions of space and time in Special Relativity (the Lorentz-Fitzgerald Contraction a nd Time Dilatation) is ex- tended by considering observations, from a stationary fram e, of : (i) objects moving with constant velocity and uniformly illuminated during a s hort time τL(their ‘Lu- minous Proper Time’) in their rest frame; these may be called ‘Transient Luminous Objects’ and (ii) a moving, extended, array of synchronised ‘equivalent clocks’ in a common inertial frame. Application of the Lorentz Transfo rmation to (i) shows that such objects, observed from the stationary frame with c oarse time resolution in a direction perpendicular to their direction of motion ar e seen to be at rest but longer in the direction of the relative velocity /vector vby a factor 1 //radicalbig 1−(v/c)2(Space Dilatation) and to (ii) that the moving equivalent clock at a ny fixed position in the rest frame of the stationary observer is seen to be runnin gfaster than a similar clock at rest by the factor 1 //radicalbig 1−(v/c)2(Time Contraction). All four space-time ‘effects’ of Special Relativity are simply classified in term s of the projective geome- try of space-time, and the close analogy of these effects to li near spatial perspective is pointed out. PACS 03.30+p Published in: American Journal of Physics 68(2000), 267-274.1 Introduction In his 1905 paper on Special Relativity [1]Einstein showed t hat Time Dilatation (TD) and the Lorentz-Fitzgerald Contraction (LFC), which had pr eviously been introduced in a somewhat ad hoc way into Classical Electrodynamics, are simple consequenc es of the Lorentz Transformation (LT), that is, of the geometry of spa ce-time. As an example of the LFC Einstein stated that a sphere moving w ith velocity vwould, ‘viewed from the stationary system’, appear to be contracte d by the factor/radicalBig 1−(v c)2in its direction of motion where cis the velocity of light in free space. It was only pointed out some 54 years later that if ‘viewed’ was interpreted in th e conventional sense of ‘as seen by the eye, or recorded on a photograph’ then the sphere d oes not at all appear to be contracted, but is still seen as a sphere with the same di mensions as a stationary one and at the same position [2, 3, 4] ! It was shown in general [ 3, 4] that transversely viewed moving objects subtending a small solid angle at the o bserver appear to be not distorted in shape or changed in size, but rather rotated, as compared to a similarly viewed and orientated object at rest. This apparent rotatio n is a consequence of three distinct physical effects: (i) The LFC. (ii) Optical Aberration. (iii) Different propagation times of photons emitted by diffe rent parts of the moving object. The effect (ii) may be interpreted as the change in direction o f photons, emitted by a moving source, due to the LT between the rest frames of the sou rce and the stationary observer. Correcting for (ii) and (iii), the LFC can be deduc ed as a physical effect, if not directly observed. It was also pointed out by Weinstein [5] t hat if a single observer is close to a moving object then, because of the effect of light pr opagation time delays, it will appear elongated if moving towards the observer and con tracted (to an extent greater than the LFC) if moving away. Only an object moving strictly t ransversely to the line of sight of a close observer shows the LFC. However, the LFC itself is a physical phenomenon similar in m any ways to (iii) above. The human eye or a photograph taken with a fast shutter record , as a sharp image, the photons incident on it during a short resolution time τR. That is, the image corresponds to a projection at an almost fixed time in the frame S of observa tion. This implies that the photons constituting the observed image are emitte d at different times from the different parts, along the line of sight , of an extended object. As shown below, the LFC is similarly defined by a fixed time projection in the frame S. T he LT then requires that the photons constituting the image of a moving object are als o emitted at different times, in the rest frame S’ of the object, from the different parts along its direction of motion . In the following S will, in general, denote the reference fra me of a ‘stationary’ observer (space-time coordinates x,y,z,t) while S’ refers to the res t frame of an object moving with uniform velocity vin the direction of the positive x axis relative to S ( space-t ime coordinates x’,y’,z’,t’). 1The purpose of this paper is to point out that the t=constant projection of the LFC (see Section 2 below) and the x′=constant projection of TD (see Section 3 below) are not the only physically distinct space-time measurements poss ible within Special Relativity. In fact, as will be demonstrated below, there are two others: Space Dilatation (SD), the t′=constant projection and Time Contraction (TC), the x=constant projection. All four ‘effects’ are pure consequences of the LT. The additiona l effects of Optical Aberration and Light Propagation Delays on the appearance of moving obj ects and synchronised clocks have been extensively discussed elsewhere [6]. Although each of the four effects may be simply derived from th e projective geometry of the space time LT, the LFC and TD give rise to more easily obs ervable physical effects, so it is not surprising that they are better known, For exampl e the LFC is essential for the physical interpretation of the Michelson-Morley exper iment, and TD is necessary to describe the observed lifetimes of unstable particles deca ying in flight. In contrast, the two new effects SD and TC seem to have no similar simple observa tional consequences. As pointed out below, the most interesting effects are likely to result from SD, which is necessary to describe observations of, for example, a rot ating extended object moving with a relativistic transverse velocity. It is easy to conce ive a simple experiment involving the observation of two synchronised clocks in space, to test the TC effect. Although it is clearly of interest to work out in more detail such example s, there is no attempt to do so in the present paper, which is devoted to the precise defi nition of the four possible space-time projections of the LT and a discussion of their in terrelations. Thet=constant projection of the LFC is the space-time measurement appropr iate to the ‘moving bodies’ of Einstein’s original paper and to th e photographic recording technique. This medium has no intrinsic time resolution and relies on that provided by a rapidly moving shutter to provide a clear image. The LFC ‘wo rks’ as a well defined physical phenomenon because the ‘measuring rod’ or other ph ysical object under obser- vation is assumed to be illuminated during the whole time int erval required to make an observation, and so constitutes a continuous source of emit ted or reflected photons, such that some are always available in the different space (∆ x′) and time (∆ t′) intervals in S’ for every position of the rod corresponding to the time inter val ∆t=τRaround the fixed timetin the observer’s frame S during which the observation is mad e. If, however, the physical object of interest has internal motion (rotation, expansion or contraction) or is only illuminated, in its rest frame S’, during a short time in terval, the above conditions, that assure that the t=constant projection gives a well defined space-time measurement no longer apply. All such objects, uniformly illuminated fo r a restricted time τL(their ‘Lu- minous Proper Time’) in their rest frames, may be called ‘Tra nsient Luminous Objects’. For such objects it is natural to define a length measurement b y taking the t′=constant projection in S’. The observation, from the stationary fram e S, of such objects is discussed in Section 2 below. In Section 3 time measurements other than the conventional T D of Special Relativity are considered. The TD phenomenon refers only to a local cloc k, in the sense that its position in the frame S’ is invariant (say at the spatial orig in of coordinates x′= 0). However, the time recorded by any synchronised clock in the s ame inertial frame is, by definition, identical. Einstein used such an array of ‘equ ivalent clocks’ situated at different positions in the same inertial frame in his origina l discussion of the relativity 2of simultanaeity [1]. The question addressed in Section 3 is : What will an observer in S see if he looks not only at a given local clock in S’, but also a t other, synchronised, equivalent clocks at different positions in S’, in compariso n to a standard clock at rest in his own frame? It is shown that such equivalent clocks may be s een to run slower than, or faster than, the TD prediction for a local clock. In partic ular they may even appear torun faster than the standard clock . This is an example of the Time Contraction effect mentioned above. In Section 4 the analogy between the Lorentz-Fitzgerald Con traction effect and linear perspective in two spatial dimensions is described. The fina l Section points out how all four space-time ‘effects’ (observed distortions of space-t ime) in Special Relativity may be described in a unified way in terms of projective geometry, in close analogy with the effect of linear perspective in the perception of space. 2 Observation of Transient Luminous Objects in Mo- tion: the Space Dilatation Effect Consider a square planar object centered at the origin of the moving frame S’ as shown in Fig 1a. The points P’( x′=−L/2,y′= 0) and Q’( x′=L/2,y′= 0) lie on the vertical edges of the square of side Lwhose boundary is shown in Fig 1a as the short dashed lines. Suppose now that the square is uniformly illum inated in the time interval −τL/2< t′< τL/2 to give the ‘Transient Luminous Object’ indicted by the zig -zag lines. The proper time interval τLis the ‘Luminous Proper Time’ of the object. For example, the surface of the square may be covered with a mosaic of light -emitting diodes that are simultaneously switched on during a time τL. The object as seen by an observer, at rest in the stationary system S, viewing the object in a direction perpendicular to the plane O′x′y′, is given by the LT connecting space time points in the frame S ’ to those in S: x=γ(x′+vt′) (2.1) t=γ(t′+βx′ c) (2.2) where β≡v/c, γ ≡1√1−β2. It is assumed that the stationary observer is sufficiently dis tant from the object that the effects of light propagation times are negligible, and that t he object is diffusely illuminated so that Optical Aberration effects may be neglected [6]. In th is case any changes in the appearance of the moving object when viewed from the frame S a re due solely to the LT. The results of the transformation for t′= 0 and x′=−L/2,0, L/2 are given in Table 1. It can be seen that the points P’,O’,Q’ are observed at different times in the frame S. This is the well known effect of the relativity of simultanaeity fir st pointed out in Einstein’s classic paper [1]. It can also be seen from Table 1 that the dis tance between the positions of P’ and Q’ as observed in S is γL; that is, the object will appear to be elongated if it is viewed with a time resolution larger than the difference in time, γβL/c , between the 3y' ya) b)0' 0P' Q' x' x γ Lvelocity = c / βL δ Figure 1: a) A square ‘Transient Luminous Object’ (indicate d by the zig-zag outline) as viewed in the frame S’ in which it is at rest. b) The same object as viewed at a fixed time from the frame S moving with velocity - βcparallel to the Ox’ axis. It is assumed that the Luminous Proper Time τLof the object is small: τL≪βL/c. The actual outlines of the objects are shown as short dashed lines. The long dashed r ectangle of width γLin b) shows the outline of the object when viewed with coarse time r esolution: τR≫γβL/c . The observer in S is assumed to be sufficiently distant from the object that the effects of light propagation delays may be neglected. The object is a lso assumed to constitute a diffuse photon source so that Optical Aberration effects are n egligible. 4observations in S of the space time points P’ and Q’ that are si multaneous in the frame S’. This is the ‘Space Dilatation’ (SD) effect. It will now be d iscussed in more detail, taking into account the non zero Luminous Proper Time τLof the Transient Luminous Object as well as the Resolution Time τRof the observer, so that the general conditions under which the SD effect occurs are established. Space time points of the Transient Luminous Object may be obs erved at the fixed timetin S provided that: x′ MIN< x′< x′ MAX where t=γ(−τL 2+βx′ MAX c) =γ(τL 2+βx′ MIN c) (2.3) In (2.3) it is assumed that x′ MIN>−L/2,x′ MAX< L/2. The general condition relating τL,L,vandcensuring the validity of this assumption will be discussed b elow. Using (2.1) the co-ordinates in S corresponding to x′ MINandx′ MAXare found to be: xMAX =c β(t+τL 2γ) (2.4) xMIN =c β(t−τL 2γ) (2.5) Thus the width δof the Transient Luminous Object observed at time tin S (indicated by the zig-zag lines in Fig1b; the actual boundary is shown by th e short dashed lines) is: δ≡xMAX−xMIN=cτL βγ, (2.6) while, as can be seen from (2.4) and (2.5), the observer in S se es a luminous object that moves with velocity c/β, i.e. faster than than the velocity of light. In the case of continous illumination of the object ( τL→ ∞) the upper and lower limits of the object observed at the fixed time tin S will correspond to the physical boundaries x′ MIN=−L/2, x′ MAX=L/2. Denoting by t′ MIN,t′ MAXthe times in S’ corresponding to the observation of these boundaries at time tin S, then, instead of (2.3), the following relation is obtai ned: t=γ(t′ MIN+βL 2c) =γ(t′ MAX−βL 2c) (2.7) Using (2.1), the boundaries of the object observed in S at tim etare then: xMAX =L 2γ+vt (2.8) xMIN =−L 2γ+vt (2.9) The width of the object as seen in S is then xMAX−xMIN=L/γ, the well known LFC effect. As can be seen from (2.8) and (2.9) the object is now obs erved to move in S with velocity v. Thus, in the limit τL→ ∞ (continous illumination of the object) the usual results of Special Relativity are recovered. Using (2.1) and (2.2) the upper (U) and lower (L) limits of the space time region in the stationary frame S swept out by the moving Transient Lumi nous Object in Fig 1b 5Point x’ t’ x t P’−L 20 -γL 2-γβL 2c O’ 0 0 0 0 Q’L 20γL 2γβL 2c Table 1: Space-time points on the object at rest in S’ (see Fig . 1), at time t′= 0, as observed in the frames S’,S. are: xU=γ 2(L+vτL) (2.10) xL=γ 2(−L−vτL) (2.11) tU=γ 2(τL+βL c) (2.12) tL=γ 2(−τL−βL c) (2.13) Taking account of the inequality: βL c<L v, it can be seen that if τL≪βL/cthe terms containing τLin (2.10)-(2.13) may be neglected, so that: xU−xL≃γL (2.14) tU−tL≃γβL c(2.15) Thus the Space Dilatation effect of Table 1 is recovered in the limitτL→0. On the other hand, because of the inequality: vτL<cτL β, then, if τL≫L/v, the terms containing Lin (2.10)-(2.13) may be neglected, leading to the relations: xU−xL≃γvτL (2.16) tU−tL≃γτL. (2.17) These are the well-known equations of Special Relativity de scribing the motion of a small continously illuminated object as observed in the fra me S. The time interval tU−tL corresponds to the TD effect and the observed velocity is: (xU−xL)/(tU−tL) =γvτL/γτL=v. (2.18) The conclusions of this Section are now summarised. When τL≪βL/c a stationary observer in S with a time resolution τR≪γβL/c , viewing the object in the direction 6transverse to the relative velocity, sees the square object at rest in S’ illuminated during the proper time interval τLas a narrow rectangular object of width δ=cτL/(βγ) moving with velocity c/βand sweeping out during the time γβL/c a region of total length γL. If however the resolution time τRof the observer is much larger than γβL/c the object will appear at rest but elongated by the factor γin the direction of motion . This is the Space Dilatation effect. In the contrary case that the Lumino us Proper Time τLis large (τL≫L/v), the object observed from S moves with velocity vand has an apparent length L/γdue to the well known LFC effect. Also, in this case, the elapse d times in S and S’ are related by the TD effect (Eqn. 2.17). It should be noted that the ‘narrow rectangular object’ refe rred to above corresponds to the case of uniform illumination of the square object. Act ually, because of the relativity of simultaneity, different parts of the square are seen at diff erent times and positions by the stationary observer. If the square were illuminated usi ng different colours: red, yellow, green, blue in four equal bands parallel to the y′axis, in the direction of increasing x′, then the moving object in Fig 1b would appear red during the ti me interval −γβL/2c < t < γβL/ 4c, yellow during the time −γβL/4c < t < 0, and so on. The colours will, of course, be seen shifted in frequency according to the relati vistic transverse Doppler effet. If the square is rotated about the y′axis by an angle α, a subtle interplay occurs between the effects of the LT and light propagation time delay s. Depending on the values ofvandαthe rectangular object may be seen, by an observer at rest in t he frame S, to move parallel to /vector v(as in the case α= 0 described above), antiparallel to /vector v, or may even even be stationary and of length γLcosα. In all cases the total length swept out by the object in the direction of motion is γLcosα. These effects have been described in detail elsewhere [6]. 3 Observation of an Array of Equivalent Moving Clocks: The Time Contraction Effect In this Section space time measurements of an array of synchr onised clocks situated in the inertial frame S’ will be considered. These clocks may be synchronised by any convenient procedure [7] (see for example Ref.[1]). For an o bserver in S’ all such clocks are ‘equivalent’ in the sense that each of them records, inde pendently of its position, the proper time τ′of the frame S’. For convenience, the array of clocks is assum ed to be placed on the wagons of a train which is at rest in S’, as shown i n Fig.2a. The clocks are labelled Cm, m=...−2,−1,0,1,2, ...and are situated (with the exception of the ‘magic clock’ CM, see below) at fixed distances Lfrom each other, along the Ox’ axis, which is parallel to the train. It is assumed that the observers in the frames S and S’ view the train transversely at a sufficiently large distance that the e ffects of light propagation time delays may be neglected. It is clear that by considering the l imitL→0 an Equivalent Clock may be associated with each position on the train and, b y extending the ‘lattice’ of clocks to 3 dimensions, to any spatial position in S’. The observer in S’ will note that each Equivalent Clock (EC) i ndicates the same time, as shown in Fig.2a. It is now asked how the array of EC will appe ar to an observer at 7L L/γ L/γLS', t'=0 S, t=0C-1 C-1CM CMC0 C0 CSC1 C1y' L/(1+γ)0' x' 0'x'a) b) 0 xβ=0.6, γ=1.25 Figure 2: a) Positions and times of equivalent clocks on the w agons of a train as seen by a distant observer in the rest frame S’ of the train at time t′= 0. b) The postions and times of the same clocks as seen by a distant observer in S at ti met=t′= 0. The same remarks concerning light propagation time and Optical Aber ration effects as made in the caption of Fig.1 apply. 8M M Figure 3: a) Times of equivalent clocks on the train ( C−1,CM,C0,C1) and the stationary clockCS, as seen by a distant observer in S at time t= 0. b) The same, at time t=τ. It is assumed that β= 0.6,γ= 1.25. 9CSC−2 C−1 CMC0 C1 C2 02(γ2−1) γτ(γ2−1) γτ(γ−1) γτ0−(γ2−1) γτ−2(γ2−1) γτ τ(2γ2−1) γτ γτ ττ γ−(γ2−2) γτ−(2γ2−3) γτ Table 2: Times observed in S of Equivalent Clocks on the movin g train in Fig.2, at the times t= 0 and t=τof the stationary standard clock CS. a fixed position in the frame S when the train is moving with vel ocityβcparallel to the direction Ox in S (Fig.2b). It is assumed that the EC C0is placed at x′= 0 and that it is synchronised with the Standard Clock CS, placed at x= 0 in S, when t=t′= 0. All the clocks are similar, that is CSand each Cmrecord exactly equal time intervals when they are situated in the same inertial frame. The appearence of the moving array of EC to an observer in S at t= 0 is shown in Fig.2b, and in more detail in Fig.3 for both t= 0 and t=τ. The period τis the time between the passage of successive EC past CS. The big hand of CSin Fig.3 rotates through 180◦during the time τ. Explicit expressions for the observed times are presented in Table 2. In Fig.2b,3 the times indicated by the clocks are shown for β= 0.6. These times are readily calculated using the LT equations (2.1),(2.2). Con sider the time indicated by C1 att= 0. The space-time points are: S′: (L, t′) ;S: (x,0) Hence, Eqns.(2.1),(2.2) give: x=γ(L+vt′) (3.1) 0 = γ(t′+βL c) (3.2) which have the solution [ C1(t= 0) ]: t′=−βL c(3.3) x=L γ(3.4) As shown in Fig 2b, the wagons of the train appear shorter due t o the LFC effect (Eqn.(3.4)) and also the wagons at the front end of the train are seen at an earlier p roper time than those at the rear end . Thus a t= 0 snapshot in S corresponds, not to a fixedt′in S’ but one which depends on x′:t′=−βx′/c. This is a consequence of the relativity of simultaneity of space-time events in S and S’, as first pointed out by Einstein in Ref.[1]. Here it appears in a particularly graphic and str iking form. Consider now the time indicated by C−1att=τ, i.e. when C−1is at the origin of S. The space-time points are: S′: (−L, t′) ;S: (0, τ) 10Hence, Eqns.(2.1),(2.2) give: 0 = γ(−L+vt′) (3.5) τ=γ(t′−βL c) (3.6) with the solutions [ C−1(t=τ) ]: t′=L v(3.7) τ=L γv=t′ γ(3.8) so that t′=γτ (3.9) The EC at the origin of S at t=τshows a later time than CSi.e. it is apparently running faster thanCS. This is an example of Time Contraction (TC). The Time Contraction effect is exhibited by the EC observed at any fixed position in S . In fact, if the observer in S can see the EC only when they are near to CShe (or she) will inevitably conclude that the clocks on the train run fast, not slow as in the classi cal TD effect (see below). Suppose that the observer is sitting in a waiting room with th e clock CSand notices the time on the train (the same as CS) by looking at C0as it passes the waiting room window. If he (or she) then compares C−1as it passes the window with CSit will be seen to be running fast relative to the latter. In order to see the TD effe ct the observer would (as will now be shown), have to note the time shown by, for example ,C0, at time t=τas recorded by CSin comparison with that shown by the same clock C0att= 0. Using Eqn.(3.8),Eqn.(3.3) may be written as [ C1(t= 0)]: t′=−β2γτ=−(γ2−1)τ γ(3.10) This is the formula for the observed time reported in Table 2. Now consider C0at time t=τ. The space-time points are: S′: (0, t′) ;S: (x, τ) Hence, Eqns.(2.1),(2.2) give: x=γvt′(3.11) τ=γt′(3.12) with the solutions [ C0(t=τ) ]: t′=τ/γ (3.13) x=vτ=L/γ (3.14) Sothe EC C0at time t=τindicates an earlier time, and so is apparently running slow er thanCS. This is the classical Time Dilatation (TD) effect. It applie s to observations of alllocal clocks in S’ ,(i.e. those situated at a fixed value of x′) as well as any other EC that has the same value of x′. 11As a last example consider the ‘Magic Clock’ CMshown in Fig 2a at time t=τ. With the space-time points: S′: (−L/(1 +γ), t′) ;S: (x, τ) Eqns.(2.1),(2.2) give: x=γ[−L/(1 +γ) +vt′] (3.15) τ=γ[t′−β cL/(1 +γ)] (3.16) with the solutions [ CM(t=τ) ]: t′=τ (3.17) x=γvτ/(1 +γ) (3.18) where the relation L=γvτfrom Eqn.(3.8) has been used. ThusCMshows the same time asCSatt=τ. Similar moving ‘Magic Clocks’ can be defined that show the sa me time as CSat any chosen time tin S. Such a clock is, in general, situated at x′=−ct(γ−1)/βγ. All of the other clock times presented in Table 2 and shown in F igs. 2b, 3 are calculated in a similar way to the above examples by choosing appropriat e values of x′and t. The combined effects of the LT and light propagation delays fo r light signals moving parallel to the train (corresponding to observations of the array of Equivalent Clocks by observers on, or close to the train) have been described in detail elsewhere [6]. The observed spatial distortions of the train in this situation were previously considered by Weinstein [5]. 4 Analogy with Linear Perspective in Two Dimen- sional Space The analogy between the observed distortions of space-time in Special Relativity and linear spatial perspective is illustrated in Fig.4. The ‘Ob ject Space’ on the right is sepa- rated from the ‘Image Space’ on the left by a plane partition c ontaining a small aperture (pin hole). Light reflected from the rod PQ in the Object Space , can pass through the pin hole and produce an image on a screen located in the Image S pace. To facilitate the comparison with the Lorentz Transformation the cartesian a xes in the Object [Image] space are denoted by (X’,T’) [ (X,T)] respectively (see Fig. 4). The T, T’ axes are per- pendicular to the plane of the partition and pass through the pin hole. The Object Space is now compared to the rest frame S’ of the moving object, with the correspondences: X′⇔x′, T′⇔t′, while the Image Space is compared to the frame S of the station ary observer with the correspondences: X⇔x, T⇔t, 12IMAGE SPACE OBJECT SPACE SCREENPLANE PARTITION WITH PIN-HOLE AT 0X T T'X' P' PQ' Qτ'τ 0ROD Figure 4: An example of linear spatial perspective analagou s to the Lorentz-Fitzgerald Contraction effect. An arbitary point with coordinates (X’,T’) on the rod will pr oject into the Image Space the line: X=−X′T τ′(4.1) which may be compared to the LT equation: x=1 γx′+vt (4.2) Taking the T=τ=constant projection in (4.1), i.e. setting the screen in the image space parallel to the planar surface at the distance τfrom it, gives for the length LIof the image of the rod: LI=X2−X1=τ τ′(X′ 1−X′ 2) =τ τ′L (4.3) where the points 1,2 denote the ends of the rod or of its image. Similarly taking the t=constant projection in (4.2) gives, for the apparent length lIof a rod, parallel to the x axis, of true length l: lI=x1−x2=1 γ(x′ 1−x′ 2) =l γ(4.4) corresponding to the LFC effect. The rˆ ole of the factor 1 /γin the LT is replaced, in the case of linear perspective, by the ratio τ/τ′that specifies the relative position and orientation of the object and the screen on which it is observ ed. 135 Discussion The different space-time effects (observed distortions of sp ace or time) in Special Rel- ativity that have been discussed above are summarised in Tab le 3. These are the well- known LFC and TD effects, Space Dilatation (SD) introduced in Section 2 above, and Time Contraction (TC) introduced in Section 3. Each effect is an observed difference ∆ q (q=x, x′, t, t′) of two space or time coordinates (∆ q=q1−q2) and corresponds to a constant projection ˜ q=constant , i.e. ∆˜ q= 0 (˜q/negationslash=q), in another of the four variables x,x′,t,t′of the LT. As shown in Table 3, the LFC, SD, TC and TD effects corr espond, respectively, to constant t,t′,xandx′projections. After making this projection, the four LT equations give two relations among the remaining thr ee variables. One of these describes the ‘space-time distortion’ relating ∆ t′and ∆ tor ∆x′and ∆ xwhile the other gives the equation shown in the last column, (labelled ‘Comp lementary Effect’) in Table 3. These equations relate either ∆ xto ∆t(for SD and TD) or ∆ x′to ∆t′(for LFC and TC). It can be seen from the Complementary Effect relations th at the two space-time points defining the effect (of space-time distortion) are spa ce-like separated for LFC and SD and time-like separated for TC and TD. For example, for the LFC when t1=t2=t, the LT equations for the two space-time points are: x′ 1=γ(x1−vt) (5.1) x′ 2=γ(x2−vt) (5.2) t′ 1=γ(t−βx1 c) (5.3) t′ 2=γ(t−βx2 c) (5.4) Subtracting (5.1) from (5.2) and (5.3) from (5.4) gives: ∆x′=γ∆x (5.5) ∆t′=−γβ c∆x (5.6) Eqn.(5.5) describes the LFC effect, while combining Eqns.(5 .5) and (5.6) to eliminate ∆ x yields the equation for the Complementary Effect. By taking o ther projections the other entries of Table 3 may be calculated in a similar fashion. It i s interesting to note that the TD effect can be derived directly from the LFC effect by using th e symmetry of the LT equations. Introducing the notation: s≡ct, the LT may be written as: x′=γ(x−βs) (5.7) s′=γ(s−βx) (5.8) These equations are invariant [8] under the following trans formations: T1 : x↔s, x′↔s′(5.9) T2 : x↔x′, s↔s′, β→ −β (5.10) Writing out the LFC entries in the first row of Table 3, replaci ngt,t′bys/c,s′/c; gives ∆x∆s= 0 ∆ x=∆x′ γ∆x′=−∆s′ β 14Applying T1 to each entry in this row results in: ∆s∆x= 0 ∆ s=∆s′ γ∆s′=−∆x′ β Applying T2: ∆s′∆x′= 0 ∆ s′=∆s γ∆s=∆x β Replacing ∆ s, ∆s′byc∆t,c∆t′yields the last row of Table 3 which describes the TD effect. Similarly TC can be derived from SD (or vice versa) by s uccessively applying the transformations T1,T2. The ‘Complementary Effects’ listed in Table 3 have the follow ing geometrical inter- pretations: •LFC (∆ x′=−(c/β)∆t′). This is the locus of all the points in S’ that are observed at the same time (∆ t= 0) in S. •SD (∆ x= (c/β)∆t). The locus of the moving object as observed in S (see Fig1b). •TC (∆ x′=−cβ∆t′). The locus of the position of the local clock in S’ observed a t a fixed position (∆ x= 0) in S. •TD (∆ x=cβ∆t). The locus of the position of the moving local clock observe d in S. A remark on the ‘Observed Quantities’ in Table 3. For the LFC, SD effects the observed quantity is a length interval in the frame S. The obs erved space distortion occurs because this length differs from the result of of a similar mea surement made on the same object in its own rest frame. ∆ x′is not directly measured at the time of observation of the LFC or SD. It is otherwise with the time measurements TD , TC. Here the time intervals indicated in their own rest frame by a local moving clock (TD), or different equivalent clocks at the same position in S (TC), are suppose d to be directly observed and compared with the time interval ∆ tregistered by an unmoving clock in the observer’s rest frame. Thus the effect refers to two simultaneous observ ations by the same observer not to separate observations by two different observers as in the case of the LFC and SD. Einstein’s first paper on Special Relativity [1] showed, for the first time, that the LFC and TD effects could be most simply understood in terms of the g eometry of space time, in contrast to the previous works of Fitzgerald, Larmor, Lor entz and Poincar´ e where dy- namical and kinematical considerations were always mixed [ 9]. However it can also be argued that Special Relativity has a dynamical aspect due to the changes in the electro- magnetic field induced by the LT. Indeed, by calculations of t he equilibrium positions of an array of point charges in both stationary and uniformly mo ving frames Sorensen has shown that the LFC may be derived from dynamical considerati ons [10]. By considering several different ‘electromagnetic clocks’ either at rest o r in uniform motion, Jefimenko has demonstrated that the TD effect may also be dynamically de rived [11]. Similar con- siderations, emphasising the ‘dynamical’ rather than the ‘ kinematical’ aspects of Special Relativity, have been presented in an article by Bell [12]. S uch calculations, based on the 15Name Observed Quantity Projection Effect Complementary Effect Lorentz-Fitzgerald Contraction (LFC)∆x ∆t= 0 ∆x=1 γ∆x′∆x′=−c β∆t′ Space Dilatation (SD)∆x ∆t′= 0 ∆x=γ∆x′∆x=c β∆t Time Contraction (TC)∆t′∆x= 0 ∆t′=γ∆t ∆x′=−cβ∆t′ Time Dilatation (TD)∆t′∆x′= 0 ∆t′=1 γ∆t ∆x=cβ∆t Table 3: Different observed distortions of space-time in Spe cial Relativity (see text). properties of electromagnetic fields under the LT, demonstr ate the consistency of Clas- sical Electromagnetism with Special Relativity, but as poi nted out by Bell [12], in no way supersede the simpler geometrical derivations of the eff ects. It is not evident to the present author how similar ‘dynamical’ derivations of the n ew SD and TC effects could be performed. In conclusion the essential characteristics of the two ‘new ’ space-time distortions dis- cussed above are summarised : •Space Dilatation (SD): If a luminous object lying along the Ox’ axis, at rest in the frame S’, is uniformly illuminated for a short time τLin this frame it will be observed from a frame S, in uniform motion relative to S’ para llel to Ox’ at the velocity - βc, in a direction perpendicular to the relative velocity, as a narrow strip of width cτL/(βγ), perpendicular to the x-axis, moving with the velocity c/βin the same direction as the object. The total distance swept out al ong the x-axis by the strip during the time βL/(c√1−β2), for which it is visible, is L/√1−β2where L is the length along Ox’ of the object as observed in S’. Thus th e apparent length of the object when viewed with a time resolution τRmuch larger than βL/(c√1−β2) isL/√1−β2. •Time Contraction (TC): The equivalent clocks in the moving frame S’, viewed at the same position in the stationary frame S, apparently run f aster by a factor 1/√1−β2relative to a clock at rest in S. Acknowledgements I thank G.Barbier and C.Laignel for their valuable help in th e preparation of the figures, and an anonymous referee whose pertinent and constr uctive criticism has allowed me to much improve the presentation of Section 2. 16References [1] A.Einstein,‘Zur Elektrodynamik bewegter K¨ orper’, An nalen der Physik 17(1905), 891. [2] J.Terrell, ‘Invisibility of the Lorentz Contraction’ P hys. Rev. 116(1959), 1041-1045. [3] R.Penrose, Proc. Cambridge Phil. Soc. 55(1959), 137. [4] V.F.Weisskopf, ‘The Visual Appearance of Rapidly Movin g Objects’, Physics Today, Sept. 1960 pp24-27. [5] R.Weinstein,‘Observation of Length by a Single Observe r’, Am. J. Phys. 28(1960), 607-610. [6] J.H.Field,‘Space Time Measurements in Special Relativ ity’ University of Geneva pre- print UGVA-DPNC 1998/04-176 April 1998, physics/9902048. Published in the Pro- ceedings of the XX Workshop on High Energy Physics and Field T heory, Protvino, Russia, June 24-26 1997. Edited by I.V.Filimonova and V.A.P etrov pp214-248. [7] If an observer in S’ knows the distance Dto any of the clocks then the clock is synchronised relative to a local clock at the same position a s the observer, when it is observed to lag behind the latter by the time D/cwhen viewed across free space. [8] Actually the transformation T2 yields the inverse of the LT (5.7),(5.8). The inverse equations may then be solved to recover (5.7) and (5.8). [9] A detailed discussion of the important differences betwe en Einstein’s theory of Special Relativity, as presented in Reference[1] above, and relate d work of Fitzgerald, Lorentz and Poincar´ e is given in Chapters 7 and 8 of: A.Pais, ‘Subtle is the Lord, the Science and Life of Albert Einstein’, Oxford University Press (1982 ). [10] R.A.Sorensen, ‘Lorentz contraction, a real change of s hape’, Am. J. Phys. 63(1995), 413-415. [11] O.D.Jefimenko,‘Direct calculation of time dilation’, Am. J. Phys. 64(1996), 812-814. [12] J.S.Bell,‘How to teach special relativity’, in ‘Speak able and Unspeakable in Quantum Mechanics’, Cambridge University Press, (1987), pp67-80. 17

1 Efficient Methods for Handling Long-Range Forces in Particle-Particle Simulations Hans Fangohr† ‡ fangohr@soton.ac.uk Andrew R. Price‡ arp97r@ecs.soton.ac.uk Simon J. Cox‡ sc@ecs.soton.ac.uk Peter A.J. de Groot† pajdeg@phys.soton.ac.uk Geoffrey J. Daniell† gjd@phys.soton.ac.uk †Department of Physics and Astronomy ‡Department of Electronics and Computer Science University of Southampton, Southampton, SO17 1BJ, UK Keywords: Infinite lattice summation, Cut-off, Long-range forces, Molecular Dynamics, Monte Carlo, Periodic boundary conditions. Subject Classification: 65C05, 70F10, 82B80 ABSTRACT A number of problems arise when long-range forces, such as those governed by Bessel functions, are used in particle-particle simulations. If a simple cut-off for the interaction is used, the system may find an equilibrium configuration at zero temperature that is not a regular lattice yet has an energy lower than the theoretically predicted minimum for the physical system. We demonstrate two methods to overcome these problems in Monte Carlo and molecular dynamics simulations. The first uses a smoothed potential to truncate the interaction in a single unit cell: this is appropriate for phenomenological characterisations, but may be applied to any potential. The second is a new method for summing the unmodified potential in an infinitely tiled periodic system, which is in excess of 20,000 times faster than previous na ïve methods which add periodic images in shells of increasing radius: this is suitable for quantitative studies. Finally we show that numerical experiments which do not handle the long-range force carefully may give misleading results: both of our proposed methods overcome these problems. FANGOHR, PRICE, COX, DE GROOT, AND DANIELL 2 1. INTRODUCTION Considerable effort has been invested in handling long-range forces for particle-particle simulations. The conventional cut-off approach truncates the potential in a single unit cell for separations greater than half the system dimension. In general it is better to sum the potential over a number of repeats of the unit cell. Infinite summation methods include the Ewald summation [1, 2, 3], multipole methods [4], lattice summation methods [5], the Lekner summation method [6, 7] and a novel method for logarithmic interactions [8]. In this paper we review some of the problems which can occur when the potential is na ïvely truncated, which have not previously been widely reported in the literature. We then derive two methods which overcome these problems. The first is suitable for phenomenological studies of systems and smooths the potential within a single unit cell. The second is a new real-space summation method appropriate for potentials governed by Bessel functions. This provides a speed-up of at least 20,000 compared to the current method of summing in a series of shells of increasing radius [9]. In section 2 we introduce our model system, which is a simulation of a layered superconductor. We discuss the problems which arise with cutting off this potential in a single unit cell in section 3, and give a simple method of smoothing the potential which overcomes these problems in section 4. In section 5 we consider an infinitely tiled periodic system and derive our new summation method. Section 6 describes a simulation of shearing a superconductor lattice using our new methods and constrasts it with the results obtained when the potential is cut-off. We draw our conclusions in section 7. 2. MODEL SYSTEM We will consider the long-range forces which arise in the simulation of pancake vortices in layered high-temperature superconductors [10]. The potential is governed by [9, 11, 12, 13, 14]:   =λrKcrU 0)( , (1) where λ is the penetration depth of the magnetic field, r is the distance between the particles and c is a constant. This may be approximated as ()() () λλλπλ <<∞→  +−=rr crU rr r 12.0lnexp )(21 2 . (2) Since λ can be several orders of magnitude larger than r [9], the K0 potential has a very long range character. It is therefore necessary to either (i) only consider the interaction inside a single unit cell which contains a large number of particles, or (ii) sum the interaction over period repeats of the unit cell. Our findings are also of relevance to the simulation of other systems governed by long-range forces such as the interaction of electrically charged rods [8]. We will show results for Monte Carlo and Molecular Dynamics simulations where the two-dimensional unit cell geometry can be chosen to be a rectangle, a parallelogram or a hexagon. In all cases periodic boundary conditions are employed. 3. CUT-OFF POTENTIAL The standard approach is to cut-off the potential to be constant outside a circle of radius equal to min( Lx / 2, Ly / 2), where Lx and Ly are the lengths of the sides of the unit cell. Since the force is the gradient of the potential, it is zero outside the cut-off radius. We then define the distance between particles, r, to be the minimum image distance [3]. In figure 1 the real force dependence F(r) is compared to that for a simulation system with a simple geometrical cut- off. For vortices in superconductors, Abrikosov [15] demonstrated theoretically that the lowest energy configuration for an infinite lattice is the hexagonal lattice, or so-called Abrikosov lattice, with an associated Abrikosov lattice energy. However, when using a sharp cut-off in our simulations we find many configurations with energies lower than the Abrikosov lattice energy. Figure 2 shows the results from a Molecular Dynamics simulation of a small number of particles in which the temperature in the system is cycled from 0K to half the melting temperature of the vortex solid and is then returned to 0K. The temperature is introduced via a stochastic noise term. The Delaunay triangulation of the vortex configuration at the end of the simulation in figure 2 is elastically deformed. Detailed examination of the EFFICIENT METHODS FOR HANDLING LONG-RANGE FORCES 3 triangulation shows that the elastic deformations arise due to particles gathering on the boundaries of the cut-off circles. In this position they minimise their contribution to the energy (or force) in the system. This gives rise to the “wavy lines ” visible in figure 2, with a curvature characterised by the cut-off radius. To demonstrate this, we have shown the cut-off circles corresponding to two of the particles. The wavy lines are less evident in larger systems, since their curvature is inversely proportional to the cut-off radius. If the system is heated above its melting temperature and then annealed slowly the final equilibrium state (i) has an energy lower than the Abrikosov energy, and (ii) contains topological defects. A topological defect is a particle which does not have six nearest neighbours in the Delaunay triangulation. We have repeated these results for Molecular Dynamics and Monte Carlo simulations with up to 2000 particles. The result in figure 3 for a Monte Carlo simulation of a system annealed from a liquid state exhibits low energy and contains defects. We have verified that our results are independent of the geometry of the unit cell (rectangular, parallelogram, or hexagonal). These problems are clearly artificial, and are caused by imposing a sharp cut-off on the very long range nature of the interaction. Since the penetration depth, λ, is generally much larger than the lattice spacing it would require systems with several hundred thousand particles before the effects of this finite size problem began to become less significant. Methods to deal with such large systems with the Bessel function interaction potential are currently being developed [16]. In studies of high temperature superconductors, interest has recently developed in the formation of topologically ordered states which exhibit quasi-long range translational order: the so-called Bragg glass. These states occur when the vortices are weakly pinned and have been investigated both theoretically and experimentally [17, 18]. Other studies have focussed on the structural properties of the dynamics of vortex systems [19, 20]. In both cases it is important that the ground state for an unpinned system should be a hexagonal lattice without topological defects. Furthermore for the calculation of numerical phase diagrams as a function of disordering pinning, it is vital that the disorder is not introduced by the model itself. We therefore propose two methods which avoid the problems described above. The first involves modifying the potential near to the cut-off, and allows qualitative simulation of small systems using only a single unit cell. The second is a new fast summation method to allow the infinitely tiled periodic system to be considered and allow quantitative simulations to be performed. 4. SMOOTHED POTENTIAL In figure 4 (left) we show the force field experienced by a vortex due to its surrounding particles in a hexagonal configuration. The discontinuities are caused by the artificial step in the force function shown in figure 1. It is natural to introduce a smoothed potential, which reduces the force smoothly to zero over a region from r fade to rcut-off, and we impose C1 continuity of the force at r = rfade and r = rcut-off. The smoothed potential is shown in figure 1, with the resulting smooth force field in figure 4 (right). The smoothing distance rcut-off - rfade is a free parameter which should be kept as small as possible to maintain the original force over the largest possible range. Numerical experiments show that three lattices spacings is sufficient. Figure 5 shows the results of a Monte Carlo simulation using a similarly smoothed energy. Simulations using this modified potential do not find configurations below the Abrikosov energy and topological defects only occur when the system is annealed very rapidly. The interpretation is that due to the slow force change at the cut-off (enforced by the derivative being zero) a particle pair separated by a distance of roughly r cut-off experiences continuous and small changes in force if their positions are perturbed. This is in contrast to the large discontinuous fluctuations, which can enable the system to discover configurations with energies less than the Abrikosov energy. We have also used interpolating polynomials of higher order and an exponential function in the smoothing region: in all cases the system does not discover energy states below the Abrikosov energy. It is important to consider whether the modification of the original force with the smooth cut-off affects the system ’s behaviour. Using a cut-off to the long-range interaction is a major change of the long-range interaction. However, introducing the smoothing distance and altering the force in the region between r fade and rcut-off cannot be worse than using a slightly smaller system with r’cut-off = rfade. The enormous advantage of using a smooth cut-off is that the structural properties of the system can be simulated correctly and that the lowest energy configuration is identical to the theoretical ground state. For studies of the dynamics of vortices, recent results show that the precise details of the long-range particle interaction are not crucial [13]. We therefore recommend the smoothed potential for phenomenological characterisation of superconductors. FANGOHR, PRICE, COX, DE GROOT, AND DANIELL 4 5. FAST INFINITE SUMMATION An alternative approach to modifying the potential is to sum the potential function over periodic repeats of the unit cell, which provides the best representation of the system given only a finite number of particles. We write the potential (1) in the form: [9] ∑     ++=   = yxmmyy xx ymLxmLrKrKcrU ,0* 0ˆˆ )( λ λ, (3) where mx and my are integers and Lx and Ly are the lengths of the edges of the simulation cell. This is truncated such that mx2 + my2 ≤ Nm2; we sum the potential in shells of increasing radius, Nm, until it has converged. Following Ryu et al [9] we will use a value for the penetration depth, λ, at 0K of 7700 ∆ for Mo 77Ge23. We will return to the temperature dependence of λ later. In figure 6, we show the exponentially fast convergence of the energy between two particles in a simulation of 300 vortices in the Abrikosov lattice state as more image cells are included. We also show the time taken to perform this calculation on a 450 MHz Pentium II using Compaq (Digital) Visual Fortran under Windows NT 4.0. For the particle-particle energy to converge to a relative error better than 1 ×10-8 requires Nm ~ 300, which takes ~ 300,000 calls to the K0 function. This ensures that the total system energy is accurate to better than 0.01%. We now derive a new method to perform this infinite summation. In figure 7 we have: ()() () () () () xxyyjiji LmLmyyxxji jiyy xx yyxxzLmLmZ 1212 2 22 2 2 tantan −−−− =+ =−+−=+ = ϕθπ , (4) which yields )cos(222 2φϕθφ zZzZw −+=+= . (5) We may use the Gegenbauer addition formulae [21] to write () ()()∑∞ −∞== kz kZ kwk IK K )cos(0 φλλ λ (6) for the energy between a particle i and one of the periodic images of j, where Ik and Kk are modified Bessel functions. This formula requires z ≤ Z, which is automatically satisfied since z is the minimum image distance between i and j. We can therefore write the total energy (3) of two particles i and j summed over all periodic images in the form: ∑∑∞ −∞= ==    +  =  =    kk k mmmmkzIZKzKwKrK yxyx)cos( 0 not ,0* 0* 0 φλλ λ λ λ , (7) where the case mx = my = 0, for which z ⇔ Z, is the contribution to the energy from the unit cell which must be explicitly included as a separate term. Further re-arrangement and use of (5) gives []∑∞ −∞=−   +  =   kk k k kskczIzKwK )sin()cos(0* 0 θ θλ λ λ, (8) where EFFICIENT METHODS FOR HANDLING LONG-RANGE FORCES 5 ∑ ∑ == ==  =   = 0 not , 0 not ,)sin( and )cos( yxyx yxyx mmmmk k mmmmk k kZK s kZK c ϕλϕλ . (9) Equations (8) and (9) have the remarkable property that the coefficients corresponding to the infinite summation over the periodic repeats of the unit cell can be pre-computed. This reduces the double summation in (3) to a single summation. Furthermore, due to the exponential convergence of the Gegenbauer addition formulae, the sum may be truncated at ktrunc ~ 5 – 20 terms. A further factor of two in performance can be obtained by using symmetry to convert the summation from k = -∞ …∞ to the range k = 0 …∞. The form (8) closely resembles a Fourier type summation method, yet the whole calculation proceeds in real space in contrast to the Ewald summation method [22]. Our proposed method couples directly to a multipole method for computing the interaction energy inside the unit cell in O( N) time [16], which is based on the Gegenbauer addition formulae, rather than a Taylor series expansion. Our O( N) method provides further speedup when there are more than ~1200 particles in the unit cell. This is analogous to the method described in [5], which couples a lattice summation method with a multipole method based on Taylor series. It is certainly not appropriate to use the method proposed in [8], which sums a genuinely logarithmic potential over infinite repeats of the unit cell, since the logarithmic approximation to the K0 potential is only valid for small r as shown in (2). The convergence of the energy between two particles in the Abrikosov lattice is identical to the convergence shown in figure 6 as we add more terms to the calculation of the coefficients ck and sk. We have chosen the case of two nearest neighbours, which yields the slowest convergence of (8) since z takes its smallest value. In a superconductor, λ is a function of the temperature. For our model system (Mo 77Ge23) λ(T) = λ(0) / (1-T / Tc)1/2 [9], where Tc = 5.63K is the critical temperature at which the material loses its superconducting properties. Hence the coefficients ck and sk need to be re-computed at each temperature. As the temperature increases additional image cells need to be included in both (3) and the pre-computation (9). The crucial difference, however, between (3) and (8) is that the time taken to evaluate the energy using (8) remains constant once the coefficients are available, whereas the naïve summation requires considerable numbers of additional image cells to converge to the solution. In figure 8 we show the speedup of our method when computing the energy between two particles at a fixed accuracy of 1 ×10 -5 (relative to the energy computed to machine accuracy by either method). In all cases the resulting energies are shown to be identical to the stated accuracy. At 0K and using 5 terms in the truncation of (8), we have a speedup of 20,000 over the na ïve summation method. This rises to 400,000 for temperatures approaching Tc. If the particle energy is required to be accurate to 1 ×10-8, then, using 30 coefficients, the speedups are between 50,000 ( T = 0K) and 1,000,000 (T ~ Tc). Since the coefficients ck and sk depend on λ (and hence temperature); the method may appear to be costly if the temperature is changed at every Molecular Dynamics or Monte Carlo step. We now discuss several ways to overcome this. Firstly, it is possible to perform simulations at a small number of temperatures and use the data from these to obtain information about the behaviour of the system as a continuous function of temperature [23, 24]. Thus improving the sophistication of the analysis of the results can reduce the number coefficients c k and sk which need to be pre-calculated. Secondly it is possible to compute the ck and sk at a small set of temperatures and use interpolation to derive their values at other temperatures. Finally, since only ~5-20 coefficients are needed, it is straightforward to compute once and store on disk the values of ck and sk for each temperature to be explored. These values will be re- used a large number of times in a typical set of numerical simulations. We implement (8) using a recurrence relation [25] for the trigonometric terms and a vendor-optimised vector Bessel function. Goertzel ’s algorithm [26] could be employed for additional efficiency, though the improvement is likely to be marginal. The remarkable speedup obtained is due to the fixed work equivalent to roughly five calls to a Bessel function routine required for (8), compared to ~100,000 calls required for (3) (at 0K). The five calls are: two to initialise the Bessel recurrence, one to evaluate the contribution from the unit cell, and the equivalent of roughly a further two for the remaining trigonometric terms. Our infinite summation is correspondingly five times slower than using the smoothed potential in a single unit cell, which requires evaluation of a single Bessel function or a polynomial. This is confirmed by experiments. For simulations using the fast infinite lattice summation, results are similar to those of figure 5. The infinite lattice summation method is suitable for quantitative studies of superconductors. FANGOHR, PRICE, COX, DE GROOT, AND DANIELL 6 6. RESULTS In the previous sections we have demonstrated that the phenomenological potential and the infinitely summed potential ensure that the Abrikosov lattice is the minimum energy configuration for our system. We now show that the presence of dislocations, which also results from incorrect handling of the long-range potential, seriously affects study of the elastic properties of a lattice. For superconductors the structure of the lattice determines the static and dynamic properties of the vortex lattice. This is known from experimental [27, 28] and theoretical work [29]. The simulation potential should not introduce dislocations, since this will affect the onset of plasticity in the lattice which is directly related to characterising current-voltage behaviour, and thus to applications. We have considered a simulation of shearing of a hexagonal lattice, which is a simplified version of the simulations required to perform current-voltage characterisations. Inset (a) in figure 9 shows a Delaunay triangulation for half the simulation cell demonstrating the experimental set-up: a shearing force is applied to the central row of particles marked by black points, and the particles marked by open circles are not allowed to move in the x-direction. The main diagram shows the resulting change in energy as a response to the shearing force. The upper part of the figure shows data for the smooth cut-off, with the lower part showing the results for the sharp cut-off. The smooth cut-off and the infinite lattice summation produce the expected behaviour: with increasing shear stress the energy increases. The slope of the energy-change as a function of the displacement characterises the shear elastic modulus of the crystal. Inset (b1) shows a triangulation of a system which has been slightly tilted by the applied force. In contrast, employing the sharp cut-off the energy decreases for applied shear stress, i.e. the material appears to collapse after applying a shearing force (inset b2)! Insets (c1) and (c2) show the time evolution of the local hexatic order, ∑=Ψ kk ni bond)6exp(1 6 θ, where the sum runs over all bond angles k in the Delaunay triangulation. Every 50, 000 time steps the system starts as a hexagonal lattice (6Ψ= 1) and a new shearing force is applied for the next 50,000 time steps. In (c1), which shows the smoothed potential, 6Ψdecreases continuously until a static state is reached, reflecting the shearing of the system. The energy data is taken from these static states. In (c2) (sharp cut-off) 6Ψ drops suddenly to a much smaller value, representing the sudden change to configurations similar to those shown in (b2). Thus, the mechanical properties of the lattice using a sharp cut-off are severely affected by the incorrect handling of the long-range potential: this would seriously affect numerical simulations aimed at studying elastic properties of superconductors. The smooth cut-off and the infinite lattice sum produce the correct physical behaviour and can be used in more complex numerical simulations for phenomenological (smoothed potential) or quantitative (infinite summation) study of the dynamic phase diagram of the superconductor lattice [30, 31]. 7. CONCLUSIONS For Monte Carlo and Molecular Dynamics simulations using long-range interactions subject to periodic boundary conditions, a sharp cut-off for the interaction energy (or force) can yield misleading results. We have considered the case of superconductors, in which the potential is governed by a Bessel function. Monte Carlo simulations are often used to study phase diagrams numerically and it is vital that the phase behaviour of the system is not affected by the model itself. We find that using a sharp cut-off the system can find irregular lattice configurations with an energy below the theoretical ground state of a regular hexagonal lattice. In Molecular Dynamics study of the dynamical phase diagram of the material can be dramatically affected by incorrect handling of the long-range potential. We have presented two methods which overcome these problems. The first is suitable for phenomenological studies of systems and uses a smoothed potential, but still truncates the interaction over a single unit cell. Annealing a system governed by this modified potential yields a perfect hexagonal lattice which is the global energy minimum. This is the least computationally expensive option and is applicable to any potential. The second sums the interaction over the infinitely tiled unit cell and is suitable for quantitative system studies. Previous methods for performing this add the tiled images in a series of shells of increasing radius. We have shown that with the pre-computation of a set of Fourier type coefficients, the whole infinite summation can be computed using a summation which converges exponentially fast and results in a speedup of between 20,000 and 1,000,000 over the na ïve summation, depending on the range of the interaction and the desired accuracy. The derivation of the summation proceeds in real space, and the results converge exactly to those obtained from other summation methods. This is roughly five times as slow as using the smoothed potential, but is the most accurate method for systems of finite size. We will report elsewhere on the EFFICIENT METHODS FOR HANDLING LONG-RANGE FORCES 7 results of systems we have studied using our methods [30, 31] and also on a method for evaluating the energy within the unit cell in O( N) time [16]. ACKNOWLEDGEMENTS The authors thank Ken Thomas for helpful discussions. REFERENCES 1. P. P. Ewald. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 64, 253 (1921). 2. S. W. de Leeuw, J. W. Perram, and E. R. Smith. Simulation of electrostatic systems in periodic boundary conditions. Proc. R. Soc. London A 373, 27 (1980). 3. M. P. Allen and D. J. Tildesley. Computer simulations of liquids . (Clarendon Press, Oxford, 1989). 4. L. Greengard and V. Rohklin. A fast algorithm for particle simulations. J. Comp. Phys. 73, 325 (1987). 5. C.L. Berman and L. Greengard. A Renormalization Method for the Evaluation of Lattice Sums. J. Math. Phys. 35 (11) 6036-6048 (1994). 6. J. Lekner. Summation of dipolar fields in simulated liquid vapor interfaces. Physica A 157, 826 (1989). 7. J. Lekner. Summation of coulomb fields in computer-simulated disordered-systems. Physica A 176, 485 (1991). 8. N. Gr ønbech-Jensen. Summation of logarithmetic interactions in periodic media. Int. J. Mod. Phys. C 7, 873 (1996). 9. S. Ryu, M. Hellerqvist, S. Doniach, A. Kapitulnik, and D. Stroud. Dynamical phase transition in a driven disordered vortex lattice. Phys. Rev. Lett . 77, 5114 (1996). 10. J.R. Clem. Anisotropy and two-dimensional behaviour in the high-temperature superconductors. Supercond. Sci. Technol. 11, 909 (1998). 11. H. J. Jensen, A. Brass, and A. J. Berlinsky. Lattice deformations and plastic flow through bottlenecks in a two- dimensional model for flux pinning in type-II superc onductors. Phys. Rev. Lett. 60, 1676 (1988). 12. C. Reichhardt, C. J. Olson, and Franco Nori. Dynamic phases of vortices in superconductors with periodic pinning. Phys. Rev. Lett . 78, 2648 (1997). 13. B. Y. Zhu, D. Y. Xing, J. Dong and B. R. Zhao. Dynamical phase transition of a driven vortex lattice with disordered pinning. Physica C 311, 140 (1999). 14. M. Tinkham. Introduction to Superconductivity , 2nd ed. (McGraw-Hill International, Inc., New York 1996). 15. A.A. Abrikosov. On the magnetic properties of superconductors of the second group. Sov. Phys. - JETP 5, 1174 (1957). 16. S.J. Cox, G.J. Daniell, and A.M. Robinson. An O( N) Multipole Algorithm for the 2D Helmholtz Equation. Submitted to J. Comput. Phys. 17. P. Le Doussal and T. Giamarchi. Moving glass theory of driven lattices with disorder. Physical Review B 57, 11356 (1998). 18. S. Kokkaliaris, P.A.J. de Groot, S.N. Goordeev, A.A. Zhukov, R. Gagon, and L. Taillefer. Onset of Plasticity and Hardening of the Hysteretic Response in the Vortex System of YBa 2Cu3O7-δ. Physical Review Letters 82, 5116 (1999). 19. M.J. Higgins and S. Bhattacharya. Varieties of dynamics in a disordered flux-line lattice. Physica C 257 , 232 (1996). 20. S. Spencer and H.J. Jensen. Absence of translational ordering in driven vortex lattices. Phys. Rev. B 55, 8473 FANGOHR, PRICE, COX, DE GROOT, AND DANIELL 8 (1997). 21. G. N. Watson. Theory of Bessel functions . (Cambridge University Press, Cambridge, 1944). 22. E. Olive and E.H. Brandt. Point Defects in the Flux-line Lattice of Superconductors. Phys Rev. B. 57(21) 13861-13871. (1998). 23. A.M. Ferrenberg and R.H. Swendsen. New Monte Carlo Technique for Studying Phase Transitions. Phys. Rev Letts. 61(23) 2635-2638 (1988). 24. A.M. Ferrenberg and R.H. Swendsen. Optimized Monte Carlo Data Analysis. Phys. Rev Letts. 63(12) 1195- 1198 (1989). 25. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in Fortran , 2nd ed. (Cambridge University Press, Cambridge, 1992). 26. G. Goertzel. An Algoithm for the Evaluation of Finite Trigonometric Series. American mathematical monthly . 34 (Jan 1958). 27. R. W ördenweber and P. H. Kes. Peak and history effects in two-dimensional collective flux pinning. Phys. Rev. B 33, 3172 (1986). 28. B. Khaykovich, M. Konczykowski, E. Zeldov, R. A. Doyle, D. Majer, P. H. Kes, and T. W. Li. Vortex-matter phase transitions in Bi 2Sr2CaCu2O8: Effects of weak disorder. Phys. Rev. B 56, R517 (1997) 29. J. Kierfeld and V. Vinokur. Dislocations and the critical endpoint of the melting line of vortex line lattices. Cond-mat preprint. 9909190 (1999) 30. H. Fangohr, S.J. Cox, P.A.J. de Groot. Vortex Dynamics in Two-Dimensional Systems at High Driving Forces. Submitted to Phys. Rev. B. 31. H. Fangohr, S.J. Cox, P.A.J. de Groot. Critical Transverse Forces in Weakly Pinned Driven Vortex Systems. Submitted to Phys. Rev. Letts. EFFICIENT METHODS FOR HANDLING LONG-RANGE FORCES 9 Figure 1 A long-range force (i) Full force (ii) Force cut-off at a distance rcut-off (iii) Smoothed force. Distances are measured in multiples of the ground-state lattice spacing. Figure 2 Left: Molecular Dynamics simulation of 90 particles using a cut-off potential, which start in a hexagonal configuration at 0K (with Abrikosov lattice energy, Ea), are heated to half their melting temperature ( Tm) and then returned to 0K. Temperature is introduced via a stochastic noise term. The system finds a new configuration with energy lower than the energy of the regular lattice. Right: Delaunay triangulation of the final configuration of the particles at time step 5000. Two cut-off circles are shown to demonstrate that particles align along these circles. FANGOHR, PRICE, COX, DE GROOT, AND DANIELL 10 Figure 3 Monte Carlo simulation of 432 particles using a conventional cut-off potential. The system starts in a regular hexagonal Abrikosov lattice and is heated above its melting point to ~ 3 Tm then annealed slowly to zero temperature in steps of 0.015 Tm each of 5000 sweeps. Left: The energy of the system drops below the Abrikosov lattice energy, Ea. Right: Delaunay triangulation of the final disordered configuration. The topological defects are circled. Figure 4 The magnitude of the force field a particle at position (0,0) experiences from a system of 418 particles using (left) the sharp cut-off and (right) the smooth cut-off. The effect of smoothing the potential is to remove the discontinuities in the force. EFFICIENT METHODS FOR HANDLING LONG-RANGE FORCES 11 Figure 5 Monte Carlo simulation of 432 particles using a potential smoothed over three lattice spacings. Left: The energy of the system never drops below the Abrikosov lattice energy, Ea. Right: Delaunay triangulation of the final configuration shows the system has a hexagonal ground state. Figure 6 Fractional error ( E∞ - En)/E∞ and time taken to compute the energy En between two particles separated by a single lattice spacing in an infinitely tiled periodic system when n image cells are used. E∞ is estimated by allowing the summation to converge to machine accuracy. FANGOHR, PRICE, COX, DE GROOT, AND DANIELL 12 Figure 7 Two particles in a unit cell with infinite periodic repeats. Figure 8 Speedup of fast infinite summation method over na ïve implementation when the relative error in the energy between each pair of particles is fixed to be 1 ×10-5: both methods yield identical results. EFFICIENT METHODS FOR HANDLING LONG-RANGE FORCES 13 Figure 9 /G38/G75/G68/G81/G74/G72/G3 /G76/G81/G3 /G72/G81/G72/G85/G74/G92/G15/G3 E, (in simulation units) as a function of a shearing force, fshear, (in simulation units) for the smooth and the sharp cut-off. For the infinite lattice summation we obtain qualitatively similar results. Insets (a), (b1) and (b2) show different snap shots of vortex configurations. Insets (c1) and (c2) show the local hexagonal order, 6Ψ, as the experiment progresses (see text for details).

arXiv:physics/0004014v1 [physics.class-ph] 10 Apr 2000Ermakov-Lewis angles for one-parameter supersymmetric fa milies of Newtonian free damping modes Haret C. Rosu†and Pedro B. Espinoza‡ †Instituto de F´ ısica de la Universidad de Guanajuato, Apdo P ostal E-143, Le´ on, Guanajuato, M´ exico and International Center for Relativistic Astrophysics, Rome -Pescara, Italy ‡Centro Universitario de los Altos, Universidad de Guadalaj ara, Lagos de Moreno, Jalisco, M´ exico We apply the Ermakov-Lewis procedure to the one-parameter d amped modes ˜ yrecently introduced by Rosu and Reyes, which are related to the common Newtonian free damping modes yby the general Riccati solution [H.C. Rosu and M. Reyes, Phys. Rev. E 57, 2850 (1998)]. In particular, we calculate and plot the angl e quantities of this approach that can help to distinguish these modes from the common ymodes. PACS number(s): 03.20.+i In a previous paper hereafter denoted as I [1], the non- uniqueness of the factorization of linear second-order dif - ferential operators has been exploited on the example of the classical Newtonian free damped oscillator, i.e. Ny≡/parenleftbiggd2 dt2+ 2βd dt+ω2 0/parenrightbigg y= 0. (1) The coefficient 2 βis the friction constant per unit mass andω0is the natural frequency of the oscillator (SI units assumed all over the work). The more general supersymmetric partner equation ˜Ng˜y≡/parenleftbiggd2 dt2+ 2βd dt+ω2 0−2γ2 (γt+ 1)2/parenrightbigg ˜y= 0 (2) has been obtained in I. This new second-order linear damping equation contains the additional last term with respect to its initial partner (1), which may be thought of as the general Darboux transform part of the frequency [4].T= 1/γoccurs as a new time scale in the Newtonian damping problem. If this time scale is infinite, the ordi- nary free damping is recovered unless for the critical case which is special even in ordinary damping. As explained in I, the ˜ ymodes can be obtained from the ymodes by operatorial means. In the following we shall call them γ modes. For the three types of free damping they have been obtained in I as follows: (i) For underdamping, β2< ω2 0, denoting ωu=/radicalbig ω2 0−β2the underdamped γmodes are ˜yu=−˜Aue−βt/bracketleftBig ωusin(ωut+φ) +γ γt+ 1cos(ωut+φ)/bracketrightBig . (3) (ii) For overdamping, β2> ω2 0andωo=/radicalbig β2−ω2 0, the overdamped γmodes are ˜y0=−˜Aoe−βt/bracketleftBig ωosinh(ωot+φ)−γ γt+ 1cosh(ωot+φ)/bracketrightBig . (4) (iii) For critical damping, β2=ω2 0. The critical γ solutions are given by ˜yc=/bracketleftBig−Acγ γt+ 1+Dc γ2(γt+ 1)2/bracketrightBig e−βt. (5)These are the onlypossible types of one-parameter damping modes related to the free damping ones by means of Witten’s supersymmetric scheme [2] and the general Riccati solution [3]. In practice the new parameter γcan be very close to zero. In this case, it is very difficult to diferentiate the γmodes from the ordinary ones. The only means we can think of is by recording somehow the geometric an- gle associated to the γmodes and compare it with the same quantity in the ordinary damping cases. One is led to this conclusion noticing that the γmodes have time- dependent frequencies ω2(t) =ω2 0−2γ2 (γt+1)2and hence for them the Ermakov-Lewis (EL) procedure can be natu- rally applied [5] (for a recent review, see [6]). For ω0/negationslash=β, Eq. (2) can be reduced to a Bessel equation and the so- lutions can be written as follows Ψu=τ1/2/bracketleftBig AJ3 2(kτ) +BY3 2(kτ)/bracketrightBig e−βτ(6) and Ψo=τ1/2/bracketleftBig CI3 2(kτ) +DK 3 2(kτ)/bracketrightBig e−βτ, (7) where τ=γt+ 1 and k2=ω2 0−β2 γ2. When k→ ∞ (i.e., γ→0), we can do Hankel’s asymptotic expansions, i.e., of large Bessel argument but fixed Bessel order (we shall not reproduce these formulas here, the reader is directed to Abramowitz and Steagun [7]). The point is that one is indeed able to get the solutions obtained by operatorial means. Thus, the supersymmetric operatorial procedure gives merely the asymptotic γ→0 solutions, which how- ever could be the most relevant from the physical view- point in this context. In the EL approach the angular quantities are given by the following formulas [6,10] ∆θd=/integraldisplayT 0/bracketleftBige−2βt′ ρ2−1 2d dt′(e2βt′ ˙ρρ) +e2βt′ ˙ρ2/bracketrightBig dt′(8) and ∆θg=1 2/integraldisplayT 0/bracketleftBigd dt′(e2βt′ ˙ρρ)−2e2βt′ ˙ρ2/bracketrightBig dt′,(9) 1for the dynamical and geometrical angles, respectively. Thus, the total angle will be ∆θt=/integraldisplayT 0e−2βt′ ρ2dt′. (10) The so-called Pinney function ρis the solution of Pin- ney’s nonlinear equation [8] ρ′′(t) +p(t)ρ′(t) +q(t)ρ=C ρ3(t)exp/parenleftbigg −2/integraldisplayt p(t′)dt′/parenrightbigg (11) forC= constant (=1), p(t) = 2βandq(t) =ω2 0−2γ2 (γt+1)2. Forρ/negationslash= constant there is a definite prescription of cal- culating ρin terms of two independent solutions of the corresponding linear equation. We have followed the method of Eliezer and Gray [9] for ρ(t) in terms of linear combinations of the aforementioned Bessel functions (for A=B=C=D= 1) that satisfy the initial conditions as given by those authors. In the critical damping case, we used the modes of Eq. (5) with Ac=Dc= 1. The results of the calculations for some particular values of the parameters are plotted in Figs. 1a,b,c, 2a,b,c, 3a,b,c for the γunderdamped, overdamped, and critical cases, respectively. For comparison, the angle quantities for γ= 0 are displayed in Figs. 1a’,b’,c’, 2a’,b’,c’, 3a’,b’,c’, respectfully. ACKNOWLEDGMENT This work was partially supported by the CONACyT Project 458100-5-25844E. [1] H.C. Rosu and M. Reyes, Phys. Rev. E 57, 2850 (1998) (paper I). [2] E. Witten, Nucl. Phys. B 185, 513 (1981). [3] B. Mielnik, J. Math. Phys. 25, 3387 (1984). [4] G. Darboux, C.R. Acad. Sci. 94, 1456 (1882). [5] V. Ermakov, Univ. Izv. Kiev, Series III 9, 1 (1880); H.R. Lewis, Jr., Phys. Rev. Lett. 18, 510 (1967); J. Math. Phys.9, 1976 (1968). [6] P. Espinoza, Ermakov-Lewis dynamic invariants with some applications , MS Thesis (Le´ on, Mexico, 2000), available as math-ph/0002005. [7] M. Abramowitz and I.A. Stegun, Handbook of mathe- matical functions , see Formulas 9.2.5, 9.2.6, 9.2.9, 9.2.10, Dover 1970. [8] E. Pinney, Proc. Am. Math. Soc. 1, 681 (1950). [9] C.J. Eliezer and A. Gray, SIAM J. Appl. Math. 30, 463 (1976). [10] M. Maamache, Phys. Rev. A 52, 936 (1995); D.A. Morales, J. Phys. A21, L889 (1988); J.M. Cerver´ o and J.D. Lejarreta, J. Phys. A22, L663 (1989). Fig. 1a The dynamical angle in the underdamped case for the following set of parameters: ω0=√ 2,β= 1,γ= 0.1. Fig. 1b The geometric angle in the underdamped case and the same parameters. Fig. 1c The total angle in the underdamped case and the same parameters. 2-20-15-10-500 1 2 3 4T Fig. 2a The dynamical angle in the overdamped case for ω0= 1, β=√ 2,γ= 0.1. 00.511.522.53 0 0.5 1 1.5 2 2.5 3 T Fig. 2c The geometric angle in the overdamping case for the same parameters.00.050.10.150.20.25 0 1 2 3 4 5 6 T Fig. 2c The total angle in the overdamping case for the same parameters. -3-2-100 0.1 0.2 0.3 0.4 0.5 0.6T Fig. 3a The dynamical angle in the critical case for ω0=β= 1 and γ= 0.1. 3-505101520 0 0.2 0.4 0.6 0.8 T Fig. 3b The geometrical angle in the critical case for the same parameters. 050100150200 0 0.2 0.4 0.6 0.8 1 T Fig. 3c The total angle in the critical case for the same parameters.01020304050 0 2 4 6 8 10 T Fig. 1a’ The dynamical angle in the underdamped case for the same ω0,βparameters as in Fig. 1a and γ= 0. Fig. 1b’ The geometrical angle in the underdamped case for the same ω0,βparameters and γ= 0. Fig. 1c’ The total angle in the underdamped case for the same ω0,β parameters and γ= 0. 4Fig. 2a’ The dynamical angle in the overdamped case for the same ω0,βparameters as in Fig. 2a and γ= 0. Fig. 2b’ The geometrical angle in the overdamped case for the same ω0,βparameters and γ= 0. Fig. 2c’ The total angle in the overdamped case for the same ω0,β parameters and γ= 0. Fig. 3a’ The dynamical angle in the critical case for ω0=β= 1 and γ= 0. Fig. 3b’ The geometrical angle in the critical case for ω0=β= 1 and γ= 0. Fig. 3c’ The total angle in the critical case for ω0=β= 1 and γ= 0. 5

arXiv:physics/0004015v1 [physics.ins-det] 10 Apr 2000Tumour Therapy with Particle Beams Claus Grupen1 Department of Physics, University of Siegen Germany Abstract. Photons are exponentially attenuated in matter producing h igh doses close to the surface. Therefore they are not well suited for t he treatment of deep seated tumours. Charged particles, in contrast, exhib it a sharp increase of ionisation density close to the end of their range, the so- called Bragg-peak. The depth of the Bragg-peak can be adjusted by varying the par ticle’s energy. In parallel with the large energy deposit the increase in bio logical effectiveness for cell killing at the end of the range provides an ideal scal pel for the surgeon effectively without touching the surface tissue. Consequen tly proton therapy has gained a lot of ground for treating well localized tumours. E ven superior still are heavy ions, where the ionisation pattern is increased by the square of their charge ( ∼z2). INTRODUCTION It has been known for a long time that tissue, in particular tu mour tissue, is sensitive to ionising radiation. Therefore it is only natur al that tumours have been treated with various types of radiation, like γ-rays and electrons. γ-rays are easily available from radioactive sources, like60Co, and electrons can be accelerated to MeV-energies by relatively inexpensive linear accelerators. The disadvantage of γ-rays and electrons is that they deposit most of their energy close to the surface. To reduce the surface dose in tumour tre atment requires rotating the source or the patient so that the surface dose is distributed over a larger volume. In contrast, protons and heavy ions deposit m ost of their energy close to the end of their range (Bragg-peak). The increase in energy loss at the Bragg-peak amounts to a factor of about 5 compared to the s urface dose, depending somewhat on the particle’s energy. Heavy ions offe r, in addition, the possibility to monitor the destructive power of the beam by observing annihilation radiation by standard positron-emission tom ography techniques (PET). The annihilation radiation is emitted by β+-active nuclear fragments produced by the incident heavy ion beam itself. 1)e-mail: grupen@aleph.physik.uni-siegen.deFIGURE 1. Mass attenuation coefficient for photons in water as a functio n of the photon energy [1] ENERGY LOSS OF PARTICLES IN TISSUE [1] A photon beam is attenuated in matter according to I(x) =I0e−µx(1) where I0is the initial intensity and I(x) the beam intensity at the depth x.µ is the linear mass attenuation coefficient which depends on th e photon energy Eand the target charge Z.µ(E) is shown in figure 1 for a target composed of water, which is essentially equivalent to tissue. The mai n interaction mech- anisms which contribute to µ(E) are the photoelectric effect ( ∼Z5/E3.5), Compton scattering ( ∼(Z/E)ln E) and pair-production ( ∼Z2ln E). For energies typical for radioactive sources ( ∼MeV) Compton scattering domi- nates. The absorption profile of photons in matter exhibits a peak close to the surface followed by an exponential decay. Charged particles suffer energy loss by ionisation. This ene rgy loss is de- scribed by the Bethe-Bloch formula: dE dx= 2κ{lnEmax kin I−β2−δ 2} (2) where κ= 2π NAr2 emec2z2Z A·1 β2. (3)z – charge of the beam particle Z – charge of the absorber material A – mass number of the absorber material me– electron mass c – velocity of light NA– Avogadro’s number re – classical electron radius β – velocity of the particle divided by c Emax kin– maximum transferable energy to an atomic electron I – mean excitation energy of the target material δ – density parameter For protons ( z= 1) interacting in water (or tissue) equation (2) can be ap- proximated by dE dx= 0,16·1 β2lnEmax kin[eV] 100/bracketleftbiggMeV cm/bracketrightbigg (4) where Emax kin≈2mec2β2γ2, (5) which gives an energy loss of 4.2 MeV/cm for 200 MeV protons at the surface and∼20MeV/cm close to the end of their range. For heavy ions the energy loss is essentially scaled by z2. When charged particles reach the end of their range the energy loss first rises like 1 /β2but when they are very slow they capture electrons from the target material and their effecti ve charge decreases and hence their energy loss rapidly falls to zero. A typical energy loss curve for ions as a function of their ene rgy is sketched in figure 2 [2]. The energy loss of12C ions as a function of the depth in water is shown in figure 3 [2,3]. The tail of the energy loss beyond th e Bragg-peak originates from fragmentation products of12C ions, which are faster than the 12C ions and have a somewhat longer range. In the ionisation process a generally small fraction of the p article’s energy is transferred to the atomic electrons. In rare cases these e lectrons can get a larger amount of energy. The δ-electrons deviate from the main ionisation trail and produce a fuzzy-like track (figure 4, [2]). In addition to ionisation light particles, like electrons, can also undergo bremsstrahlung ( dE/dx ∼z2Z2E). Since the probability for this process is inversely proportional to the square of the mass of the bea m particle, bremsstrahlung can be neglected for particles heavier than the electron for energies relevant to tumour therapy [1]. The above mentioned fragmentation of heavy ions leads to the production of positron emitters. For the12C case, lighter isotopes like11C and10C areFIGURE 2. Energy loss of ions in matter as a function of their energy (af ter [2]) FIGURE 3. Energy loss of carbon-ions (12C) in water as a function of depth [2,3]FIGURE 4. Sketch of a proton and a carbon nucleus track in tissue. The fu zziness of the tracks is caused by short range δ-rays [2] produced. Both isotopes decay with short half-lives ( T1/2(11C) = 20 ,38min; T1/2(10C) = 19 .3s) to boron according to 11C→11B+e++νe (6) 10C→10B+e++νe. The positrons have a very short range, typically below 1 mm. After coming to rest they annihilate with electrons of the tissue giving off t wo monochromatic photons of 511 keVwhich are emitted back-to-back e++e−→γ+γ . (7) These photons can be detected by positron-emission tomogra phy techniques and can be used to monitor the destructive effect of heavy ions on the tumour tissue. PRODUCTION OF PARTICLE BEAMS The treatment of deep seated tumours requires charged parti cles of typi- cally 100 to 400 MeV per nucleon, i.e. 100 to 400 MeV protons or 1.2 to 4.8GeV12C ions. These particles are accelerated in either a linear ac celer- ator or in a synchrotron. As an example figure 5 shows a typical set-up for the production of heavy ions.12C atoms are evaporated from an ion source and pre-accelerated. Thin foils are used to strip off all elec trons from the ions. The12C nuclei are then injected into a synchrotron, where they are ac- celerated by radiofrequency cavities to the desired energy . The ions are keptFIGURE 5. Sketch of a typical set-up for the acceleration of heavy ions (not all compo- nents are shown) on track by dipole bending magnets and they are focussed by qu adrupoles. After having reached the final energy they are ejected by a kic ker magnet, which directs the particles to the treatment room. Their pat h is monitored by tracking chambers (multi-wire proportional counteres, io n chambers or drift- chambers). If beam losses occur veto-counters (mostly scin tillation counters) ensure that only a pencil beam is steered to the treatment roo m. Nowadays, mainly protons and heavy ions are used for tumour t herapy. Other possibilities consist of the use of negative pions [7– 9], which are pro- duced by high energy protons in a beam dump according to p+ nucleus →p+ nucleus + π−+π++π0(8) where the π−are momentum selected and collimated. After losing their en ergy by ionisation the negative pions are captured in the tumour t issue by nuclei at the end of their range and produce so-called ‘stars’ in whi ch neutrons are created. The Bragg-peak of the negative pions along with the local production of neutrons which have a high biological effectiveness leads to an efficient cell killing in the tumour at the end of the pion’s range. Neutrons are also possible candidates for tumour treatment [10]. For this purpose the tumour is sensitized by a boron compound before n eutron treat- ment. The boron compound must be selected in such a way that it is prefer- entially deposited in the tumour region. Neutrons are then c aptured by theFIGURE 6. Comparison of depth-dose curves of neutrons, γ-rays (produced by a 8 MV driven X-ray tube), 200 MeV protons, 20 MeV electrons and192Ir-γ-rays (161 keV) [4] boron according to: n+10B→7Li+α . (9) The produced α-particles (He-nuclei) have a very short range ( ∼several µm) and high biological effectiveness. Best results are obtaine d with epithermal neutrons ( ∼1keV) produced by 5 MeV protons on light targets (e.g. Be). Direct irradiation with neutrons – without sensitizing the tumour – has the disadvantage that neutrons show a similar dose depth curve l ike60Coγ-rays thus producing a high amount of biologically very effective d amage in the healthy tissue around the tumour (see figure 6 [4]). APPLICATIONS IN TUMOUR THERAPY The target for cell killing is the DNA in the cell nucleus (see figure 7 (after [2])). The size of the DNA-molecule compares favorably well with the width of the ionisation track of a heavy ion. The DNA contains two st rands con- taining identical information. A damage of one strand by ion ising radiation can easily be repaired by copying the information from the un affected strand to the damaged one. Therefore the high ionisation density at the end of a particle’s range matches well with the requirement to produ ce double strand breaks in the DNA, which the cell will not survive. Heavy ions like12C seem to be optimal for this purpose. Ions heavier than carbon woul d even be more powerful in destroying tumour tissue, however, their energ y loss in the sur- rounding tissue and in the entrance region already reaches a level where the fraction of irreparable damage is too high, while for lighte r ions (like12C)FIGURE 7. Sketch of typical dimensions of biological targets (after [ 2]) mostly repairable damage is produced in the healthy tissue o utside the tar- geted tumour. The cell killing rate in the tumour region thus benefits from two properties of protons or ions like carbon: •the increased energy loss of protons and ions at the end of the ir range and •the increased biological effectiveness of double strand bre aks at high ion- isation density. The cell killing rate is eventually related to the equivalen t dose H in the tumour region, which can be expressed by H=1 m/integraldisplaydE dxdx·RBE (10) where mis the tumour mass and RBE the increased relative biological effec- tiveness. The integral extends over the tumour region. As mentioned above the rate and location of cell killing can b e monitored by observing the annihilation photons which result from the β+-decay of frag- ments formed by the beam. These physical and biological principles are employed in an efficient way by the raster scan technique [3,5,6]. A pencil beam of heavy ion s (diameter ∼ 1mm) is aimed at the tumour. The beam location and spread is monit ored by tracking chambers with high spatial resolution. In the trea tment planning the tumour is subdivided into three-dimensional pixels (“voxe ls”). Then the dose required to destroy the tumour, which is proportional to the beam intensity,FIGURE 8. Principle of the raster scan method [5,6] is calculated for every voxel. For a fixed depth in tissue an ar eal scan is performed by magnetic deflection sweeping the beam across th e area in a similar way as a TV image is produced (see figure 8, [5,6]). The tumour volume is filled from the back by energy variation ( ∼range variation) of the beam. Typically 50 energy steps are used starting at the rear plane . For a depth profile from 2 cmto 30 cmone has to cover energies from 80 MeV/nucleon to 430 MeV/nucleon. When the beam energy is reduced the required dose for the plane under irradiation is calculated using the dama ge that the more energetic beam had already produced in its entrance region. This ensures that the lateral (caused by magnetic deflection) and longitu dinal scanning (by energy variation) covers the tumour completely. In figure 9 ( after [2]) the dose distribution for individual energy settings and the re sulting total dose is sketched and compared with the damage that X-rays from a60Co-source would produce. An artist impression of the dose distributio n for a lung and a brain tumour is given in figure 10. TREATMENT FACILITIES Berkeley was the birthplace of therapy with hadrons. Since 1 954 protons and later Helium-nuclei were used for treatment. Throughou t the world treat- ment with protons is standard (Sweden, USA, Russia, Japan, S witzerland, England, Belgium, France, South Africa). In some places neg ative pions have been used in the past (USA, Canada, Switzerland). The most pr omising re- sults have been obtained with heavy ions (Berkeley, USA; Chi ba, Japan; and Darmstadt, Germany). In total ∼25000 patients have been treated from 1954 to 1999.FIGURE 9. Superposition of Bragg-peaks by energy variation (after [2 ]) FIGURE 10. a) The position of the Bragg-peak can be adjusted by energy se - lection to produce a maximum damage at the tumour site (here i n the lung). b) Mapping of a brain tumour with ionisation from heavy ions. Some damage at the entrance region cannot be avoidedSUMMARY AND OUTLOOK The inverse ionisation dose profile of charged particles has been known for a long time, from nuclear and particle physics. The instrumen tation originally developed for elementary particle physics experiments has made it possible to design and monitor particle beams with great precision wh ich can then be used for tumour therapy. Heavy ions seem to be ideal projecti les for tumour treatment. They are suitable for well localized tumours. Th e availability of treatment facilities is increasing. Naturally such a facil ity requires an expen- sive and complex accelerator for the charged particles. For beam steering and control sophisticated particle detectors and interlock sy stems are necessary to ensure the safety of patients. ACKNOWLEDGEMENTS The author has benefitted a great deal from information provi ded by G. Kraft from GSI-Darmstadt and from discussions with him. I acknowl- edge also the help of Mrs. L. Hoppe and C. Haucke for the drawin g of the figures, Mrs. A. Wied for typing the text, Mr. Ngac An Bang for g iving the paper the final LaTeX-touch, and Mr. D. Robinson for a careful reading of the manuscript. REFERENCES 1. C. Grupen ‘Particle Detectors’ Cambridge University Pre ss 1996 2. G. Kraft ‘Radiobiology of Heavy Charged Particles’ GSI-P reprint 96-60, Nov. 1996 3. G. Kraft ‘Tumour Therapy with Ion Beams’ invited paper at t he SAMBA- Symposium at the University of Siegen 1999, to be printed in N ucl.Instr.Meth. 2000 4. Medical Radiation Group, National Accelerator Centre, S outh Africa, Internet paper 1999 5. G. Kraft ‘The Impact of Nuclear Science on Medicine’ Nucl. Phys. A 654 (1999) 1058c - 1067c 6. G. Kraft ‘Radiotherapy with Heavy Charged Particles’ http://www.gsi.de 7. N.A. Dyson ‘Nuclear Physics with Application in Medicine and Biology’ John Wiley & Sons Inc. N.Y. (1981) and ‘Radiation Physics with Application in Medicine and Biolog y’, Ellis Horwood, N.Y. (1993) 8. S.B. Curtis, M.R. Raju ‘A Calculation of the Physical Char acteristics of Neg- ative Pion Beams Energy Loss Distribution and Bragg-Curves ’ Rad. Research 34 (1968) 2399. G.B. Goodman ‘Pion Therapy for Cancer – What are the Prospe cts’ TRIUMF- Preprint TRI-PP-92-134 (1992) 10. A.J. Lennox ‘Hospital-Based Proton Linear Accelerator for Particle Therapy and Radioisotope Production’ Fermilab-Pub. 90/217 (1990)

arXiv:physics/0004016v1 [physics.comp-ph] 10 Apr 2000Fast Algorithm for Finding the Eigenvalue Distribution of V ery Large Matrices Anthony HAMS and Hans De RAEDT Institute for Theoretical Physics and Materials Science Ce ntre, University of Groningen, Nijenborgh 4, NL-9747 AG Groninge n, The Netherlands (DRAFT: February 12, 2008) Abstract A theoretical analysis is given of the equation of motion met hod, due to Alben et al., to compute the eigenvalue distribution (densi ty of states) of very large matrices. The salient feature of this method is th at for matrices of the kind encountered in quantum physics the memory and CPU re quirements of this method scale linearly with the dimension of the matri x. We derive a rigorous estimate of the statistical error, supporting ear lier observations that the computational efficiency of this approach increases with matrix size. We use this method and an imaginary-time version of it to comput e the energy and the specific heat of three different, exactly solvable, sp in-1/2 models and compare with the exact results to study the dependence of the statistical errors on sample and matrix size. PACS numbers: 05.10.-a, 05.30.-d, 0.3.67.Lx Typeset using REVT EX 1I. INTRODUCTION The calculation of the distribution of eigenvalues of very l arge matrices is a central problem in quantum physics. This distribution determines t he thermodynamic properties of the system (see below). It is directly related to the singl e-particle density of states (DOS) or Green’s function. In a one-particle (e.g., one-electron ) description knowledge of the DOS suffices to compute the transport properties [1]. The most direct method to compute the DOS, i.e. all the eigenv alues, is to diagonalize the matrix Hrepresenting the Hamiltonian of the system. This approach h as two obvious limitations: The number of operations increases as the thir d power of the dimension D ofHand, perhaps most importantly, the amount of memory require d by state-of-the-art algorithms grows as D2[2,3]. This scaling behavior limits the application of this approach to matrices of dimension D=O(10000), which is too small for many problems of interest. What is needed are methods that scale linearly with D. There has been considerable interest in developing “fast” ( i.e.O(D)) algorithms to compute the DOS and other similar quantities. One such algor ithm and an application of it to electron motion in disordered alloy models was given by Alben et al. [4]. In this approach the DOS is obtained by solving the time-dependent S chr¨ odinger equation (TDSE) of a particle moving on a lattice, followed by a Fourier trans form of the retarded Green’s function [4]. Using the unconditionally stable split-step Fast Fourier Transform (FFT) method to solve the TDSE, it was shown that the eigenvalue spe ctrum of a particle moving in continuum space can be computed in the same manner [5]. Fas t algorithms of this kind proved useful to study various aspects of localization of waves [6–8] and other one- particle problems [9,10]. Application of these ideas to qua ntum many-body systems triggered further development of flexible and efficient methods to solve the TDSE. Based on Suzuki’s product formula approach, an unconditionally stable algor ithm was developed and used to compute the time-evolution of two-dimensional S=1/2 Heise nberg-like models [11]. Results for the DOS of matrices of dimension D≈1000000 where reported [11]. A potentially interesting feature of these fast algorithms is that they ma y run very efficiently on a quantum computer [12,13]. A common feature of these fast algorithms is that they solve t he TDSE for a sample of randomly chosen initial states. The efficiency of this appr oach as a whole relies on the hypothesis (suggested by the central limit theorem) tha t satisfactory accuracy can be achieved by using a small sample of initial states. Experien ce not only shows that this hypothesis is correct, it strongly suggests that for a fixed s ample size the statistical error on physical quantities such as the energy and specific heat decr eases with the dimension Dof the Hilbert space [12]. In view of the general applicability of these fast algorithm s to a wide variety of quantum problems it seems warranted to analyze in detail their prope rties and the peculiar Ddepen- dence in particular. In Sections II and III we recapitulate t he essence of the approach. We present a rigorous estimate for the mean square error (varia nce) on the trace of a matrix. In Section IV we describe the imaginary-time version of the met hod. The statistical analysis of the numerical data is discussed in Section V. Section VI desc ribes the model systems that are used in our numerical experiments. The algorithm used to solve the TDSE is reviewed in Section VII. In Section VIII we derive rigorous bounds on t he accuracy with which all 2eigenvalues can be determined and demonstrate that this acc uracy decreases linearly with the time over which the TDSE is solved. The results of our nume rical calculations are presented in Section IX and our conclusions are given in Sect ion X. II. THEORY The trace of a matrix Aacting on a D-dimensional Hilbert-space spanned by an or- thonormal set of states {|φn∝an}bracketri}ht}is given by TrA=D/summationdisplay n=1∝an}bracketle{tφn|Aφn∝an}bracketri}ht. (1) Note that according to (1) we have Tr 1 = D. IfDis very large one might think of approx- imating Eq. (1) by sampling over a subset of K(K≪D)“important” basis vectors. The problem with this approach is that the notion “important” ma y be very model-dependent. Therefore it is better to sample in a different manner. We cons truct a random vector |ψ∝an}bracketri}htby choosingDcomplex random numbers, cn≡fn+ign, with mean 0, for n= 1...D, so |ψ∝an}bracketri}ht=D/summationdisplay n=1cn|φn∝an}bracketri}ht, (2) and calculate ∝an}bracketle{tψ|Aψ∝an}bracketri}ht=D/summationdisplay n,m=1c∗ mcn∝an}bracketle{tφm|Aφn∝an}bracketri}ht. (3) If we now sample over Srealizations of the random vectors {ψ}and calculate the average, we obtain 1 SS/summationdisplay p=1∝an}bracketle{tψp|Aψp∝an}bracketri}ht=1 SS/summationdisplay p=1D/summationdisplay n,m=1c∗ m,pcn,p∝an}bracketle{tφm|Aφn∝an}bracketri}ht. (4) Assuming that there is no correlation between the random num bers in different realizations and that the random numbers fn,pandgn,pare drawn from an even and symmetric (both with respect to each variable) probability distribution (s ee Appendix A for more details), we have lim S→∞1 SS/summationdisplay p=1c∗ m,pcn,p=E/parenleftig |c|2/parenrightig δm,n, (5) whereE(.) denotes the expectation value with respect to the probabil ity distribution used to generate the cn,p’s. In the r.h.s of (5) the subscripts of cn,phave been dropped to indicate that the expectation value does not depend on norp. It follows immediately that lim S→∞1 SS/summationdisplay p=1∝an}bracketle{tψp|Aψp∝an}bracketri}ht=E/parenleftig |c|2/parenrightig TrA=E/parenleftig |c|2/parenrightigD/summationdisplay n=1∝an}bracketle{tφn|Aφn∝an}bracketri}ht, (6) 3showing that we can compute the trace of Aby sampling over random states {ψp}, provided there is an efficient algorithm to calculate ∝an}bracketle{tψp|Aψp∝an}bracketri}ht(see Section VII). According to the central limit theorem, for a large but finite Swe have 1 SS/summationdisplay p=1c∗ m,pcn,p=E/parenleftig |c|2/parenrightig δm,n+O/parenleftigg1√ S/parenrightigg , (7) meaning that the statistical error on the trace vanishes lik e 1/√ S, a result which is all but surprising. What is surprising is that one can proof a much st ronger result as follows. Let us first normalize the cn,p’s such that, for all p, D/summationdisplay n=1|cn,p|2= 1. (8) This innocent looking step has far reaching consequences, a s we will see. First we note that the normalization renders the method exact in (the rather tr ivial) case that the matrix A is proportional to the unit matrix. The price we pay for this i s that for fixed p, thecn,pare now correlated but that is not causing problems (see Appendi x A). Second it follows that E(|c|2) = 1/D. Obviously the error can be written as TrA−D SS/summationdisplay p=1∝an}bracketle{tψp|Aψp∝an}bracketri}ht= TrRA, (9) where Rm,n≡δm,n−D SS/summationdisplay p=1c∗ m,pcn,p, (10) is a traceless (due to Eq. (8)) Hermitian matrix of random num bers. Invoking a generaliza- tion of Markov’s inequality [14] P(|X|2≥a)≤E(|X|2) a;∀a>0, (11) whereP(Q) denotes the probability for the statement Qto be true, we put X= TrRAand computeE(|X|2) to obtain a rigorous estimate of the square of the error. The result for the general case can be found in Appendix A. For a uniform distribu- tion of the cn,p’s on the hyper-sphere defined by/summationtextD n=1|cn,p|2= 1 the expression simplifies considerably and we find E/parenleftig |TrRA|2/parenrightig =DTrA†A− |TrA|2 S(D+ 1), (12) an exact expression for the variance in terms of the sample si zeS, the dimension Dof the matrixAand the (unknown) constants Tr A†Aand|TrA|. The probability that |TrRA|2exceeds a fraction aof|TrA|2is bounded by P/parenleftigg|TrRA|2 |TrA|2≥a/parenrightigg ≤1 aS(D+ 1)DTrA†A− |TrA|2 |TrA|2;∀a>0, (13) 4or, in other words, the relative statistical error eAon the estimator of the trace of Ais given by eA≡/radicaltp/radicalvertex/radicalvertex/radicalbtDTrA†A− |TrA|2 S(D+ 1)|TrA|2. (14) We see that eA= 0 ifAis proportional to a unit matrix. From (14) it follows that, i n general, we may expect eAto vanish with the square root of SD. The prefactor is a measure for the relative spread of the eigenvalues of Aand is obviously model dependent. The dependence ofeAonS,Dand the spectrum of Ais corroborated by the numerical results presented below. It is also of interest to examine the effect of notnormalizing the cn,p’s. A calculation similar to the one that lead to the above results yields eA=/radicaltp/radicalvertex/radicalvertex/radicalbtTrA†A S|TrA|2. (15) Clearly this bound is less sharp and does not vanish if Ais proportional to a unit matrix. III. REAL-TIME METHOD The distribution of eigenvalues or density of states (DOS) o f a quantum system is defined as D(ǫ) =D/summationdisplay n=1δ(ǫ−En) =1 2π/integraldisplay∞ −∞eitǫTre−itHdt, (16) whereHis the Hamiltonian of the system and nruns over all the eigenvalues of H. The DOS contains all the physical information about the equilib rium properties of the system. For instance the partition function, the energy, and the hea t capacity are given by Z=/integraldisplay∞ −∞dǫD(ǫ)e−β ǫ, (17) E=1 Z/integraldisplay∞ −∞dǫǫD(ǫ)e−β ǫ, (18) C=β2/parenleftbigg1 Z/integraldisplay∞ −∞dǫǫ2D(ǫ)e−β ǫ−E2/parenrightbigg , (19) respectively. Here β= 1/kBTandkBis Boltzmann’s constant (we put kB= 1 and /planckover2pi1= 1 from now on). As explained above the trace in the integral (16) can be estim ated by sampling over random vectors. For the statistical error analysis discuss ed below it is convenient to define a DOS-per-sample by dp(ǫ)≡1 2π/integraldisplay∞ −∞eitǫ∝an}bracketle{tψp|e−itHψp∝an}bracketri}htdt, (20) 5where the subscript plabels the particular realization of the random state |ψp∝an}bracketri}ht. The DOS is then given by D(ǫ) = lim S→∞1 SS/summationdisplay p=1dp(ǫ). (21) Schematically the algorithm to compute dp(ǫ) consists of the following steps: 1. Generate a random state |ψp(0)∝an}bracketri}ht, sett= 0. 2. Copy this state to |ψp(t)∝an}bracketri}ht. 3. Solve the TDSE for a small time step τ, replacing |ψp(t)∝an}bracketri}htby|ψp(t+τ)∝an}bracketri}ht(see Section VII for model specific details). 4. Calculate ∝an}bracketle{tψ0|ψp(t)∝an}bracketri}htand store the result. 5. RepeatNtimes from step 3. 6. Perform a Fourier transform on the tabulated result and st oredp(ǫ). In practice the Fourier transform in Eq. (16) is performed by the Fast Fourier Transform (FFT). We use a Gaussian window to account for the finite time τNused in the numerical time-integration of the TDSE. The number of time step Ndetermines the accuracy with which the eigenvalues can be computed. In Section VIII we pro ve that this systematic error in the eigenvalues vanishes as 1 /τN. Since for any reasonable physical system (or finite matrix) t he smallest eigenvalue E0 is finite, for all practical purposes dp(ǫ) = 0 forǫ < ǫ 0< E 0. The value of ǫ0is easily determined by examination of the bottom of spectrum. To comp uteZ,E, orCwe simply replace the interval [ −∞,+∞] by [ǫ0,+∞]. IV. IMAGINARY-TIME METHOD The real-time approach has the advantage that it yields info rmation on all eigenvalues and can be used to compute both dynamic and static properties without suffering from numerical instabilities. However for the computation of th e thermodynamic properties, the imaginary-time version is more efficient. We will use the imag inary-time method as an independent check on the results obtained by the real-time a lgorithm. Repeating the steps that lead to Eq. (17) we find Z= Tr exp( −βH) = lim S→∞1 SS/summationdisplay p=1∝an}bracketle{tψp|exp(−βH)ψp∝an}bracketri}ht, (22) with similar expressions for EandC. Furthermore we have ∝an}bracketle{tψp|Hne−βHψp∝an}bracketri}ht=∝an}bracketle{te−βH/2ψp|Hne−βH/2ψp∝an}bracketri}ht, (23) 6assumingHis Hermitian as usual. Therefore we only need to propagate th e random state for an imaginary time β/2 instead of β. Furthermore we do not need to perform an FFT. Disregarding these minor differences, the algorithm is the s ame as in the real-time case with τreplaced by −iτ. V. ERROR ANALYSIS Estimating the statistical error on the partition function Zis easy because it depends linearly on the trace of the (imaginary) time evolution oper ator. However the error on E andCdepends on this trace in a more complicate manner and this fac t has to be taken into account. First we define zp≡/integraldisplay∞ ǫ0dǫdp(ǫ)e−βǫ, (24) hp≡/integraldisplay∞ ǫ0dǫdp(ǫ)ǫe−βǫ, (25) wp≡/integraldisplay∞ ǫ0dǫdp(ǫ)ǫ2e−βǫ, (26) for the real-time method and zp≡ ∝an}bracketle{tψp|e−βHψp∝an}bracketri}ht, (27) hp≡ ∝an}bracketle{tψp|He−βHψp∝an}bracketri}ht, (28) wp≡ ∝an}bracketle{tψp|H2e−βHψp∝an}bracketri}ht, (29) for the imaginary-time method. For each value of βwe generate the data {zp},{hp}, and {wp}, forp= 1,... ,S . For both cases we have Z= lim S→∞z, (30) E= lim S→∞h z, (31) C= lim S→∞β2 w z−h2 z2 , (32) wherex≡S−1/summationtextS p=1xp. The standard deviations on z,h, andware given by δz=/radicaligg var(z) S−1, (33) δh=/radicaligg var(h) S−1, (34) δw=/radicaligg var(w) S−1, (35) 7where var(x)≡x2−x2denotes the variance on the data {xp}. However the sets of data {zp},{hp}and{wp}are correlated since they are calculated from the same set {|ψp∝an}bracketri}ht}. These correlations in the data are accounted for by calculat ing the covariance matrix Mk,l (k,l= 1,... ,3) the elements of which are given by xkxl−xkxl, where {x1},{x2}, and{x3} are a shorthand for {zp},{hp}, and {wp}respectively. The estimates for the errors in Z,E andCare given by δZ2=1 S−1δz2, (36) δE2=1 S−13/summationdisplay k,l=1Mk,ldE dxkdE dxl, (37) δC2=1 S−13/summationdisplay k,l=1Mk,ldC dxkdC dxl, (38) whereE=h/zand andC=β2(w/z−h2/z2). VI. EXACTLY SOLVABLE SPIN 1/2MODELS The most direct way to assess the validity of the approach des cribed above is to carry out numerical experiments on exactly solvable models. In this p aper we consider three different exactly solvable models, two spin-1/2 chains and a mean-fiel d spin-1/2 model. The former have a complicated spectrum, the latter has a highly degener ate eigenvalue distribution. These spin models differ from those studied elsewhere [11,12 ] in that they belong to the class of integrable systems. A. Spin chains Open spin chains of Lsites described by the Hamiltonian H=−JL−1/summationdisplay i=1(σx iσx i+1+ ∆σy iσy i+1)−hL/summationdisplay i=1σz i, (39) whereσx i,σy i, andσz idenote the Pauli matrices and J, ∆ andhare model parameters, can be solved exactly. They can be reduced to diagonal form by mea ns of the Jordan-Wigner transformation [15]. We have H=L/summationdisplay i,j=1/bracketleftbigg c+ iAi,jcj+1 2/parenleftig c+ iBi,jc+ j+cjB∗ j,ici/parenrightig/bracketrightbigg +hL, (40) wherec+ iandciare spin-less fermion operators and Ai,j=−J(1 + ∆)(δi,j−1+δi−1,j)−2hδi,j, (41) Bi,j=−J(1−∆)(δi,j−1−δi−1,j), (42) 8areL×Lmatrices. By further canonical transformation this Hamilt onian can be written as H=L/summationdisplay k=1Λk/parenleftbigg nk−1 2/parenrightbigg +1 2TrA+hL, (43) wherenkis the number operator of state kand the Λ k’s are given by the solution of the eigenvalue equation (A−B)(A+B)φk= Λ2 kφk. (44) In the general case this eigenvalue problem of the L×LHermitian matrix ( A−B)(A+B) is most easily solved numerically. In the present paper we co nfine ourselves to two limiting cases: The XY model (∆ = 1) and the Ising model in a transverse fi eld (∆ = 0). B. Mean field model The Hamiltonian of the mean-field model reads H=−J LL/summationdisplay i>j=1/vector σi·/vector σj−hL/summationdisplay i=1σz i, (45) and can be rewritten as H=−2J L/vectorS·/vectorS−2hSz+3 2J, (46) with /vectorS=1 2L/summationdisplay i=1/vector σi. (47) The single spin- L/2 Hamiltonian has eigenvalues El,m=−2Jl(l+ 1)/L−2hm+3 2J, (48) with degeneracy nl,m=2l+ 1 L/2 +l+ 1/parenleftigg L L/2−l/parenrightigg . (49) This rather trivial model serves as a test for the case of high ly degenerate eigenvalues. VII. TIME EVOLUTION For the approach outlined in Sections III and IV to be of pract ical use it is necessary that the matrix elements of the exponential of Hcan be calculated efficiently. The purpose of this section is to describe how this can be done. 9The general form of the Hamiltonians of the models we study is H=−L/summationdisplay i,j=1/summationdisplay α=x,y,zJα i,jσα iσα j−L/summationdisplay i=1/summationdisplay α=x,y,zhα iσα i, (50) where the first sum runs over all pairs Pof spins,σα i(α=x,y,z ) denotes the α-th component of the spin-1/2 operator representing the i-th spin. For both methods, we have to calculate the evolution of a random state, i.e. U(τ)|ψ∝an}bracketri}ht ≡exp(−iτH)|ψ∝an}bracketri}htorU(τ)|ψ∝an}bracketri}ht ≡exp(−τH)|ψ∝an}bracketri}ht for the real and imaginary time method respectively. We will discuss the real-time case only, the imaginary-time problem can be solved in the same manner. Using the semi-group property of U(t) we writeU(t) =U(τ)mwheret=mτ, the main step is to replace U(τ) by a symmetrized product-formula approximation [17]. For the case at hand it is expedient to take U(τ)≈/tildewideU(τ)≡e−iτHz/2e−iτHy/2e−iτHxe−iτHy/2e−iτHz/2, (51) where Hα=−L/summationdisplay i,j=1Jα i,jσα iσα j−L/summationdisplay i=1hα iσα i;α=x,y,z. (52) Other decompositions [11,18] work equally well but are some what less efficient for the cases at hand. In the real-time approach/tildewideU(τ) is unitary and hence the method is unconditionally stable [17] (also the imaginary-time method can be made unco nditionally stable). It can be shown that ∝bardblU(τ)−/tildewideU(τ)∝bardbl ≤sτ3(s >0 a constant) [19], implying that the algorithm is correct to second order in the time step τ[17]. Usually it is not difficult to choose τso small that for all practical purposes the results obtained c an be considered as being “exact”. Moreover, if necessary,/tildewideU(τ) can be used as a building block to construct higher-order algorithms [20–23]. In Appendix B we will derive bounds on th e error in the eigenvalues when they are calculated using a symmetric product formula. As basis states {|φn∝an}bracketri}ht}we take the direct product of the eigenvectors of the Sz i(i.e. spin-up |↑i∝an}bracketri}htand spin-down |↓i∝an}bracketri}ht). In this basis, e−iτHz/2changes the input state by altering the phase of each of the basis vectors. As Hzis a sum of pair interactions it is trivial to rewrite this operation as a direct product of 4x4 diagonal matrices (cont aining the interaction-controlled phase shifts) and 4x4 unit matrices. Still working in the sam e representation, the action of e−iτHy/2can be written in a similar manner but the matrices that conta in the interaction- controlled phase-shift have to be replaced by non-diagonal matrices. Although this does not present a real problem it is more efficient and systematic t o proceed as follows. Let us denote byX(Y) the rotation by π/2 of each spin about the x(y)-axis. As e−iτHy/2=XX†e−iτHy/2XX†=Xe−iτH′ z/2X†, (53) it is clear that the action of e−iτHy/2can be computed by applying to each spin, the inverse of Xfollowed by an interaction-controlled phase-shift and Xitself. The prime in (53) indicates thatJz i,jandhz iinHzhave to be replaced by Jy i,jandhy irespectively. A similar procedure is used to compute the action of e−iτHx. We only have to replace XbyY. 10VIII. ACCURACY OF THE COMPUTED EIGENVALUES First we consider the problem of how to choose the number of ti me stepsNto obtain the DOS with acceptable accuracy. According to the Niquist s ampling theorem employing a sampling interval ∆ t=π/max i|Ei|is sufficient to cover the full range of eigenvalues. On the other hand the time step also determines the accuracy of t he approximation/tildewideU(τ). Let us call the maximum value of τwhich gives satisfactory accuracy τ0(for the imaginary-time method, this is the only parameter). For the examples treate d hereτ0<∆t), implying that we have to use more steps to solve the TDSE than we actually use to compute the FFT. Eigenvalues that differ less than ∆ ǫ=π/N∆tcannot be identified properly. However since ∆ǫ∝N−1we only have to extend the length of the calculation by a facto r of two to increase the resolution by the same factor. At first glance the above reasoning may seem to be a little opti mistic. Indeed it ap- parently overlooks the fact that if we integrate the TDSE ove r longer and longer times the error on the wave function increases too (although it remain s bounded because of the un- conditional stability of the product formula algorithm). I n fact it has been shown that in general [17] ∝bardble−itH|ψ(0)∝an}bracketri}ht −/tildewideUm(τ)|ψ(0)∝an}bracketri}ht∝bardbl ≤cτ2t, (54) wheret=mτ, suggesting that the loss in accuracy on the wave function ma y well com- pensate for the gain in resolution that we get by using more da ta in the Fourier transform. Fortunately this argument does not apply when we want to dete rmine the eigenvalues as we now show. As before we will discuss the real-time algorithm o nly because the same reasoning (but different mathematical proofs) holds for the imaginary -time case. Consider the time-step operator (52). Using the fact that an y unitary matrix can be written as the matrix exponential of a Hermitian matrix we ca n write /tildewideU(τ) =e−iτHz/2e−iτHy/2e−iτHxe−iτHy/2e−iτHz/2≡e−iτ/tildewideH(τ). (55) It is clear that in practice the real-time method yields the s pectrum of/tildewiderH(τ), not the one ofH. Therefore the relevant question is: How much do the spectra of/tildewiderH(τ) andHdiffer? In Appendix B we give a rigorous proof that the difference betw een the eigenvalues of/tildewiderH(τ) andHvanishes as τ2. In other words the value of m(ort=mτ) has no effect whatsoever on the accuracy with which the spectrum can be determined. Th erefore the final conclusion is that the error in the eigenvalues vanishes as τ2/NwhereNis the number of data points used in the Fourier transform of Tr e−it/tildewideH(τ). IX. RESULTS In all our calculations we take J= 1 andh= 0, except for the Ising model in a transverse field, where we take h= 0.75. The random numbers cn,pare generated such that the conditions Eqs. (A3) and (A4) are satisfied. We use two diff erent techniques to generate these random numbers: 111. A uniform random number generator produces {fn,p}and{gn,p}with−1≤fn,p,gn,p≤ 1. We then normalize the vector (see Eq. (8)). 2. Thecn,p’s are obtained from a two-variable (real and imaginary part ) Gaussian random number generator and the resulting vector is normalized. Both methods satisfy the basic requirements Eqs. (A3) and (A 4) but because the first picks points out of a 2 D-dimensional hypercube and subsequently projects the vect or onto a sphere, the points are not distributed uniformly over the su rface of the unit hyper-sphere. The second method is known to generate numbers which are dist ributed uniformly over the hyper-surface. Although the first method does not satisfy al l the mathematical conditions that lead to the error (14), our numerical experiments with b oth generators give identical results, within statistical errors of course. Also within t he statistical errors, the results from the imaginary and real-time algorithm are the same. The refore we only show some representative results as obtained from the real-time algo rithm. In Fig. 1 we show a typical result for the DOS D(ǫ) of the XY model, the Ising model in a transverse field and the mean-field model, all with L= 15 spins and using S= 20 samples. Because of the very high degeneracy we plotted the D OS for the mean-field model on a logarithmic scale. In Fig. 2 we show the relative error δZ/Z based on Eq. (36) for the three models of various size, as obtained from the simulation (symbols). For these fi gures we used the imaginary- time algorithm, because then the statistical error can be re lated toeAdirectly (see Eq. (14) withA= exp( −βH)). The theoretical results (lines) for the error estimate, obtained by a direct exact numerical evaluation of (14) are shown too. We c onclude that for all systems, lattice sizes and temperatures there is very good agreement between numerical experiment and theory. Results for the energy Eand specific heat Care presented in Fig. 3 (XY model), 4 (Ising model in a transverse field), and 5 (mean-field model). The sol id lines represent the exact result for the case shown. Simulation data as obtained from S= 5 andS= 20 samples are represented by symbols, the estimates of the statistical er ror by error bars. We see that the data are in excellent agreement with the exact results and eq ually important, the estimate for the error captures the deviation from the exact result ve ry well. We also see that in general the error decreases with the system size. Both the im aginary and real-time method seem to work very well, yielding accurate results for the ene rgy and specific heat of quantum spin systems with modest amounts of computational effort. X. CONCLUSIONS The theoretical analysis presented in this paper gives a sol id justification of the remark- able efficiency of the real-time equation-of-motion method f or computing the distribution of all eigenvalues of very large matrices. The real-time metho d can be used whenever the more conventional, Lanczos-like, sparse-matrix techniques ca n be applied: Memory and CPU re- quirements for each iteration (time-step) are roughly the s ame (depending on the actual implementation) for both approaches. We do not recommend using the real-time method if one is inter ested in the smallest (or largest) eigenvalue only. Then the Lanczos method is com putationally more efficient 12because it needs less iterations (time-steps) than the real -time approach. However if one needs information about all eigenvalues and direct diagona lization is not possible (because of memory/CPU-time) there is as yet no alternative to the rea l-time method. The matrices used in this example (up to 32768 ×32768) are not representative in this respect: The real-time method has been used to compute the distribution o f eigenvalues for matrices of dimension 16777216 ×16777216 [11]. Once the eigenvalue distribution is known the thermodynami c quantities directly follow. However if one is interested in the accurate determination o f the temperature dependence of thermodynamic (and static correlation functions) prope rties but not in the eigenvalue distribution itself, the imaginary-time method is by far th e most efficient method to compute these quantities. For instance the calculation of the therm odynamic properties for βJ= 0,... ,10 of a 15-site spin-1/2 system (i.e. implicitly solving the full 32768 ×32768 eigenvalue problem) takes 1410 seconds per sample on a Mobile Pentium II I 500 Mhz system. Finally we remark that although we used quantum-spin models to illustrate various aspects, there is nothing in the real or imaginary-time meth od that is specific to the models used. The only requirement for these methods to be useful in p ractice is that the matrix is sparse and (very) large. ACKNOWLEDGMENTS Support from the Dutch “Stichting Nationale Computer Facil iteiten (NCF)” and the Dutch “Stichting voor Fundamenteel Onderzoek der Materie ( FOM)” is gratefully acknowl- edged. APPENDIX A: EXPECTATION VALUE CALCULATION In this appendix we calculate the expectation value of the er ror squared, as defined in Section II. By definition we have E/parenleftig |TrRA|2/parenrightig =E /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 SS/summationdisplay p=1D/summationdisplay m,n=1/parenleftig δm,n−Dc∗ m,pcn,p/parenrightig Am,n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2  =1 S2S/summationdisplay p,p′=1D/summationdisplay k,l,m,n =1/parenleftig δk,lδm,n−Dδk,lE(c∗ m,pcn,p) −Dδm,nE(ck,p′c∗ l,p′) +D2E(c∗ m,pcn,pck,p′c∗ l,p′)/parenrightig A∗ k,lAm,n, (A1) wherepandp′label the realization of the random numbers cn,p≡fn,p+ign,p. First we assume that different realizations p∝ne}ationslash=p′are independent implying that E(c∗ m,pcn,pck,p′c∗ l,p′)p/negationslash=p′=E(c∗ m,pcn,p)E(ck,p′c∗ l,p′). (A2) Second we assume that the random numbers are drawn from a prob ability distribution that 13is an even function of each variable P(f1,p,g1,p,f2,p,g2,p,... ,f k,p,gk,p,... ,f D,p,gD,p) =P(f1,p,g1,p,f2,p,g2,p,... ,−fk,p,gk,p,... ,f D,p,gD,p) =P(f1,p,g1,p,f2,p,g2,p,... ,f k,p,−gk,p,... ,f D,p,gD,p), (A3) and symmetric under interchange of any two variables P(f1,p,g1,p,... ,f i,p,gi,p,... ,f j,p,gj,p,... ,f D,p,gD,p) =P(f1,p,g1,p,... ,f j,p,gi,p,... ,f i,p,gj,p,... ,f D,p,gD,p) =P(f1,p,g1,p,... ,g i,p,fi,p,... ,f j,p,gj,p,... ,f D,p,gD,p), (A4) for alli,j,k = 1,... ,D . This is most easily done by drawing individual numbers from the same even probability distribution i.e. P(f1,p,g1,p,... ,f j,p,gi,p,... ,f i,p,gj,p,... ,f D,p,gD,p) =D/productdisplay n,m=1P(fn,p)P(gn,p), (A5) whereP(x) =P(−x). Normalizing the vector ( f1,p,g1,p,... ,f D,p,gD,p) such that/summationtextD i=1|cn,p|2= 1 (forp= 1,... ,S ) does not affect the basic requirements (A3) and (A4). Making use of the above properties of P(f1,g1,... ,f D,gD) we find that E(c∗ m,pcn,p) =δm,nE(|cm,p|2) =δm,nE(|c|2), (A6) where in the last equality we omitted the subscripts of cm,pto indicate that the expectation value does not depend on morp. An expectation value of a product of two c∗’s and twoc’s can be written as E(c∗ m,pcn,pck,p′c∗ l,p′) =(1 −δp,p′)δm,nδk,lE(|cm,p|2)E(|cm,p′|2) +δp,p′δm,nδk,l(1−δmk)E(c∗ mcmckc∗ k) +δp,p′δm,kδn,l(1−δm,n)E(c∗ mcncmc∗ n) +δp,p′δm,lδn,k(1−δm,n)E(c∗ mcncnc∗ m) +δp,p′δm,lδn,kδm,nE(c∗ mcmcmc∗ m) =(1−δp,p′)δm,nδk,lE(|c|2)2 +δp,p′δm,nδk,l(1−δm,k)E(|cm,p|2|ck,p|2) +δp,p′δm,kδn,l(1−δm,n)E(|cm,p|2|cn,p|2) +δp,p′δm,lδn,k(1−δm,n)E(c∗ m,pcn,pcn,pc∗ m,p) +δp,p′δm,lδn,kδm,nE(|cm,p|4). (A7) Furthermore for m∝ne}ationslash=nwe have E(c∗ m,pcn,pcn,pc∗ m,p) =E((f2 m,p−2ifm,pgm,p−g2 m,p)(f2 n,p+ 2ifn,pgn,p−g2 n,p)) =E(f2 m,pf2 n,p) + 2iE(f2 m,pfn,pgn,p)−E(f2 m,pg2 n,p) −2iE(fm,pgm,pf2 n,p) + 4E(fm,pfn,pgm,pgn,p) + 2iE(fm,pgm,pg2 n,p) −E(g2 m,pf2 n,p)−2iE(g2 m,pfn,pgn,p) +E(g2 m,pg2 n,p) =E(f2 m,pf2 n,p)−E(g2 m,pf2 n,p)−E(f2 m,pg2 n,p) +E(g2 m,pg2 n,p) =0. (A8) 14By symmetry E(|cm,p|2|cn,p|2) does not depend on m,norpand the same holds for E(|cm,p|4). The fact that the vector ( c1,p,... ,c D,p) is normalized yields the identities E/parenleftiggD/summationdisplay n=1|cn,p|2/parenrightigg =D/summationdisplay n=1E(|cn,p|2) =DE(|c|2) =E(1) = 1, (A9) and E /parenleftiggD/summationdisplay n=1|cn,p|2/parenrightigg2 =D/summationdisplay m,n=1E(|cn,p|2|cm,p|2) =D/summationdisplay n=1E(|cn,p|4) +D/summationdisplay m,n=1(1−δm,n)E(|cn|2|cm|2) =DE(|c|4) +D(D−1)E(|c|2|c′|2) =E(1) = 1, (A10) wherecandc′refer to two different complex random variables. Therefore w e have E(|c|2) = 1/D, (A11) and E(|c|2|c′|2) =1−DE(|c|4) D(D−1). (A12) Substitution into (A7) yields E(c∗ m,pcn,pck,p′c∗ l,p′) =(1 −δp,p′)δm,nδk,lD−2 +δp,p′1−DE(|c|4) D(D−1)(δm,nδk,l(1−δm,k) +δm,kδn,l(1−δm,n)) +δp,p′δm,lδn,kδm,nE(|c|4). (A13) and the final result for the variance reads E/parenleftig |TrRA|2/parenrightig =1 S/parenleftiggD−D2E(|c|4) D−1TrA†A+1−D2E(|c|4) D−1|TrA|2 +(D+ 1)D2E(|c|4)−2D D−1D/summationdisplay n=1|An,n|2/parenrightigg . (A14) An expression for the fourth moment E(|c|4) cannot be derived from general properties of the probability distribution or normalization of random ve ctor. We can only make progress by specifying the former explicitly. As an example we take a p robability distribution such that for each realization pthe random numbers fn,pandgn,pare distributed uniformly over the surface of a 2 D-dimensional sphere of radius 1. This probability distribu tion can be written as P(f1,g1,f2,g2,... ,f D,gD)∝δ(f2 1+g2 1+f2 2+g2 2+...+f2 D+g2 D−1), (A15) 15where we omitted the subscript pbecause it is irrelevant for what follows. The even moments of|cn|= (f2 n+g2 n)1/2are defined by E(|c|2M) =/integraltext∞ −∞(f2 1+g2 1)Mδ(f2 1+g2 1+f2 2+g2 2+...+f2 D+g2 D−1)df1dg1df2dg2...df DdgD/integraltext∞ −∞δ(f2 1+g2 1+...+f2 D+g2 D−1)df1dg1...df DdgD. (A16) It is expedient to introduce an auxiliary integration varia bleXby E(|c|2M) =/integraltext∞ −∞XMδ(f2 1+g2 1−X)δ(f2 2+g2 2+...+f2 D+g2 D−(1−X))dXdf 1dg1df2dg2...df DdgD/integraltext∞ −∞δ(f2 1+g2 1+...+f2 D+g2 D−1)df1dg1...df DdgD. (A17) We can perform the integration over Xlast and regard (A17) as the M-th moment of the variableXwith respect to the probability distribution P(X) =/integraltext∞ −∞δ(f2 1+g2 1−X)δ(f2 2+g2 2+...+f2 D+g2 D−(1−X))df1dg1df2dg2...df DdgD/integraltext∞ −∞δ(f2 1+g2 1+...+f2 D+g2 D−1)df1dg1...df DdgD. (A18) The calculation of P(X) amounts to computing integrals of the form IN(X) =/integraldisplay∞ −∞δ/parenleftiggN/summationdisplay n=1x2 n−X/parenrightigg dx1dx2...dx N. (A19) Changing to spherical coordinates we have IN(X) =2πN/2 Γ(N/2)/integraldisplay∞ 0rN−1δ(r2−X)dr =πN/2 Γ(N/2)XN/2−1θ(X), (A20) yielding P(X) =I2(X)I2D−2(1−X) I2D(1) = (D−1)(1−X)D−2θ(X)θ(1−X). (A21) The moments E(|c|2M) are given by E(|c|2M) =/integraldisplay∞ −∞XMP(X)dX = (D−1)/integraldisplay1 0XM(1−X)D−2dX =Γ(D)Γ(1 +M) Γ(D+M), (A22) and the values of interest to us are E(|c|0) = 1, E(|c|2) =1 D, E(|c|4) =2 D(D+ 1), (A23) where the first two results provide some check on the procedur e used. Substituting (A23) into (A14) yields E/parenleftig |TrRA|2/parenrightig =DTrA†A− |TrA|2 S(D+ 1). (A24) 16APPENDIX B: ERROR BOUNDS Here we prove that the difference between the eigenvalues of t he Hermitian matrix A+B and those obtained from the approximate time-evolution exp (zA/2) exp(zB) exp(zA/2) (z= −iτ,−τ) is bounded by τ2. In the following we assume AandBare Hermitian matrices and takeτa real, non-negative number. We start with the imaginary-ti me case. We define the difference R(τ) by R(τ)≡eτ(A+B)−eτA/2eτBeτA/2 =1 4/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλA/2eλB{e−νB[2B,[A,B]]eνB +eνA/2[A,[A,B]]e−νA/2}eλA/2e(τ−λ)(A+B), (B1) a well-known result [21]. We have [23] ||R(τ)|| ≤1 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλA/2e(λ−ν)B[2B,[A,B]]eνBeλA/2e(τ−λ)(A+B)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +1 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλA/2eλBeνA/2[A,[A,B]]e(λ−ν)A/2e(τ−λ)(A+B)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤1 4/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλ||A||/2e(λ−ν)||B||||[2B,[A,B]]||eν||B||eλ||A||/2e(τ−λ)(||A||+||B||) +1 4/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλ||A||/2eλ||B||eν||A||/2||[A,[A,B]]||e(λ−ν)||A||/2e(τ−λ)(||A||+||B||) =1 24τ3eτ(||A||+||B||)(||[A,[A,B]]||+||[2B,[A,B]]||), (B2) and ||R(−τ)|| ≤1 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay−τ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλA/2e(λ−ν)B[2B,[A,B]]eνBeλA/2e(−τ−λ)(A+B)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +1 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay−τ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλA/2eλBeνA/2[A,[A,B]]e(λ−ν)A/2e(−τ−λ)(A+B)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle =1 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνe−λA/2e(−λ+ν)B[2B,[A,B]]e−νBe−λA/2e(−τ+λ)(A+B)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +1 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνe−λA/2e−λBe−νA/2[A,[A,B]]e(−λ+ν)A/2e(−τ+λ)(A+B)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤1 4/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλ||A||/2e(λ−ν)||B||||[2B,[A,B]]||eν||B||eλ||A||/2e(τ−λ)(||A||+||B||) +1 4/integraldisplayτ 0dλ/integraldisplayλ 0dµ/integraldisplayµ 0dνeλ||A||/2eλ||B||eν||A||/2||[A,[A,B]]||e(λ−ν)||A||/2e(τ−λ)(||A||+||B||) =1 24τ3eτ(||A||+||B||)(||[A,[A,B]]||+||[2B,[A,B]]||). (B3) Hence the bound in R(τ) does not depend on the sign of τso that we can write ||R(τ)|| ≤s|τ|3e|τ|(||A||+||B||), (B4) 17where s≡1 24||[A,[A,B]]||+||[2B,[A,B]]||. (B5) For realτwe have eτA/2eτBeτA/2≡eτC(τ), (B6) whereC(τ) is Hermitian. Clearly we have eτ(A+B)−eτC(τ)=R(τ). (B7) We already have an upperbound on R(τ) and now want to use this knowledge to put an upperbound on the difference in eigenvalues of C(τ) andA+B. In general, for two Hermitian matricesUandVwith eigenvalues {un}and{vn}respectively, both sets sorted in non- decreasing order, we have [2] |un−vn| ≤ ||U−V||,∀n. (B8) Denoting the eigenvalues of A+BandC(τ) byxn(0) andxn(τ) respectively, combining Eq. (B4) and (B8) yields |eτxn(0)−eτxn(τ)| ≤s|τ|3e|τ|(||A||+||B||). (B9) To find an upperbound on |xn(0)−xn(τ)|we first assume that xn(0)≤xn(τ) and take τ≥0. It follows from (B9) that eτ(xn(τ)−xn(0))−1≤sτ3eτ(||A||+||B||)−τxn(0), (B10) Forx≥0,ex−1≥xand we have −xn(0)≤ ||A+B|| ≤ ||A||+||B||. Hence we find xn(τ)−xn(0)≤sτ2e2τ(||A||+||B||). (B11) An upperbound on the difference in the eigenvalues between C(τ) andA+Bcan equally well be derived by considering the inverse of the exact and ap proximate time-evolution operator (B6). This is useful for the case xn(0)> x n(τ): Instead of using (B7) we start from exp( −τ(A+B))−exp(−τC(−τ)) =R(−τ) (τ≥0). Note that the set of eigenvalues of a matrix and its inverse are the same and that the matrices w e are considering here, i.e. matrix exponentials, are nonsingular. Making use of Eq. (B4 ) forR(−τ) gives |e−τxn(0)−e−τxn(τ)| ≤s|τ|3e|τ|(||A||+||B||), (B12) and proceeding as before we find τ(xn(0)−xn(τ))≤eτ(xn(0)−xn(τ))−1≤sτ3e2τ(||A||+||B||). (B13) Putting the two cases together we finally have |xn(τ)−xn(0)| ≤sτ2e2τ(||A||+||B||). (B14) 18Clearly (B14) proves that the differences in the eigenvalues ofA+BandC(τ) vanish as τ2. We now consider the case of the real-time algorithm ( z=−iτ). For Hermitian matrices AandBthe matrix exponentials are unitary matrices and hence thei r norm equals one. This simplifies the derivation of the upperbound on R(−iτ). One finds [17] ||R(−iτ)||E≤s|τ|3, (B15) where ||A||2 E≡TrA†Adenotes the Euclidean norm of the matrix A[2]. In general the eigenvalues of a unitary matrix are complex valued and there fore the strategy adopted above to use the bound on R(τ) to set a bound on the difference of the eigenvalues no longer works. Instead we invoke the Wielandt-Hoffman theorem [24]: IfUandVare normal matrices with eigenvalues uiandvirespectively, then there exists a suitable rearrangement (a permutation ̺of the numbers 1,... ,n ) of the eigenvalues so that N/summationdisplay j=1|uj−v̺(j)|2≤ ||U−V||2 E. (B16) LetUandVdenote the exact and approximate real-time evolution opera tors respectively. The eigenvalues of A+BandC(τ) arexn(0) andxn(τ) respectively. All the xn’s andxn(τ)’s are real numbers. According to the Wielandt-Hoffman theorem N/summationdisplay j=1|ei τ x j(0)−ei τ yj(τ)|2≤ ||R(−iτ)||2 E≤s2τ6. (B17) whereyj(τ) =x̺(j)(τ),̺being the permutation such that inequality (B17) is satisfie d. We see that Eq. (B17) only depends on ( τxj(0) mod 2π) and (τyj(τ) mod 2π), but so does the DOS (see Eq. (16)). Since the inequality (B17) and the DOS only depend on these “angles” modulo 2 π, there is no loss of generality if we make the choice 0≤ |τ(xj(0)−yj(τ))| ≤π. (B18) Rewriting the sum in (B17) we have N/summationdisplay j=1|ei τ x j(0)−ei τ yj(τ)|2=N/summationdisplay j=1(2−2 cos(τ(xj(0)−yj(τ)))) = 4N/summationdisplay j=1sin2(τ/2 (xj(0)−yj(τ))). (B19) As we have sin2x≤4x2 π2,for 0≤ |x| ≤π/2, (B20) the restriction Eq. (B18) allows us to write N/summationdisplay j=1(xj(0)−yj(τ))2≤π2s2 4τ4, (B21) 19implying |xj(0)−yj(τ)| ≤πs 2τ2. (B22) In summary, we have shown that in the real-time case there exi sts a permutation of the approximate eigenvalues such that the difference with the ex act ones vanishes as τ2. Finally we note that both upperbounds (B22) and (B14) hold fo r arbitrary Hermitian matricesAandBand are therefore rather weak. Except for the fact that they p rovide a sound theoretical justification for the real and imaginary -time method, they are of little practical value. 20REFERENCES [1] G.D. Mahan, Many-Particle Physics , (Plenum Press, New York, 1981). [2] J.H. Wilkinson, The Algebraic Eigenvalue Problem , (Clarendon Press, Oxford, 1965). [3] G.H. Golub and C.F. Van Loan, Matrix computations , (John Hopkins University Press, Baltimore, 1983). [4] R. Alben, M. Blume, H. Krakauer, and L. Schwartz, Phys. Re v. B12, 4090 (1975). [5] M.D. Feit, J.A. Fleck, and A. Steiger, J. Comput. Phys 47, 412 (1982). [6] H. De Raedt and P. de Vries, Z. Phys. B 77, 243 (1989). [7] T. Kawarabayashi and T. Ohtsuki, Phys. Rev. B 53, 6975 (1996). [8] T. Ohtsuki and T. Kawarabayashi, J. Phys. Soc. Jap. 66, 314 (1997). [9] T. Iitaka, RIKEN Rev. 19, 136 (1998). [10] T. Iitaka and T. Ebisuzaki, Mic. Eng. 47, 321 (1999). [11] P. de Vries and H. De Raedt, Phys. Rev. B 47, 7929 (1993). [12] H. De Raedt, A. Hams, K. Michielsen, S. Miyashita, and K. Saito, Prog. Theor. Phys., in press. [13] D.S. Abrams, and S. Lloyd, Phys. Rev. Lett. 83, 5162 (1999). [14] G.R. Grimmet and D.R. Stirzaker, Probability and Random Processes , (Clarendon, Oxford, 1992). [15] E. Lieb, T. Schultz, and D.C. Mattis, Ann. of Phys. 16, 407 (1961). [16] P. Shor, in Proc. 35th Annu. Symp. Foundations of Computer Science , S. Goldwasser ed., 124 (IEEE Computer Soc., Los Alamitos CA, 1994). [17] H. De Raedt, Comp. Phys. Rep. 7, 1 (1987). [18] M. Suzuki, S. Miyashita, and A. Kuroda, Prog. Theor. Phy s.58, 1377 (1977). [19]∝bardblX∝bardbldenotes the spectral norm of the matrix X, see [2,3]. [20] H. De Raedt and B. De Raedt, Phys. Rev. A 28, 3575 (1983). [21] M. Suzuki, J. Math. Phys. 26, 601 (1985). [22] H. De Raedt and K. Michielsen, Comp. in Phys. 8, 600 (1994). [23] M. Suzuki, J. Math. Phys. 61, 3015 (1995). [24] A.J. Hoffman and H.W. Wielandt, Duke Math. J. 20, 37 (1953). 21FIGURES 0102030405060708090 -15 -10 -5 0 5 10 15DOS ε0123456789 -15 -10 -5 0 5 10 15DOS ε10-1210-810-41 -7 -6 -5 -4 -3 -2 -1 0 1 2DOS ε FIG. 1. The density of states (DOS) as obtained from the real- time algorithm for spin chains of length L= 15 and for S= 20 random initial states. Left: XY model; middle: Ising mod el in a transverse field; right: Mean-field model. For the mean-field model a logarithmic scale was used to show the highly-degenerate spectrum more clearly. 0.0010.010.11 0 1 2 3 4 5δZ/Z T0.0010.010.11 0 0.5 1 1.5 2 2.5 3δZ/Z T0.0010.010.1 0 0.5 1 1.5 2 2.5 3δZ/Z T FIG. 2. The relative error δZ/Z (see Eq. (36)) on a logarithmic scale as a function of temper- ature T≡1/βand for various system sizes. Left: XY model; middle: Ising m odel in a transverse field; right: Mean-field model. Solid lines: eA(with A=e−βH, see Eq. (14)) for L= 6; dashed lines: eAforL= 10; dash-dotted line: eAforL= 15. Crosses: Simulation data for S= 20 and L= 6; squares: Simulation data for S= 20 and L= 10; circles: Simulation data for S= 20 and L= 15. 22-7-6-5-4-3-2-1 0 1 2 3 4 5E T-13-12-11-10-9-8-7-6-5-4-3 0 1 2 3 4 5E T-20-18-16-14-12-10-8-6-4 0 1 2 3 4 5E T 00.511.522.5 0 1 2 3 4 5C T00.511.522.533.5 0 1 2 3 4 5C T0123456 0 1 2 3 4 5C T FIG. 3. Energy (top) and specific heat (bottom) of the XY-mode l (see (39)) with ∆ = 1, h= 0 and J= 1. Left: L= 6; middle: L= 10; right: L= 15. Solid lines: Exact result. crosses: Simulation data using S= 5 samples; squares: Simulation data using S= 20 samples. Error bars: One standard deviation. -6.5-6-5.5-5-4.5-4-3.5-3-2.5-2 0.5 1 1.5 2 2.5 3E T-11-10-9-8-7-6-5-4 0.5 1 1.5 2 2.5 3E T-17-16-15-14-13-12-11-10-9-8-7-6 0.5 1 1.5 2 2.5 3E T 00.511.522.5 0.5 1 1.5 2 2.5 3C T00.511.522.533.5 0.5 1 1.5 2 2.5 3C T00.511.522.533.544.55 0.5 1 1.5 2 2.5 3C T FIG. 4. Energy (top) and specific heat (bottom) of the Ising mo del in a transverse field (see (39)) with ∆ = 0, h= 0.75 and J= 1. Left: L= 6; middle: L= 10; right: L= 15. Solid lines: Exact result. crosses: Simulation data using S= 5 samples; squares: Simulation data using S= 20 samples. Error bars: One standard deviation. 23-3-2.5-2-1.5-1-0.50 0 0.5 1 1.5 2 2.5 3E T-5-4.5-4-3.5-3-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3E T-7-6-5-4-3-2-10 0 0.5 1 1.5 2 2.5 3E T 00.20.40.60.811.21.41.61.82 0 0.5 1 1.5 2 2.5 3C T00.511.522.533.54 0 0.5 1 1.5 2 2.5 3C T012345678 0 0.5 1 1.5 2 2.5 3C T FIG. 5. Energy (top) and specific heat (bottom) of the mean-fie ld model (see (45)) with J= 1 andh= 0. Left: L= 6; middle: L= 10; right: L= 15. Solid lines: Exact result. crosses: Simulation data using S= 5 samples; squares: Simulation data using S= 20 samples. Error bars: One standard deviation. 24

arXiv:physics/0004017v1 [physics.class-ph] 11 Apr 2000Theory of acoustic analog of magneto-optic polar Kerr effect under magnon-phonon resonance A. M. Burkhanov, K. B. Vlasov, V. V. Gudkov, and B. V. Tarasov February 2, 2008 1 Introduction The acoustic analog of magneto-optic Kerr effect consists of variation of the polarization of the elastic wave after its reflection from an interface between magnetic medium and isotropic non-magnetic one. Quantitat ive character- istics of the effects are polarization parameters: ε– the ellipticity which modulus is the ratio of the minor and major ellipse axes and φ– the angle of rotation of the polarization plane or, more correctly, of the major ellipse axis if ε/negationslash= 0. There are three possible variants according to directions o f the static magnetic induction B, plane of incidence of the wave and the unit vector q normal to the interface plane: polar ( B/bardblq); meridional (or longitudinal) (q×k⊥B); and equatorial (or transverse) ( q×k/bardblB). The incident waves are called p-type if the elastic displacement vector uis perpendicular toq×k, and s-type if uis parallel to this vector. The acoustic analog of the Kerr effect was predicted by Vlasov and Kuleev [1]. Theory of the effect for an inclined incidence of a wave wa s developed by Vlasov and Babushkin [2]. In this paper an isotropic magne tic medium (atB= 0) was discussed. Here we consider the following model. The medium I is a semi-i nfinite isotropic non-magnetic and non-dissipative while the medi um II is a fer- romagnetic one with cubic lattice and easy axis along [111]- type crystallo- graphic direction. The interface is perpendicular to the [0 01] axis and the plane of incidence is given by the equation x= 0. The medium II has a form of a sphere with a small (respectively the radius of the spher e) plane area contacting with the medium I. 1In this case the elastic displacements of the reflected shear wavesu′may be represented by a sum of mutually orthogonal vectors u′ xandu′ ⊥, where u′ x=u′ xexand u′ ⊥=u′ ⊥e⊥, (1) andexande⊥are unit vectors. By introducing reflection coefficients R± nfor circular components defined by R± n≡u′ x/un±iu′ ⊥/unand expressing them in the form R± n=|R± n|exp(iρ± n), we obtain expressions for the ellipticity and the rotation of the polarization of an elastic wave of n-type upon reflection from the interface as ε=|R+ n| − |R− n| |R+n|+|R−n|and φ=1 2/parenleftBig ρ− n−ρ+ n/parenrightBig . (2) In definition (1) and later on a symbol with a prime relates to c haracteristics of a reflected wave, with two primes – to refracted (transmitt ed to the second medium) and without primes – to the incident wave; nmay refer to shear mode of arbitrary linear polarization. 2 Boundary conditions We will consider classical approach given in Ref. 3 and gener alize it to the case of magnetoelastic interaction in one of the contacting media. Thus, the boundary conditions require continuity of the displacemen ts at the interface, and of the force acting on the interface: un+3/summationdisplay i=1u′ i−m/summationdisplay j=1u′′ j= 0, (3) (τmz)n+3/summationdisplay i=1(τmz)′ i−m/summationdisplay j=1(τmz)′′ j= 0, (4) The sums in Eqs. (3)–(4) contain contributions of all the nor mal (eigen) modes that have elastic displacements. In harmonic approxi mation the bound- ary conditions can be presented as ones for complex amplitud es and phases. The last one leads to the following conclusions: frequences of all the waves (incident, reflected, and refracted) are equal, all the wave vectors belong to a common plane, namely, the plane of incidence, and, besides , sinθ s=sinθ′ i s′ i=sinθ′′ j s′′ j, (5) 2where θis the angle of incidence, θ′ ithat of reflection, corresponding to the wave with phase velocity s′ i, and θ′′ jthat of refraction, corresponding to the wave with phase velocity s′′ j. The boundary conditions given by Eqs. (3)–(4) should be appe nded with ones for variable magnetization those were formulated in Re f. 1. Remind: all the eigen modes that have elastic displacements should be represented in boundary conditions and the tensions due to t he magnetoe- lastic interaction should be in Eq. (4). There are three such solutions for an isotropic dielectric m edium: one longitudinal and two degenerate transverse. For the magnetic medium, in general, there should be five solu tions. How- ever, here we will consider a weak-coupling approximation. In this approx- imation the interaction between the subsystems results in s mall variation of the dispersion curves of normal modes which become couple d phonon- magnon (or phonon-like) and magnon-phonon (or magnon-like ) ones). Only three elastic-like modes can be accounted in Eqs. (3)–(4) an d boundary con- ditions for magnetization are not used for obtaining R± n. 3 System of equations for determination of normal modes To obtain the wave vectors and complex amplitudes of the eige n modes it is necessary to solve the equations of the elasticity theory fo r the first medium and for the second one – the system consiting of Maxwell’s equ ations, equa- tion of motion for magnetization, and equations of elastici ty theory. We propose that the crystal is in magnetically saturated state and only small variations of magnetization can occur. Thus, for the first medium we have ρ1¨ui=∂τij ∂xj, (6) with tensions τijdefined by τij=∂W ∂εij, (7) where ρ1is the density of the medium I, εij=1 2(∂ui/∂xj+∂uj/∂xi) are components of strain (or deformation) tensor, and the densi ty of free energy Wcontains only the elastic energy WL. For the second medium we have equation of motion for volume el ement similar to (6). However, the free energy which defines the ten sions and effec- 3tive field Heshould be presented by elastic, magnetic, and magnetoelast ic terms. Formally, we should use the superscript ′′on all of the parameters and variables related to the magnetic medium, but we will defer t his until such terms for all of the waves, incident, reflected, and transmit ted, appear in the same expression. It is preferable to write the equation of motion for magnetiz ation in the form of the Gilbert equation: ˙M=γM×He+α0 M0M×˙M, (8) where α0is the relaxation parameter, M0is the modulus of static magneti- zation in homogeous state, and Mis total magnetization. The density of total free energy of a magnetic crystal in addi tion to elastic and magnetoelastic energy, contains Zeeman energy W0, exchange energy Wex, energy of magnetic anisotropy Wa, and magnetostatic energy Wd. The last two terms determines the field of magnetic anisotropy Haand the static H0 dand variable hdemagnetizing fields. We will use demagnetizing factor corresponding to a spherical specimen considering the plan e areas as small. Expressions for the types of energy and fields can be found in R ef. 4. Since we are limiting ourselves to the case where all wave vec tors lie in they−zplane, the complete system has the form a11u1+a14m1= 0, a22u2+a23u3+a25m2= 0, a32u2+a33u3+a35m2= 0, (9) a42u2+a43u3+a44m1+a45m2= 0, a51u1+a54m1+a55m2= 0, where miare components of variable magnetization, a11=ρω2−c1313(k2 2+k2 3), a22=ρω2−c1111k2 2−c1313k2 3, a33=ρω2−c1111k2 3−c1313k2 2, a44=a55=iω, a14=a25=−ib2k3 M0, a23=a32=−(c1122+c1313)k2k3, a35=−ib2k2 M0, a42=−a51=iγb2k3, a43=iγb2k2, a54=γ/bracketleftbigg2A M0(k2 3+k2 2) +Hi/bracketrightbigg −iα0ω, a45=−γ/bracketleftBigg2A M0(k2 3+k2 2) +Hi+4πk2 2M0 k2 2+k2 3/bracketrightBigg +iα0ω, 4ρis the density of the medium II, Hi=H+H0 d+Ha,cijklare elastic moduli, b2is magnetoelastic constant. It was assumed that all variabl es are proportional to exp [ i(ωt−k·r)]. Existing of non-trivial solutions of the system (9) require s its determinant to be equal to the zero. It is dispersion equation since it giv es expressions for the wave vectors of eigen modes. As the angle of incidence is given and therefore, according to Eq. (5), Re( k2) for all of the eigenvectors is defined, the unknown values are Re( k3), Im( k3), and Im( k2). 4 Solution of dispersion equation Using the last two equations of (9), we may express m1andm2as functions ofu1,u2, and u3: mi=bijuj, (10) substitute these expressions into the first three equations , and set the deter- minant of the new system (containing these three equations) equal to zero: (a11+a14b11) ×[(a22+a25b22)(a33+a35b23)−(a32+a35b22) (a23+a25b23)] −a14b12[a25b21(a33+a35b23)−a35b21(a23+a25b23)] +a14b13[a25b21(a32+a35b22)−a35b21(a22+a25b22)] = 0, (11) where b11=a45a51 a44a55−a54a45, b12=−a55a42 a44a55−a54a45, b13=−a55a43 a44a55−a54a45, b21=−a51a44 a44a55−a54a45, (12) b22=a54a42 a44a55−a54a45, b23=a43a54 a44a55−a54a45, We will solve the dispersion equation (11) by an iterative me thod. The zero order approximation will be obtained by letting the ele ments describing the magnon-phonon interaction (i.e., a14,a25,a35,a41,a42,a43,a51,a52, and a53) vanish. One of the solutions is k2 3=ρω2c−1 1313−k2 2, (13) whereas the next two are k2 3=−P±/parenleftBig P2−Q/parenrightBig1/2, (14) 5where P=/bracketleftBig/parenleftBig c2 1111+c2 1313−(c1122+c1313)2/parenrightBig k2 2 −ρω2(c1313+c1111)/bracketrightBig (2c1111c1313)−1, Q=ρ2ω4−ρω2k2 2(c1313+c1111) +c1313c1111k4 2, Eqs. (13)–(14) for k3relate to pure elastic waves: transverse s-type mode with wave vector ks, quasi-transverse (quasi- p) mode with wave vector kp, and quasi-longitudinal (quasi- l) mode with wave vector kl. At this point we have obtained real wave vectors (i.e., their imaginary part s are zero), since energy dissipation is considered to be in the magnetic subsy stem only. Next, the resulting expressions for k3should be substituted in turn into the equations which describe the magnetic subsystem and the magnon-phonon interaction. Recall that all k2in the zeroth approximation are real, equal, and known. Thus the solution for k3relating to the first weakly-coupled phonon- magnon mode (now of the quasi- stype, since it has yandzcomponents of the elastic displacement) is (k3)2 qs=−k2 2+1 c1313/bracketleftBig ρω2+a14b11+ [(a22+a25b22)(a33+a35b23) −(a32+a35b22) (a23+a25b23)]−1 ×[a14b13[a25b21(a32+a35b22)−a35b21(a22+a25b22)] −a14b12[a25b21(a33+a35b23)−a35b21(a23+a25b23)]]],(15) where all aijandbmnare functions of ( k3)s. The other two solutions relating to the coupled phonon-like modes are written in the form of (14), however the parameter Qshould now be given by Q=ρ2ω4−ρω2/bracketleftBig k2 2(c1313+c1111)/bracketrightBig +c1313c1111k4 2+a35a22b23+a25a33b22 −a32a25b23−a35a23b22−a14[a11+a14b11]−1 ×[b12[a25b21(a33+a35b23)−a35b21(a23+a25b23)] −b13[a25b21(a32+a35b22)−a35b21(a22+a25b22)]].(16) Note that the solution of Eq. (14) with the + sign before the ro ot describes the coupled quasi- pmode for which all aijandbknin Eq. (16) are functions of (k3)p, whereas that with the −sign describes the coupled quasi- lmode for which all aijandbknare functions of ( k3)l. Having obtained the complex zcomponents of the wave vectors, we can now determine the imaginary part of the ycomponents for a particular k= 6kek, where kis a complex quantity and ekis a unit vector in the kdirection. Thus, the definition Re( ki) = Re[ kcos(k·ei)] should also be correct for the imaginary parts of the wave vector components. For every kqjit follows that Im(k2)qj= Im( k3)qjcos(k·e3) cos(k·e2)= Im( k3)qjRe(k2)qj Re(k3)qj, where the index qstands for quasi- andjfors, p,orl. To illustrate the differences in the positions of magnon-pho non resonances for waves of different polarizations and to compare them with data for θ= 0, Fig. 1 shows the resonant frequency ωrversus magnetic field calculated for a YIG single crystal. This frequency is determined by requiri ng that the real part of the resonant denominator a44a55−a54a45in the definitions of bijgiven by expresions (12) vanish. Recall that θis the angle of incidence, while the angle of refraction for θ/negationslash= 0 depends on Hat a particular frequency and therefore cannot be a fixed property of curves B–D. It can be seen that in general one can observe manifestation of three resonances i n the properties of a reflected wave. 5 Complex amplidutes of vibrations and ten- sions The complex amplitudes of the elastic vibrations and tensio ns (elastic and magnetoelastic) can be written by inserting the eigen wave v ectors into the system (9). In both cases magnetization is expressed in term s ofu1,u2, and u3according to Eq. (10) . The first three equations of (9), written in the form cmnun= 0 ( m, n= 1,2,3) (17) yield complex amplitudes expressed in terms of those which d o not vanish for b2→0 and θ→0 (i.e., in terms of U′′ 1sfor quasi- smode, U′′ 2pfor quasi- p, and U′′ 3lfor quasi- l; we omit the qindex here and later on for complex amplitudes and wave vectors in magnetic medium), c11=a11+a14b11, c12=a14b12, c13= a14b13, c21=a25b21, c22=a22+a25b22, c23=a23+a25b33, c31=a35b21, c32= a32+a35b22, c23=a33+a35b33,bijare defined by (12). So in the magnetic medium we have the following expressions f or the complex amplitudes of phonon-like quasi- s, -p, and - lmodes where we denote their wave vectors by k′′ s,k′′ p, andk′′ l: U′′ 2s=c31c23−c21c33 c22c33−c32c23U′′ 1s≡f1(k′′ s)U′′ 1s≡A21U′′ 1s, 7Figure 1: Magnetic field dependences of the resonant frequen cy for waves propagating in a YIG single crystal with Hparallel to [001]. Curve Acor- responds to a transverse wave propagating along H(normal incidence at the interface), curves B,C, and D– to quasi- s, quasi- p, and quasi- lmodes, respectively, calculated for the angle of incidence θ= 22.5◦. 8U′′ 3s=c32c21−c22c31 c22c33−c32c23U′′ 1s≡f2(k′′ s)U′′ 1s≡A31U′′ 1s, U′′ 1p=c32c13−c12c33 c11c33−c31c13U′′ 2p≡f3(k′′ p)U′′ 2p≡A12U′′ 2p, U′′ 3p=c31c12−c11c32 c11c33−c31c13U′′ 2p≡f4(k′′ p)U′′ 2p≡A32U′′ 2p, (18) U′′ 1l=c23c12−c13c22 c11c22−c21c12U′′ 2p≡f5(k′′ l)U′′ 3l≡A13U′′ 3l, U′′ 2l=c21c13−c11c23 c11c22−c21c12U′′ 2p≡f6(k′′ l)U′′ 3l≡A23U′′ 3l. In the non-magnetic medium we have an incident wave of p-type and reflected waves of s-,p-, and l-types with the angles of reflection defined by Eq. (5). The complex amplitudes of the tensions associated with quas i-s, quasi- p, and quasi- lwaves can be written in the form (τ0 13)′′ s=/bracketleftBigg −ic1313(k′′ 3)s+b2 M0(b11+b12A21+b13A31)/bracketrightBigg U′′ 1s≡(T13)′′ sU′′ 1s, (τ0 23)′′ s={−ic2323[(k′′ 2)sA31+ (k′′ 3)sA21] +b2 M0(b21+b22A21+b23A31)/bracerightBigg U′′ 1s≡(T23)′′ sU′′ 1s, (τ0 33)′′ s=−i[c3322(k′′ 2)sA21+c3333(k′′ 3)sA31]U′′ 1s≡(T33)′′ sU′′ 1s, (τ0 13)′′ p=/bracketleftBigg −ic1313(k′′ 3)pA21+b2 M0(b11A12+b12+b13A32)/bracketrightBigg U′′ 2p ≡(T13)′′ pU′′ 2p, (τ0 23)′′ p=/braceleftBigg −ic2323[(k′′ 2)pA32+ (k′′ 3)p] +b2 M0(b21A12+b22+b23A32)/bracerightBigg U′′ 2p ≡(T23)′′ pU′′ 2p, (τ0 33)′′ p=−i[c3322(k′′ 2)p+c3333(k′′ 3)pA32]U′′ 2p ≡(T33)′′ pU′′ 2p, (τ0 13)′′ l=/bracketleftBigg −ic1313(k′′ 3)lA13+b2 M0(b11A13+b12A23+b13)/bracketrightBigg U′′ 3l ≡(T13)′′ lU′′ 3l, (τ0 23)′′ l=/braceleftBigg −ic2323[(k′′ 2)l+ (k′′ 3)lA23] +b2 M0(b21A13+b22A23+b23)/bracerightBigg U′′ 3l 9≡(T23)′′ lU′′ 3l, (τ0 33)′′ l=−i[c3322(k′′ 2)lA23+c3333(k′′ 3)l]U′′ 3l≡(T33)′′ lU′′ 3l. (19) Note that all of the coefficients ( Tab)′′ jare functions of k′′ j, where jindicates the type of the wave (quasi- s, -p, or -l) in the magnetic medium. The tensions in the non-magnetic medium may be written in ter ms of the Lam´ e constants λandµ. 6 Ellipticity and rotation of the polarization To obtain the amplitudes of the reflected waves, we have to wri te Eqs. (3)–(4) in the form: U′′ 1s Up+A12U′′ 2p Up+A13U′′ 3l Up−U′ p Up= 0 A21U′′ 1s Up+U′′ 2p Up+A23U′′ 3l Up−cosθU′ p Up+ sinθ′ lU′ l/Up= cos θ A31U′′ 1s Up+A32U′′ 2p Up+U′′ 3l Up−sinθU′ p Up−cosθ′ lU′ l Up=−sinθ (T13)′′ sU′′ 1s Up+ (T13)′′ pU′′ 2p Up+ (T13)′′ lU′′ 3l Up−iµω spcosθU′ s Up= 0 (20) (T23)′′ sU′′ 1s Up+ (T23)′′ pU′′ 2p Up+ (T23)′′ lU′′ 3l Up −iµω spcos 2θU′ p Up+iµω slsin 2θ′ lU′ l Up=−iµω spcos 2θ (T33)′′ sU′′ 1s Up+ (T33)′′ pU′′ 2p Up+ (T33)′′ lU′′ 3l Up−2iµω spsin 2θU′ p Up −/bracketleftbigg 2iω sl/parenleftBig λ+ 2µcos2θ′ l/parenrightBig/bracketrightbigg U′ l/Up= 2iµω spsin 2θ Finally, we have a system of six inhomogeneous linear equati ons in six unknowns. Our particular interest is in U′ s/UpandU′ p/Up(note U′ sandU′ p are complex amplitudes of u′ xandu′ ⊥, respectively, in the definition (1)), which yield the reflection coefficients in terms of complex amp litudes R± p= U′ s/Up±i U′ p/Up. Thus, after solving the system (20), we have obtained all the data required to determine εandφfrom Eqs. (2). 10To illustrate the phenomena, Fig. 2 shows the Kerr rotation a nd ellipticity which occur at a quartz-YIG interface. Calculations perfor med on the basis of the scheme given in section 4 show that variations of the wa ve vector related to a quasi- stype mode exceed those of quasi- pand quasi- lmodes by factors of 102and 103, respectively. As a consequence, only the zcomponent of ∆k′′ sis shown in this figure to compare with the behaviors of ε(H) and φ(H). Remember we assumed that the ferromagnet is in a magnetica lly homogeneous state, thus the results are valid only above the field of magnetic saturation, HS= 667 Oe. In Fig. 2 (b), (c), (e), and (f) these parts of the curves are shown as solid lines. It follows that only the high field wing of the curves can be observed at the frequency of 53 .5 MHz with the relaxation parameter α0>0.1. Acknowledgement The authors acknowledge financial support from Internation al Science Foun- dation (grant RGD000–RGD300). References [1] K. B. Vlasov and V. G. Kuleev, Fiz. Tver. Tela 10, 2076 (1968) [Sov. Phys.-Solid State 10, 1627 (1969)]. [2] K. B. Vlasov and G. A. Babushkin, Fizika Metallov i Metall ovedenie 38, 936 (1974) [Phys. Met. Metallogr. (USSR) 38, 32 (1974)]. [3] F. I. Fedorov, Theory of elastic waves in crystals (Nauka, Moscow, 1965)[Plenum Press, New York, 1968]. [4] J. W. Tucker and V. W. Rampton, Microwave Ultrasonics in Solid State Physics (North-Holland, Amsterdam, 1972). 11- Figure 2: (a), (d) Magnetic field dependences of the real ( A) and imaginary (B) parts of the zcomponent of the wave vector of a quasi- sphonon-like mode with a frequency of 53 .5 MHz propagating in a YIG single crystal for an angle of incidence θ= 22.5◦and rotation of the polarization (b), (e) and the ellipticity (c), (f) of the reflected wave calculated withα0= 0.5 for (a)–(c) and 0 .1 for (d)–(f); ∆ ks≡k′′ s(H)−k′′ s(0). 120.67 0.68 0.690300600 D C B Aω/G55/G03/G12/G15π /G03/G0B/G30/G2B/G5D/G0C /G2B/G03/G0B/G4E/G32/G48/G0C0.64 0.68-20-10010 /G0B/G44/G0C/G25 /G24Δ /G0B/G4E/G16/G05/G03/G0C/G56/G03/G03/G0B/G46/G50/G03/G10/G14/G0C /G2B/G03/G0B/G4E/G32/G48/G0C0.64 0.680.00.10.20.3 /G0B/G45/G0C/G10/G03φ /G03/G0B/G47/G48/G4A/G0C /G2B/G03/G0B/G4E/G32/G48/G0C0.64 0.68-0.0020.0000.0020.004 /G0B/G46/G0C- ε /G2B/G03/G0B/G4E/G32/G48/G0C0.64 0.68-80-40040 /G0B/G47/G0CB AΔ /G0B/G4E/G16/G05/G03/G0C/G56/G03/G0B/G46/G50/G03/G10/G14/G0C /G2B/G03/G0B/G4E/G32/G48/G0C0.64 0.68-202 /G0B/G48/G0C/G10/G03φ /G03/G0B/G47/G48/G4A/G0C /G2B/G03/G0B/G4E/G32/G48/G0C0.64 0.68/G10/G13/G11/G13/G16/G13/G11/G13/G13/G13/G11/G13/G16/G13/G11/G13/G19 /G0B/G49/G0C- ε /G2B/G03/G0B/G4E/G32/G48/G0C

arXiv:physics/0004018v1 [physics.optics] 11 Apr 2000Second-harmonic interferometric spectroscopy of the buried Si(111)-SiO 2interface A.A. Fedyanin, T.V. Dolgova, O.A. Aktsipetrov Department of Physics, Moscow State University, 119899 Mos cow, Russia D. Schuhmacher, G. Marowsky Laser-Laboratorium G¨ ottingen, Hans-Adolf-Krebs-Weg 1, D-37077 G¨ ottingen, Germany Abstract The second-harmonic interferometric spectroscopy (SHIS) which combines both am- plitude (intensity) and phase spectra of the second-harmon ic (SH) radiation is pro- posed as a new spectroscopic technique being sensitive to th e type of critical points (CP’s) of combined density of states at semiconductor surfa ces. The increased sensi- tivity of SHIS technique is demonstrated for the buried Si(1 11)-SiO 2interface for SH photon energies from 3.6 eV to 5 eV and allows to separate the r esonant contributions fromE′ 0/E1,E2andE′ 1CP’s of silicon. Second-harmonic generation (SHG) is inherently sensitive to surface and interface prop- erties of centrosymmetric materials. Recently, the spectr oscopy of the second-harmonic (SH) intensity has been proved as a promising probe of surfaces an d interfacial layers1and inten- sively employed in numerous works for oxidized2, reconstructed3and H-terminated4silicon surfaces. The resonances of the SH intensity are attributed in these cases to direct interband electron transitions. By analogy with the spectrum of the li near dielectric function ε(ω) the spectrum of the quadratic susceptibility χ(2)(2ω) of such semiconductor surfaces could be expressed as the superposition of several van Hove singular ities (critical points (CP’s) of the combined density of states) χ(2) m(2ω)∝(2ω−ωm+iΓm)nwith threshold frequencies ωmand broadenings Γ m5. The exponent nreflects the dimensionality of CP: n= 1/2,0 1(logarithmic), −1/2,−1 for 3D, 2D, 1D and excitonic CP, respectively. Although the line shapes χ(2)(2ω) are quite different for various n, in most cases a large number of adjustable parameters makes the determination of the type of CP solely f rom the SH intensity spectrum doubtful, and most of authors interpret the SHG spectroscop y data within excitonic CP line shape6. The single-beam SH interferometry traces back to the mid-19 60s8and is conventionally used for the determination of the phase of the quadratic susc eptibility of adsorbate molecules and their absolute direction9and for separation of the SH contributions from thin films and their substrates10. Another use of SH phase measurements is a homodyne mixing te chnique to improve the signal-to-noise ratio for surface SHG probe11. The use of external12and internal13homodynes for dc or low-frequency modulated electric-field -induced SHG allows to measure the in-plane spatial distribution of the electri c field vector with micron resolution or to visualize weak nonlinear contributions. Further deve lopments of the SH interferometry are the frequency-domain interferometric SH spectroscopy14exploring the broad bandwidth of femtosecond laser pulses, and the hyper-Rayleigh scatte ring interferometry15using the correlation of fluctuations in linear and nonlinear optical properties of thin inhomogeneous films. In this Letter, a modification of the SH spectroscopy - the SH i nterferometric spec- troscopy (SHIS) which combines both amplitude (intensity) SH spectroscopy and SH inter- ferometry is proposed. The combination of the phase and ampl itude SH spectra, extracted from the SHIS data, is shown to be sensitive to type of CP even f or systems with interfering SH contributions from close electronic resonances. Additi onally SHIS allows to avoid the sign uncertainty of Re( χ(2)) inherent in the conventional spectroscopy of the SHG inten - sity. The spectral dependence of the phase and amplitude of t he SH waves from the buried Si(111)-SiO 2interface is measured using the SHIS technique in the spectr al range in the vicinity of silicon E2CP. In contrast to E′ 0/E1CP revealing the excitonic type, the family of E2CP’s of the bulk silicon demonstrates the 2D type in linear re sponse7. The resonant con- tributions to the quadratic susceptibility from E′ 0/E1,E2andE′ 1silicon CP’s are extracted 2within the simple phenomenological model which accounts th e complex Green’s function corrections for the SH wave generation. The scheme of the SHIS setup is shown in Fig.1(a). The p-polar ized output of a tunable nanosecond parametric generator/amplifier laser system (S pectra-Physics MOPO 710) op- erating in the interval of 490 - 690 nm is focused onto the samp le at an angle of incidence of 45◦. The SH signal is detected by a monochromator, a photomultip lier tube (PMT) and an electronic peak-hold detector. To normalize the SH inten sity spectrum over the laser fluence and the spectral sensitivity of the optical detectio n system a SHG intensity reference channel is used with a slightly wedged z-cut quartz plate and with the detection system identical to the one in the sample channel. The (phase-)refe rence sample is chosen (i) to be thin enough to avoid Maker fringes in the SH response durin g tuning the fundamental wavelength λω, (ii) to be optical inactive for conservation the polarizat ion state of the fun- damental radiation while transmitting through it, (iii) to have no resonance features in the tuning region of both the fundamental and SH waves. Therefor e the 1 mm-thick plate of fused quartz coated with a 30 nm-thick indium tin oxide (ITO) film is chosen as a reference. The SH interferogram is obtained by translating the referen ce along the fundamental laser beam varying the distance lbetween the reference and the sample. The SH signals from the reference, I2ω r, and from the sample, I2ω s, are monitored separately by inserting appropriate filters (yellow or UV, respectively) between the reference a nd the sample. I2ω ris adjusted with the angle of incidence of the fundamental beam at the ITO phase reference. The de- tected SH intensity I2ωis the result of interference of the SH waves from the referen ce,E2ω r, and from the sample, E2ω s: I2ω=c 8π|E2ω r(l) +E2ω s|2=I2ω r(l) +I2ω s+ 2α/radicalBig I2ωr(l)I2ωscos/parenleftBigg 2πl L+ Φ rs/parenrightBigg , (1) where L=λω(2∆n)−1is the period of SH interferogram with ∆ n=n2ω−nωdescribing the air dispersion, and α <1 indicates the laser coherence. The position-dependent ph ase shift 2 πl/Lbetween E2ω randE2ω scomes from the different refractive indices of air for the fundamental and SH waves. The spectral dependences of χ(2)of the reference and the 3sample as well as the complex Green’s function corrections f or the SH wave produce a position-independent phase shift Φ rs(λω, λ2ω). The dependence I2ω r(l) is described by the conventional formula for focused Gaussian beams. The spect rum of the phase of the SH wave from the ITO film, Φ r≡Arg(E2ω r), is measured using the 1 mm-thick backside-immersed y-cut quartz as a sample since the phase of the SH wave from the quartz surface is spectrally independent in the whole used spectral region. The Φ rspectrum of the ITO film appears to be a constant within the error bars and I2ω rgradually increases with decreasing λωwithout any resonance features. The samples are natively oxidized p-doped Si(111) wafers wi th resistivity of 10Ω ·cm. SHIS has been performed at the maximum of the azimuthal SH rot ational anisotropy for the p-in, p-out polarization combination of the fundamenta l and SH waves. Figure 1(b) shows typical SH interference patterns measured for differe ntλω. The spectral dependence of the period Ldue to the air dispersion and clear changes in the contrast of the patterns due to the distance dependence of I2ω rin the focused laser beam are seen. The fit of the set of SH interference patterns by Eq.(1) with Φ rs,I2ω s,L, and αas adjustable parameters leads to the spectra of Φ rs,I2ω sandLshown in Fig. 2(a) and 2(b) and in the inset of Fig. 1, respectively. To emphasize the spectral features of I2ω s(2ω), we combine the fitted intensity spectrum with the I2ω s(2ω) dependence measured directly with a fine resolution in SH photon energy. Φ rsincreases approximately by 1.2 radians within the interval of 4.2-4.6 eV and decreases outside this energy region. A small, but rel iable, non-monotonic feature is seen at 3.8 - 4.0 eV. The I2ω sspectrum has pronounced peaks centered approximately at 3.9 eV and 4.3 eV. The position of the 4.3 eV resonance is close toE2CP and we associate the observed features of SH phase and intensity spectra for e nergies between 4.1 and 4.6 eV with direct interband electron transitions at E2CP of Si7. The relative phase Φ rsmeasured in SHIS is given by: Φrs= Φ s−(Φr+ Arg( R2ω)), (2) where R2ωis the Fresnel reflection factor of the p-polarized SH radiat ion from the Si-SiO 2 4interface. The phase Φ s≡Arg(E2ω s) originates from both complex surface χ(2),Sand bulk quadrupole χ(2),BQquadratic susceptibilities as well as from Green’s functio n corrections16: E2ω s=G/bardblχ(2) /bardbl+G⊥χ(2) ⊥, (3) where G/bardblandG⊥are the Green’s function corrections for the generation and the propagation of in-plane and normal components of E2ω s, andχ(2) /bardblandχ(2) ⊥are the corresponding effective components of χ(2).χ(2) /bardblandχ(2) ⊥are the linear combinations of χ(2),Sandχ(2),BQcomponents with nonresonant coefficients depending only on the fundamen tal wavevector and taking into account the geometry of the nonlinear interaction. Thi s allows to consider the spectral dependences of χ(2) /bardblandχ(2) ⊥as a superposition of two-photon resonances for different CP ’s: χ(2) α(2ω) =B−/summationdisplay mfα mexp(iφα m)(2ω−ωm+iΓm)n, (4) where α=⊥,∝bardbl, and mnumerates the CP resonances. The oscillator strengths fα mare supposed to be real numbers. For the sake of simplicity, a sli ght spectral dependence of the termBincluding the Si resonances with threshold energies below 1 .5 eV3is neglected, and φα mare integer multiples of π/2 defining the type of CP. The solid lines in Fig.2 show the fit of Φ rsandI2ω sspectra by Eq.(2) and |E2ω s|2from Eq.(3), respectively, with the Si dispersion data from Ref.[17] and expressions for G/bardblandG⊥from Ref.[18]. Five resonant contributions are included into the fit. The first resonance, centered at ω1= 3.45 eV, has excitonic line shape ( n=−1, φ1= 0) and corresponds to the direct electron transitions at E′ 0/E1 CP. The second resonance at 3.97 eV with 1D maximum line shape (n=−1/2, φ2= 0) has no equivalent in the band structure of crystalline bulk s ilicon. However, a resonance in the close energy interval has been recently observed at th e Si(001)-SiO 2interface6and could be associated with transition in Si atoms located at th e interface with reduced lattice symmetry. The strong resonant features in the vicinity of 4. 3 eV are formed by interference of two resonances centered at ω3= 4.12 eV and at ω4= 4.34 eV with almost equal amplitudes (f4≈0.9f3). It is mostly reasonable to attribute these peaks to transi tions at E2(X) and E2(Σ) CP’s. These resonances are fitted with 2D minimum ( φ3= 0) and 2D maximum 5(φ4=π) line shapes ( χ(2) m∝ln(2ω−ωm+iΓm)) by analogy with the linear case7. Note, thatω3andω4are approximately 0.1 eV red-shifted from resonances of lin earχ(1). This allows to interpret also 4.12 eV-resonance as a contributio n from E0CP, which is normally very weak in the linear response. The best representation of the data is obtained with a 2D minimum line shape of the last resonance, centered at 5.15 eV, which can be associated with electron transitions located near E′ 1CP7. The error bars for the central frequencies are approximately 0.03 eV and mostly attributed to the relative weight of χ(2) /bardblandχ(2) ⊥being unresolvable from our data. Excitonic line shapes for all re sonances (dotted lines in Fig.2) fit the I2ω sspectrum with almost the same quality as the CP model, but fit t he Φ rsspectrum obviously worse. Summarizing, the general scheme of the second-harmonic int erferometric spectroscopy is presented. The phase and amplitude of the SH wave from the bur ied Si(111)-SiO 2interface are measured simultaneously using the SHIS technique in the interval of SH photon energies from 3.6 eV to 5 eV. The contributions of interband transitio ns located at E′ 0/E1,E2and E′ 1Si critical points are separated and sensitivity of SHIS to C P line shapes is shown. This work was supported by the Russian Foundation for Basic R esearch (RFBR) and Deutsche Forschungsgemeinschaft (DFG): RFBR grant 98-02- 04092, DFG grants 436 RUS 113/439/0 and MA 610/20-1, RFBR grant 00-02-16253, special RFBR grant for Leading Russian Science Schools 00-15-96555; NATO Grant PST.CLG97 5264, Russian Federal Pro- gram ”Center of Fundamental Optics and Spectroscopy”, and P rogram of Russian Ministry of Science and Technology ”Physics of Solid State Nanostruc tures”. 6REFERENCES 1.T. F. Heinz, in: Nonlinear Surface Electromagnetic Phenomena , Eds. H. -E. Ponath and G. I. Stegeman (Elsevier Publ. Co., Amsterdam 1991), p. 3 53; J. F. McGilp, Phys. Status Solidi A 175, 153 (1999); G. L¨ upke, Surf. Sci. Rep. 35, 75 (1999). 2.W. Daum, H.-J. Krause, U. Reichel, and H. Ibach, Phys. Rev. Le tt.71, 1234 (1993). 3.K. Pedersen and P. Morgen, Phys. Rev. B 53, 9544 (1996). 4.J. I. Dadap, N. M. Russel, Z. Xu, X. F. Hu, J. G. Eckerd, O. A. Akt sipetrov, and M. C. Downer, Phys. Rev. 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Rubtsov, M. H. Anderson, P. T. Wilson, M. ter Beek, X. F. Hu, J. I. Dadap, an d M. C. Downer, 7Phys. Rev. B 60, 8924 (1999). 14.P. T. Wilson, Y. Jiang, O. A. Aktsipetrov, E. D. Mishina, and M . C. Downer, Opt. Lett.24, 496 (1999). 15.A. A. Fedyanin, N. V. Didenko, N. E. Sherstyuk, A. A. Nikulin, and O. A. Aktsipetrov, Opt. Lett. 24, 1260 (1999). 16.P. Guyot-Sionnest, W. Chen, and Y. R. Shen, Phys. Rev. B 33, 8254 (1986). 17.D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983). 18.O. A. Aktsipetrov, T. V. Dolgova, A. A. Fedyanin, D. Schuhmac her, G. Marowsky, Thin Solid Films 364, 91 (2000). 8FIGURES Fig. 1. Panel a: Experimental setup for the SH interferometr ic spectroscopy. BS, beam splitter; GG and UG, yellow and UV filters, respectively. Pan el b: Raw SH interferograms for different SH energies. Solid curves: The dependences giv en by Eq.(1). Inset: The spectral dependence of the period Lof the SH interferograms and its fit using a phenomenological expression for air dispersion. Open circles indicate the pe riods for the curves at the main panel. Fig. 2. Spectrum of the SH phase Φ rs(panel a) and SH intensity I2ω s(panel b). Solid curves are fits to the data within the model of CP line shapes. D otted lines are fits with excitonic line shapes for all the resonances. 920 40 60 800123 4.59 eV 3.95 eVSHIntensity(arb. un.) Reference-Sample Distance (mm)3.6 4.0 4.4 4.8102030 Period(m m ) Two-Photon Energy (eV)lenslens z-quartzBS sample polarizer monochromator monochromator PMT PMTlenslensUGUGGG GGpolarizer ITO VIS-OPO a b/G16/G19/G17/G24/G19/G17/G19/G19/G17/G24/G20/G17/G19/G22/G17/G25 /G22/G17/G27 /G23/G17/G19 /G23/G17/G21 /G23/G17/G23 /G23/G17/G25 /G23/G17/G27 /G24/G17/G19 /c3 /c3/G68/G54/G43/G3/G51/G75/G82/G87/G82/G81/G3/G40/G81/G72/G85/G74/G92/G3/G11/G72/G57/G12/G54/G43/G3/G53/G72/G79/G68/G87/G76/G89/G72/G3/G51/G75/G68/G86/G72/G3/G11/G85/G68/G71/G17/G12 /G22/G17/G25 /G22/G17/G27 /G23/G17/G19 /G23/G17/G21 /G23/G17/G23 /G23/G17/G25 /G23/G17/G27 /G24/G17/G19/G19/G17/G19/G19/G17/G24/G20/G17/G19/G20/G17/G24/G21/G17/G19/G21/G17/G24/G69/G54/G43/G3/G44/G81/G87/G72/G81/G86/G76/G87/G92/G3/G11/G68/G85/G69/G17/G3/G88/G81/G17/G12 /G3/G54/G43/G3/G51/G75/G82/G87/G82/G81/G3/G40/G81/G72/G85/G74/G92/G3/G11/G72/G57/G12 /G3

arXiv:physics/0004019v1 [physics.bio-ph] 11 Apr 2000Signal Selection Based on Stochastic Resonance Hans E. Plesser∗and Theo Geisel Max-Planck-Institut f¨ ur Str¨ omungsforschung and Fakult ¨ at f¨ ur Physik, Universit¨ at G¨ ottingen Bunsenstraße 10, 37073 G¨ ottingen, Germany (February 2, 2008) Noise aids the encoding of continuous signals into pulse sequences by way of stochastic resonance and endows the en- coding device with a preferred frequency. We study encoding by a threshold device based on the Ornstein-Uhlenbeck pro- cess, equivalent to the leaky integrate-and-fire neuron mod el. Preferred frequency, optimum noise intensity, and optimum signal-to-noise ratio are shown to be linearly related to th e AC amplitude of the input signal. The DC component of the input tunes the device either into transmission (preferred fre- quency nearly independent of signal amplitude) or selection mode (frequency rising with amplitude). We argue that this behavior may facilitate selective signal processing in neu rons. I. INTRODUCTION Our view of noise has shifted markedly over the past two decades: After it had long been seen merely as a nuisance, geophysicists first suggested in 1981 that noise may amplify the effect of weak periodic signals on dy- namic systems [1]. This effect, called stochastic reso- nance (SR), has since been found in a variety of exper- iments. The origin of stochastic resonance in both dy- namical and non-dynamical systems is well understood today, although theories are mostly confined to slow sig- nals. A recent review of the field can be found in Ref. [2]. The finding that noise may aid signal detection and transmission has spurred intense research in the neuro- sciences, where scientists have long been puzzled by the seeming irregularity of neuronal activity. The benefits of noise for signal processing in neurons have now been demonstrated in a wide range of species: neurons trans- duce a signal (or stimulus) optimally if a certain amount of ambient noise is present [3]; see Ref. [4] for a review. Recent studies by ourselves [5] and other authors [6] have revealed a further noise-induced resonance effect in model neurons: There is also an optimum signal fre- quency, for which the neuron responds with spike trains (output signals) that have a particularly high signal-to- noise ratio (SNR). In this letter, we show that this res- onance frequency and the optimum noise amplitude are linearly related to the AC component of the signal im- pinging on a neuron, while the DC component serves as tuning parameter. We argue that these relations permit the neuron to switch between a signal transmission and a signal selection mode of operation.The present work is based on the integrate-and-fire neuron model, which will briefly be reviewed in Sec. II, before the noise-induced response properties of the neu- ron are presented in Sec. III. Implications for neuronal signal processing are discussed in Sec. IV. More details are contained in Ref. [7]. II. MODEL We focus on the spike generator of neurons for the sake of both simplicity and generality. The spike gener- ator integrates the net input current I(t) impinging on the neuron like a leaky capacitor. For sinusoidal input superimposed with white noise, the potential v(t) across the capacitor (the membrane potential) is thus governed by [8] ˙v(t) =−v(t) +µ+qcos(Ωt+φ) +σξ(t). (1) The input is characterized by the DC offset µ, the sig- nal amplitude q, the frequency Ω, and the (arbitrary) initial phase φ. The noise term with root mean square amplitude σand autocorrelation /angbracketleftξ(t)ξ(t′)/angbracketright=δ(t−t′) subsumes the noise arising both from cell biochemistry [9] and background activity in the neuronal network [10]. The neuron emits a stereotyped voltage pulse (a spike) whenever the membrane potential reaches a fir- ing threshold v(tk) = 1 = Θ; the potential is reset to v(t+ k) =vr<1 immediately thereafter. Time and volt- age are measured in their natural units, i.e., the mem- brane time constant τmand the firing threshold Θ. The spike generator thus operates as an analog-to- digital converter [11], encoding the continuous input sig- nalI(t) into a pulse train f(t) =/summationtext kδ(t−tk). Even though this model is a gross simplification of biological neurons, it has proven most fruitful for investigations of the nature of the neuronal code [12,13]. For a derivation of this model from more realistic neuron models, see [14]. We restrict ourselves to sub-threshold signals, which would not elicit any spikes in the absence of noise, i.e., max t→∞v(t) =µ+q/√ 1 + Ω2<1 from integration of Eq. (1). These appear to be more relevant than super- threshold signals for the encoding of periodic signals [15] . Note that the membrane potential will oscillate about ¯v=µafter an initial transient in the absence of noise. We measure the performance of the neuron in coding the sinusoidal input by the signal-to-noise ratio 1RSN=S(Ω)/SP (2) of the output spike train at the signal frequency. Here, S(Ω) =1 πTo|/integraltextTo 0f(t)eiΩtdt|2is the power spectral den- sity of the train for a given observation time To, while the white power spectrum of a Poissonian spike train with equal mean interspike interval /angbracketleftτ/angbracketright, i.e., SP= (π/angbracketleftτ/angbracketright)−1, is used as reference noise level. To= 200 is employed throughout [5,16]. III. RESPONSE PROPERTIES The leaky integrate-and-fire neuron responds best to sinusoidal stimulation—i.e. attains a maximum signal- to-noise ratio—at a particular combination of signal fre- quency Ω and input noise amplitude σ, see Fig. 1. The location of the SNR maximum is marked by an aster- isk [16]. In what follows, we shall explore the location of and the SNR value at this maximum as a function of the remaining stimulus parameters, namely the signal amplitude q, the DC offset µ, and the reset potential vr. In the absence of noise, sub-threshold stimuli evoke membrane potential oscillations about ¯ v=µas pointed out in Sec. II. The gap between this average potential and the threshold, Θ −¯v= 1−µ, needs to be bridged by the concerted effort of signal-induced oscillations and noise. It is therefore plausible to scale both stimulus and noise amplitude by this threshold distance, i.e., to define relative amplitudes, qr=q 1−µand σr=σ 1−µ. (3) Furthermore, the reset potential vrshould enter only as the ratio of the reset distance to threshold distance, which we shall refer to as relative reset γ=µ−vr 1−µ. (4) This scaling can be established rigorously via escape noise approximations to the dynamics of the integrate- and-fire neuron [7,17]. We shall now turn to the relation of the optimum tun- ing parameters (preferred signal frequency Ωopt, scaled noise amplitude σopt r, and SNR Ropt SN) to the signal am- plitude parameters (scaled AC amplitude qr, and relative resetγ; or DC offset µand reset potential vrinstead of γ). Figure 2(a) indicates a perfect linear relation between Ropt SN, the attainable SNR, and the stimulus amplitude qr, while the optimum input noise amplitude σopt r≈0.6−0.7 is practically independent of qr, see Fig. 2(b): variations ofσopt rare about one order of magnitude smaller than the range of qrvalues. Both relations are remarkably in- dependent of the value of the DC offset µ(indicated by symbol/line type in Fig. 2, supporting the scaling given in Eq. (3). Ropt SNandσopt rare thus independent of theDC component of the signal transmitted for fixed reset potential. A different behavior is observed for the optimum fre- quency Ωoptas shown in Fig. 3(a): For large values of γ, i.e., a strong positive DC offset, the optimum frequency is nearly independent of the signal amplitude qr, while small values of γlead to a marked linear dependence of Ωoptonqr: the preferred frequency may be selected by a variation of the signal amplitude. Figure 3(a) also clearly indicates that the response of the neuron depends on the DC offset µand reset potential vronly via the relative reset γ, vindicating Eq. (4): each data point shown is a superposition of two almost perfectly coincident points obtained from different ( µ, vr)-combinations yielding the sameγ(circles, crosses). The same is found for Ropt SNand σopt r(not shown). The results presented here are thus applicable both to sensory and cortical neurons: the for- mer are best fit by the model for reset potentials vr≈0 [7], while the latter require vr≈0.7 [13]. Figure 3(a) indicates that the integrate-and-fire neu- ron may operate in two different modes: a transmission mode for large γ, which optimally encodes stimuli of a fixed preferred frequency Ωopt≈1 irrespective of their amplitude, and a selection mode , in which the preferred frequency may be chosen by variation of the stimulus am- plitude qr. The slope of the frequency-amplitude curve (linear least squares fit) as a function of the relative reset γis shown in Fig. 3(b). There is a sharp transition be- tween the selection and transmission modes at γ≈2.1. No slope could be determined for γ <1.5, since the pe- riod of the optimal stimulus 2 π/Ωopttends to the dura- tion of the observation period Tofor small amplitudes qr in this case. The two modes of operation arise through different fir- ing patterns: For large relative reset ( γ >2), less than one spike is fired on average per stimulus period, i.e., the neuron fires at most one well phase-locked spike per pe- riod, and often skips periods in between spikes, with a slight increase in spike number with qr. For small reset (γ <2), in contrast, the neuron has a bursting firing pattern for small qr, i.e., two or three spikes are fired in rapid succession near the maximum of the signal in each period, followed by silence till the next period. As qr is increased and the optimum frequency rises, the signal period becomes too short to harbor more than one spike and bursting gives way to a more regular firing pattern, with a little more than one spike per period on aver- age. For intermediate reset ( γ≈2), the neuron fires almost regularly, with about 0 .8 spikes per signal period independent of qr. Cold receptor neurons show all three kinds of firing patterns (skipping, regular, bursting) de- pending on ambient temperature [18]; their behavior is reproduced well by the integrate-and-fire neuron [7]. 2IV. FUNCTIONAL SIGNIFICANCE Let us summarize the neuronal response properties and discuss them in turn: (i) The optimal signal-to-noise ratio scales linearly with the input signal amplitude, and (ii) is attained at a constant noise amplitude, while (iii) the preferred frequency is either independent of (transmis- sion mode) or linearly related to the signal amplitude (selection mode). Property (i) means that the optimal SNR of the spike train emitted by the neuron is related to the SNR of the input signal as Ropt SN∼qr∼/radicalbig Rin SN[19], in qualita- tive agreement with recent findings in a variant model neuron [20]. This suggests a law of diminishing returns for the signal-to-noise ratio: there is no point in invest- ing valuable resources to improve Rin SNbeyond a certain level, because resulting gains in Ropt SNwould be minimal. Since the output of each neuron in turn is input to other neurons, the same argument holds for raising Ropt SN. The level of noise observed in the brain might thus reflect an evolutionary compromise between coding quality and resource consumption. (ii) The independence of the optimum input noise am- plitude σopt rfrom signal amplitude makes the integrate- and-fire neuron a useful signal processing device, as no fine-tuning of the noise to the signal is required to at- tain optimal performance. The noise level need only be adjusted relative to the DC offset, which largely reflects homogeneous background activity. The optimum noise amplitude of σopt r≈0.6−0.7 (relative to the threshold distance 1 −µ) is in good agreement with the observation that coincidence detection in the auditory system of barn owls works best for sub-threshold stimuli which raise the average membrane potential to roughly one noise ampli- tude below threshold [15]. Property (iii) is the central finding reported here: a model neuron as simple as the integrate-and-fire neu- ron may switch between two distinct modes of opera- tion, a transmission and a selection mode. Switching be- tween the two modes is achieved by variation of the tem- porally homogeneous background input to the neuron: weak background activity activates the selection mode, a strong background the transmission mode. Switching between modes requires only moderate variations of the background activity as indicated by Fig. 3(b). In the former, an input signal of particular frequency reaching the neuron through synapses far from the cell body—and thus the spike generator—may easily be (de-)selected: modulatory input through synapses closer to the spike generator need only vary the amplitude qof the net in- put current I(t) to the spike generator to tune the neu- ron’s optimum frequency Ωopteither closer to or away from the given signal frequency Ω. Selected signals are then coded into spike trains with high signal-to-noise ra- tio, i.e., trains with clear temporal structure, while de-selected signals elicit more random output. Since pulse packets can propagate through networks of neurons only if they are sufficiently strong and tight [21], variation of the SNR provides a means of gating such packets through neuronal networks. V. SUMMARY We have shown here that the filter properties of a threshold system exploiting stochastic resonance in the sub-threshold regime are linearly related to the AC am- plitude of the input signal, and may be tuned by vari- ation of the DC signal amplitude. Our results indicate that such a simple device may, with the aid of noise, pro- vide the means to selectively transmit signals in neuronal networks. It might thus harness noise for the benefit of neuronal computation. Although our study is set in the framework of neurons as the most widespread threshold detectors in nature, the results apply more generally to any threshold system that may be characterized as an Ornstein-Uhlenbeck escape process. We would like to thank A. N. Burkitt, G. T. Einevoll and W. Gerstner for critically reading an earlier version of the manuscript. [1] R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14, L453 (1981). [2] L. Gammaitoni, P. H¨ anggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998). [3] A. Longtin, A. Bulsara, and F. Moss, Phys. Rev. Lett. 67, 656 (1991). J. K. Douglass, L. Wilkens, E. Pantazelou, and F. Moss, Nature 365, 337 (1993). J. E. Levin and J. P. Miller, Nature 380, 165 (1996). J. J. Collins, T. T. Imhoff, and P. Grigg, J. Neurophysiol. 76, 642 (1996). P. Cordo et al., Nature 383, 769 (1996). F. Jaramillo and K. Wiesenfeld, Nature Neurosci. 1, 384 (1998). E. Simonotto et al., Phys. Rev. Lett. 78, 1186 (1997). R. Srebro and P. Malladi, Phys. Rev. E 59, 2566 (1999). [4] K. Wiesenfeld and F. Jaramillo, Chaos 8, 539 (1998). [5] H. E. Plesser and T. Geisel, Phys. Rev. E 59, 7008 (1999). [6] F. Liu, J. F. Wang, and W. Wang, Phys. Rev. E 59, 3453 (1999). T. Kanamaru, T. Horita, and Y. Okabe, Phys. Lett. A 255, 23 (1999). [7] H. E. Plesser, Ph.D. thesis, Georg-August-Universit¨ a t, G¨ ottingen, 1999, webdoc.sub.gwdg.de/diss/1999/plesser. [8] H. C. Tuckwell, Stochastic Processes in the Neurosciences (SIAM, Philadelphia, 1989). [9] A. Manwani and C. Koch, Neural Comput. 11, 1797 (1999). [10] Z. F. Mainen and T. J. Sejnowski, Science 268, 1503 (1995). 3σ Ω0123450.050.10.150.20.25 SNR 0481216 FIG. 1. Signal-to-noise ratio RSNas function of signal fre- quency Ω and input noise amplitude σshown as grayscale plot. The asterisk marks Ropt SN= 15.7. No SNR could be determined for the white area to the bottom-right, since the neuron is practically quiet there. Other parameters: q= 0.1, µ= 0.9,vr= 0, and thus γ= 9. [11] R. Sarpeshkar, Neural Comput. 10, 1601 (1998). [12] W. Gerstner, R. Kempter, J. L. van Hemmen, and H. Wagner, Nature 383, 76 (1996). P. Marˇ s´ alek, C. Koch, and J. Maunsell, Proc. Natl. Acad. Sci. USA 94, 735 (1997). G. Bugmann, C. Christodoulou, and J. G. Taylor, Neural Comput. 9, 985 (1997). J. Feng, Phys. Rev. Lett. 79, 4505 (1997). L. F. Abbott, J. A. Varela, K. Sen, and S. B. Nelson, Science 275, 220 (1997). [13] T. W. Troyer and K. D. Miller, Neural Comput. 9, 971 (1997). [14] W. Kistler, W. Gerstner, and J. L. van Hemmen, Neural Comput. 9, 1015 (1997). C. F. Stevens and A. M. Zador, inProceedings of the 5th Joint Symposium on Neural Computation (Institute for Neural Computation, UCSD, La Jolla, CA, 1998), pp. 172–177. P. L´ ansk´ y, Phys. Rev. E55, 2040 (1997). [15] R. Kempter, W. Gerstner, J. L. van Hemmen, and H. Wagner, Neural Comput. 10, 1987 (1998). [16]RSNwas determined for each (Ω , σ)-combination shown in Fig. 1 by numerical evaluation of Eq. (2); see [5,7] for details and [22] for a related approach. To obtain Ropt SN, RSNwas maximized with respect to Ω and σusing a Nelder-Mead direct search algorithm [23]. [17] H. E. Plesser and W. Gerstner, Neural Comput. 12, 367 (2000). [18] H. A. Braun, K. Sch¨ afer, and H. Wissing, Funkt. Biol. Med.3, 26 (1984). [19] K. Wiesenfeld et al., Phys. Rev. Lett. 72, 2125 (1994). [20] A. N. Burkitt and G. M. Clark, Synchronization of the neural response to noisy periodic input, 1999, submitted. [21] M. Diesmann, M.-O. Gewaltig, and A. Aertsen, Nature 402, 529 (1999). [22] T. Shimokawa, A. Rogel, K. Pakdaman, and S. Sato, Phys. Rev. E 59, 3461 (1999). [23]MATLAB Function Reference , The MathWorks, Inc., Natick, MA, USA, 1998. RSNopt(a) 0.4 0.6 0.8 1 1.22 6 101418 σropt qr(b) 0.4 0.6 0.8 1 1.20.40.50.60.7 FIG. 2. (a) Optimal signal-to-noise ratio and (b) optimum noise amplitude as functions of the input amplitude. Square s and solid lines mark DC offset µ= 0.6, diamonds/dashed µ= 0.667 and circles/dotted µ= 0.889; reset potential is vr= 0, yielding relative resets of γ= 1.5, 2, and 8. Lines are to guide the eye. qrΩopt(a) 0.4 0.6 0.8 1 1.20 0.250.5 0.751 ∆Ωopt / ∆ qr γ(b) 02468101214160 0.250.5 0.75 FIG. 3. (a) Optimum signal frequency as function of the input amplitude for relative resets γ= 1.5 (solid), γ= 2 (dashed) and γ= 8 (dash-dotted). Lines are least-squares fits, while symbols mark different ( µ, v r)-combinations yield- ing the same γ: circles stand for vr= 0 and crosses for vr= 0.7, with µfrom Eq. (4). (b) Slope of the least-squares fits of the frequency-amplitude relation shown in (a) as func - tion of the relative reset γ. The solid line is an empirical fit ∆Ωopt/∆qr= 0.65−0.23 arctan[7 .0(γ−2.1)]. 4

arXiv:physics/0004020v1 [physics.atm-clus] 11 Apr 2000CTMC calculations of electron capture and ionization in collisions of multiply charged ions with elliptical Rydber g atoms J. Lu1, Z. Roller-Lutz2, H.O. Lutz1 1Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, 33501 Biel efeld, Germany 2Institute of Physics, Faculty of Medicine, Rijeka Universi ty, Rijeka, Croatia Abstract We have performed classical trajectory Monte Carlo (CTMC) s tud- ies of electron capture and ionization in multiply charged ( Q≤8) ion-Rydberg atom collisions at intermediate impact veloci ties. Im- pact parallel to the minor and to the major axis, respectivel y, of the initial Kepler electron ellipse has been investigated. The im- portant role of the initial electron momentum distribution found for singly charged ion impact is strongly disminished for highe r projec- tile charge, while the initial spatial distribution remain s important for all values of Q studied. The use of coherent elliptical Rydberg states in ion-atom co llision studies (for recent papers on the subject cf. [1-3] and references therei n) has not only aided the intuitive understanding of the interaction dynamics , i t also illuminates the roles of the momentum and the spatial distributions of the ta rget electron states. In classical terms, the momentum distribution can be widely varied simply by changing the eccentricity εof the Rydberg ellipse without affecting the energy of the state. In particular, for impact perpendicular to the ma jor axis of the ellipse, the capture cross section displays a maximum if vp(the perihelion electron ve- locity) is parallel and equal to the projectile velocity v; this is believed to be due to the matching electron momenta in the initial target and th e final projectile state. In contrast, the role of the spatial distribution becomes most clearly visible if the impact velocity vector in adjusted perpendicular to t he minor axis of the Rydberg ellipse; in this case, the electrons can be located e ither between the ap- proaching ion and the target nucleus (”upstream geometry”) or behind the target nucleus, as seen from the projectile (”downstream geometry ”) without otherwise changing the momentum distribution of the Rydberg state (i. e., its angular mo- mentum land the principal quantum number n). The capture cross section in both cases turns out to be quite different: it is much larger in the upstream case as compared to the downstream case; apparently, in the corre sponding region of parameter space the spatial characteristics of the initial state determine the out- come of the collision. These investigations have so far been restricted to collisions with singly charged ions. Recently, however, it has become p ossible to employ such targets in studies involving multiply charged ions [1, 4]. In another context (electron capture by multiply charged ion in the presence of an external magnetic field) we have found indications [5] that for increasing proj ectile charge Q the dis- 1tortion of the initial state increasingly dominates over th e influence of different target electron distributions; we have therefore performe d an exploratory study of such systems which is the topic of this Letter. We employ the classical trajectory Monte Carlo (CTMC) metho d which is quite useful in particular for the description of quantum me chanically complex systems, giving a good qualitative and often fairly quantit ative agreement with experimental data ( for more recent applications to the stud y of Rydberg atom collisions cf., e.g., [1-3]). Structureless ions of charge Q between 1 and 8 col- lide with Rydberg target atoms with nuclear charge q=1 and pr incipal quantum number n= 25. The geometry is chosen such that the direction of impact is perpendicular to the angular momentum direction of the Kepl er ellipse. Specif- ically, two cases are studied: (i) the impact is parallel to t he minor axis, thus allowing to study the velocity matching phenomenon, and (ii ) impact parallel to the major axis, showing the effect of the spatial orientation of the target elec- tron (”upstream-downstream asymmetry”). A useful quantit y characterizing the electron orbit is the (generalized) eccentricity ε=±/radicalBig 1−(l/n)2. In case (i), the + sign identifies orbits with the perihelion velocity vpparallel (-, antiparallel) tov; in case (ii) it characterizes the upstream (-, downstream) geometry. The impact velocity vis scaled by 1 /n, the velocity of a circular Rydberg state, i.e., V=vn= 1 in this case; the number of MC cycles was adjusted to obtain sta- tistical uncertainties of less than 5%. Care has been taken t o assure that the projectile starts sufficiently far from the target (approxim ately 3 .5×105atomic units a.u.) to correctly describe the initial part of the tra jectory; in view of the long range Coulomb potential and the known sensitivity of Ry dberg states to l, mchanging processes, this is critical particularly for high er projectile charges Q (see also below). (i) Impact parallel to the minor axis Figure 1 shows the eccentricity-dependent charge capture c ross section σ/Qfor V= 1.66 and Q ranging from 1 to 8. Velocity matching is obtained at ε= +0.47; at low Q, the cross section displays the well-known behavior with a pronounced maximum at this eccentricity and a deep minimum for negative eccentricities which characterize a strongly elongated Kepler ellipse wit hvantiparallel to vp. For increasing Q, this structure is soon washed out; while it is still noticeable for Q=2 and 4, almost any trace of the minimum has disappeared for Q=8. Inspec- tion of electron trajectories during the approach of the pro jectile ion reveals the reason: as expected, the long range Coulomb force distorts t he original Kepler el- lipse already at quite long distances. This distortion is qu ite regular, and reminds of a Stark effect. Indeed, a simple estimate confirms this: for Q=8, an electric field strength of 5V/cm (i.e., of the order of the fields applie d to the collision region in the experiments [1]) is attained at approximately 105a.u. This initial state effect might be reduced in the experiment by applying a s trong field in the target region; however, this results also illuminates an in herent weakness of the CTMC-approach: the slow rise of the electric field may ”in rea lity” induce adi- abatic transitions between the many Rydberg states which wo uld be populated differently in the classical calculation. This distortion o f the initial state becomes quite severe at distances below 104a.u., i.e., corresponding to several revolution of the Rydberg electron about its nucleus; therefore, it is t o be believed that the 2washing out of the cross section structure is indeed a real eff ect. Finally, we may add that also the impact parameter dependence of the capture probability reflects the signature of this effect. While for Q=1 and ε= +0.47 (the velocity matching situation) the capture probability is rather concentrated about the perihelion po- sition [6], it is nearly symmetric about b=0 (the position of the target nucleus) for Q=8. (ii) Impact parallel to the major axis For this study, we choose ε= 0.96, corresponding to l= 7. This creates a quite elongated state which is not too non-classical (low l). Figure 2 shows for Q=8 the cross sections for upstream ( σu) and downstream ( σd) geometry, respectively, in- cluding the contributions of the various ”swaps” to the capt ure cross sections. A swap has been defined as a passage of the electron through the m idplane between projectile and the target nucleus; note, however, that for a symmetric collisions (q/negationslash=Qas studied here), this plane has to cut the connection line be tween target and projectile nucleus at the saddle point of the two respect ive Coulomb poten- tials (i.e, at a distance R/(1 +/radicalBig q/Q) from the projectile, with R the distance of both nuclei). In view of the discussion in section (i) above, the strong upstream- downstream asymmetry seen in these data is at first glance som ewhat surprising. Again, inspection of the electron trajectories sheds light on this point: the slowly increasing electric field of the approaching projectile cau ses a distortion and pre- cession of the Kepler ellipse, in general not strong enough, however, to revert the upstream into a downstream geometry and vice versa . Therefore, upon approach of the projectile into the actual close interaction the elec tron is still mainly fore or aft, respectively, of the target nucleus, thus qualitati vely preserving the role of the initial spatial distribution of the electron; intere stingly, even 3-swap and higher swap processes can still be discerned. The qualitati vely different behavior ofσuandσdbeyond V= 1.5 is associated with differently rising cross sections for ionization; in case of the downstream geometry it sets in at c onsiderably smaller V-values as compared to the upstream geometry. This is furth er clarified by the respective impact parameter dependencies (Fig.3a,b): In t he upstream situation, charge exchange extends out to fairly large distances and io nization is still weak; in the downstream situation, the maximum charge exchange pr obability is of the same order as in the upstream case, however, it is limited to m uch smaller impact parameters, and ionization is already quite strong. To conclude, our analysis shows that for impact of multiply c harged ions of intermediate velocities the role of the initial electron momentum distribution becomes weaker for increasing projectile charge. This is du e to the strong per- turbation of the initial state by the approaching ion which i nduces pronounced changes in the momentum distribution long before the actual close interaction occurs. In contrast, the initial spatial orientation of the electron continues to be important for all Q-values studied here. Acknowledgment This work has been supported by the Deutsche Forschungsgeme inschaft (DFG). 3References: [1 ] J. C. Day, B. D. DePaola, T. Ehrenreich, S. B. Hansen, E. Ho rsdal- Pedersen, Y. Leontiev and K. S. Mogensen 1997 Phys. Rev A56, 4700; [2 ] D. M. Homan, O. P. Makarov, O. P. Sorokina, K. B. MacAdam, M . F. V. Lundsgaard, C. D. Lin, N. Toshima 1998, Phys.Rev A58, 4565 ; [3 ] L. Kristensen, T. Bov´ e, B. D. DePaola, T. Ehrenreich, E. Horsdal- Perdersen and O. E. Povlsen submitted ; [4 ] B. D. DePaola, private communiation ; [5 ] J. Lu, S. Bradenbrink, Z. Roller-Lutz, and H.O. Lutz, 199 9J.Phys. B: At.Mol.Opt.Phys .32, L681; [6 ] S. Bradenbrink, H. Reihl, Z. Roller-Lutz, and H.O. Lutz, 1995J.Phys. B: At.Mol.Opt.Phys .28, L133; 4Figure caption Fig. 1 Eccentricity-dependent capture cross section σ/Qfor impact velocity V= 1.66 (in units of the circular n= 25 Rydberg electron velocity) and different projectile charges Q. For the initial state, velocity match ing is obtained at ε= +0.47. Fig. 2 Charge capture cross sections for (a)upstream σuand (b)downstream σdge- ometry; the eccentricity ε=±0.96. The respective ionization cross sections are also given. Fig. 3 Impact parameter dependent probabilities of capture and ionization: (a) up- stream geometry, (b) downstream geometry. Impact paramete rbin atomic units; impact velocity V= 1.5; projectile charge Q= 8. 5-1,0 -0,5 0,0 0,5 1,0 0 5 10 15 20 25 generalized eccentricity e s /Q [10 -11 cm 2 ] Q=1 Q=2 Q=4 Q=8 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 0 5 10 15 20 25 30 35 40 (a) reduced velocity V s [ 10 -10 cm 2 ] total capture swap1 swap3 swap5 swap7 total ionization 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 0 10 20 30 40 50 (b) reduced velocity V s [ 10 -11 cm 2 ] total capture swap1 swap3 swap5 swap7 total ionization 0 2000 4000 6000 8000 0 10 20 30 40 50 60 70 80 90 100 (a) b (a.u.) Probability Capture Ionization 0 2000 4000 6000 8000 0 20 40 60 80 100 (b) b (a.u.) Probability Capture Ionization

arXiv:physics/0004021 11 Apr 2000INFLUENCE OF RANDOM IRREGULARITIES ON QUASI- THERMAL NOISE SPECTRUM OF PLASMA Nikolay A. Zabotin †, Yury V. Chugunov ‡, Evgene A. Mareev ‡, Andrey G. Bronin † † Rostov State University, Rostov-on-Don, Russia ‡ Institute of Applied Physics RAS, Nizhni Novgorod, Russia Introduction In the past three decades the thermal noise spectroscopy was recognized as a fruitful tool of space plasma diagnostics. It is well-known that when a passive electric antenna is immersed into a stable plasma, the thermal motion of the ambient electrons and ions produces fluctuations of the electric potential at the antenna terminals [Rostoker, 1961; Andronov ,1966; De Passiz, 1969; Fejer and Kan, 1969]. This quasi-thermal noise may be calculated if the particle velocity distribution function is known [Rostoker, 1961]. Since the noise spectrum depends on the main characteristics of plasma, as electron density or bulk temperature, the quasi-thermal noise spectroscopy can be used for diagnostics of plasma parameters. This diagnostic method is most appropriate for in situ space plasma measurements because it does not require of additional apparatus; large scale space plasma conditions allow one to construct antennas whose impedance is very small as compared with input impedance and whose characteristics can be calculated with fair accuracy. Such antennas permit direct observation of the frequency spectrum of thermal noise. Some examples of application of this diagnostic method can be found, for example, in [Meyer- Vernet, 1979; Couturier et al., 1981; Kellog, 1981; Meyer-Vernet and Pershe, 1989]. It is also well known that random irregularities of electron density always present in real space plasma. Random irregularities of the Earth's ionosphere are studied intensively and main properties of their spatial spectrum are known [Fejer and Kelley, 1980; Szuszczewicz, 1986]. These irregularities considerably affect propagation of radio waves in space plasma changing their phase, amplitude, spatial and angular distribution [Zabotin, Bronin and Zhbankov, 1998], as well as group propagation time and pulse duration [Bronin, Zabotin and Kovalenko, 1999]. Some information about the irregularity spectra of solar wind is also available [Rickett, 1973]. For the purposes of present investigation it is possible to use twin models of shape of spatial spectrum for both ionospheric and solar wind irregularities. Though that does not relate to parameters of the spectra. Irregularities substantially change properties of the medium with relation to electromagnetic radiation and they may also influence quasi-thermal noise spectrum detected by2 antenna. What is the possible physical mechanism of this influence? It is known, that fluctuations in plasma are closely connected with plasma dissipative properties. From the viewpoint of statistical mechanics, random irregularities in plasma may be understood as non-thermal large scale (in comparison with characteristic time and scale of particle motion) fluctuations. Such fluctuations may considerably change collision term in kinetic equation [Klimontovich, 1982] and, consequently, velocity distribution function. Since the quasi-thermal noise spectrum is determined by velocity distribution function, the change in distribution function will lead to change in noise spectrum. From the viewpoint of electrodynamics, random irregularities change mean dielectric properties of the medium (see, for example, [Hagfors,1984]). It means that in the media with random irregularities the roots of dispersion equation are shifted with relation to undisturbed values and even new roots may appear. The imaginary part of these roots may also be considerably changed due to additional non-collisional attenuation caused by scattering of waves in random medium. Development of strict theory of influence of random irregularities on quasi-thermal noise spectrum is a very complicated and difficult task, involving various fields of electrodynamics and statistical physics. In the present paper we will assume that noise spectrum in the random medium may be calculated using the tensor of effective dielectric permittivity. This tensor is determined as the dielectric permittivity tensor of some imaginary "effective" regular medium in which the field of point source is the same as the mean field in the corresponding random medium [Ryzhov, Tamoikin and Tatarskii, 1965, Ryzhov 1968]. Since effective dielectric permittivity tensor determines the mean field of the source in random media it also determines the impedance of antenna, which in its turn, determines the noise spectrum. It has been shown also that correlation function of electric field fluctuations in random medium may be expressed through the effective dielectric constant tensor [Ryzhov, 1968]. (It should be noted, however, that this result is based on averaging of Kallen-Welton formula and is valid only for states near the thermal equilibrium.) Spectrum of electrostatic noise in its turn is completely determined by correlation function of longitudinal electric field [Meyer-Vernet and Pershe, 1989]. The method based on effective dielectric constant tensor was used to study thermal noise spectrum in dielectrics [Ryzhov, 1970]. We apply this approach to the irregular space plasma. Calculation of the effective dielectric permittivity tensor of plasma is also rather complex problem involving summation of infinite series of perturbation theory. Some approximation is necessary to get the tensor components in closed form. In this paper we follow Born approximation. Thus, in the present paper we intend to estimate possible changes in the thermal noise frequency spectrum in plasma with random irregularities rather then to develop complete and strict3 theory of this effect. In Section 1 of the paper we briefly discuss calculation of the noise spectrum and impedance of antenna in Maxwellian plasma (we assume that presence of irregularities does not change velocity distribution function radically and it may be approximately described by Maxwell distribution). In Section 2 the method of calculation of effective dielectric permittivity tensor is stated. Section 3 discusses specific for given problem difficulties in numerical calculations. Results of numerical calculations in application to ionospheric plasma and plasma of solar wind are collected in Section 4. 1. Quasi-thermal noise spectrum and antenna impedance Usually the noise voltage spectral density measured at the electric antenna terminals, which is immersed in a plasma may be expressed through the spatial Fourier transform of the current distribution in the antenna and autocorrelation function of electrostatic field fluctuations in the antenna frame [Rostoker, 1961]. The shot noise, drift of the plasma across the antenna as well as some other phenomena also contribute into noise spectra. We do not take these phenomena into account now. If the plasma is in the thermal equilibrium at temperature T, what will be assumed in further consideration, the voltage spectral density may be expressed through antenna impedance by the formula [Meyer-Vernet and Pershe, 1989]: ()ZTkVBRe42= , (1) where Bk is Boltzmann constant, Zis antenna impedance and Re denotes the real part. Calculation of antenna impedance in general case is rather complex problem. One must take into account various phenomena such as the disturbance of trajectories of particles, collection of electrons and ions, photoemission and so on. The common approximation is to take into account only electron plasma oscillations. In this approximation one has ()() ()∫⋅ ⋅ ωεωε ωεπ=22 2 |||| 3 03)( ,),(Im 21Re kkJk kkkd Z/G26 /G26 /G26 /G26/G26 , (2) where ),(||k/G26 ωε is the plasma longitudinal permittivity, )(kJ/G26 /G26 is the spatial Fourier transform of the antenna current distribution. For Maxwellian collisionless plasma longitudinal dielectric permittivity is given by the well-known expression [Akhiezer, 1972]: [ ])exp()(11),(2 21 22 || zziz kkkD− π+Φ−+=ωε/G26 , (3)4 where ∫−=Φz dttzzz 02 2)exp()exp(2)( , ()k kz p Dω ω= 2, DDrk1=, Dr is Debye's length, pω is plasma frequency. To calculate impedance one needs to choose the current distribution )(kJ/G26 /G26 . The frequently used model of current distribution corresponds to wire dipole antenna [Kuhl 1966; Kuhl 1967; Kellog 1981]. The geometry of wire dipole antenna is shown in Fig. 1. The antenna consists of two cylinders, each of length L and radius La<<. This model may be a good approximation for geometry of real antennas used for space missions. The antenna parameters used in our calculations are in close correspondence with real parameters of spacecraft antennae used in investigations of ionosphere or solar wind (Wind and Kosmos-1809 missions). For wire dipole antenna one can get [Meyer-Vernet and Pershe, 1989; Kuhl, 1966, Kellog, 1981]: )( ),(),(Im4)Re( 02 |||| 02kF kk dk Z∫∞ ωεωε ωεπ= , (3) where []kLkLklkLSikLSi kaJkF  − − =)(sin2)2()( )()(4 2 0 , )(0xJ is Bessel function of the first kind and )(xSi is integral sinus function (see, for example, [Abramowitz and Stegun, 1964]. To calculate quasi-thermal electromagnetic noise spectrum in the plasma with random irregularities one should substitute the longitudinal dielectric permittivity for regular plasma ),(||k/G26 ωε by effective longitudinal dielectric permittivity ),(||keff/G26 ωε [Ryzhov, 1968]. 2. Calculation of effective dielectric permittivity tensor General scheme of calculation of effective dielectric permittivity tensor starts from the Dyson equation for the mean field in random medium (see [Rytov, Kravtsov and Tatarskii, 1978]). It may be easily found that effective dielectric permittivity tensor is proportional to the Fourier transform of the mass operator [Ryzhov, 1968]. Taking into account only initial term in the mass operator series expansion (what corresponds to Born approximation for the mean field) one should obtain ∫−Φω ω −ωε=ωε pdpkpGkBkk k N ml imlj ijeff ij3 2 00)(),(),( ),(),(/G26 /G26 /G26 /G26 /G26 /G26 , (4)5 where ()()ljljimim imljB δ−εδ−ε=0 0, ),(0kij/G26 ωε is dielectric permittivity tensor of regular plasma, ),(pGml/G26ω is the Fourier transform of Green tensor, ckω=0, c is velocity of light in vacuum, )(kN/G26 Φ is spatial spectrum of irregularities. In isotropic plasma one has );,( ),( ),(||2 20k kkk k kkk kji ji ij ij/G26 /G26 /G26 ωε+ωε   −δ=ωε ⊥ (5) , ),(1 ),(1),( ||2 022 2 02kkkkk kkkkkk kGji ji ij lm /G26 /G26/G26 ωε+ −ωε   −δ=ω ⊥ (6) where longitudinal permittivity ),(||k/G26 ωε is defined by formula (3) and transversal permittivity is determined by the expression [Akhiezer et al, 1972]: ( ))exp()( 1),(2 21 22 zziz k p− π−Φ ωω−=ωε⊥/G26 . At this point of our consideration we should choose a concrete type of the irregularity spatial spectrum )(kN/G26 Φ. Ionospheric irregularities are described by complex spectrum, which properties are different for different intervals of wave numbers of irregularities, but for the certain interval of wave numbers the spectrum may be described by power law: ( )2/2222221)(µ−+++∝Φzzyyxx N klklklk/G26 . Generally the spectrum is anisotropic: irregularities may be strongly stretched along the lines of force of geomagnetic field due to difference in the diffusion coefficients for longitudinal and transversal directions. However, for simplicity of calculations, we will use the model of isotropic spectrum. This situation is less characteristic of topside ionospheric plasma, but may be considered normal for solar wind plasma. In isotropic medium with isotropic irregularities tensor of effective dielectric permittivity has the same structure as tensor in regular media (5) (in other words effective medium is also isotropic): ),( ),( ),(||2 2k kkkk kkkkeffji effji ijeff ij/G26 /G26 /G26 ωε+ωε   −δ=ωε⊥, (7) where6 []() (); ),()(1),()(1),(),(),( 3 2 2 0222 03 ||222 || || ||  −ωε−Φ    ⋅−+  ωε−Φ⋅−ωε−ωε=ωε ⊥∫∫ pd pkkpk pkpkkpd kpk pkpkk k k NN eff /G26/G26 /G26 /G26 /G26/G26/G26 /G26 /G26 /G26/G26 /G26 /G26 (8) []() (). ),()(1),()(11),(21),(),( 3 2 2 0222 03 ||222  −ωε−Φ    ⋅++  ωε−Φ    ⋅− −ωε−ωε=ωε ⊥⊥ ⊥ ⊥ ∫∫ pd pkkpk pkpkkpd kpk pkpkk k k NN eff /G26/G26 /G26 /G26 /G26/G26/G26 /G26 /G26 /G26/G26 /G26 /G26 (9) For specific spectrum index =µ4 (quite typical value both for ionospheric and for solar wind plasma) the spectrum can be written as follows: ( )()222 232 1 ) exp(12)(−+ −−πδ=Φ kl lRlkR N/G26 , (10) where π=2/mLl , mL is the outer scale of spectrum, )(2RDR=δ , )(RD is structure function of irregularities at scale length R. Substituting (10) into (8) one obtains for longitudinal effective dielectric permittivity: ( )[] ( ) () ( ). 2 11 ),(2 1),(1),() exp(1),(),( 1 12222222 02 2 02 2 01 12222222 0||22 ||32 || ||  −++− −ωε+ −++  ωε×−ωε−−πδ−ωε=ωε ∫ ∫∫∫ −∞ ⊥−∞ kptlplkldtt ppkdppkkptlplkldtt pdppklRlk kR eff /G26/G26/G26 /G26 /G26 (11) Internal integration in (11) can be done in analytical form: ( )ξξ−ξ−ξ+ − ξ−ξ= −++∫ −211ln 1 2 1221 12222222 kptlplkldtt ; () ( )ξξ−ξ−ξ+ = −++−∫ −211ln 2 111 12222222 kptlplkldtt, (12) 22222 12 plklpkl ++=ξ .7 These functions turn into zero when ξ is zero (what corresponds to limiting cases 0,,0 →∞→→ kpp or ∞→k) and provide proper convergence of integrals in (11) at both limits of integration over p. Though one could use result (11) directly, without substitution of analytic expressions for the inner integral, generally such approach may be not successful. The reason is that integrands in (11) have several peculiarities, which will be discussed below, as well as significant dependence of integrand on such parameters as Debye's radius, outer scale of the irregularity spectrum, dimensions of antenna etc. The latter difficulty, however, may be considerably reduced by introduction of dimensionless variables. 3. Details of numerical calculations Considerable difficulties under numerical calculation of quasi-thermal noise spectrum and of effective dielectric permittivity tensor using expressions (3) and (11) correspondingly are caused by the "bad" behavior of integrand at some special points. In both cases integrand contains expressions of one of the following kinds: ()()2 ||2 |||| ),(Im),(Re),(Im k kk /G26 /G26/G26 ωε+ωεωε ; (12) ()()2 ||2 |||| ),(Im),(Re),(Re k kk /G26 /G26/G26 ωε+ωεωε ; (13) ()()2 ||2 |||| ),(Im),(Re),(Im k kk eff effeff /G26 /G26/G26 ωε+ωεωε . (14) As an example, in Fig. 2 the dependence of ||Reε (solid lines) and ||Imε (dashed lines) on Dkr for four values of dimensionless frequency pωω/ (= 1.01; 1.05; 1.1; 1.2) has been plotted. Functions (12) and (14) have a peak of height ||Im1~ε and width ||Im~ε (or eff ||Im1~ε and eff ||Im~ε) when 0Re||=ε or 0 Re||=εeff. For pωω~ the values of ||Imε are very small when /G13=ε||Re and the peak is very sharp (see Figs. 4 and 5). Function (13) has two peaks of opposite signs at the left and right of the point where 0Re||=ε, and these peaks are also very sharp at the frequencies close to plasma frequency (see Fig. 3). A usual way to calculate integrals with such quasi-singular integrand is to split the integration interval at the point where the singularity happens. In opposite case the result would depend upon the position of the peak with relation to8 the grid of abscissas of integration rule which would be different for different frequencies. In other words, in this case the calculated noise spectra would contain random error component due to accumulation of inaccuracy under numerical integration. However the splitting of integration interval is not sufficient for successful integration. Additional difficulties are caused by the fact that the roots of equation 0Re||=ε may be found only approximately (15 ||10~Re−ε in the root), while the magnitude of ||Imε in the root may be several orders of magnitude smaller. To avoid this interference one should take into account the fact that in real plasma some small collisional attenuation is always present. The value of collision frequency may be chosen so that inequality || ||ImRe ε<<ε always holds in the small vicinity of the root of equation 0Re||=ε. As it has been noticed above it is useful to introduce dimensionless variables to reduce the dependence of numerical calculations on certain values of plasma parameters. It is natural to determine dimensionless frequency ω~ and wave number k~ as ωω=ω =~,~ 0 p kkk . (13) In these variables longitudinal dielectric permittivity, for example, may be written as: ()[ ])exp()(1~11)~,~(2 22 0|| zziz krkk D−π+Φ− +=ωε , (14) where krkz D~~ 21 0ω= . In (14) plasma properties are taken into account only through the dimensionless constant Drk0. Using (12) – (14), expression (11) for effective dielectric constant tensor can be transformed to the following form [] ()⋅−−δ π−ωε−ωε=ωε3 0220 ||0 || ||) exp(121)~,~()~,~()~,~( lklRk k kR eff ()()()()()() [ ]    + ++ωε  − +∫ ∫∞ ∞− 0 0222 0 36 010 ||22 02 ~~~1~~41 ~,~(~~1~~ ppklkpd klkk kplkppd () ()()()() ()()     − ++ +− + +22 022 0 222 02 0 ~~1~~1ln21 ~~1~~2 pklkpklk pklkpklk . (15) We used Brent algorithm for searching the roots of equation 0)~,~(Re||=ωεk; the integration in expression (15) was done using the adaptive integration method based on Gauss-Kronrod rule.9 4. Results of numerical calculations For calculations of quasi-thermal noise spectrum we first chose the regular parameters of plasma and spectrum of irregularities corresponding to the ionosphere F-region: plasma frequency πω2p was equal to 3 MHz, Debye's length – 5 cm; R = 1 km, l = π2/10 km. Calculations were carried out for four different values of Rδ: 3105−⋅, 210−,2102−⋅ and 2103−⋅. The first value is typical for undisturbed mid-latitude ionosphere, the last is observed in polar and equatorial ionosphere and in the experiments of plasma heating by the powerful HF radio wave. In regular plasma the spectrum has a peak just above the plasma frequency; the shift from the plasma frequency is of order of ()2~LrDωδω [Meyer-Vernet and Pershe, 1989]. For the chosen parameters of antenna 1<<LrD and 1<<arD.It means that noise spectrum does not depend upon the radius of antenna wire and have a sharp peak near the plasma frequency. The spectra for regular plasma and for irregular plasma with given Rδ are plotted in Fig. 6. In presence of irregularities we observe noticeable change in the spectrum of quasi-thermal noise. The peak of the spectrum in this case is split into two peaks. The splitting of the peak is caused by the complication of dielectric properties of plasma in the presence of random irregularities. At the relatively high magnitudes of Rδ real part of effective longitudinal dielectric constant has additional roots, while the contribution of the roots existing in regular plasma is dumped due to increase of the imaginary part caused by scattering. The distance between peaks is greater for greater irregularity level. Its magnitude is small, approximately 10 – 15 KHz for 3105−⋅=δR and ~ 100 KHz for 2103−⋅=δR . Such details of the spectrum, however, can be detected in experiment. For the solar wind plasma we chose the following parameters: plasma frequency 202=πωp KHz, 10=Drkm, π=2/106l km. Noticeable influence of random irregularities on the electromagnetic noise spectrum takes place at relatively high level of irregularities. In Fig. 7 the noise spectrum is plotted for 2 21015,105− −⋅ ⋅=δR and 21030−⋅ for normalization scale length 510=Rm. The noise spectrum for ,10152−⋅=δR21030−⋅ and 21050−⋅ for normalization scale 810=R m is plotted in Fig. 8. In the case of solar wind plasma we observe the formation of plateau in the spectrum in the vicinity of plasma frequency instead of the splitting of the peak. This difference is explained by the difference in ratio of Debye's length to wire length for ionospheric plasma and for solar wind plasma.10 Conclusions In this paper we have considered the influence of random irregularities of electron density in isotropic plasma on the quasi-thermal noise spectrum using fairly simple model of irregularities. We have found that for the small values of irregularity level modification of noise spectrum is negligibly small. However, for larger values of NN/Δ, also quite possible in natural conditions, irregularities cause some noticeable effects. In the ionospheric plasma it is the splitting of the peak in the frequency noise spectrum located just above the plasma frequency, into two peaks. Though the gap between those peaks is small it still may be detected in experiment. The magnitude of the gap depends upon the value of NN/Δ, what makes possible using measurements of noise spectra for the purpose of the irregularity diagnostics. In the solar wind plasma irregularities cause changes of the shape of the main spectrum maximum near the plasma frequency resulting in appearance of the plateau under higher irregularity level. This effect also can provide essential information about the solar wind irregularities. In this paper we used simplified model of plasma which may be significantly improved. The major possible improvement concerns the spectrum of irregularities. For example, in real topside ionosphere irregularities are stretched along the lines of force of geomagnetic field and their spectrum is anisotropic. One can also take into account drift of the plasma across the antenna, because noise spectra measurement are done on the board of satellite moving through the plasma. Though the account for magnetic field is considered to be unimportant for calculation of noise spectrum in regular plasma, in the presence of irregularities its influence on wave propagation may be important. For other kinds of space plasma, like the plasma of solar wind, the account of all these factors may be essential, first of all because such plasma may be anisotropic even in the absence of external magnetic field. Besides generally it cannot be considered as being in the thermal equilibrium, so more general approach to deriving expressions for noise spectra may be required. References Abramowitz M., Stegun I.A. Handbook of mathematical functions, National Bureau of Standarts, Applied mathematics series, 1964. Akhiezer A.I., Akhiezer I.A., Polovin R.V., Sitenko A.G., Stepanov K.N. (1974) Electrodinamics of Plasma, Moscow, ''Nauka'', (in Russian). Andronov A.A. Antenna impedance and noise in a pace plasma .Kosm. Issled., 1966, 4, 588 (in Russian).11 Bronin A. G., Zabotin N.A., Kovalenko E.S. Nonmonochromatic radiation transfer in a plane slab of magnetized plasma with random irregularities, Radio Sci., 1999, 34, 5, 1321. Couturier P., Hoang S., Meyer-Vernet N., Steinberg J.L. Quasi-thermal noise in a stable plasma at rest. J. Geophys.Res., 1981, 86, 11, 127. De Passiz 0. Shot noise in antennas, Radio Sci., 1969, 4, 91. Fejer J.A., Kan J.R. Noise spectrum received by an antenna in a plasma. Radio Sci. 1969, 4, 721. Fejer B.G., Kelley M.C. Ionospheric irregularities. Rev. Space Phys. 1980, 18,2, 401. Hagfors T. Electromagnetic wave propagation in a field-aligned-striated cold magnetoplasma with application to ionosphere. J. Atmos. Ter. Phys, 1984, 46, 3, 211. Kellog P.J. Calculation and observation of thermal electrostatic noise in solar wind plasma. Plasma Phys., 1981, 23, 8, 735. Klimontovich Yu. L. Statistical physics. – Moscow, "Nauka", 1982 (in Russian). Kuhl H.H. Resistance of a short antenna in a warm plasma, Radio Sci., 1966, 1, 971. Kuhl H.H. Computations of the resistance of a shirt antenna in a warm plasma, Radio sci., 1967, 2,73. Meyer-Vernet N., Pershe C. Tool kit for antennae and thermal noise near the plasma frequency. J. Geophys. Res., 1989, 94, A3, 2405. Meyer-Vernet N. On natural noise detection by antennas in plasmas, J. Geophys. Res., 1979, 84, 5373. Rickett B.I. Power spectrum of density irregularities in the solar wind plasma. J. Geoph. Res. !973, 78, 10, 1543. Rostoker N. Fluctuations of a plasma. Nuclear fusion, 1961, 1, 101. Rytov S.M., Kravtsov Yu. A., Tatarskii V.I. Introduction to statistical radiophysics; Vol. 2, Random Fields. Nauka, Moscow, 1978 (in Russian). Ryzhov Yu. A.., Tamoikin V.V., Tatarskii V.I. On space dispersion of inhomogeneous media. JETP, 1965, 48, 2, 656 (in Russian). Ryzhov Yu. A. Analytic prorperties of the effective dielectric permeability of randomly inhomogeneous media. JETP 1968, 55, 2(3), 567 (in Russian). Ryzhov Yu. A. Thermal radiation in a randomly inhomogeneous medium. JETP 1970, 59, 7, 218 (in Russian). Szuszczewicz E.P. Theoretical and experimental aspects of ionospheric structure: a global perspective and irregularities. Radio Sci., 1986, 21, 3, 351. Zabotin N.A., Bronin A.G. and Zhbankov G.A. Radiative transfer in a layer of magnetized plasma with random irregularities. Waves in random media, 1998, 8, 421.12 Fig. 1. Wire dipole antenna geometry13 Fig. 2. Real and imaginary components of longitudinal dielectric permittivity of a regular plasma at the frequency near plasma frequency. Fig. 3. ()()2 ||2 |||| ),(Im),(Re),(Re k kk /G26 /G26/G26 ωε+ωεωε as a function of k in the vicinity of the root of equation 0Re||=ε.14 Fig. 4. ()()2 ||2 |||| ),(Im),(Re),(Im k kk /G26 /G26/G26 ωε+ωεωε as a function of k in the vicinity of the root of equation /G13=ε||Re Fig. 5. Same as 4, but as function of k and ω.15 2.902.932.952.983.003.033.053.083.10 f, MHz0.010.101.0010.00100.00δ R = 0% δ R = 0.5% δ R = 1% δ R = 2% δ R = 3% Fig. 6. Quasi-thermal noise spectrum in the ionosphere.16 16171819202122232425 f, KHz4.0E-36.0E-38.0E-31.0E-21.2E-21.4E-21.6E-21.8E-22.0E-2 δ R = 0% δ R = 5% δ R = 15% δ R = 30% Fig. 7. Quasi-thermal noise spectrum in the solar wind; normalization scale 510=Rm. 1820222426283032 f, KHz6.0E-38.0E-31.0E-21.2E-21.4E-21.6E-21.8E-22.0E-2 d R = 0% d R = 15% d R = 30% d R = 50% Fig. 8. Quasi-thermal noise spectrum in the solar wind; normalization scale 810=Rm.

arXiv:physics/0004022v1 [physics.plasm-ph] 12 Apr 2000DICLUSTER STOPPING IN A DEGENERATE ELECTRON GAS Hrachya B. Nersisyan(1)and Amal K. Das(2) (1)Division of Theoretical Physics, Institute of Radiophysic s and Electronics, Alikhanian Brothers St. 2, Ashtarak-2, 37841 0, Republic of Armenia1 (2)Department of Physics, Dalhousie University, Halifax, Nov a Scotia B3H 3J5, Canada Abstract In this paper we report on our theoretical studies of various aspects of the correlated stopping power of two point-like ions (a diclust er) moving in close but variable vicinity of each other in some metallic target m aterials the lat- ter being modelled by a degenerate electron gas with appropr iate densities. Within the linear response theory we have made a comprehensi ve investiga- tion of correlated stopping power, vicinage function and re lated quantities for a diproton cluster in two metallic targets, aluminum and cop per, and present detailed and comparative results for three approximations to the electron gas dielectric function, namely the plasmon-pole approximati on without and with dispersion as well as with the random phase approximation. T he results are also compared, wherever applicable, with those for an indiv idual projectile. 1E-mail: hrachya@irphe.am 11 INTRODUCTION The stopping of energetic charged particles in a target mate rial is a prob- lem of long-standing theoretical and experimental interes t. Early pioneering theoretical work by Bohr who had a lifelong interest in this p roblem was fol- lowed by Bethe and others. There is by now an extensive litera ture on this topic. We refer the reader to two recent review articles [1, 2 ]. The problem in its generality is rather complex. However simplified theore tical models have been applied with considerable success in explaining exper imental data. As- suming a weak coupling between the energetic particle and a t arget material, specially a metal which is usually approximated by a degener ate electron gas, a detailed theoretical model has emerged through the works o f Bohm, Pines, Lindhard, Ritchie and other authors [3-5]. A comprehensive treatment of the quantities related to inel astic particle- solid and particle-plasma interactions, e.g. scattering r ates and differential and total mean free paths and energy losses, can be formulate d in terms of the dielectric response function obtained from the electro n gas model. The results have important applications in astrophysics [6, 7] and radiation and solid-state physics [8-11], and more recently, in studies o f energy deposition by ion beams in plasma fusion targets [12-16]. One can think of several situations in which the projectile b eam ions may be closely spaced so that their stopping is influenced by t heir mutual interactions [14, 15] and thus differs from the stopping of ch arged particles whose dynamics is independent of each other. This can happen , for example, in the case of very high density beams, or more realistically , when cluster ions are to be used instead of standard ion beams [16]. The stopping of uncorrelated or independent charged partic les in a de- generate electron gas (DEG) has been extensively studied in the literature (see, for example, [3-9, 13] and other references therein). These studies have been done mostly within the linear response formalism and fo r the projectile velocity Vcomparable or greater than vF, the electron Fermi velocity. The objective of this paper is to make a study of the stopping p ower (SP) of correlated charged particles in a DEG. The simplest a nd yet phys- ically relevant case is the SP of an ion pair (a dicluster). We report on a comprehensive investigation, which is mostly numerical, o f various aspects of a dicluster stopping in a DEG. Previous theoretical works have consid- ered this problem within the linear response theory [9-11] a nd in a simple 2plasmon-pole approximation. In our study, again in the line ar response for- malism, we have used both the plasmon-pole approximation (P PA) as well as the full Lindhard expression for the random phase approxi mation (RPA) in the DEG dielectric function. As in earlier studies we cons ider a dipro- ton cluster as a projectile and compare our theoretical resu lts with those of Basbas and Ritchie [10] who used PPA and with those obtained i n RPA. Whenever applicable, the results are also compared with tho se for an inde- pendent projectile (e.g. those due to Yakolev and Kotel’nik ov [7]). No RPA results for a diproton cluster SP and related aspects have pr eviously been reported in the literature. 2 STOPPING POWER Let us consider an external charge with distribution ρext(r, t) =Qext(r− Vt) moving with velocity Vin a medium characterized by the longitudinal dielectric function ε(k, ω). Within the linear response theory and in the Born approximation the scalar electric potential ϕ(r, t) due to this external charge screened by the medium is given by [1] ϕ(r, t) = 4π/integraldisplay dkQext(k)exp [ik(r−Vt)] k2ε(k,kV), (1) where Qext(k) is the Fourier transform of function Qext(r). The stopping power which is the energy loss of the external ch arge re- garded as a projectile, per unit path length in the medium reg arded as a target material, can be calculated from the force acting on t he charge. The latter is related to the induced electric field Eindin the medium. For a three- dimensional medium we have, for the SP, S≡ −/integraldisplay drQext(r−Vt)V VEind(r, t) =2(2π)4 V/integraldisplay dk|Qext(k)|2kV k2Im−1 ε(k,kV). (2) Eq. (2) is applicable to any external charge distribution. W e shall apply it to a cluster of two point-like ions having charges Z1eandZ2eseparated by a variable distance R. For this dicluster 3|Qext(k)|2=e2 (2π)6/bracketleftbig Z2 1+Z2 2+ 2Z1Z2cos (kR)/bracketrightbig . (3) Then the stopping power of a dicluster may be written as S=/parenleftbig Z2 1+Z2 2/parenrightbig Sind(V) + 2Z1Z2Scorr(R, V), (4) where Sind(V) andScorr(R, V) stand for individual and correlated SP, respec- tively. From Eqs. (2) and (3) Sind(V) =2e2 πV2/integraldisplay∞ 0dk k/integraldisplaykV 0Im−1 ε(k, ω)ωdω, (5) Scorr(R, V) =2e2 πV2/integraldisplay∞ 0dk k/integraldisplaykV 0Im−1 ε(k, ω)ωdω (6) ×cos/parenleftBigω VRcosϑ/parenrightBig J0/parenleftBigg Rsinϑ/radicalbigg k2−ω2 V2/parenrightBigg . J0(x) is the Bessel function of the first kind and zero order and ϑis the angle between the interionic separation vector Rand the velocity vector V. Eqs. (5) and (6) can also be obtained from the linearized Vlasov-P oisson equations for a two-ion projectile system, as has been done by Avanzo et al. [14]. In their study the target material is a dense classical electro n gas. We note that there are two contributions to the SP of a two-ion cluster. The first one, given by the first term in Eq. (4), is the uncorrel ated particle contribution and represents the energy loss of the individu al projectiles due to their coupling with the target electron gas. The second co ntribution, the second term in Eq. (4), arises due to a correlated motion of th e two ions through a resonant interaction with the excitations of the e lectron gas. Both terms are responsible for an irreversible energy transfer f rom the two-ion projectile system to the target electron gas. In many experimental situations, clusters are formed with r andom orien- tations of R. A correlated stopping power appropriate to this situation may be obtained by carrying out a spherical average over Rof the Scorr(R, V) in Eq. (4). We find 4Scorr(R, V) =2e2 πRV2/integraldisplay∞ 0dk k2sin(kR)/integraldisplaykV 0Im−1 ε(k, ω)ωdω. (7) One may consider an interference or vicinage function, whic h is a measure of the separation of single-particle contribution from its correlated counter- part, to the stopping power. This function is defined as [10, 1 4, 15] Γ(R, V) =Scorr(R, V) Sind(V). (8) Eq. (4) can then be put in the form S=/parenleftbig Z2 1+Z2 2/parenrightbig Sind(V)/bracketleftbigg 1 +2Z1Z2 Z2 1+Z2 2Γ(R, V)/bracketrightbigg . (9) Γ(R, V) describes the strength of correlation effects with respect to an un- correlated situation. The vicinage function becomes equal to unity as R→0 when the two ions coalesce into a single entity, and goes to ze ro asR→ ∞ when the two ions are totally uncorrelated. 3 THEORETICAL CALCULATIONS OF SP The key ingredient in the calculation of stopping power, as o utlined in Sec- tion II, is the linear response function ε(k, ω) of the target material. The latter is modelled in most cases by a dense electron gas neutr alized by a positive background - the so-called jellium model. The effec t of lattice is not included except perhaps through an effective mass of the elec trons. This is a reasonable model for target materials like Cu and Al. ε(k, ω) for a dense, and hence a degenerate electron gas, has been calculated in vari ous approxima- tions in the literature. Two of them are: (A) Plasmon-Pole ap proximation (PPA), and (B) the full random phase approximation (RPA). Ac tually the plasmon-pole approximation is a simplification of the RPA re sponse function. We shall consider SP in these two approximations. 3.1 SP in PPA without plasmon dispersion Here we consider the simplest model of the dielectric functi on of a jellium. In Ref. [10] (see also the Ref. [15]) a plasmon-pole approximat ion to ε(k, ω) for 5an electron gas was used for calculation of dicluster SP. In o rder to get easily obtainable analytical results, Basbas and Ritchie [10] emp loy a simplified form which exhibits collective and single-particle effects Im−1 ε(k, ω)=πω2 p 2ω/bracketleftbigg H(kc−k)δ(ω−ωp) +H(k−kc)δ/parenleftbigg ω−/planckover2pi1k2 2m/parenrightbigg/bracketrightbigg ,(10) where H(x) is the Heaviside unit-step function, /planckover2pi1ωpis the plasma energy of the electron gas and the choice kc= (2mωp//planckover2pi1)1/2allows the two δfunctions in Eq. (10) to coincide at k=kcin the k-ωplane. The first term in Eq. (10) describes the response due to nondis persive plasmon excitation in the region k < k c, while the second term describes free-electron recoil in the range k > k c(single-particle excitations). This approximate function satisfies the sum rule /integraldisplay∞ 0Im−1 ε(k, ω)ωdω=πω2 p 2(11) for all values of k. In this approximation if V >(/planckover2pi1ωp/2m)1/2≡Vp Sind(λ) =e2ω2 p V2ln/parenleftbigg2mV2 /planckover2pi1ωp/parenrightbigg =Σ0 πχ2λ2ln/parenleftBigg λ2√ 3 χ/parenrightBigg , (12) where λ=V/vF,χ2= 1/πkFa0= (4/9π4)1/3rs;rs= (3/4πn0a3 0)1/3,n0is the electron gas density and a0= 0.53×10−8cm is the Bohr radius. kFis the Fermi wave number of the target electrons and Σ 0= 2.18 GeV/cm. In our calculations χ(orrs) serves as a measure of electron density. The result in Eq.(12) agrees exactly with the Bethe SP formula, except tha t the plasmon energy of the electron gas /planckover2pi1ωpappears instead of the usual mean atomic excitation energy. Eq. (12) represents the contribution of valence/conduction electrons in a solid to the stopping of an ion. Using Eq. (10) in Eq. (6), in the high-velocity limit V > V p(orλ2> χ/√ 3≡λ2 0) one finds Scorr(R, ϑ, λ ) =Σ0 πχ2λ2/braceleftbigg cos/parenleftbigg2χ λ√ 3kFRcosϑ/parenrightbigg (13) 6×/integraldisplayλ/λ0 1dx xJ0/parenleftbigg2χ λ√ 3kFRsinϑ√ x2−1/parenrightbigg + /integraldisplay2λ 2λ0dx xcos/parenleftbiggx2 2λkFRcosϑ/parenrightbigg J0/parenleftBigg xkFRsinϑ/radicalbigg 1−x2 4λ2/parenrightBigg/bracerightBigg . If one ion trails directly behind the other ( ϑ= 0) from Eq. (13) we find Scorr(R,0, λ) =Σ0 2πχ2λ2/braceleftBigg cos/parenleftbigg2χ λ√ 3kFR/parenrightbigg ln/parenleftBigg λ2√ 3 χ/parenrightBigg + (14) ci (2λkFR)−ci/parenleftbigg2χ λ√ 3kFR/parenrightbigg/bracerightbigg , where ci ( z) is the integral cosin function ci (z) =−/integraldisplay∞ zdxcosx x. (15) One sees a characteristic oscillatory behavior for large in terionic distance R. As discussed in [16], fluctuations in the stopping power of a medium for a cluster as separation increases are due to electron densit y variation in the wake of the leading ion. The wavelength of these fluctuations is∼2πV/ω p for high-velocity projectiles. In the case of randomly oriented clusters from Eq. (7) we find Scorr(R, λ) =Σ0 πχ2λ2/bracketleftbigg si2/parenleftbigg2χ λ√ 3(kFR)/parenrightbigg −si2(2λ(kFR))/bracketrightbigg ,(16) where si2(z) =/integraldisplay∞ zdxsinx x2=sin(z) z−ci(z). (17) 3.2 SP in PPA with plasmon dispersion Plasmons without dispersion are an idealization. In real sy stems plasmons are expected to undergo a dispersion leading to a ω(k). The actual dispersion ( in RPA) can be obtained from the linear response function (s ee Sec. 3.3). 7Here we shall utilize a dispersion which is valid for small an d intermediate values of the wave vector k. Consequently we write Im−1 ε(k, ω)=πω2 p 2ωδ(ω−Ω(k)), (18) where the dispersion is given by Ω2(k) =ω2 p+3 5k2v2 F+/planckover2pi12k4 4m2. (19) In this approximation when V >/parenleftbigg3 5v2 F+/planckover2pi1ωp m/parenrightbigg1/2 ≡V0 (20) we have, for ISP and CSP, Sind(λ) =Σ0 πχ2λ2lnλ2−3/5 +/radicalBig (λ2−3/5)2−4χ2/3 2χ/√ 3, (21) Scorr(R, ϑ, λ ) =Σ0 πχ2λ2/integraldisplayx+(λ) x−(λ)dx xcos/parenleftbiggφ1(x) 2λkFRcosϑ/parenrightbigg J0/parenleftbiggφ2(x) 2λkFRsinϑ/parenrightbigg . (22) HerekFis the Fermi wave number, and φ1(x) =/radicalbigg x4+12 5x2+ 16χ2/3, (23) φ2(x) =/radicalBigg 4/parenleftbigg λ2−3 5/parenrightbigg x2−(x4+ 16χ2/3), (24) x±(λ) =/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt2 λ2−3 5±/radicalBigg/parenleftbigg λ2−3 5/parenrightbigg2 −4χ2 3 . (25) In the case of randomly oriented clusters we find Scorr(R, λ) =Σ0 πχ2λ2[si2(kFRx−(λ))−si2(kFRx+(λ))]. (26) 83.3 Stopping power in RPA Now we will derive the analytical expressions for the SP of a d icluster in a fully degenerate ( T= 0) electron gas. For this purpose we use the exact RPA dielectric response function obtained by Lindhard [4] ε(z, u) = 1 +χ2 z2[f1(z, u) +if2(z, u)], (27) where f1(z, u) =1 2−1 8z/parenleftbig U2 +−1/parenrightbig ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleU++ 1 U+−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle+1 8z/parenleftbig U2 −−1/parenrightbig ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleU−+ 1 U−−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle,(28) f2(z, u) =  π 8z/parenleftbig 1−(u−z)2/parenrightbig ;|u−1|/lessorequalslantz/lessorequalslantu+ 1 0; 0 /lessorequalslantz/lessorequalslantu−1, 0;z/greaterorequalslantu+ 1, 1 2πu; 0/lessorequalslantz/lessorequalslant1−u. (29) Here, as in Refs. [6, 7, 13, 15], we have introduced the follow ing notations z=k/2kF,u=ω/kv F,U±=u±z. With these notations Eqs. (5), (6) and (7) read Sind(λ) =6Σ0 π2χ2λ2/integraldisplay∞ 0z3dz/integraldisplayλ 0f2(z, u)udu [z2+χ2f1(z, u)]2+χ4f2 2(z, u), (30) Scorr(λ, R, ϑ ) =6Σ0 π2χ2λ2/integraldisplay∞ 0z3dz/integraldisplayλ 0f2(z, u)udu [z2+χ2f1(z, u)]2+χ4f2 2(z, u) ×cos/parenleftBig 2zu λkFRcosϑ/parenrightBig J0/parenleftBigg 2zkFRsinϑ/radicalbigg 1−u2 λ2/parenrightBigg ,(31) Scorr(R, λ) =3Σ0 πλ2a0 R/integraldisplay∞ 0sin (2kFRz)z2dz/integraldisplayλ 0f2(z, u)udu [z2+χ2f1(z, u)]2+χ4f2 2(z, u). (32) In order to evaluate the integrals by zin Eqs. (30)-(32) at λ <1 (in the low-velocity limit) we split the integration region int o two domain: 0 /lessorequalslant 9z/lessorequalslant1−uand 1−u/lessorequalslantz/lessorequalslant1 +u, where Im ε∼f2/negationslash= 0. However atλ >1 (in high-velocity limit) we need to take into account the re gion 1/lessorequalslantu/lessorequalslantλ, 0/lessorequalslantz/lessorequalslantu−1, where f2may vanish. The integration in this region includes the excitation of collective plasma modes ( plasmons) by fast charged particles. Consequently, although f2= 0 the integrals in this region are not equal to zero. A calculation of the collective part of SP is facilitated if we use the following known expression χ2f2(z, u) [z2+χ2f1(z, u)]2+χ4f2 2(z, u)→πδ/parenleftbig z2+χ2f1(z, u)/parenrightbig = (33) =πδ(z−zr(χ, u))/vextendsingle/vextendsingle/vextendsingle2z+χ2∂f1(z,u) ∂z/vextendsingle/vextendsingle/vextendsingle z=zr(χ,u), where zr(χ, u) is the solution of the dispersion equation ε(k, ω) = 0 in vari- ableszandu. Figure 19 shows the solution zr(χ, u) for various values of χ(solid line, χ= 0.5; dashed line, χ= 0.15; dotted line, χ= 0.05). It may be noted that the integration domain 0 /lessorequalslantu/lessorequalslantλ,z > u + 1, where f2= 0, does not contain the dispersion curve zr(χ, u) calculated for metallic densities χ∼0.5 (rs∼2). Consequently the SP in this region of variables zanduvanishes and there is no plasmon excitation. Let us consider the low-velocity limit ( V≪vF) of Eqs. (30) and (31). In this limit one can obtain simpler expressions for SP. From Eqs. (30) and (31) we have Sind(λ)≃Σ0 πχ2λ/integraldisplay1 0z3dz [z2+χ2f(z)]2, (34) Scorr(λ, R, ϑ )≃3Σ0 2πχ2λ/integraldisplay1 0z3dz [z2+χ2f(z)]2/bracketleftbig Φ1(zkFR) + Φ 2(zkFR) sin2ϑ/bracketrightbig , (35) where Φ1(ξ) =1 ξ3/bracketleftbigg/parenleftbigg ξ2−1 2/parenrightbigg sin (2ξ) +ξcos (2ξ)/bracketrightbigg , (36) 10Φ2(ξ) =1 ξ3/bracketleftbigg/parenleftbigg3 4−ξ2/parenrightbigg sin (2ξ)−3 2ξcos (2ξ)/bracketrightbigg , (37) f(z)≡f1(z,0) =1 2+1−z2 4zln1 +z 1−z. (38) From Eqs. (36) and (37) it follows that Φ 1(ξ)→2/3, Φ2(ξ)→0 atξ→0 and consequently, as expected, Scorr(λ, R, ϑ )→Sind(λ) when R→0. Note that (as is well-known [4, 13]) in the low-velocity limit the SP is proportional to the velocity of particle (Eqs. (34) and (35)). Thus the vic inage function Γ(λ, R, ϑ ) atλ≪1 depends only on interionic distance Rand orientation angle ϑ. 4 NUMERICAL RESULTS AND DISCUS- SION Using the theoretical results of Secs. 2 and 3, we have made ex tensive nu- merical calculations of stopping power (SP) and related qua ntities. In this section we present detailed numerical results for two targe t materials, Al and Cu. These two targets have been chosen because of their fr equent use in experiments and also because of their different electron d ensities. In our calculations χ(orrs) is a measure of electron density. As a simple but generic example of a projectile, we have consi dered a diproton cluster for which we present theoretical results for the follow- ing quantities of physical interest: stopping power (SP/2) , vicinage func- tion (VF), angle-averaged stopping power (ASP/2), angle-a veraged vicinage function (AVF) together with the dependence of SP/2 and VF on R, the inter-ionic separation distance within the cluster. The re ason why SP has been divided by a factor of 2 is that the SP results for a diprot on cluster are expected to reduce asymptotically (as Rtends to infinity) to those for two uncorrelated protons, the latter being referred to as IS P. ASP has been treated in the same way. In our calculations of these quantities we have employed the linear re- sponse approach which assumes a swift ion-cluster projecti le and also that the ion cluster presents a weak perturbation on the target pl asma. The va- lidity of the linear response approach to study ion-cluster stopping has been 11discussed in detail by Zwicknagel and Deutsch [17]. We refer the reader to their insightful discussion. We model the Al and Cu targets by a dense (degenerate) electro n gas neu- tralized by a positive background (the jellium model) with e lectron densities appropriate for the respective targets. The linear respons e of the target elec- tron gas, which couples the cluster projectile to the target , is considered at three levels of approximations to the dielectric function ε(k, ω) as discussed in sections 2 and 3. In the context of stopping power these appro ximations are subject to the following general remarks: The plasmon-pole approximation (PPA) is valid only in the high velocity regime when the mean v elocity Vof the cluster is > vF, the Fermi velocity of the target electrons. For V < v F and for velocities near the threshold of collective mode exc itations, this ap- proximation is not adequate. The RPA overcomes this limitat ion although it cannot account for short-range correlations in the electro n gas. Within PPA itself, PPA-1 (without plasmon dispersion) is more limited than PPA-2 (with plasmon dispersion). The figures we present serve as a compar ative study of how these levels of approximation affect the various physi cal quantities related to stopping power. Figs. 1-4 show cluster stopping power (CSP and its dependenc e on var- ious quantities of experimental interest). Let us first note that these figures are presented for two specific values, 0 and π/2, of the angle ϑ. Correlations between the two ions in the dicluster are maximum and minimum , respec- tively, for these two values of ϑ. The objective is then to see how, for these maximum and minimum configurations, CSP depends on RandV/vF. Fig. 1 shows CSP for Al target with R= 10−8cm, as a function of V/vFfor the two above-mentioned values of ϑ, within PPA. The lines without circles cor- respond to PPA-1 and, with circles, to PPA-2. The angular dep endence of CSP is particularly noteworthy. It is seen that in a medium ve locity range (V <2vF), CSP has a remarkably higher value for the larger value of ϑ. This is likely due to single-particle excitations in this veloci ty range. In the higher velocity range, the dicluster wake-field excitations becom e important and we find that the situation is reverse in the higher velocity rang e (V >2vF) for which CSP for ϑ= 0 is larger than for ϑ=π/2. In the low velocity range the difference between PPA-1 and PPA -2 (for bothϑ= 0 and π/2) is noticeable while in the high velocity range this differ- ence becomes negligible. This is again due to single-partic le excitations in the low velocity range. For comparison, we have also presented t he uncorrelated 12stopping power (ISP). When we increase the inter-ionic separation distance Rfrom 10−8cm to 5×10−8cm, keeping other physical parameters the same, some intere sting changes occur, as can be seen from Fig. 2. A noticeable change is that now, for V <2vF, CSP for ϑ= 0 is higher than that for ϑ=π/2. This sensitivity of CSP to the angle ϑasRis varied may be due a combination of factors. The dicluster behaves like a compact project for sm allR, and like an extended projectile for large R. This has a bearing on Scorrgiven in Eqs. (13) and (22). Correlation effects are expected to be maximum when the two ions are aligned with each other in the direction of propa gation of the dicluster projectile motion ( ϑ= 0) while they decay (at least for V > v F) when ϑtends to π/2, the latter behavior being related to the wake-field due to the leading ion. The oscillation amplitude in Scorrtends to decrease from ϑ= 0 to ϑ=π/2 (the ˇCherenkov cone). However when Ris small each ion is influenced by the unscreened field of the other ion. For m odel solid targets, the ˇCherenkov cone semivertex is ϑC= arcsin(√ 0.6vF/V) [18]. ϑC approximately equal to 22.80, 7.40, and 0.080forV= 2vF,V= 6vF, and V= 10vF, respectively. Consequently in the high velocity range the trailing ion moves inside the ˇCherenkov cone of the leading ion only for almost aligned diclusters. The behavior of CSP shown in Figs. 1 and 2 reflects these features within the linear response and for PPA. It will be noted that t he high values of SP are due to the PPA-1 approximation. PPA-2 decreases the se values to a small extent. Later, when we use a more realistic, namely RPA, for the linear response function (Figs. 20 and 21) SP considerab ly decreases in strength. Figs. 3 and 4 show SP for Cu, another commonly used metallic ta rget. These figures show patterns similar to those in Figs. 1 and 2, e xcept that CSP and ISP have lower values over the entire range of V/vF. This is because Al has a higher electron density than Cu. The vicinage function (VF) given by Eq. (8) has been plotted a s a func- tion of the beam velocity for Al target in Figs. 5 and 6. This fu nction shows an interplay between ϑandRmore strikingly than CSP. Figs. 7 and 8 display a similar behavior for Cu target. As stated in Sec. 2, an average stopping power (ASP) is of expe rimental interest. Figs. 9 and 10 show ASP for Al and Cu, respectively. ISP is also shown, for comparison. The role of PPA-1 and PPA-2 is now more clearly seen. 13In the same spirit we have plotted AVF for Al and Cu in Figs. 11 a nd 12. The role of Ris highlighted in these figures. However it will be noticed that PPA-1 and PPA-2 make practically no distinction for AVF . We have so far plotted SP or ASP (divided by a factor of 2 in both the cases) vs the beam velocity V/vF, for some values of the separation distance R. We now look for some complementary information about SP, an d plot SP as a function of RwithV= 3vF, for Al target. Fig. 13 shows an oscillatory character of SP with respect to R. The oscillations are the highest for ϑ= 0 and lowest for ϑ=π/2. The role of PPA-1 and PPA-2 is clearly seen for ϑ= 0. Fig. 14 shows a similar behavior of SP for Cu although the a mplitudes are now weaker. In the same way the vicinage function (VF) is plotted in Figs. 15 and 16, for Al and Cu targets, respectively. Fig. 17 shows ASP vs Rfor both Al and Cu targets. The difference between PPA-1 and PPA-2 is negligible and the Cu target has AS P smaller than for Al. Now, there is something interesting about Fig.1 8 which shows AVF. The difference between PPA-1 and PPA-2 is again negligib le. But let us note that data for both Al and Cu lie practically on the s ame curve! Recalling the definition of VF, Eq. (8) one can see from Eqs. (1 2), (16), (21) and (26) that AVF has a weak dependence on target density. Als o, when λ=V/vF>2,Sinddoes not noticeably depend on PPA-1 and PPA-2. These features combine to lead to the behavior of AVF as seen in Fig. 18. We have so far presented results for PPA-1 and PPA-2. A more re alis- tic linear response function, namely the exact random phase approximation (RPA) will now be used for the metallic target. The theoretic al results for SP etc. have been presented in Sec. 3.3. As part of calculatin g SP in RPA it is useful to examine the plasmon dispersion obtained thro ughε(zr, u) = 0, where zanduhave been defined in Sec. 3.3. Fig. 19 displays zr(χ, u) vsu for three electron density parameter values. Next we present ISP and CSP in RPA, for Al and Cu in Figs. 20 and 21, corresponding to R= 10−8cm and R= 5×10−8cm, respectively. For the sake of a better presentation of the data we have also separat ely displayed the Al data in Figs. 20a and 21a. The RPA results show that SP an d ISP decrease in strength with an improved linear response funct ion. This should be of relevance to experiments. Next, VF in RPA vs V/vFis presented in Fig. 22, for Al (lines without circles) and for Cu (lines with circles), corresponding to R= 10−8cm and 14forϑ= 0 and π/2. This figure may be compared with Figs. 5 and 7. For V/vF<2, the curves for VF in RPA tend toward finite values whereas th e VF-curves in PPA do not although the angular trend is similar . A similar contrast may be noted between Figs. 6 and 8, and Fig. 23, corre sponding to R= 5×10−8cm. Again, these findings are of experimental relevance. Averaged SP vs. V/vFin RPA is presented in Fig. 24, for Al (curves without circles) and for Cu (curves with circles) along with ISP, correspond- ing to R= 10−8cm and 5 ×10−8cm. This figure may be compared with Figs. 9 and 10. Fig. 24 shows an expected overall decrease in the str ength of ASP in RPA. There are similarities but also some interesting difference s if we compare Figs. 11 and 12 with Fig. 25. The latter shows AVF in RPA for Al ( curves without circles) and for Cu (curves with circles). The differ ences are more noteworthy for R= 10−8cm. SP vs Rin RPA is plotted in Fig. 26, corresponding to V= 3vF, for Al and Cu in the previously stated scheme. When Fig. 26 is compar ed with Figs. 13 and 14, the differences between PPA and RPA become par ticularly striking. A similar contrast is provided by a comparison of Fig. 27 with Figs. 15 and 16, for VF vs. Rin RPA and PPA, corresponding to V= 3vFand for ϑ= 0 and π/2. For comparison AVF is also plotted in Fig.27. This completes our extensive presentation of figures exhibi ting various aspects of the stopping power of a diproton cluster in PPA and RPA, for Al and Cu targets. 5 SUMMARY In this paper we have presented a comprehensive theoretical study of stopping power (SP) of a dicluster of protons in a metallic target. Aft er a general introduction to SP of a cluster of two point-like ions, in Sec . 2, theoretical calculations of SP based on the linear response theory and us ing PPA without and with plasmon dispersion and then with RPA are discussed i n Sec. 3. The theoretical expressions for a number of physical quantitie s derived in section lead to a detailed presentation, in Sec. 4, of a large collect ion of data through figures on correlated stopping power (CSP), vicinage functi on (VF), average stopping power (ASP) and average vicinage function (AVF) of a diproton 15cluster projectile for two metallic targets, Al and Cu. When ever relevant, we have also provided a plot of independent (i.e. single-ion ) stopping power (ISP) for comparison. With the proviso stated in Sec. 4, SP and related quantities h ave been studied within a linear response formalism; some analytica l and all numerical results have been obtained corresponding to three approxim ations to the di- electric function of the target electron gas-the plasmon-p ole approximation (PPA) without dispersion (PPA-1) and with dispersion (PPA- 2), and also with the random phase approximation (RPA). To our knowledge this is the most comprehensive calculation of the SP-related physical quantities using all the three dominant approximations to the linear respons e function. The results we have presented demonstrate that with regard to se veral physical quantities of primary interest the difference between PPA an d RPA is sub- stantial while for others, specially for average quantitie s, this difference may not be of practical significance. It will be of interest to go beyond RPA in order to include some short- range correlations in the electron gas and to study how diclu ster SP is af- fected. However calculating the linear response function b y including electron energy bands is rather involved and detailed theoretical st udies of SP with band structure effects included have not yet been reported in the literature. One can include some aspect of band structure in a rather appr oximate man- ner through an effective mass for the electrons. Another aspect we have not considered in this paper is some eff ect of disorder in the target medium. In real metals electrons suffe r collisions with impurities etc. We intend to address this issue in the contex t of stopping power in a separate study. ACKNOWLEDGMENT It is a pleasure to thank to Dr. G. Zwicknagel for useful discu ssions. We are grateful to V. Nikoghosyan for technical assistance. References 16[1] P. M. Echenique, F. Flores, and R. H. Ritchie, Solid State Phys.43, 229 (1990). [2] J. F. Ziegler, J. App. Phys., 85, 1249 (1999). [3] D. Bohm and D. Pines, Phys. Rev. 82, 625 (1951); 85, 338 (1952). [4] J. Lindhard, K. Dan. Vidensk. Selsk., Mat.-Fys. Medd. 28, No. 8 (1954); J. Lindhard and A. Winther, K. Dan. Vidensk. Selsk., Mat.-Fy s. Medd. 34, No. 4 (1964). [5] R. H. Ritchie, Phys. Rev. 114, 644 (1959). [6] Yu. N. Yavlinsky, Zh. Eksp. Teor. Fis. 80, 1622 (1981) [Sov. Phys. JETP 53, 835 (1981)]. [7] D. G. Yakovlev and S. S. Kotel’nikov, Zh. Eksp. Teor. Fis. 84, 1348 (1983) [Sov. Phys. JETP 57, 781 (1983)]. [8] R. H. Ritchie, C. J. Tung, V. E. Anderson, and J. C. Ashley, Radiat. Res.64, 181 (1975); T. L. Ferrel and R. H. Ritchie, Phys. Rev. B 16, 115 (1977); C. J. Tung and R. H. Ritchie, Phys. Rev. B 16, 4302 (1977). [9] N. R. Arista, Phys. Rev. B 18, 1 (1978). [10] G. Basbas and R. H. Ritchie, Phys. Rev. A 25, 1943 (1982). [11] P. M. Echenique, Nucl. Instrum. Methods B 27, 256 (1987). [12] N. R. Arista and W. Brandt, Phys. Rev. A 23, 1898 (1981); T. A. Mehlhorn, J. Appl. Phys. 52, 6522 (1981); N. R. Arista and A. R. Piriz, Phys. Rev. A 35, 3450 (1987). [13] G. Maynard and C. Deutsch, Phys. Rev. A 26, 665 (1982). [14] J. D’ Avanzo, M. Lontano, and P. F. Bortignon, Phys. Rev. E47, 3574 (1993). [15] C. Deutsch, Phys. Rev. E 51, 619 (1995). [16] C. Deutsch, Laser and Particle Beams 8, 541 (1990). 17[17] G. Zwicknagel and C. Deutsch, Phys. Rev. E 56, 970 (1997). [18] W. Sch¨ afer, H. St¨ ocker, B. M¨ uller, and W. Greiner, Z. Physik A, 8, 349 (1978). 18Figure Captions Fig. 1. SP/2 of a diproton cluster with R= 10−8cm, vs V/vFfor Al target ( rs= 2.07).ϑ= 0 (dotted line), ϑ=π/2 (dashed line); ISP (solid line). The lines with and without circles correspond to PPA w ith and without dispersion, respectively. Fig. 2. SP/2 of a diproton cluster with R= 5×10−8cm, vs V/vFfor Al target. ϑ= 0 (dotted line), ϑ=π/2 (dashed line); ISP (solid line). The lines with and without circles correspond to PPA with and without d ispersion, respectively. Fig. 3. SP/2 of a diproton cluster with R= 10−8cm, vs V/vFfor Cu target ( rs= 2.68).ϑ= 0 (dotted line), ϑ=π/2 (dashed line); ISP (solid line). The lines with and without circles correspond to PPA w ith and without dispersion, respectively. Fig. 4. SP/2 of a diproton cluster with R= 5×10−8cm, vs V/vFfor Cu target. ϑ= 0 (dotted line), ϑ=π/2 (dashed line); ISP (solid line). The lines with and without circles correspond to PPA with and without d ispersion, respectively. Fig. 5. VF of a diproton cluster with R= 10−8cm, vs V/vFfor Al target. ϑ= 0 (solid line), ϑ=π/2 (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectivel y. Fig. 6. VF of a diproton cluster with R= 5×10−8cm, vs V/vFfor Al target. ϑ= 0 (solid line), ϑ=π/2 (dotted line). The lines with and without circles correspond to PPA with and without dispersion, resp ectively. Fig. 7. VF of a diproton cluster with R= 10−8cm, vs V/vFfor Cu target. ϑ= 0 (solid line), ϑ=π/2 (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectivel y. Fig. 8. VF of a diproton cluster with R= 5×10−8cm, vs V/vFfor Cu target. ϑ= 0 (solid line), ϑ=π/2 (dotted line). The lines with and without circles correspond to PPA with and without dispersion, resp ectively. Fig. 9. ASP/2 of a diproton cluster with R= 10−8cm (dotted line) andR= 5×10−8cm (dashed line) vs V/vFfor Al target. Solid line, ISP. The lines with and without circles correspond to PPA with and without dispersion, respectively. Fig. 10. ASP/2 of a diproton cluster with R= 10−8cm (dotted line) andR= 5×10−8cm (dashed line) vs V/vFfor Cu target. Solid line, ISP. 19The lines with and without circles correspond to PPA with and without dispersion, respectively. Fig. 11. AVF of a diproton cluster with R= 10−8cm (solid line) and R= 5×10−8cm (dotted line) vs V/vFfor Al target. The lines with and without circles correspond to PPA with and without dispersi on, respectively. Fig. 12. AVF of a diproton cluster with R= 10−8cm (solid line) and R= 5×10−8cm (dotted line) vs V/vFfor Cu target. The lines with and without circles correspond to PPA with and without dispersi on, respectively. Fig. 13. SP/2 of a diproton cluster with V= 3vFvsRfor Al target. ϑ= 0 (solid line), ϑ=π/2 (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectivel y. Fig. 14. SP/2 of a diproton cluster with V= 3vFvsRfor Cu target. ϑ= 0 (solid line), ϑ=π/2 (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectivel y. Fig. 15. VF of a diproton cluster with V= 3vFvsRfor Al target. ϑ= 0 (solid line), ϑ=π/2 (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectivel y. Fig. 16. VF of a diproton cluster with V= 3vFvsRfor Cu target. ϑ= 0 (solid line), ϑ=π/2 (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectivel y. Fig. 17. ASP/2 of a diproton cluster with V= 3vFvsRfor Al (the lines with square symbols) and Cu (the lines with circles) targets . Dotted and solid lines, PPA with and without dispersion, respectively . Fig. 18. AVF of a diproton cluster with V= 3vFvsRfor Al (the lines with square symbols) and Cu (the lines with circles) targets . Dotted and solid lines, PPA with and without dispersion, respectively . Fig. 19. Relation between zr(χ, u) andu, as obtained from the dispersion equation ε(zr, u) = 0. Solid line: χ= 0.5, dashed line: χ= 0.15, dotted line: χ= 0.05. Fig. 20. SP/2 of a diproton cluster with R= 10−8cm in a RPA vs V/vF for Al (the lines without symbols) and Cu (the lines with circ les) targets. ϑ= 0 (dotted line), ϑ=π/2 (dashed line); ISP (solid line). Fig. 20a. SP/2 of a diproton cluster with R= 10−8cm in a RPA vs V/vF for Al target. ϑ= 0 (dotted line), ϑ=π/2 (dashed line); ISP (solid line). Fig. 21. SP/2 of a diproton cluster with R= 5×10−8cm in a RPA vsV/vFfor Al (the lines without symbols) and Cu (the lines with circ les) targets. ϑ= 0 (dotted line), ϑ=π/2 (dashed line); ISP (solid line). 20Fig. 21a. SP/2 of a diproton cluster with R= 5×10−8cm in a RPA vs V/vFfor Al target. ϑ= 0 (dotted line), ϑ=π/2 (dashed line); ISP (solid line). Fig. 22. VF of a diproton cluster with R= 10−8cm in a RPA vs V/vFfor Al (the lines without symbols) and Cu (the lines with circles ) targets. ϑ= 0 (solid line), ϑ=π/2 (dotted line). Fig. 23. VF of a diproton cluster with R= 5×10−8cm in a RPA vs V/vF for Al (the lines without symbols) and Cu (the lines with circ les) targets. ϑ= 0 (solid line), ϑ=π/2 (dotted line). Fig. 24. ASP/2 of a diproton cluster with R= 10−8cm (dotted line) and R= 5×10−8cm (dashed line) in a RPA vs V/vFfor Al (the lines without symbols) and Cu (the lines with circles) targets; ISP (solid line). Fig. 25. AVF of a diproton cluster with R= 10−8cm (solid line) and R= 5×10−8cm (dotted line) in a RPA vs V/vFfor Al (the lines without symbols) and Cu (the lines with circles) targets. Fig. 26. SP/2 of a diproton cluster with V= 3vFin a RPA vs Rfor Al (the lines without symbols) and Cu (the lines with circles ) targets; ASP (solid line), ϑ= 0 (dotted line), ϑ=π/2 (dashed line). Fig. 27. VF of a diproton cluster with V= 3vFin a RPA vs Rfor Al (the lines without symbols) and Cu (the lines with circles ) targets; AVF (solid line), ϑ= 0 (dotted line), ϑ=π/2 (dashed line). 210 2 4 6 8 V/vF0123SP in PPA (in GeV/cm)Al Target; R = 10−8cm ISP ϑ = 0 ϑ = π/20 2 4 6 8 V/vF0123SP in PPA (in GeV/cm)Al Target; R = 5 10−8cm ISP ϑ = 0 ϑ = π/20 2 4 6 8 V/vF0123SP in PPA (in GeV/cm)Cu Target; R = 10−8cm ISP ϑ = 0 ϑ = π/20 2 4 6 8 V/vF0123SP in PPA (in GeV/cm)Cu Target; R = 5 10−8cm ISP ϑ = 0 ϑ = π/20 2 4 6 8 V/vF−0.500.51VF in PPAAl Target; R = 10−8cm ϑ = 0 ϑ = π/20 2 4 6 8 V/vF−0.500.51VF in PPAAl Target; R = 5 10−8cm ϑ = 0 ϑ = π/20 2 4 6 8 V/vF−0.500.51VF in PPACu Target; R = 10−8cm ϑ = 0 ϑ = π/20 2 4 6 8 V/vF−0.500.51VF in PPACu Target; R = 5 10−8cm ϑ = 0 ϑ = π/20 2 4 6 8 V/vF0123ASP in PPA (in GeV/cm)Al Target ISP R = 10−8cm R = 5 10−8cm0 2 4 6 8 V/vF0123ASP in PPA (in GeV/cm)Cu Target ISP R = 10−8cm R = 5 10−8cm0 2 4 6 8 V/vF−0.500.51AVF in PPAAl Target R = 10−8cm R = 5 10−8cm0 2 4 6 8 V/vF−0.500.51AVF in PPACu Target R = 10−8cm R = 5 10−8cm0 10 20 30 R (in 10−8cm)00.511.5SP in PPA (in GeV/cm)Al Target; V = 3vF ϑ = 0 ϑ = π/20 10 20 30 R (in 10−8cm)00.511.5SP in PPA (in GeV/cm)Cu Target; V = 3vF ϑ = 0 ϑ = π/20 10 20 30 R (in 10−8cm)−0.500.51VF in PPAAl Target; V = 3vF ϑ = 0 ϑ = π/20 10 20 30 R (in 10−8cm)−0.500.51VF in PPACu Target; V = 3vF ϑ = 0 ϑ = π/20 10 20 30 R (in 10−8cm)00.511.5ASP in PPA (in GeV/cm)V = 3vF Al Target Cu Target0 10 20 30 R (in 10−8cm)−0.500.51AVF in PPAV = 3vF Al Target Cu Target0 2 4 6 8 100.00.10.20.30.4 χ = 0.5 χ = 0.15 χ = 0.05zr(χ χ,u) u0 2 4 6 8 V/vF0123SP in RPA (in GeV/cm)Al and Cu Targets; R = 10−8cm ϑ = 0 ϑ = π/2 ISP0 2 4 6 80123Al Target; R = 10-8cm ISP ϑ = 0 ϑ = π/2SP in RPA (in GeV/cm) V/vF0 2 4 6 8 V/vF0123SP in RPA (in GeV/cm)Al and Cu Targets; R = 5 10−8cm ϑ = 0 ϑ = π/2 ISP0 2 4 6 80123Al Target; R = 5 10-8cm ISP ϑ = 0 ϑ = π/2SP in RPA (in GeV/cm) V/vF0 2 4 6 8 V/vF−0.500.51VF in RPAAl and Cu Targets; R = 10−8cm ϑ = 0 ϑ = π/20 2 4 6 8 V/vF−0.500.51VF in RPAAl and Cu Targets; R = 5 10−8cm ϑ = 0 ϑ = π/20 2 4 6 8 V/vF0123ASP in RPA (in GeV/cm)ISP R = 10−8cm R = 5 10−8cm0 2 4 6 8 V/vF−0.500.51AVF in RPAR = 10−8cm R = 5 10−8cm0 10 20 30 R (in 10−8cm)00.511.5SP in RPA (in GeV/cm)V = 3vF ϑ = 0 ϑ = π/2 ASP0 10 20 30 R (in 10−8cm)−0.500.51VF in RPAV = 3vF ϑ = 0 ϑ = π/2 AVF

arXiv:physics/0004023v1 [physics.gen-ph] 13 Apr 2000An Extended Transformation to Reach Continuous Gauge Invariance for Gauge Fields with Static Masses Mei Xiaochun ( Institute of Theoretical Physics in Fuzhou, No.303, Build ing 2, Yinghu Garden, Xihong Road, Fuzhou, 350025, Chian, E-mail: fzbgk@pub3.fz .fj.cn ) Abstract The paper provides an extended local gauge transformation. By means of it, the Lagrangian of gauge fields with masses can keep unchanged in the continuous transformations. In this way, the Higgs mechanism becomes unnecessary and we do not need to sup pose the existence of the Higgs particles. PACS numbers: 1110, 1130 According to the Yang-Mills theory, in order to keep the Lagr angian of a system unchanged under the local gauge transformation, the transformation rules o f the fieldφand its covariant differentiation should be defined as φ(x)→φ′(x) = exp[ iθi(x)Ti]φ(x) (1) Dµ(x)φ(x)→D′µ(x)φ′(x) = exp[ −iθi(x)Ti]Dµ(x)φ(x) (2) in which Dµ(x) =∂µ+Aµ(x) Aµ(x) =−igAi µ(x)Ti(3) From Eq.(2), we can get the transformation rule of gauge unde r the infinitesimal transformation Ai µ(x)→A′i µ(x) =Ai µ(x) +fijkθj(x)Ak µ(x)−1 g∂µθi(x) (4) The intensity of the gauge field is defined as Fi µν(x) =∂µAi ν(x)−∂νAi µ(x) +gfijkAj µ(x)Ak ν(x) (5) The Lagrangian of the free gauge field with zero mass is L0=−1 4Fi µνFi µν (6) It is invariable under the gauge transformation. But the Lag rangian of the free gauges fields with static masses (suppose their masses are the same for simplification ) is L1=−1 4Fi µνFiµν−1 2m2Ai µAi µ (7) It can not keep unchanged under the transformation. However, the gauge particles just as W±andZ0have masses. In order to solve this problem, in the current theory, the Higgs mechanism has to be introduced . By means of the concept of spontaneous symmetry breaking of vacuum, the gauge particles obtain the ir masses. The current gauge theory has achieved great success but has also two basic faults. The firs t is that the Higgs particles can not be found up to now. The second is that the theory can not keep unchanged again after the Higgs mechanism is introduced, that is to say, the invariance is not thorough. B esides, there exist some other problems, for example, too many parameters have to be introduced and the re ason of spontaneous symmetry breaking of vacuum is also unclear and so on.It is proved below that we can solve these problems by introdu cing a supplement function Bi µ(x). Let Bµ(x) =Bi µ(x)Ti, similar to Eq.(2), we define the extended transformation as Dµ(x)φ(x)→D′′ µ(x)φ′(x) = exp[ −iφi(x)Ti][Dµ(x) +Bµ(x)]φ(x) (8) HereD′′µ(x) =∂µ+A′′i µ(x). It can be calculated from Eq.(8) that by the extended trans formation, the transformation rule of Ai µ(x) becomes A′′i µ(x) =Ai µ(x) +Gi µ(x) (9) in which Gi µ(x) =fijkθj(x)Ak µ−1 g∂µθi(x) +Bi µ(x) +fijkθj(x)Bk µ(x) (10) The transformation means that we let Ai µ(x)→Ai µ(x) +Bi µ(x) (11) in Eq.(4) again, or let Fi µν(x)→Fi µν(x) +Ki µν(x) (12) in Eq.(5). Here Ki µν=∂µBi ν−∂νBi µ+gfijk(Aj µBk ν+Aj νBk µ+Bj µBk ν) (13) Considering the fact that Fi µνFi µνis invariable in the current gauge transformation with (Fi µνFi µν)′=Fi µνFi µν (14) Putting Eq.(12) into the two sides of Eq.(14), we have (Fi µνFi µν)′′=Fi µνFi µν+Q1 (15) In the formula Q1= 2Fi µνKi µν+Ki µνKi µν (16) Putting Eq.(11) and Eq.(15) into Eq.(7), we get the extended Lagrangian L′′ 1=L1−1 4Q1−1 2Q2 (17) in which Q2=m2(2Ai µGi µ+Gi µGi µ) (18) We haveL′′ 1=L1as long as let Q1+ 2Q2= 0 (19) In this way, the Lagrangian is invariable under the extended gauge transformation. It is obvious that the invariance of continuous transformations can also be kept i n this way. Now let us discuss how to determine the forms of the functions Bi µ(x) from Eq.(19). Suppose i= 1.2...N, the number of Bi µ(x) is 4N. But they only satisfy one equation, that is to say, only one o f them is independent. The other 4N-1 Bi µ(x) can be chosen arbitrarily. As the simplest form, if we take Bi µ(x) =B1 1(x) =B(x) µ= 1 i= 1 Bi µ(x) = 0 µ/negationslash= 1 i/negationslash= 1 (20) Eq.(19) can be written as 4N∂µB∂µB−(/summationdisplay µ∂µB)2+Fµ∂µB+F1B+F2B2+F3B3+F4B4+F5= 0 (21)In the formula Fµ=gfjk 1(Aj µAk+AjAk µ) F1=g2fiρ1fijkAρ µ(Aj µAk 1+Aj 1Ak µ) + 2m2(A1 1+fijkθjAi 1+fjk1θjAk 1−1 g/summationdisplay µ∂µθ1+ fiρ1fijkθjθρAk 1−1 gfij1θj/summationdisplay µ∂µθi) F2=g2[fi11fijkAj 1Ak 1+fij1fiρ1(4Aj µAρ µ+Aj 1Aρ 1) + 4m2(n+ 2f1j1θj+fij1fiρ1θjθρ)] F3= 8g2fi11fij1Aj F4= 8g2fi11fi11 F5=m2(2fijkθjAi µAk µ−2 gAi µ∂µθi+fijkfiρσθjθρAk µAσ µ−2 gfijkθjAk µ∂µθi+1 g2∂µθi∂µθi) (22) If we take the most symmetric form, let Bi µ(x) =B(x) (AllBi u(x) have the same value for the different indexesµand i.)and introduce the signs θ=/summationdisplay iθiAi=/summationtext µAi µA=/summationdisplay iAi=/summationdisplay µiAi µ fjk 1=/summationdisplay ifijkfik 2=/summationtext jfijkfij 3=/summationdisplay kfijkfi 4=/summationdisplay jkfijkfj 5=/summationdisplay ikfijk(23) the parameters in Eq.(21) become Fµ=gfjk 1(Aj µAk+AjAk µ) F1=g2fiρ 3fijkAρ µ(Aj µAk+AjAk µ) + 2m2(A+fij 3θjAi+fjk 1θjAk−1 g/summationdisplay µ∂µθ+ fiρ 3fijkθjθρAk−1 gfij 3θj/summationdisplay µ∂µθi) F2=g2[fi 4fijkAjAk+fij 3fiρ 3(4Aj µAρ µ+AjAρ) + 4m2(n+ 2fj 5θj+fij 3fiρ 3θjθρ)] F3= 8g2fi 4fij 3Aj F4= 8g2fi 4fi 4 F5=m2(2fijkθjAi µAk µ−2 gAi µ∂µθi+fijkfiρσθjθρAk µAσ µ+2 gfijkθjAk µ∂µθi+1 g2∂µθi∂µθi) In the formulas, the functions Ai µ(x) can be determined by the motion equations of gauge fields. Bu t the forms ofθi(x) are arbitrary. So the form of B(x) can be determined by Eq.(21) if the forms of θi(x) are given. However, B(x) does not appear in the Lagrangian. It is only a useful tool to help us to reach the gauge invariance. The situation with interaction is discussed below. Taking t he united theory of electric-weak interaction between spinor fields and gauge fields as an example, we have tw o kinds of spinor fields called the left-hand field and right-hand field with different transformation rule s in this case ψL(x)→ψ′ L(x) = exp( −i/vectorθ·/vector τ 2+iθ))ψL(x) (25) ψR(x)→ψ′ R(x) = exp(iθ)ψR(x) (26) In the current transformation, the mass item in the Lagrangi an can not keep unchanged with the form ms(¯ψ′ Lψ′ R+¯ψ′ Rψ′ L) =ms(¯ψLψr+¯ψRψL) +ims(¯ψL/vectorθ·/vector τ 2ψR−¯ψR/vectorθ·/vector τ 2ψL) (27) The Lagrangian of the system is L=−¯ψLrµDµψ′ L+¯ψRrµDµψR−ms(¯ψLψR+¯ψRψL)−1 4Fi µνFi µν−1 2m2Ai µAi µ (28)After the extended transformation is carried out, we get L′′=L−1 4Q1−1 2Q2−Q3 (29) Q3=¯ψLrµBµψL+¯ψRrµBµψR+ims(¯ψL/vectorθ·/vector τ 2ψR−¯ψR/vectorθ·/vector τ 2ψL) (30) We haveL′′=L, so that the invariance is also reached as long as let Q1+ 2Q2+ 4Q3= 0 (31) Eq.(31) can also be written as the form of Eq.(20) as long as we let F0→F′ o=Fo+ 4im(¯ψL/vectorθ·/vector τ 2ψR¯−ψR/vectorθ·/vector τ 2ψL) F1→F′ 1=F1+ 4(¯ψLrψR+¯ψRrψL)r=/summationdisplay µrµ (32) in Eq.(22).In the formula above, ψLandψRare determined by their motion equations. Besides, if the ghost particle and fixed gauge term are also considered, the c orresponding factors have to be added into the formulas, but we do not discuss them any more here. After the extended gauge transformation is introduced, the mass items of gauge particles can be added into the Lagrangian directly. Because the Lagrangian can st ill keep unchanged in the extended gauge transformation, the Becchi-Rouet-Stora invariability an d the Ward-Takahashi equationare also tenable in this case. So the theory is still renormalizable. We can do it similar to what we have done in the current theory. In this way, the Higgs mechanism becomes unnecessar y. Therefore, according to the paper, we do not need to suppose the existence of the Higgs particles. References [1] Hu Yiaoguang, The theory of Gauge Field,183 (1984).

arXiv:physics/0004024v1 [physics.class-ph] 13 Apr 2000The role of acceleration and locality in the twin paradox Hrvoje Nikoli´ c Theoretical Physics Division, Rudjer Boˇ skovi´ c Institut e, P.O.B. 180, HR-10002 Zagreb, Croatia hrvoje@faust.irb.hr May 13, 2013 Abstract We study the role of acceleration in the twin paradox. From th e coordinate transformation that relates an accelerated and an inertial observer we find that, from the point of view of the accelerate d ob- server, the rate of the differential lapses of time depends no t only on the relative velocity, but also on the product of the acceler ation and the distance between the observers. However, this result do es not have a direct operational interpretation because an observ er at a cer- tain position can measure only physical quantities that are defined at the same position. For local measurements, the asymmetry be tween the two observers can be attributed to the fact that noninert ial frames, contrary to inertial frames, can be correctly interpreted o nly locally. 1 Introduction According to the special theory of relativity, all motions w ith a constant velocity are relative. The twin paradox consists in the fact that a twin B that travels around and eventually meets his brother Ais younger than his brother A. How the twin Bknows that he is the one who is actually moving? An often answer, especially in the older literature, is: “He knows, because he accelerates and consequently feels an inertial force.” H owever, it has been stressed many times in the literature1−5that acceleration is not an essential part of the twin paradox. The most general explanation of the twin paradox is purely ge ometrical; the proper lengths of two trajectories in spacetime, which c orrespond to times 1measured by the corresponding observers, do not need to be eq ual. This explanation is perfectly correct and is also the simplest on e, because one does not need an explicit coordinate transformation that relate s the two observers. Yet, such an explanation may not be completely satisfactory , because one may want to know where and when the different aging of the two ob servers occurs. Although, strictly speaking, this question is not r eally meaningful,4 we show in Sec. 2 that, in a certain sense, the different aging c an be attributed to instants of time when one of the observers accelerates. We find that the accelerated observer “observes” that the rate of the differe ntial lapses of time depends not only on the relative velocity, but also on th e product of the acceleration and the distance between the observers. This r esult has already been obtained in Ref. 6, but our derivation is quite different , and, we believe, more elegant. However, we also emphasize that this result do es not have a direct operational interpretation because an observer at a certain position can only measure physical quantities that are defined at the s ame position. Another way of posing the twin paradox is to ask what, if not ac celeration, is the source of asymmetry between the two relatively moving observers. When spacetime is curved1or has a nontrivial topology,3one can obtain the twin paradox completely without acceleration. In these two cases it is clear that, owing to a nontrivial geometry or topology, there is no symmetry with respect to rotations of the velocity directions. However, a flat opened universe possesses such a symmetry, so the question of the source of as ymmetry for such a case remains opened. We find in Sec. 3 that this asymmetr y can be attributed to the fact that noninertial frames, contrary to inertial frames, can be correctly interpreted only locally. Recently, this f act has been used to resolve the Ehrenfest paradox,7to give the correct interpretation of the Sagnac effect,7and to show that the notion of radiation does not depend on acceleration of an observer.8In this paper we explain how this fact helps in understanding the twin paradox. 2 The role of acceleration LetSbe the frame of an inertial observer and S′the frame of an observer that moves arbitrarily along the x-axis. Let ube the velocity of the observer inS′as seen by an observer in S. Let us also assume that the observers in S andS′do not rotate (this makes the analysis simpler, but has no infl uence on the final results). The coordinate transformation between t hese two frames is given by9,7y=y′,z=z′, and x=γ(t′)x′+/integraldisplayt′ 0γ(t′)u(t′)dt′, (1) t=1 c2γ(t′)u(t′)x′+/integraldisplayt′ 0γ(t′)dt′, (2) 2where γ(t′) = 1//radicalBig 1−u2(t′)/c2. It was assumed in this transformation that the space origins coincide at t=t′= 0. The position of the observer in S′is x′= 0. The transformation (1)-(2) is linear in x′. However, if u(t′) is not a constant, then this transformation is not linear in t′. Contrary to the case of constant u, this transformation cannot be simply inverted by putting u→ −u. This is why the inertial and the noninertial observers are n ot equivalent. Note, however, that acceleration does not appear explicitl y in (1) and (2). Let us now see how the clock in S′appears to the observer in S. Since the clock in S′is atx′= 0, from (2) we find t=/integraldisplayt′ 0γ(t′)dt′, (3) ∂t ∂t′=γ(t′). (4) Equations (3) and (4) express the fact that the observer in Ssees that the clock in S′is slower than the clock in Sand that ∂t/∂t′does not depend on the acceleration, but only on the instantaneous velocity. Let us now see how the clock in S, not necessarily at x= 0, appears to the observer in S′. By eliminating x′from (1) and (2), we find t=/integraldisplayt′ 0γ(t′)dt′+1 c2u(t′)[x−xo(t′)], (5) where xo(t′) =/integraldisplayt′ 0γ(t′)u(t′)dt′=/integraldisplayt′(t) 0u(t′(t))dt (6) is the position of the observer in S′as a function of time and t′(t) is given by the inverse of (3). From (5) and (6) we find ∂t ∂t′=1 γ(t′)+1 c2du(t′) dt′[x−xo(t′)]. (7) Comparing (7) with (4), we see that the first term in (7) corres ponds to what we expect from the relativity of motion. However, the se cond term in (7) shows that acceleration has a direct influence on what the accelerated observer will observe. Note also that the influence of accele ration does not depend only on the acceleration itself, but also on the relat ive distance be- tween the accelerated observer and the inertial clock. In pa rticular, if the inertial and the noninertial clocks are at the same instanta neous position, then acceleration has no influence. Now it seems that we understand what is the true origin of the d ifferent aging of the inertial and the noninertial clocks. For exampl e, if the observer inS′moves with a constant velocity and then suddenly reverses th e direction 3of motion, then, at this critical instant of time, it will app ear to him that the time of the inertial clock at x= 0 instantaneously jumps forward. There is no such jump of the noninertial clock from the point of view of the inertial observer. This is the reason that, when the two observers fina lly meet, they have different age. A similar conclusion, although obtained in a completely different way, was drawn also in Ref. 11. However, as we show in the next section, this is not the end of the story. 3 The role of locality The discussion of the preceding section seems to resolve the twin paradox. However, this discussion raises a new paradox. The right-ha nd side of (7) can be negative, which means that it may appear to the accelerate d observer that an inertial clock lapses backward in time. This seems to be in contradiction with the principle of causality. This paradox is an artefact of the tacit assumption that an ob server receives information from a distant clock instantaneously . In that sense, equations (3), (4), (5), and (7) do not represent what the obs ervers will really see, unless the clocks and the observers in SandS′are at the same instantaneous position. One could caculate what the observ ers would really see by assuming that the observers communicate with light si gnals, which would remove the causality paradox. However, we do not want t o introduce a new entity, such as a light beam needed for communication, b ecause such a complication could hide the real origin of the twin paradox. Instead, we insist on resolving the twin paradox using only the properties of th e transformation (1)-(2). An observer at a certain position can measure only the values of physical quantities at this same position. Therefore, equations (5) and (7) have a direct operational interpretation only for x=xo(t′). Therefore, when the observer in S′compares his clock with a clock in Sat the same instantaneous position, he sees t=/integraldisplayt′ 0γ(t′)dt′, (8) ∂t ∂t′=1 γ(t′). (9) The apparent inconsistency of equations (8) and (9) is a new w ay of viewing the twin paradox. Equations (9) and (4) correspond to the rel ativity of motion; the two observers do not agree on which clock is faste r. On the other hand, equations (8) and (3) correspond to the twin para dox; the two observers doagree that, at the same instant and the same position, the han d of the clock in S′points to a smaller number than the hand of the clock in S. But how is that possible? 4Note first that the apparent inconsistency of (8) and (9) has a simple mathematical origin. One should not derive (9) directly fro m (8), but instead one needs firstto calculate the derivative of (5) and thento put x=xo(t′). However, (8) and (9) are correct equations even for a motion w ith a constant velocity. What is the source of asymmetry between SandS′? The asymmetry lies in another tacit assumption; the conditi onx=xo(t′) corresponds to an experimental arrangement in which the mov ing clock in S′ is compared each time with another clock in S, that which, at this instant, is at the same position as the clock in S′. In other words, there is precisely one clock in S′, while there are many clocks in Sdistributed along the x-axis. One will say: “OK, but we can arrange our experiment such that there is only one clock in S, say at x= 0, while there are many clocks in S′ distributed along the x′-axis.” However, now comes the essential point of this section. It was legitimate to use many clocks at differen t positions in S and compare the clock in S′each time with another clock in S, because all these clocks in Sbelong to the same frame of reference. On the other hand, if, at least for a short time, S′is not an inertial frame, then we cannot longer say that clocks at different constant positions x′belong to the same frame of reference. Consequently, it is not legitimate to compare the inertial clock atx= 0 each time with another noninertial clock that does not mov e with respect to the noninertial clock at x′= 0 and interpret the result as something that tell us about the behavior of the clock at x′= 0. In the context of the twin paradox, a privileged role of inertial frames is not rel ated to the fact that inertial observers do not feel an inertial force, but ra ther to the fact that inertial frames in flat spacetime have a clear global interpretation, while noninertial frames only have a clear localinterpretation. This purely local interpretation of noninertial frames has already been expl ained in more detail in Refs. 7. and 8., but, for the sake of completeness, below we give a short resume of the results of these papers. Proper coordinates of an observer arbitrarily moving in arb itrary space- time are the so-called Fermi coordinates.10They are determined by the trajec- tory of the observer, as well as by the geometry of spacetime. It is convenient to define them such that the position of the observer is at the s pace origin of the Fermi coordinates. Even if there is no relative motion between two observers, they belong to different Fermi frames if they are n ot staying at the same position. In particular, the S-coordinates in (1)-(2) are the Fermi coordinates of an inertial observer in flat spacetime, while theS′-coordinates are the Fermi coordinates of an observer that moves arbitrar ily (without ro- tation) along the x-axis with respect to the inertial observer. In general, the coordinate transformation that relates the Fermi coordina tes of two different observers is a complicated transformation. However, the co ordinate trans- formation that relates the Fermi coordinates of two inertia l observers in flat spacetime that do not move relatively to each other is a simpl e translation 5of the space origin, which is a transformation that belongs t o the class of restricted internal transformations,7i.e., it is a transformation that does not mix space and time coordinates: t′=f0(t), x′i=fi(x1, x2, x3). (10) If the Fermi coordinates of two observers are related by such a transforma- tion, then they can be regarded as belonging to the same physical frame of reference. This is why one can regard the clocks at different p ositions xin (5) as belonging to the same frame of reference. At the end, let us also mention some measurable effects relate d to the local interpretation of noninertial frames. Assume that th eS′-coordinates in (1)-(2) refer to a uniformly accelerated observer at x′= 0. Assume also that another observer has such a trajectory that his positio n is given by x′=constant /negationslash= 0. Then this second observer also accelerates uniformly, b ut with a different acceleration.12If, on the other hand, two observers at dif- ferent positions move in the same direction with equal accel erations and initial velocities, then, as seen by these observers, the di stance between them changes with time, which also leads to a variant of the twin pa radox.2 Acknowledgments I am grateful to G. Duplanˇ ci´ c for asking me questions that m otivated the investigation resulting in this paper. This work was suppor ted by the Min- istry of Science and Technology of the Republic of Croatia un der Contract No. 00980102. 1B. R. Holstein and A. R. Swift, “The Relativity Twins in Free F all,” Am. J. Phys. 40(5), 746-750 (1972). 2S. P. Boughn, “The case of the identically accelerated twins ,” Am. J. Phys. 57(9), 791-793 (1989). 3T. Dray, “The Twin Paradox Revisited,” Am. J. Phys. 58(9), 822-825 (1990). 4T. A. Debs and M. L. G. Redhead, “The twin “paradox” and the con ven- tionality of simultaneity,” Am. J. Phys. 64(4), 384-392 (1996). 5R. P. Gruber and R. H. Price, “Zero time dilation in an acceler ating rocket,” Am. J. Phys. 65(10), 979-980 (1997). 6R. J. Low, “When Moving Clocks Run Fast,” Eur. J. Phys. 16(5), 228-229 (1995). 7H. Nikoli´ c, “Relativistic contraction and related effects in noninertial frames,” Phys. Rev. A 61, 032109 (2000). 68H. Nikoli´ c, “Notes on covariant quantities in noninertial frames and invari- ance of radiation in classical and quantum field theory,” gr- qc/9909035. 9R. A. Nelson, “Generalized Lorentz transformation for an ac celerated, rotating frame of reference,” J. Math. Phys. 28(10), 2379-2383 (1987). 10C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Free- man and Company, New York 1995). 11W. G. Unruh, “Parallax distance, time, and the twin “paradox ”,” Am. J. Phys.49(6), 589-592 (1981). 12H. Nikoli´ c, “Relativistic contraction of an accelerated r od,” Am. J. Phys. 67(11), 1007-1012 (1999). 7

arXiv:physics/0004025v1 [physics.bio-ph] 13 Apr 2000Symmetry breaking and coarsening in spatially distributed evolutionary processes including sexual reproduction and disruptive selection Hiroki Sayama1, Les Kaufman1,2and Yaneer Bar-Yam1 1New England Complex Systems Institute, Cambridge, MA 02138 2Boston University, Dept. of Biology, Boston, MA 02215 Sexual reproduction presents significant challenges to for - mal treatment of evolutionary processes. A starting point for systematic treatments of ecological and evolutionary p he- nomena has been provided by the gene centered view of evolu- tion. The gene centered view can be formalized as a dynamic mean field approximation applied to genes in reproduction / selection dynamics. We show that spatial distributions of o r- ganisms with local mating neighborhoods in the presence of disruptive selection give rise to symmetry breaking and spo n- taneous pattern formation in the genetic composition of lo- cal populations. Global dynamics follows conventional coa rs- ening of systems with nonconserved order parameters. The results have significant implications for ecology of geneti c di- versity and species formation. PACS: The dynamics of evolution can be studied by statisti- cal models that reflect properties of general models of the statistical dynamics of interacting systems [1]. Research on this topic can affect the conceptual foundations of evolutionary biology, and many applications in ecology, population biology, and conservation biology. Among the central problems is understanding the creation, persis- tence, and disappearance of genetic diversity. In this paper, we describe a model of sexual reproduction which illustrates mean field approaches (the gene-centered view of evolution) and the relevance of symmetry breaking and pattern formation in spatially distributed populations as an example of the breakdown of these approximations. Before introducing the complications of sexual repro- duction, we start with the simplest iterative model of ex- ponential growth of asexually reproducing populations: Ni(t+ 1) = λiNi(t) (1) where Niis the population of type iandλiis their fitness. If the total population is considered to be normalized, the relevant dynamics is only of the proportion of each type, then we obtain Pi(t+ 1) =λi/summationtext iλiPi(t)Pi(t) (2) where Piis the proportion of type i. The addition of mutations to the model, Ni(t+ 1) =/summationtext jλijNj(t), gives rise to the quasi-species model [2] which has attracted significant attention in the physics community. Recent research has focused on such questions as determining the rate of environmental change which can be followedby evolutionary change. The quasispecies model does not incorporate the effects of sexual reproduction. Sexual reproduction causes offspring to depend on the genetic makeup of two parents. This leads not only to mathematical but also to conceptual problems because the offspring of an organism may be as different from the parent as organisms it is competing against. A partial so- lution to this problem is recognizing that it is sufficient for offspring traits to be correlated to parental traits for the principles of evolution to apply. However, the gene centered view [3] is a simpler perspective in which the genes serve as indivisible units that are preserved from generation to generation [4]. In effect, different versions of the gene, i.e. alleles, compete rather than organisms. This view simplifies the interplay of selection and hered- ity in sexually reproducing organisms. We will show, formally, that the gene centered view corresponds to a mean field approximation [5]. This clar- ifies the domain of its applicability and the conditions in which it should not be applied to understanding evo- lutionary processes in real biological systems. We will then describe the breakdown of this model in the case of symmetry breaking and its implications for the study of ecological systems. It is helpful to explain the gene centered view using the “rowers analogy” introduced by Dawkins [3]. In this analogy boats of mixed left- and right-handed rowers are filled from a common rower pool. Boats compete in heats and it is assumed that a speed advantage exists for boats with more same-handed rowers. The successful rowers are then returned to the rower pool for the next round. Over time, a predominantly and then totally sin- gle handed rower pool will result. Thus, the selection of boats serves, in effect, to select rowers who therefore may be considered to be competing against each other [6]. In order to make the competition between rowers precise, an effective fitness can be assigned to a rower. We will make explicit the rowers model and demonstrate the assignment of fitness to rowers. The rowers analogy can be directly realized by con- sidering nonhomologue genes with selection in favor of a particular combination of alleles on genes. Specifically, for two genes, after selection, when allele A1appears in one gene, allele B1must appear on the second gene, and when allele A−1appears on the first gene allele B−1must appear on the second gene. We can write these high fitness organisms with the notation (1 ,1) and ( −1,−1), and the organisms with lower fitness (for simplicity, non- reproducing) as (1 ,−1) and ( −1,1). 1The assumption of placing rowers into the rower pool and taking them out at random is equivalent to assum- ing that there are no correlations in reproduction (i.e. no correlations in mate pairing) and a sufficiently dense sam- pling of genomic combinations by the population (in this case only a few possibilities). Then the offspring genetic makeup can be written as a product of the probability of each allele in the parent population. This assump- tion describes a “panmictic population” which is often used as a model in population biology. The assumption that the offspring genotype frequencies can be written as a product of the parent allele frequencies is a dynamic form of the usual mean field approximation neglect of correlations in interacting statistical systems [7]. Whil e the explicit dynamics of this system is not like the usual treatment of mean-field theory, e.g. in the Ising model, many of the implications are analogous. In our case, the reproducing parents (either (1 ,1) or (−1,−1)) must contain the same proportion of the cor- related alleles ( A1andB1) so that p(t) can represent the proportion of either A1orB1and 1 −p(t) can represent the proportion of either A−1orB−1. The reproduction equation specifying the offspring (before selection) are: P1,1(t+ 1) = p(t)2(3) P1,−1(t+ 1) = P−1,1(t+ 1) = p(t)(1−p(t)) (4) P−1,−1(t+ 1) = (1 −p(t))2(5) where P1,1is the proportion of (1 ,1) among the offspring, and similarly for the other cases. The proportion of the alleles in generation t+1 is given by the selected organisms. Since the less fit organisms (1,−1) and ( −1,1) do not survive this is given by p(t+ 1) =P′ 1,1(t+1)+ P′ 1,−1(t+1) = P′ 1,1(t+1), where primes indicate the proportion of the selected organisms. Thus p(t+ 1) =P1,1(t+ 1) P1,1(t+ 1) + P−1,−1(t+ 1)(6) This gives the update equation: p(t+ 1) =p(t)2 p(t)2+ (1−p(t))2(7) There are two stable states of the population with all organisms (1 ,1) or all organisms ( −1,−1). If we start with exactly 50% of each allele, then there is an unsta- ble steady state. In every generation 50% of the organ- isms reproduce and 50% do not. Any small bias in the proportion of one or the other will cause there to be progressively more of one type over the other, and the population will eventually have only one set of alleles. This problem is reminiscent of an Ising ferromagnet at low temperature: A statistically biased initial condition leads to alignment. This model can be reinterpreted by assigning a mean fitness (analogous to a mean field) to each allele as in Eq. (2). The fitness coefficient for allele A1orB1isλ1=p(t)with the corresponding λ−1= 1−λ1. The assignment of a fitness to an allele reflects the gene centered view. The explicit dependence on the population composition (a right handed rower in a predominantly right handed rower pool has higher fitness than one in a predominantly left handed rower pool) has been objected to on grounds of biological appropriateness [8]. For our purposes, we recognize this dependence as the natural outcome of a mean field approximation. We can describe more specifically the relationship be- tween this picture and the mean field approximation by recognizing that the assumptions of no correlations in re- production, a random mating pattern of parents, is the same as a long-range interaction in an Ising model. If there is a spatial distribution of organisms with mating correlated by spatial location and fluctuations so that the starting population has more of the alleles represented by 1 in one region and more of the alleles represented by −1 in another region, then patches of organisms that have predominantly (1 ,1) or ( −1,−1) form after several gen- erations. This symmetry breaking, like in a ferromagnet, is the usual breakdown of the mean field approximation. Here, it creates correlations / patterns in the genetic makeup of the population. When correlations become significant then the species has two types. Thus the gene centered view breaks down when multiple organism types form. Understanding the spatial distribution of organism genotype is a central problem in ecology and conserva- tion biology [9,10]. The spatial patterns that can arise from spontaneous symmetry breaking through sexual re- production, as implied by the analogy with other models, may be relevant. A systematic study of the relevance of symmetry breaking to ecological systems begins from a study of spatially distributed versions of the model just described. This model is a simplest model of disruptive selection, which corresponds to selection in favor of two genotypes whose hybrids are less viable. Assuming over- lapping local reproduction neighborhoods, called demes, the relevant equations are: p(x, t+ 1) = D(¯p(x, t)) (8) D(p) =p2 p2+ (1−p)2(9) ¯p(x, t) =1 NR/summationdisplay |xj|≤Rp(x+xj, t) (10) NR=/vextendsingle/vextendsingle{xj/vextendsingle/vextendsingle|xj| ≤R}/vextendsingle/vextendsingle (11) where the organisms are distributed over a two- dimensional grid and the local genotype averaging is per- formed over a preselected range of grid cells around the central cell. Under these conditions the organisms lo- cally tend to assume one or the other type. In contrast to conventional insights in ecology and population biol- ogy, there is no need for either complete separation of or- ganisms or environmental variations to lead to spatially varying genotypes. However, because the organisms are 2not physically isolated from each other, the boundaries between neighboring domains will move, and the domains will follow conventional coarsening behavior for systems with non-conserved order parameters. Simulations of this model starting from random ini- tial conditions are shown in Fig. 1. These initial con- ditions can arise when selection becomes disruptive af- ter being non-disruptive due to environmental change. The formation of domains of the two different types that progressively coarsen over time can be seen. While the evolutionary dynamics describing the local process of or- ganism selection is different, the spatial dynamics of do- mains is equivalent to the process of coarsening / pat- tern formation that occurs in many other systems [11]. Fourier transformed power spectra (Figs. 2–4) confirm the correspondence to conventional coarsening by show- ing that the correlation length grows as t1/2after initial transients. In a finite sized system, it is possible for one type to completely eliminate the other type. However, the time scale over which this takes place is much longer than the results assuming complete reproductive mixing, i.e. the mean field approximation. Since flat boundaries do not move except by random perturbations, a non- uniform final state is possible. The addition of noise will cause slow relaxation of flat boundaries but they can also be trapped by quenched (frozen) inhomogeneity. The results have significant implications for ecology of genetic diversity and species formation. The observation of harlequin distribution patterns of sister forms is gener - ally attributed to nonhomogeneities in the environment, i.e. that these patterns reflect features of the underly- ing habitat (=selective) template. Our results show that disruptive selection can give rise to spontaneously self- organized patterns of spatial distribution that are inde- pendent of underlying habitat structure. At a particular time, the history of introduction of disruptive selection events would be apparent as a set of overlapping patterns of genetic diversity that exist on various spatial scales. More specific relevance of these results to the theoreti- cal understanding of genetic diversity can be seen in Fig. 5 where the population averaged time dependence of pis shown. The gene centered view / mean field theory pre- dicts a rapid homogenization over the entire population. The persistence of diversity in simulations with symme- try breaking, as compared to its disappearance in mean field theory, is significant. Implications for experimental tests and methods are also important. Symmetry break- ing predicts that when population diversity is measured locally, rapid homogenization similar to the mean field prediction will apply, while when they are measured over areas significantly larger than the expected range of re- production, extended persistence of diversity should be observed. The divergence of population traits in space can also couple to processes that prevent interbreeding or doom the progeny of such breedings. These may include as- sortive mating, whereby organism traits inhibit inter- breeding. Such divergences can potentially lead to theformation of multiple species from a single connected species (sympatric speciation). By contrast, allopatric speciation where disconnected populations diverge has traditionally been the more accepted process even though experimental observations suggest sympatric speciation is important. Our concept of genetic segregation via spontaneous pattern formation differs in concept from models in which sympatric differentiation is achieved solely via either runaway sexual selection or transient or micro-allopatry. More recent papers have begun to connect the process of symmetry breaking to sympatric speciation when driven by specific models of competition [12–14]. [1] Bar-Yam, Y., Dynamics of Complex Systems, Addison- Wesley Longman (1997). [2] Eigen, M., McCaskill, J. and Schuster, P., Adv. Chem. Phys.75, 149 (1989). [3] Dawkins, R., The Selfish Gene, 2nd ed., Oxford Univer- sity Press, p.86 (1989). [4] The evolutionary indivisibility of genes is also not nec es- sarily well justified. [5] Bar-Yam, Y., Adv. Complex Sys. 2, 277 (2000). [6] For a collection of articles discussing the “levels of se lec- tion controversy” see: Brandon, R. N. and Burian, R. M., eds., Genes, Organisms, Populations: Controversies Over the Units of Selection, MIT Press (1984). [7] The general relationship between the mean-field approx- imation and the gene centered view can be shown by considering a two-step of reproduction and selection: {N(s;t)}=R[{N′(s;t−1)}] {N′(s;t)}=D[{N(s;t)}] where sis a particular genome, and N, N′are numbers of reproduced, selected organisms respectively. Using a mean field approximation for offspring, the probability of a particular genome sis the product of probabilities of the alleles ai:P(a1, . . . , a N) =/producttext P(ai). This enables the two-step update equations to be written as a one-step update equation for each allele: n′(ai;t) =˜D[{n′(ai;t−1)}] =λ({n′})n′(ai;t−1) where n′(ai) is the number of allele ai. For details see [5]. [8] Sober, E. and Lewontin, R. C., Phil. Sci. 49, 157 (1982). [9] Tilman, D. and Kariena, P., eds., Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, Princeton University Press, p.368 (1997). [10] Durett, R. and Levin, S. A., Phil. Trans. R. Soc. Lond. B 343 , 329 (1994). [11] Bray, A. J., Advances in Physics 43, 357 (1994). [12] Kondrashov, A. S. and Kondrashov, F. A., Nature 400, 351 (1999). [13] Dieckmann, U. and Doebeli, M., Nature 400, 354 (1999). 3[14] Higashi, M., Takimoto, G. and Yamamura, N., Nature 402, 523 (1999). /K74 /K20/K3D/K20/K30 /K31 /K32 /K34 /K38 /K31/K36 /K33/K32 /K36/K34 /K31/K32/K38 /K32/K35/K36 /K35/K31/K32 /K31/K30/K32/K34 FIG. 1. Spatially distributed evolution with disruptive se - lection giving rise to two types appearing in patches and coa rs- ening. The space is periodic and has 256 ×256 sites, and the mating neighborhood radius is R= 5. FIG. 2. Fourier power spectra averaged over ten simu- lations of evolutionary processes like that shown in Fig. 1 (512×512 sites and R= 1). FIG. 3. Temporal behavior of the peak of a Fourier power spectrum in the shown case. Top: The peak frequency kp(t) which follows approximately t−1/2. Bottom: The peak power S(kp) which follows approximately t1/2. FIG. 4. Collapsed version of the Fourier power spectra demonstrating the scaling form S(k) =t1/2f(kt1/2). 4FIG. 5. Comparison of the time dependence of type prob- ability in the mean field approximation and symmetry break- ing, calculated using different random number sequences. Di - versity persists much longer in the latter. In some cases, fo r- ever. 5

arXiv:physics/0004026v1 [physics.class-ph] 14 Apr 2000SOMMERFELD PARTICLE IN STATIC MAGNETIC FIELD: TUNNELING AND DELAYED UNTWISTING IN CYCLOTRON Alexander A. Vlasov High Energy and Quantum Theory Department of Physics Moscow State University Moscow, 119899 Russia Motion of a charged particle with finite size, described by So mmerfeld model, in static magnetic field has two peculiar features: 1. ) there is the effect of tunneling - Sommerfeld particle overcomes the barr ier and finds itself in the forbidden, from classical point of view, area; 2.) the untwisting of trajectory in cyclotron for Sommerfeld particle is stron gly delayed compared to that of a classical particle. 03.50.De Here we continue our investigation of peculiar features of m otion of Som- merfeld particle [1]. Let us remind that long time ago [2] Som merfeld pro- posed a model of a charged particle of finite size - sphere with uniform sur- face charge Qand mechanical mass m. In nonrelativistic approximation such sphere obeys the equation (see also [3]): m˙/vector v=/vectorFext+η[/vector v(t−2a/c)−/vector v(t)] (1) herea- radius of the sphere, η=Q2 3ca2, /vector v=d/vectorR/dt, /vectorR- coordinate of the center of the shell, /vectorFext- some external force. This model is a good tool to consider effects of radiation reac tion of a charged particle of finite size, free of problems of classica l point-like Lorentz- Dirac description. A. If Sommerfeld particle moves in the external static magneti c field /vectorH, the 1force /vectorFext=/integraltextd/vector rρ·[˙/vectorR,/vectorH] forρ=Qδ(|/vector r−/vectorR| −a)/4πa2has the form Fext=Q c[˙/vectorR,/vectorH] If magnetic field has non-zero values only in the shell of finit e size S( 0< Y < S ,/vectorHis parallel to z-axis, /vectorR= (X, Y,0) ), then, as the particle has finite size 2 a, force /vectorFextmust be multiplied by the factor f: f=  0, Y < −a; Y 2a+1 2, −a < Y < a ; 1, a < Y < S −a; S−Y 2a+1 2, S −a < Y < S +a; 0, S +a < y;(2) For dimensionless variables x=X/M, y =Y/M, τ =ct/M (M-scale factor) equation (1) takes the form ¨y=K·[ ˙y(τ−d)−˙y(τ)]−λ·˙x·f, ¨x=K·[ ˙x(τ−d)−˙x(τ)] +λ·˙y·f, (3) here f=  0, y < −d 2; y d+1 2, −d 2< y <d 2; 1,d 2< y < L −d 2; L−y d+1 2, L −d 2< y < L +d 2; 0, L +d 2< y;(4) and K=Q2M 3a2mc2, λ=QHM mc2, d=2a M, L=S M. Classical analog of equation (3) for point-like particle wi thout radiation re- action reads ¨y=−λ·˙x·g, ¨x=λ·˙y·g, (5) here g=  0, y < 0; 1,0< y < L ; 0, L < y ;(6) 2For initial conditions x(0) = 0 , y(0) = 0 ,˙x(0) = 0 ,˙y(0) = vsolution of (5) is x=−v λ+v λcos (λτ), y=v λsin (λτ) (0< y < L ) (7) We see that for initial velocities vsmaller, then the critical velocity vcr= λL, particle trajectory (half-circle) lies inside the shell, i.e. particle cannot overcome the barrier. If L= 104, λ= 10−4thenvcr= 1. We numerically investigated the particle motion governed b y equation (3) for the following values of initial velocity: v= 0.43, v= 0.44 and for L= 104, λ= 10−4, d= 1.0, K= 4/(3d2), i.e. particle is of electron size and mass, magnetic field app roximately equals 1012gauss and S≈5,6·10−9sm. The result is shown on Fig. A, compared with classical trajec tory, gov- erned by (7) with v= 0.44. Horisontal axis is xand vertical axis is y. The effect of tunneling for Sommerfeld particle is vividly se en: velocity v= 0.44 is smaller then the critical vcr= 1, but the particle overcomes the barrier and finds itself in the forbidden from classical p oint of view area y > L = 104. B. If magnetic field is parallel to z-axis for y <0 and y > L and equals to zero for 0 < y < L , and for 0 < y < L there is static electric field E, parallel toy- axis in such a way, that it is always collinear to y-component of particle velocity (i.e. particle is always accelerates in the cleara nce 0 < y < L ), then there is a model of cyclotron. Equation of motion for Sommerfeld particle in cyclotron rea ds ¨y=K·[ ˙y(τ−d)−˙y(τ)]−λ·˙x·f+ǫ·Sgn( ˙y)·(1−f), ¨x=K·[ ˙x(τ−d)−˙x(τ)] +λ·˙y·f, (8) 3here ǫ=QEM mc2 Classical analog of (8) one can construct replacing in (8) Kby zero and f byg(6): ¨y=−λ·˙x·g+ǫ·Sgn( ˙y)·(1−g), ¨x=λ·˙y·g, (9) Initial conditions are: x(0) = y(0) = ˙ x(0) = ˙ y(0) = 0 Due to classical equation of motion without radiation react ion (9) particle moves along untwisting trajectory. Total increase of kinet ic energy Wc= ( ˙x)2/2 + ( ˙y)2/2 of particle is N·e·L: Wc=N·ǫ·L where N- is the total number of passing of particle through the accel erating fieldE. IfN= 10, ǫ=λ= 10−7, L= 105, then Wc= 10−1. We numerically calculated the particle motion governed by e quation (8) with zero initial conditions for the following values of paramet ers: L= 105, λ= 10−7=ǫ, d= 0.3, K= 2.0, i.e. particle is of electron size and mass, magnetic field app roximately equals to 8.1·107gauss and electric field produces in the clearance potential differ- ence equal to 104eV. The results of calculations are shown on Fig. B.1 - classical case and on Fig. B.2 - case of Sommerfeld particle. Horisontal axis is x·λand vertical axis is y·λ. We see that for the same ”time” τ≈108(i.et≈10−4sec) classical particle (without radiation reaction) made N= 10 passings through the accelerating fieldEwith total energy increase Wc= 10−1, while Sommerfeld particle made only N= 6 passings with total energy increase Ws= 0.0375 ( Wc 4forN= 6 is equal to 0 .06 ). Thus untwisting of trajectory for Sommerfeld particle is strongly delayed compared to that of a classical one. Delay in energy increase falls mainly on the moments of passi ng through the clearance. It can be explained by difference in accelerat ions in electric field (proportional to ǫ≈10−7) and in magnetic field (proportional to v·λ≈ 10−8) as flux of radiating energy is proportional to square of acce leration. REFERENCES 1. Alexander A.Vlasov, physics/9905050, physics/9911059 . 2. A.Sommerfeld, Gottingen Nachrichten, 29 (1904), 363 (19 04), 201 (1905). 3. L.Page, Phys.Rev., 11, 377 (1918). T.Erber, Fortschr. Ph ys., 9, 343 (1961). P.Pearle in ”Electromagnetism”,ed. D.Tepliz, (Pl enum, N.Y., 1982), p.211. A.Yaghjian, ”Relativistic Dynamics of a Char ged Sphere”. Lecture Notes in Physics, 11 (Springer-Verlag, Berlin, 199 2). 5v=0.43Sommerfeld: v=0.44 Classic: v=0.44 2.61e01.80e33.60e35.40e37.20e39.00e31.08e41.26e41.44e41.62e41.80e4 0.00e01.80e33.60e35.40e37.20e39.00e31.08e41.26e41.4 4e41.62e41.80e4 Fig. A 6-6.00e-1-4.80e-1-3.60e-1-2.40e-1-1.20e-10.00e01.20e-12.40e-13.60e-14.80e-16.00e-1 -6.00e-1 -4.80e-1 -3.60e-1 -2.40e-1 -1.20e-1 0.00e01.20e -1 2.40e-1 3.60e-1 4.80e-1 6.00e-1 Fig. B.1. 7-6.00e-1-4.80e-1-3.60e-1-2.40e-1-1.20e-10.00e01.20e-12.40e-13.60e-14.80e-16.00e-1 -6.00e-1 -4.80e-1 -3.60e-1 -2.40e-1 -1.20e-1 0.00e01.20e -1 2.40e-1 3.60e-1 4.80e-1 6.00e-1 Fig. B.2 8

WHAT ARE THE HIDDEN QUANTUM PROCESSES IN EINSTEIN’S WEAK PRINCIPLE OF EQUIVALENCE? Tom Ostoma and Mike Trushyk 48 O’HARA PLACE, Brampton, Ontario, L6Y 3R8 emqg@rogerswave.ca Monday April 12, 2000 ACKNOWLEDGMENTS We wish to thank R. Mongrain ( P.Eng) for our lengthy conversations on the nature of space, time, light, matter, and CA theory.2ABSTRACT We provide a quantum derivation of Einstein’s Weak Equivalence Principle (WEP) of general relativity using a new quantum gravity theory proposed by the authors called Electro-Magnetic Quantum Gravity or EMQG (ref. 1). EMQG is manifestly compatible with Cellular Automata (CA) theory (ref. 2 and 4), and is also based on a new theory of inertia (ref. 5) proposed by R. Haisch, A. Rueda, and H. Puthoff (which we modified and called Quantum Inertia, QI). QI states that classical Newtonian Inertia is a property of matter due to the strictly local electrical force interactions contributed by each of the (electrically charged) elementary particles of the mass with the surrounding (electrically charged) virtual particles (virtual masseons) of the quantum vacuum. The sum of all the tiny electrical forces (photon exchanges with the vacuum particles) originating in each charged elementary particle of the accelerated mass is the source of the total inertial force of a mass which opposes accelerated motion in Newton’s law ‘F = MA’. The well known paradoxes that arise from considerations of accelerated motion (Mach’s principle) are resolved, and Newton’s laws of motion are now understood at the deeper quantum level. We found that gravity also involves the same ‘inertial’ electromagnetic force component that exists in inertial mass. We propose that Einstein’s general relativistic Weak Equivalence Principle (WEP) originates from common ‘lower level’ quantum vacuum processes occurring in both inertial mass and gravitational mass, in accordance with the principles of quantum field theory. Gravitational mass results from the quantum activities of both the electrical force (photon exchanges) component and the pure gravitational force (graviton exchanges) component, acting simultaneously on the elementary particles that make up a mass. However, inertial mass is strictly the result of the electrical force process only, as given by quantum inertia principle (with negligible graviton processes present). Under gravitation, the elementary particles of a test mass near the earth exchanges gravitons with the earth. However what is frequently neglected is that the surrounding (electrically charged) virtual fermion particles also exchange gravitons with the earth, causing the virtual fermion particles to accelerate (fall) towards the earth. A test mass does move under the influence of these direct graviton exchanges, but more importantly, moves under the influence of the falling (electrically charged) virtual particles of the quantum vacuum, which dominates the total quantum force process. Therefore a test mass under gravity ‘sees’ that the quantum vacuum accelerate in the same way as it did when the test mass was subjected to acceleration alone. Thus, equivalence arises from the reversal of the acceleration vectors of the quantum vacuum with respect to a test mass undergoing acceleration, as compared to a test mass subjected to a gravitational field. In accelerated frames, it is the mass that accelerates. Inside gravitational fields it is the virtual particles of the quantum vacuum that accelerates . A consequence of EMQG is that all elementary mass particles must consist of combinations of just one fundamental matter particle type, which we call the ‘ masseon’ particle. The masseon has one, fixed (smallest) quanta of mass (similar to a quanta of electric charge), which we call ‘mass charge’. The masseon also carries either a positive or negative quanta of electric charge. The masseon particle generates a fixed flux of graviton exchange particles, with a flux rate being completely unaffected by high speed motion (like electric charge). The graviton is the vector boson exchange particle of the pure gravitational force interaction. The physics of graviton exchanges is nearly identical to the photon exchanges of QED, with the same concept of positive and negative gravitational ‘mass charge’ carried by masseons and anti- masseons respectively. The ratio of the strength of graviton to photon exchange force coupling is about 10-40 for an electron. In QED, the quantum vacuum consists of virtual electrons, virtual anti-electrons, and virtual photons. In EMQG, the quantum vacuum consists of virtual masseons, virtual anti- masseons, and virtual gravitons, which also posses both positive and negative electrical charge and positive and negative ‘mass charge’. There are almost equal numbers of virtual masseon and anti- masseon particles existing in the quantum vacuum everywhere, and at any given time. This is why the cosmological constant is very close to zero.3TABLE OF CONTENTS ABSTRACT ________________________________ ____________________________ 2 1. INTRODUCTION ________________________________ _____________________ 5 2. INTRODUCTION TO ELECTROMAGNETIC QUANTUM GRAVITY _________ 8 3. THE QUANTUM VACUUM AND IT’S RELATIONSHIP EMQG _____________ 12 3.1 INTRODUCTION TO THE CASIMIR FORCE EFFECT _____________________ 14 3.2 EVIDENCE FOR THE EXISTENCE OF VIRTUAL PARTICLES _____________ 15 4. INTRODUCTION TO QUANTUM INERTIA THEORY _____________________ 17 5. GENERAL RELATIVITY, ACCELERATION, GRAVITY AND CA ____________ 20 5.1 INTRODUCTION TO THE PRINCIPLE OF EQUIVALENCE ________________ 22 6. PHYSICAL PROPERTIES OF THE GRAVITON AND THE MASSEON _______ 23 6.1 THE QUANTUM VACUUM AND VIRTUAL MASSEON PARTICLES _________ 29 7. VIRTUAL PARTICLES NEAR SPHERICAL MASS LIKE THE EARTH _______ 31 8. DERIVATION OF THE WEAK PRINCIPLE OF EQUIVALENCE ____________ 33 8.1 INTRODUCTION ________________________________ _______________________ 34 8.2 MASSES INSIDE A GRAVITATIONAL FIELD (THE EARTH) _______________ 37 8.3 MICROSCOPIC EQUIVALENCE PRINCIPLE OF PARTICLES ______________ 39 8.4 THE INERTION: AN ELEMENTARY QUANTA OF INERTIA _______________ 41 8.5 EQUIVALENCE PRINCIPLE FOR THE SUN-EARTH-MOON SYSTEM _______ 42 8.6 LIGHT MOTION IN A ROCKET: SPACE-TIME EFFECTS __________________ 44 8.7 LIGHT MOTION NEAR EARTH’S SURFACE - SPACE-TIME EFFECTS ______ 46 8.8 GENERAL CONCLUSIONS FOR THE EQUIVALENCE OF LIGHT MOTION _4749. THE EQUIVALENCE OF 4D SPACE-TIME CURVATURE _________________ 49 9.1 GENERAL RELATIVISTIC 4D SPACE-TIME CURVATURE ________________ 50 9.2 EMQG, LIGHT MOTION, AND 4D SPACE-TIME CURVATURE _____________ 51 9.3 CLASSICAL SCATTERING OF PHOTONS IN REAL MATTER ______________ 56 9.4 QUANTUM FIELD THEORY OF PHOTON SCATTERING IN MATTER _____ 57 9.5 THE SCATTERING OF PHOTONS IN THE QUANTUM VACUUM ___________ 58 9.6 THE FIZEAU EFFECT: LIGHT VELOCITY IN A MOVING MEDIA _________ 61 9.7 LORENTZ SEMI-CLASSICAL PHOTON SCATTERING ____________________ 62 9.8 PHOTON SCATTERING IN THE ACCELERATED VACUUM _______________ 63 9.9 SPACE-TIME CURVATURE OBTAINED FROM SCATTERING THEORY ____ 65 10. EXPERIMENTAL VERIFICATION OF THE EQUIVALENCE MODEL ______ 70 11. CONCLUSIONS ________________________________ ____________________ 73 12. REFERENCES ________________________________ _____________________ 74 13. ILLUSTRATIONS ________________________________ ___________________ 7751. INTRODUCTION “I have never been able to understand this principle (principle of equivalence) ... I suggest that it be now buried with appropriate honors.” - Synge: Relativity- The General Theory Imagine that you are stationary, and standing on the surface of the earth. Gravity feels like a force that is holding your gravitational mass to the earth’s surface. Yet when you are standing in a rocket undergoing accelerated motion (far from gravitational fields, moving with an acceleration of 1 g), the principle of equivalence tells us that there is an identical force exerted against the rocket floor by your inertial mass. However, this force is now caused by your dynamic accelerated motion through empty space. While standing on the earth, however, you were definitely not in motion. Why should there be such a deep connection between what appears to be two completely different physical phenomena: a static force of gravity with no apparent motion, and a dynamic force due to the accelerated motion of your mass? In other words, why should you weigh the same in the rocket as you do on the earth? The principle of equivalence encompasses this strange ‘coincidence’ , and is one of the founding postulates of general relativity theory. In it’s stronger form, it states that all the laws of physics are the same in the above thought experiment. When stated in it’s weaker form, it implies that objects of different mass fall at the same rate of acceleration in a uniform gravity field, and that only laws of motion of physics are the same . Equivalence also means that the inertial mass, i.e. the mass defined by Newton’s law of motion: mF gii = is exactly equal to the gravitational mass, which is mass defined by a completely different law given by Newton’s universal gravitational law: mFr GMgg =2 (where mi and m g are the inertial and gravitational mass of an object, Fi and Fg are the inertial and gravitational forces exerted on the object, g is the acceleration of the object, M is the mass of the earth, r is the distance from earth’s center, and G is Newton’s universal gravitational constant). Newton was well aware of this coincidence of mass equivalence, which we refer to here as the Newtonian mass equivalence principle. The equivalence principle demands that that mi = m g. Why should this be true in our universe? It has been about 85 years since the discovery of the Einstein equivalence principle, and hundreds of years since the discovery of Newton’s mass equivalence principle. Yet, it is still not understood why inertial mass exists in the first place, or why a mass opposes acceleration with a back acting inertial force. More importantly, it is also not known why there are two totally different physical definitions for inertial and gravitational mass (instead of just one).6Furthermore, masses can have different temperatures (and therefore different energy content), and may be composed of different materials like lead, wood, water. Again, why should mi = m g,, no matter what the material composition and the energy content that a mass may contain? In short, the principle of equivalence is one of the deepest, unsolved mysteries that exists in fundamental physics today! Lacking any deeper understanding of this question, most physicists prefer to accept equivalence as the fundamental way in which the universe operates. Other physicists maintain that the origin of the principle of equivalence is one of the deepest, unsolved mystery of modern physics, and deserves an explanation. In this paper we reveal for the first time, the low-level quantum processes that are hidden from our view, and provide a quantum field theoretic solution to the principle of equivalence. In other words, we show how the principle of equivalence turns out to be purely a quantum process. Equivalence results from the activities of quantum particles interacting with quantum particles, while obeying the general laws of quantum field theory. This paper is also an invitation to explore a new theory of gravity called ElectroMagnetic Quantum Gravity, or EMQG (ref. 1). EMQG is based on a new understanding of both inertia and the principle of equivalence, which exists on the distance scales of quantum particles. How is the Einstein principle of equivalence precisely defined? Physicists recognize two main formulations of the principle of equivalence (with some minor variations): the Weak Equivalence Principle (WEP), and the Strong Equivalence Principle (SEP). The strong equivalence principle states that the results of any given physical experiment will be precisely identical for an accelerated observer in free space as it is for a non-accelerated observer in a perfectly uniform gravitational field. This is the form of the equivalence principle used by Einstein to formulate his theory of general relativity in 1915. A weaker form of equivalence principle (WEP) restricts itself only to the laws of motion of masses in these two situations. In other words, the weak principle of equivalence states that only the laws of motion of a mass on the earth is identical to the laws of motion of the same mass inside an accelerated rocket (at 1g). Technically, when comparing the equivalence of a mass in a rocket to a mass on the earth, we assume that the motion is restricted to short distances (mathematically, at a point) on the earth, where gravity does not vary with height. The WEP implies that objects of different mass, different material composition, and different energy content all fall at the same rate of acceleration towards the earth (as they do in a rocket accelerating at 1g). How do we know that the WEP holds true in general? Newton was well aware that inertial mass was equivalent to gravitational mass (which we call Newtonian equivalence). In fact even before Newton, there was an early experimental demonstration of the equivalence principle at work. This historical experiment was performed by Galileo in Pisa, where two objects of significantly different mass were dropped off the leaning tower of Pisa. Galileo observed that the two masses arrived on the ground roughly at the same time. The weak equivalence principle implies that these two different masses should fall at exactly the same rate (as they obviously do inside an7accelerated rocket). Since this early experiment, the equivalence of inertia and gravitational mass has been verified to a phenomenal accuracy of about one part in about 10-15 (ref 24). Einstein is generally credited with the elevation of the equivalence principle to a fundamental symmetry of nature in 1915. Conventional wisdom in physics assumes that the strong principle of equivalence is exact , and somehow reflects a fundamental aspect of nature. It is assumed to be applicable under any physical circumstance. It is believed to hold true at the elementary particle level, and under enormously large gravitational fields such as on a neutron star. As a consequence, Einstein’s general relativity theory (which is based on this principle) is also assumed to hold true under any physical condition. The principle of equivalence has been tested under a wide variety of gravitational field strengths and distance scales. It has been tested with different material types (ref. 6 and 7). It has been tested to an extremely high precision for laboratory bodies (up to 3 parts in 10-12). It also has been checked to 1 part in 10-12 for the acceleration of the moon and earth towards the sun. It has even been tested for individual elementary particles, such as the neutron. To our knowledge, no confirmed violation of the equivalence principle has been reported to date. Yet after 80 years of close scrutiny, the principle of equivalence has still remained only a postulate of general relativity. It cannot be proven from more fundamental principles. Some of the better literature on general relativity have drawn attention to this fact, and admit that no explanation can be found as to; “why our universe has a deep and mysterious connection between acceleration and gravity” (ref. 8). One must bear in mind that mass is really nothing more than a vast collection of quantum particles, which interact with each other through forces. Forces are also ultimately the result of quantum particles called bosons, which act like the exchange particles that transmit momentum from one particle to another. Therefore, it is essential that equivalence be understood at a quantum particle level in order for a deeper understanding to emerge. We will show why there is equivalence, and also demonstrate that the principle of equivalence is just an approximation, albeit an extremely close approximation. We will also show that there is actually a tiny imbalance in the equivalence of inertial and gravitational mass, with the gravitational mass of an object being (every so slightly) larger than the inertial mass. This effect is magnified when comparing the free fall times of very large mass to that of an extremely tiny mass on the earth. Furthermore this effect may be measured experimentally in the near future. We will also show that in certain extremely rare physical circumstances, the equivalence principle does not hold at all (for example, anti-matter; described in section 8)! In order to understand the equivalence principle (and inertia) on the quantum level, we must have an understanding of the basic concepts of a new theory of gravity called ElectroMagnetic Quantum Gravity (EMQG, ref. 1, 1998). The motivation for the development of EMQG was the consideration that the universe may be a vast Cellular Automaton process (ref. 57). In the process of developing EMQG theory with this goal in mind, we discovered the hidden quantum processes that are responsible for the principle of8equivalence. Before we can present our derivation of the equivalence principle, we must first briefly review the important concepts of EMQG theory that are required to understand the principle of equivalence. Reference 1 (1998) contains the entire work. 2. INTRODUCTION TO ELECTROMAGNETIC QUANTUM GRAVITY “The interpretation of geometry advocated here cannot be directly applied to submolecular spaces … it might turn out that such an extrapolation is just as incorrect as an extension of the concept of temperature to particles of a solid of molecular dimensions” A. Einstein (1921) Two years ago we introduced a theory of gravity based solely on the activities of quantum particles, which consist of quantum mass particles ( fermions) and quantum force particles (bosons). ElectroMagnetic Quantum Gravity (EMQG, ref. 1) grew out of an attempt to understand not just gravity on a quantum scale, but also inertia and the principle of equivalence. Various attempts at the unification of general relativity with quantum theory have not been entirely successful in the past, because these theories do not grasp the true quantum nature of inertia and the hidden quantum processes behind Einstein’s principle of equivalence. In developing a theory of quantum gravity, one might ask which of the existing approaches is more relevant or fundamental; quantum field theory or classical general relativity (with conventional 4D space-time continuum)? Currently it seems that both theories are generally not compatible with each other. EMQG is based on a postulate that the universe operates like some form of a vast Cellular Automaton (CA) (ref. 2, 3, 4, 34, 57). Figure 21 shows a tentative model for the universal CA. This led us to the position that quantum field theory is actually in closer touch to the true inner workings of our universe, than general relativity. This is because many CA’s inherently produce stable, oscillating particle-like patterns that are reminiscent of elementary particles interacting. The CA requires strict locality, and therefore forces can only be possible through the exchange of ‘force particles’. On the other hand general relativity is taken as a global, classical description of space-time, gravity and matter. General relativity reveals the large-scale patterns and organizing principles that arise from the hidden, local quantum processes existing on the quantum particle distance scales. A Cellular Automaton consists of a huge array of 'cells' or memory locations that are capable of storing numerical data. The numeric state of all the cells, everywhere, changes at a regular synchronized interval called a ‘clock cycle’. The change in numeric state depends on the numeric values of a cell’s neighbors and on the same set of fixed rules or algorithm, which are located in each and every cell of the cellular automaton. It is important to note that the ‘clock cycle’ marks events related to the CA operation, and is not the same thing as events in our measure of time. Unlike the ordinary desktop computer, the cellular automata computer updates all it’s memory locations in a single ‘clock cycle’, thus making the CA computer vastly more faster and powerful than any of the common computers systems in use today. It should be9noted that there already exists computational machines modeled on the CA architecture, which are being used by physicists for various physical modeling projects. These CA machines are used to model nature, such as problems in fluid flow and turbulence, rather then for computing things like income tax or other business applications. However, it is interesting to note how well the CA computer model is suited to modeling various particle interactions, such as for elementary particle physics problems, fluid mechanics, and turbulence problems. The CA structures used for these type of simulations are known as ‘Lattice Gases’. The cellular automaton computer was discovered theoretically by Konrad Zuse and Stanislav Ulam in the late 1940’s, and later was put to use by John von Neumann to model the real world behavior of complex spatially extended structures. The best known example of a CA is the ‘game of life’ originated by John Horton Conway and published in the famous mathematical games department of Scientific American magazine in the 1970’s. A direct consequence of CA theory is that our universe must be utterly simple on the smallest scales. Furthermore, the universe must be completely deterministic, or at least be deterministic in principle. The apparent randomness of nature set forth by quantum theory must reflect our general ignorance of the exact numeric state of a system on the CA. If we know the structure of the universal CA, the exact mathematical rules that govern the cells of the CA, and the exact numeric state of all the cells in the universe at a given instant, then we have all the information needed to predict the exact future state of the system. Of course this is completely impossible to do in practice, for a variety of reasons. Even if we can somehow gather all this information, we would need a computer just as powerful as the universal CA to process the data in a reasonable period of time! We believe that Quantum Field Theory (QFT) is currently our best mathematical description of the inner fundamental workings of the universe. QFT tells us that all forces originate from a quantum particle exchange process. These particle exchanges transfer momentum from one quantum particle to another, and the huge numbers of particles exchanged produces generates what appears to be a smooth force interaction. The exchange process is universal, and applies to electromagnetic, weak and strong nuclear forces, and also (as we hope to show here) for the gravitational force. The generic name given to the force exchange particle is the ‘vector boson’ particle. ElectroMagnetic Quantum Gravity (EMQG) is basically a quantum field theory model of gravity. Figure 17 shows a block diagram of the relationship of EMQG with the rest of physics. Basically EMQG supports the view that gravity is based on particle exchange processes in accordance with the general principles of quantum field theory. EMQG is based on the graviton (the exchange particle for the pure gravitational force), and also on the photon (the exchange particle for electromagnetic force) exchange particles. What is unique about EMQG theory is that gravitation involves both the photon and graviton exchange particles operating at the same time, where now the photon plays a very important role in gravity! In fact, the photon exchange process dominates all gravitational interactions and is, in most part, responsible for the principle of equivalence. The photon10particle is also responsible for another important property that all matter possesses; Newtonian inertia . When formulating a new theory of quantum gravity, a mechanism must be found that produces the gravitational force, while somehow being linked to the principle of equivalence. In addition, this mechanism should naturally lead to 4D space-time curvature and should be compatible with the principles of general relativity theory. Nature has another long-range force called electromagnetism, which is extremely prominent in all physical interactions, and has been described quite successfully by the principles of quantum field theory. This well-known theory is called Quantum ElectroDynamics (QED), and this theory has been tested for electromagnetic phenomena to an unprecedented accuracy. Electromagnetism plays the essential role of atomic and molecular interactions, and is responsible for most observable phenomena, including virtually all of chemistry and biology. Electromagnetic force is a long range force. Gravitational force is also a long range force. Therefore we believe that it is reasonable to assume that gravitational force should, somehow, result from a similar process as electromagnetism. The similarity between the two basic forces is also evident from classical physics, which describe the Coulomb electrical force and Newton’s law for the gravitational force. However a few obstacles lie in the way for this line of reasoning. First, gravitational force is observed to be always attractive, and never repulsive! In QED, electrical forces can be both attractive or repulsive. One result of this is that there are equal numbers of positive and negative charged virtual particles in the quantum vacuum at any given instant of time. This is because virtual particles are always created in equal and opposite charged particle pairs, according to quantum field theory. Thus, there is a balance of attractive and repulsive electrical forces in the quantum vacuum, and the quantum vacuum is electrically neutral, overall. If this were not the case, the electrically charged virtual charged particles of one charge type in the vacuum would dominate over all other physical interactions involving real matter, due to the enormous number of vacuum particles involved. Secondly, QED is formulated in a relativistic, flat 4D space-time with no curvature. In QED, electrical charge is defined as the fixed rate of emission of photons (strictly speaking, the fixed probability of emitting a photon) from a given charged particle. Electromagnetic forces are caused by the exchange of photons, which propagate between the charged particles. The photons transfer momentum from the source charge to the destination charge, and travel in flat 4D space-time (assuming no gravity). From these basic considerations, a successful theory of quantum gravity should have an exchange particle or graviton, which is emitted from a mass particle at a fixed rate as in QED. This is called ‘mass charge’, and is analogous to electrical charge in QED. Therefore, the graviton transfers momentum from one mass to another, which is the root cause of gravitational force. Yet, the graviton exchanges must somehow produce disturbances of the normally flat space and time, when originating from a very large mass.11Since mass is known to vary with velocity (special relativity), one might expect that ‘mass charge’ of a particle might also vary with velocity. In QED the electromagnetic force exchange process is governed by a fixed, universal constant ( α) which is not affected by anything like motion (more will be said about this point later). Should this not be true for graviton exchange process in quantum gravity as well? As we hinted, gravity is a long- range force, governed by a similar mathematical law as found in Coulomb’s Electrical Force law. Coulomb’s Electric Force law states: F = KQ 1Q2/R2 , and Newton’s Gravitational Force law: F=GM 1M2/R2. Both are inverse square laws, both depend linearly on the product of the two masses (or charges). Both obey the principle of superposition. We believe that this suggests that there is a deep connection between gravity and electromagnetism. Yet, gravity supposedly has no counterpart to negative electrical charge. Thus might leads one to believe that there is no such thing as negative ‘mass charge’ for gravity. Furthermore, QED also has no analogous phenomena as the principle of equivalence. Why should gravity be connected with the principle of equivalence, and thus inertia, and yet no analogy of this phenomena exists for electromagnetic phenomena? To answer the question of negative ‘mass charge’, EMQG postulates the existence of negative ‘mass charge’ for gravity, in close analogy to electromagnetism. Furthermore, we claim that this property of matter is possessed by all anti-particles that carry mass. Therefore anti-particles, which are opposite in electrical charge to ordinary particles, are also opposite in ‘mass charge’. In fact, negative ‘mass charge’ is not only abundant in nature, it comprises nearly half of all the mass particles in the form of ‘virtual’ particles in the universe! The other half exists as positive ‘mass charge’, also in the form of virtual particles. Furthermore, all familiar ordinary (real) matter comprises only a tiny fraction of the total ‘mass charge’ in the universe, and is experimentally found to be almost all positive! Real anti-matter seems to be very scarce in nature, and no search for it in the cosmos has revealed any to date. Both positive and negative ‘mass charge’ appear in huge numbers in the form of virtual particles, which quickly pop in and out of existence in the quantum vacuum (section 3), everywhere in empty space. We suggest that the existence of negative ‘mass charge’ is the key to the solution to the famous problem of the cosmological constant (ref. 46), which is one of the great unresolved mysteries of modern physics. Finally, we propose that the negative energy, or the antimatter solution of the famous Dirac equation of quantum field theory is also the negative ‘mass charge’ solution for fermions . Previous attempts at quantizing the gravitational field have been made using the principles of quantum field theory. They focused on using the graviton force exchange particle alone, which is proposed as the quanta of the gravitational field in direct analogy with the quantization of electromagnetic fields with photons. The graviton particle is chosen with the right mathematical characteristic to quantize gravity in accordance with quantum field theory and general relativity. These attempts however, fail to account for the origin of space-time curvature. Specifically, how does a graviton ‘produce’ curvature when12propagating from one mass to another? Does the graviton move in an already existing 4D space-time curvature? If it does, how is the space-time produced by the graviton? If not, how is 4D space-time curvature produced? In other words, if the 4D space-time curvature is not caused by the graviton exchanges, then what is the cause? Furthermore, why do the virtual particles of the quantum vacuum not contribute a nearly infinite amount of curvature to the whole universe? After all, the force of gravity is universally attractive. According to quantum field theory, virtual gravitons should exist in huge numbers in the quantum vacuum, and should therefore contribute huge amounts of attractive forces and a large amount of space-time curvature. This infamous question is known as the problem of the cosmological constant. Does graviton exchange processes get affected by high velocity motion (with respect to some other reference frame)? In other words, do the number of gravitons exchanged increase as the velocity of the mass increases, as seems to be required by the special relativistic mass increase with velocity formula? Why does the state of motion of an observer near a gravitational field affect his local 4D space-time curvature? For example, why does an observer in free fall near the earth affect his local space-time conditions in such a way as to match an observer in far space (who lives in flat space-time)? EMQG provides answers for many of these questions. Interested readers should refer to reference 1 (this is quite lengthy) for the answers to these and other questions to do with EMQG. In the interest of being concise, this work presents only the important results from EMQG that are needed to understand the principle of equivalence. One of the key players of gravity in EQMG is the quantum vacuum. Knowledge of this important medium is essential to the understanding of the principle of equivalence. Therefore we now briefly review this subject. 3. THE QUANTUM VACUUM AND IT’S RELATIONSHIP EMQG Philosophers: “Nature abhors a vacuum. ” On pondering the vacuum, one might think that the vacuum is completely devoid of everything. In fact, the physical vacuum is really far from being empty. The recipe for creating a complete vacuum is to remove all matter from an enclosure. However, one would find that this is still not good enough. One must also lower the temperature down to absolute zero in order to remove all thermal electromagnetic radiation. However, Nernst correctly deduced in 1916 (ref. 32) that empty space is still not completely devoid of all radiation after this is done. He predicted that the vacuum is still permanently filled with an electromagnetic field propagating at the speed of light, called the zero-point fluctuations (sometimes called vacuum fluctuations). This was later confirmed by the full quantum field theory developed in the 1920’s and 30’s. Later, with the development of QED, it was realized that all quantum fields should contribute to the vacuum state, like virtual electrons and positron particles, for example.13In order to make a complete vacuum, one must remove all matter from an enclosure. However one would find that this is still not good enough. One must also lower the temperature inside the closure to absolute zero in order to remove all thermal electromagnetic radiation. Nernst correctly deduced in 1916 (ref. 32) that empty space is still not completely devoid of all radiation after this is done. He predicted that the vacuum is still permanently filled with an electromagnetic field propagating at the speed of light, called the zero-point fluctuations (or sometimes called by the generic name ‘vacuum fluctuations’). This result was later confirmed theoretically by the newly developed quantum field theory that was developed in the 1920’s and 30’s. Later with the development of QED (the quantum theory of electrons and photons), it was realized that all quantum fields should contribute to the vacuum state. This means that virtual electrons and positron particles should not be excluded from consideration. These particles possess mass and have multiples of half integer spin (such as the electron), and therefore belong to the generic class of particles known as fermions. We refer to virtual electrons and virtual anti-electrons (positron) particles as virtual fermions. We believe that ultimately all fermions can be broken down to a fundamental entity that is also electrically charged, as well as having half integer spin and mass. According to modern quantum field theory, the perfect vacuum is teeming with activity as all types of quantum virtual particles (and virtual bosons or the force carrying particles) from the various quantum fields appear and disappear spontaneously. These particles are called ‘virtual’ particles because they result from quantum processes that generally have short lifetimes, and are mostly undetectable. One way to look at the existence of the quantum vacuum is to consider that quantum theory forbids the complete absence of propagating fields. This is in accordance with the famous Heisenberg uncertainty principle. In general, it is known that all the possible real particles types (for example electrons, quarks, etc.) will also be present in the quantum vacuum in their virtual particle form. In the QED vacuum, the quantum fermion vacuum is produced from the virtual particle pair creation and annihilation processes that create enormous numbers of virtual electron and virtual positron pairs. We also have in QED the creation of the zero-point-fluctuation (ZPF) of the vacuum consisting of the electromagnetic field or virtual photon particles. Indeed in the standard model, we also find in the vacuum every possible boson particle, such as the gluons, gravitons, etc., and also every possible fermion particle, such as virtual quarks, virtual neutrinos, etc. Some estimates of the quantum vacuum particle density (ref. 11) maintain that the number of vacuum particle is a staggering 1090 particles per cubic meter of empty space! The densest places in the universe, for example neutron stars, contain nowhere near this many particles. For comparison purposes, a neutron star contains about 1045 nucleons per cubic meter, and therefore the quantum vacuum particles has on the order of 1045 times more density than a neutron star!! We believe that the quantum vacuum plays a much more prominent role in physics than is generally believed by most physicists. We maintain that14the effects of the quantum vacuum are present in virtually all physical activity. In fact, Newton’s three laws of motion can be understood to originate directly from the effects of the quantum vacuum (ref. 48). One of the most important experimental evidence for the vacuum particles is the Casimir effect, which we discuss in detail. 3.1 INTRODUCTION TO THE CASIMIR FORCE EFFECT The existence of virtual particles of the quantum vacuum reveals itself in the famous Casimir effect (ref. 49), which is an effect predicted theoretically by the Dutch scientist Hendrik Casimir in 1948. The Casimir effect refers to the tiny attractive force that occurs between two neutral metal plates suspended in a vacuum. He predicted theoretically that the force ‘F’ per unit area ‘A’ for plate separation D is given by: F Ahc D=−p2 4240 … Newton’s per square meter ( Casimir Force ‘F’) (3.1) Casimir obtained this formula by calculating the sum of the quantum-mechanical zero- point energies of the normal modes of the electromagnetic field (virtual photons) between two conductive plates. The origin of this minute force can be traced to the disruption of the normal quantum vacuum virtual photon distribution between two nearby metallic plates as compared to the vacuum state outside the plates. Certain virtual photon wavelengths (and therefore energies) are forbidden to exist between the plates, because these waves do not ‘fit’ between the two plates (which are both at a relative classical electrical potential of zero). This creates a negative pressure due to the unequal energy distribution of virtual particles inside the plates as compared to those outside the plate region. The pressure imbalance can be visualized as causing the two plates to be drawn together by radiation pressure. Note: Even in the vacuum state, the virtual photon particles do carry energy and momentum while they exist. Although the Casimir effect has been attributed to the zero-point fluctuations (ZPF) in the EM field inside the plates, Schwinger showed in the late 70’s that the Casimir effect can also be derived in terms of his source theory (ref. 50), which has no explicit reference to the ZPF of the EM field between the plates. Recently Milonni and Shih have developed a theory of the Casimir force effect, which is totally within the framework of conventional QED (ref. 51). Therefore it seems that it is only a matter of taste whether we attribute the Casimir force effect to the ZPF fields or to the matter fields in vacuum (ref. 52). Recently Lamoreaux made accurate experimental measurements for the first time of the Casimir force existing between two gold-coated quartz surfaces that were spaced on the order of a micrometer apart (ref. 53). Lamoreaux found a force value of about 1 billionth15of a Newton, agreeing with the Casimir theory to within an accuracy of about 5%. More recently, U. Mohideen and A. Roy have made an even more precise measurement in the 0.1 to 0.9 micrometer plate spacing to an accuracy of about 1% (1998, ref. 54). Therefore the experimental reality of this effect is beyond question. Can the vacuum state be disrupted by other physical processes besides the Casimir plates? One might ask what happens to the virtual particles of the quantum vacuum that are subjected to a large gravitational field like the earth? Since the quantum vacuum is composed of virtual fermions (as well as virtual bosons), the conclusion is inescapable: all the virtual fermions possessing mass must be falling (accelerating) on the average towards the earth during their very brief lifetimes . This vacuum state is definitely different from the vacuum of far outer space away from gravitational fields. Yet to our knowledge, no previous authors have acknowledged the existence of this effect, or studied the physical consequences that result from this. It turns out that the free fall state of the virtual, electrically charged fermion particles of the vacuum is actually the root cause of 4D space-time curvature and also leads to a full understanding of the principle of equivalence. In EMQG we fully study the consequences of a falling quantum vacuum in quantum gravity, which does lead to new experimentally testable predictions. The physics of the Casimir force effect implies that the quantum vacuum contains an enormous reservoir of energy (ref. 11). Although in standard quantum field theory the number density of virtual particles is unlimited, some estimates place a high frequency cut- off at the plank scale which is estimated to be a density of 1090 particles per cubic meter (ref. 11)! Generally this energy-density is not available because the energy-density is uniform and it permeates everything. It’s like the situation in the deep ocean, where deep sea fishes easily tolerate the extreme pressures in the abyss, because the pressure inside and outside the fish’s body balance. If a human goes into these depths, a great difference in pressure must be maintained to support atmospheric pressure inside the human body. Some physicists are looking at ways in which this vast energy reservoir can be tapped (ref. 11). Is there any other evidence for the existence of the quantum vacuum? 3.2 EVIDENCE FOR THE EXISTENCE OF VIRTUAL PARTICLES Aside from the Casimir force effect, there is other evidence in theoretical and experimental physics for the existence of virtual particles. We present a very brief review of some theoretical and experimental evidence for the existence of the virtual particles of the quantum vacuum. Some knowledgeable readers may wish to skip this section. (1) The extreme precision in the theoretical calculations of the hyper-fine structure of the energy levels of the hydrogen atom, and the anomalous magnetic moment of the electron and muon are both based on the existence of virtual particles in the framework of QED. These effects have been calculated in QED to a very high precision (approximately 10 decimal places), and these values have also been verified experimentally to an unprecedented accuracy. This indeed is a great achievement for QED, which is essentially16a perturbation theory of the electromagnetic quantum vacuum. Indeed, this is one of physics greatest achievements. (2) Recently, vacuum polarization (the polarization of electron-positron pairs near a real electron particle) has been observed experimentally by a team of physicists led by David Koltick. Vacuum polarization causes a cloud of virtual particles to form around the electron in such a way as to produce an electron charge screening effect. This is because virtual positrons tend to migrate towards the real electron, and the virtual electrons tend to migrate away. A team of physicists fired high-energy particles at electrons, and found that the effect of this cloud of virtual particles was reduced the closer a particle penetrated towards the electron. They reported that the effect of the higher charge for the penetration of the electron cloud with energetic 58 giga-electron volt particles was equivalent to a fine structure constant of 1/129.6. This agreed well with their theoretical prediction of 128.5 of QED. This can be taken as verification of the vacuum polarization effect predicted by QED, and further evidence for the existence of the quantum vacuum. (3) The quantum vacuum explains why cooling alone will never freeze liquid helium. Unless pressure is applied, vacuum energy fluctuations prevent its atoms from getting close enough to trigger solidification. (4) For fluorescent strip lamps, the random energy fluctuations of the virtual particles of the quantum vacuum cause the atoms of mercury, which are in their exited state, to spontaneously emit photons by eventually knocking them out of their unstable energy orbital. In this way, spontaneous emission in an atom can be viewed as being directly caused by the state of the surrounding quantum vacuum. (5) In electronics, there is a limit as to how much a radio signal can be amplified. Random noise signals are always added to the original signal. This is due to the presence of the virtual particles of the quantum vacuum as the real radio photons from the transmitter propagate in space. The vacuum fluctuations add a random noise pattern to the signal by slightly modifying the energy of the propagating radio photons. (6) Recent theoretical and experimental work done in the field of Cavity Quantum Electrodynamics suggests that the orbital electron transition time for excited atoms can be affected by the state of the virtual particles of the quantum vacuum immediately surrounding the excited atom in a cavity, where the size of the cavity modifies the spectrum of the virtual particles. In the weight of all this evidence, only a few physicists doubt the existence of the virtual particles of the quantum vacuum. Yet to us, it seems strange that the quantum vacuum should barely reveal it’s presence to us, and that we only know about it’s existence through rather obscure physical effects like the Casimir force effect and Davies-Unruh effect. This is especially odd considering that the real observable particles in our universe constitute a minute fraction of the total population of virtual particles of the quantum17vacuum in any given point in time (even when considering in the densest places in the universe). We believe that the vacuum particle plays a much more prominent role in physics, and is totally responsible for Newtonian Inertia. Furthermore, it plays a major role in the principle of equivalence. The quantum vacuum mechanism responsible for inertia is the common electrical force process originating from photon exchanges. We describe this process for inertia in more detail in the next section. 4. INTRODUCTION TO QUANTUM INERTIA THEORY “Under the hypothesis that ordinary matter is ultimately made of subelementary constitutive primary charged entities or ‘ partons’ bound in the manner of traditional elementary Plank oscillators, it is shown that a heretofore uninvestigated Lorentz force (specifically, the magnetic component of the Lorentz force) arises in any accelerated reference frame from the interaction of the partons with the vacuum electromagnetic zero-point-field (ZPF). ... The Lorentz force, though originating at the subelementary parton level, appears to produce an opposition to the acceleration of material objects at a macroscopic level having the correct characteristics to account for the property of inertia.” - B. Haisch, A. Rueda, H. E. Puthoff (1994) It has been recently proposed (1994) that Newtonian Inertia is strictly a quantum vacuum phenomenon (as the quote above suggests)! If this is true, then the existence of the quantum vacuum actually reveals it’s presence to us in all daily activities! Unlike the hard- to-measure quantum vacuum effects such as the Casimir force, the presence of the inertial force is universal and it’s presence prevails throughout all of physics. For example, when you step on the gas pedal of your car you feel an inertial force pushing you against the seat. If inertia is truly a quantum vacuum effect, then every time you are accelerating you are witnessing the effects of the quantum vacuum! This is a far cry from the rather obscure and exotic status that the quantum vacuum now holds in current physical thought. In 1994 three physicists, R. Haisch, A. Rueda, and H. Puthoff (ref. 5), were the first to propose such a vacuum theory of inertia (known here as HRP Inertia), in which the quantum vacuum played a central role in acceleration and inertial mass. They proposed that inertia is due to the strictly local electrical force interactions of charged matter particles with the immediate background virtual particles of the quantum vacuum (in particular the virtual photons, or ZPF as the authors called it). They found that inertia is caused by the magnetic component of the Lorentz force, which arises between what the author’s call the charged ‘ parton’ particles in an accelerated reference frame interacting with the background quantum vacuum virtual particles. The sum of all these tiny forces in this process is the source of the resistance force opposing accelerated motion in Newton’s F=MA. The ‘ parton’ is a term that Richard Feynman coined for the constituents of the nuclear particles such as the proton and neutron. Partons are now generally called quarks. We have found it necessary to make a small modification to HRP Inertia theory as a result of our investigation into the principle of equivalence. Our modified version of HRP inertia18is called “Quantum Inertia” (or QI). This theory also resolves the long outstanding problems and paradoxes of accelerated motion introduced by Mach’s principle, by suggesting that the vacuum particles themselves serve as Mach’s universal reference frame (for acceleration only), without violating the principle of relativity of constant velocity motion. In other words, our universe offers no observable reference frame to gauge inertial frames (a non-accelerated frame, where Newton’s laws of inertia is valid), because the quantum vacuum offers no means to determine absolute constant velocity motion. However for accelerated motion, the quantum vacuum plays a very important role by offering a resistance to the acceleration of a mass, which results in an opposing inertial force. Thus, the very existence of inertial force reveals the absolute value of the acceleration with respect to the net statistical average acceleration of the virtual particles of the quantum vacuum. In the past there have been various clues to the importance of the state of the virtual particles of the quantum vacuum, with respect to the accelerated motion of real charged particles. One example is the so-called Davies-Unruh effect (ref. 15), where uniform and linearly accelerated charged particles in the vacuum are immersed in a heat bath, with a temperature proportional to acceleration (with the scale of the quantum heat effects being very low). However, the work of reference 5 is the first place we have clearly seen the identification of inertial forces as the direct consequence of the interactions of real matter particles with the virtual particles of the quantum vacuum. It has even been suggested that the virtual particles of the quantum vacuum are somehow involved in gravitational interactions, a central concept of EMQG. The prominent Russian physicist A. Sakharov proposed in 1968 (ref. 16) that Newtonian gravity could be interpreted as a van der Waals type of force induced by the electromagnetic fluctuations of the virtual particles of the quantum vacuum. Sakharov visualized ordinary neutral matter as a collection of electromagnetically, interacting polarizable particles made of charged point-mass subparticles ( partons). He associated the Newtonian gravitational field with the Van Der Waals force present in neutral matter, where the long-range radiation fields are generated by the parton ‘ Zitterbewegung’. Sakharov went on to develop what he called the ‘metric elasticity’ concept, where space-time is somehow identified with the ‘hydrodynamic elasticity’ of the vacuum. However, he did not understand the important clues offered in understanding the equivalence principle, nor the important role that the quantum vacuum plays in inertia and in Mach’s principle. We will see that the quantum vacuum also make it’s presence felt in a very important way in all gravitational interactions. After Sakharov there has been further hints that the quantum vacuum is involved in gravitational physics. In 1974 Hawkings (ref. 17) announced that black holes are not completely black. Black holes emit an outgoing thermal flux of radiation due to gravitational interactions of the black hole with the virtual particle pairs created in the quantum vacuum near the event horizon. At first sight, the emission of thermal radiation19from a black hole seems paradoxical (since nothing can escape from the event horizon). However, the spontaneous creation of virtual particle and anti-particle pairs in the quantum vacuum near the event horizon can be used to explain this effect (ref. 18). Heuristically, one can imagine that the virtual particle pairs (created with wavelength λ are approximately equal to the size of the black hole) ‘tunnel’ out of the event horizon. For a virtual particle with a wavelength comparable to the size of the hole, strong tidal forces operate to prevent re-annihilation. One virtual particle escapes to infinity with positive energy to contribute to the Hawking radiation, while the corresponding antiparticle enters the black hole to be trapped forever by the deep gravitational potential. Thus, the quantum vacuum is important in order to properly understand the Hawking radiation. According to Quantum Inertia theory, the property which Newton called the inertial mass of an object, is caused by the vacuum resistance to acceleration of all the individual, electrically charged masseon particles (section 6) that make up the mass. This resistance force is caused by the electromagnetic force interaction (where the details of this process are still unknown at this time) occurring between the electrically charged virtual masseon/anti- masseon particle pairs created in the surrounding quantum vacuum (section 6.1), and all the real masseons particles making up the accelerated mass. Therefore inertia originates in the photon exchanges with the electrically charged virtual masseon particles of the quantum vacuum. The total inertial force Fi of a mass is simply the sum of all the little forces fp contributed by each of the individual masseons, where the sum is: Fi = (Σ fp) = MA (Newton’s law of inertia). Figure 1 shows a schematic diagram of this process for a mass accelerated by a rocket motor.. Any physicist that believes in the existence of the virtual particles of the quantum vacuum and accepts the fact that many virtual particles carry mass (in the form of virtual fermions such as virtual electrons and virtual quarks), will have no trouble in believing that virtual fermion particles are falling in the presence of a large gravitational mass like the earth. Figures 4 and 5 show the falling state of the quantum vacuum, and it’s effect on test masses. Yet no one has examined the full ramifications of this phenomena, which is extremely important to EQMG theory. We believe the existence of the downward accelerating virtual particles (which are accelerating during their very brief lifetimes) under the action of a large gravitational field turns out to be the missing link between inertia and gravity. It leads us more or less directly towards a full understanding of the principle of equivalence. Although the quantum vacuum has been studied in detail in the past, to our knowledge no one has examined the direct consequences of a quantum vacuum in a state of free-fall near the earth. Before we continue studying these concepts, we present a rather brief review of the important results from general relativity.205. GENERAL RELATIVIT Y, ACCELERATION, GRAVITY AND CA “The general laws of physics (and gravitation) are to be expressed by equations which hold good for all systems of coordinates.” - Albert Einstein From the perspective of EMQG, Einstein’s gravitational field equations are a set of observer dependent equations, for observers that are subjected to gravity and/or to acceleration. These equations are based on measurable relativistic 4D space-time. The core of Einstein’s theory is the principle of equivalence and the principle of general covariance, which allows an observer in any state of motion (and coordinate system) to describe gravity and acceleration. We very briefly review these basic postulates, and important concepts of Einstein’s general theory of relativity. POSTULATES OF GENERAL RELATIVITY General relativity is a classical field theory founded on all the postulates and results of special relativity. It is also based on Newtonian inertia and Newton’s laws of motion, as well as on the following new postulates introduced by Einstein: (1) PRINCIPLE OF EQUIVALENCE (STRONG) - The results of any given physical experiment will be precisely identical for an accelerated observer in free space as it is for a non-accelerated observer in a perfectly uniform gravitational field. A weaker form of this postulate states that: objects of different mass fall at the same rate of acceleration in a uniform gravity field. (2) PRINCIPLE OF COVARIANCE - The general laws of physics can be expressed in a form that is independent of the choice of space-time coordinates and the state of motion of an observer. As a consequence of postulate 1, the inertial mass of an object is equivalent to it's gravitational mass. Einstein uses this principle to encompass gravity and inertia into his single framework of general relativity in the form of a metric theory of acceleration and gravity, based on quasi- Riemann geometry (where time also enters as a coordinate, unlike pure Riemann geometry). These postulates, and the additional assumption that when gravitational fields are present nearby, space-time takes the form of a quasi- Riemannian manifold endowed with a metric curvature of the form ds2 = gik dxi dxj, led Einstein to discover his famous gravitational field equations given below: R gRG cT ik ik ik−=1 28 2p … Einstein’s Gravitational Field Equations (5.1)21where, gik is the metric tensor, Rik is the covariant Riemann curvature tensor. The left-hand side of the above equation is called the Einstein tensor or Gik, which is the mathematical statement of space-time curvature that is reference frame independent, and generally covariant. The right hand side Tik is the stress-energy tensor which is the mathematical statement of the special relativistic treatment of mass-energy density, G is Newton’s gravitational constant, and c the velocity of light. Einstein’s law of gravitation ( eq 5.1) cannot be derived from any ‘rigorous’ proof. The famous physicist S. Chandrasekhar writes in regards to this ( ref 37): “... It seems to this writer that in fact no such derivation exists and that, at the present time, no such can be given. ... It is the object of this paper to show how a mixture of physical reasonableness, mathematical simplicity, and aesthetic sensibility lead one, more or less uniquely, to Einstein’s field equations.” In Einstein’s field equation ( eq. 5.1) the principle of equivalence (in its strong form) is incorporated in the above framework by the assertion that all accelerations are caused by either gravitational or inertial forces, and are metrical in nature. In other words, the space- time is equivalent in inertial frame of a rocket at 1g and gravitational field on the earth (in a small vicinity). More precisely, the presence of acceleration caused by either an inertial force or a gravitational field modifies the geometry of space-time such that it is a quasi- Riemannian manifold endowed with a metric. Furthermore, point particles move in gravitational fields along geodesic paths governed by the equation: dx dsdx dsdx dsi jkij k 2 20 + = Γ ... Equation for the geodesics (5.2) The most striking consequence of general relativity is the existence of curved 4D space- time specified by the metric tensor gik. We will see that in EMQG theory, the meaning of the geodesic is very simple; it is the path taken by light or (force free) point particles through the falling, electrically charged, virtual particles undergoing acceleration in the absence of any other external forces, that maintains a relative, average acceleration of zero with the vacuum particles. In other words for light traveling through the falling quantum vacuum, the photons frequently scatter with the electrically charged virtual particles of the quantum vacuum causing a deflected path. We will see that curvature can be completely understood at the particle level, as a pure vacuum-particle process. Furthermore, we will see that the principle of equivalence is a pure particle interaction process, and not a fundamental law of nature. Before we can show this, we must carefully review the principle of equivalence from the context of general relativity theory.225.1 INTRODUCTION TO THE PRINCIPLE OF EQUIVALENCE “I have never been able to understand this principle (principle of equivalence) ... I suggest that it be now buried with appropriate honors.” - Synge: Relativity- The General Theory Again we would like to emphasize that Einstein did not explain the origin of inertia in general relativity. Instead he retained the classical Newtonian theory of inertia. Inertia is described by Newton in his famous law: F=MA; which states that an object resists being accelerated by a force (F). In other words, a force is required to accelerate an object of mass (M) to an acceleration (A) in order to overcome the inertial back-acting force. Since acceleration is a form of motion, it would seem that a reference frame is required in order to gauge this motion. But this is not the case in Newtonian physics. All observers agree as to which frame is actually accelerating (without observing the actual motions of other observers), by finding out which frames has a force associated with it. Only non- accelerated frames are relative. If the observers are placed in three rockets with no windows available, and only one observer has the rocket motor running, it is easy for each of the observer to determine if it is their rocket engine that is turned on. The observer simply looks for the presence of a force on his body. Two observers will feel no forces, and the third will. However, the observers who’s rocket engine is turned on might conclude that one of the observers who’s rocket engine is turned he is accelerating by measure x and t through relative measurements of space and time. So constant velocities are always relative. Accelerations may be absolute (because we do not need to know x or t to determine a=d2x/dt2, and can determine acceleration by measuring F and m, where a=F/m), or acceleration may be relative (if we decide to measure a by direct determination of x and t) in special relativity. To us this seems very paradoxical. Einstein did not solve this anomaly (which relates to Mach’s principle), nor did he provide a reason why the inertial and gravitation masses of an object are equal. This also still remains as a postulate in his theory. It is widely known that the principle of equivalence has been tested to great accuracy. The equivalence of inertia and gravitational mass has been verified to an accuracy of one part in about 10-15 (ref 24). Einstein's general theory of relativity is considered a "classical" theory, because matter, space, and time are treated as continuous classical variables. It is known however, that matter is made of discrete particles, and that forces are caused by particle exchanges as described by quantum field theory. A more complete theory of gravity should encompass a detailed quantum process for gravity involving particle interactions only. In the next section we introduce another important player in EMQG, the masseon particle. This particle is required in order to understand the principle of equivalence. The masseon turns out to be the fundamental particle that makes up all fermions.236. PHYSICAL PROPERTIES OF THE GRAVITON AND THE MASSEON The theory that best describes the quantization of the electromagnetic force field is called Quantum Electrodynamics (QED). Here the charged particles (electrons, positrons) act upon each other through the exchange of force particles, which are called photons. The photons represent the quantization of the classical electromagnetic field. In classical electromagnetic theory, the force due to two charged particles decreases with the inverse square of their separation distance (Coulomb’s inverse square law: F = kq 1q2/r2, where k is a constant, q 1 and q 2 are the charges, and r is the distance of separation). QED accounts for this inverse square law by postulating the exchange of photons between the charged particles. The number of photons emitted and absorbed by a given charge (per unit of time) is fixed and is called the charge of the particle. Thus, if the charge doubles, the force doubles because twice as many photons are exchanged during the force interaction. This force interaction process causes the affected particles to accelerate either towards or away from each other depending on if the charge is positive or negative (because positive and negative charges transmit photons with slightly different wave functions that induce attraction or repulsive accelerations). The strength of the electromagnetic force varies as the inverse square of the distance of separation between the charges in the following way: each charge sends and receives photons from every direction. But, the number of photons per unit area, emitted or received, decreases by the factor 1/4 πr2 (the surface area of a sphere) at a distance ‘r’ due to the photon emission pattern spreading in all directions. Thus, if the distance doubles, the number of photons exchanged decreases by a factor of four. Imagine that an electron particle is at the center of a sphere sending out virtual photons in all directions. Imagine that another electron is on the surface of a sphere at a distance ‘r’ from the emitter, which absorbs some of these photons. The absorption of these photons causes an outward acceleration, and thus a repulsive force. If the charge is doubled on the central electron, there is twice as many photons appearing at the surface of the sphere, and twice the force acting on the other electron. This accounts for the linear product of charge terms in the numerator of the inverse square law. In QED, photons do not interact with each other (through force exchanges). As a result, in-going and out-going photons do not affect each other during the exchange process, thorugh the exchange of other force particles. For gravitational forces, it is experimentally observed that the force originating from two particles possessing mass decreases with the inverse square of their separation distance, and is given by Newton’s inverse square law: F = Gm 1m2 / r2, where G is the gravitational constant, m 1 and m 2 are the masses, and r is the distance of separation. We have seen that the two force laws are very similar in form. QED theory accounts for Coulomb’s law by the photon exchange process. Following the lead from the highly successful QED, EMQG replaces the concept of electrical ‘charges’ exchanging ‘photons’ with the idea that ‘mass charges’ exchange gravitons. Hence, gravitational mass at a fundamental level is simply24the ability to emit or absorb gravitons, and pure low-level gravitational mass is interpreted as ‘mass charge’ of a fermion. For gravity there are gravitons instead of photons, which are the force exchange particles of gravity. Like charge, it is the property called mass-charge that determines the number of exchange gravitons. The larger the mass, the greater the number of gravitons exchanged. Like electromagnetism, the strength of the gravity force decreases with the inverse square of the distance. This conceptual framework for quantum gravity has been around for some time now, but how are we to merge these simple ideas to be compatible with the framework of general relativity? We must be able to explain the Einstein's Principle of Equivalence and the physical connection between inertia, gravity, and curved space-time all within the general framework of graviton particle exchange. General Relativity is based on the idea that the forces experienced in a gravitational field and the forces due to acceleration are equivalent, and both are due to the space-time curvature. In classical electromagnetism, if a charged particle is accelerated towards an opposite charged particle, the rate of acceleration depends on the electrical charge value. If the charge is doubled, the force doubles, and the rate of acceleration is doubled. If quantum gravity were to work in the exact same way, we would expect that the rate of acceleration of a mass near the earth would double if the mass doubles. The reason for this expectation is that the exchange process for gravitons should be very similar to electromagnetism. In other words, if the ‘mass-charge’ is doubled, the gravitational force is doubled. The only difference between the two forces is that gravity is a lot weaker by a factor of about 10-40. The weakness of the gravitational forces might be attributed either to the very small interaction cross-section of the graviton particle as compared to the photon particle, or to a very weak coupling constant (the absorption of a single graviton causing a minute amount of acceleration), or both. Unfortunately, if the graviton exchange process worked exactly like QED, it would not reproduce the known nature of gravity. First, there is the problem of variation of mass with velocity as described by special relativity as m= m 0 (1-v2/c2)-1/2 . At face value, this would mean that the number of gravitons exchanged depends on velocity of the gravitational mass, which does not easily fit into the framework of a QED type approach to quantum gravity. Secondly, if two masses are sitting on a table with mass ‘M’ and mass ‘2M’, the forces against the tabletop varies with the mass, just as you would expect in a QED-like exchange of graviton particles. If the mass doubles, the force on the table doubles. Yet, the rate of acceleration is the same for these two masses in free fall. Why? Since twice the number of gravitons is exchanged under mass ‘2M’ in free, you would expect twice the force, and therefore mass ‘2M’ should arrive early. Matter has inertia, and this complicates everything. In almost all quantum gravity theories inertia appears as a separate process that is ‘tacked’ on in an ad hoc manner. As we said, the principle of equivalence merely raises this relationship between inertia and gravitation to the status of a postulate as in Einstein’s theory of general relativity.25All test masses accelerate at the same rate (g=9.8 m/sec2 on the earth) no matter what the value of the test mass is. This is a direct consequence of the principle of equivalence. Mathematically, this follows from Newton's two different force laws: inertia and gravity as follows: Fi = ma ..... (Inertial force) (6. 1) Fg = GmM / r2 ..... (Gravitational force) (6. 2) If in free fall, an object (mass m) in the presence of the earth's pull (mass M) is force free, then Fi = Fg (since no other forces are present). Note that the same mass value ‘m’ appears in the two mass definition formulas for equations 6.1 and 6.2. Therefore, ma = GmM/ r2 or a = GM/ r2 for the mass, independent of the value of the mass m. Thus we see that the rate of acceleration does not depend on the test mass m. All test masses accelerate at the same rate. Thus, inertia and gravity are intimately connected in a deep way because the measure of mass m is the same for acceleration as for gravity. What is mass? In EMQG, gravitational mass originates from a low-level graviton exchange process that results from ‘mass charge’, where there the emission rate is constant. In fact, mass is quantized in exactly the same way as electric charge in QED. (There exists a fundamental unit of mass charge that is carried by the masseon particle, the lowest quanta of mass). The graviton ultimately responsible for the gravitational force. We have seen that the graviton particle is the fundamental boson of the pure gravitational force. In EMQG, the graviton is very similar to the photon in physical characteristics. Both particles move at the speed of light, and have the same spin 1 (contrary to conventional wisdom). Both bosons have rest mass of zero. The following table compares the properties of the graviton with the photon particle: TABLE #1 PHYSICAL PROPERTIES OF THE PHOTON AND GRAVITON NAME BOSON PROPERTIES Symbol Spin Rest Mass Electrical/Mass Charge Photon Particle γ 1 0 0 0 Graviton Particle G 1 (not 2) 0 0 0 Gravitons are emitted and absorbed by the ‘ masseon’ particle, which is the most elementary form of matter or anti-matter. The masseon carries the lowest possible quanta of gravitational ‘mass charge’. This elementary particle is called the masseon particle (and also comes in anti- masseon varieties, which is the corresponding anti-particle). The masseon is postulated to be the smallest fundamental mass particle and readily combines26with other masseons through a new, hypothetical force coupling, which we call the ‘primal force’. Presumably, the primal force comes in positive and negative ‘primal charge’ types. The proposed mediator of this force is called the ‘ primon’ particle. Since the masseon has not yet been detected by particle collisions, we can safely assume that the primal force is very strong. It is not necessary to understand the exact nature of the primal force to achieve and understanding of the principle of equivalence. Suffice it to say that the primal force binds together masseon particles to make all the known fermion particles of the standard model. The masseon carries the lowest possible quanta of positive gravitational ‘mass charge’. Gravitational ‘mass charge’ is defined as the fixed rate of emission of graviton particles in close analogy to electric charge in QED (actually the probability of emission/absorption). Recall that the graviton is the vector boson of the pure gravitational force. Gravitational ‘mass charge’ is a fixed constant in EMQG, and is analogous to the fixed electrical charge concept. Gravitational ‘mass charge’ is not governed by the ordinary physical laws of observable mass, which includes Einstein’s energy and mass-velocity variation formulae: E=mc2 and m = m 0 (1 - v2/c2)-1/2 . Gravitational ‘mass charge’ is the low-level mass of a particle, should not be confused with the ordinary observable inertial or observable gravitational mass properties of a particle. These are not the same. It will be assumed that when we use the term ‘low level mass charge’ in this paper, we are talking about the low level gravitational ‘mass charge’ property of a particle, and the associated graviton exchange process. Gravitational mass is the observable mass property that is measured in actual experiments. Another important property exhibited by the graviton particle is the principle of superposition . This property works exactly the same way as it does for photons. The action of the gravitons originating from several different sources acts to yield a net acceleration vector sum that determines the state of acceleration for a receiving particle. EMQG treats graviton exchanges by the same successful methods developed for the behavior of photons in QED. The dimensionless coupling constant that governs the graviton exchange process is what we call ‘ β‘ in close analogy with the dimensionless coupling constant ‘ α‘ in QED, where β ≈ 10-40 α, tremendously weaker than electrical forces. As we said, the masseon particle is the fundamental fermion that carries both electrical charge and mass charge. It is very similar to the electron in physical characteristics, where the electron is a composite of many masseons. The following tables compares the properties of the masseon particle with the electron particle:27TABLE #2 PHYSICAL PROPERTIES OF THE ELECTRON AND MASSEON NAME FERMION PROPERTIES Symbol Spin Rest Mass Electrical/Mass Charge Electron particle e 1/2 5.11x10-4 Gev/c2-1 +k Masseon particle m 1/2 unknown + k or -k * +1 * The constant k less than 1/3 (quark electrical charge), and who’s value is not known at this time. Masseons simultaneously carry a positive gravitational ‘mass charge’, and either a positive or negative electrical charge (defined exactly as in QED). Therefore, masseons also exchange photons with other masseon particles. It is important to note that the graviton exchange process is responsible for the low-level gravitational interaction only, which is not directly accessible to our measurements (as we will see later), and is also masked by the presence of the electromagnetic force component in all gravitational measurements. Masseons are fermions with half integer spin, which behave according to the rules of quantum field theory. Gravitons have a spin of one (not spin two , as is commonly thought), and travel at the speed of light. This paper addresses the gravitational and electromagnetic force interactions only, and the strong and weak nuclear forces are ignored here. ( Masseons also carry the strong and weak ‘nuclear charge’ as well, which is outside the scope of this work). Anti- masseons carry the lowest quanta of negative gravitational ‘mass charge’. Anti- masseons also carry either positive or negative electrical charge, with electrical charge being defined according to QED. An anti- masseon is always created with ordinary masseon in a particle pair, as required by quantum field theory. The negative energy, or the antimatter solution to the famous Dirac equation for fermions in quantum field theory is also the negative ‘mass charge’ solution for that fermion . Therefore we propose that the symmetry between matter and anti-matter is perfect in Dirac’s equation. Every physical parameter is reversed in the anti-matter solution of Diarc’s equation, where previously mass remained positive, and the other physical quantities such as, energy, electrical charge, spin, etc. were reversed when going from matter to anti-matter. Ordinarily the standard model requires that the mass of any anti-particle is always positive, in order to comply with the Einstein principle of equivalence, or M i= Mg. In EMQG, the principle of equivalence is not taken to be an absolute law of nature, and is definitely grossly violated for anti-particles (the reasons will become clear in section 8). The anti- particles can have both positive inertia mass and negative gravitational mass, or Mi = -M g. This violates equivalence and general relativity, and so is an experimentally testable consequence of EMQG theory (section 10, and ref. 1). There exists a beautiful symmetry between EMQG and QED for gravitational and electromagnetic forces. The masseon-graviton interaction becomes almost identical to the electron-photon interaction. There are only two differences between these forces. First, the ratio of the strength of the electromagnetic over the gravitational forces is on the order28of 1040 for an electron. Secondly, there exists a difference in the nature of attraction and repulsion between positive and negative gravitational ‘mass-charges’ (as detailed in the table #3 and 4 below). The reason for this slight asymmetry is still an unresolved problem in EMQG theory. We would normally expect that a perfect symmetry should exist between the photon and the graviton. In QED, the quantum vacuum, as a whole, is electrically neutral over macroscopic distance scales, because the virtual electron and positron (negative electron) particles are always created in particle pairs with equal numbers of positive and negative electrical charge. In EMQG, the quantum vacuum is also gravitationally ‘neutral’ for the same reason . At any given instant of time, there is a 50-50 mixture of positive and negative virtual gravitational ‘mass charge’, which is carried by the virtual masseon and anti- masseon pairs. These masseon / anti- masseon particle pairs are created with equal and opposite gravitational ‘mass charge’. This is the reason why the cosmological constant is zero (or very close to zero, ref. 1). Half the graviton exchanges between quantum vacuum particles result in attraction, while the other half result in gravitational repulsion. The result is a neutral masseon vacuum. To see how this works, we will closely examine how masseons and anti- masseons interact. The following tables compares the fundamental attractive and repulsive characteristics of two electrical charged electrons or anti-electrons, and two masseons or anti- masseons possessing ‘mass charge’, all undergoing electrical or gravitational force interactions: TABLE #3 EMQG MASSEON - ANTI-MASSEON GRAVITON EXCHANGE (DESTINATION) MASSEON ANTI-MASSEON (SOURCE) MASSEON attract attract ANTI-MASSEON repel repel TABLE #4 QED ELECTRON - ANTI-ELECTRON PHOTON EXCHANGE (DESTINATION) ELECTRON ANTI-ELECTRON (SOURCE) ELECTRON repel attract ANTI-ELECTRON attract repel29In QED, if the source particle is an electron, it emits photons whose wave function induce repulsion when absorbed by the destination electron, and induces attraction when absorbed by a destination anti-electron. Similarly, if the source is an anti-electron, it emits photons whose wave function induce attraction when absorbed by the destination electron, and induces repulsion when absorbed by a destination anti-electron. In EMQG, if the source particle is a masseon, it emits gravitons whose wave function induces attraction when absorbed by a destination masseon, and induces attraction when absorbed by a destination anti- masseon. If the source is an anti- masseon, it emits gravitons whose wave function induces repulsion when absorbed by a destination masseon, and induces repulsion when absorbed by a destination anti- masseon. This subtle difference in the nature of graviton exchange process is responsible for some major differences in the way that low-level gravitational ‘mass charge’ and the electrical charges operate. It is convenient to think of the photon as occurring in photon and anti-photon varieties (the photon is its own anti-particle). Similarly, the graviton comes in graviton and anti- graviton varieties. Thus, we can say that the masseons emit gravitons, and anti- masseons emit anti-gravitons. The absorption of a graviton by either a masseon or anti- masseon induces attraction. The absorption of an anti-graviton by either a masseon or anti- masseon induces repulsion. Similarly, we can say the electrons emit photons and anti-electrons emit anti-photons. The absorption of a photon by an electron induces repulsion, and the absorption of a photon by an anti-electron induces attraction. The absorption of an anti- photon by an electron induces attraction, and the absorption of an anti-photon by an anti- electron induces repulsion. Now that we have characterized the masseons and gravitons, let us reexamine the nature of the quantum vacuum from the perspective of EMQG. 6.1 THE QUANTUM VACUUM AND VIRTUAL MASSEON PARTICLES In this paper we talk extensively about virtual particles of the quantum vacuum. Precisely what kinds of virtual particles are present in the quantum vacuum? In QED, we talk about the vacuum being populated with virtual electrons and anti-electrons (and virtual muons and tauons in the second and third generation of elementary particles), along with the associated virtual photons. In the standard model of particle physics the quantum vacuum consists of all varieties of virtual fermion and virtual boson particles representing the known virtual matter and virtual force particles, respectively. These include fermions such as virtual electrons, virtual quarks, virtual neutrinos. This also includes bosons such as the virtual photons, virtual gluons, and virtual W and Z bosons. In EMQG, we restrict ourselves to the study of gravity and electromagnetism. Therefore, the EMQG quantum vacuum consists of the virtual masseons and virtual anti- masseons, and the associated virtual photons and virtual graviton particles (virtual masseon can combine to form virtual electrons, etc.). Recall that ordinary matter consists only of real masseons bound together in certain combinations to form the familiar elementary particles.30We now ask how the virtual electrons and positrons of the QED vacuum behave in the vicinity of a real electrical charge. We want to compare this behavior with virtual masseon and virtual anti- masseon near a real and very large mass-charge like the earth. First, we review how the QED quantum vacuum is affected by the introduction of a real negative electrical charge. According to QED, the nearby virtual particle pairs become polarized around the central charge. This means that the virtual electrons of the quantum vacuum are repelled away from the central negative charge, while the virtual positrons are crowded towards the central negative charge. For real electrons this process is called ‘vacuum polarization’ , and produces the well known phenomena of electric charge screening. Electric charge screening reduces the apparent electric charge of a real electron, when measured over relatively long distances away from the charge. According to QED, each electron is surrounded by a cloud of virtual particles that winks in and out of existence in pairs (lasting tiny fractions of a second), and this cloud is always present and acts like an electrical shield against the real charge of the electron. Recently, a team of physicists led by D. Koltick of Purdue University in Indiana reported charge screening for an electron, experimentally, at the KEK collider (ref. 33). They fired high- energy particles at electrons and found that the charge screening effect of this cloud of virtual particles was reduced the closer a high-energy charged particle penetrated towards the electron. They report that the effect of the higher charge for an electron, that was penetrated by particles accelerated to an energy 58 giga-electron volts, was equivalent experimentally to a fine structure constant of 1/129.6. This agreed well with the theoretical prediction of 1/128.5, based on QED. Now we study how the EMQG quantum vacuum is affected by the introduction of a large mass like the earth. According to EMQG, the quantum vacuum virtual masseon particle pairs are not polarized near a large mass, as we found for electrons (as can be seen from table #3 above). The virtual masseon and anti- masseon pairs are both attracted towards the mass. It is this lack of vacuum polarization that results in the main difference between electromagnetism and gravity. This is an extremely important result of EMQG theory. The earth does not produce vacuum polarization of virtual particles (as far as mass charge is concerned). In large gravitational fields, all the virtual masseon and virtual anti- masseon particles of the vacuum have more or less the same net average statistical acceleration directed towards a large mass, and produces a net inward flux of quantum vacuum virtual masseon/anti- masseon particles (acceleration vectors only) that can, and does, affect other masses placed nearby. In contrast to this, a large electrically charged object does produce vacuum polarization in QED; where the positive and negative electric charges accelerates towards and away, respectively from the charged object. Hence in QED there is no energy contribution to other electrical test charges placed nearby (from the vacuum particles only), because the charged vacuum particles contributes equal amounts of force contributions from both directions, i.e. towards and away from the large charge.31On the earth, the lack of vacuum polarization leads us towards the weak equivalence principle. This is because both the electrically charged positive and negative virtual masseons and anti- masseon particles can act in unison against the electrically charged particles that make up a test mass dropped on the earth. Had there been vacuum polarization for the masseons as far as mass-charge was concerned, the virtual and anti- virtual vacuum particles would accelerate in the two opposite directions from the earth, and hence no net vacuum force would result against a test mass from the vacuum. We have introduced all the major players in EMQG, and we are now ready to state our basic postulates. At this time, these postulates cannot be derived from more basic principles. 7. VIRTUAL PARTICLES NEAR SPHERICAL MASS LIKE THE EARTH We now have enough background material to determine the quantum nature of the gravitational field for a spherically symmetrical, large mass like the earth. In general relativity, Einstein’s field equation has been solved exactly for this special case (ref. 39), and the solution is called the Schwarzchild metric (after the discoverer) and is given by: dsdr GM rccdtGM rcrd22 22 2 22 2 1212= −−−+ ( ) Ω where d Ω2 = dθ2 + sin2 θ dφ2 (7. 1) This is a complete mathematical description of the 4D space-time curvature according to general relativity, and also describes the motion of mass near the earth. The metric is given in spherical coordinates. This equation describes the path that light (or matter with additional considerations) takes through the curved 4 D space-time. We will find that this solution is a very good approximation to the quantum gravitational field. There are, however, hidden quantum processes involving the virtual particles in EMQG that are responsible for this curvature, and for the very tiny inaccuracy of this metric due to a slight violation of the principle of equivalence (section 8). We will also see that the metric has a limited range of applicability, and cannot be used on the individual particle scale as r à 0 (due to quantization). A large spherical mass turns out to have a very simple motion associated with the electrically charged virtual particles that makes up the surrounding quantum vacuum. The normal vacuum condition of the virtual particles near a massive spherical distribution of matter is disrupted, compared to the similar vacuum state in far empty space. Surrounding a large spherically symmetrical mass like the earth, the virtual particles created in the quantum vacuum have a net average acceleration vector that is directed downward towards the earth's center along radius vectors. ( Note: We are ignoring the mutual interactions of the vacuum particles, which are why this statement must remain as a statistical statement of an average virtual particle ). The cause of this downward acceleration of the vacuum is the direct graviton exchanges between the real masseons that constitute the earth and the virtual masseon particles of the32quantum vacuum, which propagate at the speed of light. At any one instant, the vacuum particles have random velocity vectors, which point in all directions (even including the up direction). However, the acceleration vectors are generally pointing in the downward direction, on the whole. The closer the virtual particles are to the earth, the greater the acceleration, as you would expect from the inverse square law. On the quantum scale is due to the geometric spreading of the graviton flux (ref. 1), which gives less graviton flux at greater distances. We conclude that the net average acceleration of the falling virtual particles is directed towards the earth’s center, and the magnitude of the acceleration vector varies with the inverse square of the height ‘r’. This accelerated vacuum ‘flow’, which permeates all matter on the earth, plays an important role in the dynamics of gravity in EMQG. It also naturally ties in with the problem of inertia and the equivalence principle. We suggest that the average net acceleration vector of the electrically charged vacuum particles, which varies in magnitude and direction at different points near the earth, interacts with test masses and with light in such a way as to be equivalent to Schwarzschild 4D space-time metric concept above. This is because the average net acceleration vector of the charged virtual particles at each point in space surrounding the mass, guides the motion of the real photons electromagnetically, and also guides the motion of real electrically charged masseon particles that make up a mass. From simple considerations of the inverse square law of gravity, we easily obtain the general motion of the falling vacuum particles near the earth, and is given by: The magnitude and direction of the acceleration of an average, falling, electrically charged virtual masseon particle of the surrounding quantum vacuum at point ‘r’ above the earth of mass M e is given by: r raGM rre =− 3(7.2) where rr is the radius vector and ra is the acceleration vector. The direction of the virtual particle acceleration is along the earth’s radius vectors. It is also possible to calculate this from the basic EMQG field equations for a general, slow mass (ref. 1). However, this complex calculation of the state of motion of the vacuum particles is not required in the case of large spherically symmetrical masses like the earth, because of the symmetry and simplicity of the inverse square law for a spherically large mass like the earth. When a small test mass moves through the space surrounding the earth, the electrical interactions between the real charged masseon particles in the mass with respect to the virtual charged particles quantum vacuum dominates over the pure graviton exchange process between the test mass and the earth. This electrical component plays the major role in the dynamics of motion of a nearby test mass. The real masseon particles that constitute the mass of the earth exchanges gravitons with the virtual masseons of the33quantum vacuum, causing a downward acceleration of the quantum vacuum with a magnitude of 1g at the earth’s surface. If we now introduce a test mass near the earth, the real masseons making up the test mass will fall at the same average rate as that of the net statistical average of virtual particles of the vacuum. This is due to the vastly stronger electrical forces acting between the electrically charged virtual masseons of the surrounding vacuum and the real, electrically charged masseons of the test mass. In some sense, the vacuum acts like a much stronger constraining force that guides the motion of a masseon so that it follows a ‘geodesic’ path through space. Note: We have not proved that equation (7.2) is correct. Instead, it is based on the general observations of the motion of a test mass falling near the earth based on the inverse sqaure law, where the electrical forces from the falling vacuum constrains the fall rate of a point-like test mass.. However, this equation can be derived from first principles using the semi-classical EMQG equations of motion (ref. 1). Because equation 7.2 involves space and time, a small correction (for the earth) needs to be applied to compensate for space-time effects as well, since equation 7.2 involves distance and time (ref. 1). To fully understand the gravitational field around a spherically symmetrical massive object like the earth, the motion of light near the earth must also be accounted for in EMQG. We will find that the altered behavior of light near a massive object drastically modifies the nature of space-time in equation. This is because the acceleration a=dv/ dt, and the velocity v=dx/ dt involves distance and time measurements. We will return to this important issue of the meaning of curved 4D space-time in section 9, after deriving the weak principle of equivalence for the motion of matter, from the fundamental postulates of EMQG. 8. DERIVATION OF THE WEAK PRINCIPLE OF EQUIVALENCE “The principle of equivalence performed the essential office of midwife at the birth of general relativity, but, as Einstein remarked, the infant would never have got beyond its long clothes had it not been for Minkowski’s concept [of space-time geometry]. I suggest that the midwife be now buried with appropriate honors and the facts of absolute space-time faced.” - Synge We now derive the principle of equivalence from the motion of the virtual particles in the vacuum surrounding the earth. As we stated, the equality of inertial and gravitational mass is known to be true only through physical observation and through experience. Yet until recently, it has been generally believed that it cannot be derived from more fundamental principles. However is the equivalence principle truly an exact statement of the nature of physical reality?348.1 INTRODUCTION Again, there are two main formulations of the principle of equivalence. The strong equivalence principle states that the results of any given physical experiment will be precisely identical for an accelerated observer in free space as it is for a non-accelerated observer in a perfectly uniform gravitational field. A weaker form of this postulate restricts itself to the laws of motion of masses only. In other words, the laws of motion of identical masses on the earth are identical to the same situation inside an accelerated rocket (at 1g). We have shown (section 5.1) that objects of different mass all fall at the same rate of acceleration in a uniform gravity field. In regards to the strong equivalence principle, Synge writes: “… I never been able to understand this Principle … Does it mean that the effects of a gravitational field are indistinguishable from the effects of an observer’s acceleration? If so, it is false. In Einstein’s theory, either there is a gravitational field or there’s none, according as the Riemann tensor does not or does vanish. This is an absolute property. It has nothing to do with any observer’s world line … The principle of equivalence performed the essential office of midwife at the birth of general relativity, but, as Einstein remarked, the infant would never have got beyond its long clothes had it not been for Minkowski’s concept [of space-time geometry]. I suggest that the midwife be now buried with appropriate honors and the facts of absolute space-time faced.” Few physicists would doubt the validity of his statement. Synge has hit on an important point in regards to the nature of the equivalence principle and space-time. He is right to say that “ either there is a gravitational field or there’s none, according as the Riemann tensor does not or does vanish. This is an absolute property (of space near a large mass)”. What he means is that the Riemann tensor describing curvature is there, or is not there, depending on whether or not there is a large mass present to distort space-time. (in other words, whether there exists a global space-time curvature or not). There is nothing relative about this fact. The existence of a global space-time curvature reveals whether you are in a gravitational field. In an accelerated frame, the space-time curvature is local to your motion only, and is not global property of 4D space-time. For accelerated motion from gravitational fields, the global 4D space-time is flat, but the local 4D space-time for the accelerated observer is curved. There is no mystery here according to EMQG. If a large mass is present, the mass emits huge numbers of graviton particles, and distorts the surrounding virtual particles of the quantum vacuum. This is an absolute statement about the physical nature of the vacuum there. In an accelerated frame, there are very few gravitons, and the quantum vacuum is not affected. However, an observer in the accelerated frame ‘sees’ the quantum vacuum accelerating with respect to his frame in the same way, and hence the background vacuum state is the same. However, globally the quantum vacuum still remains (mostly) undisturbed for accelerated motion .35In other words, on a global scale, i.e. on a scale larger that the accelerated observer and his local frame, the state of relative acceleration of the quantum vacuum is uniform. Thus we conclude that the equivalence principle can be regarded as being a coincidence, due to local quantum vacuum acceleration state happening to appear the same for accelerated observers as it does for a specific observer on the earth. Recently, we uncovered theoretical evidence that suggests that the strong equivalence principle does not hold in certain circumstances. First, if gravitons could be detected experimentally with a new and sensitive graviton detector (which is not likely to be possible in the near future), we would be able to distinguish between an inertial frame and a gravitational frame with this detector. This would be possible because inertial frames would have virtually no graviton particles present, whereas the gravitational fields like the earth have enormous numbers of graviton particles. We would be able to construct a device with an indicator light that reads ‘ GRAVITATIONAL FRAME ’ or ‘ACCELERATED FRAME ’ by setting a certain threshold on the graviton count, and lighting the appropriate electronic indicator. Thus, we would have performed a physics experiment that can easily detect whether you are in a gravitational field or an accelerated frame in clear violation of the strong equivalence principle. Secondly, recent theoretical considerations of the emission of electromagnetic waves from a uniformly accelerated charge, and the lack of radiation from the same charge subjected to a static gravitational field leads us to the conclusion that the strong equivalence principle also does not hold for radiating charged particles. Stephen Parrott ( ref 23) has done an extensive analysis of the electromagnetic energy released from an accelerated charge in Minkowski space and a stationary charge in Schwarzchild space. He writes in his paper on “Radiation from a Uniformly Accelerated Charge and the Equivalence Principle”: “It is generally accepted that any accelerated charge in Minkowski space radiates energy. It is also accepted that a stationary charge in a static gravitational field does not radiate energy. It would seem that these two facts imply that some forms of Einstein’s Equivalence Principle do not apply to charged particles. To put the matter in an easily visualized physical framework, imagine that the acceleration of a charged particle in Minkowski space is produced by a tiny rocket engine attached to the particle. Since the particle is radiating energy, that can be detected and used, conservation of energy suggests that the radiated energy must be furnished by the rocket. We must burn more fuel to produce a given accelerated world line than we would to produce the same world line for a neutral particle of the same mass. Now consider a stationary charge in Schwarzchild space-time, and suppose a rocket holds it stationary relative to the coordinate frame (accelerating with respect to local inertial frames). In this case, since no radiation is produced, the rocket should use the same amount of fuel as would be required to hold stationary a similar neutral particle. This gives an experimental test by which we can determine locally whether we36are accelerating in Minkowski space or stationary in a gravitational field - simply observe the rocket’s fuel consumption.” He does a detailed analysis of the energy in Minkowski and Schwarzchild space-time, and concludes that strong principle of equivalence does not hold for charged particles in general. This is because the readout of the fuel consumed by the two rocket could be used to drive an indicator light to reveal whether you are in a gravitational field or in an accelerated frame. As for the weak equivalence principle, so far we can now only specify the accuracy as to which the two different mass types have been shown experimentally to be equal in an inertial and gravitational field. In EMQG, we show that the equivalence principle follows from lower level physical processes, and the basic postulates of EMQG. We will see that mass equivalence arises from the equivalence of the electrical force generated between the net statistical average acceleration vectors of the matter particles inside a mass interacting with the surrounding quantum vacuum virtual particles inside an accelerating rocket . The same force occurs between the matter particles and virtual particles for a mass near the earth. We will show that weak equivalence is not perfect, and breaks down when the accuracy of the measurement approaches forty decimal places of accuracy! Basically mass equivalence arises from the reversal of the net statistical average acceleration vectors between the charged matter particles and virtual, electrically charged, particles in the famous Einstein rocket, with the same matter particles and virtual particles process occurring near the earth (figures 2 through 5). To fully understand the hidden quantum processes in the principle of equivalence on the earth, we will detail the behavior of test masses and the propagation of light near the earth. Equivalence is shown to hold for the motion of masses in two cases: stationary test masses on the floor, and for freely falling test masses. First we derive mass equivalence for stationary and falling masses. Next, we show that a quantum principle of equivalence holds for individual elementary particles. Next, we will demonstrate that equivalence also holds for large spherical masses with considerable self- gravity (and self-energy) such as the earth with a hot molten core, and the moon with a considerably colder core, with respect to the motion of third mass which is the sun. We will see that if both the earth and the moon fall towards the sun, they would arrive at the same time to a high degree of precision in the framework of EMQG. Finally, we examine the principle of equivalence for curved Minkowski 4D space-time curvature in a rocket, and on the earth, which is the 4D space-time equivalence. 8.1 MASSES INSIDE AN ACCELERATED ROCKET AT 1g Figure #2: There are two different masses at rest on the floor of a rocket which is accelerated upwards at 1 g far from any gravitational sources. The floor of the rocket37experiences a force under the mass ‘2M’ that is twice as great as for the mass ‘M’. In Newtonian physics, the inertial mass is defined in precisely this way, the force ‘F’ that occurs when a mass ‘M’ is accelerated at rate ‘g’ as given by F=Mg. The quantum inertia explanation for this is that the two masses are accelerated with respect to the net average statistical motion of the virtual particles of the vacuum by the rocket. Since mass ‘2M’ has twice the masseon particle count as mass ‘M’, the sum of all the tiny electrical forces between the virtual vacuum and the masseon particles of mass ‘2M’ is twice as great as compared to mass ‘M’, i.e. for mass ‘M’, F 1=Mg and for mass ‘2M’, F 2=2Mg=2F 1. Because the particles that make up the masses do not maintain a net zero acceleration with respect to the virtual particles, a force is always present from the rocket floor. Figure #3 : There are two different masses (M and 2M) that have just been released, and are in free fall inside the rocket. According to Newtonian physics, no forces are present on the two masses since the acceleration of both masses is zero (the masses are no longer attached to the rocket frame). The two masses hit the rocket floor at the same time. The quantum inertia explanation for this is trivial. The net acceleration between all the real masseons that make up both masses and virtual masseon particles of the vacuum is a net (statistical average) value of zero. The rocket floor simply reaches up to the two masses at the same time, and thus unequal masses fall at the same rate inside an accelerated rocket and arrive at the floor at the same time. 8.2 MASSES INSIDE A GRAVITATIONAL FIELD (THE EARTH) Figure #4: There are the same two masses (2M and M) which are at rest on the surface of the earth. The surface of the earth experiences a force under mass ‘2M’ that is twice as great as for that under mass ‘M’. The reason for this is that the two stationary masses do not maintain a net acceleration of zero with respect to the net statistical average acceleration of the virtual masseons in the neighborhood. This is because the virtual particles are all (falling) accelerating towards the center of the earth ( a=GM/ r2) due to the graviton exchanges between the real masseons consisting of the earth and the virtual masseons of the vacuum. Since mass ‘2M’ has twice the masseon particles as mass ‘M’, the sum of all the tiny electrical forces between the virtual masseon particles of the vacuum and the real masseon particles of mass ‘2M’ is twice as great as that for mass ‘M’. Thus, a force is required from the surface of the earth to maintain these masses at rest, with mass ‘2M’ having twice the force of mass ‘M’. The physics of this force is the same as for figure #3 in the rocket, but now the relative acceleration frames of the virtual charged masseons and the real charged masseon particles of the mass are interchanged (with the exception of the direct graviton induced forces on the masses, which is negligible and discussed later). Figure #5 : There are two different masses (M and 2M) that have just been released, and are in free fall towards the earth (no external forces are present on the two masses). The two masses hit the earth at the same time. The relative average acceleration of the real masseon particles that make up the two masses with respect to the electrically charged38virtual masseon particles of the vacuum is zero, because both masses and the quantum vacuum are accelerated at the same rate through graviton exchanges with the earth. The electrical forces between the virtual particles and the matter particles of the test mass dominates the local interactions, because the electrical force is 1040 times stronger than the graviton component. Although mass ‘2M’ has twice the pure gravitational force (i.e. forces contributed from direct graviton exchanges with the earth) due to twice the number of graviton exchanges, this is totally swamped out by the electrical interaction, and the accelerated virtual particles and the test masses are in a state of electrical force equilibrium, as far as acceleration vectors are concerned. Both masses fall at the same rate (neglecting the slight imbalance of the graviton component). In fact, in both cases (i.e. accelerated and gravitational) there is a net equilibrium between the accelerated state of masseons in the mass and virtual masseons of the quantum vacuum (as far as their relative acceleration is concerned, which is zero). Therefore, the two unequal masses fall to the surface of the earth at the same acceleration, and arrive at the same time. NEWTONIAN MASS EQUIVALENCE PRINCIPLE Newton was aware of the mass equivalence principle, i.e. the inertial mass and the gravitational mass of an object are the same. We now revisit the equivalence principle from Newton’s perspective, bearing in mind these hidden quantum processes we just described. Now the elegant picture developed above can even be seen to be hidden in Newton’s formulation of accelerated motion and gravity. Equivalence between the inertial mass ‘M’ on a rocket moving with acceleration ‘A’, and gravitational mass ‘M’ under the influence of a gravitational field with acceleration ‘A’ can be seen to follow from Newton’s laws if we look at them in the slightly different way: Fi = M (A) ... Stationary vacuum opposes acceleration A of the mass ‘M’ in rocket. Fg = M (GM e/r2) ... Accelerated vacuum ( A= GMe/r2) opposes the stationary mass ‘M’. Viewed in this way, even Newton’s law of gravity looks a lot like his famous third law of inertia , i.e. Newton’s ‘F = GM eM/r2’ looks a lot like ‘F = M A’. Under gravity, the magnitude of the gravitational acceleration is ‘A=GM e/r2’, which is the same as the magnitude of the acceleration of the rocket. Switching to the reference frame of an average electrically charged masseon in the mass ‘M’ in both cases, it ‘sees’ a typical virtual masseon particle in the vacuum near the earth in exactly the same way as a typical virtual masseon in the rocket. In other words, the accelerated quantum vacuum particle state appears the same from the perspective of the mass in an accelerated frame and the same mass on the earth. This is the basis of Newtonian mass equivalence. This example shows why both the masses of figure 2 and 3 are equivalent to the masses in figure 4 and 5. The force magnitude is the same because the calculation of the force involves the same sum of all the tiny electrical forces between the virtual charged masseon particles and the real masseon particles of the mass. The only difference in the physics of39the masses is that the relative motions of all the tiny electrical force vectors are interchanged. The other difference is that large numbers of graviton particles (that originate from the earth’s mass) are absorbed by the masses in figures 4 and 5 for the earth case, while for the rocket there are negligible gravitons present. To summarize, two unequal masses in free fall hit the surface of the earth at the same time. The net statistical average acceleration of the real masseon particles that make up the masses and virtual charged masseon particles of the vacuum is zero, because this process is dominated by the electrical force (where the direct graviton exchanges are negligible). The electrical forces between the virtual particles and the matter particles of the test mass dominates the interactions, because the electrical force is 1040 times stronger than the graviton component. Although mass ‘2M’ has twice the pure gravitational force (i.e. forces contributed from direct graviton exchanges only with the earth) due to twice the number of graviton exchanges, this is totally swamped out by the electrical interaction, and the accelerated virtual particles and the test masses are in a state of electrical equilibrium as far as acceleration vectors are concerned. Both masses fall at the same rate. In fact, in both cases (i.e. accelerated and gravitational) there is a net equilibrium between the accelerated state of masseons in the mass and virtual masseons of the quantum vacuum as far as their relative acceleration is concerned. IT IS IMPORTANT TO NOTE THAT MASS EQUIVALENCE IS NOT PERFECT IN THE ROCKET AND IN THE EARTH-BOUND EXPERIMENTS ABOVE! First, the processes in figures 2 and 3 are physically different then in figure 4 and 5. The most important difference is that gravitons are present on the earth, but not in the rocket (strictly speaking there are negligible numbers of gravitons from the rocket structure). Secondly, the force contributed by the direct graviton exchanges between the mass and the earth slightly imbalance the equivalence of the fall rate for the two unequal masses. This discrepancy in the free fall rate of test masses near the earth is extremely minute in magnitude because there is a ratio of about 1040 in the field strength existing between the electromagnetic and gravitational forces. In principle it could be measured by extremely sensitive experiments, if two test masses are chosen with a very large mass difference, which would amplify this effect (ref. 1). Does the weak equivalence principle hold for an individual elementary particle? 8.3 MICROSCOPIC EQUIVALENCE PRINCIPLE OF PARTICLES We have been talking about macroscopic masses until now. What about the case for elementary particles? For example, does a proton and an electron simultaneously dropped on the surface of the earth fall at the same rate? In other words, if both particles are dropped inside a rocket that is accelerating at 1 g and also on the earth, would they arrive40on the floor at the same time? The answer is yes. In fact, the equivalence principle has actually been experimentally verified for the case of the neutron (ref. 40). An astute observer may have questioned our reasoning all along, by asking why all the virtual particles (for example, virtual neutrons, virtual electrons, virtual quarks, etc., which all consist of different mass values) are falling at the same rate. Certainly inside an accelerated rocket an observer who is stationed on the floor will view all the virtual particles of the quantum accelerating with respect to him at the same rate, no matter what the masses of the virtual particles are. This is simply because the floor of the rocket accelerates upwards at 1g to meet them, giving the illusion (to an observer on the rocket) that virtual particles of different mass are falling at the same rate. Since the masses of the different types of virtual particles are all different according to the standard model of particle physics, why are they all falling at the same rate on the earth? Here the cause of the downward acceleration is due to graviton exchanges with the earth. Shouldn’t the virtual particles with with greater mass charge fall at a faster rate? Since we are trying to derive the equivalence principle from fundamental concepts, we cannot resort to this principle itself to state that the virtual particles must be accelerating at the same rate on the earth. In order to solve this puzzle, we have to postulate the existence of a fundamental mass charge unit for all fermion particles, which is carried by the masseon particle. As we have indicated, all particles with mass (virtual or not) must be composed of combinations of the fundamental “ masseon” particle, which carries just one fixed quanta of mass charge (postulate #2). Since all virtual masseon particles exchange the same fixed flux of gravitons with the earth, all the virtual masseons are all falling at the same rate. However, we must bear in mind that virtual masseons can bind momentarily combine together to form all the familiar virtual particles, such as virtual electrons, virtual positrons, virtual quarks, etc. or even currently unknown species of virtual particles. Recall that masseons carry both gravitational ‘mass charge’ and ordinary electrical charge. However, the electrical interactions (photon exchanges) will work to equalize the fall rate of virtual masseons that combine to make virtual particles such as virtual electrons and virtual quarks. If a virtual quark consists of say 100 bound masseons (the actual number is not known), the graviton exchanges would normally be cumulative. We would expect to have 100 times more acceleration imparted to the virtual quark, compared to a single virtual masseon. However, virtual masseons dominate the quantum vacuum since they are the fundamental mass particle, and do not have to bound with other masseons to exist. Therefore the lone, unbound virtual masseon is by far the most common virtual mass particle in the quantum vacuum (this is illustrated in figures 10). No matter how many virtual masseons combine to give other virtual particles such as quarks, the local electrical interaction between the far more numerous lone virtual masseons and the virtual quark will tend to equalize the fall rate in exactly the same as we discussed above for macroscopic masses. This process works like a microscopic version41of the EMQG weak principle of equivalence for the wide variety of falling virtual particles. In other words, the same action occurring on the particle level for large falling masses is happening on the microscopic scale for elementary particles. Figure 10 shows the microscopic equivalence principle at work for free falling virtual particles in a gravitational field. To summarize, the electrical forces from the vastly numerous, free virtual masseons of the quantum vacuum (which all fall at the same rate due to equal mass charge) dominates over the action of the more familiar virtual particles that consist of combinations of virtual masseons. The virtual quark would normally fall faster than the virtual electron and the virtual electron faster than an individual virtual masseon if there were no quantum vacuum . 8.4 THE INERTION: AN ELEMENTARY QUANTA OF INERTIA Recall that a real neutron particle with no net electrical charge has inertial mass because it is composed of real electrically charged masseon particles, which interact electromagnetically with the charged virtual masseon particles of the quantum vacuum (recall that the number of positive and negative electrically charged masseons inside a neutron are equal). Each real masseon inside the neutron contributes a fundamental unit or ‘quanta’ of inertia to the neutron, because of the electrical force interaction with the immediate surrounding virtual masseons of the vacuum. Therefore, the sum of the real masseon force contribution to the inertial mass of the neutron defines the neutron’s total inertial mass. We propose that the electrical force that exists between one real masseon particle accelerating at a rate of 1g with respect to the surrounding quantum vacuum be given the status of a new universal constant. We call this new constant the ‘ inertion ’ constant, or “i”. The inertion has measurement units of force. Thus the inertion represents the lowest possible quanta of inertia force. In order to determine the numerical value of the inertion in the lab, one must determine the number of masseons inside a neutron for example, and divide this into the observable (inertial) mass of the neutron. These numbers are currently unknown at this time. How does the background quantum vacuum look like from the frame of reference of the neutron particle sitting on the surface of the earth? The answer is that it is the same as it appears for a neutron on a rocket (1g). As we have found, the virtual masseon and anti- masseon particles of the quantum vacuum are accelerated by the direct graviton exchanges with the earth. Therefore, we can conclude that the ‘ inertion’ also represents the lowest possible quanta of gravitational force on the earth.428.5 EQUIVALENCE PRINCIPLE FOR TH E SUN-EARTH-MOON SYSTEM So far we only considered the weak equivalence principle for medium sized masses and elementary particle. Does the weak equivalence principle hold for the following imaginary scenario: the earth and the moon are simultaneously in free fall towards the sun with an acceleration of gravity equal to ‘ Gsun? In other words, would the earth and moon arrive at the same time on the surface of the sun? Would they also arrive on the floor at the same time when free falling inside a huge rocket undergoing acceleration Gsun in space (far from any other large masses)? This question is important because large astronomical bodies have their own internal gravity, and are able to significantly disrupt the virtual particles around them. In other words, the earth’s gravity further modifies the acceleration vectors of the virtual particles in it’s neighborhood, and adds to contribution from the sun’s gravity. This question also turns out to be at the heart of the so-called metric theories of gravity. In metric theories such as general relativity, all objects with any kind of internal composition follow the natural curvature of space-time. This includes objects with considerable internal energy sources and self-gravity. Any deviation from perfect equivalence would constitute what is generally called the Nordtvedt effect (ref. 8), after the discoverer in 1968. Because the earth’s core is molten and very hot, the earth contains a significant internal energy. The earth is also a fairly large source of gravitational energy as well, which significantly contributes to the acceleration to the nearby virtual particles of the quantum vacuum in the earth’s vicinity. Contrast this with the moon, which is relatively cool and less energetic, with considerably less gravitational energy. To see if the weak equivalence works, we start with the fantastic situation where the earth and moon are ‘dropped’ simultaneously from a height ‘h’ inside a huge rocket (with negligible mass) accelerating with the same Gsun as exists on the surface of the sun (figure 12). The result of this experiment should be obvious. They both arrive on the floor of the rocket at the same time. Actually, it is the floor of the rocket that accelerates upwards and greets both bodies at the same time! However, one should be aware that the virtual particles of the quantum vacuum are disturbed near these large bodies (by direct graviton exchanges between these bodies and the nearby vacuum), particularly the acceleration vectors of the virtual particles which are in close proximity with the earth and the moon. The vacuum acceleration vectors now tend to point towards the centers of the two bodies, near these regions. This, however, does not affect the results of this experiment. The results are no different than if two small masses are dropped inside the rocket; again because it is the floor that accelerates up to meet them at the same time. However, one must note that the shape of the nearby virtual particle distribution around the bodies in this case (figure 12). Now we turn to another fantastic situation near the surface of the sun, where the earth and moon are ‘dropped’ on the sun from the same height ‘h’ (figure 11). This situation is far more complex than the same experiment inside the rocket. Now all three bodies disturb43the virtual particles of the quantum vacuum! The sun sets up a strong GM sun/r2 acceleration field of virtual particles, which applies over very long distances, and points towards the center of the sun. The earth and moon also produce their own vacuum particle acceleration fields GM earth /r2 and GM moon /r2 , respectively in their vicinity, although much weaker over large distances. The sun’s acceleration dominates over the surrounding space, except near the moon and near the earth; where some of the virtual particles actually are moving away from the sun (as happens near the surface of the earth on the night side). How can equivalence with the rocket possibly hold in this scenario? Recall that the electrical action of the virtual particles of the vacuum on the real particles inside the bodies that determines the motion of the earth and the moon undergoing gravitational acceleration. However, in different regions of the earth, the virtual particles are accelerating in different directions! Part of the answer to this problem comes from an important property exhibited by the masseon-graviton particles, which is the principle of superposition . This is a property that is also shared by the electron-photon particle. The action of the gravitons originating from all three sources on a given virtual particle of the quantum vacuum yields a net acceleration that is the net vector sum of the action of all the gravitons received by that virtual particle. To explain equivalence, we must first recall that equivalence only holds in a sufficiently small region of space (technically, at a given point above the sun) when compared to the equivalent accelerated reference frame. This is because the acceleration of the sun varies with the distance ‘r’ from the sun’s center, whereas inside an accelerated rocket it does not vary with height. Secondly, we must recall that the motion of the virtual particles in the rocket is also disturbed near the vicinity of the earth and the moon. In fact, inside the rocket the virtual particles are also directed along the radius vectors of both the earth and the moon in their vicinity (figure 12). Yet we believe that the sun and the moon still reach the floor of the rocket at the same time. Therefore, the quantum vacuum can be disturbed in the case of the free fall of the earth and the moon in a rocket, and yet can still cause both bodies to fall at the same rate in this case. A close examination of figures 11 and 12 reveals that the quantum vacuum pattern is the same for the two different experiments, when viewed from the correct reference frame. In figure 11, the observer is stationed on the surface of the sun, so that we can see the reason why both bodies are attracted to the sun. Recall that the electrical interaction between the acceleration vectors of the falling virtual masseons and the real masseons in the earth and moon is the primary reason for the attraction. The direct graviton action is negligible in comparison. Now in order to compare the two experiments of figure 11 and 12, the reference frame for the sun experiment should be equivalent to the rocket. In the rocket experiment, the frame chosen for our observer is outside the rocket (the quantum vacuum has a relative acceleration of zero) in order to understand the results. Therefore for the sun, the observer’s frame should be in free fall, thus restoring the relative acceleration of the quantum vacuum to zero just as for the observer outside the44rocket. When this is done, we have to correct the acceleration vectors of the virtual particles near the earth and moon in figure 12. It is easy to show that the result of this operation gives an identical result as figure 12. Therefore, equivalence holds in both experiments, because the vacuum acceleration patterns are the same! Next we address the important, and more complex issue of light motion equivalence in accelerated and gravitational frames. 8.6 LIGHT MOTION IN A ROCKET: SPACE-TIME EFFECTS According to general relativity, the strong equivalence principle demands that all physical phenomena are equivalent in a rocket (1g) and on the earth, for a sufficiently small volume. This includes the motion of light . Furthermore, there ought to be equivalent curvature of 4D space-time. We now review the Einstein famous thought experiment in regards to the motion of light in accelerated and gravitational frames. First we carefully examine three scenarios for the motion of light in a rocket (figures 6 and 7), which is accelerated upwards at 1 g (far from any gravitational sources) and the same situation on the earth (figure 8 and 9). The 1 g acceleration is purposely chosen to match the earth’s acceleration. First we study light moving from the floor of the rocket to the ceiling where it is detected by an observer there. Next we look at light moving from the ceiling of the rocket to the floor where it is detected by an observer there. Finally, we examine more closely to what happens to light moving parallel with the floor of the rocket (figure 14) and the same situation on the earth (figure 13), where it follows a curved path. (A) LIGHT MOVING FROM THE FLOOR TO THE CEILING OF THE ROCKET FIGURE 6 - Here the light is positioned on the floor of the rocket which is being accelerating upwards at 1 g, and propagates in a straight line up to the observer on the ceiling. Meanwhile, the rocket has accelerated upwards while the light is in flight. What happens to the light? Experimentally, we observer a red shift in the color of the light. Furthermore, one can also conclude that there is a Doppler shift towards the red end of the spectrum. This conclusion is based on to the motion of the observer on the ceiling, who obtains an additional velocity over the light source during the time of flight of the light, when velocities are compared to a third observer outside the rocket. What happens to the velocity of light? To answer this, we must consider the nature of space and time, since velocity is defined as distance/time.. According to general relativity, all observers measure the light velocity as being the standard vacuum value, where each use their measuring instruments that are identically constructed, and calibrated inside each of their reference frame. In other words, the speed of light does not vary under all these circumstances. Closer examination reveals that the45clocks in the ceiling differ from the clocks stationed on the floor. In particular, the clock on the floor of the rocket runs slower than the one on the ceiling. Distances measurements are also affected. General relativity explains all these observations with the 4D space-time curvature existing inside accelerated frames. We will return to this example with our EMQG interpretation of these measurements. (B) LIGHT MOVING FROM THE CEILING TO THE FLOOR OF THE ROCKET FIGURE 6 - Here the light is positioned on the ceiling of the rocket which is accelerating upwards at 1 g, and propagates in a straight line down to the observer on the floor. Meanwhile, the rocket has accelerated upwards while the light is in flight. What happens to the light? Experimentally, we observer a Doppler blue shift in the color of the light. According to general relativity, all observers measure the light velocity as being the standard vacuum value, where each use their measuring instruments that are identically constructed, and calibrated inside each reference frame. Closer examination reveals that the clocks in the ceiling differ from the clocks stationed on the floor. Distances measurements are also affected. General relativity explains all these observations with the 4D space-time curvature existing inside accelerated frames. We will return to this example with our EMQG interpretation of these measurements. (C) LIGHT MOVING PARALLEL TO THE FLOOR OF THE ROCKET FIGURE 7 - Here the light leaves the light source on the left wall of the rocket which is accelerating upwards at 1 g, and propagates in a straight line towards the observer on the right wall. Meanwhile, the rocket has accelerated upwards while the light is in flight. Therefore an observer in the rocket observes a curved light path. An observer outside the rocket sees a straight light path. With the introduction of the third observer outside the rocket, the reason for curvature becomes readily apparent. Outside the rocket the observer sees the light path go straight across the rocket. Meanwhile, inside the rocket the light leaves the source and is totally de-coupled from the source and moves straight (as seen outside). However, the observer on the floor is drawn upwards by the accelerated motion of the floor. This causes the inside observer to see a curved path. According to general relativity, the space-time inside the rocket is curved (in the direction of motion), and light moves along the natural geodesics of curved 4D space-time. Meanwhile, the observer outside the rocket lives in flat 4D space-time, and therefore observes light moving in a perfect straight line, which is the geodesic path in flat 4D space-time. We will return to this example with our EMQG interpretation of these measurements. Next we will examine the same scenarios for light motion on the surface of the earth..468.7 LIGHT MOTION NEAR EARTH’S SURFACE - SPACE-TIME EFFECTS Now we carefully examine the same three scenarios for the motion of light, but now we are on the surface of the earth. We will ignore the variation of acceleration with height found on the earth, by selecting a room that is sufficiently low height and narrowness as to ignore the slight change in the direction of acceleration caused by acceleration vectors being directed along earth’s radius vectors. First, light moves from the floor of the room on the surface of the earth to the ceiling, where it is detected. Next, light is moving from the ceiling of the room to the surface of the earth where it is detected by an observer. Finally, light is moving parallel with the earth’s surface from the left side of the room to the right, and follows a curved path. (A) LIGHT MOVING FROM THE FLOOR TO THE CEILING ON EARTH FIGURE 8 - Light is positioned on the floor of a room on the surface of the earth, and propagates in a straight line up to the observer on the ceiling. What happens to the light? According to the principle of equivalence, the light behaves the same way as in the rocket, i.e. it is red-shifted. However, one cannot claim a Doppler red shift, since there is no relative motion of the source and the detector according to general relativity. Is there any other physical justification for the red shift, other the principle of equivalence? Since light is moving through a gravitational potential, it ought to lose energy as it gains height. According to E=hν, if the energy ‘E‘ goes down, then the frequency ν decreases, and we have a red shift. Therefore, when we look at the physics behind accelerated and gravitational frames for this situation (without resorting to the principle of equivalence), we find two different explanations; i.e. Doppler shift and gravitational energy loss. In EMQG, we find that the virtual particle dynamics alone gives an adequate explanation for this. What happens to the velocity of light? To answer this, we must again consider the nature of space and time, since velocity is defined as distance/time. According to general relativity, all observers measure the light velocity as being the standard vacuum value, where each use their measuring instruments that are identically constructed, and calibrated inside each reference frame. In other words, the speed of light does not vary under all circumstances. Closer examination reveals that the clocks in the ceiling differ from the clocks stationed on the floor. In particular, the clock on the floor on the earth runs slower than the one that is on the ceiling. Distances measurements are also affected. General relativity explains all these observations with the 4D space-time curvature existing inside accelerated frames. We will return to this example with our EMQG interpretation of these measurements.47(B) LIGHT MOVING FROM THE CEILING TO THE FLOOR ON EARTH FIGURE 8 - Here the light is positioned on the ceiling of the room on the surface of the earth, and propagates straight down to the observer on the floor. What happens to the light? Again to answer this, we must again consider the nature of space and time, since velocity is defined as distance/time. According to general relativity, all observers measure the light velocity as being the standard vacuum value, where each one uses their measuring instruments that are identically constructed, and calibrated inside each reference frame. In other words, the speed of light does not vary under all circumstances. Closer examination reveals that the clocks in the ceiling differ from the clocks stationed on the floor. In particular, the clock on the floor on the earth runs slower than one that is on the ceiling. Distances measurements are also affected. General relativity explains all these observations with the 4D space-time curvature existing inside accelerated frames. According to general relativity, an observer outside the room in free fall observes the light moving downward at the speed of light in a straight path. Meanwhile, according to general relativity, an observer inside the room stationed on the floor also observes the light moving downwards in a straight line. He also observes that the light is blue-shifted. He makes a measurement of the light velocity with his measuring instruments (which were calibrated within his reference frame) and observes that the velocity of light is the same on the floor as he found when he measured received light speed in his internal reference frame with the same instruments. In other words, the speed of light does not vary in all cases. Closer examination reveals that clocks measured in his reference frame differ from the clocks on the ceiling. In particular, the clock on the ceiling of the room runs faster than the one on the floor. Distances are also affected. In general relativity, all these conclusions follow directly from 4D space-time curvature. (C) LIGHT MOVING PARALLEL TO THE SURFACE OF THE EARTH FIGURE 9 - Here the light leaves the light source on the left wall of the room on the earth and propagates in a curved path towards the observer on the right wall. Meanwhile, an observer in free fall towards the earth’s surface sees light moving in a straight path. According to general relativity, and light moves along the natural geodesics of curved 4D space-time in the room. Meanwhile, an observer in free fall lives in flat 4D space-time, and hence an observer sees straight-line paths for light. In general relativity, all these conclusions are identical as for the observer accelerated in the rocket at 1g in accordance with the principle of equivalence. Now we look at 4D space-time curvature from the perspective of EMQG. 8.8 GENERAL CONCLUSIONS FOR THE EQUIVALENCE OF LIGHT MOTION The three elementary observables discussed above are:48(1) The red-shift for light moving straight up in a rocket, or on the earth. (2) The blue-shift for light moving straight down in a rocket, or on the earth. (3) The curved light path when light moves parallel to the earth, or in a rocket. One can easily conclude that there is a Doppler shift for light moving up or down in the rocket, because the detector obtains an additional velocity with respect to the source during the travel time of the photons (as seen by the observer outside the rocket). It is also clear that one cannot conclude that a Doppler shift is responsible for the shift in color of the light as it climbs or descends in the case of light motion in a gravitational field. This conclusion is based on the fact that there is no relative motion between the light source and the detector during the light travel time. The Doppler shift requires motion. However, in general relativity the actual motion of the light is identical on the earth and inside the accelerated rocket. It does not say anything about the low level, quantum processes responsible for the light, and why they give that same results! In other words, general relativity prescribes a common cause for the light motion inside the accelerated rocket on the earth; that is the curvature of space and time in both reference frames. We maintain that the actual low level physical causes cannot be the same, however because the physics at the quantum level are different for accelerated and gravitational frames! This is a central idea in EMQG, where there actually is no strong equivalence principle. We acknowledge that the physical details of the low level processes are different, and we provide an explanation at the quantum scale in this section. We can view the wave crests of light moving upwards against a gravitational field as a true representation of the emitted history of the light wave. This is because the light crests are conserved during propagation. No wave crests are created on route. Also, no wave crests are lost during the motion of light. For each wave crest that enters the space between the light source and the detector, there must another wave crest leaving that same space. Einstein himself gave a simple and elegant explanation of this in his 1911 paper. The question to ask is why does an atom that emits a photon upwards in a gravitational field produce a photon of different frequency then the same atom in far empty space? Furthermore why does the atom in an accelerated rocket (1g) that emits light upwards, produce the same frequency as the same atom in the earth’s gravitational field? Einstein’s answer to these questions is to resort to the strong principle of equivalence and the metric (although in 1911 he used the considerations of gravitational potential energy and arrived at an answer that was off by a factor of two). The strong equivalence principle states that time is slowed in the same way in the rocket and on the earth, thus slowing all atomic processes (without recourse to lower level quantum phenomena). EMQG provides a simple answer to these questions. The accelerated state of the electrically charged quantum vacuum appears the same from the perspective of both the atom in a rocket and an atom on the earth that emits photons. That is, the vacuum background conditions for the atom in an accelerated rocket (1g) that emits photons49upwards is identical to the same vacuum background for the same atom that is on the earth’s surface (neglecting the graviton background for the case of the earth)! Furthermore, the electrically charged virtual particles affect the motion of photons . We will elaborate on this after reviewing photon scattering theory, and the effects of accelerated electrical charges on the propagation of light. We would like to point out an important problem with the standard general relativistic interpretation of the blue shift of light (when light travels straight down on the surface of the earth) that has recently been pointed out to us by V. Petkov (ref. 47, 1999). In this scenario, light propagates downwards from the ceiling to the floor of the gravitational field and light becomes blue-shifted rather than red-shifted (figure 6). Why is this so? V. Petkov makes a strong case for what he calls the anisotropic light velocity in non- inertial frames. He argues that the proper velocity of light changes in a gravitational field. Light moves faster when propagates downwards, and light moves slower when it propagates upwards compared to the vacuum value. The proper light velocity is defined by the proper length and proper time measurements. It turns out that Petkov’s approach is very similar to the approach we adopted in EQMG, which we now examine fully in the next section (with the exception of a difference in opinion on the nature of the 4D space- time). 9. THE EQUIVALENCE OF 4D SPACE-TIME CURVATURE Here we will contrast the two very different approaches to the problem of 4D space-time curvature, and the equivalence of 4D space-time in accelerated and gravitational frames: General Relativity: Light velocity is constant, and we have 4D space-time curvature. EMQG&Scattering :Photons scatter with the electrically charged falling vacuum. The central theme in general relativity is the constancy of light velocity. The curvature of 4D space-time is the root cause of all the observable aspects of light motion in the rocket and on the earth. A deflected light path in a gravitational field would seem to violate the constancy of light velocity, but it is the curving of 4D space-time that is the culprit. The central theme in EMQG is that photons scatter with the electrically charged, and falling (accelerated) virtual particles of the quantum vacuum. Photon scattering accounts for the illusion of 4D space-time curvature! Furthermore, matter also reacts to the falling vacuum, and together a 4D space-time curvature phenomena emerges from the low level activities of quantum particles on a kind of absolute space and time background (the CA). We have already presented a unified picture of the equivalence of real matter motion in an accelerated rocket and on the earth, with the model of the falling electrically charged quantum vacuum. We now wish to extend this model to cover the motion of light under50the action of the falling vacuum in an accelerated frame and on the surface of the earth, which we claim encompasses the relativistic standard model of 4D space and time. First we will derive the gravitational time dilation equation in the conventional way, using the general relativistic solution to Einstein’s gravitational field equations for a large spherical, non-rotating mass. This solution is called the Schwarzschild metric. Next, we fully develop the EQMG theory of light scattering and 4D space-time curvature. From this, we calculate the quantity of space-time curvature using EMQG theory, and show that the results are the same as relativity for spherical masses. In EMQG the light velocity is anisotropic as Petkov has suggested, but the changes in light velocity (compared to the vacuum value) occur not in proper units of length and time ( Petkov, ref. 47), but in the CA absolute units of cells and clock cycles. For this work, we will not go into the details of the absolute CA space and time, and reference 1 gives a complete account. 9.1 GENERAL RELATIVISTIC 4D SPACE-TIME CURVATURE It is well known that general relativity accounts for the motion of light near a large spherical mass, using the concept of 4D space-time curvature. General Relativity postulates space-time curvature in order to preserve the constancy of the light velocity (a notion first introduced in special relativity) in an accelerated frame or in a gravitational field. The 4D space-time metric must satisfy Einstein’s Gravitational field equations: R gRG cT ik ik ik−=1 28 2p … Einstein’s Gravitational Field Equations (9.11) The solution of this equation for the case of spherical mass distribution was given by the Schwarzchild in spherical coordinates (ref. 39): dsdr GM rccdtGM rcrd22 22 2 22 2 1212= −−−+ ( ) Ω where d Ω2 = dθ2 + sin2 θ dφ2 (9.12) This is a complete mathematical description of the 4D space-time curvature near the large spherical mass in spherical coordinates in differential form called the 4D space-time metric. It is of the form of the element of distance ‘ ds’ between two closely spaced points expressed in terms of the chosen coordinates, where ds2 = gik(x) dxi dxk, and the time coordinate is g 00. From this, it is easy to show (ref. 39) that the comparison of time measurements between a clock outside a gravitational field (called proper time t( ∞ ) to a clock at distance r from the center of a spherical mass distribution (called the coordinate time t( r ) is given by:51trt()()=∞ 1 - 2GM rc2 … which follows from Schwarzchild metric directly. (9.13) Using the relationship (1 - x)-1/2 ≈ 1 - x/2 when x << 1, and realizing the quantity 2GM/rc2 is very small, we can write this as: trGM rct ()( )() ≈−∞ 1 2(9.14) This gives the amount of time dilation between a clock on the earth “t( r )” compared to a clock positioned at infinity “t( ∞ )”. From this, we see that clocks on the earth run slower then at infinity. The deeper a clock is in a gravitational potential the slower it runs, as compared to it’s identically constructed cousin at infinity. So does the frequency of light that is emitted from an atom on the earth. Similarly, from this metric, we find that the distance Δr at point s( r ) compared to the same distance Δr at infinity follows as: ssr()()∞= 1 - 2GM rc2(9.15) or we can also write this as: srs GM rc()()≈∞ −1 2(9.16) This gives the amount of space distortion for a ruler of length Δr on the earth “s( r )” compared to the identically constructed ruler of length Δr positioned at infinity “s( ∞ )”. Observationally, equation 9.13 has been verified experimentally by atomic clocks mounted on board an airplane (ref. 6 and 7). In these experiments, an atomic clock had spent many hours at a high altitude inside and was returned to the ground and compared to an atomic clock in the laboratory. After correcting for various background effects, the laboratory clock lagged behind the airplanes atomic clock by the predicted amount. 9.2 EMQG, LIGHT MOTION, AND 4D SPACE-TIME CURVATURE The general principles of EMQG are applied here in order to understand 4D space-time curvature and the principle of equivalence of all light motion in an accelerated rocket (1g)52on the surface of the earth. To do this we must examine in detail the effects of the background electrically charged, and falling, virtual particles of the quantum vacuum on the propagation of light. The big question to consider is: Is the general downward acceleration of the electrically virtual particles of the quantum vacuum near a large mass responsible for the motion of photons in a gravitational field, or is this motion truly the result of an actual 4D space-time geometric curvature that exists near the earth (and which holds to the absolute tiniest distance scales)? The answer to this very important question hinges on whether our universe is truly a curved, geometric Minkowski 4D space-time on the smallest of distance scales, or whether curved 4D space-time results merely from the activities of quantum particles such as bosons and fermions (living in an absolute CA space and time) interacting with other virtual quantum particles in the background quantum vacuum. EMQG takes the second approach, in order to be compatible with quantum inertia and CA theory. The second approach also leads to an in- depth understanding of the principle of equivalence on the quantum scale. According to EMQG theory, light takes on the same general acceleration component as the net statistical average value of the falling, electrically charged, virtual particles of the quantum vacuum, through an electrical ‘ Fizeau-like’ scattering process. By this we mean that the photons are frequently absorbed and re-emitted by the electrically charged virtual particles, which are (on the average) accelerating towards the center of the large mass. When a virtual particle absorbs the real photon, a new photon is re-emitted after a small time delay in the same average downward direction that the original photon takes. This process is called photon scattering. Photon scattering is well understood for real matter, and the effects of scattering on light velocity are also well known in field of optics. We will see that photon scattering in an accelerated quantum vacuum (where the vacuum can be considered as a fluid medium) is also central to the understanding of 4D space-time curvature. One of the most important (and completely undisputed) results of classical optics is that light moves slower in water than it does in air. Furthermore it is recognized that the velocity of light in air is slower than that of light’s vacuum velocity. This effect is described by the index of refraction ‘n’, which is defined as the ratio of light velocities in the two different media. What is also an interesting result from classical physics is that the velocity of light in an ordinary transparent and moving medium (such as moving water) is known to differ from its value in the same stationary medium (such as stationary water). Fizeau (1851) is credited for demonstrated this experimentally in the laboratory with light propagating through pipes containing currents of water, flowing with a constant velocity in two opposite directions. In 1915 Lorentz identified the physics of this phenomena as being due to his microscopic semi-classical electromagnetic theory of photon propagation. Einstein was also aware of53these results and attributed this to the special relativistic velocity addition rule for the addition of the light and water velocities, without consideration of the low level quantum processes involved. In EMQG, we propose that in gravitational fields (and for accelerated motion) the moving water of Fizeau’s experiment is now replaced by the accelerated and electrically charged virtual particle flow of the quantum vacuum. Like in the Fizeau experiment, photons scatter with the accelerated virtual particles of the quantum vacuum and vary their motion. Imagine what would happen if you place a clock inside a stream of moving water in the Fizeau experiment. Would the clock keep time properly, when compared to an identically constructed clock placed in stationary water. Of course not! The very idea of this seems almost ridiculous. Of course one could argue that the nature of the hydrodynamic forces in moving and stationary water would alter the dynamics of how the clock operates. We suggest that you should also not be surprised to find that the accelerated flow of virtual particles affects both clocks and rulers placed in a gravitational field, as compared to the identical measuring instruments in far space where the vacuum is undisturbed. Another important result from EMQG and scattering theory is that the velocity of light in a vacuum, without the existence of any virtual particles of the quantum vacuum, should be much greater than the observed average light velocity in the vacuum. The electrically charged virtual fermion particles of the quantum vacuum frequently scatter photons, which introduce many tiny delays for the photon propagation. This causes a great reduction in the total average light velocity in the vacuum that is populated by countless numbers of virtual, electrically charged, particles. In other words; the low level light velocity (between virtual particle scattering events) is much greater than the measured average light velocity after vacuum scattering in the normal vacuum . A similar effect is known to occur when light propagates through glass, where photons scatter with electrons in the glass molecules, which subsequently reduces the average light velocity through glass compared to the normal vacuum light velocity. This result is frequently overlooked by physicists. In our study of special relativity, we have seen the importance of the propagation of light in understanding the nature of space and time measurements. Recall that according to standard text books (for example, Serway ref. 19) the definition of an inertial frame in space is a vast rigid 3D grid of identically constructed clocks placed at regular intervals with a ruler (figure 20). These clocks are viewed by an observer with ordinary light, i.e. one can simply ‘look’ at a clock and determine the time. Since light has a finite speed, we are forced to conclude in this scheme of things that two clocks cannot be simultaneous! If light is somehow affected by gravitational fields, this is bound to distort this concept of space and time, using the above definition of an inertial frame. Therefore, we will closely examine the behavior of light near the earth and in an accelerated rocket. We must remind the readers that the constancy of light velocity in gravitational frames still remains a postulate of general relativity, in spite of 90 years of close scrutiny . General54relativity incorporates all the results of special relativity, including the second postulate of light constancy for all inertial observers. In other words, it is impossible experimentally to distinguish between whether the light velocity changes when light moves upwards, or whether 4D space-time changes with height. What is lacking is a standard to gauge these alternatives. In other words, what measurement standard is assumed in discussions regarding the rates of clocks, the lengths of rulers, and the speed of light. The measurements of local observers cannot be used to answer this question, because we are considering changes to the actual measurement standards that are used by different observers! First we must carefully understand what is meant by light velocity. Velocity is defined as distance divided by time, or c=d/t. Light has very few observable characteristics in this regard: we can measure velocity c (the ratio of d/t); frequency ν; wavelength λ; and we can also measure velocity by the relationship c= νλ. It is important to note that all these observables are closely interrelated. We know that ν = 1/t (t is the period of one light cycle) and λ=d (the length of one light cycle). Thus, c=d/t and c= νλ are equivalent expressions of space and time. If we transmit green light to an observer on the ceiling of a room on the earth, and he claims that the light is red shifted, it is impossible for him to tell if the red shift was caused by the light velocity changing, or by space and time distortions which causes the timing and length of each of the light cycles to change. This is very important, and we illustrate this with an example. Imagine that the frequency of light is halved, or νf = (1/2) νi and the wavelength doubles λf = 2λi , and that you were not aware of both of these changes. Then you would conclude that the velocity of light remains unchanged (c= νλ). However, if the velocity of light is halved, and you were not aware of it, then you could conclude that the frequency is halved, νf = (1/2) νi and the wavelength doubles λf = 2λi. To illustrate this point, we will now examine what happens if an observer on the floor feeds a ladder (which represents the wave character of light) with equally spaced rungs to an observer on the ceiling, where each observer cannot see what the other observer does with the ladder. A ladder is used as an analogy for the wave characteristic of light, where the distance between rungs represent the wavelength of light (figure 15). Imagine a perfect ladder with equally space rungs of known length being passed up to you at a known velocity (figure 15), such that it is impossible to tell the motion of the ladder other than by observing the rungs moving past you. If the rung spacing are made larger, you would conclude that either the ladder is slowing down, or that the spacing of the ladder rungs was increased. But it would be impossible to tell which is which. Let us assume that you make a measurement on the moving rungs, and observe a spacing of 1 meter between any two rungs. Then you observe that two rungs move past you every second. You therefore conclude the velocity of the ladder is 2 m/sec. Now suppose that the ladder is fed to you at half speed or at 1 m/sec, and that you are not aware of this change in velocity. You could conclude that the velocity halved from your55measurements, because you now observe that one rung appears in view for every second that elapses instead of two rungs, and that the velocity was thus reduced to 1 m/sec. However, you could just as well concluded that your space and time was altered, and that the velocity of the ladder is constant or unaffected. Since you observe only one rung in view per second instead of the usual two rungs, you could claim that the rung spacing on the ladder is enlarged (red-shifted) or doubled to 2 m by distortions of your measuring equipment, and that the velocity still remained unaltered. From this, you might conclude that the frequency is halved, and that time measurements that might be based on this ladder are now dilated by a factor of two. Which of these two approaches to the ladder problem is truly correct? It is impossible to say by measurement, unless you know before hand what trait of the ladder was truly altered. For photons, the same problem exists. No known measurement of photons in an accelerated rocket or on the surface of the earth can reveal whether space and time is affected, or whether the velocity of light has changed. This is why the constancy of light remains a postulate in general relativity. In EMQG theory, the variable light velocity approach is chosen for several important reasons. First, the equivalence of light motion in accelerated and gravitational frames now becomes fully understood as a dynamic process having to do with motion rather then a fundamental principle of physics. For gravity, it is the hidden virtual particle motion, and for accelerated frames it is the motion of the frame itself. Furthermore, this model retains the same basic behavior as for ordinary matter in motion in accelerated frames and gravitational frames. The equivalence principle now becomes understood on a quantum scale for both light and matter, using the same background vacuum concept. Secondly, the physical basis of the curvature of Minkowski 4D space-time near a large mass now becomes clear. It arises from the interaction of light and matter with the background accelerated virtual particle processes. This process can be visualized as an accelerated quantum vacuum fluid flow in both frames types. However, in gravitation the vacuum fluid flow is caused by direct graviton exchanges acting between the earth and the electrically charged, virtual fermions of the quantum vacuum. In accelerated frames, the vacuum fluid flow is apparent and caused by the actual acceleration of the observer’s frame and his light sources (and the measuring instruments he possesses). Finally, and most importantly, the mysterious physical action that occurs between the earth and the surrounding 4D space-time curvature now becomes very clearly understood . The earth acts on the virtual particles of the quantum vacuum through graviton exchanges, causing them to accelerate towards the earth. The accelerated virtual particles in turn act on light and matter to produce the effects of curved 4D space-time. Since photon scattering is essential to our understanding of the details of 4D space-time curvature, we will examine scattering in some detail in the following sections. Readers that are familiar with photon scattering theory may wish to skip ahead to section 9.9. First we review the conventional physics of light scattering in a real media such as water or glass,56introducing the concept of the index of refraction and Snell’s Law of refraction. Next we introduce photon scattering for a real media moving at a constant velocity, where the velocity of light varies in the moving media (known as the Fizeau effect). Next we introduce an accelerated motion for the real medium and examine how the photons scatter there. This is an important step towards understanding EMQG theory. Finally we generalize these arguments to examine photon scattering with the electrically charged virtual particles of the quantum vacuum. In the next sub-section we show the relationship between photon scattering and 4D space-time curvature. 9.3 CLASSICAL SCATTERING OF PHOTONS IN REAL MATTER It is a well-known result of classical optics that light moves slower in glass than in air. This effect is described by the index of refraction ‘n’, which is the ratio of light velocities in the two different media. The Feynman Lectures on Physics gives one of the best accounts of the classical theory for the origin of the refractive index and the slowing of light through a transparent material like glass (ref. 42, chap. 31). When light passes from a vacuum into glass, with an incident angle of θ0 it deflects and changes it’s direction and moves at a new angle θ1 , where the angles follow Snell’s law: n = sin θ0 / sin θ1 (9.31) This follows geometrically because the wave crests on both sides of the surface of the glass must have the same spacing, since they must travel together (ref. 42). The shortest distance between crests of the wave is the wavelength divided by the frequency. On the vacuum side of the glass surface it is λ0 = 2πc/ω, and on the other side it is given by λ = 2πv/ω or 2πc / ωn since we define v=c/n. If we accept this, then Snell’s law follows geometrically (ref. 42). In some sense, the existence of the index of refraction in Snell’s law is confirmation of the change in light speed going from the vacuum to glass. Snell’s law does not tell us why we have a change in light velocity, nor does it give us any insight into the phenomena of dispersion and back scattering of light in refraction. A good classical account of the derivation of the index of refraction is given by Feynman himself in ref. 42. Feynman derives the index of refraction for a transparent medium by accepting that the total electric field in any physical circumstance can be represented by the sum of the fields from all charge sources, and by accepting that the field from a single charge is given by it’s acceleration evaluated with a retardation speed ‘c’ (the propagation speed of the exchanged photons). We only summarize the important points of his argument below, and the full details are available in reference 42: (1) The incoming source electromagnetic wave (light) consists of an oscillating electric and magnetic field. The glass consists of electrons bound elastically to the atoms, such that if a force is applied to an electron the displacement from its normal position will be proportional to the force.57(2) The oscillating electric field of the light causes the electron to be driven in an oscillating motion, thus acting like a new radiator generating a new electromagnetic wave. This new wave is always delayed, or retarded in phase. These delays result from the time delay required for the bound electron to oscillate to full amplitude. Recall that the electron carries mass and therefore inertia. Therefore some time is required to move the electron. (3) The total resulting electromagnetic wave is the sum of the source electromagnetic wave plus the new phase-delayed electromagnetic wave, where the total resulting wave is phase-shifted. (4) The resulting phase delay of the electromagnetic wave is the root cause of the reduced velocity of light observed in the medium. Feynman goes on to derive the classic formula for the index of refraction for atoms with several different resonant frequency ωk which is given by: nq emN ie k k k k=+  −+∑122 02 2ww gw(9.32) where n is the index of refraction, qe is the electron charge, m is the electron mass, ω is the incoming light frequency, γk is the damping factor, and Nk is the number of atoms per unit volume. This formula describes the index of refraction for many substances, and also describes the dispersion of light through the medium. Dispersion is the phenomenon where the index of refraction of a media varies with the frequency of the incoming light, and is the reason that a glass prism bends light more in the blue end than the red end of the spectrum. If the medium consists of free, unbound electrons in the form of a gas such as in a plasma (or as the conduction electrons in a simple metal) then the index of refraction with the conditions γk << ω and ωk = 0 is given by (ref. 42): n ≈ 1 - [ Nk qe2 / (2e0m)] / ω2≈ {1 - [ Nk qe2 / (e0m)] / ω2 }1/2 (9.33) where we recall that (1-x)1/2 ≈ 1 - x/2 if x is much less than 1. The quantity Ω = [ Nk qe2 / (e0m) ] 1/2 is sometimes called the Plasma frequency Ω, where there is a transition to the transparent state at Ω = ω. 9.4 QUANTUM FIELD THEORY OF PHOTON SCATTERING IN MATTER Although the classical account of scattering predicts the experimentally confirmed results, the correct account must be a quantum mechanical account.58To quote R. Feynman: “ … yes, but the world is quantum not classical dam-it ”. The propagation of light through a transparent medium is a very difficult subject in QED. It is impossible to compute the interaction of a collection of atoms with light exactly. In fact, it is impossible to treat even one atom’s interaction with light exactly in QED. However the interaction of a real atom with photons can be approximated by a simpler quantum system. Since in many cases only two atomic energy levels play a significant role in the interaction of the electromagnetic field with atoms, the atom can be represented by a quantum system with only two energy eigenstates. In the text book “Optical Coherence and Quantum Optics” a thorough treatment of the absorption and emission of photons in two-level atoms is given (ref. 43, Chap. 15, pg. 762). When a photon is absorbed, and later a new photon of the same frequency is re- emitted by an electron bound to an atom, there exists a time delay before the photon re- emission. The probabilities for emission and absorption of a photon is given as a function of time Δt for an atom frequency of ω0 and photon frequency of ω1 : Probability of Photon Absorption is: K [ sin (0.5( ω1 - ω0) Δt) / ( 0.5( ω1 - ω0)) ]2 Probability of Photon Emission is: M [ sin (0.5( ω1 - ω0) Δt) / ( 0.5( ω1 - ω0)) ]2 (9.41) (where K and M are complex expressions defined in ref. 43) The important point we want to make from eq. 16.41 is that the probability of absorption or emission depends on the length of time Δt, where the probability of the emission is zero, if the time Δt = 0. In other words according to QED, a finite time is required before re-emission of the photon. There are other factors that affect the probability, of course. For example, the closer the frequency of the photon matches the atomic frequency, the higher the probability of re-emission in some given time period. We maintain that these delays are the actual route cause of the index of refraction in a medium. We believe that a similar thing happens when photons propagate through the quantum vacuum. Therefore, we want to address the effect of the virtual particles of the quantum vacuum on the propagation velocity of real (non-virtual) photons, a subject that is largely ignored in the physics literature. 9.5 THE SCATTERING OF PHOTONS IN THE QUANTUM VACUUM In section 9.3 we discussed photon scattering in a real matter medium and in a real negatively charged electron gas. The electron gas model is the closest model we have towards understanding photon scattering of the quantum vacuum. However, there are several important differences between the charged electron gas medium and the electrically charged virtual fermion particles of the quantum vacuum as a medium.59First, and most importantly, virtual particles do not carry any net average energy. Instead an individual virtual particle ‘borrows’ a small amount of energy during it’s brief existence, which is then paid back quickly in accordance to the uncertainty principle. It is because quantum mechanics forbids knowing the value of two complementary variables precisely (in this case energy and existence time) for a virtual particle that virtual particles are allowed to exist at all. Therefore unlike the electron gas, the vacuum is incapable of permanently absorbing light that propagates through it. Thus the quantum vacuum does not absorb any light over macroscopic distance scales. This statement seems trivial, but it is never-the-less important when considering the quantum vacuum as a medium. On microscopic scales real photons are absorbed and re- emitted by individual virtual particles, in accordance with QED. Photon energy is lost in some collisions and regained in others so that on the average the energy loss is zero. This is because during the brief existence time of a virtual fermion particle, the virtual particle does possess energy, which is paid back almost immediately. This quantum process happens an enormous number of times as light travels through macroscopic distance scales. The energy balances out to zero over sufficiently large distance scales. Furthermore unlike the electron gas, there can be no dispersion of light in the quantum vacuum. In other words all frequencies of electromagnetic radiation move at the same speed through the quantum vacuum in spite of the incredible numbers of virtual particle interactions that occur for any particular frequency of photon. Zero dispersion follows experimentally from many astronomical observations of distant supernova, where there is a dramatic change in light and electromagnetic radiation with time. Observations have been made of specific events in the light curves of supernovae light curves that range from the radio band frequencies to the X-ray / Gamma Ray frequencies. All the different frequencies are observed to arrive on the earth at the same time. With distances of thousands or millions of light years away, any discrepancy in the photon velocity of supernovae at different frequencies would be very apparent. For example with the relatively nearby supernova 1987A (which exploded about 160,000 years ago in the Large Magellanic cloud) all the different frequencies of EM waves has reached us very much at the same time. If there had been a dispersion of only 0.01 m/sec in light velocity (i.e. 3 parts in 10-11) between two different frequencies, then the light of one frequency would arrive on the earth: 160000 x 365.25 x 24 x 60 x 60 x 10-2 / (3x10-8) =170 seconds or 2.8 minutes later! A result like this obviously disagrees with observations made of the spectrum of supernova 1987A. Spectra have been obtained for very distant supernovae up to a few billion light years away in other galaxies. One study places the maximum allowed dispersion to be on the order of 1 part in 10-21. Thus we conclude that there is no dispersion of light in the vacuum.60Is there a possibility for an index of refraction in the vacuum, as we have in an electron gas? Remember that an index of refraction requires two different media in which to compare the relative velocities of light. However the vacuum particle density must nearly uniform, with no transitions in density. Let us imagine a situation where somehow we have removed all the virtual particles in half of an empty box in vacuum, and the other half has the normal population of virtual particles in the normal quantum vacuum state. Would there be an index of refraction as light traveled from one side of the box to the other? This is a very important question because the validity of special relativity at the sub- microscopic distance scales comes into question here. You might think that if the vacuum has no energy, there should no effect on the propagation speed of photons. However we believe that the virtual particles in the quantum vacuum do indeed delay the progress of photons through electrically charged vacuum particle scattering effects. Thus we believe that photon scattering reduces the light velocity on the half of the box with electrically charged virtual particles. How can we justify this belief, in spite of the contradiction to special relativity? Special relativity is a classical theory, and was developed in the macroscopic domain of physics. It is almost impossible to measure light velocities over the extremely short distance scales that we are talking about. The electrically charged virtual particles in the quantum vacuum all have random velocities and move in random directions. They also have random energies ΔE during their brief life time Δt, which satisfies the uncertainty principle: ΔE Δt > h/(2 π). Imagine a real photon propagating in a straight path through the electrically charged virtual particles in a given direction. The real photon will encounter an equal number of virtual particles moving towards it as it does moving away from it. The end result is that the electrically charged quantum vacuum particles do not contribute anything different than the situation where all the virtual particles in the it’s path were at relative rest. Thus we can consider the vacuum as some sort of stationary crystal medium of virtual particles with a very high density, where each virtual particle is short-lived and constantly replaced (and carry no net average energy as discussed above). The progress of the real photon is delayed as it travels through this quantum vacuum ‘crystal’, where it meets uncountable numbers of electrically charged virtual particles. Light travels through this with no absorption or dispersion. Based on our general arguments above, we postulate that the photon is delayed as it travels through the quantum vacuum. We can definitely say that the uncertainty principle places a lower limit on the emission and absorption time delay, and forbids the time delay from being exactly equal to zero. Therefore we conclude that the electrically charged virtual particles of the quantum vacuum frequently absorb and re-emit the real photons moving through the vacuum by introducing small delays during absorption and subsequent re-emission of the photon, thus reducing the average propagation speed of the photons in the vacuum (compared to the light speed of photons between absorption/re-emission events).61Our examination of the physics literature has not revealed any previous work on a quantum time delay analysis of photon propagation through the quantum vacuum, presumably because of the precedent set by Einstein’s postulate of light speed constancy in the vacuum under all circumstances. We will take the position that the delays due to photon scattering through the quantum vacuum are real. These delays reduce the much faster and absolutely fixed ‘low-level light velocity cl’ (defined as the photon velocity between vacuum particle scattering events) to the average observed light velocity ‘c’ in the vacuum (300,000 km/sec) that we observe in our actual experiments. Furthermore, we propose that the quantum vacuum introduces a sort of Vacuum Index of Refraction ‘ nvac’ (compared to a vacuum without all virtual particles) such that c = cl / nvac. If this is true, what is the low-level light velocity? It is unknown at this time, but it must be significantly larger than 300,000 km/sec. In fact we believe that the vacuum index of refraction ‘ nvac’ must be very large because of the high density of virtual particles in the vacuum. This concept is required in EMQG theory, and has become central to understanding the equivalence principle and 4D space-time curvature in accelerated frames and in gravitational fields. 9.6 THE FIZEAU EFFECT: LIGHT VELOCITY IN A MOVING MEDIA It also has been known for over a century that the velocity of light in a moving medium differs from its value in the same stationary medium. Fizeau demonstrated this experimentally in 1851 (ref. 41). For example, with a current of water (with refractive index of the medium of n=4/3) flowing with a velocity V of about 5 m/sec, the relative variation in the light velocity is 10-8 (which he measured by use of interferometry). Fresnel first derived the formula (ref. 41) in 1810 with his ether dragging theory. The resulting formula relates the longitudinal light velocity ‘ vc’ moving in the same direction as a transparent medium of an index of refraction ‘n’ defined such that ‘ c/n’ is the light velocity in the stationary medium, which is moving with velocity ‘V’ (with respect to the laboratory frame), where c is the velocity of light in the vacuum: Fresnel Formula: vc = c/n + (1 – 1/n2) V (9.61) Why does the velocity of light vary in a moving (and non-moving) transparent medium? According to the principles of special relativity, the velocity of light is a constant in the vacuum with respect to all inertial observers. When Einstein proposed this postulate, he was not aware that the vacuum is not empty. However he was aware of Fresnel’s formula and derived it by the special relativistic velocity addition formula for parallel velocities (to first order). According to special relativity, the velocity of light relative to the proper frame of the transparent medium depends only on the medium. The velocity of light in the stationary medium is defined as ‘ c/n’. Recall that velocities u and v add according to the formula: (u + v) / (1 + uv/c2) Therefore:62vc = [ c/n + V ] / [ 1 + ( c/n) (V)/c2 ] = ( c/n + V) / ( 1 + V/( nc) ) ≈ c/n + (1 – 1/n2) V (9.62) The special relativistic approach to deriving the Fresnel formula does not say much about the actual quantum processes going on at the atomic level. At this scale, there are several explanations for the detailed scattering process in conventional physics. We investigate these different approaches in more detail below. 9.7 LORENTZ SEMI-CLASSICAL PHOTON SCATTERING The microscopic theory of the light propagation in matter was developed as a consequence of Lorentz’s non-relativistic, semi-classical electromagnetic theory. We will review and summarize this approach to photon scattering, which will not only prove useful for our analysis of the Fizeau effect, but has become the basis of the ‘ Fizeau-like’ scattering of photons in the accelerated quantum vacuum near large gravitational fields. To understand what happens in photon scattering inside a moving medium, imagine a simplified one-dimensional quantum model of the propagation of light in a refractive medium. The medium consisting of an idealized moving crystal of velocity ‘V’, which is composed of evenly spaced, point-like atoms of spacing ‘l’. When a photon traveling between atoms at a speed ‘c’ (vacuum light speed) encounters an atom, that atom absorbs it and another photon of the same wavelength is emitted after a time lag ‘ τ’. In the classical wave interpretation, the scattered photon is out of phase with the incident photon. We can thus consider the propagation of the photon through the crystal is a composite signal. As the photon propagates, part of the time it exists in the atom (technically, existing as an electron bound elastically to some atom), and part of the time as a photon propagating with the undisturbed low-level light velocity ‘c’. When the photon changes existence to being a bound electron, the velocity is ‘V’. From this, it can be shown (ref. 41, an exercise in algebra and geometry) that the velocity of the composite signal ‘ vc’ (ignoring atom recoil, which is shown to be negligible) is: vc = c [1 + (V τ/l) (1 - V/c)] / [1 + (c τ/l) (1 - V/c)] (9.71) If we set V=0, then vc = c / (1 + c τ/l) = c/n. Therefore, τ/l = (n – 1)/c. Inserting this in the above equations give: vc = [( c/n) + (1 – 1/n) V (1 - V/c)] /[1 - (1 – 1/n)(V/c)] ≈ c/n + (1 – 1/n2) V (to first order in V/c). (9.72) Again, this is Fresnel’s formula. Thus the simplified non-relativistic atomic model of the propagation of light through matter explains the Fresnel formula to the first order in V/c through the simple introduction of a scattering delay between photon absorption and subsequent re-emission. This analysis is based on a semi-classical approach. What does63quantum theory say about this scattering process? The best theory we have to answer this question is QED. 9.8 PHOTON SCATTERING IN THE ACCELERATED VACUUM Anyone who believes in the existence of virtual fermion particles in the quantum vacuum that carry mass, will acknowledge the existence of a coordinated general downward acceleration of these virtual particles near any large gravitational field. In EMQG gravitons from the real fermions on the earth exchange gravitons with the virtual fermions of the vacuum (which carry electric charge), causing a downward acceleration. The virtual particles of the quantum vacuum (now accelerated by a large mass) acts on light (and matter) in a similar manner as a stream of moving water acts on light (and matter) in the Fizeau effect. How does this work mathematically? Again, it is impossible to compute the interaction of an accelerated collection of virtual particles of the quantum vacuum with light exactly. However, a simplified model can yield useful results. We will proceed using the semi-classical model proposed by Lorentz, above. We have defined the raw light velocity ‘ cr’ (EMQG, ref. 1) as the photon velocity in between virtual particle scattering. Recall that raw light velocity is the shifting of the photon information pattern by one cell at every clock cycle on the CA, so that in fundamental units it is an absolute constant. Again, we assume that the photon delay between absorption and subsequent re-emission by a virtual particle is ‘ τ’, and the average distance between virtual particle scattering is ‘l’. The scattered light velocity vc(t) is now a function of time, because we assume that it is constantly varying as it moves downwards towards the surface in the same direction of the virtual particles. The virtual particles move according to: a = gt, where g = GM/R2. Therefore we can write the velocity of light after scattering with the accelerated quantum vacuum: vc(t) = cr [1 + ( gtτ/l) (1 - gt/cr)] / [1 + ( crτ/l) (1 - gt/cr)] (9.81) If we set the acceleration to zero, or gt = 0, then vc(t) = cr / (1 + crτ/l) = cr/n. Therefore, τ/l = (n – 1)/c r. Inserting this in the above equation gives: vc(t) = [( cr/n) + (1 – 1/n) gt (1 - gt/cr)] / [1 - (1 – 1/n)( gt/cr)] ≈ cr/n + (1 – 1/n2) gt …. to first order in gt/cr. (9.82) Since the average distance between virtual charged particles is very small, the photons (which are always created at velocity cr) spend most of the time existing as some virtual charged particle undergoing downward acceleration. Because the electrically charged virtual particles of the quantum vacuum are falling in their brief existence, the photon effectively takes on the same downward acceleration as the virtual vacuum particles (as an average acceleration over macroscopic distances). In other words, because the index of64refraction of the quantum vacuum ‘n’ is so large (compared to no vacuum particles), and because c = cr/n and we can write in equation 9.61: vc(t) = cr/n + (1 – 1/n2) gt = c + gt = c (1 + gt/c) if n >>1. (9.83) Therefore for photons going with the flow (downwards): vc(t) = c (1 + gt/c) (9.84) Similarly, for photons going against the flow (upwards): vc(t) = c (1 - gt/c) (9.85) as the refractive index of the quantum vacuum n ® ¥. Remember that this is a semi-classical derivation, and does not constitute an actual proof of the scattering effects on photons. An important limitation of these results is the issue of space-time, which we cover in the next section. This issue limits the accuracy of these expressions to small local regions of distance and time. These formulas are used by EMQG as a starting point for the expression for the variation of light velocity near a large gravitational field at a point, and we show that this leads to the correct amount of general relativistic 4D space-time curvature, taking into account some additional assumptions. To see the connection between variable light velocity and space-time effects consider the following thought experiment. Imagine that two clocks that are identically constructed, and each calibrated with a highly stable monochromatic light source in the same reference frame. These clocks keep time by using a high-speed electronic divider circuit that divides the light output frequency by “n” such that an electrical voltage pulse is produced every second. For example, the light frequency used as the clock is precisely calibrated to 1015 Hz; this light frequency is converted in to an electronic pulse train of the same frequency, where it is divided by 1015 to give one electronic pulse every second. Another counter in this clock increments every time a pulse is sent, thus displaying the total time elapsed in seconds on the clock display. Now, let us place these two clocks in a gravitational field on earth with one of them on the surface, and the other at a height “h” above the surface. Imagine that the clocks are compared every second to see if they are still running in unison in the gravitational field by exchanging light signals. We would find that as time progresses, the clocks loose synchronism, where the lower clock appears to run slower than the higher. According to general relativity, light always maintains a constant speed, and 4D space-time curvature is responsible for the difference in the timing of the two clocks, where the lower clock runs slower. We argue that the accelerated Fizeau-like quantum vacuum fluid affects the light velocity of the exchanged light signals, and it also affects the atomic photon emission process. In the next section we derive the same time dilation effect predicted by general relativity using eq. 9.85, which assumes that the light velocity has exactly the same downward acceleration component of the falling electrically charged virtual particles of the quantum vacuum (and a very high vacuum index of refraction).659.9 SPACE-TIME CURVATURE OBTAINED FROM SCATTERING THEORY In this section we are in a position to mathematically formulate EMQG 4D space-time curvature for the specific example of spherically symmetric mass. We derive the expected amount of space-time curvature effects near a spherical, non-rotating, massive object using our results from scattering theory in the previous section. We compare our results to those obtained from the Schwarzchild metric in section 9.1, which is the standard textbook method of general relativity. We now take a bold step (that we justify later, with certain qualifiers to do with space and time that is inherent in these formulas) and assume that for the case on the surface of the earth, equation 9.85 holds. Therefore: ch = c(1 - gt/c) = c(1 - gh/c2), where t=h/c. This describes the propagation of photons moving upwards for a distance h, but we assume it holds only for very short distances , where h ®0. Technically this is true only at a point, which means that this equation must be written in differential form. The reason that we cannot say that it holds for large distances is because light velocity happens to also depend on the very nature of space and time, because the expression ‘c(1 - gh/c2)’ involves both space and time, implicitly. Einstein himself derived a similar expression in 1911 for the velocity of light in a gravitational field (ref. 55), before developing his general theory of relativity (ref. 56): cccso 0 21=+  ΔΦ … Einstein’s Formula (9.90) where ΔΦSO is the difference of the gravitational potential of the source and the observation points S and O. This, however, leads to incorrect experimental values. Later this was corrected by himself in 1915. However, Einstein was not aware of the existence of the quantum vacuum at that time, and attributed this result to the degradation of the photon energy as it climbs out of a gravitational potential. In what follows, we ignore the special relativistic postulate of the constancy of light velocity for now, and return to this issue later. We will take the position that photons continuously vary their velocity (remember that on the CA, the velocity of light is still an absolute constant in between vacuum particle scattering events) by scattering with the falling, electrically charged virtual particles. In the scattering picture of light propagation, we found that the velocity of light takes on a different value when traveling straight upwards, compared to straight downwards, an effect that was independently proposed by V. Petkov (ref. 47, 1999). According to Petkov:66“Up to now little attention has been paid to an expression for the velocity of light in a gravitational field derived by Einstein in 1911. … it is shown that the proper velocity of light is anisotropic in non-inertial frames …” Petkov believes that this can actually be measured experimentally (ref. 47). However, we disagree! Light plays a central role in defining the nature of space and time. If light velocity varies on the earth, then why is it that we do not actually observe this variation in light velocity in real experiments? Part of the answer, of course, lies in the incredibly small variation of light velocity on the earth (on the order of one part in 10-15). More importantly, it involves the deep connection between the nature of space-time and the propagation of light itself, first discovered by Einstein himself in his work on special relativity (1905)! Einstein showed that many concepts related to space and time are traceable to the behavior of light. For example, the concept of simultaneous events is deeply related to the motion of light (ref. 19, 20, 21). The derivation of the Lorentz transformations and the resulting time dilation and Lorentz contraction effects are solely based on the behavior of light. These considerations, and the results of section 9.2 lead us to believe that variations of light velocity will result in space-time variations on actual clocks and rulers. EMQG provides a plausible mechanism for light velocity variation: the scattering of photons with the falling vacuum near a large gravitational mass. This concept is based on the concept that the vacuum acts like Fizeau-like quantum vacuum fluid. We will assume that light velocity varies, (but only for short distances) as it moves upward from the surface of the earth. As the photon moves upward from point r to point r+ Δr it decelerates at -1g according to equation 16.54, and obtains a new velocity a short time Δt later: cr rcrgt c( )() +=−   ΔΔ1 (9.91) Since Δt = Δr /c for small distances (in the limit as Δr→0), we can then write: cr rcrgr c( )() +=−   ΔΔ1 2(9.92) Since, g = GM/r2 at point r above the center of the earth, we can write this as: cr rcrGM r rc( )() +=−   ΔΔ1 22(9.93) Since, the only observable property of light that we can be sure about is the red shift, as discussed in section 9.2, and with c = ν λ, it follows that the change in frequency is:67n n ( ) () r r rGM r rc+=−   ΔΔ1 22(9.94) from which the corresponding wavelength appears longer by the same factor, or l l ( ) () r r rGM r rc+=+   ΔΔ1 22(9.95) To find the total change in frequency from point r on the earth’s surface to infinity (no gravity), we integrate for the change in frequency GM Δr / r2 c2, with respect to r as follows: GM rcdrGM rcr22 20∞ ∫=+  (9.96) n n () () ∞=−   rGM rc1 2(9.97) But, since ν = 1/t by definition, therefore time must be affected as follows: 1 11 2t trGM rc () () ∞=−   (9.98) Finally, we have: trtGM rc()() =∞−  1 2(9.99) which is the same expression for time dilation in a gravitational field we obtained from the Schwarzchild metric from equation 9.14, i.e. trGM rct ()( )() ≈−∞ 1 2. Similarly, wavelength received at infinity is increased by the following expression: l l () () ∞=−   rGM rc1 2 (9.991) Now, an observer at infinity can use the light signal from the surface of the earth to make measurements of distance in his reference frame at infinity. For example, suppose that in his own reference frame, a reference laser light source is used to measure a given reference length, and say that this corresponds to 1,000,000 wavelengths or 106 λr, where λr is the68reference wavelength. Subsequently, he uses the light received from the surface of the earth from an identically constructed reference laser light source ( λr) to measure the same length, and finds that when he counts the standard 1,000,000 wavelengths the reference length has shortened (because of the wavelength increase). In general he concludes that the distances at infinity s( ∞ ) are contracted by the amount: s srGM rc()() ∞=−  1 2(9.992) compared to distances s( r ) on the surface of the earth. Finally, we can write: srs GM rc()()=∞ −1 2(9.993) which is again, exactly the same expression for length that we obtained from the Schwarzchild metric in eq. 16.15. This equation specifies the amount of distortion for rulers on the earth “s( r )” compared to rulers positioned at infinity “s( ∞ )”. Thus by postulating that it is the light velocity that is actually varying (and not space-time curvature), we are led to the same amount of red shift, and the same amount of space-time curvature. DISCUSSION So why insist that it is light velocity rather than space-time that is responsible for these effects on earth, especially when it becomes impossible experimentally to tell the difference between the two results? With the space-time approach we have to assume, without any prior reason, that a large mass curves space-time. We have to accept that it just does this, without understanding the action principle! With the variable light velocity approach the physical action that exists between matter and space-time becomes well understood! Matter acts on the vacuum particles, and the vacuum particles act on matter and light, to give a curved space-time. The other advantage with the EMQG approach is that the equivalence of space-time in accelerated and gravitational fields also becomes clear. Therefore, the principle of equivalence is seen to hold from a common cause, the action of the quantum vacuum which appears to be the same to a light particle or a mass in accelerated frames and in gravitational frames. In short, we have closure as to why the principle of equivalence holds for space-time! We can now see that in order to formulate a theory of gravity involving observers with measuring instruments (such as clocks and rulers) we must take into account how these69measurements are affected by the local conditions of the quantum vacuum. Our analysis above shows that quantum vacuum can be viewed as a Fizeau-like fluid undergoing downward acceleration near a massive object, which affects the velocity of light. Indeed, not only is the velocity of light affected, it is all the particle exchange processes including graviton exchanges. Therefore, we find that the accelerated Fizeau-like ‘quantum vacuum fluid’ effects all forces. This has consequences for the behavior of clocks, which are constructed with matter and forces. After all, nobody questions the fact that a mechanical clock submerged in moving water cannot keep proper time with respect to an external clock. Similarly, a clock near a gravitational field (with a Fizeau-like, quantum vacuum fluid flow through the clock) also cannot be expected to keep proper time with respect to an observer outside the gravitational field. The accelerated Fizeau-like ‘quantum vacuum fluid’ moves along radius vectors directed towards the center of the earth, and thus has a specific direction of action. Therefore, the associated space-time effects should also work along the radius vectors (and not parallel to the earth). For the case of light moving parallel to the earth's surface, the light path is the result of a tremendous number of photon to virtual particle scattering interactions (figure 10). Again in between virtual particle scattering, the light velocity is constant and ‘straight’. The total path is curved as shown in figure 10. The path the light takes is called a geodesic in general relativity. In EMQG, this path simply represents the natural path that light takes through the accelerated vacuum, which affects the motion of light. For the case of light moving parallel to the floor of the accelerated rocket (figure 11), the path for light is also the result of virtual particle scattering, but now the quantum vacuum is not in a state of relative acceleration. Therefore, the path is straight for the observer outside the rocket. The observer inside the rocket sees a curved path simply because he is accelerating upwards. We now see why Einstein’s gravitational theory takes the form that it does. Because of the continuously varying frequency and wavelength of the light with height, Einstein interpreted this as a variation of space and time with height. We postulated that the scattering of light with the falling vacuum changes the light velocity in absolute CA units, which cause the measurements of space and time to be affected. As we have already seen, these two alternative explanations cannot be distinguished by direct experimentation. This is why the principle of the constancy of light velocity is still a postulate in general relativity (through the acceptance of special relativity). We are now in a position to understand the concept of the geodesic proposed by Einstein. The downward acceleration of the virtual electrically charged masseons of the quantum vacuum serves as an effective ‘electrical guide’ for the motion of light (and for test masses) through space and time . This ‘electrical guide’ concept replaces the 4D space-time geodesics that guide matter in motion in relativity. For light, this guiding action is through the electromagnetic scattering process of section 9.4. For matter, the electrically charged virtual particles guide the particles of a mass by the electrical force70interaction that results from the relative acceleration. Because the quantum vacuum virtual particle density is quite high, but not infinite (at least about 1090 particles/m3), the quantum vacuum acts as a very effective reservoir of energy to guide the motion of light or matter. The relative nature of 4D space-time can now be easily seen. Whenever the background virtual particles of the quantum vacuum are in a state of relative acceleration with respect to an observer, the observer lives in curved 4D space-time. Why should the reader accept this new approach, when both approaches give the same result? The reason for accepting EMQG is that the action between a large mass and 4D space-time curvature becomes quite clear. The reason that 4D space-time is curved in an accelerated reference is also clear, and very much related to the gravitational case. The relative nature of curved 4D space-time also becomes very obvious. An observer inside a gravitational field would normally live in a curved 4D space-time. If he decides to free-fall, he cancels his relative acceleration with respect to the quantum vacuum, and 4D space-time is restored to flat 4D space-time for the observer. The principle of general covariance no longer becomes a principle, but merely results for the deep connection between the quantum vacuum state for accelerated frames and gravitational frames. Last, but not least, the principle of equivalence is completely understood as a reversal of the (net statistical) relative acceleration vectors of the charged virtual masseons of the quantum vacuum, and real masseons that make up a test mass. 10. EXPERIMENTAL VERIFICATION OF THE EQUIVALENCE MODEL We have suggested that the strong principle of equivalence does not hold at all in section 8. Mass Equivalence is also not perfect (section 8). Imagine that a very large test mass and a very small test mass are dropped simultaneously on the earth (in a vacuum), We predict that there will be an extremely small difference in the arrival time of the masses on the surface of the earth, which is in slight violation of the principle of equivalence. This occurs because on the earth there are direct graviton exchanges between the test mass and the earth, which tend to unbalance perfect equivalence. The larger test mass has a much larger excess of gravitons exchanged compared to the tiny mass, resulting in a greater pure gravitational force of attraction. However, the electrically charge quantum vacuum dominates over these pure graviton forces, and stabilizes the fall rate and causes the masses to reach the ground almost at the same time. This is because the electrical force from the vacuum is on the order of 1040 times greater than the pure graviton generated force for these test masses. We propose several new experimental tests of the principle of equivalence that gives results that are different from the conventional general relativistic physics. These experiments are designed to show that the equivalence principle is not a perfect symmetry of nature, and contains has a few flaws.71(1) ANTI-MATTER GRAVITATIONAL PHYSICS - EMQG opens up a new field of investigation, which we call anti-matter gravitational physics. We propose that if two sufficiently large pieces of anti-matter are manufactured to allow measurement of the mutual gravitational interaction, then the gravitational force will be found to be repulsive! The force will be equal in magnitude to -GM2/r2 where M is the mass of each of the equal anti-matter masses, r is their mutual separation, and G is Newton’s gravitational constant). This is in clear violation of the principle of equivalence, since in this case Mi = - M g , instead of masses Mi = M g. Antimatter that is accelerated in far space has the same inertial mass ‘ Mi’ as ordinary matter, but when interacting gravitationally with another antimatter mass it is repelled (M g). (Note: The earth will attract bulk anti-matter because of the large abundance of gravitons originating from the earth of the type that induce attraction). This means that no violation of equivalence is expected for anti-matter dropped on the earth, where anti-matter falls normally (recall that virtual masseons and anti- masseons are both attracted to the earth). However, an antimatter earth will repel an antimatter mass dropped on the earth. Recent attempts at measuring earth’s gravitational force on anti-matter (e.g. anti-protons will not reveal any deviation from the principle of equivalence). (2) VIOLATION OF THE WEAK EQUIVALENCE PRINCIPLE - For an extremely large test mass and a very small test mass dropped simultaneously on the earth (in a vacuum free of air resistance), there will be an extremely small difference in the arrival time of the masses, in slight violation of the principle of equivalence. This effect is on the order of ≈ ΔN x δ, where ΔN is the difference in the number of masseon particles in the two masses, and δ is the ratio of the gravitational to electric forces for one masseon. This experiment is very difficult to perform on the earth, because δ is extremely small ( ≈10-40), and ΔN cannot be made sufficiently large. To achieve a difference of ΔN =1030 particles between the small and large mass requires dropping a molecular-sized atomic cluster and a large military tank simultaneously in the vacuum in order to give a measurable deviation. Note: For ordinary objects that might seem to have a large enough difference in mass (like dropping a feather and a tank), the difference in arrival time may be obscured by background interference, or by quantum effects like the Heisenberg uncertainty principle which restrict the accuracy of time measurements. (3) DETECTION OF THE VIOLATION OF THE STRONG PRINCIPLE OF EQUIVALENCE USING A GRAVITON DETECTOR - The strong equivalence principle does not hold at all, as we suggested in section 8! To see this, we suggest a thought experiment involving a hypothetical graviton detector. If gravitons can be detected by the invention of a graviton detector/counter in the distant future, then there will be easy experimental proof for the violation of the strong principle of equivalence. The strong equivalence principle states that all the laws of physics are the same for an observer situated on the surface of the earth as it is for an accelerated observer on a rocket (1 g). The graviton detector will find a tremendous difference in the graviton count in these two cases, because gravitons are vastly more numerous here on the earth due to the vast numbers of masseons in the earth. On the rocket, the72graviton count would be negligible. Therefore a visual indicator could be placed on the graviton detector box that would easily distinguish between accelerated frames and gravitational frames. This is a gross violation of the strong equivalence principle. (4) REDUCTION OF GRAVITATIONAL MASS ELECTROMAGNETICALY - Since mass has a strong electromagnetic force component, a sensitive gravitational mass measurement near the earth might be disrupted by experimentally manipulating the electrically charged virtual particles of the nearby quantum vacuum through electromagnetic means . If a rapidly fluctuating magnetic field (or rotating magnetic field) is produced under a mass it might effect the instantaneous virtual charged particle spectrum, and disrupt the tiny electrical forces contributed by each electrically charged masseon of the mass. This may reduce the measured gravitational mass of an object in the vicinity (this would also affect the inertial mass). In a sense, this device would act like a primitive weak “anti-gravity” device. The virtual particles are constantly being “turned-over” in the vacuum at different rates depending on the energy, with the high frequency particles (and therefore, high-energy particles) being replaced the quickest. If a magnetic field is made to fluctuate fast enough so that it does not allow the new virtual particle pairs to replace the old and smooth out the disruption, the spectrum of the vacuum will be altered. According to conventional physics, the energy density of virtual particles is infinite, which means that all frequencies of virtual particles are present. In EMQG there is a definite upper cut-off to the frequency, and therefore the highest energy according to the Plank’s law: E=hυ, where υ is the frequency that a virtual particle can have. This frequency cutoff is very roughly on the order of the plank distance scale. We can therefore state that the smallest wavelength that a virtual particle can have is on the order of about 10-35 meters, e.g. the plank wavelength (or a corresponding maximum Plank frequency of about 1043 hertz for very high velocity ( ≈c) virtual particles). Unfortunately for our “anti-gravity” device, it is technologically impossible to disrupt the highest frequencies. According to the uncertainty principle, the relationship between energy and time is: ΔE x Δt < h. This means that the high frequency end of the spectrum consists of virtual particles that “turns-over” the fastest. To give measurable mass change the higher frequencies of the vacuum must be disrupted, which requires magnetic fluctuations on the order of at least 1020 cycles per seconds. Therefore, only lower frequencies virtual particles of the vacuum can be practically affected, and only small changes in the measured mass can be expected with today’s technology. As a result of this, a relationship should exist between the amount of gravitational (or inertial) mass loss and the frequency of electromagnetic fluctuation or disruption. The higher the frequency the greater the mass loss. Work on the Quantum Hall Effect (ref. 29) by Laughlin has suggested that the electron density in a two-dimensional sheet under the influence of a strong magnetic field causes the electrons to move in concert, with very high speed swirling vortices created in the resulting 2D electron gas. In ordinary magnetic fields, electrons are merely ‘pushed’ around, while a strong magnetic field causes the electrons to swirl in high-speed ‘whirlpools’. There is also a possibility that this ‘whirlpool’ phenomena holds for the virtual particles of the quantum vacuum under the influence of a strongly fluctuating magnetic field. These high-speed73whirlpools might disrupt the high frequency end of the spectral distribution of electrically charged virtual particles in small pockets. Therefore, there might be a greater mass loss under these circumstances. Recent experiments on mass reduction with rapidly rotating magnetic fields are inconclusive at this time. Reference 30 gives an excellent and detailed review of the various experiments on reducing the gravitational force with superconducting magnets. 11. CONCLUSIONS Using a newly developed quantum field theory of gravity theory called EMQG, we have illustrated the hidden quantum processes involved in Einstein’s principle of equivalence. We found that almost the same quantum processes occurring in inertial mass are also happening in gravitational mass. We found that gravity involves two pure, quantum force particle exchange processes. Both the photon and graviton particle exchanges occur simultaneously for a test mass in a large gravitational field, whereas for inertial mass only photon exchanges are involved. We modified a new theory of inertia first introduced by Haisch, Rueda, and Puthoff, which we call HRP inertia. In HRP inertia, inertia is the resulting electromagnetic force interaction of the charged ‘ parton’ particles making up a mass with the background virtual photon field, which is called the zero point fluctuations (or ZPF). We modified HRP inertia, (which we call Quantum Inertia, or QI), which involves the introduction of a new particle of nature called the masseon. The masseon contains the smallest possible quanta of electric charge as well as the smallest possible quanta of ‘mass-charge’. Masseons combine with other masseons to produce all the known fermion mass particles of the standard model. The masseon is electrically charged, as well as possessing a new form of charge called ‘mass-charge’. Mass-charge is analogous to electric charge, where gravitons take the place of photons as the exchange particle for the pure gravitational force, and masseons take the place of electrons as the charge source and destination. The physics of graviton exchanges between masseons is virtually the same as photon exchanges for electrons in QED. Gravitons have spin 1, just as the photon (not spin 2). Quantum Inertia is based on the idea that inertial force is due to the tiny electromagnetic force interactions originating from each charged masseon particle of real matter undergoing relative acceleration with respect to the vast swarm of virtual, electrically charged masseon particles of the nearby quantum vacuum. These tiny forces add up to the total resistance force opposing the accelerated motion in Newton’s law ‘F = MA’, where the sum of each of the tiny masseon forces equals the total inertial force. When masseons move through the vacuum at a constant velocity (i.e., an inertial frame), the sum of the total vacuum forces is zero, and the vacuum does not oppose the motion. Therefore, the virtual masseons of the quantum do not act as a form of an ether for inertial observers.74We proposed that gravity also involves this very same ‘inertial’ electromagnetic force component found inside an accelerated mass above. This is the source of the deep connection between inertia and gravity, which is at the heart of Newtonian mass equivalence. Since virtual masseons possesses mass-charge, and the earth (composed of real masseons) possesses mass-charge, the result is that virtual masseons of the quantum vacuum fall during their very brief life-times, which has profound effects on test masses. Newtonian mass equivalence has been explained as a consequence of the falling quantum vacuum. Inside a test mass subjected to a large gravitational fields, there exists a similar quantum vacuum process that occurs for an inertial mass, where the roles of the real charged masseon particles of the mass and the virtual, electrically charged masseons of the quantum vacuum are reversed. Now it is the electrically charged virtual masseon particles of the quantum vacuum that are accelerating (falling), while the mass particles are at relative rest. The reason why the virtual particles of the quantum vacuum fall in a large gravitational field is the huge numbers of graviton particles that are exchanged between the earth and the surrounding virtual masseon particles of the quantum vacuum. Furthermore, the general relativistic Weak Equivalence Principle (WEP) also results from this common physical process existing at the quantum level in both gravitational mass and inertial mass. The falling, electrically charged, virtual masseons of the quantum vacuum affect both the motion of real test masses, and also the motion of light. The action of the falling quantum vacuum on light is very much reminiscent of the action of flowing water on the motion of light in the Fizeau experiment. The path that light takes while moving parallel to the surface of the earth, through the falling quantum vacuum, is curved . This implies space-time curvature. The action of falling ‘ Fizeau-like’ quantum vacuum on clocks and rulers is responsible for the origin Riemannian curved 4D space-time geometry near the earth, and is the basis of a quantum theory of general relativity. Therefore based on a new theory of quantum gravity called EMQG, we have discovered the hidden quantum interactions that occur in Newtonian inertia and the quantum machinery responsible for Einstein’s Weak Equivalence Principle. 12. REFERENCE S (1) ELECTROMAGNETIC QUANTUM GRAVITY : On the Quantum Principle of Equivalence, Quantum Inertia, and the Meaning of Mass , by Tom Ostoma and Mike Trushyk, Sept. 11, 1998, LANL E-Print Server, http:// xxx.lanl.gov document #: physics/9809042, and for a brief summary of EMQG see physics/9902066 (24 pages). (2) CELLULAR AUTOMATA: THEORY AND EXPERIMENT Edited by H. Gutowitz, 1991. Contains many reprints from Physica D. See pg. 254 for an excellent article by Edward Fredkin titled ‘DIGITAL MECHANICS’. (3) SPECIAL RELATIVITY DERIVED FROM CELLULAR AUTOMATA THEORY: The origin of the universal speed limit by Tom Ostoma and Mike Trushyk, Oct. 7, 1998, LANL E-Print Server, physics/9810010 and the APS E-Print server.75(4) WHAT HAS AND WHAT HASN’T BEEN DONE WITH CELLULAR AUTOMATA by by H. V. McIntosh, Nov 10, 1990, LANL Archives. (5) INERTIA AS A ZERO-POINT-FIELD LORENTZ FORCE by B. Haisch, A. Rueda, and H.E. Puthoff; Physical Review A, Feb. 1994. This landmark paper provides the first known proposal that inertia can be understood as the interactions of matter with the surrounding virtual particles. (6) THEORY AND EXPERIMENT IN GRAVITATIONAL PHYSICS by C. M. Will, Chap. 2, pg. 24, Cambridge University Press, 1985. (7) GENERAL RELATIVITY AND EXPERIMENT by T. Damour, Dec. 8, 1994, LANL Archives. (8) ESSENTIAL RELATIVITY: SPECIAL, GENERAL, AND COSMOLOGICAL , 2nd Edition, Springer-Verlag, 1977. Chap. 1. One of the best reviews of the status of inertia, the principle of equivalence, and Mach’s principle. (9) THE RECURSIVE UNIVERSE: CHAOS, COMPLEXITY, AND THE LIMITS OF SCIENTIFIC KNOWLEDGE by W. Poundstone, 1988, Oxford Univ. Press. Chap. 2 contains a very good survey of the Game of Life. (10) CAN WE SCALE THE PLANK SCALE? by D. J. Gross, Physics Today, June 1989, pg.9. (11) THE ENERGETIC VACUUM by H.E. Puthoff, Speculations in Science and Technology, vol. 13, No. 4, pg. 247-257, 1990. (12) Physical Review Letters , Vol. 78, pg5. (13) THE DETECTION OF GRAVITATIONAL WAVES edited by D.G. Blair, Chap. 1. (14) RELATIVITY OF MOTION IN VACUUM by M. Jaekel, .., LANL archives, quant-ph/9801071, Jan.30 1998. (15) QUANTUM FIELDS IN CURVED SPACE by N.D. Birrell & P.C.W. Davies, Cambridge Monographs, chap. 3.3, pg. 48 (16) SOV. PHYS. – DOKL. 12, 1040 by A.D. Sakharov, 1968 and THEOR. MATH. PHYS. 23, 435 (1975) by A.D. Sakharov. (17) PARTICLE CREATION BY BLACK HOLES by S. W. Hawking, Commun. Math. Phys. 43, 199-220 (1975). (18) QUANTUM FIELDS IN CURVED SPACE by N.D. Birrell & P.C.W. Davies, Cambridge Monographs, chap. 8.2, pg. 264. (19) PHYSICS FOR SCIENTISTS AND ENGINEERS by R. Serway, Chap. 5. (20) ESSENTIAL RELATIVITY by W. Rindler, Chap. 5. (21) SPECIAL RELATIVITY by A.P. French, Chap. 7, pg.214. (22) GENERAL RELATIVITY by I.R. Kenyon, Chap. 2. (23) RADIATION FROM A UNIFORMLY ACCELERATED CHARGE AND THE EQUIVALENCE PRINCIPLE by S. Parrott, LANL Archives, GR-QC/9303025V4, Jan. 23, 1996. (24) GENERAL RELATIVITY AND EXPERIMENT by T. Damour, LANL Archives, GR-QC / 9412024, Dec. 8, 1994. (25) PHYS. Z. 19 , 156-63 Lense J and Thirring H (English Translation 1984 Gen. Rel. Grav. 16 711- 50). (26) TEST OF LENSE-THIRRING ORBITAL SHIFT DUE TO SPIN by I. Ciufolini, F. Chieppa, D. Lucchesi, and F. Vespe, Class. Quantum Grav. (1997) 2701-2726. (27) NEAR ZERO Edited by J.D. Fairbank, Chap. VI.2 – Gravitomagnetism by Kip S. Thorne. (28) NEAR ZERO Edited by J.D. Fairbank, Chap. VI.3(A) – The Stanford Relativity Gyroscope Experiment, C. W. F. Everitt. (29) SPLITTING THE ELECTRON by B.Daviss, New Scientist, Jan. 31, 1998. (30) SUPERCONDUCTORS AND GRAVITY SHIELDING: REFERENCES AND EXPERIMENTS, Internet Web address: Error! Reference source not found. . (31) RELATIVITY: THE GENERAL THEORY by J.L. Synge, 1971, North-Holland, Amsterdam, p. IX.76(32) VERH. DEUTSCH. PHYS. GES. 18, 83 , 1916, W. Nernst. (33) KOLTICK , Measurement of electron charge screening. (34) DIGITAL MECHANICS: An Informational Process based on Reversible Cellular Automata by Edward Fredkin, Physica D 45 (1990) 254-270. (35) INTRODUCTION TO THE THEORY OF RELATIVITY by P.G. Bregmann, Chap. IV, pg.33. (36) UNIVERSITY PHYSICS by H. Benson, Wiley, Chap. 39, pg. 797 (37) ON THE “DERIVATION” OF EINSTEIN’S FIELD EQUATIONS by S. Chandrasekhar, AJP Volume 40, pg. 224 (1972). (38) ESSENTIAL RELATIVITY by W. Rindler, Chap. 1, pg. 10 (Rise and fall of Absolute Space). (39) GRAVITATION AND COSMOLOGY by S. Weinberg, Chap. 8, pg. 179. (40) VERIFICATION OF THE EQUIVALENCE OF GRAVITATIONAL AND INERTIAL MASS FOR THE NEUTRON by L. Koester, Physical Rev. D, Vol. 14, Num. 4, pg.907 (1976). (41) DOES THE FIZEAU EXPERIMENT REALLY TEST SPECIAL RELATIVITY by G. Clement, Am. J. Phys. 48(12), Dec. 1980. (42) THE FEYNMAN LECTURES ON PHYSICS by Feynman, Leighton, and Sands, Vol. 1, Chap. 31 The Origins of the Refractive Index. (43) OPTICAL COHERENCE AND QUANTUM OPTICS by L. Mandel and E. Wolf., Cambridge (44) POTENTIAL THEORY IN GRAVITY AND MAGNETIC APPLICATIONS by R.J. Blakely, Cambridge. (45) THE COSMOLOGICAL CONSTANT IS PROBABLY ZERO by S.W. Hawking, Physics Letters, Vol. 134B, Num. 6 (1984), pg. 403. (46) THE PROBLEM OF THE COSMOLOGICAL CONSTANT by T. Ostoma, M. Trushyk, March 1999, LANL E-Print Server, #physics/9903040. (47) DOES GRAVITATIONAL REDSHIFT SUPPORT THE CURVED-SPACETIME INTERPRETATION OF GENERAL RELATIVITY by Vesselin Petkov, Nov 1999, LANL E- Print Server, #gr-qc/9810030 v8. (48) WHAT ARE THE HIDDEN QUANTUM PROCESSES BEHIND NEWTON’S LAWS? By T. Ostoma and M. Trushyk, LANL Archive, http://xxx.lanl.gov/physics/9904036 April 1999. (49) H.B.G. CASIMIR, Proc. K. Med. Akad. Wet. 51 793 (1948) (50) J. SCHWINGER, L. DeRaad, K. Milton, Ann. Phys. (NY) 115 1, 1978. (51) P. W. MILONNI and M. Shih, Phys. Rev. A 45, 4241 (1992). (52) MATTER FIELD THEORY OF THE CASIMIR FORCE, M. Koashi and M. Ueda , LANL Archive, cond-matt/9809031 , Sept. 2, 1998. (53) DEMONSTARTION OF THE CASIMIR FORCE in the 0.6 to 6 um Range by S.K. Lamoreaux, Physical Review Letters Vol. 78, Num 1, 6 Jan. 1997. (54) PRECISION MEASUREMENT OF THE CASIMIR FORCE from 0.1 to 0.9 um by U. Mohideen and A. Roy, Physical Review Letters Vol. 81, Num 21, 23 Nov. 1998. (55) A. EINSTEIN, Ann. Phys, 35 , 898 (1911). (56) A. EINSTEIN, Ann. Phys, 49 , 769 (1916). (57) CELLULAR AUTOMATA THEORY AND PHYSICS: A new paradigm for the Unification of Physics by Tom Ostoma and Mike Trushyk, July, 1999, LANL E-Print Server http://xxx.lanl.gov/physics/9907013.7713. ILLUSTRATIONS The captions for the figures are shown below: Figure 1: The Quantum Mechanism Behind Newton’s Law of Inertia F=MA Figures 2 to 5: Principle of equivalence for Stationary Mass on the Earth and in a Rocket Figures 6 to 9: Principle of equivalence for Light Motion inside a rocket and on the Earth Figure 10: Microscopic Equivalence Principle for Falling Virtual Particles Figure 11: Virtual Particle Pattern for the Earth and Moon in Free Fall near the Sun Figure 12: Virtual Particle Pattern for the Earth and Moon in Free Fall in a Rocket Figure 13: Motion of Real Photons in the Presence of Virtual Particles Near Earth Figure 14: Motion of Real Photons in Rocket Accelerating at 1g Figure 15: Impossible to Distinguish between Space-Time and Light Velocity Effects Figure 16: EMQG and Space-Time Effects from Photon Scattering Figure 17: Block Diagram of Relationship of CA and EMQG with Physics Figure 18: Simplified Motion of a Photon Inf ormation Pattern Figure 19: Light Velocity Measurement from two Different Observers Figure 20: Definition of an Inertial Reference Frame Figure 21: Schematic Diagram of what Space Looks Like on a CA78.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. 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FIGURE 1 - The Quantum Mechanism behind Newtonian Inertia Fi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .An elementary electrically charged fermion particle that makes up the accelerated mass Photon exchanges with the electrically charged virtual particlesA mass 'M' is accelerated by the rocket motor The sum of all the tiny electrical forces caused by the photon exchanges between the electrical charged particles that make up the mass and electrically charged virtual particles of the quantum vacuum is equal to the total inertial massThe electrical bonding forces that bind the atoms together in the massEMPTY VACUUM OF OUTER SPACE . . . .The undisturbed virtual particles of the quantum vacuum In Quantum Inertia, Newton's law is written as: = MA The force f i is the back acting electrical force opposing acceleration, that is contributed by the 'i' th electrically charged particle that builds up the mass M. 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............Acceleration of the Rocket is 1 g Figure #2 - Masses '2M' and 'M' at rest on the floor of the rocketFigure #3 - Masses '2M' and 'M' in free fall inside of a rocket Figure #4 - Masses '2M' and 'M' at rest on Earth's surfaceFigure #5 - Masses '2M' and 'M' in free fall above the Earth LEGEND: I = Relative downward acceleration (1g) of a virtual particle i = Relative downward acceleration (1g) of a real matter particle . = A real stationary matter particle (with respect to the earth's center)LEGEND : . = A virtual particle of the quantum vacuum (taken as the rest frame) = A real mass particle undergoing relative upward acceleration of 1g = A real matter particle at relative rest (acceleration) with vacuum FIGURES 2 TO 5 - THE PRINCIPLE OF EQUIVALENCE FOR A STATIONARY MASS ON THE EARTH AND INSIDE A ROCKET1 g 1 g . . . . i i i i . . i iIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 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IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIAcceleration of Box is 1 g Surface of the Earth where gravity produces a 1 g accelerationSNAPSHOT OF MASSES IN FREE FALLSNAPSHOT OF MASSES IN FREE FALL UNEQUAL MASSES AT REST ON SURFACEUNEQUAL MASSES AT REST ON THE FLOOR EquivalenceFORCE EQUIVALENCE FORCE EQUIVALENCE Electrical Force InteractionElectrical Force Interaction...................... ...................... ...................... ...................... ...................... ............................................ ...................... ...................... ...................... ...................... ......................Electrical Force InteractionElectrical Force Interaction80................................................................................... ................................................................................... 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................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ..................................................................................Acceleration of the Rocket is 1 g Figure #6 - Light Moving Up and Down in a rocketFigure #7 - Light Moving Parallel to the floor of a rocket Figure #8 - Light Moving Up / Down on Earth's surfaceFigure #9 - Light Moving Parallel to the Earth's Surface FIGURES 6 TO 9 - THE PRINCIPLE OF EQUIVALENCE FOR LIGHT MOTION ON THE EARTH AND INSIDE A ROCKET1 g 1 g IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 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Surface of the Earth where gravity produces a 1 g accelerationLIGHT MOVING PARALLEL TO EARTH SURFACELIGHT MOVING PARALLEL TO ROCKET FLOOR LIGHT MOVING UP/DOWN ON EARTHLIGHT MOVING UP AND DOWN IN ROCKET EquivalenceLIGHT MOTION EQUIVALENCE LIGHT MOTION EQUIVALENCE81................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ 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IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIFIGURE #10 - MICROSCOPIC PRINCIPLE OF EQUIVALENCE FOR VIRTUAL PARTICLES Close-up of a falling Virtual Electron Virtual Particles of various typesVirtual Masseon Electrical Vacuum forces (blue) and Temporary Bonding(red) to form an electron I I I I I I I I I I II I I I I I I I I I I I I I II II II I I IHuge numbers of lone, falling virtual masseons I I I II I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I IVirtual Quark82SUNIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 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IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIEARTH MOONFigure #11- VIRTUAL PARTICLE PATTERN FOR THE EARTH AND MOON IN FREE FALL NEAR THE SUN I I I I I iVirtual Particles accelerated by the SunThe Observer is stationed on the surface of the Sun. The virtual particle pattern is disrupted by the Earth and Moon. It is difficult to see that the resulting virtual particle pattern causes the two bodies to reach the Sun at the same time. However, if the observer is placed in free fall, the virtual pattern is the same as Figure #8, where the results are obvious.83................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ..................................................................................................................................EARTH MOONFigure #12 - VIRTUAL PARTICLE PATTERN FOR THE EARTH AND MOON IN FREE FALL IN A ROCKET Acceleration of Rocket equals Sun's accelerationAn Observer outside the rocket 'sees' the virtual particles of the vacuum at a relative acceleration of zero The Earth and Moon arrive on the floor of the huge rocket at the same time (the floor simply moves up to meet these bodies). But now, the virtual masseon particles near these two bodies are distorted and interacting with the real masseons in these two bodies.84EARTHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIFigure #13 - MOTION OF REAL PHOTONS IN THE PRESENCE OF VIRTUAL PARTICLE NEAR EARTH I I IElectrically charged virtual particles are accelerated (1g) through graviton exchanges with Earth.The Observer is stationed on the surface of the Earth. The virtual particles are accelerating downwards at1g through graviton exchanges with the earth. Light scatters with the electrically charged virtual particles, thus altering the path. I I I I I I I I I I I IFIZEAU-LIKE SCATTERING OF LIGHT THROUGH THE ACCELERATED VACUUM The photons scatter with the electrically charged virtual particles, which are accelerating downward at 1g. Photons travel perfectly straight, and at a much higher velocity (in absolute units) than the measured light velocity (which is constant). The scattering deflects the photon path, and reduces it's velocity through Fizeau-like vacuum scattering.This segment of the light beam travels in a perfect straight path and at an absolute constant speed, and represents the actual low-level CA photon velocity (Cabs) between the electrically charged virtual particle scattering, and is not affected by anything. Light Source85................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................ ................................................................................................................................Figure #14 - MOTION OF REAL PHOTONS IN A ROCKET ACCELERATING AT 1g Virtual Particles at relative rest with respect to our observer oustside the rocket. No gravitons are present here. Observer A is stationed outside the rocket at rest (non-accelerated). The virtual particles are at relative rest with respect to him (accelerated state). Light travels perfectly straight for Observer A outside the rocket, but appears to curve for observer B stationed on the floor of the rocket ............. ............. ........................................ ........................... ........................... ........................... ........................... ........................... ..............The real photons still scatter with the electrically charged virtual particles of the quantum vacuum, but results in a straight path for our outside observer. However, the Fizeau-like scattering reduces the absolute light velocity. Rocket Acceleration = 1gACTUAL VACUUM PROCESSObserver A Observer B. . . .86Observer A Observer BObserver B can choose among three different ladder types, and he can also choose the velocity that the ladder moves upwardsClockRulerOnly the rungs of the ladder are discernible to observer A Observer A wishes to measure both the velocity of the ladder and the spacing of the rungsLadder Velocity = c Ladder Velocity = cIf Observer A and B agree on the spacing of the rungs on the ladder and Observer B moves the ladder upward at velocity c. Observer A can measure the velocity of the ladder by measuring the rung spacing 'd' and by measuring the time 't' for a pair of rungs to pass the fixed distance 'd' on his ruler. The ladder velocity is given by c = d/t. Figure #15 - It is impossible to distinguish between the change in the velocity of light versus space-time curvature when light propagates from a lower to a higher gravitational potential. Light is represented by a ladder where the peaks of the wave are replaced by rungs on a ladder.The ladders represent different wavelengths of light, where the wavelength of the light is represented by the spacing of the rungs. IT IS IMPOSSIBLE TO DISTINGUISH BETWEEN THE CHANGE IN VELOCITY OF THE LIGHT AND A CHANGE IN FREQUENCY AND WAVLENGTH DUE TO 4D SPACE-TIME CURVATURECannot see the actions of Observer B87Photon at t1Photon at t2Photon at t3Photon at t4x0 x0x0x0 1 pdu 1 ptu = t2 - t11 pvu = Photon Velocity (or Cabs) between scattering events Cellular Automata array - Black squares signify the occupation of a cell with the numeric state '1', otherwise it is zeroPHOTON Figure 16 - EMQG and Space-Time effects from photon scattering by falling electrically charged virtual particlesVIRTUAL PARTICLE ENCOUNTER IN EMPTY SPACE, PHOTONS FREQUENTLY ENCOUNTER ELECTRICALLY CHARGED VIRTUAL PARTICLES WHICH REDUCE THE AVERAGE LIGHT VELOCITY TO THE MEASURED VALUE OF C A Falling Electrically Charged Virtual Particle in the Vacuum Photons in the light beam move upwards, and encounter downward accelerating, electrically charged virtual particles in the vacuum where there is frequent electrical scatteringA closeup of a small segment of the light beam moving upwards on the Plank scale, where details of the CA process are revelaedThe electrically charged virtual particle absorbs the photon here, and the photon momentarily ceases to exist for a brief time. Meanwhile the virtual particle is falling. In this region the photon propagates at an absolute constant rate, determined by the CA cells and clock cycle. It is unaffected here.In the vacuum far from earth, we illustrate a short segment on the Plank scale. Light velocity reduces to the average light velocity of 'c' we measure in the lab over macroscopic distance scales by virtual particle scattering. If the CA absolute velocity is 'Cabs' then: c = Cabs / n , where n is the 'vacuum index of refraction' of light Unlike the case illustrated above for light travel in the far vacuum away from gravity, the electrically charged virtual particle encounters are modified, where now the virtual particles are in free fall. This reduces the velocity of light further by c (r +delta r) = c (1-g delta t / c) where this is applicable only for infinitesimal distances and time.The simple shifting of the photon CA information pattern from cell to cell CabsPhoton does not exist when absorbed by the virtual particle NOTE: It is important to realize that a photon is created at velocity Cabs at each virtual particle scattering event, and propagates at an absolute constant speed until the encounter with the next virtual particle scatteringGraviton exchanges between fermion particles in the earth and the virtual fermion particles of the vacuum causing them to fallA Virtual Particle falls at an acceleration given by: GM / r2 Gravitons88* CELLULAR AUTOMATA PARADIGM The fastest known Parallel Computer Model. Here strict locality prevails, and there exists a maximum limiting speed for the transfer of information. There automatically exists an absolute, quantized 3D space in the form of 'cells', and quantized time. Quantum Field Theory and Quantum Electrodynamics (QED) All Forces result from Particle Exchanges. Dirac equ. predicts particle-antiparticle pair creation, with all charge types reversedQuantum Mechanics The links between this and Cellular Automata theory are not fully known.Special Relativity This theory follows as a direct consequence of Cellular Automata. * VIRTUAL PARTICLES OF THE VACUUM The existence of the 'Electrically-Charged' and 'Mass- Charged' Virtual Particles (Masseons) of the Vacuum. These are responsible for inertia. Their existence automatically resolves the Cosmological ConstantBOSON PARTICLE EXCHANGE PARADIGM * ElectroMagnetic Quantum Gravity (EMQG) Theory This theory is based on both Photon and Graviton exchanges occuring with the virtual particles. In Inertia, only Photon exchanges occur between matter particles and the Virtual Particles. In Gravitational Fields, this process still occurs with the addition of graviton exchanges with the vacuum particles. The Equivalence Principle is derived from this.Classical Electro- Magnetism General Relativity* QUANTUM INERTIA This is based on the Photon Exchanges between matter particles and Virtual Particles.Mach's Principle Deep connection with the vacuum.Newton's Laws of Motion Deep Connection with the quantum vacuum. A Finalized Quantum Gravity Theory Curved Riemann 4D Space-Time CurvatureGRAVITON PARTICLE Responsible for gravity. Principle of Equivalence * Newly Developed TheoryFigure #17 - BLOCK DIAGRAM OF RELATIONSHIP OF CA AND EMQG WITH PHYSICS89Photon at t1Photon at t2Photon at t3Photon at t4 Figure #18 - Simplified model of the motion of the photon information pattern on the CA. The photon information pattern moves 1 plank unit to the right at every plank 'clock cycle' (Note: The photon is actually an oscillating wavepattern shown highly simplifiedx0 x0x0x0 Photon motion is decoupled from the source, and propagates on the CA as shown above. Green Light source Green Light source Detector The Relative velocity is v = vb - vavavb Figure #19 - Light velocity measurement from two observers with identical measuring apparatus with different absolute CA velocities va and vbObserver A Observer B1 pdu 1 ptu = t2 - t11 pvu = Photon Velocity Absolute CA units: 1 pdu is the shifting of information by 1 cell; 1 ptu is the time to shift 1 cell; 1 pvu = photon velocity d d Figure #20 - Definition of an inertial reference frame. Identically constructed clocks are spaced apart at regular intervals by a rulerCellular Automata array - Black squares signify the occupation of a cell with the numeric state '1', otherwise it is zeroPHOTON (A 2 dimenional arrangement is shown for simplicity)MEASUREMENTS ARE MADE USING ABSOLUTE CA UNITS OF DISTANCE AND TIME RELATIONSHIP BETWEEN THE CA MODEL, LIGHT MOTION ON THE CA, AND SPECIAL RELATIVITY90In a 3D Geometric Cellular Automata, the numeric content of Cell Ci,j,k is uniquely determined by the numeric contents of each of the surrounding 26 neighbouring cells (and possibly with it's own numeric state). On the next CA 'clock cycle' the contents of cell Ci,j,k is determined by a function (or algorithm) Fi,j,k such that the contents of the cell Ci,j,k = F(Ci+x,j+y,k+z) where x,y,z take on all the following values: -1,0,1. This same function F is programmed into each and every cell in the entire CA. In the figure below, the binary number system is chosen for illustration purposes (any number system can be used). The dotted lines indicate what cells affect cell Ci,j,k. Figure #21 - Schematic Diagram of what space looks look on the Cellular AutomataCi,j,k The Dimensionality of the CA space is simply determined by the number of connections that exist between a cell and the surrounding neighborsThe numeric content of thiscell is being updated by it's neighbors91

arXiv:physics/0004028v1 [physics.ed-ph] 14 Apr 2000Revitalization of an undergraduate physics program John W. Norbury Physics Department,University of Wisconsin-Milwaukee, P .O. Box 413, Milwaukee, Wisconsin 53201 , e-mail: norbury@csd.uwm.edu G. R. Sudhakaran Physics Department, University of Wisconsin-La Crosse, La Crosse, Wisconsin 54601 , e-mail: sudhak@physics.uwlax.edu This article describes the successful revitalization of an undergraduate physics program. The areas of curriculum development, undergradua te research ex- periences and advising and retention, to name a few, are emph asized in this interconnecting and systematic approach whereby each and e very effort com- bines to get results. The program can be used by other physics departments wishing to improve and expand undergraduate education in ph ysics. 1 Introduction Undergraduate physics programs in the United States and oth er countries seem to be in a period of decline. There are less students taki ng the physics major and less students entering graduate study [1]. Some de partments are being closed down and others are under the threat of closure. In such cir- cumstances demoralization sets in and spirals into every as pect of a program discouraging faculty and students. Five years ago the physics department at the University of Wi sconsin - La Crosse had a total of 5 physics majors, 5 faculty and a gradu ation rate of about one physics major every two years. The department ha d received a poor review and was in danger of being phased out, but instea d of taking this easy option the dean of the college decided instead to hi re a new chair in an attempt to turn the department around. Five years later th e department is one of the best on campus, has received an excellent review and currently has a total of about 85 physics majors and 7 faculty. The prese nt article describes how this was achieved. It is hoped that the informa tion presented here can be used by other physics departments to revitalize t heir programs. 12 Program outline 1. Academic Programs. The first thing was to change the academic programs being offered and re-package them in attractive ways directi ng students, par- ents and teachers to expand their typical view of what a physi cs degree could do for the student. We still continued the core subjects of mo dern physics, mechanics, electrodynamics, quantum mechanics, thermody namics and op- tics. We also continued two popular astronomy courses and th e introductory year long sequences of algebra and calculus based physics co urses. However several new courses were added to make the elective list a lot more interest- ing and useful for the students. Some of the electives added w ere quantum optics, electronics, seminar (for credit), research (for c redit), computational physics and advanced computational physics, general relat ivity and cosmol- ogy, astrophysics, advanced quantum mechanics and particl e physics. 2. Emphases and Concentrations. One of the important additions in at- tracting new majors was the introduction of a set of emphasis programs that could be packaged along with course and career information. These included physics major with business concentration, physics major w ith astronomy emphasis, physics major with computational physics emphas is and physics major with optics emphasis. The physics major with business concentration basically co nsisted of a physics major with a business minor for a total of about 55 cre dits. Why go to the trouble of simply re-packaging an already existing pr oduct (physics major and business minor) into a ’new’ product (physics majo r with business concentration)? This is an important point and should not be lost. Physics now needs to be marketed as does any other product, and the dep artment has to have the products to suggest and then deliver to studen ts. Physics programs now need to be attractive not only to students, but a lso to parents and teachers, who heavily influence the students. Often thes e clients sim- ply don’t think of physics and business as going together, ye t most of us in the field know that this is an excellent combination for stude nts wishing to obtain employment with a bachelors degree. Having a formal p rogram such as a physics major with business concentration highlights t he career oppor- tunities available to students and simply makes the overall physics program look a better match in today’s job market. A quick perusal of j ob sections in newspapers show marketing/sales in technical areas, com puter and tech- nology skills being needed in the business sector. These are all skills taught 2in a physics degree (with business concentration). In order to be able to offer these emphases the department had t o add quite a few courses to the catalog, as mentioned above. The ar eas of op- tics, computational physics and astronomy were chosen deli berately. Optics is very important for industry and is a good area for job seeke rs. Com- putational physics is also an excellent area for both job see kers and those wishing to go to graduate school. Astronomy was chosen simpl y because so many students have an interest in this area. It was also impor tant to make sure that the department had expertise in these areas. Aside from offering the regular physics major, three areas of emphasis were intr oduced namely optics, astronomy and computational physics. The total num ber of credits for these three programs was similar to the regular physics m ajor, but the elective options were eliminated. Instead, the electives w ere chosen for the students. For the astronomy emphasis the student was requir ed to take the core physics courses plus 3 astronomy courses and a research project in as- tronomy. Similarly for the optics and computational physic s emphasis. An optics experimentalist was hired to help with this. One can e asily imag- ine other departments with different areas of expertise deve loping different emphasis programs. There are many departments that already have an ex- tensive listing of electives. From the student, parent, tea cher point of view however, the existence of these electives and what they can d o is often lost. It is very worthwhile to package some strong elective progra ms into emphasis areas so that it is clear as to what specializations are avail able to students and how this relates to the real world task of getting a job. 3. Honors Program. A physics honors program was also introduced, in which students are required to submit a formal application, maintain a cer- tain GPA, complete a research project with distinguished pe rformance, give a seminar and be recommended by two faculty members. 4. Dual degree in physics and engineering. One of the most important programs introduced was a dual degree program in physics and engineer- ing. Such programs are starting to gain popularity and are an excellent way to revitalize an undergraduate physics department. The pro gram introduced was a collaborative program between our own department and t wo engineer- ing schools (University of Wisconsin - Madison and Universi ty of Wisconsin - Milwaukee). An essential feature of this program is that a s tudent is guar- anteed acceptance into the engineering school upon complet ion of a set of physics and other required courses with a specified grade poi nt average. The 3students spend 3 years in the department at the University of Wisconsin - La Crosse studying selected physics courses and then trans fer to one of the engineering schools for 2 years to study an area of engine ering. After the first year at the engineering school the student receives a physics degree from La Crosse and after the second year receives an engineer ing degree, thus graduating with two degrees that complement each other. Thi s program has been extremely attractive to students, parents and teacher s. We strongly recommend such a program to undergraduate physics departme nts. 5. Laboratory Upgrades. As part of improving the academic programs, a lot of attention was paid to upgrading the laboratory facili ties. During the past five years approximately $200,000 in laboratory modern ization funds were spent in upgrades. One cannot expect students first and f acilities later. They only come together. The freshman physics labs were comp letely over- hauled using computer based ”workshop physics” style labor atories. The students went from hating lab work to actually enjoying it. I n addition the modern physics lab, optics lab and electronics lab were comp letely re-done with a full complement of modern experiments and equipment. 6. Quality Instruction. The quality of instruction in all courses (but es- pecially the introductory courses) was improved by trying v ery hard to use the best instructors. This seems mundane, but is extremely i mportant in building up a physics program. If the majors have a couple of, or even one, poor instructor then the program suffers tremendously. It is absolutely vital to have high quality instruction and every effort must be made to make sure this happens. 7. Undergraduate Research. One of the major factors that lead to high student satisfaction with our new program was a strong set of research expe- riences for the undergraduate physics majors. Before we cam e to the depart- ment, research was almost non-existent. As an incentive to f aculty the chair allowed supervision of undergraduate research to count for one course for the faculty member and as an incentive for students we introduce d research for credit that a student could take. Further to this a research e xperience was a requirement for each of the 3 emphasis programs described a bove. Three out of the six faculty became actively involved in student re search projects immediately. Research was offered in all three areas of exper iment, theory and computation in the areas of optics, condensed matter phy sics, particle physics, nuclear physics and astronomy. Several students w orked only on semester-long projects, but the most successful experienc es were with stu- 4dents who would work for two semesters and one summer. The adv antage of this work is many-fold. Fliers to schools, brochures, campu s news, depart- ment annual reports, student and faculty resumes all are enh anced. Many good undergraduate departments have such an undergrad uate re- search program in place. When talking to administrators abo ut physics we often emphasize the triad of theory, experiment and computa tion. When talking about physics education we emphasize the triad of le cture, labora- tory and research. It is vitally important for undergraduat es to have a good research experience during their education. It also helps t he department at- mosphere tremendously and is very good to display when givin g tours to students, parents, teachers and administrators. 8. Student Presentations. The research outlined above could be show- cased to other students or advertised, and so attracted furt her students. Students were encouraged and trained to present the results of their work at department seminars and at conferences such as the Argonn e symposium [2-9] and also at national and international meetings such a s the Ameri- can Astronomical Society and the International Symposium o n Molecular Spectroscopy [10-13]. Many students and faculty published papers together [14-21]. Faculty benefitted also by being able to place any of the work related to student research in their promotion files. 9. Funding for Students. Funding was obtained so that students could work on research over the summer. This also gave the departme nt the op- portunity to give students and parents the promise of moneta ry support and see the immediate connection between learning physics and m onetary gain. Again any student getting such support was used for real prom otional ad- vantages in the department literature and annual reports. 10. Scholarships and Internships. We went to great efforts to have the students apply for scholarships and internships. Several s tudents won very prestigous scholarships (e.g Barry Goldwater scholarship , Council on Under- graduate Research Felllowship, American Physical Society Summer Fellow- ship) and this had a strong effect on the motivations of the oth er students. Summer interns were also arranged. One of the best programs h ere is the ‘Research Experiences for Undergraduates’ run by the Natio nal Science Foun- dation. Again a great deal of busy work is involved in arrangi ng scholarships and interships but the work is certainly worthwhile. It also helps a lot with recruitment in being able to give examples of the successes o f previous stu- dents. 511. Seminar Program for Credit. There are several other elements that went into building up the physics program. One was the establ ishment of a department seminar program. This was specifically designe d to provide a meeting place for the majors and faculty. We introduced a pro gram where students could sign up for 1 credit of course work. The requir ement was to attend all the seminars and to either write a report on one o f them or present a seminar. What was interesting about this was that m any students outside of physics also signed up. Many physics majors did no t sign up but attended anyway and the group grew. Speakers included facul ty from physics and other departments, physics majors and outside speakers . The physics majors would often talk about their research projects and th is was a great way for other students to see what opportunities were availa ble. Students also talked about their summer internship experiences. Out side speakers gave talks primarily on research topics, but there were also talks on careers and engineering programs. 12. Recruitment, Advising, Retention. Recruitment and advising appears at first to be another area that seems to be very mundane. Howev er our experience is that the role of the undergraduate physics maj or advisor is absolutely essential for a successful physics program. The advisor should be very knowledgeable about the employment situation, sala ries, current job openings, scholarships, internships, summer jobs, tutori ng jobs, housing, in- ternational opportunities, graduate record exam, graduat e schools, etc.The physics advisor needs to be constantly available, always ha ppy and willing to spend lots of time with the students, have a friendly perso nality, and do many other tasks too numerous to mention. Not all faculty find all this busy work palatable, but in our experience it is one of the most imp ortant factors inretaining students once they sign up for the major. 13. Advertising and Brochures. Advertising is another extremely impor- tant area that needed attention. One can have the best physic s program in the world, but if no one knows about it, then not much is goin g to hap- pen. The primary way of advertising was to be in touch with phy sics high school teachers and counselors and to let them know of the new programs that were available with regular mail outs. Teachers were se nt information about actual student work as well as general programs so they could give this immediately to their own students. Teachers were also i nvited regularly to the department seminars and social gatherings. We believ e that letting teachers know about the unique aspects of a physics program i s one of the 6best ways to bring in new majors. 14. Presenting a Plan and Cooperating with Administration. Another as- pect of building up the physics program was cooperation and i nteraction with the university administration. This included not only the d eans, provost and chancellor, but also people in the international office, the c areer center, the counseling center, the affirmative action office, the library, the computer cen- ter, etc. It is vitally important to have a good relationship with all of these areas and to explain your plan and future directions. The dea n, vice chancel- lor and chancellor were especially important. When buildin g up a program it is essential to obtain financial commitments and to have th ese commit- ments followed through. Often these groups were invited to t he department seminars or demonstrations or we provided a tour for adminis tration visitors. 15. Department Team Work and Priority Mission. Finally we should men- tion the obvious, that all of the above cannot be done by one person as every aspect needs attention. No one idea is a quick fix that will wor k but a sus- tained concerted effort is needed over several years. We were very fortunate to have a few faculty members who really cared about the progr am and were willing to work very hard as a team to make it succeed. Once it s ucceeded then we moved into maintenance. REFERENCES (underlined names refer to students) [1] P.J. Mulvey, E. Dodge and S. Nicholson, Enrollments and degrees report R-151.33 , (American Institute of Physics, April 1997). [2] M. Waldsmith and J.W. Norbury, Uranium Beam Lifetimes at RHIC and LHC, 5th Annual Argonne Symposium for undergraduates in Scienc e, Engi- neering and Mathematics, Argonne National Lab, (1994). [3] B. Soller , G. Sudhakaran, and M. Jackson, Far-Infrared Laser Stark Spec- troscopy of13CH3OH, 5th Annual Argonne Symposium for Undergraduates in Science, Engineering & Mathematics, Argonne National La boratory, 1994. [4] P. Valentine , G. Sudhakaran, and M. Jackson, Far-Infrared Laser Stark Spectroscopy of CH3OD, 5th Annual Argonne Symposium for Undergradu- ates in Science, Engineering & Mathematics, Argonne Nation al Lab, 1994. 7[5] K.J. Cook , G.R. Sudhakaran, and M. Jackson, Far-Infrared Water Vapor Laser, 5th Annual Argonne Symposium for Undergraduates in Scienc e, En- gineering and Mathematics, Argonne National Laboratory, 1 994. [6] J.T. Dobler , G.R. Sudhakaran, and M. Jackson, Stark Spectroscopy using a Far-Infrared Laser , 7th Annual Argonne Symposium for Undergraduates in Science, Engineering and Mathematics, Argonne National La boratory, 1996. [7] E.J. Gansen , G.R. Sudhakaran and M. Jackson, Far-Infrared Laser Stark Spectroscopy of PH3, 7th Annual Argonne Symposium for Undergraduates in Science, Engineering and Mathematics, Argonne National La boratory, 1996. [8]Regge Trajectories for Mesons (M. Pruse , J.W. Norbury) 5th Annual Ar- gonne Symposium for undergraduates in Science, Engineerin g and Mathe- matics, Argonne National Laboratory, 1994. [9]Parameterization of spectral distributions for pion and ka on production in proton-proton collisions (J. Schneider , J.W. Norbury and F.A. Cucinotta) 5th Annual Argonne Symposium for undergraduates in Science , Engineering and Mathematics, Argonne National Lab, 1994. [10] J.P. Schneider , J.W. Norbury and F.A. Cucinotta, Parameterization of spectral distributions for pion and kaon production from pr oton-proton colli- sions, Bulletin of the American Astronomical Society 26, 873 (1994). [11] M. Jackson, B.J. Soller , G.R. Sudhakaran, R.M. Lees, and I. Mukhopad- hyay,Far-Infrared Laser Stark Spectroscopy of13CH3OH, 50th International Symposium on Molecular Spectroscopy, Ohio State Universit y, 1995. [12] M. Jackson, G.R. Sudhakaran, and E.J. Gansen ,Far-Infrared Laser Stark Spectroscopy of PH3, 51st International Symposium on Molecular Spec- troscopy, Ohio State University, 1996. [13] M. Jackson, G.R. Sudhakaran, and E.J. Gansen ,Far-Infrared Laser Stark Spectroscopy of13CD3OD, 52nd International Symposium on Molecu- lar Spectroscopy, Ohio State University, 1997. 8[14] J.W. Norbury and C.M. Mueller ,Cross Section parameterizations for Cosmic Ray Nuclei II. Double Nucleon Removal , Astrophys. J. Suppl. 90, 115-117 (1994). [15] R. Wheeler and J.W. Norbury, Higher order corrections to Coulomb fis- sion, Phys. Rev. C 51, 1566-1567 (1995). [16] G.R. Sudhakaran, E.K. Coulson , and M. Jackson, Laser Stark Spec- troscopy of13CH3F, Int. J. Infrared Millimeter Waves, 16, 1329-1333 (1995). [17] G.R. Sudhakaran, B.J. Soller , M. Jackson, I. Mukhopadhyay, and R.M. Lees,Far-Infrared Laser Stark Spectroscopy of CH3OHand13CH3OH, Int. J. Infrared Millimeter Waves, 16, 2111-2131 (1995). [18] M. Jackson, G.R. Sudhakaran, and E.J. Gansen ,Far-Infrared Laser Stark Spectroscopy of13CD3OD, J. Mol. Spectrosc., 176, 439-441 (1996). [19] M. Jackson, G.R. Sudhakaran, and E.J. Gansen ,Far-Infrared Laser Stark Spectroscopy of PH3, J. Mol. Spectrosc., 181, 446-451 (1997). [20]Parameterized spectral distributions for meson productio n in proton- proton collisions (J.P. Schneider , J.W. Norbury and F.A. Cucinotta) NASA Technical Memorandum 4675 (1995). [21]Parameterization of spectral distributions for pion and ka on production in proton-proton collisions (J.P. Schneider , J.W. Norbury and F.A. Cucinotta) Astrophysical Journal Supplement 97571-574 (1995). 9

arXiv:physics/0004029v1 [physics.ed-ph] 14 Apr 2000Lagrangians and Hamiltonians for High School Students John W. Norbury Physics Department and Center for Science Education, Unive rsity of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 5 3201, USA e-mail: norbury@uwm.edu Abstract A discussion of Lagrangian and Hamiltonian dynamics is pres ented at a level which should be suitable for advanced high school stu dents. This is intended for those who wish to explore a version of mechani cs beyond the usual Newtonian treatment in high schools, but yet who do not have advanced mathematical skills. 1 Introduction Newtonian dynamics is usually taught in high school physics courses and in college level freshman physics class [1]. Lagrangian and Hamiltonian dy- namics [2, 3] is usually reserved for an upper division under graduate physics course on classical dynamics. This is all as it should be, par ticularly since one needs the technique of calculus of variations for the Lagran gian formulation. However it is always nice to be able to whet the appetite of the advanced high school student for a taste of things to come. For those st udents who have successfully mastered the contents of the typical high school physics course, one can give an extra lesson on Lagrangian and Hamilt onian dynamics without having to use calculus of variations. The idea is sim ply to present some new formulations of dynamics that an advanced high scho ol student will find enjoyable and intellectually interesting. (The st udents can be told that a rigorous formulation will be presented in college cou rses.) For simplicity, consider only the one-dimensional problem . Write New- ton’s equation F=ma (1) and define the potential energy U(x), which is a function only of position, as F≡ −dU dx(2) 1where−dU dxis the spatial derivative of the potential energy. Thus re-w rite Newton’s equation as −dU dx=m¨x (3) where ˙ x≡dx dt=vfor the speed and ¨ x≡d2x dt2=afor the acceleration. 2 Lagrangian Dynamics To introduce Lagrangian dynamics define a Lagrangian as a function of the two variables of position xand speed ˙ x L(x,˙x)≡T(˙x)−U(x) =1 2m˙x2−U(x) (4) where the kinetic energy T( ˙x)≡1 2m˙x2is a function only of the speed variable and the potential energy again is only a function of position U(x). Now introduce the idea of a partial derivative. This is very e asy. For a function of a single variable f(y) the notationd f dyis used for the derivative. For a function of twovariables g(y, z) there are two possible derivatives for each variable yorz. In this case one simply introduces a different notation for derivative, namely∂g ∂yfor the yderivative (where yis changing but zis constant) and∂g ∂zfor the zderivative (where zis changing but yis constant). Even though high school students won’t see partial derivati ves until they are in college, nevertheless the idea is very simple and can easi ly be explained to the advanced student who is taking a course in calculus. From (4) one can easily see that ∂L ∂x=−dU dx(5) and∂L ∂˙x=m˙x≡p (6) which is called the momentum p. Obviously then d dt/parenleftBigg∂L ∂˙x/parenrightBigg =m¨x (7) 2Combining (5), (7) and (3), Newton’s equation (3) becomes ∂L ∂x=d dt/parenleftBigg∂L ∂˙x/parenrightBigg (8) which is the Euler-Lagrange equation in one dimension. It ca n be explained to the students that it is this equation in Lagrangian dynami cs which replaces F=main Newtonian dynamics. 2.1 Lagrangian example Students will obviously want to see some examples of how the L agrangian formulation works. A simple example is the one-dimensional harmonic oscil- lator with F≡ −dU dx=−kx. (9) Newton’s equation is −kx=m¨x. (10) The potential U(x) is obtained by integrating (9) to give U(x) =1 2kx2. (11) Thus the Lagrangian is L(x,˙x) =1 2m˙x2−1 2kx2(12) giving ∂L ∂x=−kx (13) and substituting into (8) and (7) gives exactly back the same equation of motion (10) as in the Newtonian case. Many teachers will have had the students work out the equatio n of motion from Newtonian dynamics for other types of forces, such as a p article in a uniform gravitational field. Students can be encouraged to p rove that the same equations of motion result from the Lagrangian formula tion. Students can also be encouraged to think about three-dimensional pro blems and to derive, on their own, the three Euler-Lagrange equations (c orresponding to 3the three component equations Fx=m¨x,Fy=m¨y,Fz=m¨z) which result from the three dimensional Lagrangian L(x, y, z, ˙x,˙y,˙z) =1 2m( ˙x2+ ˙y2+ ˙z2)−U(x, y, z ). (14) 3 Hamiltonian Dynamics Now consider the Hamiltonian formulation of dynamics. Defin e aHamilto- nianas a function of the two variables, momentum pand position x, H(p, x)≡p˙x−L(x,˙x) (15) which can be seen to be just the total energy T+UasH=p˙x−L=m˙x2− 1 2m˙x2+U=1 2m˙x2+U=T+U. Hamilton’s equations follow immediately. Lis not a function of pand therefore ∂H ∂p= ˙x. (16) ButLis a function of xand thus ∂H ∂x=−∂L ∂x. (17) However (6) and (8) give∂L ∂x= ˙pso that −∂H ∂x= ˙p. (18) Equation (16) and (18) are Hamilton’s equations which repla ceF=main Newtonian dynamics. 3.1 Hamiltonian example For the harmonic oscillator example, the Hamiltonian is H(p, x) =p˙x−1 2m˙x2+1 2kx2=p2 2m+1 2kx2(19) 4where we have had to replace ˙ xbyp mbecause H(p, x) is supposed to be a function of pandxonly. Thus Hamilton’s equations (16) and (18) give p m= ˙x (20) and −kx= ˙p. (21) These are shown to give the equation of motion (10) by differen tiating (20) as˙p m= ¨x (22) and substituting (21) for ˙ pgives back equation (10). Once again students can be encouraged to use other examples t hat they have already studied in Newtonian dynamics and to show that H amilton’s equations result in the same equation of motion. Again stude nts can work out the three-dimensional generalization of Hamilton’s eq uations using H(px, py, pz, x, y, z ) =px˙x+py˙y+pz˙z−L(x, y, z, ˙x,˙y,˙z). (23) Finally teachers can emphasize to students that Newtonian m echanics is based on forces, whereas Lagrangian and Hamiltonian dynami cs is based on energy. In summary, a discussion of Lagrangian and Hamiltonian dyna mics has been presented which should be suitable for advanced high sc hool students, who are interested in exploring some topics not normally pre sented in the high school physics curriculum. It is also hoped that this ar ticle can be given to students to read on their own. References [1] R. A. Serway, Principles of Physics , (Saunders, New York, 1998), pp. 80-141. [2] T. L. Chow, Classical Mechanics , (Wiley, New York, 1995), pp. 99-175. [3] G. R. Fowles and G. L. Cassiday, Analytical Mechanics , 5th ed., (Saun- ders, New York, 1993), pp. 340-373. 5

arXiv:physics/0004030v1 [physics.gen-ph] 15 Apr 2000An Underpinning for Space Time B.G. Sidharth Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India) Abstract We argue, that from a background pre space-time Zero Point Fi eld, physical space-time emerges on averaging over unphysical C ompton scales. 1 Introduction In a previous communication[1] it was shown how from the frac tal dimension two of a Quantum Mechanical path, the space time divide emerg es: The real coordinate x, shows up as a complex number x+ix′, and it was de- duced that x′=ct. The underpinning for the complex coordinate is the double Weiner process, which in the Nelsonian formulation l eads to a com- plex velocity (Cf.ref.[1] and [2]). It also appears as zitte rbewegung and the resulting complex or non Hermitian position operator. It wa s also argued[3] that this Quantum Mechanical indeterminism gives birth to t ime. Indeed it was pointed out that the above fractal behaviour has an immed iate analogue in Richardson’s coastline[4], while the Compton scale is th e thick brush that delineates the coastline[5]. We will now see that all this im plies the emer- gence of space-time from a pre space-time background Zero Po int Field or Quantum Vacuum. 0Email:birlasc@hd1.vsnl.net.in 12 Space-time In the light of the above comments, we can now notice that with in the Comp- ton time, for example we have a double Weiner process leading to non differ- entiability with respect to time. That is, at this level time in our usual sense does not exist. To put it another way, within the Compton scal e we have the zitterbewegung effects - these are non local and chaotic. As pointed out by Dirac[6], it is only after an averaging over these scales t hat meaningful physics emerges. Indeed in the stochastic formulation of th e Schrodinger, Klein-Gordon and Dirac equations[5, 7], we again encounter minimum space time units, the Compton scale within which there are unphysi cal effects. It is only outside the Compton scale that physics emerges. This is a Quantum Mechanical and an experimental fact. It exp resses the Heisenberg Uncertainty Principle - space time points imply infinite momenta and energies and are thus not meaningful physically. Howeve r Quantum The- ory has lived with this contradiction[8]. To put it simply to measure space or time intervals we need units which can be to a certain exten t and not indefinitely subdivided - but already this is the origin of di screteness. So physical time emerges at values greater than the minimum uni t, which has been shown to be at the Compton scale[9]. Going to the limit of space-time points leads to the well known infinities of Quantum Field The ory (and clas- sical electron theory) which require renormalization for t heir removal. The conceptual point here is that time is in a sense synonymou s with change, but this change has to be tractable or physical. The non differ entiability with respect to time, due to the double Weiner process, within the Compton time, precisely highlights time or change which is not tractable, that is is unphys- ical. However Physics, tractability and differentiability emerge from this indeterminism once averages over the zitterbewegung or Com pton scale are taken. It is now possible to track time physically in terms of multiples of the Compton scale. To elaborate on this point it may be mentioned that in[9, 10] a nd elsewhere, it was argued that given nparticles in the universe, within the Compton timeτ,√nparticles would be fluctuationally or unphysically created (out of a background Zero Point Field), so that we would have, dn dt=√n τ(1) 2On integrating (1) from n= 0 ton=N∼1080, as we have in the present universe, it was shown that we recover the flow of time, infact the correct age of the universe, as indeed can be easily verified from (1). The above considerations can be looked at from another point of view. Time is essentially an ordering or sequencing of events. We would like to know the basis on which this ordering takes place and which leads t o our physical universe. The question is, can we liberate this sequence of e vents from any ordering at all, even though this would on the face of it lead t o a totally chaotic universe without any physics. However within the Co mpton scale this should certainly be possible: As noted in [9], let us start with hypothetical instantaneou s point particles (or Zero Point Energies or Ganeshas as they were designated) . Their states could be denoted by φn, which form a basis, so that a general state could be written as ψ=/summationdisplay ncnφn, (2) φncould be eigen states of for example the position or energy, w ith eigen valuesλn. It is known that, owing to the Random Phase Axiom, viz., (cn,cm) = 0, n∝negationslash=m. (3) wherencould stand not for a single state but also for a set of states niand so alsom, and where the bar denotes an average over a suitable interva l we have[11] ψ=/summationdisplay nbnφn, (4) where |bn|2= 1 ifλ<λ n<λ+ ∆λand = 0 otherwise. What all this means is the following. We take a totally random sequence of states likeφn. Such a sequence averaged over the interval τ, the Compton time and there by using (3), constitutes a particle that has p hysical existence, as expressed by (4). Without such an average, that is within t he Compton time, we have instead of equation (4), equation (2), wherein there is no or- dering or sequencing whatsoever, that is time in the physica l sense has no meaning. This is a pre time (or pre space time) perfectly rand om background Zero Point Field scenario. It is only on averaging that we rec over the physi- cally meaningful equation (4). Identical arguments apply to the case of space, except that t here is the space 3time divide arising out of the fractal two dimensionality of the Quantum path as noted above: We have x+ict. Indeed it was pointed out[1] that this is the origin of Special Relativity. This apart we can proceed i n terms of the Compton wavelength lto get this time instead of (1), dn dx=√n l(5) Integrating (5) exactly as we did (1), we recover this time, t he well known hitherto empirical Eddington formula, R∼√ Nl (whose time analogue from (1) is, T∼√ Nτ). Once again within the Compton wavelength we have non local un physical effects. However as we will now see, in the case of space, we recover thr ee dimensions. The reason is that as pointed out in[12, 13] and elsewhere the zitterbewe- gung or unphysical effects represent charge and double conne ctivity or spin half which in terms of spin networks[14] or similar argument s[15] immedi- ately leads to three dimensionality. Thus the the three dime nsionality is not apriori, but rather is a holistic property arising from seve ral particles. Another way of looking at this is when two particles interact without any spe- cific reference to electromagnetism and three dimensionali ty, their potential energy is given by, as is well known[16] V=−α¯hc r(6) whereαis the fine structure constant and rthe distance between the two particles, without reference to any dimensionality. We would now like to point out that the significance of (6) is th at it gives the inverse square Coulumb Force Law and therefore three dim ensionality. Indeed as Barrow [8] points out ”Interestingly the number of dimensions of space which we experience in the large plays an important rol e.... it also ensures that wave phenomena behave in a coherent fashion. We re there four dimensions of space, then simple waves would not travel at on e speed in free space.... in any world but one having three large dimensions of space, waves would become distorted as they travel...” Also as pointed ou t in[17], in the 4case of two elementary particles separated by a distance r, as the spectral density of the background Zero Point Field is given by ρ(ω)∝ω3 whereωis∝1 r, the electromagnetic force is given by Force ∝/integraldisplay∞ rω3dr,∝1 r2 which in effect is the same as (6). 3 Discussion The picture that presents itself is the following: There is a perfectly random, incoherent background Quantum Vacuum or Zero Point Field. F rom what we term as fluctuations of this Quantum Vacuum, particles are created at the Compton scale. This is a transition from an equation like (2) which represents total incoherence, to (4), where we have particl es occupying a coherent space time with physics. The coherence or link betw een the various particles is interaction provided by the background Zero Po int Field within the Compton scales, or in more conventional language by the v irtual photons linking the various constitutents. In the transition from ( 2) to (4), there has been a totally random sequencing within the Compton scale, a s expressed by the Random Phase equation (3). In other words within the Comp ton scale, there is no physics - indeed this is the Zitterbewegung regio n of non local effects. References [1] B.G. Sidharth, ”Space Time at a Random Heap”, to appear in Chaos Solitons and Fractals. [2] L. Nottale, Chaos, Solitons & Fractals, 4, 3, 361-388, 19 94, and refer- ences therein. [3] Sidharth, B.G., ”Comment on the Paper ’Unification of Fun damen- tal....’”, to appear in Chaos Solitons and Fractals. 5[4] B.B. Mandelbrot,”The Fractal Geometry of Nature”, (198 2) W.H. Free- man, New York, pg.2,18,27. [5] B.G. Sidharth, ”The Chaotic Universe”, CSF 11 (2000), pp .1171-1174. [6] P.A.M. Dirac, The Principles of Quantum Mechanics , Clarendon Press, Oxford, 1958, p263. [7] B.G. Sidharth, ”The Universe of Chaos and Quanta”, CSF 11 (2000), pp1269-1278. [8] J.D. Barrow, ”Theories of Everything”, Vintage, London , 1992. [9] B.G. Sidharth, ”Universe of Fluctuations”, Int.J.of Mo d.Phys. A 13(5), 1998, pp599ff. [10] B.G. Sidharth, International Journal of Theoretical P hysics, 37 (4), 1307-1312, 1998. [11] K. Huang, ”Statistical Mechanics”, Wiley Eastern, New Delhi, 1975. [12] B.G. Sidharth, ”Quantum Mechanical Black Holes:Towar ds a Unifica- tion of Quantum Mechanics and General Relativity”, IJPAP, 3 5, 1997. [13] B.G. Sidharth, Gravitation & Cosmology, Vol.4, No.2, 1 998. [14] R. Penrose, ”Angular Momentum: An approach to combinat ional space- time” in, ”Quantum Theory and Beyond”, Ed., Bastin, T., Camb ridge University press, Cambridge, 1971. [15] C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitatio n”, W.H. Free- man, San Francisco, 1973. [16] T. Jacobsen, European Journal of Physics, 17 , 1996, p.92. [17] B.G. Sidharth, in Instantaneous Action at a Distance in Modern Physics: ”Pro and Contra” , Eds., A.E. Chubykalo et. al., Nova Science Publish- ing, New York, 1999. 6

arXiv:physics/0004031v1 [physics.ins-det] 17 Apr 2000The rf control and detection system for PACO the parametric converter detector Ph. Bernard1, G. Gemme2∗, R. Parodi2and E. Picasso3 1)CERN, CH-1211, Geneva 23, Switzerland 2)INFN-Sezione di Genova, via Dodecaneso 33, I-16146 Genova, Italy 3)Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126, Pisa, Italy Abstract In this technical note the rf control and detection system fo r a detector of small harmonic displacements based on two coupl ed mi- crowave cavities ( PACO ) is presented. The basic idea underlying this detector is the principle of parametric power conversion be tween two resonant modes of the system, stimulated by the (small) harm onic modulation of one system parameter. In this experiment we ch ange the cavity length applying an harmonic voltage to a piezo-el ectric crystal. The system can achieve a great sensitivity to small harmonic displacements and can be an interesting candidate for the de tection of small, mechanically coupled, interactions (e.g. high freq uency gravi- tational waves). 1 Introduction In this technical note we describe the rf control and detecti on system for a detector of small harmonic displacements based on two coupl ed microwave cavities ( PACO ). This experimental configuration was initially proposed b y Bernard, Pegoraro, Picasso and Radicati [1], [2] and later p ut in practice by Reece, Reiner and Melissinos [3], [4], and has been discusse d in some detail ∗e-mail: gianluca.gemme@ge.infn.it 1in previous papers [5], [6], [7], [8]. Here we just remind the basic principles underlying the detector operation. The detector consists of an electromagnetic resonator with two resonant frequencies ωsandωaboth much higher than the characteristic frequency of the harmonic perturbation Ω, and which satisfy the resona nce condition |ωs−ωa|= Ω. In the scheme proposed in [1], [2] the two resonant modes are obtained coupling two identical resonators; ωsis the frequency of the symmetric (even) resonant mode, while ωais the frequency of the antisym- metric (odd) one. If some energy is initially stored in the sy mmetric mode, an harmonic perturbation can induce the transition of some e nergy to the initially empty level that can be extracted at frequency ωa. The electromag- netic power in the antisymmetric mode is proportional to the square of the amplitude of the perturbation. In this experiment we use two cylindrical cavities coupled t rough an ax- ial iris and we change the cavity length applying an harmonic voltage to a piezo-electric crystal. To increase the sensitivity of the detector a resonator geometry and field configuration with high geometrical facto r and high qual- ity factor Qare preferred. To avoid electron field emission from the cavi ty surface rf modes with vanishing electric field at the surface are mandatory. For these reasons we have chosen TE mode (TE 011) superconducting cavi- ties. The choice of frequency was imposed by the maximum dime nsion of the resonator that can be housed in a standard vertical cyost at; in our case the inner diameter is 300 mm, giving us enough room for a 3 GHz, TE011 resonator. The system can achieve a great sensitivity to small harmonic displace- ments (up to δl/l≈10−20@ Ω =1 MHz, and can be an interesting candidate for the detection of small, mechanically coupled, interact ions (e.g. high fre- quency gravitational waves). 2PACO rf system The three main functions of rf control and measurement syste m of the PACO experiment are shown in figure 1. The system performs three separate functions listed in the f ollowing. 1. The first task of the system is to lock the rf frequency of the master oscillator to the resonant frequency of the even mode of the c avity and to keep constant the energy stored in the mode. 2Figure 1: PACO rf system 2. The second task is to increase the detector’s sensitivity by driving the coupled resonators purely in the even mode and receiving onl y the rf power up- converted to the odd mode by the perturbation of t he cavity walls. This goal can be obtained by rejecting the sign al at the even mode frequency tacking advantage of the symmetries in t he field distribution of the two modes. 3. The third task is the detection of the up-converted signal pushing the detector’s sensitivity to the limit set by the contribution of the noise sources at the operating frequency. The various noise sourc es have been described and discussed in a previous paper [6]. 3 The rf control loop The output of the master oscillator (HP4422B) is fed into the cavity trough a TWT amplifier giving a saturated output of 20 Watt in the frequ ency range of 2-4 GHz. The stored energy in the cavity is adjusted at the ope rating level 3by controlling the output of the master oscillator via the bu ilt in variable attenuator. The output signal is sampled via a 3 dB power splitter. The Aoutput of the splitter is sent to the TWT amplifier, the Boutput is sent, trough the phase shifter (PS), to the local oscillator (LO) input of a rf mixer acting as a phase detector (PD). The output of the rf power amplifier i s fed to the resonant cavity trough a double directional coupler, and a 1 80ohybrid ring acting as a magic tee. The rf power enters the magic tee via the sum arm, Σ, and is split in two signals of same amplitude and zero relativ e phase, coming out the tee co-linear arms 1 and 2. The rf signal, reflected by the input ports of the cavity, ente rs the magic tee trough the co-linear arms. The two signals are added at th e Σ arm and sampled by the directional coupler to give information a bout the energy stored in cavity allowing for the measurement of the couplin g factor, quality factor, stored energy. While driving the cavity on the even m ode no reflected signal is shown at the ∆ port of the magic tee where the signals coming from the co-linear arms are algebraically added to zero due to the 180ophase shift between the two signals coming by the co-linear arms. To get the maximum of the performances of the magic tee we need to have equal reflected signals (phase and amplitude) at the cav ity input ports. The equal amplitude goal can be achieved by a careful design o f the input couplers; a good design guarantees us also very nice phase eq uality at the ports. To preserve the signal integrity we use matched outpu t lines (in phase and amplitude) inside the cryostat. The cable used for the ou tput lines is the best rf cable (as far as attenuation and phase stability a re concerned) money can buy, nevertheless no characterisation at cryogen ic temperatures exists. Because the phase shift is very sensitive to tempera ture inhomo- geneities between the two cables and the phase difference bet ween the two co-linear arms of the magic tee gives a quite strong signal at the ∆ port, we need to compensate for differential thermal contractions of the cables inside the cryostat, leading to phase unbalance in the feed lines. T o do that we insert a phase shifter in one of the lines to reduce to a minimu m the leakage of the unwanted modes on the two ports. As we will show in the fo llowing section, mode leakage of the even mode to the ∆ port sets a limi t to the system sensitivity increasing the overall noise level of th e detector. Mode leakage of the odd mode to the Σ port reduces the system se nsitivity by reducing the signal level available for detection. The co nverted rf power (odd mode) coupled to the input Σ port is lost forever. The rf s ystem is 4symmetric on both the input and output ports of the detector c avity. The output ports of the cavity are coupled for a maximum output si gnal on the odd mode (detection mode) and the magic tee is used to reject t he rf power at the frequency of the even mode. The up-converted signal (odd mode) comes out at the ∆ port of the magic tee. A fraction of the signal at th e Σ port is fed to the rf input of the phase detector PD via a low noise rf am plifier. The intermediate frequency (IF) output of the phase detector PD is fed back to the rf master oscillator to lock the output signal to the reso nant frequency of the resonator. The total phase shift around the loop is set trough the phase shifter PS, to have the maximum of energy stored in the d etector. A carefully design of the servo loop amplifier (SLA) guarantee s the stability of the system and the rejection of the residual noise of the mast er oscillator up to one MHz. The same fraction of the Σ output of the output magi c tee is used to keep constant, to 100 ppm, the energy stored in the cav ity feeding back an error signal to drive the electronically controlled output attenuator of the master oscillator. Great deal of care is needed in tailoring the frequency respo nse of both controls because the two loops can interact producing phase -amplitude os- cillations in the rf fields stored in the cavities. 4 Sensitivity enhancement using the mode sym- metry The two modes of the detector cavity have (as in the case of two coupled pendulums) opposite symmetries of the fields. The rf detecti on system can greatly be improved if the mode symmetry information is used in the rf feed and detection system to reduce the residual components of the noise produced by the master oscillator line width. A further impr ovement is obtained tacking advantage of the very high selectivity of t he resonator’s modes having a quality factor Q≈1010. Using two separate sets of ports to drive the cavity and to rec eive the up-converted signal (at the frequency of the odd mode), the c avity acts as a very sharp filter, with an high rejection of the signal noise ( coming from the master oscillator) at the frequency of the up-converted sig nal. The resulting attenuation is given by the shape of the cavity resonance, a L orentz shape with a full width half maximum FWHM =f/Q, where fis the resonant 5Figure 2: PACO rf control loop frequency. This already low residual noise, can be even more reduced with the two magic tees. Using two equal couplers at the cavity inp ut, driven by the two co-linear arms of a magic tee fed via the Σ port, we stor e energy in the cavity only on the even mode. Receiving the up-converted signal at the ∆ port the of the second magic tee, rejects the even mode compo nents from the cavity by an amount given by the magic tee insulation. In the case of an ideal magic tee the mode rejection is infinite as the tee insulation. No even mode component is transmitted trough th e system and there will be no signal at the output port if the cavity is driv en purely in the even mode. In the ideal case this results is obtained also in t he more simple scheme used by Melissinos and Reece [3], [4], measuring the u p converted power coming out of the detector along the input lines. Our scheme gives better sensitivity and performances in the real case. The first obvious gain is the sum of the ∆ to Σ port insulation of the two tees, plus the possibility of adjusting separately the inpu t and output lines to get better mode rejection. In a commercial magic tee the in sulation is specified to be ≈25−35 dB over the whole bandwidth of the device (usually 6Figure 3: Sensitivity enhancement circuit an octave in frequency). The reason for this quite low insula tion is mainly due to the difficulty of balancing on a large range of frequency the phases of the signals coming from the two co-linear arms of the tee. Consider two equal signals in phase V=V0cos(ωt+φ) (1) entering the co-linear arms of the magic tee; suppose a phase unbalance of α degrees between the two path to the ∆ port : the resulting sign als at the ∆ are V∆=V0cos(ωt+φ)−V0cos(ωt+φ+α) = =V0cos(ωt+φ)−V0cos(ωt+φ) cosα+V0sin(ωt+φ) sinα (2) or, for a small value of α V∆=α V0sin(ωt+φ) (3) Now a phase unbalance as small as five degrees reduces the insu lation from ∆ to Σ port to only 25 dB. This fairly low insulation in magic te es comes 7from the difficulty of designing a 90 degrees rf broad band phas e shifter. A custom designed magic tee can be optimized to give a better in sulation (35- 40 dB) on a more limited frequency interval. Since the bandwi dth-insulation product is roughly constant, pushing the requirements to hi gh insulation (70- 80 dB) reduces too much the useful bandwidth, giving not so mu ch flexibility in the rf system design . Any mismatch between the frequencie s of the cavity modes and of the magic tees results in a very severe degradati on of the noise rejection for the whole system at the detection freque ncy, spoiling the ultimate sensitivity of the detector. Our electronic scheme allows for an independent compensati on of the magic tee phase mismatch either at the feed frequency and at t he detection frequency in a flexible way: the phase mismatch is compensate d using a variable phase shifter at the input of one of the co-linear ar ms. Getting the optimum phase at the input side will results in a pure excitat ion of the even (drive) mode of the two cavity system, keeping the power at th e frequency of the odd (detection) mode 70 dB (the tee insulation) below t he level of the drive mode. Adjusting the phase at the output will couple to t he output only the odd mode components rejecting the even mode component by 70 dB. The total even-odd mode rejection of the system is the sum of the a ttenuation we can obtain from the two 90ohybrids; in the real world a practical limit is set by the cross talk effects between input and output limit ing to 120-130 dB the maximum achievable insulation. The price we pay for the mode selectivity improvement in our s et-up is affordable and gives us the possibility of a separate and mo re controlled tuning of the input and output circuits. The use of two separa ted ports for the input and output (with separated magic tees) add some com plication to the rf system. The coupling coefficient of the input and outp ut port of the two cell cavity need to be critically coupled ( β= 1) to the rf source and the rf detection system. In this way we will have the optim um transfer of power to the even mode (a maximum of stored energy) and to th e odd mode (a maximum in the detector output). Because the frequen cy and field distribution of the two modes are quite close, the input and o utput ports will be critically coupled to both modes. For that reason 50% of the even mode signal will be coupled to the idle Σ port at the output mag ic tee, and symmetrically 50% of the odd mode signal is coupled to the idl e ∆ port at the input magic tee. The new couplings will greatly affect the detector’s performances, affecting the loaded quality factor of the cav ity and changing in a substantial way the way to couple the cavity to the rf sour ce. 8At the glance it is clear that closing the two idle ports with a matched load will worsen the detector performance due to the followi ng reasons: 1. Half of the converted power at the odd mode frequency will b e lost worsening of a factor 2 the detector sensitivity. 2. The loaded quality factor of the cavity is reduced by a fact or roughly 2 with the same reduction in the system sensitivity. 3. The coupling to the rf generator is reduced with a reductio n of the detector efficiency (due to the input mismatch the incoming po wer is partially reflected and more power is needed from the rf syste m to store the same energy in the cavity). A detailed analysis showed that closing the idle ports with a n impedance giving a reflection coefficient amplitude one and phase zero so lves the problem reflecting back to the cavity all the rf power coupled to the id le ports. Using that termination of the idle ports the two rf arms at the input and output are de-coupled (at least by the mode selectivity factor equa l to the sum of the ∆ and Σ ports insulation of the two hybrids). An rf impedan ce having reflection coefficient equal to one is an open circuit or a short circuit plus a 180ophase shift. For this reason we close the two idle ports using a phase shifter and a short. Again this solution gives the possibili ty to fine tune the system and compensate for the effect of non-ideal elements in the rf system. At the Σ port of the detection arm we insert a directional coup ler to sample a tiny amount of the even mode power coming from the cav ity. This transmitted power is fed to the frequency-amplitude servo l oop used to lock the frequency of the master oscillator to the cavity frequen cy and to keep constant the energy content in the cavity. This choice gives to our rf control system a better reliability over the Melissinos-Reece cont rol system using the reflected power from the cavity for the frequency control loo p. The effect of this sampling behaves in our system as the sampling antenna u sed to monitor and control the rf power fed to a superconducting cavity. As a final remark our system, despite some complexity, guaran tees the following improvement over the one used in the previous expe riment: •A better rejection of the phase amplitude noise of the master oscillator obtained using the sharp resonance of the resonator itself. •A better insulation of the drive and detector ports obtained by using separate drive and detection arms of the rf system. 9Figure 4: PACO detection system •The possibility of an independent adjustment of the phase la g in the two arms giving a better magic tee insulation at the operating fr equencies. •A greater reliability for the frequency amplitude loop usin g the trans- mitted power, instead than the reflected, coming from the cav ity. 5 Detection of the converted signal The signal converted to the odd mode by the interaction betwe en the mechan- ical perturbation and the rf fields is coupled to the ∆ port of t he detection arm of the rf system and amplified by the low noise rf amplifier L NA. Great deal of care must be used in the choice of the LNA, because the n oise figure of the amplifier greatly affects the detector’s sensitivity i n the detection re- gion above the mechanical resonances of the cavity (typical ly spanning the frequency interval 1-10 kHz) The LNA we choose is a commercia l, room tem- perature, low noise amplifier with a 48 dB gain and quite large bandwidth of 500 MHz centered on our operating frequency of 3 GHz. The LN A Noise figure is 0.8 dB corresponding to a noise temperature of 360 K. The converted rf signal amplified by the LNA is fed to the rf inp ut port (RF) of a low noise double balanced mixer M1; the local oscill ator port (LO) 10of M1 is driven by the even mode rf power (at frequency ωs) transmitted by the cavity. The LO input level is adjusted for a minimum noise contribution from the mixer. As shown in the previous section the input spe ctrum to the RF port of the mixer is composed by two signals: the first at fre quency ωs coming from the rf leakage of the even mode trough the detecti on system (even if greatly reduced by the aforementioned double rejec tion of the care- fully tuned magic tees); the second is the converted energy o n the odd mode at frequency ωa. Both signal are down converted by the M1 mixer giving to the IF port a DC signal proportional to the even mode leakag e and the signal at frequency Ω proportional to the odd mode excitatio n. The down-converted IF output is further amplified using a low noise audio preamplifier (Stanford Research SR560). The combination of tunable built in filters of the SR560 amplifier allows us to further reject (i f necessary) the DC component coming from the even mode leakage. Last the outp ut of the audio amplifier is fed to the lock in amplifier (Stanford Resea rch SR844) used as synchronous detector driven by the low frequency syn thesizer used to drive the detector cavity trough a piezoelectric ceramic at frequency Ω. The detection electronics for the detection of harmonic, me chanically cou- pled interactions, at frequency Ω (as gravitational waves) is slightly different. Since the exact frequency and phase of the driving source is n ot known, we can’t perform a synchronous detection; we need to perform a a uto correla- tion of the detector output - or to cross-correlate the outpu ts of two different detectors - to detect the down converted component at Ω. The o utput of the SR560 preamplifier is fed to a fft signal analyzer (Stanford Re search SR780) able to average the input signal on a bandwidth as small as 0.1 mHz, giving us a very comfortable margin to get the ultimate sensitivity foreseen for the feasibility study of our detector. The outlined scheme of electronic detection gives us the non -marginal benefit of being (at the first order at least) self compensatin g against per- turbations changing in the same way the frequency of the two m odes of the detector. This type of perturbation is usually produced by c hanges of the cavity walls due to changes of pressure of the helium bath use d to keep the cavity at the operating temperature or by the thermal contra ction of the cav- ity walls produced by changes of the operating temperature. Furthermore similar effects are produced by the radiation pressure of the electromagnetic energy stored in the even mode; this pressure acts in a symmet rical way on the cavity walls producing equal deformation of both the det ector’s cells. The result of the deformation is a frequency shift of both the mod es of a same 11amount of frequency; this kind of deformation does not affect (at the first order at least) the coupling of the cells fixing the amount of s plitting between the even and odd modes. As a result the frequencies of the two m odesωs andωawill change under the effect of pressure changes on the cavity walls but the difference Ω (related to the coupling) will not. In our scheme the local oscillator (used for the down convers ion of the rf signal coming from the detector) is the rf signal transmitte d by our cavity at the frequency of the even mode. In this way the common mode f requency drift of the resonators is automatically cancelled (at the fi rst order at least) and the mixer M1 output is a signal exactly at the perturbatio n frequency Ω. This effect gives us the possibility of using a quite simple refrigeration scheme (without any complex and cumbersome servo loop) to ke ep constant the helium bath temperature and pressure. 6 Experimental results The electromagnetic properties of the cavity have been meas ured in a vertical cryostat after careful tuning of the two cells frequencies. The symmetric mode frequency was ωs/(2π) = 3.03431 GHz and the mode separation was Ω = 1 .3817 MHz. The unloaded quality factor at 4.2 K was Q= 5·107, and no significant improvement was found lowering the helium bat h temperature at 1.8 K. Even after a second chemical polishing, performed a t CERN, which removed approximately 300 µm of niobium from the surface, no improvement was observed. We believe that this very low Qvalue is due to hot spots on the surface caused by welding problems occurred during the cavi ty fabrication. Adjusting the phase and amplitude of the rf signal entering a nd leaving the cavity, the arms of the two magic tees were balanced to launch the even mode at the cavity input and to pick up the odd mode at the cavit y output. With 30 dBm of power at the Σ port of the first magic tee, -90 dBm w ere detected at the ∆ port of the second one, giving an overall att enuation of the symmetric mode of 120 dB. The energy stored in the cavity w ith 30 dBm input power was approximately 1 mJ. The signal emerging from the ∆ port of the output magic tee was amplified by the LNA and fed into a sp ectrum analyzer. The signal level at frequency ωawas 120 dBm in a 1 Hz bandwidth. System sensitivity at this stage is given by hmin≈6.55·10−18Hz−1/2(4) 12This value is quite far from our goal of hmin= 10−20(Hz)−1/2. Since, to our knowledge, this is the first example of a parametric detec tor operated in trasmission, and since this configuration requires very car eful adjustments of the input and output ports balancing, we believe that signifi cant improve- ments are obtainable. Furthermore the new cavity under cons truction should show the high quality factor needed to reach high sensitivit y. References [1] F. Pegoraro, L.A. Radicati, Ph. Bernard and E. Picasso, P hys. Lett. 68A, 165, (1978). [2] F. Pegoraro, E. Picasso and L.A. Radicati, J. Phys. 11A, (10), 1949, (1978). [3] C.E. Reece, P.J. Reiner and A.C. Melissinos, Phys. Lett. 104A , 341, (1984). [4] C.E. Reece, P.J. Reiner and A.C. Melissinos, Nucl. Instr . Meth. A245 , 299 (1986). [5] Ph. Bernard, G. Gemme, R. Parodi and E. Picasso, Eigth Wor kshop on Rf Superconductivity, Abano Terme, (1997); Particle Accel erators 61, [343]/79, (1998) [6] Ph. Bernard, G. Gemme, R. Parodi and E. Picasso, Infn Inte rnal Note, INFN/TC-98/17, (1998). [7] Ph. Bernard, G. Gemme, R. Parodi and E. Pi- casso, Infn Internal Note, INFN/TC-99/21, (1999), http://wwwsis.lnf.infn.it/pub/INFN-TC-99-21.pdf . [8] Ph. Bernard, G. Gemme, R. Parodi and E. Picasso, Ninth Wor kshop on Rf Superconductivity, Santa Fe (NM), (1999), gr-qc/9911024 . 13

arXiv:physics/0004032v1 [physics.ins-det] 17 Apr 2000SUPERHEATED DROP AS A NEUTRON SPECTROMETER Mala Das, B. K. Chatterjee, B. Roy1and S. C. Roy Department of Physics, Bose Institute 93 / 1 A. P. C. Road, Calcutta 700009, India Abstract Superheated drops are known to vaporise when exposed to ener getic nu- clear radiation since the discovery of bubble chamber. The a pplication of su- perheated drops in neutron research specially in neutron do simetry is a subject of intense research for quite sometime. As the degree of supe rheat increases in a given liquid, less and less energetic neutrons are require d to cause nucleation. This property of superheated liquid makes it possible to use it as a neutron spectrometer. Neutron detection efficiency of superheated d rops made of R12 exposed to Am-Be neutron source have been measured over a wid e range of temperature -17oC to 60oC and the results have been utilized to construct the energy spectrum of the neutron source. This paper demonstra tes that a suit- able neutron spectrometer may be constructed by using a sing le liquid and varying the temperature of the liquid suitably at a closer gr id. PACS No. : 29.30 Hs, 29.40.-n Keywards : SDD, R12, temperature dependence, neutron, efficiency, spe c- trometry. 1Author for communication : Tel.no.-350 2402/03, Ext.-305, Fax no.- 91 033 350 6790, e-mail- biva@boseinst.ernet.in 11. INTRODUCTION A fluid kept in the liquid state above its boiling temperature is called super- heated. Application of superheated liquid to detect ionizi ng radiation is well known from the times of bubble chamber[1]. The resurgence of its us e has been observed from the late seventies[2,3] and the investigations on the s ubjects in the last two decades turned into an almost maturing technology especial ly to detect neutrons and more recently to detect photons. The suitability of using su perheated drops as a neutron dosimeter[4,5,6,7,8] has already been establishe d. The superheated drops are now commercially available as neutron and photon dosime ter in the trade name superheated drop detector (SDD)2and bubble detector (BD).3In a superheated liquid minimum energy (threshold) required to nucleate dec reases as the degree of superheat increases. The degree of superheat of a liquid cou ld be defined simply by the difference of ambient temperature above the boiling poin t of the liquid. There- fore, liquids with lower boiling points possess higher degr ee of superheats at a given ambient temperature (above their boiling points) and as the ambient temperature increases the given liquid becomes more and more superheate d. This property of the superheated liquid are being utilised to develop neutro n dosimetry and neutron spectrometry[2,3,9,10]. There are two distinct types of me thodologies used in de- veloping neutron spectrometer. In one, a collection of supe rheated samples made of liquids with different boiling points (i.e. with different th reshold neutron energies) are utilised[11], while in the other, two liquids are chosen and the temperature of the liquids are varied at four different temperatures to obta in eight sets of threshold energies[12,13,14] (equivalent to eight different samples with different boiling points). The temperature variation method is superior than using sam ples with different boil- ing points. By controlling the temperature of the sample one can, in principle, change the threshold neutron energy at any desired level (equivale nt to using ’finer’ windows to scan the spectrum), while in the other method one is limite d by the availability of liquids with lower boiling points (equivalent to using ’c oarser’ windows). In the present work we measured the detection efficiency of a single s ample made of R- 12(Dichlorodifluoromethane : C Cl2F2), which is known to be sensitive to neutrons of energies from thermal to tens of MeVs, by (almost) continuou sly changing tempera- tures over a wide range. The response of the sample to Am-Be ne utrons have been measured at about thirty different temperatures in the range which is equivalent to using thirty different samples with thirty different boiling points (the experiment has been actually performed more than 50 different temperatures in the range -17oCto about 60oCbut it has been observed that the sample started responding f rom about 0.5oCto Am-Be neutrons). In addition to the advantage of using sin gle liquid, con- 2Superheated Drop Detector is the registered trademark of Ap fel Enterprises Inc, NewHaven, CT, USA 3Bubble Detector is the registered trademark of Bubble Techn ology Industries Ltd. 2trolling the temperature enables one to scan the energy spec trum by finer ’windows’ thereby improving the inherent energy resolution of the spe ctrometer when compared with other such spectrometers using superheated liquid. 2. PRINCIPLE OF OPERATION The superheated state of the liquid is a metastable state and the nucleation in this state can be initiated by the presence of heterogeneo us nucleation sites such as air bubbles, solid impurites, gas pockets etc. or by radia tion interactions. The nucleation in superheated state starts with the formation o f a critical sized vapour embroy. The free energy required to form a spherical vapour b ubble of radius r in a liquid is given by G= 4πr2γ(T)−4 3πr3(pv−po) (1) whereγ(T) is the liquid-vapour interfacial tension, pvis vapour pressure of the super- heated liquid and pois the ambient pressure. The difference pv-pois called the degree of superheat of a given liquid. One can see from equation (1) that G is maxim um at r= 2γ(T)/(pv−po) =rc (2) wherercis called the critical radius . When a bubble grows to the size of the critical radius it becomes thermodynamically unstable and grows ver y fast till the entire liquid droplet vaporises. The minimum amount of energy (W) needed to form a vapour bubbl e of critical size rcas given by Gibbs[15] from reversible thermodynamics is W= 16πγ3(T)/3(pv−po)2(3) whereγ(T) = C (Tc−T−d) withTcthe critical temperature of the liquid and C and d are constants [16]. With increase in temperature, since the degree of superheat (pv-po) increases andγ(T) decreases, the minimum energy ( W) required for vapour bubble nucleation will be less. The variation of Wwith temperature for superheated drops of R-12 is shown in figure 1. Therefore W, the threshold energy for nucleation depends on the type and the temperature of the liquid. When a neutron of ener gyEninteracts with a nucleus of atomic weight A, the maximum energy that can be tr ansferred to the nucleus from the neutron is through the elastic head on colli sion and is given by, Ei= 4AEn/(A+ 1)2(4) 3After receiving the energy, the nucleus is scattered from it s atom and moves through the liquid losing its energy through Coulombic inte raction until it comes to rest. For a given neutron energy, different nuclei of the liqu id will receive different amount of energy, depending on their atomic weight. The ion w ith the highest value of linear energy transfer (LET) or ( dE/dx ) in the liquid, will play the major role in vapour nucleation[17]. The energy deposited along that par t of the ion’s path (L) corresponding to about twice the critical radius contribut es significantly to bubble formation[17,18]. For nucleation to occur this deposited m ust exceedWthe minimum energy required for bubble formation. Usually most of the energy is lost into heat and a very small fr action of the deposited energy is utilised in nucleation and W/ Ecis called the thermodynamic efficiency (ηTof nucleation [17]. W=krcdE/dx = 2ηTrcdE/dx (5) wherek(constant) equals twice the thermodynamic efficiency ( η). Hence, in the equation W/r c=kdE/dx (6) relates the threshold energy (corresponding to the dE/dx ) for nucleation to the am- bient temperature (corresponding to W/r c). This enables us to convert the tempera- ture scale of superheated drops to the (threshold) energy sc ale of incident neutrons. Therefore by varying the ambient temperature of the superhe ated drops, one can observe the variation of the SDD response at different neutro n energies which has been used in neutron spectrometry. 3. EXPERIMENT The experiment was performed with superheated drops of R12 a t different temperatures by using the volumetric method, described by D aset al.[19]. The vial containing the sample was connected to a graduated horizont al glass tube with a small coloured water column as marker. The vapourization of a liquid drop displaces the water column by the distance corresponding to the volume of vapour formed. The details of the preparation of the sample is given elsewhe re[20]. If neutrons of flux ψare incident on superheated drops of volume V, liquid of densityρLand molecular weight Mthe vaporization rate is given by dV dt=VψNAρL Mηd/summationdisplay niσi (7) 4where N A= Avogadro Number d = average droplet volume ni= weight factor of the ith element in the molecule whose neutron nucleus elastic scattering cross section is σi η= efficiency of neutron detection. Due to nucleation by neutrons, the displacement of the water column along a horizontal glass tube was measured as a function of time. The procedure of calculating ηfrom the measured displacement of water column has been expl ained in detail in one of our recent publications[19]. The temperature of the sample was controlled by an indigeneo usly made tem- perature controller. For low temperature measurements the sample was placed in an alcohol bath sitting on the top of a cold finger dipped in liq uid nitrogen. The upper part of the finger was wrapped with heating tape and by ap plying different voltages to the tape, different steady temperatures of the ba th can be achieved. For measurements of higher temperatures the same setup was used without the liquid nitrogen. The fluctuation of temperature in these measureme nts was found to be within ±0.1oC. In the experiment the ambient temperature of the sample was i ncreased slowly from low to any desired higher value and the nucleation rate w as measured at each temperature. The measurement was performed from in a temper ature range of -17oC to about 60oCin a close grid necessary to obtain the energy spectrum of the source from temperature. The nucleation due to background radiati on and due to other fluctuations has been subtracted. The liquid was observed to become unstable due to spontaneous nucleatation at about 60oC. 4. EXPERIMENTAL RESULTS AND COMPUTATION OF THE NEU- TRON ENERGY The variation of neutron detection efficiency ( η) for R12 with temperature in presence of neutrons from Am-Be neutron source is presented in figure 2. The solid line in figure 2 is the spline smoothing of the efficiency data at different temperatures. The uncertainties presented in the figure are the total exper imental uncertainties of estimating η. The derivative of efficiency, d η/dT against temperature is shown in figure 3. Now one has to estimate the equivalence of the energy of the detected neutrons with the temperature of the SDD sample. One way to do it is to expose the sample at different temperatures to different monoenerge tic neutron sources and to note the threshold neutron energies for nucleation [12,1 4]. A novel approach has been used in this work. 5As has been presented in Section 2, the different nuclei of the superheated liquid would receive different amount of energy and they must have different dE/dx. In case of R-12 containing C, Cl and F, dE/dx of these ions with different neutron energy are presented in Figure 4. From Figure 4, it is clear th at the dE/dx values of C, Cl and F are comparable in the neutron energy of our inter est and we take the average value of dE/dx of all the ions using the equation belo w (dE/dx )average =/summationdisplay niσi(dE/dx )i//summationdisplay niσi (8) whereniis the number of ions of the i-th element in a R12 molecule, σiis the neutron- nucleus elastic scattering cross section and ( dE/dx )iis the LET of the i-th ion. The variation of average dE/dx as a function of neutron energy is shown in figure 5. From equations (3) and (5), we obtain (dE/dx ) = 8πγ2(T)/3k(pv−po) (9) From the equation above the dE/dx has been plotted against te mperature for differ- ent arbitrary values of kof which only four such plots ( k=1, 0.1, 0.05, 0.0195) are presented in figure 6 using the equivalence between the tempe rature of the sample and the incident neutron energy from figure 5. The variation o f threshold neutron energy for nucleation with temperature of the sample for diff erentkhas been studied of which four such variations are shown in figure 7. With the op timum value of k the temperature axis of the figure 3 has been converted to neut ron energy and the resulting spectrum was fitted with the peak neutron energy of the241Am-Be neutron spectrum. The best fit is obtained for k equals 0.0195. The ana lysis has been per- formed upto a maximum temperature of 42.5oC. The final neutron energy spectrum of241Am-Be source obtained from our analysis is shown in figure 8. If L = 2rcis taken as the distance in the ion’s path which contributes s ignif- icantly in nucleation of superheated drops [17,18] in calcu lating the thermodunamic efficeincy of nucleation, our experimental analysis produce s the value close to 0.01. It may be noted here that Apfel et al.[17] obtained this value ranging from 0.03 to 0.05. 65. DISCUSSION The result shows that ηincreases with temperature. At low temperature, the threshold energy for nucleation (W) is high, which indicate s that a larger energy is required to cause nucleation. According to equation 3 as tem perature increases, W decreases and more and more neutrons from the low energy rang e of the spectrum are taking part in nucleation. So ηincreases with temperature. The sharp increase ofηnear 25oC corresponds to detection of neutrons with energies rangin g from high- est available to those at the peak of the spectrum. At high tem perature when all the neutrons of the spectrum contribute in nucleation, ηshould be constant with temperature. But at about 45oC,ηincreases again. We suspected that the sample becomes sensitive to gamma rays coming out of the Am-Be sourc e. In a separate experiment, we indeed observed that R-12 becomes sensitive to gamma rays at about 45oC. Since, as mentioned before, the analysis has been performe d upto a maximum temperature of 42.5oC, the contribution due to gamma rays is absent in the measured energy spectrum in this work. Figure 3 shows that the d η/dT vs. T graph resembles the neutron energy spectrum of241Am-Be where the second peak corresponds to the gamma sensiti vity of the sample. The ambient temperature of the superheated dr ops was converted to the energy of the neutrons following the method described in Section 4. So by using superheated drops at different temperatures, it is pos sible to obtain the neutron energy spectrum. This indicates important use of SDD in neut ron spectrometry. The maximum uncertianty in neutron energy as could be found from figure 7 is within 5% in the entire region of our investigation. This method can be used to determine any other neutron energy outside the present range, only then on e has to consider the ion with maximum dE/dx and the rest of the analysis is same as t his. The present study also helps to select the suitable material (liquid) fo r a given neutron energy spectrum. It has been observed in this experiment that the nucleation r ate of superheated drops rapidly changes for samples exposed to thermal shock compared to samples whose temperature was changed slowly. The liquid appeared t o be more fragile when temperature was changed rapidly. Though this is not quite un expected in the exact physics of this phenomenon, why the liquid becomes more frag ile under heat shock, requires further investigation. 7REFERENCES 1. D.A. Glaser, Phys. Rev. A 87, 665 (1952). 2. R.E. Apfel, US Patent 4,143,274 (1979). 3. R.E. Apfel, Nucl. Inst. Meth. 162, 603(1979). 4. R.E. Apfel and S.C. Roy, Nucl. Inst. Meth. 219, 582 (1984). 5. R.E. Apfel and Y.C. Lo, Health Phys. 56, 79 (1989). 6. R.E. Apfel, Rad. Prot. Dos. 44, 343 (1992). 7. S.C. Roy, R.E. Apfel and Y.C. Lo, Nucl. Inst. Meth. A255 , 199 (1987). 8. H.Ing, Nuclear Tracks. 12, 49 (1986). 9. R.E. Apfel, Nucl. Inst. Meth. 179, 615 (1981). 10. K. Chakraborty, P. Roy, S.G. Vaijapurkar and S.C. Roy, Proc. of 7th National Conference on Particles and Tracks , Jodhpur pp 133 (1990). 11. H. Ing, R. A. Noulty and T. D. Mclean, Rad. Meas. 27, 1 (1995). 12. F. d’Errico, W. G. Alberts, G. Curzio, S. Guldbakke, H. Kl uge, and M. Matzke, Rad. Proc. Dos. 61, 159 (1995). 13. F. d’Errico, R. E. Apfel, G. Curzio, E. Dietz, G. F.Gualdr ini, S. Guldbakke, R. Nath, B. R. L. Siebert, Rad. Proc. Dos. 70, 1 (1997). 14. F. d’Errico, W. G. Alberts and M. Matzke, Rad. Proc.Dos. 70, 103 (1997). 15. J. W. Gibbs, Translations of the Connecticut Academy III , p.108 (1875). 16. F. H. Newman and V. H. L. Searle, The general properties of matter(fifth ed.), p.189 (1985). 17. R. E. Apfel, S. C. Roy and Y. C. Lo, Phys. Rev. A 31, 3194 (1985). 18. M. J. Harper and M.E. Nelson, Radiat. Prot. Dosim. 47, 535 (1990). 19. Mala Das,B. Roy, B. K. Chatterjee and S. C. Roy, Rad.Meas. 30, 35 (1999). 20. B. Roy, B. K. Chatterjee and S. C. Roy, Rad. Meas. 29, 173 (1998). 8FIGURE CAPTIONS Fig. 1: Variation of threshold energy (W) required for nucleation i n R12 as a function of temperature (T). Fig. 2: Observed variation of neutron detection efficiency ( η) as a function of temperature (T) in R12. Fig. 3: Variation of the derivative of neutron detection efficiency ( dη/dT) as a function of temperature (T). Fig. 4: Variation of stopping power (dE/dx) of different ions (C, Cl, F) in Freon-12 as a function of neutron energy. Fig. 5: Variation of average stopping power ( dE/dx )average over three different ions in Freon-12 as a function of neutron energy. Fig. 6: Variation of stopping power (dE/dx) of ion in R12 as a functio n of temper- ature (T) of the sample, for different arbitrary values of k. Fig. 7: Variation of neutron energy as a function of temperature (T) of the sample, for different k. Fig. 8: The neutron energy spectrum of241Am-Be obtained from the experiment. 9

arXiv:physics/0004033v1 [physics.ins-det] 17 Apr 2000EFFICIENCY OF NEUTRON DETECTION OF SUPERHEATED DROPS OF FREON-22 Mala Das, B. Roy, B. K. Chatterjee and S. C. Roy Department of Physics, Bose Institute 93 / 1 A. P. C. Road, Calcutta 700009, India Abstract Neutron detection efficiency of superheated drops of Freon-2 2 for neutrons obtained from a 3 curie Am-Be neutron source has been reporte d in this paper. Although Freon-22 having lower boiling point than many othe r similar liquids (e.g.Freon-12, Freon-114, Isobutane ) is expected to be mor e sensitive to neu- trons ,it has not been reported so far and therefore this pape r constitutes the first report on the subject. Neutron detection efficiency of bo th Freon-22 and Freon-12 have been determined from the measured nucleation rate using the volumetric method developed in our laboatory. The result sh ows that neutron detection efficiency of Freon-22 for the neutron energy spect rum obtained from an Am-Be source, is almost double, while the life time is 58.6 % smaller than that of Freon-12, for a particular neutron flux of that source . 1. INTRODUCTION A liquid maintained at the same state above its boiling point is said to be superheated. It is a metastable state of the liquid and can be nucleated to form vapour by the deposition of small energy by ions, charged par ticles or by any het- erogeneous nucleation sites such as gas pockets, impuritie s etc. The superheated drops,suspended in gel can be used to detect neutrons throug h the nucleation in- duced by the recoil nuclei in the medium . The recoil nuclei ar e produced by collision of neutrons with the nuclei constituting the superheated dr ops. The application of superheated drop detector (SDD) in neutron dosimetry has al ready been established (Apfel et al., 1984, 1989; Ing, 1986) and several other potential applica tions of SDD in neutron research has been discussed (Apfel, 1979a, 1979b , 1981; Chakraborty et al., 1990 ). Apfel (1992) has developed and characterised a pass ive superheated drop dosemeter using a volumetric technique for neutron monitor ing of personnel and in accelerartor applications. These type of dosimeters are al so commercially available from Apfel Enterprises Inc.,USA. Practical application of the SDD demands that the 1sample must be reasonably stable against spontaneous nucle ation due to background radiation and other environmental effects, and at the same ti me as much as possible sensitive to neutrons with energy spectrum of interest. Amo ng the different impor- tant features of the detector, the systematic quantitative evaluation of the sensitivity of SDD has been studied for some liquids (e.g. Freon-114, Fre on-12, Isobutane, Freon 142B) (Roy et al., 1987) and the response function was report ed (Lo et al., 1988). This paper is an attempt to investigate more sensitive liqui d for neutron detection. Of the liquids investigated so far, Freon-12 is considered t o be the most sensitive liquid for neutron detection. Since the boiling point of Fre on-22 is much lower than Freon-12, we expect Freon-22 to be more sensitive to neutron s than Freon-12. To the best knowledge of the authors, no such investigation has been reported with Freon-22. An accurate method of determining the nucleation rate of SDD has been developed by Roy, et al., (1997b) using a relative manometer. Some other studies on the neutron detection sensitivity and detector response we re made by Ing and Birn- boim (1984), Ing (1986), Ipe et al., (1988) and Biro et al., (1990). Nath et al.,(1993) measured the neutron dose equivalent to patients undergoin g high energy x-ray and electron radiotherapy beams using a SDD device. They employ ed a passive method for measuring the total volume of neutron induced bubbles by displacing an eqiva- lent volume of gel into a graduated pipette. The method for th e determination of efficiency of detection of neutrons by vapour nucleation of su perheated drops using volumetric method has been developed in our laboratory by Ro yet al., (1997a). We in this work, present the efficiency of neutron detection, max imum nucleation rate and life-time of SDD of Freon-22 irradiated by neutrons from an Am-Be source for a particular neutron flux and compare it with those of Freon-12 . Some of the physical parameters of SDD of Freon-12 and Freon-22 are presented in t able-1. Comparison is made with Freon-12, since Freon-12 is the most well studie d liquid in neutron de- tection. The paper has been organised to present a brief outl ine of the method of bubble formation, description of volumetric method and mea surement, results and discussion. 22. THEORY OF BUBBLE FORMATION The free energy required to form a spherical vapour bubble of radius r in a liquid is given by (Roy et al., 1987) G= 4πr2γ(T)−4 3πr3(pv−po) (1) whereγ(T) is the liquid-vapour interfacial tension, Pvis vapour pressure of the super- heated liquid and Pois the ambient pressure. The difference pv-pois called the degree of superheat of a given liquid. One can see from equation (1) that G is maxim um at r= 2γ(T)/(pv−po) =rc (2) wherercis called the critical radius . When a bubble grows to the size of the critical radius it becomes thermodynamically unstable and grows ver y fast till the entire liq- uid droplet vaporises. The minimum amount of energy (W) need ed to form a vapour bubble of critical size rcas given by Gibbs (1875) from reversible thermodynamics is W= 16πγ3(T)/3(pv−po)2(3) which is supplied by the energy deposition dE/dx being the en ergy deposited per unit distance travelled by the nuclei in the liquid by the recoil nucleus in a path length of 2rcinside the droplet. 3. PRINCIPLE OF THE VOLUMETRIC METHOD The present method utilizes the superheated drops suspende d in a dust free viscous elastic gel. The excess pressure required to form a v apour bubble of diameter 1mm inside this gel matrix is usually less than 1mm of mercury (as observed in a separate experiment). Hence the volume of the bubble trappe d inside the gel and that at atmospheric pressure are almost equal. This is why th e volume change upon nucleation would be the same whether the bubble formed is tra pped inside the gel or liberated from it. Upon nucleation the increase in volume of the droplet would displace the trapped air inside a vial containing sample. In the present volumetric method described in detail in the next section, the measurem ent of rate of change of volume has been made using such an air displacement system . The superheated drops suspended in the gel do not touch each other physically and they nucleate in a random manner, independent of each other. Under such condit ions it can be shown that the number of the drops and hence the volume of superheat ed liquid would decay 3exponentially with time. The rate of change of nucleated vol ume varies exponentially with a time constant τ(lifetime). The life time of these droplets in presence of a neutron flux and the efficiency of neutron detection has been st udied by measuring the volume of vapour formed upon nucleation. If neutrons of flux ψare incident on superheated drops of volume V, liquid densityρLand molecular weight M the vaporization rate is given by dV dt=VψNAρL Mηd/summationdisplay niσi (4) where N A= Avogadro Number d = average droplet volume ni= number of nuclei of the ith element of the molecule whose neutron nucleus elastic scattering cross section is σi η= efficiency of neutron detection. Due to nucleation by neutrons, the change in position (h) of t he water column along a horizontal glass tube which is connected to the sampl e vial (arrangement is given in next section), could be measured with respect to tim e (t). The present set up is made completely free from any leakage. So the rate of inc rease of the volume of vapour during nucleation should be equivalent to the rate of decrease of the volume of the superheated liquid. So, we have this equation ρVAdh dt=−ρLdV dt=−ρLVψNAρL Mηd/summationdisplay niσi (5) where A = cross section of the horizontal tube ρV= density of Freon vapour V = volume of superheated liquid at any instant of time t. Integrating and solving equations (3) one can obtain hA m=a[1−exp−b(t−to)] (6) where a =ρLVo ρVmand b=1 τ=ψNAρL Mηd/summationdisplay niσi (7) hA mis the volume of accumulated vapour in time t per unit mass of t he sample containing Superheated drops and gel, m being the mass of tot al sample (gel + superheated drops). Therefore the efficiency of neutron dete ctionηis given by η=bM ψNAρLd/summationtextniσi(8) 4Vo= initial volume of the Freon drops. t o= initial time at which the experiment has been started. τ=life-time of SDD in presence of source. and the product ab gives the maximum nucleation rate. The equation (6) has been scaled to per unit mass of the sample for standardization in making comparison of two samp les. Fitting equation (6) for different values ofhA mand t, constants a, b are obtained. Knowing b, the life timeτ(=1/b) in presence of neutron flux ψis obtained. The neutron detection efficiencyηcan be found out from equation (8) for a known flux of neutrons 4. EXPERIMENT The experiment was performed with 3Ci Am-Be neutron source, which has an energy distribution of neutrons with peak near 3.5 MeV. The s ource was placed inside a chamber through which neutrons coming in a fixed range of dir ection were used to irradiate the SDDs. Two separate sets of experiments were do ne with Freon-12 and Freon-22 respectively at about 30oC and at atmospheric pressure. The superheated drops are usually suspended in an immiscible gel. The gel tha t we used here, was a homogeneous mixture of some ultrasonic gel and glycerol in suitable proportion. The detail of the preparation of the sample was given elsewhe re (Roy et al., 1997b). The experimental apparatus consists of long glass tube of cr oss section 0.1573 sq.cm, placed horizontally on a graduated platform. The tube conta ined a coloured water column as an indicator of the volume of the vapour formed on nu cleation. One end of the glass tube was connected to the glass vial containi ng the sample, by means of rubber tube. The glass vial and the horizontal tube w ere at the same height. So the pressure inside and outside the tube were equa l, e.g. both were at the same atmospheric pressure. Therefore the displacement of the water column of 1 cm length due to nucleation would be directly related to the volume of the vapour formed. These displacements of the water column along the gl ass tube were measured as a function of time. The gamma sensitivity of sample of Freo n-22 was tested with 241Am (59.54keV),60Co (1170keV,1332kev),137Cs (662keV) gamma sources. The sample was placed either just in contact or at a close distanc e to the source. The sensitivity of the sample for 4.43MeV gamma ray present in241Am-Be source has also been tested by substantially reducing the neutron dose . 5. RESULTS The chemical formula and other physical parameters includi ng critical radius (rc) and minimum energy required for nucleation (W) as calculat ed using equations 5(2) and (3) respectively, for Freon-12 and Freon-22 have bee n listed in Table-1. The experimentally observed variation of volume of vapour form ed,hA m(per unit mass of the sample) during nucleation as a function of time for Freon -12 and Freon-22 are shown in Fig.1 and Fig.2 respectively. The measured data on r ate of nucleation and life-time of superheated drops of Freon-12 and Freon-22 in p resence of neutrons from a 3 Ci Am-Be source and the efficiency of neutron detection are g iven in table-2. The samples irradiated by gamma sources did not show any noticeb le nucleation at the experimental temperature. 6. DISCUSSION From the tabulated results it is seen that the degree of super heat (defined by the difference of vapour pressure of the superheated liqui dpvand the ambient pressurepo) attained for Freon-22 at room temperature (30oC) and at atmospheric pressure is about 38% larger than that of Freon-12. This indi cates that in presence of neutrons the life time of Freon-22 should be smaller than t hat of Freon-12. Our experimental results show that the maximum nucleation rate of Freon-22 is about 38% larger and the life time of Freon-22 is about 58.6% smaller th an that of Freon-12 for a fixed neutron flux from an Am-Be source. As can be seen from the c hemical formula that both of Freon-12 and Freon-22 contain carbon, chlorine and fluorine while Freon- 22 contains one hydrogen replacing one chlorine in Freon-12 . The neutron-nucleus elastic scattering cross-section for Freon-12 is 1.89 barn s while that of Freon-22 is 1.69 barns at neutron energy 3.5 MeV. Although the probabili ty of interaction of a neutron with nuclei of Freon-12 is larger than Freon-22, ou r experimental results (table-2) show that the present prepared sample of Freon-22 is about twice as efficient to detect neutrons than Freon-12. This is due to the fact that as seen from equation (3), the energy required for nucleation (W) decreases with d egree of superheat and as is evident fron table-1, the saturation vapour pressure of F reon-22 at 30oC is higher than that of Freon-12, therefore at this temperature Freon- 22 will attain a higher degree of superheat which means a smaller amount of energy de position is required for nucleation than that in Freon-12. So, although less numb er of recoil nuclei are available in Freon-22 from neutron nuclei elastic scatteri ng, the percentage of nuclei capable of deposition of energy greater than W for Freon-22 m ust be larger than that in Freon-12. As a result the efficiency of neutron detecti on of SDD of Freon- 22 is larger than that of Freon-12. From the experiment with g amma sources it is clear that the SDD based on Freon-22 is insensitive to gamma a t the experimental temperature. The high efficiency of neutron detection and the insensitivity towards gamma makes Freon-22 a suitable superheated drop detector f or neutron detection. 6REFERENCES Apfel R.E. (1979a) Detector and dosimeters for neutrons and other radiations. US Patent 4,143,274. Apfel R.E. (1979b) The superheated drop detector. Nucl. Inst. Meth. 162, 603. Apfel R.E. (1981) Photon-insensitive,thermal to fast neut ron detector. Nucl. Inst. Meth. 179, 615. Apfel R.E. (1992) Characterisation of new passive superhea ted drop (bubble) doseme- ters.Rad. Prot. Dos. 44, 343. Apfel R.E. and Lo Y.C. (1989) Practical neutron dosimetry wi th superheated drops. Health Phys. 56, 79. Apfel R.E. and Roy S.C.(1984) Investigation on the applicab ility of superheated drop detector in neutron dosimetry. Nucl. Inst. Meth. 219, 582. Biro T.,Kelemen A and Pavlicsek I. (1990) Acoustic Detectio n of neutrons by bubble detectors. Nucl. Tracks Radiat. Meas. 17, 587. Chakraborty K.,Roy P.,Vaijapurkar S.G. and Roy S.C. (1990) Study on neutron spectrometer using superheated drop detector. Proc. of 7th National Conference on Particles and Tracks , Jodhpur pp 133. Gibbs J.W. (1875) Translations of the Connecticut Academy I II, p.108. Ing H. (1986) The status of the bubble-damage polymer detect or.Nuclear Tracks. 12, 49. Ing H. and Birnboim H.C. (1984) A bubble damage polymer detec tor for neutrons. Nucl. Tracks and Radiat. Meas. 8, 285. Ipe N.E., Busick D.D. and Pollock R.W. (1988) Factors affecti ng the response of the bubble detector BD-100 and a comparison of its response to CR -39.Rad. Prot. Dos. 23, 135. Lo Y.C. and Apfel R.E. (1988) Prediction and experimental co nfirmation of the re- sponse function for neutron detection using superheated dr ops.Phys. Rev. A 38, 5260. Nath R, Meigooni A.S., King C.R., Smolen S. and d’Errico F. (1 993) Superheated drop detector for determination of neutron dose equivalent to patients undergoing high energy x-ray and electron radiotherapy. Medical Phys. 20, 781. Roy B, Chatterjee B.K., Das Mala and Roy S.C. (1997a) Study on nucleating effi- ciency of superheated droplets by neutrons. Radiation Physics and Chemistry (ac- cepted for publication). Roy B., Chatterjee B.K. and Roy S.C.(1997b) An accurate meth od of measuring life time of superheated drops using differential manometer. Radiation Measurements (accepted for publication). Roy S.C., Apfel R.E. and Lo Y.C. (1987) Superheated drop dete ctor: A potential tool in neutron research. Nucl. Inst. Meth. A255 , 199-206. 7Table-1 : A comparison of the physical parameters of Freon-12 and Freo n-22 Freon-12 Freon-22 1. ChemicalFormula CCl 2F2 CHClF 2 2. Molecular weight 120.91 80.47 3. Boiling Point -29.79oC -40.75oC 4. Surface tension ( γ) (at 30oC) dyn/cm 9 8 5. Vapour Pressure (p v) dyn / sq. cm. 7.4556 ×1061.14777 ×107 6. Density ( ρL) gm / cc 1.293 1.175 7. Degree of superheat (pv- po) dyn / sq. cm 6.441639 ×1061.0463739 ×107 8. Critical radius rc=2γ(T) (pv−po)cm 2.79 ×10−61.53×10−6 9. Minimum energy required (W) keV to form a vapour bubble of size r c 0.184 0.049 (W =16πγ3(T) 3(pv−po)2) 8Table - 2: Observed results on nucleation (Neutron flux = 2.5374 ×107/cm2/s; Peak neutron energy = 3.5 MeV). Freon-12 Freon-22 1.Initial Nucleation rate1(cm3/gm/s) 3.0268 ×10−44.9205 ×10−4 2. Lifetime τ(s) 3089.59 1279.13 3. Neutron detection efficiency20.2825% 0.5588% 1Initial nucleation rate is/parenleftBig 1 m/parenrightBig dV dtat the initial time, where m is the mass of the sample anddV dtis the rate of volume change upon nucleation. 2neutron detection efficiency is the percentage of neutrons ca using nucleation. 9FIGURE CAPTIONS Fig. 1: Observed variation of volume of vapour formed as a function o f time for Freon-12;hA m= vapour volume per unit mass of the sample. Fig. 2: Observed variation of volume of vapour formed as a function o f time for Freon-22;hA m= vapour volume per unit mass of the sample. 10

arXiv:physics/0004034 17 Apr 2000On the Physical Cause and the Distance of Gamma Ray Bursts and Related Phenomena in the X-Rays and the Ultra-Violet Ernst Karl Kunst Im Spicher Garten 5 53639 Königswinter Germany The modified Lorentz transformation of a distance-dependent special theory of relativity - which will be briefly summarized - predicts the possibility of superluminal velocity of very distantly moving material bodies to be connected with the generation of Cerencov radiation off the quantum vacuum. It is shown that vacuum Cerencov radiation due to the superluminal propagation of extraterrestrial spaceprobes in the interstellar space would account for all known properties of gamma ray bursts (GRBs) and the “afterglow” at lower frequencies. Distances and other parameter prove to be calculable and the theoretical results on these grounds to be in good accord with experiment. Key words: Far range transformation - superluminal velocity - vacuum Cerencov radiation - gamma ray bursts Introduction As widely known, were those bursts detected by the Vela military satellites in the late 1960s. Currently the most precise findings come from the Burst and Transient Source Experiment (BATSE) on board the Compton Gamma Ray Observatory (CGRO), which has collected nearly threethousand GRBs since 11 April 1991 at a rate of /G11 0.8/day . The most important results, established by BATSE, are that the GRBs are distributed fairly uniformly on the sky, and their spatial distribution is finite. These results taken together rule out sources confined to the galactic disc and nearby sources, in any case if neutron stars or other known galactic objects are considered to be those sources. Therefore, it has been proposed GRBs to be of extra galactic origin. Far away galaxies and other cosmic objects as sources of the GRBs seem to explain their finite distribution, their obvious attenuation with increasing distance. The successful identifications made by the BeppoSAX satellite in X-rays, followed by the detection of counterparts in visual light as well as radio waves, seem to support this point of view, which further has been stressed by the measurement of absorption lines in the spectrum of some counterparts. Thus, GRBs seem to be put definitely into the cosmological distance. On the other hand if GRBs are really at cosmological distances, this leads to the energy and to the compactness problem in the respective astrophysical models. Furthermore, there remain other puzzling open problems in GRB studies, as for instance the unseen “afterglows” in most cases or the obvious lack of “host galaxies”. In the following the well-known properties of the GRBs are briefly summarized. 1) Burst durations range from 10 ms to hundreds of seconds (s), with a distribution maximum at /G11 30 s (see 5) and Fig. 1); 2) GRBs occur randomly in time and po sition on the sky; They are2 distributed isotropically in direction and do no t repeat, which means that after one ceases, it vanishes entirely. In most cases no steady object remains detectable; 3) GRBs tend to have similiar non thermal energy spectra; Photon energies are between /G11 10 keV and some GeV, typically peaking in the MeV region; 4) The distribution of burst brightness follows a -3/2 power law for a narrow range and falls off elsewhere which implies that the population is confined in space with the distance limit unknown [1]; 5) A subclass of short GRBs seems to be distributed anisotropically [2]; 6) Burst durations show evidence for a bimodal distribution with a first event maximum at /G11 0.3 - 0.4 s, a minimum at 2 s and an absolute maximum of burst events at /G11 30 s [3]. But rather it seems to be a more trimodal distribution if attention is given to the but only apparently small second maximum at /G11 1.35 s. If in Fig. 1 onto the latter the same scale is applied as valid for the maximum at /G11 0.4 s, it becomes evident that it spans over a duration time of /G11 0.5 s as compared with only /G11 0.25 s of the first maximum; Fig. 1 7) Some light curves of GRBs show significant count rate variations on time scales as short as ms; 8) It also has been found that there may exist GRBs the soft emission of which has a time delay relative to the high energy emission [4]; 9) The range of intensities for all GRBs ever detected covers a factor of /G11 10000, from 10 erg/cm down to 10 erg/cm;-3 2 -7 2 10) GRBs seem to rise rapidly to a peak of intensity and then decay exponentially, which phenomenon is known as “FRED” (Fast Rise, Exponential Decay) [3]; 11) In the present number of GRBs with X-ray prompt counterpart detected by BeppoSAX none is classified as short (sub second) event [5]; 12) In the light of the afterglow of some GRBs a system of absorption lines has been found - first, with z = 0.835 in the spectrum of the optical x/G0C 2/G0A/G0A/G0B/G0B 0x/G0B 1/G09v0t/G0B 1 R,y/G0C 2/G0A/G0Ay/G0B 1,z/G0C 2/G0A/G0Az/G0B 1,t/G0C 2/G0A/G0A/G0B0t/G0B 1/G09v0x/G0B 1 c2R, x1°/G0B/G0A/G0A/G0B/G0B 0x/G0C 2/G08/G08v0t/G0C 2 R,y1°/G0B/G0A/G0Ay/G0C 2,z1°/G0B/G0A/G0Az/G0C 2,t1°/G0B/G0A/G0A/G0B0t/G0C 2/G08/G08v0x/G0C 2 c2R, /G0B/G0B 0/G0A/G0A1 1/G09v2 0 c2R2,/G0B0/G0A/G0A1 1/G09v2 0 c2. R/G0A/G0AS/G0B 1S/G0C 2. /G0DR/G0D/G0A/G0A/G0DS/G0B 1S/G0C 2/G0D»/G0Dc/G0Ct/G0D, /G0Cx/G0B 1 /G0Ct/G0B 1/G0A/G0A/G0Cx/G0C 2 /G0Ct/G0C 2/G0B03 counterpart to GRB 970508 - which show remarkably small velocity dispersion, suggesting that the lines arise from a single cloud [6]. Superluminal Propagation of Material Bodies in Vacuo as the Cause of Cerencov Radiation In the previous study [7] on relativistic kinematics among others has been shown that the transformations are valid between an inertial system S* considered to be at rest and a very distantly1 moving inertial system S' , where2 and From this follows that if the velocity of an object resting in the coordinate source of S' relative to S* must be2 1ux1/G0B/G0A/G0Aux1/G0B0/G0A/G0Av0/G0B0. V0/G0A/G0Av0/G0B0 V0/G0Ct1/G0A/G0Av0/G0Ct2, C/G0Ct1/G0A/G0Ac/G0Ct2. p/G0A/G0A/G0CE/G0Ct h/G0A/G0A/G0CE h/G0C/G1F/G07V4 0 c4/G061, p/G0AhV4 0 Ec4/G0A/G0A1/G0A/G0Aconst, /G0Ah/G0B4 0 E/G0A/G0A1/G0A/G0Aconst4 (1) (1a) (2)or, as expressed in the coordinates and the time parameter of the rest frame S - the1 conventional system of special relativity of the observer considered at rest: The dilation of time in S' is compensated for by the symmetrical inertial (far range)2 velocity of this system relative to S* also implying the velocity of light in S' to be C = c/G0B - if1 2 0 both systems are spatial far apart. As a result if S' approaches S* observers resting in2 1 the coordinate sources of either system will meet or receive light signals emitted from the respective other system after the same amount of time has elapsed: Any velocity even exeeding that of light in any amount is possible, allowing in principle the superluminal propagation of solid bodies and thereby superluminal transfer of information between very distant systems. Furthermore, it has been found that the probability p to encounter virtual dipoles or photons for a particle traversing the fluctuating quantum vacuum at subluminal symmetrical velocity V < c according to the0 uncertainty relation is given by Hence, as bodies propagate superluminally through vacuo they generate Cerencov radiation because the energy of virtual dipoles becomes real and stable as the far range velocity V exceeds that of light and the above probability relation becomes0cos/G05/G0A/G0Ac V0/G0A/G0A1 /G0B0/G070/G0A/G0A1 4/G1F, 22cos/G05/G0A/G0Ac V0/G0A/G0A1 /G0B0/G070/G0A/G0A1 4/G1F, /G0A/G0A1 /G0B0 22/G0B0cos/G05/G0A/G0Ac V0/G0A/G0A1 /G0B0/G070/G0A/G0A1 4/G1F, /G0A/G0A1 /G0B05 (3) (4)if V/c » 1, where h means Planck’s constant and E = (2E) “natural energy”0 phot1/2 acording to symmetrically modified special relativity [8]; E denotes conventionalphot photon energy. For this radiation the relation has been proposed, where V /G07 c, /G07 = v/c, /G05 halfangle of the radiation cone and /G1F0 00 frequency of the radiated photon. But subsequently I come to the conclusion that this derivation of the vacuum Cerenkov relation is incomplete and to begin with must be corrected to if /G07 /G19 1. The reason is that sin/G05 of the Cerencov angle of a very distant superluminally0 moving system S' must equal the relation c/V = c/(v/G0B). According to (2) is the2 0 00 generation of Cerenkov radiation of lowest energy to expect if c/V = c/(v/G0B) = 10 00 implying v/c = 1//G082 and, thus, sin/G05 /G06 1. Furthermore, according to (1a) the radiation of0 Cerenkov light of lowest energy will be observed by an observer based at S' if v/c /G0720 1//G082 and of highest energy if v/c /G19 1.This implies that the radiation can only occur0 within a cone of /G05 = /G25/4. Accordingly an observer based at S* will this radiation1 observe in a cone ranging from /G05 /G07 0 to /G05 /G19 /G25/4. But (3) can be valid only if 2/G082cos/G05 /G19 1 and V only slightly > c for the following0 reasoning. Suppose the very distant system S' is heading toward the observer resting in the2 coordinate source of S* at rest. As observed from the latter will all dimensions normal1 to the velocity vector of S' remain unaltered. We introduce a system S at rest relative2 1 to S* and also very far from S' implying the latter to move parallel relative to the first.1 2 As observed from S must (1a) be valid and, thus, the space-time of S' elongated by a1 2 factor of /G0B. Accordingly the angle under which the Cerenkov light from the path of the0 superluminally moving system S' is emitted toward the resting system S* must be2 1 foreshortened by the factor of 1//G0B so that (3) eventually attains the form0 if /G07 /G19 1. This implies that an observer based at S* at rest will observe the Cerencov0 16 radiation from the path of the superluminally moving system S' under a very narrow2 angle nearly head on if V » c. As S' propagates relative to S with the velocity V = v/G0B0 2 1 000 » c the Cerenkov radiation from its flight path will arrive at S* from ever more distant1 points with decreasing frequency or increasing wavelength, respectively, and, considering the enormous velocity it is clear that it is generated practically simultaneously, as observed from the latter system. As it turns out that the order of arrival of the photons coincides with the decrease of frequency, this implies that some (arbitrary) time interval /G0Ct in the frequency band must also be directly coincident with the travel time of light C/G0Ct = c/G0B/G0Ct. Accordingly the apparent motion of the Cerenkov0 image of S' will be slowed, as is shown below. 2 Application of the Relativistic Vacuum Cerenkov Effect on the Phenomenon of Gamma Ray Bursts In the following it is proposed to apply these findings on the long standing enigma of the GRBs. Considering the introductory remarks, namely that the isotropic distribution of the GRBs on the sky leads to the rejection of the galactic halo model, we argue here that this reasoning reversely forces one to abandon the cosmic (distant galaxies) model, too, although measured red shifts in some cases seem to fully support the latter. Suppose the cosmic model to be correct. Then, as seen from our vantage point /G11 8.5 kpc from the center of the Galaxy, nearly the whole galactic matter - aside from the “dark matter” - is concentrated in the central galactic bulge and in the extreme disc in the form of stars, gas and dust. Thus, a considerable amount of the gamma photons coming from point sources concealed by the galactic plane and bu lge must be completely blocked by the amassed stars of the Milky Way. For photons of GRBs coming from the other side of the Milky Way especially the center of the galactic bulge must be a totally invincible hindrance. Therefore, if the cosmic model would be correct, in the direction of the galactic bulge and extreme disc, but at least the bulge, considerably less GRBs should be detectable. The BATSE results show the contrary. Therefore, because there appears no absorption at all of GRBs by intervening material of the galactic disc, but especially the bulge (indeed, BATSE recorded at the very center of the Galaxy several GRBs, the sources of which owing to the known compact mass at this location impossibly can be located beyond the bulge) the conclusion must be drawn that the sources of GRBs are located in front of the disc and bulge, thus, ruling out any extragalactic origin. This result, the GRBs mainly to end before the galactic center and, therefore, their sources to be of galactic origin, also is an obvious constraint to any distribution of any known source candidates for GRBs within the Galaxy. None of the known astronomical objects in the Milky Way come into question as emitters of GRBs. So, what could they be? It is here predicted that the sources of GRBs are no known astronomical, but rather objects propagating with superluminal speed of about /G0710 c according to (1) through4 the galactic space and be yond, thereby generating Cerenkov gamma and other radiation according to (2) and (4). Presumably these superluminal bodies ared/G0A/G0Al×sin/G07 sin/G05,7 (5)spaceprobes (perhaps for communication purposes) of galactic super civilisations, though, of course, the underlying physics of the propulsion and its power source remains completely unknown to say the least. In the following it is shown that the distances of the GRBs as well as the distribution of burst durations (Fig. 1) are calculable and that the main features of GRBs summarized above do exactly coincide with this hypothesis. Consider an extraterrestrial spaceprobe hurtling in a yet unknown distance relative to Earth with a velocity of 2.637 × 10 c in accordance with the relativistic far range4 transformation and (1) through the galactic space. Be this so, then it must according to (2) generate Cerenkov radiation, which according to (4) is emitted along and in a very narrow cone practical in the direction of its flight path with the highest conventional photon energy to be E = 2 MeV /G3C E = 2 keV. Thus, an observer in the vicinity ofphot Earth will observe a GRB exactly then, when the spaceprobe crosses the line of sight in a very narrow angle (Fig. 2) and the generated Cerenkov radiation is bright enough to be observed. It is clear that the intensity of the Cerenkov light must strongly depend on the size and shape of the spacecraft. But at the present nothing can be said to this point and must be treated on a pu rely phenomenological level. Of course, not only photons of E = 2 keV are generated, but a radiation cone with ones of lower energy till down to the radio waveband, too. Suppose BATSE is detecting the GRB on all 4 channels (1, 2, 3, 4), which correspond to energy (E) ranges of 20 - 50, 50 - 100, 100 - 300 and 300 - 2000 keV,phot respectively, with a duration of 80 s. From the top energy E = 2 keV according to (2)top the mentioned superluminal velocity V/c = /G0B = 2.637 × 10 follows. In connection with0 04 the half angle of the radiation cone of the /G0B-photons of lowest (E) and highest (E)l h energy, respectively, of the respective BATSE-channel the distance of the GRB is calculable as follows: where d means distance to the GRB, l = B¯C¯ = length of spacecraft track - as “seen” from the vicinity of Earth in gamma light -, /G07 the half angle of the radiation cone of the /G0B-photons of lowest energy and /G05 the angle under which l = B¯C¯ is to observe on the sky from the vicinity of Earth (see Fig.2). Fig. 2 shows (exaggerated) that the ”visible” gamma light track on the sky constitutes the side B¯C¯ of the triangle ABC, where A¯B¯ and A¯C¯ are the “lines of sight” to the location of generation of the photons of lowest (B) and top (gamma) energy (C), respectively, and the distance “d” coincides with A¯C¯ of this triangle. According to (4) will the Cerenkov point source not be observed to recede backward in time along B¯C¯ with the steady velocity 2.637 × 10 c, but rather with ever lower velocity depending on the4 cosine of the Cerenkov angle of emission (Fig. 2). Obviously the relation holdscos/G05/G0B cos/G05/G1F/G0A/G0AV0/G0B/G0Ct/G0B V0/G1F/G0Ct/G1F, /G0Ct/G0B /G0Ct/G1F/G0A/G0AV2 0/G1F V2 0/G0B, C/G0Ct/G0B/G0A/G0Ac/G0B0/G0Ct/G0B/G0A/G0Avapp/G1F/G0Ct/G1F, vapp/G1F/G0A/G0ACV2 0/G1F V2 0/G0B, /G0A/G0Ac/G0B0/G0B2 0/G1F /G0B2 0/G0B, cos/G05min/G0A/G0A/G1F/G091 4 max/G0A/G0A/G2D1 4 1/G0A/G0A1.079×10/G096, vapp/G1F/G0A/G0AC/G0B2 0/G1F /G0B2 0max/G0A/G0Ac/G0B0/G0B2 0/G1F /G0B2 0max.8 (6) (6a) (6b)which results in where /G0B denotes the respective value of higher and /G1F the one o f lower energy or frequency. Furthermore, it must be valid wherefrom in connection with (6) is derived where v means apparent velocity of the Cerenkov image on the sky in the light of app lower frequencies (/G1F) than gamma. From (2) follows that the photon energy of the Cerenkov radiation tends to a maximum value if cos /G05 /G3C 0 implying the existence of a maximum superluminal velocity V if0max only cos /G05 would attain a minimum value > 0. That is the case, because according to [8] a quantum of time exists with the numerical value /G2D = 1.357628 × 10 s [9]. Thus,1-24 we have wherefrom follows V = 0.926 × 10 c and the highest possible Cerenkov photon0max6 energy E = 3.43 GeV /G3C E = 5.9 × 10 TeV. So eventually (6a) can be writtenmax photmax6 asd/G0A/G0Al×sinarccos/G07 sin(arccos/G0B/G0C/G09arccos/G07). d/G0A/G0A/G0Ct×/G0B0/G1F×sinarccos/G0B/G091 0/G1F 22/G0B0 sinarccos/G0B/G091 0/G0B/G09arccos/G0B/G091 0/G1F 22/G0B0×3.1536×107×/G0B0/G1F /G0B0max2 ,9 (7)The Cerenkov half angle /G0B’ follows from (4). Thus, we have /G0B = 180° - /G0B’ and, therefrom, /G05 = 180° - /G07 - (180° - /G0B’) = /G0B’ - /G07 so that (5) attains the form In connection with (2), (4) and (6b) we finally obtain the GRB distance (here in light- years (lys)) where /G0Ct denotes duration of the GRB, /G0B = (E/h), /G0B = (E/h) and /G0B = (E/h) -0 /G0Bh 0 /G1Fl 0top1/4 1/4 1/4 E highest and E lowest /G0B-photon energy of the respective detector channel of BATSEh l or other satellites (the following calculations are solely based on data from channel 2 and 3 of BATSE). Applied straightforwardly on our model spacecraft with a GRB duration of 80 s (7) delivers a distance of /G07 4.3 lys and for some of the most energetic and longest GRBsvapp/G1F/G0A/G0Asin/G07×C×/G0B2 0/G1F /G0B2 0max.10 (8)ever detected the following values ( E and E taken from channel 2 (50 - 100 keV) ofh l BATSE): burst durationE E velocity distance s GeV keV V/c = /G0B lystop 0 0 84 10 141 .42 76472 /G07 4.3 200 0.314 25 .06 49616 /G07 10.7 100 1.2 48 .99 58668 /G07 5.3 30 1.6 56 .57 60816 /G07 1.6 These results reveal GRBs to be a very local phenomenon and confirm the introductory conclusion that the GRBs are confined to a space with a radius ending far before the galactic center. For the first maximum at /G11 0.4 s in the established tri(bi)modal distribution of burst durations (No. 6 of the synopsis and Fig. 1) a distance of 7.8 light- days and for the second maximum at /G11 1.35 s of 27.3 light-days are computed. The distance of the absolute GRB maximum at /G11 30 s is /G07 1.6 lys, whereas the millisecond bursts are only a couple of light-hours away. Besides, the model GRB with a duration of 80 s coincides about with the real GRB 970228 (although in the literature I could not find any hint to the E of this burst -top BATSE could not observe the burst). Thus the calculated distance of /G07 4.3 lys must approximately be valid for this burst, too. The accurate position determination of this GRB by BeppoSAX for the first time led to the discovery of a fading X-ray source and an optical counterpart of the burst. About a month after the GRB event the proper motion of the optical counterpart was detected [10]. For the angular displacement the transverse velocity-distance relation v(km/s) = 2.7 × d(pc) has been derived (Caraveo et al. 1997), where pc denotes parsec. Our theory delivers for the recession of the Cerenkov point source on the projection C¯D/G09 of the sky the apparent velocity (see Fig. 2 and (6b)) which gives 3,6 km/s, where /G0B = 5400, the highest posible value for the optical. The0 /G1F phenomological relation also delivers a transversal velocity of 3.6 km/s for the distance of 4.3 lys of GRB 970228. Thus, theory and experiment seem to be in very good accord. But possibly the exerimental result is somewhat to high. If in (8) /G0B = 4900 - the0 /G1F mean /G0B of the optical - is inserted, v = 3.0 km/s results. 0 /G1F app /G1F According to this theory the nebulosity or “fuzz” connected with the point source of the optical “afterglow” of this GRB (which has been interpreted as its host galaxy) must be streaming interstellar gas, which the superluminal spaceprobe propelled about transversal to its flight path into the interstellar space and, therewith, also normal to the line of sight from Earth to the Cerenkov track. We see the receding Cerenkov image of/G0Ct/G1F/G0A/G0A/G0Ct/G0BV2 0/G0B V2 0/G1F11 (9)the spaceprobe track through this column of streaming gas. The uniform velocity of the latter explains the very small dispersion of the absorption lines (No. 12 of the synopsis). Accordingly, the fuzzy and cloudy appearance is due to the scattering of the Cerenkov light within the cloud of streaming gas. If we compare the images of the X-ray counterpart of GRB 970228 taken by BeppoSAX on February 8 and March 3 1997 [11] with the first HST observations 26 and 38 da ys after the burst [12] we clearly see that the fuzz is already visible in X-rays with exactly the same morphology as in the optical, although somewhat less extended. With the X-ray counterpart the “X-ray fuzz” vanished, too, and it is predicted that the optical nebulosity will share the same fate, which implies that it disappeared already (as well as the optical point source) since the last HST observation in september 1997. This conclusion is further backed by the comparison of ground-based measurements taken on March 6 and April 5 and 6, which clearly indicate that within a month the fuzz faded below detection level of the telescope [13]. From the foregoing it is clear that the observed red shifts of the afterglow of some GRBs are not of cosmological, but rather special relativistic origin, where the measured red shift is but only the imprint of the transversal Doppler-shift z = /G0B - 1. This implies all0 red shift measurements other than from the putative afterglows to be erronous. From the theory it is clear that repeaters do not exist (although recurrence cannot be excluded if those spaceprobes fly the same routes in some time interval) and why all attempts to associate the GRBs with known astronomical objects will be futile (see below). This implies that the identification of “host galaxies” as e. g. in the case of GRB 971214 is erronous and must be due to the chance superposition of the GRB respectively its afterglow and the putative host. Furthermore, follows that the random distribution of GRBs on the sky results from the accidental crossing of the line of sight by spacecraft or -probes criss-crossing the galactic space in all possible directions relative to Earth with superluminal velocities exceeding /G11 10 c. From this directly results that far more though invisible (from Earth)4 spacecraft tracks in Cerenkov gamma light must be generated than the stated 0,8/day. As already mentioned, with growing time Cerenkov radiation of ever lower frequency will be observed from the spaceprobe track. Thus, an observer really sees the evolution of the track in Cerenkov light of ever lower frequency backward in time (see below). This is valid for all GRBs. But in the case of very short bursts the duration of the following Cerenkov radiation of lower energy is also proportionally shortened and therefore hard to detect (No. 11 of the synopsis). According to (6) can the duration time of the Cerenkov afterglow at any frequency be approximated by the expression (see Fig. 2): where /G0Ct means duration of the respective waveband (X-ray, optical, radio) and /G0Ct/G1F /G0B duration of GRB. Applied on GRB 970228 and GRB 970508 (9) delivers about the correct results of some days for the Cerencov X-ray glow and a couple of months up toFig. 312 /G07 one year for the visual Cerenkov light. GRB Properties as a Consequence of the Track Geometry Due to the extreme beaming of the Cerenkov radiation without any deflection whatsoever from its generation by nonhuman made superluminal spaceprobes till its detection by human made satellites, the gamma light does not expand isotropically but cylindrically around the “visible” gamma track. Because the “hight” of the “cylinder” l = B¯C¯ (Fig. 2) does not alter on the way of the photons to the detector, only the “cylinder barrel” expands with the superluminal velocity C of light. This implies that the intensity does not decrease as 1/d - as in the case of normal astronomical objects - but rather2 as 1/d instead. Therefore, according to this theory, the intensity of GRBs is exactly dependen t on their distance, which again is proportional to the burst duration. If the mean of the durations (distances) of the longest lasting GRBs (table) is divided by the duration (distance) of the shortest bursts of about 0.01 s, 10350 results as compared with the empirical value /G11 10000 (No. 10 of the synopsis). These findings suggest that the absolute maximum of GRB durations at /G1130 s /G3C d /G11 1.6 lys and their rapid decline is not real but faked by the pesumable smallness of the sources and the limited sensitivity of current detectors, analogous to the limited number of stars visible to the naked eye. This implies that the maxima at /G11 0.4 s /G3C d /G11 7.8 light- days and /G11 1.35 s /G3C d /G11 27.3 light-days owe their existence to a real increase of superluminal space travel activities in these distances (this result will be confirmed below). Thus, it can be said that the limitation for the detection of GRBs due to the geometry of the spaceprobe tracks is further and decisive reduced by the fact that beyond burst durations of /G0Ct /G11 30 s /G3C d /G11 1.6 lys the GRBs drop rapidly below the range of visibleness. The duration or distance of the GRBs also is the key of understanding the noteworthy trimodal distribution of burst durations (No. 6. of the synopsis and Fig. 1). From Fig. 1 and (6) it is clear that at a given E the burst duration is a function of the distance (andtop vice versa), whereas the length of the spaceprobe track l = B¯C¯ can be thought of as the13 projection C¯D/G09 onto a circle, the radius of which is the distance d (see Fig. 2). Consequently, under the provision that GRBs occur randomly in space, the probability of bursts to be observed from the vicinity of Earth must be proportional to the circumference of the said circle or its radius = distance d. Therefore, the number of GRBs must be directly dependen t on their distance d being equivalent to the duration time /G0Ct. Applied on Fig. 1 this means that the number of GRBs must steadily decrease toward our vantage point in the system of the sun from the absolute maximum of 29 bursts in d /G11 1.6 lys or burst duration /G11 30 s, proportional to shorter burst times or distances. To the resulting curve the difference to the local maxima at /G0Ct /G11 1.35 s /G3C d /G11 27.3 light-days (11 - 29 × 1.35/30) and /G0Ct /G11 0.4 s /G3C d /G11 7.8 light-days (9 - 29 × 0.40/30) have to be added . The steady increase of GRBs toward both local maxima follows the same law, but other than the apparent decrease of GRBs beyond the absolute maximum owing to the drop benea th the range of visibleness, those local maxima must also decrease proportional to the burst duration or increasing distance. The crosses in Fig. 3 mark this theoretical derivation. The result that the small maxima at burst duration time /G0Ct /G11 0.4 s /G3C d /G11 7.8 light-days and /G0Ct /G11 1.35 s /G3C d /G11 27.3 light-days are due to a real increase of bursts or superluminal space travel activities in the vicinity of the sun is strongly supported by the finding short GRBs to be distributed anisotropically (No. 6 of the synopsis). Especially has been found (Balázs et al. 1998) that GRBs with duration times < 2 s and <1 s, respectively, depart significantly from isotropy, whereas long bursts > 2 s do not. As already mentioned, it is from the track geometry (Fig. 2) clear that the observer in A at first will receive /G0B-photons of top and subsequently of ever lower energy, which effect also causes the observed delay of low energy /G0B-photons (No. 9 of the synopsis). Thus, each waveband must have its own characteristic peak, which leads in the /G0B-region to the observed FRED phenomenon, and it is clear that this is valid for the following X-ray, optical and radio Cerenkov emission as well, with a rise to a plateau value and a subsequent decay. But this smooth evolution of the Cerenkov track can only be valid if the track geometry remains unaltered. If the spaceprobe changes its course anywhere during the “visible” track, then this also will lead to a drastic change in the evolution of the subsequent Cerenkov radiation. Suppose a spaceprobe moves on a head ing, which allows the Cerenkov radiation to be observed from Earth. At a point of its track, wherefrom visual light or radio emission would reach the observer, it turns slightly to a new course leading away from Earth - comparable to an ascending aircraft. Due to this turn in the track the beam of the more energetic (than visual light or radio) Cerencov radiation arrives earlier at the observer as in the normal case with no alteration of the flight path (see Fig. 2). Thus, the observer would register the incoming /G0B-rays, X-rays and visual light nearly simultaneously. Possibly due to the turn the point of the Cerenkov track with /G0B-rays of medium energy or X-rays will be nearest to the observer with the consequence that this radiation arrives first (pre-cursor), whereas the most energetic photons appear delayed. Obviously was GRB 990123 such an event. It was observed in optical during the course of the GRB itself, yet the most energetic (MeV) emission did not rise significantly until 18 s after BATSE’s lower energy burst onset [14]. The rapid fluctuations in the ms region (No. 7 of the synopsis) are presumably due toEtop El/G0A/G0AV0 V0l4 .14 the basically stochastic nature of the Cerenkov gamma radiation. From (2) directly follows that the probability for the generation of photons of top energy E (astop compared with photons of lowest energy E) as a function of the velocity V varies asl 0 Thus, with increasing energy E the time difference between two gamma photontop generating events is growing and it comes to the observed fluctuations. This basically also must be true at the onset of the Cerenkov radiation of any frequency. From the theory it is clear that the more distant a spacecraft track, the longer and smoother the “visible” GRB with ever lower intensity Further Implications It seems highly improbable that all spaceships of the unknown galactic super civilisations propagate so fast through the near vacuum of the Milky Way and its vicinity that always Cerenkov gamma radiation is generated along their flight paths. More likely also spacecraft cruise the galactic space with velocities sufficing to generate photons with the top energy E in the optical, ultraviolett or X-ray region. Probably are most iftop not all transient optical phenomena found on a rchival astronomical photographic plates such Cerenkov events in visual light. In the extreme ultra-violet (EUV) waveband transients have been ob served, too. For instance revealed ROSAT WFC observations a transient EUV source with a flux amplitude variability by a factor of /G11 7000 over a maximum period of presumably /G11 one year, which has not been found in earlier EUV observations nor in follow-up /G0B-ray, X- ray, optical and radio observations. If (7) is straightforwardly applied on this EUV transient (date of observation: 25.06. - 07.07.1997; energy: 62 - 110 eV) this results (/G0Ct = 13 days) in a minimum distance of 2538 lys. Short-duration X-ray transients, which are not related to known astronomical objects, have also been ob served long since. Typically they range from seconds to less than few hours [15]. BeppoSAX observed at least 9 fast X-ray bursts during its operation life time, which are not counterparts of GRBs. Nevertheless they show spectral characteristics typical of the GRB class, but do not present relevant gamma emission [16]. If e. g. (7) is applied on the WATCH fast X-ray transient GRS 2037-404 with a duration of 110 min in the 8 - 15 keV band (Tirado et. al 1999) a distance of 155 lys is calculated. From the foregoing it is clear, especially because E = 3.43 GeV /G3C E = 5.9 × 10max photmax6 TeV far exceeds the top energy of the presently observable most energetic GRBs, that we also have to expect burst phenomena in the TeV range - although presumably far more seldom than at lower energies . It is predicted that all these transient phenomena in the different wavebands are of common origin: Vacuum Cerenkov radiation due to the extreme superluminal 15 propagation of interstellar spaceships or -probes of various extraterrestrial civilisations in our Galaxy. Dedicated to my friend Tom Witte. References [1] Meegan, C. A. et al., Nature 355, 143 (1993) [2] Balázs, L. G. et al., Astron. Astrophys. 339, 1 - 6 (1998) [3] Fishman, G. J. et al., Astrophys. J. Suppl. Ser. 92, 229 (1994) [4] Cheng, L. X. et al., Astron. Astrophys. 300, 746 (1995) [5] Gangolfi, G. et al.: What can BeppoSAX tell us about short GRBs: An Update from the Subsecond GRB project, astro-ph/0001011 [6] Metzger, M. R. et al., Nature 387, 878 (1997) [7] Kunst, E. K.: Is the Lorentz Transformation Distant-Dependent? physics/9911022 [8] Kunst, E. K.: Is the Kinematics of Special Relativity incomplete? physics/9909059 [9] Kunst, E. K.: On the Origin of Time, physics/9910024 [10] Caraveo, P. A. et al.: HST Data Suggest Proper Motion of the Optical Counterpart of GRB 970228, astro-ph/9707163 [11] Costa, E. et al., Nature 387, 783 (1997) [12] Sahu, K. C. et al., Nature 387, 476 (1997) [13] Metzger, M. R. et al., C. 1997, IUE Circ. 6631 [14] http://www.gro.unh.edu/bursts/cgrbnewb.html [15] http://www.ias.rm.it/ias-home/sax/xraygrb.html [16] Castro-Tirado, A. J. et al., Astron. Astrophys. 347, 927 - 931 (1999)

THERMODYNAMIC EQUILIBRIUM IN OPEN CHEMICAL SYSTEMS. The R-modynamics of chemical equilibria. B. Zilbergleyt, Independent Scholar E-mail: livent1@msn.com ABSTRACT. The article presents new model of equilibrium in open chemical systems suggesting a linear dependence of the reaction shift from equilibrium in presence of the externalthermodynamic force. Basic equation of this model contains traditional logarithmicterm and a non-traditional parabolic term.. At isolated equilibrium thenon-traditional term equals to zero turning the whole equation to the traditional formof constant equation. This term coincides with the excessive thermodynamicfunction revealing linear relationship between logarithm of the thermodynamicactivity coefficient and reaction extent at open equilibrium. Discovered relationshipprompts us to use in many systems a combination of the linearity coefficient andreaction shift from true equilibrium rather then activity coefficients. The coefficientof linearity can be found by thermodynamic computer simulation while the shift isan independent variable defining the open equilibrium state. Numerical dataobtained by various simulation techniques proved premise of the method ofchemical dynamics. INTRODUCTION: BACK TO CHEMICAL DYNAMICS. Nowadays we know that chemical self-organization happens in a vaguely defined area far-from-equilibrium [1], while classical thermodynamics defines what is frozen at the point of true equilibrium. What occurs in between? True , or internal thermodynamic equilibrium is defined by current thermodynamic paradigm only for isolated systems. That s why applications to real systems often lead tosevere misinterpretation of their status, bringing approximate rather than precise results. A fewquestions arise in this relation. Is it possible to expand the idea of thermodynamic equilibriumto open systems? How to describe and simulate open equilibrium in chemical systems? Is thereany relationship between deviation of a chemical system from true equilibrium andparameters of its non-ideality?Traditional methods use excessive thermodynamic functions to account external interaction ofsome system s components. It is noteworthy that the functions and related coefficients ofthermodynamic activity were introduced rather for convenience [2], first playing a role of figleave for the lack of our knowledge of what s going on in real systems.One of the current methods in equilibrium thermodynamics of open systems, to a certain extentinfluenced this work, was offered by D. Korzhinsky [3]. Considering interaction of open systemswithin the multisystem , the method distinguishes between the common, or mobile components and specific for each subsystem inert components, which cannot be present in any other subsystem. The mobile components are responsible for the subsystems interaction and carryintensive thermodynamic characteristics, thus contributing the subsystem s Gibb s potential. It isimportant that coefficients of thermodynamic activity of the mobile components may vary whilefor the inert components they do not have any physical sense [4]. The model successfully resultedin well developed theory of multisystems with extensive application output [5]. In the2 Korzhynsky s model openness of the system is simulated with two-level component stratification and appropriate expression for change of Gibbs potential. Interaction with other parts of themultisystem in this model may be simulated via series of consecutive titrations of the subsystemby mobile components.To answer the above questions more consistently, we used currently almost neglected deDonder s method which has introduced the thermodynamic affinity, interpreting it as athermodynamic force and considering the reaction extent a chemical distance [6]. For greaterconvenience and universalization of the method, we have redefined the reaction extent as dξ j=dn kj/ηkj, instead of d ξj=dn kj/νkj by de Donder, or ∆ξj =∆nkj/ηkj in increments. Value of ∆nkj equals to amount of moles, consumed or appeared in j-reaction between its two arbitrary states, more often one of them is the initial state. The ηij value equals to a number of moles of k-component, consumed or appeared in an isolated j-reaction on its way from initial state to true equilibrium and may be considered a thermodynamic equivalent of chemical transformation . Thus redefined value of the reaction extent remains the same being calculated for any componentof a simple chemical reaction; the only (and easily achievable by appropriate choice of the basisof the chemical system) condition for this is that each chemical element is involved in only one substance on each side of the reaction equation. Now, in our definition ∆ξ j is a dimentionless chemical distance ( cd ) between initial and running states of j-reaction, 0 = ∆ξj=1, and thermodynamic affinity A = - ( ∆G/∆ξ)p,T turns into a classical force by definition, customary in physics and related sciences. Chemical reaction in isolated system is driven only by internal force (eugenaffinity, A ij). True thermodynamic equilibrium occurs at A ij = 0 and at this point ∆ξj = 1. Reactions in open system are driven by both internal and external (Aej ) forces [7] where the external force originates from chemical or, in general, thermodynamic (also due to heat exchange, pressure, etc.) interaction ofthe open system with its environment. Linear constitutional equations of non-equilibriumthermodynamics at zero reaction rate give us the condition of the open equilibrium withresultant affinity A* ij + a ie A* ej = 0, (1) where a ie is the Onsager coefficient [7]. The accent mark and asterisk relate values to isolated ( true ) or open equilibrium correspondingly.In this work we will use only one assumption which in fact slightly extends the hypothesis oflinearity. Taking as given that there must be a relation between the reaction shift from equilibrium δξ j =1 - ∆ξj and external thermodynamic force causing this shift, we will suppose at the first approximation that the reaction shift in the vicinity✩ of true thermodynamic equilibrium is linearly related to the shifting force δξj = αie Aej . (2) Recalling that A i =-(∂Gi /∂ξi ), or A i =-(∆Gi /∆ξi ) and substituting (2) into (1), we will have after a simple transformation and retaining in writing only ∆j for ∆ξj and δj for δξj ∆G*ij + b ie δ*j ∆*j = 0, (3) where b ie = a ie /αie. Corresponding constant equation is ∆G0 ij + RTln Π*j (η, ∆*j) + b ie (1−∆ * j )∆*j = 0, (4) where Π*j (η, ∆*j) is the activities product with mole fractions expressed using rection extent. So, as soon as chemical system becomes open, given the above assumption its Gibbs potential and the appropriate constant equation include a non-linear, non-classical term originated due to __________________________________________________________________________ ✩ Vicinity in this case is certainly not less vague than far-from-equilibrium . Some discussion will take place later on.3 system s interaction with its environment . What opens up immediately is a similarity between the non-classical term of (4) and the well known product r ⋅x⋅(1-x) from the chaotic equation [8]. To get more symmetric shape of (4) we may change it introducing a new value - the non-thermodynamic , or alternative temperature of the open system Ta = b ie / R, (5) where R is universal gas constant. The value of T a is introduced in this work for convenience and symmetry; we cannot give any explanation of its physical meaning at this moment.The logarithmic term contains well defined thermodynamic temperature T t , and (4) turns to ∆G0 ij + RT t ln Π*j + RT a ∆*j (1−∆ *j) =0. (6) Recall well known classical expression ∆G0 ij = - RTln K i . Now, dividing (6) by (-RT t ), presenting the activity product at open equilibrium as Π*j (ηkj, ∆*j)= Π{[( n0 pj + ηpj ∆*j)/Σ]νpj/ Π{[( n0 rj - ηrj, ∆*j)/Σ]νrj and equilibrium constant as K i = Π`(ηkj,1) due to ∆`j =1, and defining reduced temperature as τ = T a /Tt we transform equation (6) into ln [Π`(ηkj,1)/ Π(η kj , ∆*j)] + τ j ∆*j δ*j = 0. (7) Being divided by ∆*j , this equation still expresses linearity between the thermodynamic force and reaction shift {ln [Π`(ηkj,1)/ Π(η kj , ∆*j)]}/ ∆*j = - τ j δ*j, (8) while numerator of the left part is a new expression for the thermodynamic force. Containing parameters η and τ, and variable ∆*j (or δ*j), equation (7) in general can be written as φ* = φ (∆*, η, τ). (9) It is easy to see that in case of isolated system δ*= 0, τ = 0 as well as the thermodynamic force equals to zero, and (7) turns to the normal constant equation g* = g (P, T, n*). (10) We distinguish between them calling (9) the Greek and (10) - the Gibbs (Latin) equations. For better understanding of internal relations between (9) and (10) one should recall that η, serving as a parameter in the Greek equation, is the only output from the Gibbs equation (because η = n`- n0, where right side contains equilibrium and initial mole amounts). INVESTIGATION OF THE FORCE-SHIFT RELATIONSHIP. First, consider the force expression from equation (8). Its numerator is a logarithm of acombination of molar parts products for a given stoichiometric equation. The expression underthe logarithm sign is the molar parts product for ideal system divided by the same product where η kj replaced by a product ( ∆*jηkj) due to the system s shift from true equilibrium. Table 1 represents functions Π`(ηkj,1)/ Π(η kj , ∆*j) for some simple chemical reactions with initial amounts of reactants A and B equal to one mole. Graphs of the reaction shifts vs thermodynamic forces are shown at Fig. 1. One can see well expressed linearity on shift-force curves. The linearity extent depends on the η value. Going down to real objects, consider a model system containing a double compound A•R and an independent reactant I (for instance, sulfur) such that I reacts only with A•, while •R restricts reaction ability of A• and releases in the reaction as far as A• is consumed. Symbol A• relates to reactant A, pertaining to the open to an interaction with R system (A,I). Two competing processes take place in the system - decomposition of A•R, or control reaction (C): A•R = A• + R, and leading reaction (L): A• + I = Σ*L, the right side in the last case represents a sum of products. Resulting reaction in the system is A•R + I = Σ*L + R. To obtain numbers for real substances, we used thermodynamic simulation (HSC Chemistry for Windows) in the model set of substances.4 Table 1. Thermodynamic forces {ln[ Π`(ηkj,1)/ Π(η kj , ∆*j)]}/ ∆ for some simple chemical reactions. Initial amounts of reactants are taken equal to 1 mole and products to zero for simplicity. Reaction equation. Thermodynamic force from eq. (8). A + B = AB [(2η-η2)/(1-2 η-η2)] / [(2 ∆η-∆2η2)*(1-2 ∆η+∆2η2)] A + 2B = AB 2 (1- ∆η) / ( ∆− ∆η) 2A + 2B = A 2B2 [(2-3 η)/( 2-3∆η)]3 * [(1-2 ∆η)/(1−2η)]4 * (1/ ∆) 0.000.200.400.600.801.00 0 5 10 15 20 0.000.200.400.600.801.00 0 5 10 15 20 0.000.200.400.600.801.00 0 5 10 15 20 Fig. 1. Shift of some simple chemical reactions from true equilibrium (ordinate) vs. shifting force (abscissa). Reactions, left to right, values of η in brackets: A+B=AB ( 0.1, 0.3, .. , 0.9), A+2B=AB 2 (0.1, 0.2, 0.3,.., 0.9), 2A+2B=A 2B2 (0.1, 0.2, 0.3, 0.4) . One can see light delay along the x-axis for bigger η. Also, linear areas on the curves give an estimation of how far the vicinity of equilibria extents. The Is were S, C, H 2, and MeO•Rs were double oxides with symbol Me standing for Co, Ni, Fe, Sr, Ca, Pb and Mn. As restricting parts •R were used oxides of Si, Ti, Cr, and some others. Chosen double compounds had relatively high negative standard change of Gibbs potential to providenegligible dissociation in absence of I. In chosen systems the C-reactions were (MeO)•R=(MeO)•+•R, and L-reactions - (MeO)•+I. Amount of the MeO moles consumed in isolated (MeO+I) reaction between initial state and true equilibrium was taken as value of ηkL. Reaction extents for open L-reactions with different •Rs have been calculated as quotients of consumed amounts of (MeO)• (that is ∆nkj) by ηkL. As numerator for the thermodynamic force we used traditional ∆GC (or even ∆G0 C which does not make a big difference at moderate temperatures), and the force was equal to (- ∆G0 C / ∆*L). Some of the results for reactions (MeO•R+S) are shown on Fig.2. In this group of reactions value of (- ∆G0 C / ∆*L) plays role of external thermodynamic force regarding the (MeO+S) reaction. 00.250.50.75 0 200 400 600FeO*R CoO*R CaO*Rδ Fext.*5 Fig.2. δ*L vs. force (= - ∆G0 C / ∆*L ), kJ/m cd, 298.15K, direct thermodynamic simulation. Points on the graphs correspond to various •Rs. One can see a delay along x-axis for CaO•. The most important is the fact that in both cases the data, showing the reality of linear relationship, have been received using exclusively current formalism of chemical equilibriumwhere no such kind of relationship was ever assumed at all. It is quite obvious that lineardependence took place in some cases up to essential values of deviation from equilibrium.Results shown on Fig.1 and Fig.2 prove the basics and some conclusions of the method ofchemical dynamics. FROM CHEMICAL DYNAMICS TO CHEMICAL THERMODYNAMICS: THERMODYNAMIC ACTIVITY AND REACTION SHIFT AT OPEN EQUILIBRIUM. It was already mentioned that classical thermodynamics has no idea of thermodynamic force.Instead, the impact of the system s interaction results in its non-ideality and usually isaccounted by means of excessive thermodynamic functions and coefficients of thermodynamicactivity Qj = - RT t ln Πγkj. (10) Within current paradigm of chemical thermodynamics, constant equation for non-ideal system with γkj ≠1 is ∆G0 j = - RT t ln Π*γkj - RT t⋅ln Π*xkj, (11) where power values, equal to stoichiometric coefficients, are omitted for simplicity, and x kj are molar fractions. The non-linear term of the Greek equation also belongs to a non-ideal system,and comparison of (6) and (11) leads to following equality in open equilibrium τ j δ*j = (− ln Π* γkj)/ ∆*j. (12) This result is quite understandable. For instance, in case of A•R the chemical bond between A and R reduces reaction activity of A; the same result will be obtained for reaction (A + I) with reduced coefficient of thermodynamic activity of A. Now, to avoid complexity and using only one common component A• in both subsystems, the relationship between the L-shift and activity coefficient of A• is very simple δ*L = (1/ τL) [(-ln γ*)/ ∆* L], (14) where [(-ln γ*)/ ∆*L] represents divided by RT t external thermodynamic force acting against L-reaction.. This expression for the force as well as the total equation (14) are new . This equation connects values from chemical dynamics with traditional values of classical thermodynamics. Yet again, at δ*L= 0 we have immediately γ*=1, and vice versa, a correlation, providing an explicit and instant transition between open and isolated systems. In case of multiple interactions one should expect additivity of the shift increments, caused byinteraction with different reaction subsystems, which follows the additivity of appropriate logarithms of activity coefficients . It was also proved by simulation. Data for Fig. 3 were obtained using two different methods of thermodynamic simulation. I-simulation relates to an isolated (A •R+I) system with real R and A and γA•R =1 in all cases. In O-simulation a combination of |A•+Y2O3+I| represented the model of open system where •R was excluded and replaced by neutral to A and I yttrium oxide to keep the same total amount of moles in the system as in I-simulation and avoid interaction between A and R. Binding of Ainto double compounds with R, resulting in reduced reaction ability of A, was simulated varying γ Α. I-simulation provided a relationship in corresponding rows of the δ*L - γ* values, and O-simulation - with δ*L - ∆G0 A •R correspondence. S tandard change of Gibbs potential ∆G0 C, determining strength of the A•R bond, was considered an excessive thermodynamic function to the L-reaction.6 0.000.250.500.751.00 02 5 5 0 7 5 1 0 0 Fig. 3. δ* vs. (-ln γΑ/∆*) (I-simulation, x) and vs. ( ∆G0 A •R/ ∆*) (O-simulation, o), (MeO•R+S). From left to right, PbO• and CoO• at 298K, SrO• at 798. Curve for SrO• shows light delay along the x-axis. We have calculated some numeric values of the factor τL∆ from the data used for plotting Fig. 2. They are shown in Table 2. Table 2. Reduced temperatures, standard deviations and coefficients of determination between δ*L and ( -ln γ*) in some A•R-S systems. Initial reactants ratio S/A *= 0.1. CoO*R SrO*R PbO*R T t , K 298.15 798.15 298.15 τL∆ 40.02 6.54 3.93 St. deviation, % 8.99 2.99 6.80 Coeff. of determination 0.98 0.99 0.97 It is worthy to mention that the range of activity coefficients usable in equilibrium calculations seems to be extendible down to unusually low powers (see Fig.1). Strong relation between reaction shifts and activity coefficients means automatically strong relation between shifts and excessive thermodynamic functions, or external thermodynamic forces. Along with standard change of Gibbs potential we also tried two others - the QL which was calculated by equation (14) with γ*, used in the O- simulation, and another, ∆G*L , found as a difference between ∆G0 L and equilibrium value of RT t⋅lnΠ*xiL. Referring to the same ∆*L, all three should be equal or close in values. Almost ideal match, illustrating this idea, was found in the CoO•R - S system and is shown on Fig.4. In other systems all three were less but still enough close. Analysis of the values, which may be used as possible excessive functions, shows that the open equilibrium may be defined using both external (like ∆G0 C) and internal (the bound affinity, see [4]) values as well as, say, a neutral, or general value like a function calculated by (14) at given activity coefficient. In principle all three may be used to calculate or evaluate τL∆. This allows us to reword more explicitly the problem set in the beginning of this work and explain the alternative temperature more clear. It is easy to see that equation (14) represents another form of the shift-force linearity. Error! Not a valid link. Fig. 4 (left). δ*L vs. shifting forces, kJ/(mole*ched). CoO•R-S system, 298K (∆-Q L/∆* L, -∆G0 C /∆* L, ο-∆G*L /∆* L.). Fig. 5 (right). δ*L vs. (-ln γ/∆*). TDS. CoO•R-C system, 298K, different reactant ratio. Numbers at plots identify the initial value of the CoO•/C ratio. Error! Not a valid link.7 Multiplying nominator and denumerator of its right side by RT t and recalling that τL∆ = (T ch /Tt)∆*L one can receive δ*L = [1/(R Ta)] (QE / ∆*L), (15) where Q E is a general symbol for excessive thermodynamic function. It means that the shifting force is unambiguously related to the excessive function, and the alternative temperature is justinverse to the coefficient of proportionality between the force and the shift it causes. The product R Ta has dimension of energy while ∆ and δ stay dimentionless. Because we ran thermodynamic simulation within a certain range of initial ratios between components of the reaction mixtures it was interesting to see what a difference it made. Fig. 7 shows no essential dependence of the τL∆ value on this ratio in the CoO•R - C system . A POSSIBILITY OF MULTIPLE STATES AT OPEN CHEMICAL EQUILIBRIUM. Recall that equations (7) and (8) describe open equilibrium, first in terms of Gibbs potential ofthe open system, second - in terms of the thermodynamic force applied to it. While the force inthe vicinity of equilibrium is linear regarding reaction shift, the potential depends upon itparabolically thus leading to bifurcations [1] or multiple possible states of chemical system inopen equilibrium. To obtain graphs of the Greek equation, we varied τ, η i and ∆*j given reaction stoichiometric equation with reasonable coefficients and initial mole amounts of reactants. One of such sets for reaction A+B = AB and ηkj = 0.1 m. is shown on Fig.1. More complicated reaction equations do not bring essential difference in the curve shapes. Open equilibrium state in this particular model is defined essentially by standard change of Gibbs potential ∆G0 C of A•R formation from A and R as major restricting factor. Some results of the thermodynamic simulation are plotted in coordinates ∆*L- ∆G0 C on Fig.2 (marked as Sim. ). vs. calculated with equation (7) data (marked by Calc. ). 00.51 03 . 57∆*L φ* Fig.6. ∆*L vs. φ*. Curves for ηkj = 0.1 m, τ = 0, 5, 10, 15, 25 (left to right). Occasional scaling was applied. Points along the curves correspond to different •Rs. The qualitative coincidence between the curves on Fig.2 seems to be quite satisfactory. 00.51 0 40 80 120 160Calc. Sim.SrO*R, S, 798.15K. 00.51 03 0 6 09 0CoO*R, S, 298.15 Sim. Calc. 00.51 03 06 0Calc. Sim.CoO*R, C, 898.15 Fig.7. ∆*L vs. ∆G0 C of A•R formation, kJ/mol. Solution to the Greek equation is not unique if τ and the external thermodynamic force exceed certain values . That leads to a very important conclusion that the Zel`dovich s theorem [9],8 declaring the uniqueness of the state of the chemical equilibrium, is not valid beyond a certain extent of openness of chemical system. This part of the work was touched only slightly. One statement can be done for sure - the less is the value of η, that is the "weaker" is the chemical reaction, the less external force is necessary to bring to the situation with multiple states at open equilibrium. It is well recognized that the bifurcations are more probable for weak reactions [1]. PRACTICAL APPLICATIONS. We are unable to simulate and compute equilibrium composition of most complex chemicalsystems if we don t know appropriate coefficients of thermodynamic activity, and their numericvalues are very expensive. The method of chemical dynamics offers an easier and involving muchless efforts way to run that kind of research. Indeed, equilibrium of complex chemical system perthis method may be interpreted as equilibrium of subsystems shifts from their true equilibrium states, explicitly defined by the basic equation for j-subsystem. Now, having the η kj value from solution for the isolated state and τj as a characteristic for subsystem response to external thermodynamic perturbation (as it was above described in details), we have equation containing only δ*j as variable and τj as a parameter of the theory −lnΚj = ln Π(ηkj,δ*j) - τj ∆* δ*j. (20) Below the critical value of φ* this equation has only one solution (see Fig. 6). Current methods of simulation of complex equilibria use the constant equations (or equivalent expressions if minimizing Gibbs potential of subsystems) in the same form as if the subsystemswere isolated. Their joint solution is only to restrict consuming the common participants and thusachieve material balance within the system playing an accounting role. Application of currentmethods to real systems leads to some errors in simulation results originated due tomisinterpretation of their status [10]. Here developed method, assuming subsystems states asopen equilibria within a complex equilibrium, must give more correct numerical output.A principal feature of application following from here developed method consists in usage ofreaction shift (as the the system s response to external impact) multiplied by proportionality coefficient δ*j rather than activity coefficient γkj. Due to an easy way to obtain value of τj by thermodynamic simulation within minutes (not hours!), the method of chemical dynamics brings new opportunities into analysis and simulation of complex chemical systems. DISCUSSION. The new basic equation received in this work links equilibrium and non-equilibriumthermodynamics and may be rewritten more generally as ∆G = ∆G 0 j + RT t f t (∆*j) − RT a f a (∆*j). (21) The found relationship between reaction shift and external force resembles to a great extent the well known Hooke s law [11] with its linearity at low elongations of a stretched material and its yield point. In our case t he yield point, where the curve sharply deviates from the straight line or in some cases just changes the slope, was very distinctive on all plots. By analogy, the value of 1/ τL∆ may be considered a coefficient and the yield point - a limit of thermodynamic proportionality . This limit of the force-shift linearity may help to conceive the meanings of in the vicinity of and far from equilibrium areas. We cannot tell to what extent the coefficient and the limit of thermodynamic proportionality maybe considered characteristics of a chemical elasticity thus providing complete return of chemical reaction back to initial point when the chemical force returns to zero value, that is without or with a sort of chemical hysteresis in force-shift coordinates. This problem could be investigated in the future research. It must be clearly understood that despite the universality of the basicequation it would be wrong to state that any system may occur in the classical or non-classical areas depending on external conditions.9 We call this method, including the original de Donder s approach, a method of chemical dynamics, or a force-shift method for explicit usage of chemical forces, originally introduced as thermodynamic affinities. The method treats true, isolated thermodynamic equilibrium of asystem as a reference state for its open equilibrium when the system becomes a part of a supersystem. This reference state is memorized in η kj. Such approach well matches interpretation of equilibrium at zero control parameters as origin of the chaosity scale (S-theorem, [12]). Based on a very simple and quite natural assumption, t he basic equation of the present work naturally and smoothly drags non-linearity into thermodynamics of open systems thus bridging a gapbetween classical and non-classical thermodynamics. Addressing to a skeptical reader, we d like to underline that all new results of this work have been received and proved numerically strictly within the current paradigm of chemical thermodynamics. Our non-traditional term of the basic equation already existed in chemical thermodynamics in form of excessive thermodynamic function. This work offers alternativedescription of its origin and its relation to an external impact on the chemical system. From thispoint of view, we consider results of this work neither revolutionary nor contradictory. We justtried to find out what has been lost or hidden when chemical system, the major object of chemicalthermodynamics, has been idealized as an isolated entity. REFERENCES. [1] Prigogine, I. R. From Being to Becoming; W.H. Freeman: San Francisco, 1980. [2] Alberty, R. A. Physical Chemistry; Wiley & Sons: New York, 1983. [3] Korzynsky, D. S. Physico-Chimicheskie Osnovy Analiza Paragenezisa Mineralov (Physico-chemical Basics of Analysis of Mineral Paragenesis); AN USSR : Moscow, 1956. [4] Zilbergleyt, B. J. Russian J. Phys. Chem., 1985, 59 (7), 1059. [5] Karpov, I. K. Physico-Chimicheskoje Modelirovanie na EVM v Geochimii (Physico-chemical Computer Simulation in Geochemistry); Nauka: Novosibirsk, 1981. [6] de Donder, T.; van Risselberge, T. Thermodynamic Theory of Affinity; Oxford University Press; Stanford, 1936. [7] Gyarmati, I. Non-Equilibrium Thermodynamics; Springer-Verlag; Berlin, 1970. [8] Devaney, R. Introduction to Chaotic Dynamic Systems; Benjamin/Cummings: New York, 1986. [9] Zel`dovich, Y. A. J. Phys. Chem. (USSR), 1938, 3 , 385. [10] Zilbergleyt, B. J. Russian J. Phys. Chem., 1985, 59 (10), 1574. [11] Parton, V. Z.; Perlin, P. I. Mathematical Methods in Theory of Elasticity; MIR Publishers, Moscow, 1984. [12] Klimontovitch, Yu. J. Tech. Physics Letters (USSR), 1983, 8 , 1412.

arXiv:physics/0004036v1 [physics.gen-ph] 18 Apr 2000INTERSTELLAR HYDROGEN AND COSMIC BACKGROUND RADIATION B.G. Sidharth∗ Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Hyderabad 500 063 (India) Abstract It is shown that a collection of photons with nearly the same f re- quency exhibits a ”condensation” type of phenomenon corres ponding to a peak intensity. The observed cosmic background radiati on can be explained from this standpoint in terms of the radiation d ue to fluctuations in interstellar Hydrogen. In a previous communication[1] it was suggested that the ori gin of the Cosmic Background Radiation is the random motion of interstellar H ydrogen. We will now deduce the same result from a completely different po int of view. We start with the formula for the average occupation number f or photons of momentum /vectorkfor all polarizations[2]: /angbracketleftn/vectork/angbracketright=2 eβ¯hω−1(1) Let us specialize to a scenario in which all the photons have n early the same energy so that we can write, /angbracketleftn/vectork/angbracketright=/angbracketleftn/vectork′/angbracketrightδ(k−k′), (2) where /angbracketleftn/vectork′/angbracketrightis given by (1), and k≡ |/vectork|.The total number of photons N, in the volume Vbeing considered, can be obtained in the usual way, N=V[k] (2π)3/integraldisplay∞ odk4πk2/angbracketleftnk/angbracketright (3) 0E-mail:birlasc@hd1.vsnl.net.in 1where Vis large. Inserting (2) in (3) we get, N=2V (2π)34πk′2[ǫθ−1]−1[k], θ≡β¯hω, (4) In the above, [ k]≡[L−1] is a dimensionality constant, introduced to com- pensate the loss of a factor kin the integral (3), owing to the δ-function in (2): That is, a volume integral in /vectorkspace is reduced to a surface integral on the sphere |/vectork|=k′,due to our constraint that all photons have nearly the same energy. We observe that, θ= ¯hω/KT ≈1,since by (2), the photons have nearly the same energy ¯ hω. We also introduce, υ=V N, λ=2πc ω=2π kand z=λ3 υ(5) λbeing the wave length of the radiation. We now have from (4), u sing (5), (e−1) =υk′2 π2[k] =8π k′z[k] Using (5) we get: z=8π k′(e−1)=4λ (e−1)[k] (6) From (6) we conclude that, when λ=e−1 4= 0.4[L] (7) then, z≈1 (8) or conversely. Though equation (7) is quite general, as it stands, we have to assign suitable units to it depending on the particular physical situation: We must get an additional input, in the order of magnitude sense, from the s ystem under consideration to fix the units. Let us now consider the case of radiations due to fluctuations of the cold interstellar Hydrogen as considered from an alternative vi ew point in ref.[1]. In this case it is infact known that1 υ∼1 molecule per c.c.[2, 3]. On the 2other hand, the energy range for these cold molecules is smal l, so that the above considerations apply. So from (7), owing to the fact th atυ1/3∼λcm., it follows that, λ= 0.4cm. (9) Remembering that from (5), λis the wave length and υis the average volume per photon, the condition (8) implies that all the photons ar e very densely packed as in the case of Bose condensation. This means that fr om (9), we con- clude that at the wave length 0 .4cm,in the micro-wave region, the radiation has a peak intensity. It is remarkable that the cosmic backgr ound radiation has the maximum intensity exactly at the wave length given by (9)[4]. So, even without the Big Bang event it is possible to receive the o bserved cosmic background radiation due to fluctuations in interstellar Hy drogen (cf.ref.[3] and[5] for Hoyle and Wickramasinghe’s attempt to explain th e background radiation in terms of Helium synthesis, without invoking th e Big Bang.) The same conslusion can be obtained from yet another argumen t also[6]. In- terestingly, El Naschie has studied the Cosmic Background R adiation from the point of view of fractal quantum space time[7, 8]. References [1] B.G. Sidharth, Chaos Solitons & Fractals 11, 2000, p.147 1-1472. [2] K. Huang, ”Statistical Mechanics”, (Wiley Eastern, New Delhi, 1975). [3] J.V. Narlikar, ”Introduction to Cosmology”, (Cambridg e University Press, Cambridge, 1993). [4] L.Z. Fang, R. Ruffin, ”Basic Concepts in Relativistic Astr ophysics”, (World Scientific, Singapore, 1983). [5] F. Hoyle, J.V. Narlikar, G. Burbidge, Physics Today, Apr il 1999, pp.38ff. [6] B.G. Sidharth, Int.J.Mod.Phys.A, (1998) 13 (15), p.259 9ff. [7] M.S. El Naschie, Chaos Solitons & Fractals, 10(11), 1999 , pp1807-1811. [8] M.S. El Naschie, Chaos Solitons & Fractals, 8(5), 1997, p p847-850. 3

arXiv:physics/0004037v1 [physics.atom-ph] 18 Apr 2000Coherent X-Ray Generation with Laser Driven Ions Massimo Casu, Carsten Szymanowskia, Suxing Huband Christoph H. Keitel Theoretische Quantendynamik, Fakult¨ at Physik, Universi t¨ at Freiburg Hermann-Herder-Str. 3, D-79104 Freiburg, Germany email: casu@physik.uni-freiburg.de or keitel@uni-freib urg.de aNew Address: Department of Physics and Astronomy, Universi ty of Rochester, Rochester NY 14627 bNew Address: Max Born Institut, Rudower Chaussee 6, D-12489 Berlin, Germany (June 30, 2011) Abstract Small parts of the electronic wavepacket of a multiple charg ed ion may still tunnel through the high Coulomb barrier for a sufficiently int ense laser field of 1015W/cm2and higher. Solving numerically the corresponding Schr¨ od inger equation we find that the periodic recollisions of such a wave packet in the oscillating high power laser field will give rise to coherent X-ray radiation in the multiple keV regime, i.e. substantially higher than pre dicted or observed before. 1Since the invention of the laser, scientists have tried to ge nerate coherent light also far below the optical wavelength [1]. The striking advantages i n high resolution measurements and imaging are doubtless. All X-ray lasers up to date howeve r still lack the efficiency, size, price and in particular coherence properties of optic al laser schemes. In one of the most promising mechanisms for coherent high frequency gene ration, high power near optical laser fields were imposed on atomic systems. Parts of the elec tron wavepackets are then oscillating through the ionic core and due to the nonlinear i nteraction with the nucleus give rise to multiples of the applied laser frequency, i.e. harmo nics [2]. In the so called tunneling regime only very small parts of the wavepackets tunnel throu gh the laser modified Coulomb barrier but when returning the so far highest orders of harmo nics were observed [3,4]. The highest possible achievable photon energy has been determi ned theoretically to give the so called cut-off law [5–7] nmax¯hω=Ip+ 3.2Up (1) with the ionisation potential Ip, the ponderomotive energy Up=E2/4ω2, ¯hthe Planck constant, nmaxthe maximal order of harmonics and Eandωbeing the maximal amplitude and angular frequency of the applied laser field. The above simple formular was then explained in a simple thre e step model [8]: (i) parts from the bound electron wavepacket tunnel through the Coulo mb barrier at the time when the laser field is close to its maximum and is sufficiently narro w for a substantial tunnel rate (ii) the born wavepackets oscillates freely and classi cally in the laser field and when returning has picked up the kinetic or ponderomotive energy 3.2Upand (iii) at the return with the nucleus may at best drop back to the ground state of en ergy−Ipand release in form of radiation the ionisation energy Ipand all the gained kinetic energy 3.2 Up. If this process is repeated several times, interference among thos e processes leads to coherence of the generated light [9]. Many attempts have been made to enhance the above cut-off rule for the benefit of even higher coherent high frequency generation. By preparing th e atomic ensemble appropriately 2prior the interaction with the laser field or turning to more c omplex systems as molecules, the factor 3.2 could be enhanced to at best the order of 10 [9]. In principle the Upcould be strongly increased by several orders of magnitude with pres ent laser technology by increasing the driving laser field strength up to the relativistic regim e [10]. However, the Coulomb barrier of neutral atoms is then modified so strongly that the above described three step model does not apply anymore and instead often only Bremsstr ahlung harmonics arise. A large experimental enhancement was recently achieved ind ependently by the groups of Krausz [11] and by that of Murnane and Kapteyn [12] where ul tra short pulses were employed. With at about 500 eV they have generated the shorte d coherent wavelength so far and have entered the so called water window where optical imaging of living species in water is possible. The break through was possible essential ly because of the use of very short pulses which allowed to move quickly to the highest int ensity of the laser pulse before too much of the ground state wavepacket was lost. In spite of t his dramatic improvement a draw back of the short pulses is that only few recollisions o f born wavepackets with the atomic core are possible and thus the coherence properties o f the generated light are limited. In this Letter we present a way to enhance high harmonic gener ation towards the hard X-ray regime, maintaining still high coherence properties . For this purpose we propose to use multiply charged ions [13,14] rather than atoms because they possess a considerably larger ionization potential Ipand according to the cut-off rule Eq.(1) promise a far larger maximal harmonic order. Since relativistic corrections tu rned out negligible for the low ion charges of interest here, we solve numerically the correspo nding Schr¨ odinger equation. For each effective ion charge we evaluate the appropriate laser e lectric field strength to allow for tunneling and recollision harmonic generation. For a ch arge of Z= 3 of the ionic core as sensed by the active electron, harmonics up to the order 25 00 arise, i.e. with a photon energy of more than 4 keV. The parameter that has been employed so far to estimate if the laser and atomic param- eters are such that the three step tunnel mechanism takes pla ce is the Keldysh parameter γK[9] 3γK=/radicaltp/radicalvertex/radicalvertex/radicalbtZ2I(1) p 2Up=Zω E/radicalBig 2I(1) p=Zω E(2) withIp=Z2I(1) pbeing the ionization potential of the ion of interest with eff ective charge Z andI(1) pthe ionization potential for hydrogen. If this parameter is well below unity, one is in the tunneling regime. To have significant tunneling and th us harmonic intensity, we need a small γ. However to avoid substantial ionization this parameter sh ould not be too small. Following ref. [16] tunneling decreases exponentially wit hZ3and a Upincreasing with Z6 is necessary to compensate for this. As this is only an estima tion for our model potential we have considered and optimized the tunneling dynamics als o via numerical studies prior calculating the spectrum. We show in fact that we have to decr ease the γparameter with increasing charge of the ion strongly to ensure a sufficiently large tunneling rate and thus high harmonic output. The laser field strength employed do not exceed the order of 2 ·1016W/cm2so that due the laser field we do not enter the relativistic regime. Conce rning the interaction of the electronic wavepacket with the ionic core we employ a net cha rge of up to Z= 3. This can also be assumed as non relativistic; a corresponding cal culation via the Dirac equation will thus give essentially equal results. For more highly ch arged ions the velocities close to the nucleus can be significant as compared to the speed of ligh tcso that the laser Lorentz force term can be nonnegligible as compared to the laser elec tric field [15]. This term would then give rise to complex modifications to the recollision dy namics and thus to the harmonic spectrum but is not of interest to the nonrelativistic regim e of interest here [17]. With those assumptions in mind we consider the following dyn amical equation for the electronic wavefunction Ψ( x, t) as a function of the spatial coordinate xin the direction of the linearly polarized laser light and the time t: i∂tΨ(x, t) =HSΨ(x, t) (3) where HSstands for the Schr¨ odinger Hamiltonian with: HS=1 2/parenleftBigg p−A(t) c/parenrightBigg2 +V(x, Z) (4) 4where the vector potential of the laser pulse is defined via A(t) =E·h(t)·sin(ωt). The electric field E·h(t) involves a pulse shape h(t) with turn-on phase and a phase with constant amplitude during which we evaluate the radiation spectrum. We use the regularized nuclear potential, V(x, Z) =−k√ x2+s(5) where s= 2 is a smoothing parameter of the nucleus to compensate for t he neglect of the second and third dimension [18]. kis a direct function of the net charge Zof the ion and can be adjusted to recover the experimental values for the bi nding energies. In our numerical analysis for hydrogen (H) and hydrogen like ions (He+, Li2+) interacting with short intense laser pulses, we employed a laser wave len gth of λ= 800nm (TiSapphire), an electric field amplitude with intensity I∼1014−1016W/cm2and pulse lengths between 20 and 200 cycles with a turn-on phase of 5 to 20 cycles. In fig. 1 we displayed the effective potentials for H, He+and Li2+in the laser pulse at the time of maximal electric field and as a horizontal line the corresponding ground state energy. Fo r the case of hydrogen we have chosen the laser parameters ( γK= 0.89) such that we are well in the tunneling regime but simultaneously such that ionization is still small. When mo ving to He+we first increased the electric field strength by a factor of two only to obtain th e same Keldysh parameter and note from the corresponding dashed line for the effective potential in Fig. 1 that the tunneling barrier is then very large and that we appear to be i n the intermediate regime to the multiphoton regime. While the ionization potential sca les with Z2, the tunneling rate reduces expotenially with Z3/E[16]. In the solid line for He+we have further increased the electric field strength such that we have similar ionization as for Habove, however though with a smaller γK. Before addressing Li2+, we discuss Fig. 2 where we have displayed the radiation spec trum corresponding to the situations in H (a) and the two cases for He+displayed in Fig. 1. For H we are in the tunneling regime and thus find as well known a plateau and cut- off following Eq. (1). In Fig 2(b) for increasing IpandUpby a factor of two we find a 5spectrum with the expected cut-off enhanced by a factor of Z2= 4. However, even though γKis unchanged we have clearly moved towards the multi photon r egime. The harmonic structure has detoriated and we find even resonance structur es in the low frequency part of the spectrum. The enhancement of about the 12th harmonic cor responds to the transition from the first excited to the ground state. Also we note a begin ning of a tilting of the spectrum in the plateau area. To reenter the tunneling regim e, we have increased the E field further, such that the ionization rate becomes comparable t o that for Hin Fig. 1 a) and found an up-conversion of the cut-off frequency by a factor of 10. We continue the procedure for Li2+as depicted in the lowest entry of Fig. 1. We have increased the electric field with respect to Hnot only by a factor 3 but substantially more to secure sufficient tunneling. However, for such high harmon ics as expected here and the corresponding low efficiency we chose a long pulse for a clear r esolution of the very high harmonics. This requires us to maintain a sufficiently wide io nization barrier somewhat larger than in the two previous cases and thus accept a spectr um with a less pronounced cut-off frequency. Still we achieve at about 2500 harmonics a s visible in Fig. 3, i.e. hard X-ray harmonics with an energy of at about 4 keV. We note that s ometimes in the literature those energies are still counted to the upper end of soft X-ra ys. However, proceeding to higher charged ions, even higher harmonics are achievable, though with an increasingly small efficiency. Given the available parameters we note a scaling l aw of at about Z3for the energy of the cut-off harmonics as a function of the ion charge and the correspondingly chosen laser field intensity. We note that for practical reasons we have in creased the intensity with rising charge Zrather than modifying the frequency which is less controlla ble experimentally. A constant wavelength of the applied laser field may however be problematic, since relative to the modified characteristic length of the ionic core due to the variable Z, one may need adapt the time available for tunneling. Moving towards the relativistic regime [10] with higher las er intensities and ionic charges, higher harmonics are possible. However, the harmonic spect rum deviates further from the traditional structure in the tunneling regime including a h orizontal plateau and a well pro- 6nounced cut-off. The magnetic component of the laser field ind uces a significant momentum transfer in the propagation direction of the laser field. Thi s makes recollisions more diffi- cult, especially involving long return times after many fre e oscillations in the laser field. As a consequence we find a reduction of harmonics between the per turbative and the cut-off regime. This goes beyond the scope of this letter and will be d iscussed in detail in future work [17]. In conclusion multiply charged ions still allow for tunneli ng and recollisions of electron wave packets with the parent ionic core with laser field inten sities well above those employed before for high harmonic generation with atoms. Consequent ly coherent X-rays become feasible in the multiple keV regime. The efficiency is below th at for coherent high harmonic generation via laser driven atoms. This work has been funded by the German Science Foundation (N achwuchsgruppe within SFB 276). CS and SXH acknowledge present funding from the Ale xander von Humboldt foundation. 7REFERENCES [1] see e.g. R.C. Elton, X-ray Lasers , Academic Press (1990); P. A. Norreys et al., Phys. Rev. Lett. 761832 (1996). [2] B. W. Shore and P. L. Knight, J. Phys. B20, 413 (1987). [3] A. L’Huillier, K. Sch¨ afer and K. C. Kulander, Phys. Rev. Lett. 66, 2200 (1991); A. L’Huillier, and P. Balcou, Phys. Rev. Lett. 70, 774 (1993). [4] J. Macklin, J. Kmetec, and C. Gordon III, Phys. Rev. Lett. 70, 766 (1993). [5] J.L. Krause, K.J. Sch¨ afer and K.C. Kulander, Phys. Rev. Lett. 68, 3535 (1992) and Phys. Rev. A45(1992) 4998. [6] W. Becker, S. Long and J. McIver, Phys. Rev. A41(1990) 4112 and Phys. Rev. A50 (1994) 1540. [7] P. B. Corkum, Phys. Rev. Lett. 711994 (1993). [8] M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier a nd P. B. Corkum, Phys. Rev. A492117 (1994). [9] M. Protopapas, C. H. Keitel, and P. L. Knight, Rep. Progr. Phys. 60, 389 (1997). [10] C. H. Keitel, P. L. Knight, and K. Burnett, Europhys. Lett. 24, 539 (1993); U.W. Rathe et al,J. Phys. B30, L531 (1997); N. J. Kylstra, A. M. Ermolaev, and C. J. Joachai n, J. Phys. B30, L449 (1997); C. H. Keitel et al,J. Phys. B31, L75 (1998); R. Taeib, V. Veniard and A. Maquet, Phys. Rev. Lett. 81, 2882 (1999); C. Szymanowski, C. H. Keitel and A. Maquet, Las. Phys. 9, 133 (1999). [11] Ch. Spielmann, et al,Science 278, 661 (1997); M. Schn¨ urer , et al,Phys. Rev. Lett. 80, 3236 (1998); Ch. Spielmann, et al, IEEE 4, 249 (1998). [12] Z. Chang, et al,Phys. Rev. Lett. 79, 2967 (1997); H. Kapteyn and M. Murnane, Phys. World 12, 31 (1999). 8[13] T. Ditmire, et al,Nature (London) 386, 54 (1997); T. Ditmire, et al,Phys. Rev. A57, 369 (1998). [14] P.H. Kokler and Th. Stoehlker, Adv. At. Mol. Opt. Phys. 37, 297 (1996); J. Ullrich et al,J. Phys. B30, 2917 (1997). [15] S.X. Hu and C.H. Keitel, Europhys. Lett. 47318 (1999) and Phys. Rev. Lett. 83,4709 (1999). [16] M. V. Ammosov, N. B. Delone and V. P. Krainov, Zh. Eksp. Te or. Fiz. 91, 2008 (1986); V. P. Krainov, J. Phys. B 32, 1607 (1999); S. J. McNaught, J. P. Knauer and D. D. Meyerhofer, Phys. Rev. A58(1998) 1399. [17] S.X. Hu and C.H. Keitel, in preparation. [18] J.H. Eberly, Phys. Rev. A42(1990) 5750. 9Figure Captions Fig.1. The effective potentials for Z= 1 (H, k=1), Z= 2 (He+, k=3.48) and Z= 3 (Li2+, k=7.35) interacting with electric fields corresponding to t he intensities 1 .4·1014W/cm2 (→γK= 0.89 for H), 2 .19·1015W/cm2(→γK= 0.456 for He+) and 2 ·1016W/cm2 (→γK= 0.255 for Li2+). The corresponding ground state energies are indicated by the horizontal lines. The laser wavelength is 800nm, which c orresponds to an atomic frequency ω=0.057au. He+with the same γKasHis represented by the dashed line. Fig.2. The harmonic spectrum for H (a) and He+(b/c) in appropriately intense laser fields as given in Fig. 1. The pulses consist of 20 cycles full intens ity and 5 cycles turn-on. With Eq. 1 the cut-off is placed at nmax,H = 27 for H, nmax,He+=Z2nmax,H = 108 for He+with the same γKas for H and nmax,He+= 305 with the enhanced Up(the ionisation rate here is comparable to that of H and is about 20 %). Fig.3. a) The harmonic spectrum for Li2+for the laser parameters given in Fig. 1. b) is an enlargement of the high frequenct range of figure a). The full intensity was applied here for 150 cycles with an 18 cycle turn-on. After 100 cycles we have ∼60% ionised and just after 150 cycles essentially 100% is ionised. 10FIGURES −2 0 2 4 6 8 10 12 14 x [a.u.]−6−5−4−3−2−10V(x) [a.u.]H He+ Li2+ FIG. 1. M. Casu et al., “Coherent ...” 110 5 10 15 20 25 30 35 40 45 5010−3100103 0 10 20 30 40 50 60 70 80 90 100 110 120 130 14010−410−1102105 0 50 100 150 200 250 300 350 harmonic order10−710−410−1102105harmonic signal (arb. units)a) b) c) FIG. 2. M. Casu et al., “Coherent ..” 120 500 1000 1500 2000 250010−1210−810−4100104 2350 2370 2390 2410 2430 2450 2470 2490 harmonic order10−1210−1110−1010−910−8harmonic signal (arb. units) FIG. 3. M. Casu et al., “Coherent ..” 13

arXiv:physics/0004038v1 [physics.chem-ph] 19 Apr 2000Hyperfine spectra of CH 3F nuclear spin conversion Pavel L. Chapovsky Institute of Automation and Electrometry, Russian Academy of Sciences, 630090 Novosibirsk, Russia; and Laboratoire de Physique des Lasers, Atomes et Mo lecules, Universit´ e des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex, France (July 22, 2013) Abstract A theoretical model of hyperfine spectra of nuclear spin conv ersion of molec- ular isomers has been developed. The model takes into accoun t the nu- clear spin-spin and spin-rotation interactions, as well as the saturation of in- tramolecular mixing of molecular ortho and para states. The model has been applied to hyperfine spectra of nuclear spin conversion in13CH3F molecules subjected to an external electric field. Conditions under wh ich the hyperfine structure in the spectra can be resolved have been determine d. 34.30.+h, 35.20.Sd, 33.50.-j Key words: Molecular spin isomers, conversion, hyperfine sp ectra. Typeset using REVT EX 1I. INTRODUCTION Recently, much attention has been devoted to the investigat ion of the level-crossing resonances in the spin isomer conversion ( conversion spectra ) in CH 3F [1–7]. The goal was to test the mechanism of the ortho-para conversion in this mo lecule. The importance of these efforts is clear. CH 3F is the first and until now the only polyatomic molecule in whi ch a spin conversion mechanism was identified. On the other hand , one may expect a similar conversion mechanism in other symmetrical molecules too. The essence of the isomer conversion spectra consists of the following. An external homogeneous electric field splits the molecular states. At s ome electric field the ortho and para states of the molecule cross which, speeds up the rate of conversion. Scanning of the electric field gives the field dependence of conversion rate w hich is the conversion spectrum. Measurements of the conversion spectra are presently perfo rmed at room temperature. The usefulness of the conversion spectra should not be limit ed to the investigation of the conversion itself. In fact, this is a new physical effect whic h can find various applications. For example, it was proposed [3] to measure with the help of the co nversion spectra characteris- tics of molecular spin-rotation interaction which are hard ly accessible by ordinary methods. Many of these applications require a deeper understanding o f the phenomenon. Up to now, only the rotational structure in the conversion spectra res ulting from M-splitting of molec- ular states has been considered. This is sufficient for the con version spectra of ∼1 MHz resolution. Higher resolution needs an account of the hyper fine structures in the conversion spectra, which has not been done before. In this paper we perf orm such an analysis and determine the conditions at which these hyperfine structure s can be observed. We check also an accuracy of the simplified approach to the conversion spectra in which hyperfine splitting of molecular states is neglected. The calculatio ns are done for13CH3F because for this molecule the rotational structure in the conversion sp ectra was experimentally observed recently [5]. The spectral resolution achieved already in [ 5] was better than 6 MHz. It makes the13CH3F molecule the best candidate for the observation of the hype rfine structures in the conversion spectra. II. QUANTUM RELAXATION OF SPIN ISOMERS Ordinary gaseous relaxation processes are insensitive to a tiny hyperfine splitting of molecular states. This is a consequence of the fact that the k inetic energy of colliding particles is much larger than the hyperfine level splitting. For the nuclear spin conversion in molecules the situation can be different if the conversion is induced by intramolecular state mixing and collisional interruption of this mixing (we will refer to this process as quantum relaxation [8]) which is rather sensitive to the ortho-para level split ting. The CH 3F molecules exist in the form of two nuclear spin isomers [9]. The total spin of the three hydrogen nuclei in the molecule can have the magn itude either I= 3/2 (ortho isomers), or I= 1/2 (para isomers). Angular momentum projections ( K) on the molecular symmetry axis divisible by 3 are allowed only for ortho isome rs. For para isomers all other K values are allowed. Consequently, the quantum states of the CH3F molecule are divided into 2two subspaces which are schematically shown in Fig. 1. Quali tatively, CH 3F spin conversion by quantum relaxation can be described as follows. Suppose t hat at the beginning of the process a test molecule is placed into the ortho subspace. Du e to collisions with surrounding gas particles, which cannot change the spin state of the mole cule, the test molecule starts to perform fast migration along rotational states inside th e ortho subspace. This running up and down along the ladder of the ortho states continues unt il the molecule jumps into the state mwhich is mixed by an intramolecular perturbation ˆVwith the energetically close para state n. During the free flight after this collision, para state nwill be admixed with the ortho state m. Consequently, the next collision can transfer the molecul e into other para states and thus localize it inside the para subspace. More de tails on quantum relaxation of nuclear spin isomers can be found in Refs. [10–12]. Quantum relaxation of nuclear spin isomers of molecules can be quantitatively described in the framework of the density matrix formalism. The result of this description is the following [12]. One has to divide the molecular Hamiltonian into two parts ˆH=ˆH0+ˆV , (1) where the main part of the Hamiltonian, ˆH0, has pure ortho and para states as the eigen- states, and ˆVis a small perturbation which mixes the ortho and para states . In the Ref. [12] the hyperfine contribution to the term ˆH0was neglected and it was assumed that the molecule is not subjected to an external field. Suppose that at the instant t= 0 a nonequilibrium concentration of, e.g., ortho molecule s, δρo(0), was created. the solution of the problem gives an expone ntial isomer conversion: δρo(t) =δρo(0)e−γt, with the rate γ=/summationdisplay α∈o,α′∈p2Γ|Vαα′|2 Γ2+ω2 αα′(Wα+Wα′). (2) Here Γ is the collisional decay rate of the off-diagonal densi ty matrix elements, (∂ραα′/∂t)coll=−Γραα′;α∈ortho, α′∈para, (3) assumed to be the same for all ortho-para level pairs; ωαα′is the gap between the states |α >and|α′>;WαandWα′are the Boltzmann factors. Solution (2) was obtained under the assumption that collisions do not transfer molecules di rectly from ortho to para states, i.e., the cross-section σ(ortho|para) = 0. The validity of this assumption has been discussed in detail in the review [13]. At low gas pressure, when Γ ≪ω, the conversion rate is proportional to Γ, thus being proportional to the gas pressure. In this limiting case, the pressure depe ndence of the conversion rate is quite similar to ordinary gaseous relaxa tion which is linear in pressure too. On the other hand, in this pressure limit the conversion rate is sensitive to the ortho- para level splitting (note ωin the denominator of (2)) thus having a distinctive signatu re, unusual for gaseous relaxation processes. This strong depe ndence on ωis at the heart of the level-crossing effect at the spin isomer conversion. 3III. STARK LEVEL SPLITTING Calculation of the ortho-para state mixing is rather compli cated, especially if one has to account for both the Stark and hyperfine level splitting. I n this case special care should be taken about proper symmetrization of states and their tra nsformation under these per- turbations. Let us consider first the quantum states of the fr ee molecule and ignore the hyperfine perturbation. Spin-rotational states in the grou nd electronic and vibrational state of CH 3F can be constructed as follows [14,15]. To determine the mol ecular position in space, one introduces a system of coordinates ( xyz) fixed in the molecule (Fig. 2). By combining rotational and nuclear spin states, one introduces the stat es|β >which are invariant under cyclic permutation of the three hydrogen nuclei |β >≡ |J, K, M > |I, σ, K > ;|β >≡ |J,−K, M > |I, σ,−K >;K≥0, (4) where |J, K, M > are the quantum states of symmetric top characterized by the angular momentum ( J), its projection ( K) on the zaxis of the molecular system of coordinates and the angular momentum projection ( M) on the quantization Zaxis of the laboratory system of coordinates. Iandσare the total spin of the three hydrogen nuclei and its projec tion on the Zaxis, respectively. The explicit expressions for the spin s tates|I, σ, K > are given in [14]. These expressions specify the allowed K-quantum numbers for the ortho and para spin isomers as was explained in Section II. Permutation of any two hydrogen nuclei in CH 3F inverts the zaxis of the molecular system of coordinates. Such permutations, e.g., the permut ationP23, acts on the state |β > asP23|β >= (−1)J|β >. Because the states |β >are invariant under cyclic permutation of the three hydrogen nuclei, a similar result is valid for the o ther two permutations of hydrogen pairs: P12andP31. States |β >and|β >generate a two-dimensional representation of the molecula r sym- metry group. Spin-rotation states, which generate one-dim ensional representations are |β, κ > ≡1√ 2[(−1)κ+P23]|β >;κ= 0,1. (5) From this definition it is easy to conclude that the symmetry o f states |β, κ > is determined by the rule: P23|β, κ > = (−1)κ|β, κ > and by similar relations for the permutations of the other two pairs of protons. Further, one has to take into account the molecular inversio n states. Let us designate the antisymmetric and symmetric inversion states of the mol ecule as |s= 0>and|s= 1>, respectively. Permutation of two protons, e.g., P23, acts on the inversion states as [15] P23|s= 0>=−|s= 0>;P23|s= 1>=|s= 1> . (6) The total quantum states of CH 3F have to be antisymmetric under the permutation of any two hydrogen nuclei because protons are fermions. Con sequently, the only allowed molecular states are |β, κ=s >|s >. 4The last step is to account for the spin states of the fluorine a nd carbon (13C) nuclei, both having spin 1/2. Finally, the molecular states are |αs>≡ |β, κ=s >|s >|σF>|σC>, (7) where σFandσCdenote, respectively, the projections of the fluorine and ca rbon nuclear spins on the laboratory quantization Zaxis. The states (7) of the free molecule will be denoted as the α-basis. For rigid symmetric tops, like CH 3F is, the states |αs>are degenerate in the quantum numbers M,σ,σF,σCands. An external electric field due to the Stark perturbation, ˆVSt=−ˆdE, mixes the states having |∆s|= 1 and |∆J| ≤1. In the following, we will consider relatively weak electr ic fields which produce Stark splitting much smaller than the J-splitting. Consequently, the Stark mixing of states having |∆J|= 1 can be neglected and one has to solve the Schr¨ odinger equation for the doubly degenerate states |α0>;|α1>. In a standard way [9], one can find the energy of the new states, E(β, ξ) =Efree(J, K) + (−1)ξK|M| J(J+ 1)dE;ξ= 0,1, (8) where Efree(J, K) is the energy of a free molecule; dis the so-called permanent electric dipole moment of the molecule; Eis the electric field strength; ξis the quantum number of the new molecular states which are |β, ξ > =1 2/bracketleftBigg (−1)ξM |M||s= 0>(1 +P23) +|s= 1>(1−P23)/bracketrightBigg |β > . (9) The magnitude of the electric dipole moment, d, is determined by the molecular spatial and electronic structure [14] Similar to the case of a free molecule, one has to take into acc ount the spin states of fluorine |σF>and carbon |σC>nuclei which remain intact by Stark perturbation. Finally, the states of the CH 3F molecule subjected to an external electric field are |µ >≡ |β, ξ > |σF>|σC> . (10) The manifold of these states will be denoted as the µ-basis. The Stark effect partially lifts the degeneracy of the α-states. But the µ-states remain degenerate in sign of M, and in quantum numbers σ,σFandσC. IV. HYPERFINE LEVEL SPLITTING In accordance with the general rules, the molecular Hamilto nian, ˆH, which contains now the hyperfine and Stark perturbations, can be expressed in th eµ-representation as ˆH=/summationdisplay µ1/summationdisplay µ2|µ2>< µ 2|ˆH|µ1>< µ 1| ≡ ˆH(o) +ˆH(p) +ˆV , (11) 5where the sums run over the complete set of states. We recall t hat the operator/summationtext|µ >< µ |is the identity operator if the summation includes all states o f the complete set. Eq.(11) shows how the molecular Hamiltonian can be split into three operat ors:ˆH(o) and ˆH(p) which have only diagonal matrix elements in the ortho and para subspace s, respectively, and operator ˆV, which has only matrix elements off-diagonal in the ortho and para quantum numbers. When considering the combined level splitting by the three p erturbations: Stark effect, hyperfine spin-rotation and spin-spin perturbations it is h elpful to start with the Stark effect. Taking into account the Stark effect is not difficult because St ark perturbation does not mix the ortho and para states of a free molecule. In fact, transfo rmation from the α-basis of a free molecule to the µ-basis gives the level splitting by the Stark effect. A. Spin-rotation interaction The next perturbation to be considered is the spin-rotation coupling which is due to the interaction of nuclear spins with the intramolecular ma gnetic field induced by molecular rotation. The spin-rotation interaction Hamiltonian is gi ven by the operator [14,16] ¯hˆHSR=−/summationdisplay kˆI(k)•C(k)•ˆJ≡¯h/summationdisplay kˆH(k) SR, (12) where ˆI(k)andC(k)are, respectively, the spin operator and the spin-rotation tensor for the k-th nucleus; ˆJis the molecular angular momentum operator and kdenotes all nuclei in the molecule. The magnitude of the spin-rotation tensor Cdepends on the molecular spatial structure and motion of the molecular electrons [14]. Spin-rotation interaction is of the order 10 −100 kHz which is much smaller than the energy gaps between the states having different JandK(>102GHz), and is much smaller than the Stark splitting of states different in |M|andξ(>10 MHz). Consequently, only matrix elements of ˆHSRdiagonal in quantum numbers J,K,M,ξandIare important for the calculation of the spin-rotation splitting of state s [17]. Because the operator ˆHSRis a scalar, the only non-zero matrix elements will be the eleme nts diagonal also in the spin projections σ,σFandσC. Explicit expressions for these matrix elements can be foun d in [16]. For example, the diagonal matrix elements for the spin -rotation interaction of the fluorine nucleus are MσF/bracketleftBigg cF α+K2 J(J+ 1)(cF β−cF α)/bracketrightBigg , (13) where constants cF αandcF βare the diagonal elements of the tensor CF, calculated in the molecular system of coordinates. The spin-rotation tensor of fluorine and hydrogen nuclei are given in [16]. The tensor CCfor the carbon nucleus is absent in [16]. We will use for its estimation the relation: CC=CFmC/mF, where mCandmFare the magnetic moments of 13Cand F nuclei, respectively. One can conclude, that the spin-rotation splitting of state s appears to be rather simple in the µ-basis because this perturbation only shifts the states but does not mix them. As an example, the spin-rotation splitting of the state ( J=9,K=3,M=9) is shown in Fig. 3b. 6B. Spin-spin interaction Nuclear spin-spin interaction in molecules is composed of d ipole-dipole interaction of pairs of nuclei. The spin-spin interaction Hamiltonian for the two magnetic dipoles m1and m2separated by the distance rhas the form [9] ¯hˆH12=P12ˆI(1)ˆI(2)• •T(12); T(12) ij=δij−3ninj;P12=m1m2/r3I(1)I(2), (14) where ˆI(1)andˆI(2)are the spin operators of the particles 1 and 2, respectively ;nis the unit vector directed along r;iandjare the Cartesian indices. The total spin-spin interaction in13CH3F (ˆHSS) consists of the interactions between the three hydrogen nuclei ( ˆHHH), hydrogen - fluorine nuclei ( ˆHHF), hydrogen - carbon nuclei (ˆHHC), and fluorine - carbon nuclei ( ˆHFC). Consequently, the total spin-spin Hamiltonian in13CH3F is ˆHSS=ˆHHH+ˆHHF+ˆHHC+ˆHFC. (15) Explicit expressions for all these terms can be written on th e basis of Eq.(14) for one pair of nuclei. For example, for the spin-spin interaction between the three hydrogen and fluorine nuclei one has ¯hˆHHF=PHF/summationdisplay nˆI(n)ˆIF• •TnF;n= 1,2,3. (16) Here the sum runs over all hydrogen nuclei in the molecule. Co mplete definition of the spin-spin interaction requires knowledge of the dimension al factors P(see Eq.(14)) and the spatial structure of the molecule. These data can be found in Refs. [12,18]. Without hyperfine interactions, quantum states of the molec ule subjected to an external electric field form the µ-basis. The intramolecular spin-spin perturbation ˆHSShas non-zero matrix elements diagonal and off-diagonal in quantum number sJ,K,M,σ,σF, and σC. TheJ, K-splitting of states is much larger than the spin-spin split ting which is on the order of 10−100 kHz. Therefore, one can neglect the matrix elements of ˆHSSoff-diagonal in J andK. Under our conditions the Stark splitting of states different in|M|is much larger than the spin-spin interaction. Thus, only the matrix elements of ˆHSSdiagonal in Mare important if |M|>1 because of the selection rule |∆M| ≤2 for the spin-spin interaction. Off-diagonal in Mmatrix elements of ˆHSSare important only for the states having |M|= 1. This particular case will be considered elsewhere. One can conclude, that th e spin-spin perturbation of µ- states is determined by the matrix elements of ˆHSSdiagonal in quantum numbers J,K,M, ξ,Iand in the values of the sum σ+σF+σC. To summarize, the hyperfine structure of molecular states in our case is determined by the hyperfine Hamiltonian ˆHSR+ˆHSSspanned by the states different only in the projections of nuclear spins σ,σF, and σC. There are 4 ×2×2 = 16 such states for ortho molecules: 7|βortho, ξ >|σF>|σC>, (17) and 2 ×2×2 = 8 states for para molecules: |βparaξ >|σF>|σC> . (18) Matrix elements of ˆHSRare give by Eq. (13). Calculation of the matrix elements of al l terms of ˆHSScan be done in a way similar to that explained in the Appendix. After all necessary matrix elements of ˆHSRandˆHSSare determined one can find the eigenstates and eigenvalues of the operator ˆHSR+ˆHSS. This will give the hyperfine level splitting of the molecular states under joint action of the spin-spin and spi n-rotation interactions. In the case of ortho states one has to diagonalize the 16x16 matrix. For the para states one has to diagonalize the 8x8 matrix. Ortho and para quantum states of the molecule which take into account the Stark effect and both types of hyperfine inter actions will be designated as theτ-basis. As an example, splitting of the ortho state J=9,K=3,M=9 by the spin-spin and spin-rotation interactions is shown in Fig. 3c. These ca lculations were done numerically. One can see from these data that the hyperfine interaction lif ts completely degeneracy of the nuclear spin states. V. CONVERSION SPECTRA A. Rotational structure in the conversion spectra For the calculation of the conversion spectra it is importan t to know the positions of ortho and para states in13CH3F. The nuclear spin conversion in13CH3F is dominated by mixing of the two level pairs: ( J=9,K=3)–( J′=11,K′=1) and (20,3)–(21,1) [12,19]. Of these two level pairs the former has the smaller energy gap (1 30 MHz) and contributes nearly 65% to the conversion rate of a free molecule. The stat es (11,1)–(9,3) are mixed by the spin-spin interaction only [12]. The spin-rotation int eraction does not mix this pair of states because of the selection rule for the spin-rotation i nteraction |∆J| ≤1 [20]. To describe the rotational structure in the conversion spec tra [1] one has to include the Stark perturbation ˆVSt=−ˆdEinto the ˆH0term of the splitting (1). As a result, an equation similar to Eq.(2) is obtained but in the µ-basis which accounts for the Stark effect. Note that the modeling of the off-diagonal elements of the collisi on integral (3) will also have the same form in the µ-basis. Further, one needs to know the matrix elements of the spin- spin perturbation which mixes ortho and para states of13CH3F. From the total spin-spin perturbation (15) only the part, ˆV=ˆHHH+ˆHHF+ˆHHC, produces the ortho-para state mixing. The matrix elements Vµ′µin the µ-basis can be calculated using the matrix elements ofˆVin the α-basis [12] and the relation between these two bases given in Section III. An overview of the rotational structure in conversion spect ra of13CH3F is shown in Fig. 4. The spectrum was calculated using Γ = 1 .75·107s−1which corresponds to the gas pressure 0.1 Torr. (Here and below we use the decoherence rat e 1.75·108s−1/Torr [5]). For such a value of Γ the hyperfine structure of molecular states i s not important because the 8line broadening is much larger than the hyperfine splitting. At lower pressures the hyperfine level splitting does play an important role, as is shown in th e next Section. B. Hyperfine spectra To calculate the hyperfine spectra of spin conversion one has to include both the Stark terms and the hyperfine terms into the operators ˆH(o) and ˆH(p) (see Eq.(11)). This will result in transformation of the µ-basis of molecular states to the τ-basis. Matrix elements of the perturbation ˆVbetween ortho and para states should be calculated in the τ-basis also. The calculations were done by using the matrix elements of ˆVin the α-basis from [12] and the matrix which transforms the α-basis to the τ-basis. The new expression for the conversion rate obtained in this w ay, which is exactly Eq.(2) withτindices instead of αindices, needs further modification. The point is that hyper fine structure in the spin conversion spectra can be revealed onl y at rather low pressures. The gas pressure should be low enough to make the decoherence rat e Γ smaller than the hyperfine splitting. The latter is estimated to be of the order of V. Consequently, the condition Γ <∼V should be fulfilled. As was shown in [8], calculation of the spin conversion rate i n first order perturbation theory is not valid if Γ <∼V. This is a consequence of the level population saturation by intramolecular ortho-para states mixing. The saturation e ffect can be accounted for by using the new expression for the conversion rate γ=/summationdisplay τ′∈p,τ∈o2Γτ′τ|Vτ′τ|2 Γ2 τ′τ+ω2 τ′τ+ 4Γτ′τ ν|Vτ′τ|2(Wτ′+Wτ), (19) where νis the rotational relaxation rate which was assumed equal fo r ortho and para isomers. Expression (19) is a straightforward generalization of the result [8], where the theory was developed for the mixing of only one pair of ortho and para sta tes. As is clear from Eq. (19), the saturation effect is most import ant for “strong resonances”, which have large values of the mixing matrix elements Vτ′τ. As an example of the saturation effect at work, we show the hyperfine structure of the stronges t peak in the spectrum of Fig. 4 which results from the crossing of magnetic sublevels M′=11 and M=9. This hyperfine spectrum is given in Fig. 5. Calculation of the spectrum was p erformed for the gas pressure 0.1 mTorr, and the ratio Γ /ν= 10. From the spectrum of Fig. 5 one concludes that the saturation effect can produce substantial broadening of the spectral lines. Consequently, the hyperfine structure in Fig. 5 remains unresolved. Decrea sing the gas pressure does not result in a decrease in the width of the lines. Nevertheless, well-resolved hyperfine spectra of spin conv ersion can be obtained if one chooses “weak resonances” having small magnitude of Vτ′τ. As an example, the hyperfine structure produced by the “weak crossing” of the M′=7 and M=9 states is shown in Fig. 6. One can see that in this case the hyperfine structure is well re solved. Decreasing the gas pressure makes the lines even more narrow. 9It is interesting to understand the accuracy of the simplifie d approach to the conversion spectra in which hyperfine splitting of molecular states is n ot taken into account. In Fig. 7 we show the pressure dependence of the amplitude of the “strong resonance” (9,3,9)–(11,1,11) calculated using the simplified and the present model. One ca n see that a 10% difference in the two amplitudes already appears at 5 mTorr. At the press ure 1 mTorr, the present model gives one-third the value of the old model. VI. CONCLUSIONS A theoretical model of hyperfine spectra of nuclear spin conv ersion has been developed. Calculations have been performed for the conversion of spin isomers of13CH3F molecules subjected to an external electric field. The spin-rotation a nd the spin-spin intramolecular interactions were taken into account in the calculations. It has been shown that the hyperfine structure of spin convers ion spectra can be sub- stantially distorted by the saturation effect. In the case of “strong resonances” this effect hides the hyperfine structure of the spectra completely. It h ad been proposed to observe the hyperfine spectra by using “weak resonances” which have smal l ortho-para matrix elements. Such weak resonances have been found for the13CH3F molecules and hyperfine spectra for them have been calculated. Calculations performed in the present paper have revealed t he conditions at which ex- perimental observation of the hyperfine spectra of nuclear s pin conversion can be performed. These conditions include the choice of suitable level cross ings and gas pressure. The pressure at which the hyperfine structure of the spectra is resolved ap peared to be rather low. This implies strong limitations to the experimental setup, viz. , molecular collisions with the walls should contribute to the decoherence rate Γ not much more tha n collisions in the bulk. The latter should be on the order of 104s−1only. The theoretical model proposed in this paper is rather gener al. It can be applied to molecules having various symmetries without changing its e ssence. Another possible exten- sion of the model can be related to the use of the Zeeman effect f or the level splitting instead of the Stark effect considered in the paper. VII. ACKNOWLEDGMENT This work was made possible by financial support from the Russ ian Foundation for Basic Research (RFBR), grant No. 98–03–33124a and from the Region Nord Pas de Calais, France. VIII. APPENDIX As an example, we give here the calculation of the matrix elem ents of ˆHHF. First, let us express the ˆHHFthrough the spherical tensors [9]: 10ˆHHF≡/summationdisplay nˆH(n) HF=PHF/summationdisplay n/summationdisplay q1,q2,q(−1)q<1, q1,1, q2,|2,−q >ˆI(n) 1,q1ˆIF 1,q2TnF 2q. (20) Here ˆH(n) HFis the spin-spin operator of the interaction between fluorin e and the n-th hydrogen nuclei; n= 1,2,3 denotes the hydrogen nuclei; < . . .|. . . > stands for the Clebsch-Gordan coefficient. Calculations of the matrix elements of ˆHHFcan be substantially simplified if one takes into account that the matrix elements < µ1|ˆH(n) HF|µ2>are equal for all n. This can be proven by applying to these matrix elements the cyclic permutation ,P123, of the three equivalent protons. Taking this simplification into account one has < µ1|ˆHHF|µ2>= 3< µ1|ˆH(1) HF|µ2> . (21) The matrix elements of the operator ˆHHFwhich contribute to the spin-spin level splitting (see Section IV) are given by the expression < µ2|HHF|µ1>= 3PHF/summationdisplay q<1, q,1,−q|2,0>< I, σ 2, K|ˆI(1) 1,−q|I, σ1, K > × < σF 2|ˆIF 1,q|σF 1>< J, K, M |T1F 2,0|J, K, M > δσC 2,σC 1. (22) The matrix elements in Eq.(22) can be evaluated using the fol lowing relations. The matrix elements of the spatial tensor T1F 2,0are given in Ref. [9]. In our particular case one has < J, K, M |T1F 2,0|J, K, M > =< J, K, 2,0|J, K >< J, M, 2,0|J, M > T1F 2,0, (23) where T1F 2,0is the spherical tensor component of T1Fcalculated in the molecular frame. Finally, the matrix elements of the spin operators ˆI(1) 1,−qandˆIF 1,qcan be evaluated using the Wigner-Eckart theorem which states for the spherical tenso r of rank κ[9]: < J′, M′|fκ,q|J, M > =iκ(−1)Jmax−M′/parenleftBigg J′κ J −M′q M/parenrightBigg < J′||fκ||J >, (24) where (: : :) stands for a 3j-symbol and < J′||fκ||J >is the reduced matrix element. Eqs.(22)-(24) allow us to perform the calculations of the ma trix elements of ˆHHF. 11REFERENCES [1] B. Nagels, M. Schuurman, P. L. Chapovsky, and L. J. F. Herm ans, Chem. Phys. Lett. 242, 48 (1995). [2] P. L. Chapovsky, in 12th Symposium and School on High-Resolution Molecular Spe c- troscopy , edited by L. N. Sinitsa, Y. N. Ponomarev, and V. I. Perevalov (SPIE, Wash- ington, 1997), Vol. 3090, pp. 2–12. [3] K. Bahloul, M. Irac-Astaud, E. Ilisca, and P. L. Chapovsk y, J. Phys. B: At. Mol. Opt. Phys.31, 73 (1998). [4] K. Bahloul, Ph.D. thesis, Univ. Paris-7, Denis Diderot, 1998. [5] B. Nagels, N. Calas, D. A. Roozemond, L. J. F. Hermans, and P. L. Chapovsky, Phys. Rev. Lett. 77, 4732 (1996). [6] B. Nagels, Ph.D. thesis, Leiden University, 1998. [7] J. Cosl´ eou, F. Herlemont, M. Khelkhal, J. Legrand, and P . L. Chapovsky, The Eur. Phys. J. D 10, 99 (2000). [8] P. L. Chapovsky, Physica A (Amsterdam) 233, 441 (1996). [9] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon Press, Oxford, 1981). [10] R. F. Curl, Jr., J. V. V. Kasper, and K. S. Pitzer, J. Chem. Phys.46, 3220 (1967). [11] P. L. Chapovsky, Zh. Eksp. Teor. Fiz. 97, 1585 (1990), [Sov. Phys. JETP. 70, 895 (1990)]. [12] P. L. Chapovsky, Phys. Rev. A 43, 3624 (1991). [13] P. L. Chapovsky and L. J. F. Hermans, Annu. Rev. Phys. Che m.50, 315 (1999). [14] C. H. Townes and A. L. Shawlow, Microwave Spectroscopy (McGraw-Hill Publ. Comp., New York, 1955), p. 698. [15] P. R. Bunker, Molecular symmetry and spectroscopy (Academic Press, New York, San Francisco, London, 1979). [16] S. C. Wofsy, J. S. Muenter, and W. Klemperer, J. Chem. Phy s.55, 2014 (1971). [17] The conclusion concerning Mis the consequence of the selection rule for the spin- rotation interaction |∆M| ≤1 [20]. [18] T. Egawa, S. Yamamoto, M. Nakata, and K. Kuchitsu, J. Mol . Structure 156, 213 (1987). [19] P. L. Chapovsky, D. Papouˇ sek, and J. Demaison, Chem. Ph ys. Lett. 209, 305 (1993). [20] K. I. Gus’kov, Zh. Eksp. Teor. Fiz. 107, 704 (1995), [JETP. 80, 400-414 (1995)]. 12V m nmn para ortho FIG. 1. Schematic of the ortho and para states of the CH 3F molecule. It is assumed that only one pair of ortho and para states ( m−n) is mixed by an intramolecular perturbation ˆV. The bent lines indicate transitions induced by collisions with gas p articles. These collisions do not produce direct transitions between molecular ortho and para states . 13Czx yF 1 3 2 FIG. 2. Orientation of the molecular system of coordinates. It has i ts origin in the molecular centre of mass and is oriented by the numbered hydrogen nucle i in such a way that the xyplane is parallel to the hydrogen plane, the xaxis is directed to the H1atom and the yaxis is between the H1and H2atoms. The zaxis is directed along the molecular symmetry axis. 14-40-20 02040c b a Level splitting (kHz) FIG. 3. Splitting of the state J=9,K=3,M=9 of13CH3F: (a) no hyperfine perturbation; (b) splitting under the spin-rotation perturbation; (c) sp litting by joint action of the spin-rotation and spin-spin perturbations. Note that three components ne ar 19 kHz are accidentally nearly degenerate. 1505001000 1500 2000 25000.000.020.040.060.080.10Conversion rate (s -1) Electrical field (V/cm) FIG. 4. Overview of the level-crossing resonances in the conversio n (conversion spec- tra) of13CH3F nuclear spin isomers. The decoherence rate in this example was taken equal Γ = 1.75·107s−1(gas pressure 0.1 Torr). 16640.5641.0641.5642.0642.5643.00.00.20.40.60.8 0.1 mTorrConversion rate (s -1) Electrical field (V/cm) FIG. 5. Hyperfine structure of the “strong resonance” resulting fro m the crossing of magnetic sublevels M′=11 and M=9. The gas pressure is 0.1 mTorr. 17562.4 562.6 562.8 563.00.0000.0010.0020.003 0.1 mTorrConversion rate (s -1) Electrical field (V/cm) FIG. 6. Hyperfine structure of the “weak resonance” resulting from t he crossing of magnetic sublevels M′=7 and M=9. The gas pressure is 0.1 mTorr. 180.1 1 10 1000.1110 (b)(a)Conversion rate (s-1) Pressure (mTorr) FIG. 7. Pressure dependence of the amplitude of the “strong resonan ce” (9,3,9)–(11,1,11) (a) without and (b) with accounting of the hyperfine splitting of molecular states. 19

arXiv:physics/0004039v1 [physics.bio-ph] 19 Apr 2000Quasispecies evolution on a fitness landscape with a fluctuat ing peak Martin Nilsson Institute of Theoretical Physics, Chalmers University of T echnology and G¨ oteborg University, S-412 96 G¨ oteborg, Sw eden martin@fy.chalmers.se Nigel Snoad Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501 USA The Australian National University, ACT 0200, Australia nigel@santafe.edu (February 2, 2008) A quasispecies evolving on a fitness landscape with a single p eak of fluctuating height is studied. In the approximation that back mutations can be ignored, the ra te equations can be solved analytically. It is shown that the error threshold on this class of dynamic l andscapes is defined by the time average of the selection pressure. In the case of a periodically fluct uating fitness peak we also study the phase-shift and response amplitude of the previously docum ented low-pass filter effect. The special case of a small harmonic fluctuation is treated analytically . I. INTRODUCTION Ever since Eigen’s work on replicating molecules in 1971 [1], the quasispecies concept has proven to be a very fruitful way of modeling the fundamental behavior of evolution. A quasispecies is an equilibrium distribu- tion of closely related gene sequences, localized around one or a few sequences of high fitness. The kinetics of these simple systems has been studied in great detail, and the formulation has allowed many of the techniques of statistical physics to be applied to replicator and evo- lutionary systems, see for instance [1–8]. The appearance in these models of an error-threshold (or error-catastrophe) as an upper bound on the muta- tion rate above which no effective selection can occur, has important implications for biological systems. In partic- ular it places limits on the maintainable amounts of ge- netic information [1,2] which puts restrictions on theorie s for the origins of life. Until now studies of quasispecies have mainly focused on static fitness landscapes. However, many organisms in nature live in a quickly changing environment. In this pa- per we will study how a population responds to changes in the fitness landscape. More precisely we will study the population dynamics on a fluctuating single peaked fitness landscape. Since the full theory turns out to be impossible to solve analytically, we introduce a simple ap- proximation that makes the rate equations analytically tractable. The expression for the error threshold is then obtained from the expression in the static case by replac- ing the height of the fitness peak by the time average of the height of the fluctuating peak. We also study how the phase-shift between fitness oscillations and popula- tion dynamics depends on the frequency in the case of a small harmonic fluctuation.II. QUASISPECIES IN DYNAMIC ENVIRONMENTS A quasispecies consists of a population of self- replicating genomes, where each genome is represented by a sequence of bases sk, (s1s2· · ·sν). We assume that the bases are binary, sk∈ {1,0}and that all sequences have equal length ν. Every genome is then given by a binary string (011001 · · ·), which also can be represented by an integer k=/summationtext jsj2j(0≤k <2ν). To describe how mutations affect a population we de- fineWl kas the probability that replication of genome l gives genome kas offspring. We only consider point mu- tations, which conserve the genome length ν. We assume that the point mutation rate µ= 1−q (where qis the copying accuracy per base) is constant in time and independent of the position in the genome. We can then write down an explicit expression for Wl kin terms of the copying fidelity: Wl k=µhklqν−hkl=qν/parenleftbigg1−q q/parenrightbigghkl (1) where hklis the Hamming distance between genomes k andl. The Hamming distance hklis defined as the num- ber of positions where the genomes kandldiffer. The equations that describe the dynamics of the pop- ulation take a relatively simple form. Let xkdenote the relative concentration and Ak(t) the time-dependent fit- ness of genome k. We then obtain the following rate equations: ˙xk(t) =/summationdisplay lWl kAl(t)xl(t)−e(t)xk(t) (2) where e(t) =/summationtext lAl(t)xl(t), and the dot denotes a time derivative. The second term ensures the total normal- ization of the population (/summationtext lxl(t) = 1) so that xk(t) describe relative concentrations. 1In the classical theory introduced by Eigen and cowork- ers [1,9,10], the fitness landscape is static. The rate equa- tions (2) can then be solved analytically by introducing a change of coordinates that makes them linear, and then solving the eigenvalue system for the matrix Wl kAl. The equilibrium distribution is given by the eigenvector cor- responding to the largest eigenvalue. If the fitness landscape is time-dependent, this method cannot be applied. A time-ordering problem occurs when we define exponentials of time-dependent matrices, since in general the matrix Wl kAl(t) does not commute with itself at different points in time. Later in this paper we make a simple approximation that makes the rate equa- tions one-dimensional; time-ordering is then no longer necessary. Much of the work on quasispecies has focused on fitness landscapes with one gene sequence (the master sequence) with superior fitness, σ, compared to all other sequences. These are viewed as a background with fitness 1. These landscapes are referred to as single peaked landscapes. The master sequence is denoted x0. In this paper we fo- cus on single peaked landscapes where the height of the fitness peak is time-dependent. The fitness landscape is then given by Ak(t) =/braceleftbigg σ(t) ifk= 0 1 otherwise(3) This class of time-dependent landscapes was studied by Wilke and co-workers [11,12]. They investigated the be- havior of a periodically fluctuating single peak landscape by numerically integrating the dynamics to find the limit cycle of the concentrations for a full period. Fig. 1 shows how the concentration of the master se- quence responds to a sudden, sharp jump in its fitness. When the fitness changes it takes some time for the pop- ulation to reach the new equilibrium. It is this delay that causes a phase shift between a periodically changing fitness function and the response in the concentrations. The relaxation time of the population to the appropri- ate equilibrium distribution depends on both the fitness values of the landscape and the mutation rate. For ex- tremely slow and smooth changes in the fitness the popu- lation will effectively reach equilibrium at every point in time. Thus the continued existence of a quasispecies will depend on the local dynamics of the landscape. When the landscape changes quickly the population will fail to follow the changes adequately and thus responds to the landscape dynamics in a way that is typical of a low pass filter. The following section examines the fluctuat- ing single peak landscape in some detail. In particular, we introduce an approximation that lets us find an ana- lytic form for the relaxation time of the population, and the phase lag it introduces in a periodic landscape.10 11 12 13 14 15t0.630.640.650.66xmas FIG. 1. The concentration of the master sequence when the fitness peak makes a sudden jump. The fitness is given byσ= 10 when t∈[10,12],σ= 5 otherwise. The genomic copying fidelity is given by Q= 0.7. III. APPROXIMATE QUASISPECIES DYNAMICS We now introduce a simple approximation of the model presented above. In this approximation we can solve the rate equations and find an expression for the concentra- tion of the master sequence x0(t). In the limit of long chain-length ( ν≫1) we can neglect back-mutations from the background to the master sequence. This gives a sim- plified one-dimensional version of the rate equation of the following form: ˙x0(t) =Qσ(t)x0(t)−e(t)x0(t) (4) where Q=qνis the copying fidelity of the whole genome ande(t) = (σ(t)−1)x0(t) + 1. Fig. 2 compares the concentration of the master se- quence calculated by solving approximation 4 and by nu- merically integrating the full rate equation 2. The figure shows that the approximation is quite accurate. 7.5 8 8.5 9 9.5 10t 0.6450.6550.660.6650.670.675x0 FIG. 2. The concentration of the master sequence calcu- lated using the approximation 4 (dashed) and by numerically solving the rate equations 2 (solid). The fitness is given by σ= 10 + 5sin(4 t). The genomic copying fidelity is given by Q= 0.7 and the genome length ν= 25. Since this equation is one-dimensional there is no time- ordering problem and it can be solved analytically for 2non-periodic peak fluctuations. Equation 4 can be trans- formed to a linear form by introducing a new variable y(t) =Q−x0(t) (1−Q)x0(t). This gives ˙y(t) = 1 −(Qσ(t)−1)y(t) (5) which can be solved. Substituting back gives the concen- tration of the master sequence x0(t) =x0e/integraltextt 0(Qσ(s)−1)ds 1 +x0/integraltextt 0e/integraltexts 0(Qσ(u)−1)du(σ(s)−1)ds(6) Since we are only interested in the long time behavior of the system we can ignore transients carrying memory from initial values. Assuming e/integraltextt 0(Qσ(s)−1)ds≫1 gives x0(t) =Q 1 + (1 −Q)/integraltextt 0e−/integraltextt s(Qσ(u)−1)duds(7) This is a generalization of the static expression for the asymptotic concentration: xs 0=Qσ−1 σ−1(8) On a static single peaked fitness landscape there is a phase transition in the concentration distribution when the copying fidelity decreases below a critical value [2,13] . At high mutation rate the selective advantage of the master sequence due to its superior fitness is no longer strong enough for the gene sequences to be localized in sequence space. Instead they diffuse over the entire sequence space, and the distribution becomes approxi- mately uniform. This is generally referred to as the error catastrophe or error threshold and is one of the main im- plications of the original quasispecies model. By making the same approximation as above, i.e. assuming no back- mutations onto the master sequence, the static landscape error threshold can be shown to occur when Q=1 σ. In other words, the transition occurs when the selective ad- vantage of the master-sequence no longer is able to com- pensate for the loss of offspring due to mutations. This can also be seen from Eq. 8 which defines the stationary distribution of the master sequence in the static case. One has to be careful when discussing the error thresh- old on a fluctuating peak. The fitness can, for example, slowly move from being strong enough to localize the population around the peak, to beibg so weak that the population delocalizes, and then back again. If we how- ever consider an average over a time scale much longer than the fluctuation time of the fitness peak, a sensible definition of the error threshold can be made based on the average concentration of the master sequence. The time average of the concentrations can be found by rewriting equation 4 as differentials /integraldisplaydx0 x0=/integraldisplay (Qσ(t)−1−(σ(t)−1)x0(t))dt (9)The concentration of the master-sequence is positive. The left hand side of Eq. 9 is therefore positive and the last term in the integral, −(σ(t)−1)x0(t), on the left hand side is negative. This implies that for x0(t) to be positive as time goes to infinity, we must assume/integraltext (Qσ(t)−1)dt > 0. The fluctuating time dependent equivalent to the static error threshold is therefore given by Qfc=1 /an}bracketle{tσ/an}bracketri}htt(10) This shows that the error threshold on a fluctuating fit- ness peak is determined by the time average of the fitness, if the fluctuations are fast compared to the response time of the population. Eq. 7 indicates that the response time of the system is approximately given by ( Qσ(t)−1)−1, i.e. the rela- tive growth of the mastersequence compared to the back- ground. For the time average mentioned above to be an interesting parameter the fluctuations of the fitness peak must therefore be faster than this response time; only for this kind of environmental dynamics is it sensible to talk in terms of the average concentration of the master- sequence. Thus if the fluctuations occur on a time-scale faster than the response-time of the quasispecies, then the error-threshold is defined by Eq. 10. For extremely slow changes the system will effectively be in equilibrium around the current value of the fitness. For slightly faster changes the response of the population will lag somewhat behind the changes in selective environment. In these cases it is more interesting to study the minimal concen- tration of the master sequence, which occurs when the fitness peak has a minimum (as we shall see later the phase-shift decreases when the fluctuation frequency de- creases). When the full replicator equations for a rapidly fluctu- ating peak are numerically integrated, the time-averaged quasispecies distribution displays an error catastrophe a t high error rates µ= 1−q. In figure 3 the fitness peak fluctuates periodically with σ(t) = 10 + 5 sin( t). The av- erage fitness is given by /an}bracketle{tσ/an}bracketri}htt= 10 and the genome length ν= 25 and thus Eq. 10 predicts the error-threshold to occur at µ= 0.088, which agrees with the value found by numerically integrating the equations of motion di- rectly. The analysis in this section demonstrates that by making the error tail approximation and reducing the dynamics to one-dimensional form, an analytic form ex- ists for the error-threshold on fast moving landscapes. This one-dimensional formulation removes the need to time-order the changes in selective advantage of the land- scape. This allows the integrals for the time history of the master-sequence concentration to be solved explicitly in equation 6. 30.025 0.05 0.075 0.1 0.125 0.15 0.1 0.2 0.3 0.4 0.5 k = 0 k = 1 k = 12,13 mutation rate µxs k FIG. 3. The time-averaged quasispecies distribution is shown as a function of the error rate µ=q−1. The figure shows the numerical solution to the full rate equations. The fitness peak is defined as σ(t) = 10 + 5 sin( t) and the genome length is ν= 25. The error threshold is located at µ≈0.085, corresponding to Qfc= 0.109 which can be compared to the approximate value Qfc= 0.1 predicted by Eq. 10. IV. PHASE-SHIFTS ON PERIODIC LANDSCAPES To study how the master sequence responds to changes in the height of the fitness peak it is convenient to assume that the fluctuations are periodic. It then makes sense to speak of the amplitude of the oscillations in concen- tration and of the phase-shift between the concentration and the fitness. It is intuitively clear that when the fit- ness peak is oscillating slowly (compared to the response time ( Qσ(t)−1)−1) there will be a very small phase-shift; the population will have time to reach an equilibrium about every value of σ(t). The amplitude of changes in the master-sequence concentration will, for the same rea- son, be as large as possible. This result, together with the time-averaging effect found in the previous section, indicates that the population responds to the driving of the environment with a low pass filter effect. In one- dimensional population genetic models this phenomenon has been noted for some time [14–16]. Wilke et al. [11] demonstrated via simulations that the same filtering oc- curred to quasispecies evolution on a periodically fluctu- ating single peak. Noting that the maxima and minima in concentration occurs when ˙ x0= 0, we can find a rela- tion between the phase-shift (between the concentration and fitness fluctuations), and the amplitude of the fitness fluctuations. Let txmaxbe the time when the concentra- tion has a maximum. Similarly the fitness is at a max- imum at time tσmax. Thus the phase-shift between the two is δ=txmax−tσmax. From equation 4 the condition for the maximum value of x0during a full cycle can be derived max t(x0(t)) =Qσ(tσmax+δ)−1 σ(tσmax+tδ)−1(11)0.5 1 1.5 2 2.5 3 0.665 0.666 0.667 0.668 0.669 x0 time (t) FIG. 4. Thhe response in concentration of the master se- quence (solid line) as the fitness peak oscillates according to σ(t) = 10 + sin(4 t). The genomic copying fidelity is Q= 0.7. The dashed line shows σ(t), scaled to fit in the plot. Note the phase-shift between the fitness function and the concentrat ion response. In general there is no closed analytic expression for this phase-shift ( δ), or the response amplitude of the master- sequence concentration. When the fluctuations of the fitness peak is a small harmonic oscillation equation 11 becomes analytically tractable. For such fluctuations ˙y(t) = 1 −(Qσ(t)−1)y(t) (12) σ(t) = ¯σ+ǫsin(ωt) (13) From equation 7 it is reasonable to assume the solution to be of the form y(t) =1 Q¯σ−1+u(t), where u(t) is small compared to the average. Ignoring higher order terms equation 4 can be written in terms of the perturbation u(t) as ˙u(t) = (1 −Q¯σ)u(t)−ǫQsin(ωt) Q¯σ−1(14) This differential equation can be solved to obtain u(t) =−ǫQ (Q¯σ−1)/radicalbig (Q¯σ−1)2+ω2sin(ωt−δ) (15) tan(δ) =ω Q¯σ−1(16) In eq. 15 and 16 transients have been ignored since they decay exponentially as e−(Q¯σ−1)t. Thus the frequency of the oscillations is normalized by the (average) response rate of the population Q¯σ−1. 0.5 1 510 50T0.020.050.10.20.51d 4FIG. 5. The phase shift as a function of the period T=2π ω. The dashed line is a prediction using Eq. 16 and the solid is derived by numerically solving the reate equations 2. Param - eters used are σ= 10 + sin( ωt),Q= 0.7 and ν= 25. Substituting this back into the expression for x0(t) gives x0(t) =¯x 1−ǫ(1−Q) sin(ωt−δ) (¯σ−1)√ (Q¯σ−1)2+ω2(17) where ¯ x=Q¯σ−1 ¯σ−1. The characteristic behavior of a low pass filter is clearly shown in equation 16 and 17. As the frequency of the fluctuations increases, the amplitude of the concentra- tion response decreases and the phase shift converges to π 2. Figure 4 shows how a population responds to har- monic oscillations of the fitness peak. The phase-shift makes the concentration of the mastersequence reach its maximum when the actual fitness has already decreased below maximum.V. CONCLUSIONS In this paper we have shown that the time dynam- ics of a quasispecies on a fluctuating peak can be stud- ied under the standard no back-mutation approximation. The general time ordering problem stemming from a time dependent landscape disappears since the rate equation becomes one–dimensional. We show that the time depen- dent equivalent to the static error threshold is determined by the time average of the fluctuations of the fitness peak. An expression for the typical response time for a popu- lation is given in terms of copying fidelity and selection pressure. We also show that for small periodic fluctua- tions the time dynamics of the population has a phase shift and a low pass filter amplitude response. Analytic expressions for the phase shift and the amplitude are de- rived in the special case of small harmonically oscillating fluctuations. When doing this work Nigel Snoad and Martin Nilsson were supported by SFI core funding grants. N.S. would also like to acknowledge the support of Mats Nordahl at Chalmers University of Technology while preparing this manuscript. We also Mats Nordahl for valuable com- ments and discussions. [1] M. 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Error thresholds of replicatio n in finite populations: Mutation frequencies and the onset of M¨ uller’s ratchet. J. theor. Biol. , 137:375–395, 1989. [9] M. Eigen and P. Schuster. The hypercycle – a principle of natural selforganization . Springer, Berlin, 1979. [10] M. Eigen, J. McCaskill, and P. Schuster. The molecular quasispecies. Adv. Chem. Phys. , 75:149–263, 1989. [11] C.O. Wilke, C. Ronnewinkel, and T. Martinetz. Molecu- 5lar evolution in time dependent environments. In H. Lund and R. Kortmann, editors, Proc. ECAL’99 , Lecture Notes in Computer Science, page 417, Heidelberg, 1999. Springer-Verlag. LANL e-print archive: physics/9904028. [12] C.O. Wilke, C. Ronnewinkel, and T. Martinetz. Dynamic fitness landscapes in the quasispecies model. LANL e- print archive: physics/9912012, December 1999. [13] J. Maynard Smith and E. Szathm´ ary. The Major Tran- sitions in Evolution . W.H. Freeman, Oxford, 1995. [14] K. Ishii, H. Matsuda, Y. Iwasa, and A. Saskai. Evolu- tionarily stable mutation rate in a periodically changing environment. Genetics , 121:163–174, January 1989. [15] B. Charlesworth. The evolution of sex and recombinatio n in a varying environment. J. Hered. , 84:345–350, 1993. [16] R. Lande and S. Shannon. The role of genetic variation in adaptation and population persistence in a changing environment. Evolution , 50:434–437, 1996. 6

arXiv:physics/0004040v1 [physics.bio-ph] 19 Apr 2000The Physical Origin of Intrinsic Bends in Double Helical DNA Alexey K. Mazur Laboratoire de Biochimie Th´ eorique, CNRS UPR9080 Institut de Biologie Physico-Chimique 13, rue Pierre et Marie Curie, Paris,75005, France. FAX:+33[0]1.58.41.50.26. Email: alexey@ibpc.fr (February 2, 2008) The macroscopic curvature induced in the double helical B-DNA by regularly repeated adenine tracts (A-tracts) is a long known, but still unexplained phenomenon. This effect plays a key role in DNA studies because it is unique in the amount and the variety of the available experimental infor- mation and, therefore, is likely to serve as a gate to the un- known general mechanisms of recognition and regulation of genome sequences. We report the results of molecular dy- namics simulations of a 25-mer B-DNA fragment with a se- quence including three A-tract phased with the helical scre w. It represents the first model system where properly directed static curvature emerges spontaneously in conditions excl ud- ing any initial bias except the base pair sequence. The effect has been reproduced in three independent MD trajectories of 10-20 ns with important qualitative details suggesting tha t the final bent state is a strong attractor of trajectories for m a broad domain of the conformational space. The ensemble of curved conformations, however, reveals significant micr o- scopic heterogeneity in contradiction to all existing theo retical models of bending. Analysis of these unexpected observatio ns leads to a new, significantly different hypothesis of the poss i- ble mechanism of intrinsic bends in the double helical DNA. INTRODUCTION The intrinsic sequence dependent curvature of the DNA molecule is likely to be involved in fundamen- tal mechanisms of genome regulation. The possibility of strong static bends in the B-DNA double helix has been proven for sequences containing regular repeats of AnTm,with n+m >3, called A-tracts1. This effect plays an important role in DNA studies because it is unique in the amount and the variety of the available experimental information and, possibly, it can serve as a gate to the un- known general mechanisms of recognition and regulation of genome sequences. Every A-tract deviates the helical axis in a locally fixed direction by approximately 18◦, and, if the A-tracts are repeated in phase with the heli- cal screw, a macroscopic curvature emerges. The effect was first noticed and identified in restriction fragments from the kinetoplast body of Leishmania tarentolae2,3, and confirmed by electric birefringence decay4and elec- tron microscopy5. A large variety of interesting infor- mation has been obtained by biochemical methods. It appeared that the double helix bends towards the minor grooves of A-tracts6,7. The curvature is reduced with thetemperature above 40◦and in high salt, but for some se- quences it is increased in presence of divalent metal ions8. It depends upon the length and composition of A-tracts as well as on sequences between them6. Detailed analy- sis of these results can be found in comprehensive reviews published in different years1,9,10,11,12,13. According to many independent experimental obser- vations, the structure of A-tract sequences should differ significantly from the “random” B-DNA. It is well es- tablished that, in solution, the poly-dA double helix is overwound to a twist of around 36◦from around 34◦ of a random sequence14,15,16. The models constructed from fiber diffraction data suggest consistently that the poly-dA double helix is characterized by a very narrow minor groove and a high propeller twist17,18,19. Yet an- other distinction is an apparently large negative inclina- tion of base pairs18. Several A-tracts avaliable in sin- gle crystal structures of B-DNA oligomers have irreg- ular conformations, but exhibit similar trends toward their centers20,21,22,23,24. Even though the curvature is apparently caused by A-tracts, in Xray structures, A- tracts look generally less prone to bending than other sequences. Some indirect observations also support this view, notably, poly-dA fragments move faster than ran- dom DNA in gel migration assays6and avoid wrapping around nucleosome particles25,26. In spite of a large body of the experimental infor- mation accumulated during the last twenty years, the possible physical mechanism of this effect remains un- clear. Since every base pair in a stack interacts only with the two neighbors, any sequence specificity in the DNA structure should mainly depend upon the stacking interactions in one base pair step. Non-local effects are also possible, however, due to base-backbone interactions and propagation of correlations along the backbone. The initial experimental data on A-tract bending were inter- preted in terms of two alternative mechanisms, namely, the wedge model27and the junction model28. Both had to be modified significantly as and when new experimen- tal data appeared and some other theories were discussed as well. The possible mechanisms of bending considered in the literature will be discussed below. Here we note only that none of them explains all experimental data and can be definitely preferred1. The overall pattern has been additionally complicated when it was found that certain non A-tract sequences also exhibit distinguish- able curvature29. 1Here we report new results obtained in free unbiased molecular dynamics simulations of a DNA oligomer with phased A-tracts. We managed to find computational con- ditions in which stable sequence dependent static cur- vature emerges spontaneously in good agreement with experimental observations in terms of both the bending direction and magnitude. It is found that three inde- pendent long time trajectories converge to conformations with similar bends, but rather different local structural parameters, in evident contradiction with the common views of the origin of curvature. Analysis of these dis- crepancies leads to a new, significantly different hypoth- esis of the possible mechanism of intrinsic bends in the double helical DNA. Theoretical Background As a theoretical problem, the phenomenon of the se- quence dependent DNA bending presents a challenge in many aspects similar to the protein folding problem. In order to understand the underlying physical mechanism, one has to analyze terribly noisy experimental data, with the noise being due to the biological diversity and, there- fore, unremovable. If and when the physical mechanism is understood one will have to tackle this diversity again, because it will be necessary to analyze specific DNA se- quences. The atom level molecular modeling is virtually the only theoretical approach that is potentially able to treat these difficulties. Ideally, we would like to have a model where the base pair sequence represents the only initial bias towards a specific conformation. If it could reproducibly yield curved DNA conformations in agree- ment with experimental data we should be able disclose the mechanism of bending in the model, and hope that a similar mechanism takes place in the nature. The foregoing scheme, however, is too difficult and, un- til now, most of the modeling studies used other strate- gies. Much has been learned about the DNA bending mechanics by using energy minimization30,31,32,33and Monte Carlo34. Unfortunately, the possibilities of such studies are limited by the multiple minima problem es- pecially when it is necessary to take into consideration specific interaction with solvent molecules. The proposed alternative strategies commonly involved some bias to- wards specifically bent conformations introduced either explicitly, by imposing restraints, or implicitly, by choo s- ing particular initial conditions, which made impossible unequivocal conclusions concerning the possible mech- anism of bending. In the recent years, owing to the progress achieved in improving the full atom force fields, multi-nanosecond free MD simulations of DNA became feasible35,36. A few such studies of phased A-tract se- quences have been already reported37,38. It has been demonstrated that, without any a priori bias, the DNA double helix bends anisotropically, and certain sequence specific features of A-tracts were at least qualitatively re -produced. At present, the free MD simulations represent the most promising line of research in this field, and we continue it here by using the recently proposed minimal model of B-DNA39,40. The minimal B-DNA consists of a double helix with the minor groove filled with explicit water. Unlike the more widely used models, it does not involve explicit counterions and damps long range electrostatic interac- tions in a semi-empirical way by using distance scaling of the electrostatic constant and reduction of phosphate charges. We have earlier found that the minimal model gives B-DNA conformations which better compare with experimental structures than DNA structures obtained with other computational methods. Notably, it is free from a systematic negative bias of the average twist ob- served in simulations with full hydration and explicit counterions39. This factor is likely to be involved in DNA bending because, as noted above, A-tracts are over- wound with respect to the average B-DNA. For the stan- dard test case of the EcoRI dodecamer structure, the dy- namics of the minimal model reproducibly converged to structures slightly bent towards the minor groove which was narrowed in excellent agreement with the single crys- tal conformation39,40. All these preliminary observations suggested that the minimal model was a good choice for studying the DNA bending induced by A-tracts. We report here simulations of the bending dynam- ics of a 25-mer B-DNA fragment. Its sequence, AAAATAGGCTATTTTAGGCTATTTT, has been con- structed after many preliminary tests with shorter se- quence motives, and it includes three A-tracts separated by one helical turn. Our general strategy came out from the following considerations. Although the A-tract sequences that induce the strongest bends are known from experiments, probably not all of them would work in simulations. There are natural limitations, such as the precision of the model, and, in addition, the lim- ited duration of trajectories may be insufficient for some A-tracts to adopt their specific conformation. Also, there is little experimental evidence of static curvature in short DNA fragments, notably, one may well expect the specific A-tract structure to be unstable near the ends. That is why we did not simply take the strongest experimental “benders”, but looked for sequence mo- tives that in calculations readily adopt the character- istic local structure, with a narrow minor groove pro- file and high propeller twist, both in the middle and near the ends of the duplex. The complementary du- plex AAAATAGGCTATTTTAGGCTATTTT has been constructed by repeating and inverting one such motive. Our goal was to find sequences that would appear stat- ically bent in these conditions. It means that, in dy- namics, the structure should fluctuate around a state with a distinguishable bend and a definite bending direc- tion. Since any MD simulation is limited in time, there is no way to prove rigorously that some specific confor- mation is representative. Some degree of confidence can be achieved, however, if independent trajectories are able 2converge to the same state, and this is exactly what we tried to obtain. We found important to place A-tracts at both ends because of the two reasons. First, we could study only short DNA fragments, therefore, it was prefer- able to place A-tracts at both ends in order to maximize the possible bend. Second, in calculations with short se- quences, it is important to be sure that the boundary conditions are correct. Since the A-tracts have a char- acteristic local structure, the boundary conditions could be at least qualitatively verified, which would not be the case for GC-rich sequences. RESULTS Three long MD trajectories were computed for a complementary DNA duplex with the sequence AAAATAGGCTATTTTAGGCTATTTT. The model system employed was same in all three simulations, with only the starting states varied. The first trajectory re- ferred to below as TJBa started from the fiber canoni- cal B-DNA structure41and continued to 10 ns. When it was found that TJBa converged to a statically bent conformation, in good qualitative agreement with expec- tations based upon experimental data, another trajec- tory (TJBb) was computed in order to verify the repro- ducibility of the results. It started with random velocitie s from a re-minimized straight conformation taken from the initial phase of TJBa and was also continued to 10 ns. Simultaneously, in order to remove any initial bias implicitly involved in the choice of the starting state, the third trajectory (TJA) was obtained which started form the fiber canonical A-DNA conformation41. Initially, we computed 10 ns of TJA and found that it sampled con- formations rather dissimilar from those observed in TJBa and TJBb. A careful analysis revealed, however, certain slow structural trends and prompted us to continue TJA to 20 ns. The first two trajectories (TJBa and TJBb) have been the subject of our initial report42. Therefore, below we describe in detail only TJA and use the cor- responding data from TJBa and TJBb in comparisons. The structures referred to as the final MD states are the conformations averaged over the last nanosecond of the corresponding trajectory. The detailed computational protocols are described in Methods. All Three Trajectories Converge to Similar Structures within B-DNA Family Table I presents atom rmsd comparison between the final MD states of the tree trajectories and canonical A and B-forms of this 25-mer duplex. All computed struc- tures are clearly different from the standard A-DNA, even though for TJA it was the starting point. Moreover, the TJA final state appears to be the less similar of the three,TABLE I. Nonhydrogen Atom rmsd ( ˚A) between standard and computed structures. The upper and the lower triangles show results for all and for the middle 11 base pairs, respec- tively A-DNAaB-DNAaTJAbTJBabTJBbb A-DNAa0.0 8.62 9.18 8.62 7.98 B-DNAa5.84 0.0 3.13 2.85 3.28 TJAb6.15 1.96 0.0 2.90 2.92 TJBab5.41 1.35 1.44 0.0 1.79 TJBbb5.66 1.83 1.39 1.32 0.0 aFiber canonical DNA conformations constructed from the published atom coordinates41. bMD conformations averaged over the last nanosecond of the corresponding trajectory. 0 5 10 15 20 Time (ns)1234567RMSD (A) FIG. 1. Kinetics of the structural convergence for TJA. The time dependence of the nonhydrogen atom rmsd from the canonical B-DNA conformation (blue) and the final TJBa state (red) are shown. Both traces were smoothed by averag- ing with a window of 75 ps. demonstrating that the trajectories were not trapped ki- netically in the vicinities of their starting points. At the same time, the final MD states are evidently close to the canonical B-DNA. When only the central undecamer is considered, the rmsd values are in the same range as those observed in our earlier MD simulations of dode- camer duplexes39,40. They are low, and the three com- puted conformations seem to form a single cluster around the canonical B-DNA. The rmsd naturally increases with the helix length, but it should be noted that the some- what larger values obtained for the whole structures are much lower than ever observed in free MD simulations of long DNA helices37. It appears, however, that, if taken as a whole, the TJA state is as close to the canonical B- DNA as to the TJBa and TJBb states, while the latter two are yet closer to each other. The kinetics of the structural convergence in terms of atom rmsd is illustrated in Fig. 1 for TJA. It is seen that the trajectory rapidly went from the initial A-DNA con- formation towards the B-DNA form and, after the equi- libration, the rmsd from B-DNA have already lost a half of the initial 8.6 ˚A. Starting from the second nanosecond it fluctuated between 2 and 4 ˚A. The corresponding ki- netics for TJBa and TJBb were very similar except for the initial fall of the rmsd value42. The rmsd from the TJBa state also shown in Fig. 1 falls down to similar final values, but exhibits a somewhat different kinetics. Namely, an overall negative drift occurred during the first 3TABLE II. Some Structural Parameters of Standard and Comput ed DNA Conformations XdispaInclinaRiseaTwistaBendbBendingc angle direction A-DNAd-5.4 +19.1 2.6 32.7 0.0 – B-DNAd-0.7 -6.0 3.4 36.0 0.0 – TJAe-0.7—-0.3 +0.0—-1.2 3.5—3.5 34.3—34.8 35.8—30.4 -136.8—+ 110.4 TJBae-1.0—-0.3 -6.9—-4.5 3.6—3.5 34.1—34.8 13.2—29.5 22.3—42. 9 TJBbe-1.0—-0.7 -4.6—-1.8 3.5—3.5 34.3—34.2 18.4—28.4 35.6—53. 7 aSequence averaged values computed with program Curves43. bThe angle between the two terminal vectors of the optimal hel ical axis. cThe bending direction is characterized by the angle between the plane of the optimal helical axis and the xzplane of the local DNA coordinate frame constructed in the center of the duplex . According to the Cambridge convention44the local xdirection points to the major DNA groove along the short axis of the base -pair, while the local zaxis direction is adjacent to the optimal helicoidal axis. Thus, a zero angle between the two planes co rresponds to the overall bend to the minor groove exactly at t he central base pair. dFiber canonical DNA conformations constructed from the pub lished atom coordinates41. eMD conformations averaged over one nanosecond intervals. T he two numbers correspond to the first and the last nanosecond , respectively. ten nanoseconds followed by random fluctuations during the second half of the trajectory. One may say, therefore, that TJA first quickly traveled from A-DNA towards the B-DNA family and next slowly refined its position within this family coming closer to other computed structures. This refinement was not complete, however, since, ac- cording to Table I, the final TJA-TJB difference is still larger than that between TJBa and TJBb. Figure 1 sug- gests that a more accurate convergence, if possible, would require much longer time. Table II compares a few representative structural pa- rameters of MD conformations with the corresponding standards. Already after the first nanosecond even TJA gave the helicoidals corresponding to the B-DNA fam- ily, and they exhibited no systematic change afterwards. All three final MD states have an overall bend of around 30◦. The bending direction is somewhat different be- tween TJA and TJB, which is the main cause of the corresponding residual difference in terms of atom rmsd. Table II indicates that in all three trajectories both the magnitude and the direction of the bends changed sig- nificantly, and that very large variations in the bending direction apparently occurred in TJA. Thus, the slow rmsd kinetics considered above appear to be largely due to the bending of the double helices whereas the contri- bution from the variations of the helical parameters looks minor. Figure 2 shows the three last nanosecond average struc- tures superimposed. They all are evidently curved, with the bends being nearly planar in each structure. In agree- ment with Table II, the TJA bending plane slightly de- viates from the other two. The bending planes intersect the minor groove in five points which alternate between the inside and the outside edges of the bend, and in each case the the three A-tracts appear at the inside edge. The tracts are approximately phased with the helical turn, but, since the lower one is inverted with respect to theFIG. 2. The final MD states of the three trajectories super- imposed with optimal helical axes shown by curved lines. Two perpendicular views are shown. The superimposed structure s were rotated to minimize the divergence of the projections o f the helical axes in the left hand view. Different nucleotides are coded by colors, namely, A - green, T - red, G - yellow, C - blue. 4other two, its 5’ end is phased with the 3’ ends of the other. The three inside intersection points are shifted within the A-tracts from their middle towards the 3’ end of the upper two and the 5’ end of the lower one. On the other hand, the minor grooves of the two AGGC tetraplets appear at the outside edge of the curved axis, and it is readily seen that the minor groove is widened here, especially at the upper tetraplet. Quasi-Regular Rotation of The Bending Plane in TJA The two surface plots in Fig. 3 exhibit the time evo- lution of the shape of the helical axis for TJA. It is seen that the molecule was strongly bent after the initial equi- libration, which was not observed in case of TJBa and TJBb42. One should note that considerable initial de- formation of the double helix is common for trajectories starting from the A-DNA conformation. Apparently, the molecule is stressed because the transition to the B-form occurs in these conditions during unphysically short time with much energy released. During the next few nanosec- onds the bend reduced and the axis acquired a more com- plex shape with wound profiles in both projections. After the fifth nanosecond the bending became more planar, with much smaller curvature in Y projection. A pla- nar bent may just mean that the helical axis is kinked in a single point or, alternatively, a lager number of lo- cal bends are properly directed. Figure 3 indicates that there is probably a mixture of these two effects. During the last few nanoseconds the axis had one stable bend- ing point shifted upwards from the middle while another bend in the lower half emerged from time to time. The two bends were slightly misaligned, therefore, the over- all bending plane rotated a little when the second bend emerged, and, in the Y projection, one sees alternation of straight and S-shaped profiles. Figure 4 displays kinetics of several quantitative mea- sures of the magnitude of bending. The three parame- ters used, namely, the total angle, the shortening, and the average shift of the curved axis, all exhibit a coher- ent pattern of fluctuations, which locally correlates also with the rmsd from the canonical B-DNA (see Fig. 1). This indicates that they all are produced by the same motion, namely, the axis bending. Comparison of the data in Figs. 3 and 4 with similar plots earlier reported for TJBa and TJBb42reveals little difference except the already mentioned initial deformation and the absence of a stable bend between the two lower A-tracts. Ac- cordingly, TJBa and TJBb showed a somewhat stronger bending, with one-nanosecond average values usually be- yond 35◦. In TJA, after the initial strong temporary bending, a comparable magnitude has been reached only during the last four nanoseconds. There is, however, a striking difference between TJA and the other two trajectories in the dynamics of theAAAATAGGCTATTTTAGGCTATTTT 0ns4ns8ns12ns 16ns02.557.510 0ns4ns8ns12ns 16ns AAAATAGGCTATTTTAGGCTATTTT 0ns4ns8ns12ns 16ns-202 0ns4ns8ns12ns 16nsX Y FIG. 3. The time evolution of the overall shape of the op- timal helical axis in TJA. A best fit axis of coaxial cylindri- cal surfaces passing through sugar atoms was computed for conformations stored with a 2.5 ps interval. In all cases pre - sented here this axis was close to that produced by the Curves algorithm43. The curved DNA axis is rotated with the two ends fixed at the OZ axis to put the middle point at the OX axis. Note that this procedure does not keep the structures superimposed, that is the same curved axis can correspond to different bending directions. The axis is next character- ized by two perpendicular projections labeled X and Y. Any time section of the surfaces shown in the figure gives the axis projection averaged over a time window of 150 ps. The hor- izontal deviation is given in angstr¨ oms and, for clarity, i ts relative scale is two times increased with respect to the tru e DNA length. Shown on the right are the two perpendicular views of the last one-nanosecond-average conformation in t he orientation corresponding to that in the surface plots at th e end of the trajectory. 50 5 10 15 20 Time ns10203040500123456789012345678Angle Axshift Shortening(%) FIG. 4. The time evolution of the magnitude of bending in TJA. The bending angle is measured between the two ends of the helical axis. The shift parameter is the average deviati on of the helical axis from the straight line between its ends. The shortening is measured as the ratio of the lengths of the curved axis to its end-to-end distance minus one. bend direction which is exhibited in Fig. 5. Both in TJBa and TJBb the final bending direction occurred early in the trajectories and remained quite stable although the molecule sometimes straightened producing broad scat- tering of points in Fig. 5. In contrast, during the first ten nanoseconds of TJA, the bending plane made almost a half turn with respect to the coordinate system bound to the molecule. It means that a transition occurred be- tween the oppositely bent conformations, but, as seen in Fig. 3, the straight one was avoided. This rotation was very steady, almost regular. It gradually slowed down becoming indistinguishable in the last five nanoseconds. After this transition the directions of the bends in the three trajectories became much closer, and this quasi- regular motion is apparently responsible for the slow drift of the rmsd from the TJBa state in Fig. 1. The overall amplitude of this motion was around 150◦, that is the initial strong bend noticed in Fig. 3 was nearly opposite to that finally established. The Rotation of the Bending Plane is Not Energy Driven The overall character of motion revealed in Fig. 5, namely, the steady rotation of the bending plane, looks strange and counter-intuitive. A priori , we would rather expect to obtain random sampling of different bending di-0 2 4 6 8 10 Time ns-180-120-60060120180TJBb0 2 4 6 8 10 Time ns-180-120-60060120180TJBa0 5 10 15 20 Time ns-180-120-60060120180TJA FIG. 5. The time evolution of the bending direction in the tree trajectories. The bending direction is characterized by the angle (given in degrees) between the X-projection plane in Fig. 3 and the xzplane of the local DNA coordinate frame at the center of the duplex. It is constructed according to the Cambridge convention44, namely, the local xdirection points to the major DNA groove along the short axis of the base-pair, while the local zaxis direction is adjacent to the optimal helicoidal axis. Thus, a zero angle between the two planes corresponds to the overall bend to the minor groove exactly at the central base pair. 0 5 10 15 20 Time (ns)-5300-5275-5250-5225-5200-5175-5150Energy (kcal/mol) FIG. 6. The time dependence of the potential energy in three different trajectories. The color coding is: TJA – red, TJBa – green and TJBb – blue. 6rections, with the correct one statistically preferred due to its lower energy. The apparent quasi-regular dynam- ics exhibited in Fig. 4 might mean that our trajectory represents a downhill motion along a valley on a poten- tial energy surface. Its steep borders would separate bent conformations from the straight one, with the bottom of this valley slightly inclined towards the preferred bendin g direction. In this case all bent conformations, including incorrect bends, should have been lower in energy than the straight one. Figure 6 displays the time evolution of the potential energy in all three trajectories. It is seen that the energy dropped during the first nanosecond and later remained stable. No clear correlation is seen between the instanta- neous magnitude of bending displayed in Fig. 4 and the potential energy, therefore, one cannot say that straight states have significantly different energies than the bent ones. Neither can we claim that the preferred bending direction is characterized by lower energy values than other bends. In Fig. 6, a slight decrease in energy is observed during the second half of TJA, but it occurred when the regular rotation of the bending plane has essen- tially finished. On the other hand, the lowest energy dur- ing the first half of the trajectory was observed at around 3.2 ns when the bending direction was completely differ- ent. Note also that, during the first ten nanoseconds, the traces of TJBa and TJBb go above the last one, al- though in these cases the correct bending direction has already established. We have to conclude, therefore, that the simple energetic al explanation of the observed effect does not work. The Minor Groove Profiles of Converged Structures Are Similar But Not Identical The surface plot in Fig. 7 exhibits the evolution of the profile of the minor groove during TJA. The initial A- DNA conformation is characterized by a uniformly wide minor groove of 13.6 ˚A. It is seen that after the equilibra- tion period the groove was much narrower, but still wider than in the canonical B-DNA model. Moreover, a com- plex profile have emerged with three local widenings at A-tracts, which is exactly opposite to the expectations. The two terminal widenings reduced during the first ten nanoseconds whereas the maximum of the middle one gradually shifted from its 3’ end to 5’ end. This shift evidently accompanies the rotation of the bending plane described above. One can note that the maximal widening of the mi- nor groove moved for only 2-3 base pair steps, which is less than a 150◦rotation seen in Fig. 5. It appears that, in fact, the maximal widenings and narrowings in the minor groove profile do not always correspond to the direction of local bends. The initial bend was directed towards the minor groove of the upper TAGG tetraplet where the minor groove was narrowed. The two neigh- boring widenings are shifted by three base pairs only andAATAGGCTATTTTAGGCTATT0 4 8 12 16ns 468101214A0 4 8 12 16ns FIG. 7. The time evolution of the profile of the minor groove in TJA. The surface is formed by 150 ps time-averaged successive minor groove profiles, with that on the front face corresponding to the final DNA conformation. The groove width is evaluated by using space traces of C5’ atoms as de- scribed elsewhere45. Its value is given in angstr¨ oms and the corresponding canonical B-DNA level of 7.7 ˚A is marked by the straight dotted lines on the faces of the box. AAATAGCCTAAAATAGCCTATTT468101214 Groove width (A) AAATAGGCTATTTTAGGCTATTT FIG. 8. The profiles of the minor groove in the last nanosecond average conformations from TJA, TJBa and TJBb. The groove width is evaluated by using the C5’ traces as described elsewhere45. The dotted line marks the groove width of the canonical B-DNA structure41. The color coding is: TJA – red, TJBa – green and TJBb – blue. they appear at the opposite sides of the bending plane which is approximately collinear to the pseudodiad axis at the center of the middle ATT triplet. In contrast, in the last structure shown in Figs. 2 and 3 the bending plane passes exactly through the maximum widening of the minor groove. The overall rotation, therefore, corre- sponds to approximately four base pair steps which gives the observed turn by 150◦. It can be noted, finally, that although Fig. 5 indicates that the bending stabilized af- ter ten nanoseconds, the profile of the minor groove in Fig. 7 continues to evolve slowly till the very end of the trajectory. Figure 8 displays the minor groove profiles of the last average structures from the three trajectories. For TJBa and TJBb their kinetics was detailed in our first report42, and we only note here that the corresponding profiles shown in Fig. 8 established during the first two nanosec- onds and showed little variations afterwards42. The three traces evidently exhibit a certain similarity, but do not coincide. The TJBa and TJBb grooves have the same number of local narrowings and widenings which differ slightly between the two both in amplitude and in posi- 7tion. The TJA profile is similar in the right-hand half of the figure. One can notice that the change from TJBa to TJBb and next to TJA involves the growing widen- ing at the TTAG tetraplet accompanied by a shift of the secondary maximum, and looks rather regular and con- certed. At the opposite half of the structure, the TJA conformation shows a narrow minor groove without sig- nificant modulations of the width. This difference may be related with the smaller magnitude of the bending in the case of TJA where the second bending point appeared from time to time only and was less significant than in the other two trajectories. Key Helicoidal Parameters Exhibit Consistent Regular Patterns Only after Window Averaging Figure 9 shows variation of some helicoidal parameters along the duplex in the three structures. The two inter base pair parameters, namely, roll and tilt, are most of- ten quoted in the literature in relation to the static DNA curvature. If one first takes an ideal straight column of stacked parallel base pairs and next introduce a non-zero roll value at a certain step, the structure will bend at this step towards the major groove, if the roll is positive, and to the minor groove if it is negative. A similar experi- ment with the tilt value would result in bending in the perpendicular direction. It seems obvious that, whatever the physical origin of the curvature, in a bent double heli- cal DNA, the roll and tilt values must exhibit systematic variations phased with the helical turn. Moreover, it is often assumed that for some short DNA sequences cer- tain non-zero roll and tilt values are strongly preferred energetically, which produces static bending when they are repeated appropriately. However surprising, although all three average struc- tures are smoothly curved, only a few supporting signs for the foregoing paradigm are readily seen in Fig. 9. For the tilt, the three traces are very dissimilar and the only feature that repeats is the alteration of its values between consecutive steps. Namely, if the tilt is low at a given step it normally goes up at the next one, and vice versa. In the three average structures, however, these al- terations are sometimes oppositely phased even in TJBa and TJBb where the overall structures look particularly similar. The same is true for the roll and twist although, in these cases, some clear sequence preferences do exist. Note, for instance, that, in all four TpA steps, the roll is almost always positive and larger than in the neighbor- ing steps. Paradoxically, two of these TpA steps occur almost exactly at the inside edge of the curved axis, that is a high positive roll accompanies the bending in an ex- actly opposite direction. This paradox is readily resolved when one looks at the roll values at the neighboring steps. A TpA step with a high positive roll is normally preceded or followed by a step with a low negative roll. The higherAAAATAGCCTAAAATAGCCTATTTT-26-21-16-11-6-14910141822263034384246-11-9-7-5-3-113579-21-16-11-6-1491419Propeller Twist Tilt RollAAAATAGGCTATTTTAGGCTATTTT FIG. 9. Sequence variations of some helicoidal parameters in the final states of TJA, TJBa and TJBb. The sequence of the first strand is shown on the top in 5’ – 3’ direction. The complementary sequence of the second strand is written on the bottom in the opposite direction. All parameters were evaluated with the Curves program43and are given in degrees. The color coding is: TJA – red, TJBa – green and TJBb – blue. AAAATAGCCTAAAATAGCCTATTTT-12-10-8-6-4-20246810-27-22-17-12-7-232123252729313335373941-6-4-20246-15-10-50510Inclination Propeller Twist Tilt RollAAAATAGGCTATTTTAGGCTATTTT FIG. 10. Sequence variations of some helicoidal parame- ters in the final states of TJA, TJBa and TJBb, with window smoothing applied to the tilt, roll, twist, and propeller pa ram- eters. A sliding window of two base pair steps was applied to the data in Fig. 9, with the resulting average value assigned to the middle point. Notation as in Fig. 9. 8is the maximum, the lower is the neighboring minimum, so that the two nearly cancel each other. The other two TpA steps are found at the outside edge of the helical axis and their high roll probably contributes to bend- ing. However, while a more or less repetitive pattern is observed around the third TpA step, the first one ex- hibits rather dissimilar pictures even for TJBa and TJBb structures which both have a widened minor groove here. Also, the roll values at the third TpA step differ consid- erably between the structures, but do not correlate with the bending magnitudes. The twist, tilt, and roll values used for the plots in Fig. 9 are the so called “global” parameters from the outputs of the Curves program43. One may argue that they are not appropriate in the present context since they are computed by using local directions of already curved optimal helical axis. However, when “local” values are used instead, the amplitudes of the alternations in these profiles are only increased. The last plot in Fig. 9 exhibits the variations of the propeller twist. Again one sees that its value alternates between consecutive base pairs, with little phase similar- ity between the three structures. At the same time, in this case, a consistent repetitive pattern is evident, with strong negative propeller values in all A-tracts. These regular patterns look even more similar than the struc- tures themselves. For instance, there is no evident differ- ence between the three traces that would correspond to that in the minor grove profiles in Fig. 8. The apparent jumping alterations of the helicoidal pa- rameters along the double helix naturally suggests that one should try to smooth them out by averaging the traces in Fig. 9 with a sliding window. Figure 10 shows the results of such treatment and also includes the corre- sponding data for the inclination which was, however, used without the smoothing. The difference between Figs. 9 and 10 is rather significant. Now all four he- licoidal parameters considered in Fig. 9 exhibit regular, sometimes almost sinusoidal, oscillations. The phasing of these oscillations with the helical turn, however, is not always evident. The propeller and the inclination both exhibit approximately 2.5 periods, that is the dominating Fourier component has a wave length of approximately ten base pairs corresponding to one helical turn. For the roll, the dominating wave length apparently corresponds to 5-6 base pairs, that is a half of a helical turn. When different structures are compared, however, it is seen that only for propeller the maxima and minima coincide well. A more complex pattern is observed for the twist, and one can notice a correlation between its traces and the minor groove profiles shown in Fig. 8. Namely, the twist is lower in the widenings of the groove and higher in its narrowings. These results suggest that the relationship between the helicoidal parameters and the bending is complex and cannot be reduced to simple models of roll-like or tilt- like bends outlined above. Accumulation of the regular variations revealed in Fig. 10 probably gives the correctAAAATAGCCTAAAATAGCCTATTTT0 4 8 12 16 20Time (ns)AAAATAGGCTATTTTAGGCTATTTT -207.056140.563 FIG. 11. (a) AAAATAGCCTAAAATAGCCTATTTTTJBbTJBaTJAAAAATAGGCTATTTTAGGCTATTTT FIG. 11. (b) overall bend angles and directions, but neither can be easily predicted just by looking at these traces. The Distributions of B Iand B IIBackbone Conformers in Bent Structures are Surprisingly Dissimilar In all three trajectories, dynamics of B I↔BIIback- bone transitions was qualitatively similar in a few as- pects. Consider Fig. 11a, for instance, where the results are shown for TJA. The overall pattern reveals rather FIG. 11. Dynamics and final distributions of B Iand B II backbone conformers. The B Iand B IIconformations are dis- tinguished by the values of two consecutive backbone torsio ns, εandζ. In a transition they change concertedly from (t,g−) to (g−,t). The difference ζ−εis, therefore, positive in B Istate and negative in B II, and it is used in as a monitoring indicator, with the corresponding gray scale levels shown on the right in plate (a). Each base pair step is characterized by a column consisting of two sub-columns, with the left sub-columns re - ferring to the sequence written at the top in 5’-3’ direction from left to right. The right sub-columns refer to the com- plementary sequence shown at the bottom. (a) Dynamics of BI↔BIIbackbone transitions in TJA. (b) The distributions of BIand B IIconformations in the final states of the three trajectories 9slow dynamics, suggesting that MD trajectories in the 10 ns time scale are not long enough to sample all rele- vant conformations. A somewhat higher B I↔BIIactiv- ity was observed during the first half of the trajectory, when the rotation of the bending plane occurred. It is seen that, in A-tracts, the B IIconformers are preferably found in ApA steps and that they tend to alternate with BIwithin the same strand. There are many examples of concerted B I↔BIItransitions, when a given step switches from B IIto BIsimultaneously with an opposite transition in one of the neighboring steps. Sometimes three consecutive steps are involved and, less often, the opposite strand as well. Many B I↔BIItransitions are reversed within hundreds of picoseconds, but there are also very long-living conformers and sites where either BIor BIIstates are preferred. A strong preference of B I state is observed for all TpT steps, for example. How- ever, it seems to be the only case when the effect repeats at a base pair step level. In a few steps where the B II conformation is preferred this is apparently determined by a broader sequence context. The corresponding data for TJBa and TJBb were in- cluded in our first report and they revealed the same qualitative features42. It was very surprising for us, how- ever, that, in spite of the good convergence in terms of the overall bent shape of the molecule, the three trajec- tories gave rather dissimilar distributions of B Iand B II conformers along the sequence. Fig. 11b compares these distribution in the final backbone conformations in the three trajectories. There are 14 non TpT steps where the BIconformation is found in all three structures. How- ever, since our trajectories started from B Istates, this number hardly tells us something. On the other hand, the number of B IIconformers found in each structure and in each strand is similar and roughly corresponds to 25% of phosphate groups. Assuming that the B IIstates are evenly distributed in the sequence one gets the expecta- tion value of 0.75 for the number of cases when the B II conformer should be found in the same base pair step in all three structures. The observed number of such sites is three. Note, however, that they all are found in A-strands of A-tracts where, as noted above, the B IIcon- formers tend to alternate. This, together with the strong preference of TpT steps for B I, increases the probability of matching. These results suggest that the relationship between the bending of the DNA double helix and the B I↔BIIback- bone transitions, if any, is loose in the sense that a given bent shape does not impose a fixed B IIdistribution upon the backbone. DISCUSSION This study gives the first example of a successful im- plementation of the general strategy outlined in Theoret- ical Background. Namely, we showed that the minimalmodel of B-DNA, which is biased only by the nucleotide sequence, in dynamics, reproducibly converges to a single state characterized by an ensemble of similar statically bent conformations. The effect has been demonstrated here for one sequence only. Moreover, this sequence was specifically constructed rather than taken from ex- perimental studies. Nevertheless, the sequence motive AnTAG used in construction was found in the center of the first bent DNA fragment studied experimentally3. In addition, the character of bending in the computed conformations, notably, its direction with respect to the A-tracts, and modulations of the groove width, quali- tatively agree with the rules derived from experiments. These observations validate an attempt to make the next step of the above strategy, namely, below we try to dis- close the mechanism responsible for the bending within the framework of the minimal model. We believe that, in spite of the obvious limitations of this model, its main features responsible for the bending correspond to real- ity. At the same time, the real situation is certainly more complex. Results Poorly Agree with Earlier Theories of Bending All theories proposed during the last 20 years to ex- plain intrinsic bends in DNA double helices agree with some experimental observations and disagree with the other and, probably, each of them continues to attract some proponents. Here we compare our results with the most popular models of bending regardless of their ex- perimental validation. Comparisons with experimental data have been the subject of many reviews1,9,10,11,12,13. The wedge model of DNA bending27resulted from merging of ideas developed in seventies to explain the ability of a double helix to wrap around nucleosome par- ticles. The first idea was that this can occur due to kinks of the helical axis phased with the helical screw46,30, with kinks implying destacking of base pairs in fable points in order to maintain perfect stacking elsewhere. The second idea was that the double helix can be smoothly bent, without destacking, by small deformations in ev- ery base pair step47. The wedge model merges the two by postulating that, in every specific dinucleotide, the preferred stacking of bases is slightly non-parallel and this causes bending in the same way as kinks do. It can be further developed by increasing the number of wedge degrees of freedom, by considering triplets, tetraplets, and so forth instead of dinucleotides, and by assuming that the non-zero average wedges result from random sampling from asymmetrical energy valleys around lo- cal energy minima, rather than from minimum energy configurations48. Depending upon the specific wedge pa- rameters, this model can place the curvature inside A- tracts or between them48,49and also explain bending in non A-tract sequences29. For the present discussion, it is convenient to unite all such mechanisms in one group 10characterized by the tacit emphasis upon the specific base pair stacking preferences as the source of the DNA bend- ing. The results shown in Figs. 9 and 10 obviously dis- agree with these views. There is little similarity between matching dinucleotides in the same structure and, more- over, base pair steps put in the same sequence context in three closely similar bent conformations exhibit broadly different helical parameters. The last observation means that even a generalized wedge model with dinucleotide blocks replaced by triplets, tetraplets, and so forth, woul d disagree with our results. The junction model28postulates that there is a distinct specific A-tract form of the double helix characterized by a stronger inclination of base pairs with respect to the helical axis than in the normal B-DNA. In this case, pla- nar stacking at the junction between the two structures would result in a kink of the helical axis. Formally, such geometry can also be obtained with the generalized wedge model above, but the junction theory puts an emphasis upon the specific A-tract form of DNA as the principal physical cause of bending. Its structure can be due to co- operative interactions in long DNA stretches and its en- vironment. Within the framework of the junction model, particular roles were sometimes attributed to bifurcated hydrogen bonds17, the water spine in the minor groove32, or the NH 2groups of adenines31. It is evident that the junction model also poorly agrees with our results. In dynamics, conformations, both smoothly bent and kinked at the two insertions between the A-tracts, are observed periodically. The kinks, how- ever, are not centered at the boundaries between A-tracts and the flanking sequences, and they are not sharp. In such conformations, A-tracts are less bent than regions between them, that is the bend is localized but still smooth. In Fig. 10 the inclination shows smooth oscil- lations, even without window averaging, with no kinks. It decreases from 5’ to 3’ ends of A-tracts, and since the 3’ ends of A-tracts are dephased and positioned dif- ferently with respect to the bending plane, no evident relationship to bending can be readily seen. All helical parameters vary along the sequence so that there is no A-tract fragment where they repeat at two consecutive steps. Thus, although the structures are bent, the spe- cific regular A-tract structure is not seen, as well as the “random B-DNA”, though, which are the two key com- ponents of the junction model. The third model, which was first mentioned in the con- text of the junction theory50, but became popular only in the recent years51, attributes the cause of bending to solvent counterions. If they are specifically bound by mi- nor grooves of A-tracts, in a phased sequence, phosphate groups would appear partially neutralized at one side of the double helix, and the repulsion between the oppo- site phosphates would bend DNA towards minor grooves of A-tracts. The very fact that the minimal model of B-DNA, without explicit counterions, produces static bends, in good agreement with experiments, stronglysuggests that the counterions hardly play a key role in the A-tract induced bending. At the same time, our results do not contradict less specific non-local theories of A-tract bending. The mod- ified junction model52assumes that the deformations at the boundary between the two conformations can prop- agate for several base pair steps. The A-tracts in the sequence studied here may be too short for their inge- nious structure to establish. The second such theory53 proposed that the bending is caused by the modulations of the minor groove. Really, the double helix is usu- ally bent towards the major groove at the minor groove widenings, and in the opposite direction at its narrow- ings. In TJA, for instance, this relationship is maintained during the rotation of the bending plane. Sometimes, however, the double helix straightens and remains un- bent during nanoseconds, while the minor groove profile does not change42. It is understood, however, that the last two non-local models are incomplete. Actually, they cannot be veri- fied or disproved because the issue of the physical origin of bending is tacitly dropped. Simple geometrical con- siderations dictate that the grooves must be narrower at the inside edge of the bend54. One may postulate, therefore, that groove modulations cause bending or, vice versa, that it is bending that causes groove modulations, but the physical origin of the phenomenon remains ob- scure. Similarly, the modified junction model essentially discards the essence of the original theory, which consid- ers the specific poly-dA structure as the source of the bend. If the “boundary deformations” can exist with- out the structures and boundaries themselves one should look for another force that maintains these deformations. Generally speaking, the results presented here are best interpreted if one assumes that there is an external force that imposes a bent shape upon the double helix as a series of mechanical constraints. The double helix is al- lowed to move, but so that these constraints would re- main fulfilled. Thus, the bases can change their mu- tual orientation and the backbone can switch between BIand B IIconformations, but the overall proportion of the B IIconformers remains constant, and fluctuations of helical parameters in the neighboring base pair steps tend to compensate. Possible Physical Origin of Spontaneous Static Bends in Double Helices The hypothesis outlined below is based upon our com- putational results as well as upon analysis of well-known experimental data. Although it does not answer all un- clear questions concerning DNA bending we consider it most likely and describe it here for discussion and further investigation. Let us first ask the following simple geometric ques- tion: “What is the shortest line that joins two points on 11a surface of a cylinder?” To answer it, one should first cut the cylinder parallel to its axis, unfold its surface onto a plane, join the two points by a straight line and then fold the surface back upon the cylinder. The resultant curve represents an interval of a spiral trace with a constant inclination to the cylinder axis. Now consider an ideal canonical B-DNA model of a double helix. The stacked base pairs form the core of a cylinder and the sugar- phosphate backbone forms an ideal spiral trace on its surface, that is the shortest line that joins the “surface” nitrogens N 1/N9of the bases. If we now assume that the backbone is a stiff elastic that can be characterized by a certain specific length, we are obliged to conclude that this model implicitly assumes that the backbone is stretched and tends to reduce its length on the surface of this cylinder. Our last question is: “What would happen if the preferred backbone length appeared to be longer that allowed by the canonical model?” A simple answer is: it would try to extend by pushing bases. They can ac- commodate this extention within the framework of a reg- ular helical structure by increasing the helical twist and changing other helical parameters. This option, however, is opposed by the loss in the stacking energy and, when it becomes difficult to extend in this way, the backbone will try to deviate from the ideal spiral trace. In this case the the grooves on the surface of the Watson-Crick double helix can no longer maintain a constant width. It seems possible to assume that, in physiological con- ditions, the equilibrium specific length of the DNA back- bone is actually larger than that allowed by a regular B-DNA structure with the average helical twist of 34.5◦. Its further extention is opposed by the limit of the tol- erance of the pase pair stacking and, as a result, the backbone appears “frustrated” and is forced to wander on the cylindrical surface formed by base pair stack. The ideal parallel stacking has to be perturbed and we be- lieve that it is this effect that eventually bends the dou- ble helix. Modulations of the DNA grooves, which is a well-known ubiquitous feature of the single crystal B- DNA structures, is a natural indicator of this particular state of the backbone. It is observed for very different se- quences as an apparently general attribute of the B-DNA form in physiological conditions. Thus, if we had to de- cide whether the DNA backbone in stretched, relaxed or compressed by looking only at the single crystal B-DNA structures, we would be obliged to conclude that the first option looks unlikely, the second is possible, while the third is the most probable. A compressed backbone is more likely to cause smooth groove modulations found in experimental structures than a relaxed one, which would rather be controlled by the local sequence effects. Let us consider an ideal B-DNA model, with planar base pairs perpendicular to the helical axis, and try to imagine how wandering of the backbone traces can emerge. For simplicity, we first consider the helical twist as the only variable parameter. Obviously, by smoothly increasing and decreasing the twist we obtain, respec- tively, narrowings and widenings of the minor groove.The desired backbone waving emerges, and a larger its length can be accommodated on the same cylindrical surface. It is easy to see, however, that, if the paral- lel stacking is maintained, the backbone must be com- pressed when the twist is reduced and stretched in the opposite phase. In reality, however, the backbone is stiff and it cannot be compressed significantly, therefore, it is the stacking that suffers when the twist is reduced. Al- though this description is simplistic, and other base pair degrees of freedom in addition to the twist can contribute to the wandering, it captures the essence of the under- lying mechanics. In the widenings of the minor groove, where the twist is reduced, the backbone pushes off the neighboring base pairs at C1’ atoms, causing various per- turbations of the parallel stacking. On average, they are likely to deviate the helical axis towards the major groove because C1’ atoms are at the minor groove side. These perturbations are delocalized and involve rolling, tiltin g, as well as other relative motions of base pairs, and there is an ensemble of orientations that fulfill the constraints imposed by the backbone lengths, rather than a single preferred bent conformation. The modulations of the minor groove width and the bending of the double helix appear related, as was suggested earlier53, because they represent two consequences of a single cause. They are related as brothers rather than as a parent and a child and, probably, are not bound to always appear together. The major component of the backbone stiffness is the electrostatic repulsion between the charged phos- phate groups. Even though this repulsion is shielded by water and counterions, the experiments where bend- ing in B-DNA was induced by specific neutralization of phosphates55proved that they are not shielded even when separated by two helical turns. Complete neutral- ization, therefore, is hardly imaginable. The local elec- tric field around a pair of neighboring phosphates in the same strand is created by all surrounding charges, in- cluding the phosphates of the opposite strand, and it fa- vors maximal possible separation between P nand P n+1. In B-DNA, this distance is close to the absolute maxi- mum, which is achieved by putting all relevant backbone torsions except one in the trans configuration56. The maximal extention gives the ground energy state with the P n−Pn+1distance around 7.7 ˚A. The corresponding thermodynamic average for a free backbone in solution isD(T, ǫ)<7.7˚A, where Tis the temperature, and ǫis the effective dielectric constant that depends upon the concentration of counterions. In the single crystal structures, the largest P n−Pn+1distances observed are in the range of 7.3 – 7.6 ˚A suggesting that there are no prohibitive steric obstacles for a completely extended backbone. At the same time, the distances most com- monly found are around 6.7 ˚A while in the fiber canon- ical structure it is 6.5 ˚A. Apparently, with normal tem- perature in physiological conditions D(T, ǫ)>∼6.7, and the backbone is forced to wander. D(T, ǫ) should be a decreasing function of both arguments, therefore, the backbone stiffness and, accordingly, the curvature should 12decrease as the temperature grows and/or as the phos- phate shielding is improved by increasing the ionic force. These two non sequence-specific effects have been found in experiments8. With D(T, ǫ)≈6.5 the backbone re- laxes and the phenomenon of DNA bending vanishes. According to this hypothesis, the transition of A-tracts in a specific DNA form is not an indispensable prerequi- site of the bending. Moreover, it suggests that the regular poly-dA structure may not exist in solution because, with the average twist increased to 36◦, the backbone is, pos- sibly, still compressed and continues to wander, although with a longer characteristic wave lengths. The structures of short A-tracts computed here are rather variable and it is not clear how they can be extended to longer poly- dA double helices. We believe that the A-tracts rather label the regions where higher twist values are allowed by the base stacking interactions. The backbone prefers to compress the minor groove here, thus fixing the phase of its modulations along the double helix. In random and homopolymer sequences the minor groove probably also narrows and widens, but the corresponding maxima and minima can migrate along the double helix in a way similar to that observed here during the rotation of the bending plane in TJA. Our model considers the bending of a DNA double helix as a deformation imposed upon the stem of the stacked base pairs by interactions external to it. The bases are forced to “forget” their preferred stacking ori- entations and look for a possibility to maintain the over- all structure by sampling the orientations at the limit of destacking. At the same time, it is the broad “tolerance” of the base pair stacking that makes all this game pos- sible. If true, this theory gives a slightly different over- all view of the DNA molecule in physiological conditions and entails important biological consequences. Notably, it suggests that the local DNA structure is not simply determined by the stacking preferences of base pairs in dinucleotides, trinucleotide, and so forth. The two wav- ing backbone profiles on the surface of the helix impose a medium range context upon the local base pair stacking because the phases of these modulations can well corre- late over several DNA turns. This makes possible mutual dependence of local conformations in sites separated by considerable DNA stretches. Fine tuning of phases of these modulations may be the function of single short A- tracts as well as of some regulatory proteins. The degree of the backbone compression is connected with super- coiling and can be controlled in this way, which gives yet another possible instrument of structural regulation. One may note also that this theory offers a unified model which explains static bends in A-tract and non A-tract sequences as well as the bending induced by the negative supercoiling in circular DNA.Conclusions The static curvature spontaneously emerges in free MD simulations of an atom level molecular model of B-DNA double helix, with the nucleotide sequence as a single structural bias. Convergence to similar statically bent states have been demonstrated in three independent MD trajectories of 10-20 ns. The bending direction and its magnitude are in good agreement with experimental ob- servations. Unexpectedly, the three computed MD struc- tures exhibit a striking microscopic heterogeneity as re- gards variations of helical and conformational parameters along the molecule. Based upon the computational re- sults as well as the literature experimental data a new possible mechanism of bending in a double helical DNA is proposed. It postulates that, in physiological condi- tions, the equilibrium specific length of the DNA back- bone is larger than is admissible in the regular B-DNA form, which forces it to fold in a wavy trace on the cylin- drical surface of the double helix. This results in mod- ulations of the minor groove width, slight asymmetrical destacking of base pairs at the groove widenings and, eventually, in bending of the DNA molecule. Methods Molecular dynamics simulations have been performed with the internal coordinate method (ICMD)57,58includ- ing special technique for flexible sugar rings59. The so- called minimal B-DNA model was used39,40which con- sists of a double helix with the minor groove filled with explicit water. It does not involve explicit counterions and damps long range electrostatic interactions in a semi- empirical way by using linear distance scaling of the elec- trostatic constant and reduction of phosphate charges. The DNA model was same as in earlier reports,39,40 namely, all torsions were free as well as bond angles cen- tered at sugar atoms, while other bonds and angles were fixed, and the bases held rigid. Molecular dynamics cal- culations were carried out with a time step of 10 fsec and the effective inertia of planar sugar angles increased by 140 amu ·˚A2as explained elsewhere39. The coordinates were saved once in 2.5 ps. AMBER9435,60force field and atom parameters were used with TIP3P water61and no cut off schemes. The initial conformation for TJBa was prepared by vacuum energy minimization starting from the fiber B-DNA model constructed from the published atom coordinates41. The subsequent hydration protocol to fill up the minor groove39normally adds around 16 water molecules per base pair. The starting state for TJBb was obtained by energy minimizing an equilibrated straight structure taken from the initial phase of TJBa. The ini- tial conformation for TJA was prepared by hydrating the minor groove of the corresponding A-DNA model41 without the preliminary energy minimization. In TJA, 13the necessary number of water molecules was added after equilibration to make it equal to that in TJBa and TJBb. The heating and equilibration protocols were same as before39,40. During the runs, after every 200 ps, water positions were checked in order to identify those pene- trating into the major groove and those completely sepa- rated. These molecules, if found, were removed and next re-introduced in simulations by putting them with zero velocities at random positions around the hydrated du- plex, so that they could readily re-join the core system. This procedure assures stable conditions, notably, a con- stant number of molecules in the minor groove hydration cloud and the absence of water in the major groove, which is necessary for fast sampling40. 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Jorgensen, Transferable intermolecular potential functions for water, alcohols and ethers. application to li q- uid water., J. Am. Chem. Soc. 103, 335 (1981). APPENDIX This section contains comments from anonymous ref- erees of a peer-review journal where this and a closely related paper entitled “Molecular Dynamics Studies of Sequence-directed Curvature in Bending Locus of Try- panosome Kinetoplast DNA” has been considered for publication, but rejected. A. Journal of Molecular Biology 1. First referee These companion manuscripts describe a series of molecular dynamics trajectories obtained for DNA se- quences containing arrangements of oligo dA - oligo dT motifs implicated in intrinsic DNA bending. Unlike pre- vious MD studies of intrinsically bent DNA sequences, these calculations omit explicit consideration of the role 15of counterions. Because recent crystallographic studies of A-tract-like DNA sequences have attributed intrinsic bending to the localization of counterions in the minor groove, a detailed understanding of the underlying ba- sis of A-tract-dependent bending and its relationship to DNA-counterion interactions would be an important con- tribution. Although the MD calculations seem to have been car- ried out with close attention to detail, both manuscripts suffer from some troubling problems, specifically: The DNA sequence in question is a 25-bp deoxy- oligonucleotide that contains 3 A/T tracts. Two of these are arranged in phase with the helix screw with the third tract inverted with respect to the other two. Extrapo- lating from available experimental data, this sequence is expected to confer some degree of intrinsic bending. The main focus of this manuscript is the comparison of data obtained for an MD trajectory computed from an A-form starting conformation (TJA) with two other trajectories that begin with B-form structures (TJBa and TJBb). Significant differences in behavior and in time-averaged helical parameters are observed for the TJA trajectory compared with both TJBa and TJBb, suggesting that the structures are not fully equilibrated. This is partic- ularly evident in the computed bending direction, which varies dramatically during early times in the TJA trajec- tory. Even after 15 ns, when the orientations of bend- ing planes appear to have approached asymptotic values, the TJA plane is displaced by between 30 and 60 degrees from those of TJBa and TJBb, which are quite similar to one another. This fact strongly suggests that the MD- simulation results depend nontrivially on initial condi- tions, even after 15-20 ns, which calls into question most of the results obtained from the computed trajectories. 2. Second referee Dr. Mazur reports the results of MD simulations of DNA 25-mers sequences that contain three phased A-tracts. He believes that he has obtained the first model system in which properly directed static curvature emerges spontaneously in conditions excluding any ini- tial bias except for base pair sequence. He observes that the ensemble of curved conformations reveals significant microscopic heterogeneity, which he believes is in contra- diction to existing theoretical models of DNA bending. In CAM110/00 he performs a series of simulations on a DNA fragment that has not been shown experimentally to bend in solution. In this case the DNA sequence was chosen based on its propensity to adopt a characteris- tic structure during simulations. In CAM167/00 he per- forms a similar investigation on B-DNA fragment com- posed of a sequence that has been shown experimentally to bend. My view is these two papers should be combined as one, and the review will treat them as one. I found this paper to be interesting and possibly wor-thy of publication in JMB, even as I took issue with a substantial portion of it. The basic premise of the pa- per is that a model lacking realistic electrostatics can provide meaningful information about long range DNA conformation. In Mazur’s model, long range electrostatic interactions are dampened and phosphate charges are at- tenuated. I had some concerns about the basic rationale for the non-electrostatic model. I initially assumed that it’s greater simplicity would allow longer trajectories, etc. But the trajectories of Dr. Mazur are not substantially longer than those described by Beveridge, Pettitte, etc. And in fact that seems not to be the rationale. Dr. Mazur believes that full atom force fields, with explicit ions, give less realistic results than his electrostatic-attenu ated model. In particular he says that full atom force fields give slightly overwound DNA, which camouflages DNA bending. What is the cause of this? A problem in the force field? Why not fix that instead of going the non- electrostatic route? I am not comfortable enough with the world of MD pass judgment on this issue, but think someone who is should evaluate that prior to publication. I just went back and re-read Diekmann’s classic 1985 JMB paper [Diekmann, S., & Wang, J. C. ”On the se- quence determinants and flexibility of the kinetoplast DNA fragment with abnormal gel electrophoretic mobil- ities” (1985) J. Mol. Biol. 186, 1-11.] Diekmann shows clearly that the electrophoretic anomaly of kinetoplast DNA decreases with increasing Na, and increases very dramatically with increasing Mg. His experiments seem well-conceived , well-conducted and well-analyzed. For example he implants a temperature sensor within his gels to insure constant, fixed temperature. One has to be- lieve Diekmann’s results. My fundamental problem with Mazur’s model is that it cannot account for experimen- tal data. How can bending be cation-dependent, but the mechanism not be electrostatic in nature? Mazur does concede that experimentally ”curvature is reduced in high salt, but for some sequences it is in- creased in the presence of divalent metal ions” (cites Diekmann). [page 2 MS CODE CAM110/00]. But the implication here is that the observation of Diekmann is not general to all A-tracts. The next sentence of the manuscript may be read as confirming that the cation effect is not general, but is length and composition de- pendent (the text is a little confusing here). However the Woo and Crothers citation, used as support for that, does not discuss the cation effect. If there are data some- where suggesting that the cation effects are not general, they should be cited and discussed. That would really increase the strength of Mazur’s argument. If not, the text should be clarified. An additional issue that is not illuminated much here is the comparison of Mazur’s model with the results of x-ray crystallography. In crystals of oligonucleotides, A - tracts are straight (”less prone to bending than other se- quences” is rather understating it). To accommodate this observation, Dickerson (JMB 1994) proposed a model in 16which A-tract DNA curvature results from roll-bending of non-A-tracts, and linear A-tracts. Crothers (JMB 1994) is contemptuous of that model, and believes that the linear conformations of A-tracts observed thus far in crystals are not those associated with the curvature ’observed’ in gel mobility experiments. In fact such a discrepancy between dilute solution (where intramolec- ular forces would dominate) and condensed states (such as crystals, where intermolecular forces dominate) is ex- pected if long-range electrostatics play a key role in cur- vature. Those long range forces are turned off in Mazur’s model. (He does seem to allude to the crystallogra- phy/solution discrepancy on page 19). So again his model does not account for experimental data. As an aside: Mazur believes that groove narrowing and bending are coupled. How does one then explain the observation that A-tracts in crystals have narrow minor grooves, yet are not bent? Finally, some aspects of Mazur’s (combined) model seems to be inconsistent and self-contradictory. In his model (as I understand it), (1) electrostatic repulsion be- tween adjacent phosphate groups drives helical twisting, (2) A-tracts are regions where higher helical twist is facil - itated by lower stacking energies in comparison to those of G-C base pairs, (3) higher helical twist narrows the mi- nor groove, and (4) groove narrowing is somehow related to axial curvature (this is a little unclear; the descrip- tion ”as brothers rather than a parent and a child” did not enlighten me). This model has certain attractive fea- tures, [the idea that electrostatic repulsion between adja - cent phosphate groups drives helical twisting while stack- ing opposes it was previously presented by Alex Rich in 1992 in a chapter of Structure & Function, Volume I: Nucleic Acids pp. 107-125 (from a Sarma meeting)] but some deficiencies also. If electrostatic repulsion be- tween adjacent phosphate groups drives helical twisting, then how can correct values of helical twist be obtained with attenuated phosphate charges? Or restated: Does this model not ascribe electrostatic forces as the ultimate cause of static bends, contradicting the non-electrostati c assumption? And I just checked in one crystal struc- ture and found a place where OP to OP (phosphate oxy- gens, where the negative charge resides) across the minor groove are less than those between adjacent phosphates. How can electrostatic repulsion between adjacent phos- phate groups drive other phosphate groups together like that, especially if stacking forces are working in opposi- tion? How can one understand such phenomena without explicitly considering electrostatic interactions? Although the bulk of this review might appear rather critical, a model can be useful even if it does not account for all data. And that may be the case here. If Mazur has indeed obtained the first model system where properly di- rected static curvature emerges spontaneously, then his model clearly has utility. If a reviewer who specializes in MD simulations (not this reviewer) would confirm that, and support the utility of the approach, then publication may be in order. However I would like the paper more ifit were reformulated as an exploration of possible mod- els rather than the last word on the physical origin of intrinsic bends. Re: measurement of the groove width: Is the some rea- son that an old version of Curves was used? The newer versions fit a surface to the groove, rather than just mea- sure phosphate-phosphate distances, and provide a much finer view of groove width. 17

arXiv:physics/0004041v1 [physics.atm-clus] 19 Apr 2000Structures and Stabilities of (CaO) nCa2+(n=1–29) Cluster Ions: An alternative interpretation of the experimental mass spectra. Andr´ es Aguado∗and Jos´ e M. L´ opez Departamento de F´ ısica Te´ orica, Universidad de Valladol id, Valladolid 47011, Spain The structures and relative stabilities of doubly-charged nonstoichiometric (CaO) nCa2+(n=1–29) cluster ions and of neutral stoichiometric (MgO) nand (CaO) n(n=3,6,9,12,15,18) clusters are studied through ab initio Perturbed Ion plus polarization calculations. The large co ordination-dependent polarizabilities of oxide anions favor the formation of sur face sites, making the critical cluster size where anions with bulk coordination first appear larger than that expected for purely ionic systems. Thus, we show that there are substantial structural differen ces between alkali halide and alkaline- earth oxide cluster ions, contrary to what is suggested by th e similarities in the experimental mass spectra. An alternative interpretation of the magic number s for the case of oxides is proposed, which involves an explicit consideration of isomer structu res different from the ground states. A comparison with the previously studied (MgO) nMg2+cluster ions suggests a less ionic behaviour of CaO compared to MgO. Nevertheless, the structures of the d oubly charged clusters are rather similar for the two materials. On the contrary, the study of t he neutrals reveals interesting structural differences between MgO and CaO, similar to those found in the case of alkali halides. PACS numbers: 36.40.c; 61.46.+w; 61.50.Lt; 61.60.+m; 79.6 0.Eq I. INTRODUCTION Small clusters are of great interest both to the phys- ical and chemical communities because of their numer- ous potential applications (for example, in nanoelectron- ics or catalysis), and also because one can gain impor- tant insight into the evolution from atomic and molec- ular properties to bulk and surface properties. To have a knowledge of the structures adopted by the clusters is of paramount importance, as many interesting clus- ter properties are largely determined by them. From the theoretical side, to find the lowest energy structure for each cluster is a complicated matter, being the main rea- son that the number of isomers increase exponentially with cluster size. Another reasons are that one has to treat bulklike and surfacelike ions on an equal foot, and that the number of ions to be explicitely considered in a cluster is larger than in a bulk or surface study, where symmetry restrictions impose a number of useful atomic equivalences. From the experimental side, the problem is so important that during the approximately 30 years of intensive cluster research, the main source of structural information has been theory. Very recently, experimental techniques like electron diffraction from trapped clusters1 or measurements of cluster mobilities2–6have been suc- cesfully applied to study the structures of covalent and ionic clusters. Photoelectron spectroscopy has also been applied to study isomerization transitions in small al- kali halide clusters,7and measurements of ionization po- tential to detect structural transitions in barium oxide clusters.8At the moment, however, these techniques need parallel theoretical calculations to make a definite assign - ment of the observed diffraction pattern, mobility or ion- ization potential to a specific isomer geometry. A large amount of theoretical work has been devoted to metallic, semiconductor and noble gas clusters. The work on ionic materials has been centered mostly in the family of alkali halides, and studies of metal oxide clus-ters have been comparatively scarce, despite their impor- tance in many branches of surface physics, like heteroge- neous catalysis or corrotion. Saunders9,10reported mass spectra and collision induced fragmentation data for stoi- chiometric (MgO)+ nand (CaO)+ ncluster ions, Martin and Bergmann11published mass spectra of (CaO) nCa2+clus- ter ions, and Ziemann and Castleman12–15performed ex- perimental measurements of several singly- and doubly- ionized cluster ions of MgO and CaO by using laser- ionization time-of-flight mass spectrometry. Theoretical calculations have been performed at different levels of ac- curacy: simple ionic models based on phenomenological pair potentials were used by Ziemann and Castleman13,14 to explain the global trends found in their experi- ments; Wilson16has studied neutral (MgO) n(n≤30) clusters by using a compressible-ion model17that in- cludes coordination-dependent oxide polarizabilities;18,19 semiempirical tight-binding calculations for MgO cluster s were reported by Moukouri and Noguera;20,21finally, ab initio calculations on stoichiometric MgO clusters have been presented recently by Recio et al.,22,23Malliavin and Coudray,24and de la Puente et al,25and calculations on stoichiometric (Li 2O)nclusters have been reported by Finocchi and Noguera.26Regarding the nonstoichiomet- ric cluster ions, Aguado et al.27have studied the struc- tures and stabilities of (MgO) nMg2+. Trying to find an interpretation of the obtained mass spectra, Ziemann and Castleman14performed some simple pair potential calculations of the structures of (MgO) nMg2+cluster ions by using a rigid ion model. The conclusion of those calculations was that the magic numbers can be explained in terms of highly compact structures that can only be obtained for certain cluster sizes, an interpretation very similar to that found in the closely related case of alkali halides.28–30In our previous work,27we showed that the structures of (MgO) nMg2+ cluster ions were quite different from those of alkali halides. Specifically, the influence of the large and coor- dination dependent polarizabilities of oxide anions (not 1included in the rigid ion model) favors the formation of surface oxide sites, and thus structures with bulk ox- ide anions (coordination 6) are not energetically com- petitive until large values of the number of molecules n are attained. For example, a highly compact 3 ×3×3 cube (where the notation denotes the number of atoms along three perpendicular edges) is not the ground state of (MgO) 13Mg2+. Nevertheless, the agreement between the magic numbers obtained through an examination of the stabilities of the clusters against the loss of an MgO molecule and the experimental ones is complete. It is just the interpretation of them in terms of structures that is different, that is, model-dependent. It is interesting to study a similar system like calcium oxide in order to as- sess whether those trends are a general feature of alkaline- earth oxide clusters or not. Moreover, the experiments of Saunders9,10suggest interesting structural differences between both materials, as the main fragments observed after collisions with inert gas ions were (MgO) 3in one case and (CaO) 2in the other, and the mass spectra of Ziemann and Castleman14,15show different stabilities in the small size regime (magic numbers at n=5,8,11 for (MgO) nMg2+and at n=5,7,9,11 for (CaO) nCa2+), pro- viding further motivation for our study. From the theo- retical point of view, Ca2+is larger than Mg2+, so we can expect ionic size packing effects to play an important role in determining structural differences. Besides, Ca2+has a polarizability approximately 6 times larger than Mg2+, and the polarizabilities of the oxide anions are also larger in CaO because the bonding is weaker than in MgO. In this work we present the results of an extensive and systematic study of (CaO) nCa2+cluster ions with n up to 29. The rest of the paper is organized as follows: in Section II we give a brief resume of the theoretical model employed, as full exposition have been already reported in previous works.28The results are presented in Section III, and the main conclusions to be extracted from our study in Section IV. II. THE AIPI MODEL AND POLARIZATION CORRECTIONS The theoretical foundation of the ab initio per- turbed ion model31lies in the theory of electronic separability,32,33and its practical implementation in the Hartree-Fock (HF) version of the theory of electronic separability.34,35Very briefly, the HF equations of the cluster are solved stepwise, by breaking the cluster wave function into local group functions (ionic in nature in our case). In each iteration, the total energy is min- imized with respect to variations of the electron den- sity localized in a given ion, with the electron densities of the other ions kept frozen. In the subsequent itera- tions each frozen ion assumes the role of nonfrozen ion. When the self-consistent process finishes,28the outputs are the total cluster energy and a set of localized wave functions, one for each geometrically nonequivalent ion of the cluster. These localized cluster-consistent ionic wav e functions are then used to estimate the intraatomic cor- relation energy correction through Clementi’s Coulomb- Hartree-Fock method.36,37The large multi-zeta basis setsof Clementi and Roetti38are used for the description of the ions. At this respect, our optimizations have been performed using basis sets (5s4p) for Mg2+and (5s5p) forO2−, respectively. Inclusion of diffuse basis functions has been checked and shown unnecessary. One important advantage coming from the localized nature of the model is the linear scaling of the computational effort with the number of atoms in the cluster. This has allowed us to study clusters with as many as 59 atoms at a reasonable computational cost. In our previous work on alkaline-earth oxide clusters,27 we concluded that the aiPI model is equivalent to a first- principles version of the semiempirical breathing shell model.39The binding energy of the cluster can be written as a sum of deformation and interaction terms Ebind=/summationdisplay RER bind=/summationdisplay R(ER def+1 2ER int). (1) where the sum runs over all ions in the cluster. The interaction energy term is of the form ER int=/summationdisplay S/negationslash=RERS int=/summationdisplay S/negationslash=R(ERS class+ERS nc+ERS X+ERS overlap ), (2) where the different energy contributions are: the clas- sical electrostatic interaction energy between point-lik e ions; the correction to this energy due to the finite exten- sion of the ionic wave functions; the exchange interaction energy between the electrons of ion R and those of the other ions in the cluster; and the overlap repulsive energy contribution.33The deformation energy term ER defis the self-energy of the ion R. It is an intrinsically quantum- mechanical many-body term that accounts for the energy change associated to the compression of the ionic wave functions upon cluster formation, and incorporates the correlation contribution to the binding energy. As the model assumes, for computational simplicity, that the ion densities have spherical symmetry, the only relevant terms that are lacking from the ab initio description are the polarization terms. In a polarizable point-ion approx- imation, the polarization contribution to the deformation and interaction energies is ERS,pol int =−qR(/vector µS/vector rRS) r3 RS−qS(/vector µR/vector rRS) r3 RS −3(/vector µR/vector rRS)(/vector µS/vector rRS) r5 RS+(/vector µR/vector µS) r3 RS, (3) ER,pol def=µ2 R 2αR, (4) where αRis the polarizability of the ion R, and /vector µRthe dipole moment induced on ion R. The new terms added to the interaction energy are the monopole-dipole and dipole-dipole interaction energy terms. The term added to the deformation energy represents the energy cost of deforming the charge density of the ion to create the dipole moment. The point-ion approximation provides just the asymptotic part of the polarization interaction 2energy, that is, it is exact only for large ionic separa- tions. As soon as the ions begin to overlap, there is an important short-range contribution to the induced dipole moments,40–43which is of opposite sign to the asymptotic limit for anions and may in some specific cases reverse the sign of the asymptotic value. These effects can be easily acomodated in the formalism by substituting the asymptotic value of the induced dipole moments by the following expression: µtotal.R α =µasymp,R α +µsr,R α, (5) with µasymp,R α =αR/summationdisplay S/negationslash=RrRS,α r3 RSqR, (6) µsr,R α=αR/summationdisplay S/negationslash=RrRS,α r3 RSf(rRS), (7) where srstands for “short-range” and µtotal.R α is the α component of the dipole moment vector induced on ion R. The physics behind the short-range polarization cor- rection has been explained in Ref. 40, and is associated to the finite extents of the electron densities of the an- ions and cations. Then, fis a short-range function that switchs on as the cation-anion overlap becomes apprecia- ble. Madden and coworkers43have employed the Tang and Toennies dispersion damping function44as a suitable form for f: f(rRS) =−ckmax/summationdisplay k=0bk k!e−brRS. (8) This is a smoothed step function passing from zero for large rto−cfor r=0. The range of rvalues at which fbecomes significantly different from zero is primarily determined by the range parameter b. We have included the polarization terms in the self- consistent process with this parameterised method, that calculates the induced dipole moments from eq. (5) and the correction to the deformation and interaction ener- gies from eqs. (3) and (4), respectively. The “enlarged” aiPI+polarization model thus obtained accounts for all the relevant physical interactions. The relaxation of the assumption of spherical symmetry being computation- ally expensive, the price to be paid is the inclusion in the model of a set of parameters, namely, the polarizabili- tiesαRand the range parameter b. Appropriate values for the other two constants candkmaxcan be taken equal to the bulk values ( c=-3 and kmax=4).43Given the meaning of the range parameter b, inversely related to half the interionic distance between first neighbors, one might expect different values of bfor clusters as com- pared to bulk materials if the interionic distances are substantially different. As a matter of fact, the evolu- tion of those distances with cluster size is not too com- plicated in the case of ionic clusters.25,28–30Specifically, the average interionic distance dinitially increases quite abruptly with the number of molecules n, and then slowly approaches the bulk limit. As a consequence of this be- haviour, we will see that the bulk value ( b=0.75 a.u.)43is appropriate for all (CaO) nCa2+clusters with n≥4. Different values of bare needed just for n <4 to avoid overpolarization problems.16Regarding the polarizabili- ties, oxide anions have the interesting property of show- ing strongly coordination-dependent values. In fact, the O2−anion does not exist as a free ion, which is equiv- alent to an infinite polarizability; in the solid phase it is stabilized by the crystal environment and has a finite material-dependent polarizability. Wilson16has interpo- lated between those two limits and gives values for the coordination-dependent values of α(O2−) in MgO. We have assumed that the ratio of the bulk oxide polarizabil- ities for MgO and CaO ( αbulk(O2−:CaO)/αbulk(O2−: MgO)=1.469)43is independent of the oxide coordina- tion, and have deduced the αvalues for CaO from those of MgO. This procedure is justified because the cation size does not change appreciably with coordination num- ber. For the calcium cation we take the bulk polarizabil- ity (3.193 a.u.)45 We close this section with a consideration of several criticisms that could be raised against (and of the ad- vantages of) the employed methodology. We have cho- sen a mixed ab initio /semiempirical energy model in or- der to obtain a good compromise between computational efficiency and accuracy. All the relevant energy terms excluding polarization are described with an ab initio methodology. To include polarization, we have used an accurate model,43where the parameters have been fitted by a comparison to ab initio calculations.19Special care has been devoted to the separation of all the indepen- dent physical factors that influence a given quantity, thus avoiding a mixing of different effects in a single parameter and enhancing the transferability of the model. The good parameterisation is reflected in the fact that parameters can be transfered between closely related systems (like, for example, different metal oxides) by simple scaling ar- guments involving ionic radii.46Thus, we think that the reliability of our calculations is reduced just a little com - pared to full ab initio methodologies. To support this expectation, we made a comparison with DMOL calcu- lations performed on neutral (CaO) nclusters by Malli- avin and Coudray.24All the interionic distances were in agreement to their calculations up to differences of 4 %. The energetic ordering of the isomers, as well as the specific energy differences, are reproduced with a maxi- mum error of 5 %. We believe that this is a very rea- sonable agreement, even more if we realize that we are neglecting dispersion interactions, and polarization in- teractions beyond the dipolar terms. The solid MgO is excellently described with the aiPI model (at least in its static properties).47The model is then expected to trans- fer properly between both limits. The larger computa- tional simplicity has been exploited to study large cluster sizes (up to 59 ions) with full relaxations of the geome- tries. Moreover, for each cluster size, a large number of isomers (between 10 and 15) have been investigated. The generation of the initial cluster geometries was ac- complished by using a pair potential, as we explained in our previous publication.27The optimization of the ge- ometries has been performed by using a downhill simplex algorithm.48,49 3III. RESULTS AND DISCUSSION A. Structural Trends in (CaO) nCa2+Cluster Ions In Fig.1 we present the optimized aiPI+polarization structures of the ground state (GS) and lowest lying iso- mers or (CaO) nCa2+(n=4–29) cluster ions. Below each isomer we show the energy difference (in eV) with re- spect to the ground state. For n <4, the clusters are not detected in the experiments, probably because they un- dergo a Coulomb explosion driven by the excess charge and the small cluster size, but we are not interested here in this aspect of the experiments. From n=4 to n=10, there is a predominance of n ×2×2 fragments (n=2–5), that is, the (CaO) 2subunit appears as the basic building block. The total number of ions in these nonstoichiomet- ric clusters is an odd number, and thus those structures are never perfectly compact. There is either an extra cation added to or a missing anion removed from the perfect structure. Less compact structures as for example planar fragments are not energetically competitive. The structures in this size range tend to be elongated as a di- rect consequence of the excess cluster charge. When n=9, a 3×3×2+1 fragment is more stable than that based on the (CaO) 2building block, and n=10 is the largest clus- ter size for which a fragment of this kind is the ground state. Another thing to be pointed out is that in this size range, the extra cation present in the n ×2×2+1 struc- tures induces a larger cluster distortion than the missing anion in the n ×2×2-1 structures. We will see that this feature has important implications in the stability of the clusters. From n=11 to n=15, the dominant fragments are based on n×3×2 units. For n=16 and 17, the most stable iso- mers are n ×4×2 fragments. None of these structures has still developed an anion with full bulk coordination. In particular, the 3 ×3×3 isomer for n=13, which is particu- larly stable in the case of nonstoichiometric alkali halide cluster ions,30does not even appear in Fig. 1. The large coordination-dependent values of the polarizabilities of the oxide anions favors the formation of surface sites, and gives rise to somewhat less compact ground state structures, for which the increase in dipolar energy com- pensate for the decrease in Madelung energy. The 3 ×3×3 (CaO) 13Ca2+is specially unfavored by the dipolar energy terms because it has a central oxide anion with bulk co- ordination (so with a comparatively low polarizability), and another 12 anions with coordination 4. On the con- trary, the largest coordination in the GS structure is five, and some three-coordinated anions (in corner positions) also appear, inducing a large dipolar energy stabilization . For n=18 and 19 there is a glimpse of a transition to more compact cluster structures. The important feature of the GS structures of these two cluster sizes, compared to the 3×3×3 for n=13, is that now there are oxide anions in corner positions. These make a large contribution to the polarization energy term, that added to the increased Madelung energy of a compact fragment, gives a total GS energy more negative than that of n ×3×2 or n×4×2 structures. Nevertheless, the energy differences between isomers are small, and from n=20 on, ground state iso-mers without bulk anions are again obtained (n=24 and 27 are the only relevant exceptions, because the ground states of n=26 and n=29 can be considered degenerate within the accuracy of our theoretical model). A general feature of (CaO) nCa2+cluster ions with n ≥8 is that a ×b×c+1 fragments are specially stable compared to other isomers whenever they can be formed. In Ta- ble I we show all the fragments of that kind relevant to the cluster size range considered in this study. Each se- ries has a typical periodicity that could in principle be reflected in different portions of the mass spectra, given the high stability of these fragments. Some sizes can be accomodated in several families, that is, the classificatio n is highly redundant, but useful anyway to our purposes. If for a given cluster size, a cluster with that formula can be formed, it is always the ground state structure. If it is possible to build up two different isomers with that formula (n=12, 18, 24), the more compact structure is energetically favored. This rule works as long as we do not consider structures that are not energetically compet- itive anymore (the isomer based on the (CaO) 2building block of (CaO) 14Ca2+is an example), and can be helpful in guessing specially stable structures for clusters large r than those studied here. For nearly all those cluster sizes with no competitive a ×b×c+1 structure, a ×b×c-1 frag- ments are obtained as the ground state or specially stable isomers (examples are found for n=5, 7, 11, 19, 23 and 29). The special stability of a ×b×c+1 structures is some- times reflected in high stabilities for the corresponding a×b×c+3 structures, comparable indeed to the stabili- ties of a ×b×c-1 fragments; this occurs for n=13, 17, 19 and 26. With the only exceptions of n=13,14,22,26 and 29, all (CaO) nCa2+GS structures are explained in terms of those three kinds of fragments. Comparing to the results of our previous paper on (MgO) nMg2+cluster ions,27we can see that from n=4 to n=20 the GS structures are basically the same in both systems (the only exceptions are n=7 and n=13). Inter- esting structural differences between both materials ap- pear in the size range n >20. Specifically, the transition to bulklike structures, containing inner anions with bulk coordination, is slower in the case of (CaO) nCa2+. An analysis of the several energy components shows that the net effect of polarization is more important in calcium oxide. Although the polarizability of Ca2+is larger than that of Mg2+and the coordination-dependent values of α(O2−) are larger in CaO than in MgO, this is not a triv- ial conclusion, because the interatomic distances are also larger in CaO, and so the electric fields acting on each ion are correspondingly smaller. If we consider that a highly ionic material is that one for which the Madelung en- ergy term is almost completely dominant in determining structural and several other properties, we would con- clude that the ionic character of (CaO) nCa2+is smaller that that found for (MgO) nMg2+. Indeed, being the po- larization contribution more important, the structures of calcium oxide clusters have a larger directionality degree , a feature that is usually associated to covalency (opposite to the natural tendency of purely ionic systems to form isotropic structures). However, we think that the term “covalency” should be employed just in those situations 4where charge transfer between different atomic centers is important. As Madden and coworkers have discussed,41 polarization terms in ionic systems are responsible for a lot of properties traditionally attributed to “covalency” . One point that deserves further investigation, however, is whether the directional properties induced by polar- ization effects (and reflected in a lower average coordi- nation) can be responsible for a larger charge transfer between different centers. This would be reasonable be- cause the saturation of the bonds is less complete. B. Relative stabilities and connection to experimental mass spectra In the experimental mass spectra,11,15the populations observed for some cluster sizes are enhanced over those of the neighboring sizes. These socalled “magic numbers” are a consequence of the evaporation events that occur in the cluster beam, mostly after ionization.50A magic clus- ter of size n has an evaporation energy that is large com- pared to that of the neighboring sizes (n-1) and (n+1). Thus, on the average, clusters of size n undergo a smaller number of evaporation events and this leads to the max- ima in the mass spectra. As our main concern in this sec- tion is to compare with the experimental mass spectra, we calculate the evaporation energy as a function of clus- ter size. To do this, we assume that the dominant evap- oration channel is the loss of a neutral (CaO) molecule, something supported by the experiments of Ziemann and Castleman.15In the size range n <11, some other chanels seem to be opened in the experiments,15and indeed for n<4 Coulomb explosion is dominant, that is the reason why we do not consider clusters with n <4. With that as- sumption, the evaporation energy of (CaO) nCa2+reads Eevap(n) =Ecluster [(CaO)n−1Ca2+] +E(CaO) −Ecluster [(CaO)nCa2+].(9) Maxima in the evaporation energy curve do not al- ways coincide with maxima in the experimental mass spectra.27There are two main processes that contribute to enhance the cluster population for size n: a)A small evaporation energy for size (n+1); b)A large evaporation energy for size n. Thus, a most convenient quantity to compare with experiment is the second energy difference ∆2(n) =Eevap(n+ 1)−Eevap(n). (10) A negative value of ∆ 2(n) indicates that the n-population increases by evaporations from the (n+1)–clusters more rapidly than it decays by evaporation to the (n-1)– clusters. Specifically, the specially stable cluster sizes will be reflected as minima in the ∆ 2(n) curve. Now, the evaporation energy E evap(n) of eq. (9) can be calculated in two different ways. In the first one, energy differences are always taken between the ground state structures of sizes n and (n-1). This procedure, which we call (by obvious reasons) adiabatic evapora- tion, reflects the stability of the clusters in the limit of small energy barriers between isomers or alternatively of large experimental times of flight. The stabilities calcu- lated in this way are shown in the upper part of figure2. Magic numbers are found for n=5,8,12,15,18,20,24,27. The only a ×b×c-1 structure that shows a special stabil- ity is that of n=5. The rest of magic clusters belong to the a×b×c+1 family of structures. If n is a magic size, and both (n+1) and (n-1) GS structures do not belong to the a ×b×c+1 family, a deep minima is found in the ∆2(n) curve (this happens for n=12 and 18). For the rest of magic sizes, the (n+1) GS structure has also the formula a ×b×c+1, and has a correspondingly high sta- bility reflected in a negative value of ∆ 2(n+ 1). In these cases the stability of size n is just slightly enhanced over that of size (n+1). One can appreciate the increasing rel- evance of the Madelung term in determining the cluster stabilities: when n <20, the most stable a ×b×c+1 struc- tures are the less compact ones (n=8 and 15 more stable than n=9 and 16, respectively); if n ≥20, that trend is reversed (n=20, 24 and 27 more stable than n= 21, 25, and 28, respectively). The special relevance of a ×b×c+1 structures in explaining the cluster stabilities does not conform to the initial experimental expectations of high stabilities for a ×b×c-1 structures.15Analysing the en- ergy components, we find that the polarization contri- bution stabilizes the a ×b×c+1 structure more than the corresponding a ×b×c-1 structure for all values of a,b,c. For the smallest cluster sizes, however, the extra cation present in a ×b×c+1 structures induces a large cluster distortion compared to that induced by the missing anion in a×b×c-1 structures, and the Madelung contribution favors these last structures in a larger amount, making them more stable for some sizes. The second kind of calculation of E evap(n) proceeds as follows: we consider the optimized GS structure of (CaO) nCa2+and identify the CaO molecule that contributes the least to the cluster binding energy. Then we remove that molecule and relax the resulting (CaO) n−1Ca2+fragment to the nearest local minimum. This process can be termed locally adiabatic because both fragments are allowed to relax to the local min- imum energy configuration after the evaporation. For some cluster sizes, the fragment of size (n-1) left when a CaO molecule is removed from (CaO) nCa2+does not lie on the catchment basin of the (CaO) n−1Ca2+GS iso- mer, so that the locally adiabatic evaporation energies are larger than the energy differences between adjacent ground states minus E(CaO) in those cases. The locally adiabatic evaporation energies are plotted as a function of n in the lower part of Fig. 2. These will reflect the clus- ter stabilities in the limit of large energy barriers betwee n isomers or of short experimental times of flight. Magic numbers are obtained for n=5,7,9,11,13,16,19,22,25 and 27, in complete agreement with the experiments of Zie- mann and Castleman.15 The main message to be extracted from these consider- ations is that the magic numbers obtained in the experi- ments might be dominated by the effects of kinetic traps occuring in the course of the evaporation process. Being our calculations static, we can not rigorously assert that this is the only possible explanation, but a plausibility argument based on a comparison to the closely related and more thoroughly studied case of alkali halides sup- ports our expectations. The mobility experiments per- 5formed by the group of Jarrold5,6show that the relax- ation dynamics to the ground state structure for sodium chloride clusters involves drift times of almost one sec- ond. The importance of kinetic traps in explaining these interesting results is shown in the theoretical works of Doye and Wales.51Specifically, these authors show that the potential energy landscape of alkali halide clusters, calculated by using a phenomenological pair potential to describe the interactions, is structured in several funnel s, separated from each other by high free-energy barriers. When a cluster evaporates a molecule, it cools in the process, so trapping kinetic effects are expected when- ever parent and product GS structures belong to different funnels. Given the close similarities between halide and oxide systems, one expects similar effects in the evap- oration kinetics of alkaline-earth oxides to be relevant. The main structural differences are due to the effects of polarization, and these could also affect the mechanisms of structural transitions. In the case of alkali halides, Doye and Wales find that a highly cooperative process is energetically less impeded by energy barriers than se- quential ionic diffusion,51with interesting implications for the mechanical properties of these clusters. Perhaps the same is true for the clusters studied here, but one has to keep in mind that polarization tends to lower the barriers against diffusion,52and those effects are more important for oxides. We think that further calculations of this kind for oxide clusters would be very interesting. Mobility experiments on (MgO) nMg2+or (CaO) nCa2+ could conclusively confirm the structural trends found in the present work. C. Neutral Stoichiometric (MgO) nand (CaO) n clusters The experiments performed by Saunders9,10show that both (MgO)+ nand (CaO)+ nstoichiometric cluster ions with a number of molecules n=6,9,12 and 15 are expe- cially abundant in the mass spectra. However, when these clusters are allowed to collide with inert gas ions, the fragmentation channels are different: (MgO) 3 fragments are predominantly observed in one case and (CaO) 2fragments in the other. These results suggest that the basic cluster building blocks are different for the two materials, but not so different as to lead to different magic numbers. We found a similar scenario in the case of alkali halide clusters.29Specifically, a universal set of magic numbers n=4,6,9,12,... was found for the whole family of (AX) n clusters, with A=Li,Na,K,Rb and X=F,Cl,Br,I. However, the cluster structures were not found to be the same for all the different materials. When the cation size is much smaller than the anion size (all lithium halides and sodium iodide),28ground state structures based on the stacking of hexagonal (AX) 3rings are obtained. For the rest of materials, the ground state structures are mostly obtained by stacking of rectangular (or double- chain) (AX) 3planar fragments. This is just a packing ef- fect: when the ratio of cation to anion size is very small, anion-anion overlap repulsive interactions are large, for c- ing an opening of the (AX) 3rectangular fragments intohexagons. The magic numbers are the same for both structural families because it is for those cluster sizes th at specially compact structures can be formed. When we studied (AX) nA+alkali halide cluster ions,30we found that the structures were much more similar irrespective of packing considerations. The ring structures are not competitive in this case because it is not possible to build up a perfect hexagonal fragment with an odd number of ions. On the contrary, perfect cubic structures can be formed (as for example the 3 ×3×3 structure for n=13). From our study on doubly-charged clusters, we have not found important structural differences between (CaO) nCa2+and (MgO) nMg2+,27at least in the small size regime. We have performed aiPI+polarization cal- culations on the structures of (MgO) nand (CaO) nwith n=3,6,9,12,15 and 18. Specifically, we have considered just those structures based on staking of hexagonal and rectangular (AO) 3units, and those based on stacking of (AO) 2units, with A=Mg or Ca. We find that the ground state structures of (MgO) nclusters are based on (MgO) 3 units, while those of (CaO) nclusters have a rectangular (CaO) 3building block, being this the same packing ef- fect found in the case of alkali halides. The structure of (CaO) 6could be alternatively viewed as the stacking of three (CaO) 2units, but for n=9,12,15 and 18, the tubu- lar shapes obtained by stacking (CaO) 2units are not competitive anymore. Were all the ground state struc- tures of (CaO) nclusters based on the (CaO) 2building block, we would expect a periodicity of 2 in the magic numbers observed in the mass spectra. Saunders shows the collision induced fragmentation spectra of (CaO) n, with n=4,6,8,9which are certainly based on stacking of (CaO) 2units, but does not show those for (CaO) 9or (CaO) 12, for example. Our main conclusion is that the special stability of (CaO) nclusters is also explained in terms of (CaO) 3units, but with rectangular instead of hexagonal shape. This explains the same periodicities observed in the magic numbers of both materials. IV. SUMMARY Theab initio perturbed ion model, supplemented with a parameterised treatment of dipolar terms, has been employed in order to study the structural and energetic properties of (CaO) nCa2+(n=1–29) cluster ions. Po- larization effects favor the formation of surface sites, and reduce the stability of highly compact structures contain- ing anions with bulk coordination. Thus, despite many similarities in the experimental mass spectra, the struc- tures of alkaline-earth oxide and alkali halide cluster ion s are shown to be different. Most of the lowest energy structures have the formula a ×b×c+1. The structures of (CaO) nCa2+and (MgO) nMg2+cluster ions are very similar for n <20, irrespective of differences in cationic size and polarization. It is just for n ≥20 that structural differences emerge, showing a slower convergence to bulk properties for CaO compared to MgO. The analysis of the stabilities suggests that the experimental mass spec- tra could be dominated by the effects of kinetic traps. Specifically, if we consider locally adiabatic evaporation 6events, complete agreement is found with the experi- mental stabilities. The neutral stoichiometric (MgO) n and (CaO) nclusters (n=3,6,9,12,15,18) show structural differences similar to those observed in neutral stoichio- metric alkali halide clusters: the basic building block is an (MgO) 3hexagonal fragment in the case of MgO and a (CaO) 3rectangular (or double-chain) fragment in the case of CaO. This is just a packing effect due to the larger overlap repulsion between anions when the cation size is very small. While the structures of (CaO) nclus- ters, with n=4,6,8 are certainly based on (CaO) 2units, as suggested by collision-induced fragmentation experi- ments, the specially stable (CaO) nclusters are based on a (CaO) 3unit. This explains the same periodicity of 3 observed in the experimental magic numbers of both (MgO)+ 3and (CaO)+ 3clusters. Captions of Figures and Tables. Figure 1 . Lowest-energy structure and low-lying iso- mers of (CaO) nCa2+cluster ions. Dark balls are Ca2+ cations and light balls are O2−anions. The energy dif- ference (in eV) with respect to the most stable structure is given below the corresponding isomers. Figure 2 . Adiabatic (a) and locally adiabatic (b) evaporation energies required to remove a neutral MgO molecule from (MgO) nMg2+cluster ions as a function of n. The local minima in the evaporation energy curve are shown explicitely. Table I Possible different a ×b×c+1 structures, with their inherent periodicities. Those cluster sizes n that are actually observed as ground state structures of (CaO) nCa2+clusters are written in boldface. 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Phys. 105, 11209 (1996). 8[ 0 5 10 15 20 25 30 n−4−2024∆2(eV)−4−2024∆2(eV)(a) (b) 9This figure "4to8.xfig.gif" is available in "gif" format from: http://arxiv.org/ps/physics/0004041v1This figure "9to13.xfig.gif" is available in "gif" format from: http://arxiv.org/ps/physics/0004041v1This figure "14to18.xfig.gif" is available in "gif" format from: http://arxiv.org/ps/physics/0004041v1This figure "19to23.xfig.gif" is available in "gif" format from: http://arxiv.org/ps/physics/0004041v1This figure "24to28.xfig.gif" is available in "gif" format from: http://arxiv.org/ps/physics/0004041v1This figure "29to30.xfig.gif" is available in "gif" format from: http://arxiv.org/ps/physics/0004041v1

arXiv:physics/0004042v1 [physics.bio-ph] 19 Apr 2000Optimal Mutation Rates in Dynamic Environments Martin Nilsson Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501 USA Institute of Theoretical Physics, Chalmers University of T echnology and G¨ oteborg University, S-412 96 G¨ oteborg, Sw eden martin@fy.chalmers.se Nigel Snoad Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501 USA The Australian National University, ACT 0200, Australia nigel@santafe.edu (February 2, 2008) In this paper we study the evolution of the mutation rate for simple organisms in dynamic environments. A model with multiple fitness coding loci tracking a moving fitness peak is developed and an analytical expression for the optimal muta - tion rate is derived. Surprisingly it turns out that the opti - mal mutation rate per genome is approximately independent of genome length, something that also has been observed in nature. Simulations confirm the theoretical predictions. W e also suggest an explanation for the difference in mutation fr e- quency between RNA and DNA based organisms. I. INTRODUCTION In any given environment the vast majority of muta- tions that have any effect on the fitness of a biological organism are deleterious. One might expect the damag- ing effect of non-zero mutation rates to imply that when under evolutionary control the lowest mutation rate com- patible with physiological constraints should be selected for. However, when examined experimentally bacteria and viruses (and indeed all organisms) have significant non-zero rate, the magnitude and diversity of which have failed to find satisfactory theoretical explanation. Some results from a number of experiments measuring the mu- tation rates of a selection of small DNA-based organisms are shown in Table I. Organism ν µ b µG Bacteriophage M13 6 .4·1037.2·10−70.0046 Bacteriophage l 4.9·1047.7·10−80.0038 Bacteriophage T2 &T4 1 .7·1052.4·10−80.0040 E. coli 4.6·1064.1·10−100.0025 S. cerevisiae 1.2·1072.2·10−100.0027 N. crassa 4.2·1077.2·10−110.0030 TABLE I. Spontaneous mutation rates (per base µband per genome µG) in DNA-based microbes with different genome lengths ν. (Data reproduced from Drake et al. [1])Despite the huge variation in genome length over four orders of magnitude the mutation rate per genome and replication µGremains constant roughly within a fac- tor of roughly 2 (which is at the same level as the esti- mated accuracy of the figures). As pointed out by Drake and others [1,2] this constancy in µGis surprising since DNA/RNA repair and transcription are primarily local processes that act on individual bases. Thus the data strongly suggest that point mutation rates for the differ- ent organisms have evolved towards individual optimal values that result in almost constant genomic copying fidelity. In this paper we develop a model of the evolution of mutation rates based on changing environments. The evolved point mutation rate of this model scales so that the genomic copying fidelity is approximately indepen- dent of genome length and insensitive to other parame- ters in the model. The evolved mutation rates are also of the same magnitude as observed in Table I for biologically plausible parameter settings. We also suggest a possible explanation for the high mutation rates of RNA viruses. Simulations confirm the predictions of the model. II. EVOLVING MUTATION RATES It is impossible to perfectly maintain and copy genetic information. All molecules, including DNA and RNA are thermodynamically unstable, and their physical struc- ture and hence the information they encode changes over time. In addition the binding sites of enzymes such as DNA polymerase are not perfectly specific and errors will be introduced during replication. Lowering the error rate requires the use of increasingly complex proof-reading and repair mechanisms, all of which ultimately impose an energetic, and hence fitness, cost on the organism. We can expect a balance to develop between the pressure to lower mutation rates due to the fitness cost of deleterious mutants and the physiological cost of high copying accu- racy [3–5]. Such a balance certainly provides an ultimate lower limit to the mutation rate of all organisms but ex- plaining the concstancy in genomic copying fidelity using such arguments causes unnatural assumptions on the re- lation between cost of local copying fidelity and genome 1length. There is also little experimental evidence that mutation rates are actually determined by such a bal- ance. When viewed as a whole the genome encodes not only proteins that directly influence its reproductive or sur- vival ability, but also the copying fidelity with which the genome reproduces. For example some viroids contain genes that are translated into surface coat proteins while others genes code for the replicase enzymes that perform the copying of its genetic material. In more complex or- ganisms additional genes may encode for modifiers of the accuracy of copy and repair enzymes, usually increas- ing mutation rates [6–9], but sometimes resulting in a decrease [10]. These modifiers can have large or small ef- fects on mutation rate and affect individual bases or the entire genome [11–13]. One consequence of this flexibility of mutation rates and their encoding is that if there are random changes (mutations) in genes determining the mutation rate then the copying fidelity will itself undergo Darwinian evolu- tion. III. POPULATION GENETICS IN CHANGING ENVIRONMENTS When comparing two haploid genomes, the one with lower mutation frequency will produce offspring that are on average more closely related to itself. This means that for an asexual haploid replicator evolving on a static fit- ness landscape the optimal mutation rate for a sequence whose fitness is already globally maximal is zero. If the fitness peak moves, however, the situation changes: to avoid extinction a genome with an initially superior fit- ness is forced to accept a non-zero mutation rate to sur- vive. This leads to a non-trivial optimal copying fidelity. Kimura formalized the evolutionary effect of a chang- ing environment by considering the genetic load of a pop- ulation [3]: the proportion by which the population fit- ness is decreased in comparison with an optimum geno- type. Genetic load results from a number of competing factors; most notably the mutational load due to the dele- terious effects of most mutations and the segregational loaddue to the temporary reduction in fitness that occurs whenever the selective environment changes. Assuming that a population minimizes the genetic load, the opti- mal mutation rate can be calculated. Using a descrete time model, i.e. a model where there is no overlap be- tween generations, with one fitness determining locus the optimal mutation rate becomes: µopt=1 τ(1) where τis the number of generations between environ- mental changes. This model only considers the effect of mutations on the population and is therefore based on group selection.Later population genetic models that examined com- petition between genetic modifiers of the mutation rate demonstrated that (for haploids with a single fitness de- termining locus) a non-zero mutation rate comes to dom- inate a population evolving in an oscillating environ- ment [14–17]. These models are not built on group selec- tion. However a general and simple to interpret multi- locus modifier model does not exist. IV. THE MODEL We will explore a more general model of the evolution of mutation rates in a dynamic environment. Consider a population of haploid genomes where a genome con- sists of two separated parts, one coding for the fitness and one coding for the probability per base µof an er- ror occurring during copying. There is complete linkage (no recombination) between the sections of the genome that encode the mutation rate and those that determine the fitness. We also assume that the fitness determining region is of fixed length ν. In general we are interested in the fates of certain genomes giwhich have a (possibly time-dependent) fitness advantage σ(t) over all other se- quences. We call these genomes master-sequences. The genomic copying fidelity of the fitness determining re- gion of each strain giisQi= (1−µi)ν, the index irefers to the mutation rate of the strain, different strains have different mutation rates but identical fitness σ. We as- sume that mutations do not affect the copying fidelity, only the fitness. Changes to the mutation rates occur on a time-scale significantly slower than the time it takes for the population to reach equilibrium. During a period when a specific sequence has superior fitness compared to the background (i.e. between environmental shifts) the changes in the relative concentrations xiof the master- sequences are described by the replicator equation ˙xi(t) =Qiσ(t)xi(t)−f(t)xi(t) (2) where f(t) =σ(t)/summationtext jQjxj(t) normalizes the relative concentrations of the master-sequence strains. Mutations from background sequences onto the strains with optimal fitness are ignored. Since we are only interested in com- petition between master-sequences the background is not explicitly expressed in these equations. The environment changes as follows: for a time t∈ [0, τ1] one genotype has superior fitness, followed by a new gene-sequence for time t∈[τ1, τ1+τ2], etc. The no- tation is chosen so that τdenotes lengths of time inter- valls. We assume that the initial concentration of the new master-sequences xiimmediately after the shift (at time ta=/summationtextm i=1τi+ǫ, where mdenotes shifts of the fitness- peak and ǫis am infinitely small time-period) are propor- tional to the concentrations of the old master-sequence before the shift (at tb=/summationtextm i=1τi−ǫ) xi(ta) =h(µi)xi(tb) (3) 2It is reasonable to assume that h(µi) is a function with Taylor-expansion in the mutation rate µ h(µ) =∞/summationdisplay j=kmαjµj(4) where kmis a measure of the environmental change, i.e. the number of point mutations needed to transform the old superior sequence into the new. This basically means thatkmis the Hamming distance from the old peak to the new at shift m. The constants αjare combinatorial factors. It will turn out that the optimal mutation rate is independent of these factors. To analyze the long term behavior of this system we make a change of variables yi(t) =e/integraltextt 0f(s)dsxi(t). The new system of differential equations is linear and the equations are decoupled (due to the assumption that the selective dynamics is significantly faster than the changes in mutation rate), it is therefore easy to find the analyt- ical solution: yi(t) =yi(0)eQi/integraltextt 0σm(s)ds(5) Since xiis propotional to yi, maximizing the growth ofyiandxiare equivalent. After a suitably long time interval the population will be completely dominated by genomes that have a mutation rate closest to the optimal value µoptwhich maximizes the long term growth of the strain max µ/parenleftBig Πmh(µ)e(1−µ)ν/angbracketleftσ/angbracketrightmτm/parenrightBig (6) where /angbracketleft·/angbracketrightmdenotes a time average during time-period m. Setting the derivative of this expression to zero and using Eq. 4 we find the optimal copying fidelity to be approximately µopt=/angbracketleftk/angbracketright ν/angbracketleftσ/angbracketright/angbracketleftτ/angbracketright(7) where /angbracketleft·/angbracketrightdenotes a time average over all time periods. We also assume no correlation between /angbracketleftσ/angbracketrightmandτm. Since the genome lengths is large ν≫1, the optimal copying fidelity and mutation rate per genome become: Qopt=e−/angbracketleftk/angbracketright /angbracketleftσ/angbracketright/angbracketleftτ/angbracketright (8) µG=/angbracketleftk/angbracketright /angbracketleftσ/angbracketright/angbracketleftτ/angbracketright(9) Thus we find that the genomic optimal copying fidelity is independent of the genome length for fairly general types of environmental change in both the advantage of the fittest genotype σ(t) and the size of environmental shifts h(µ).V. SIMULATIONS To confirm the theoretical derivations we simulated the evolution of replicators in continuous time on a mov- ing single peaked landscape using a birth-death process. Each time unit in the continuous time replicator equation is the mean replacement time of the population and could therefore be identified as a generation. In the simulation each generation is devided into Ntime-steps (where N is the population size). At each of these time-steps a single individual is selected to copy and mutate. Individ- uals are selected wita h probability proportional to their relative fitness, which is given by σor 1 on the single- peaked landscape. Thus a master-sequence of strain gi (with mutation rate µi) is chosen with probabilityxiσ /angbracketleftf/angbracketright. This copy replaces a randomly chosen individual in the existing population which is then discarded. Thus the population is replaced one by one in discrete birth-death events. In the limit of large population size the dynamics of this simulation approaches the continuous time repli- cator equation. The fitness peak is changed every τgenerations to one of its nearest neighbors. For the binary genomes used here it accomplished by flipping a randomly chosen bit in the definition of the fitness peak. The population was first seeded with a diverse range of mutation rates and the population was allowed to evolve while these rates were kept fixed. This is a true test ofµopt, since the fastest growing sequence should come to dominate. In general the population converged to the strain with mutation rate closest to the theoretically pre- dicted µopt. Figure 1 shows the mean mutation rate of the population ¯ µevolving down towards the theoretically predicted optimum µopt≈1 νστ= 0.00445. From about generation 800 the variance in mutation rates in the pop- ulation is larger than the fluctuations in the mean and the evolution of rates has effectively ended. Simulations were also made to study the effects of more rapidly changing mutator dynamics. In these simulations errors in the copying process not only introduce changes in the fitness determining genotype, but also result in offspring with slightly different mutation rates than their parents, i.e. the mutation rate is allowed to evolve. The mutation rate was treated as a continuous variable which had Gaussian noise introduced during the copying pro- cess. 300.0050.010.0150.020.0250.03 0100 200 300 400 500 600 700 800 900 1000 Generationsµ FIG. 1. Mean mutation rate evolving towards the optimal rate of µopt= 0.00445. Error bars are one standard deviation about the mean. σ= 5, τ= 2, ν= 25, N= 104 Fig. 2 shows the evolution of mutation rates in detail in a population with a reasonably fast rate of change of mutation rates. This simulation has the same landscape parameters as Fig. 1. The mean mutation rate fluctuates around the optimum. For mutation rates close to the optimum fluctuations in selection are significantly larger than the selective advantages of one mutation rate over another. In this region the evolution of mutation rates is effectively neutral and thus the mean mutation rate conducts a random walk about the optimum. We also note that the population typically spends more time with mutation rates above the optimum than below. This is mainly a finite population size effect. When the peak moves and the population size is limited there is a rela- tively large probability that there will be no individuals representing a master-sequence with very low mutation rate on the new peak. This leads to a temporary increase in mutation rate in the population after an environmental shift. FIG. 2. Evolution of mutation rates of mutationally diverse population. µopt= 4.45×10−3,σ= 5, τ= 2, ν= 25, N= 104VI. BIOLOGICAL IMPLICATIONS In nature the existence, and value, of an optimum mu- tation rate that results from a changing environment de- pends on many different parameters: the time between shifts in the selective environment, the complex struc- ture of the fitness-landscape, the genome length, co- evolutionary effects, the strength of selection, neutral- ity in the fitness landscape and fluctuations due to finite population sizes etc. One must therefore be careful when comparing the results of a simple model, such as the one we have presented in this paper, and biological measure- ments. Nonetheless it is this range of possible differences between organisms and the complexity of their evolution- ary environments that leads us to consider the possibility that simple laws of biology — such as the scaling of point mutation rates with genome length — are likely to have quite simple explanations that do not depend on the de- tails of the particular organism. It is therefore worth comparing the results of the model presented in this pa- per with the biological data. 100 50 150 200 τ0.00250.0050.00750.010.01250.015 σ=2 σ=5 =10σ=2 =5 =10genomic mutation rate ( µ) FIG. 3. The shaded region shows the genomic mutation rates for DNA based organisms listed in Table I. For low average fitness advantage σthe mutation rate is relatively insensitive to the frequency of changes in the environment. For clarity we have assumed /angbracketleftk/angbracketright= 1 in this figure. For low mutation rates Eq. 9 is relatively insensitive to changes in the average fitness or size and frequency of environmental changes, as shownin Fig. 3. This insen- sitivity of the optimal genomic mutation rates to evolu- tionary parameters is important, since the bacteria and phages illustrated in table I are most unlikely to live in environments with the same types of time-dynamics and time-scales. In Fig. 3 we see that the sensitivity to one of the parameters in the model, σorτ, depends strongly on in which region the other parameter is. For most realistiv populations we may expect the selective advantage σto be weak, maybe on average less than 2. The predicted mutation rate is then highly insensitive to the average time between shifts in the fitness landscape, e.g. σ= 2 gives τ∈[110,200] for the organisms listed in Table I. It is also reasonable to assume the fitness landscapes of the organisms listed in Table I to be more similar to each other than to higher eukaryotes and since our predictions 4as toQoptare rather insensitive to the details of σ(t),τ andh(µ) we would expect many organisms to have ap- proximately the same mutation rate per genome (within an order of magnitude). This is what we observe for sim- ple DNA-based organisms. VII. RNA VIRUSES The lytic RNA viruses consistently show an extremely high mutation rate — orders of magnitude larger than that of any DNA viruses of similar size. This rate of around one substitution per genome per generation is inconsistent with the analysis conducted above for muta- tion rates evolving in a changing selective environment. Such high rates imply implausible values for the dynamic environment parameters. As an explanation for the high mutation rates ob- served in many RNA viruses and the mutation rate scal- ing with genome length it has been suggested that these viruses have evolved the highest mutation rate possible to be able to adapt to a rapidly changing environment. The maximal mutation rate is then given by the error- threshold, which was first discussed in a model by Eigen et al. [18]. It basically states that on a singled peaked fitness landscape an organism must have high enough copying fidelity so that its relative superiority in repro- duction rate multiplied by the probability of reproduc- ing onto a perfect copy of itself must be larger than one, otherwise there will be no effective selection for the geno- type. It has later been shown that the error-threshold can rather easily be generalized to include effects of a dynamic environment [19]. From this argument it is how- ever not clear why RNA viruses should evolve towards the error-threshold while DNA based organism tend to have much lower mutation rates (by orders of magnitude). In this section we will combine the error-threshold with the model presented in this paper to suggest a possible ex- planation to the difference in observed mutation rates between DNA and RNA based organisms. The dynamic environment model presented in this pa- per applies to organisms where the copying fidelity is en- coded in a part of the genome that has little or no effect on fitness. In many viruses this may not be appropri- ate, partly because the proteins involved in mutagenesis may have a multitude of functions but also because the relatively high selective pressure towards short genome lengths will result in the overlap and multiple use of ge- netic material where possible. This give rise to a differ- ent possibility for the evolution of optimal mutation rates and might help explain the large differences between the observations for RNA and DNA based organisms. We suggest that for organisms which have strong over- laps between genes coding for the mutation rate and genes coding more directly for reproductive advantage there is no effective selection for lower mutation rates, as long as the mutation rate is below the error threshold.This argument is based on the assumption that most mutations are deleterious in terms of fitness, and that the relative fitness advantage on the local peak results in stronger selection pressure than the pressure towards lower mutation rates. We also assume that evolution of mutation rates usually affect regions of the genome where the organism need mutations to be able to adapt ot changes in the environment. If these assumptions ap- ply we expect a population to have mutation rates close to the error-threshold. Changes to mutation rate is tran- sient, assuming that the organism is not pushed beyond the error-threshold. For this hypotheses to apply, viruses with high mu- tation rate (mainly RNA viruses) should have overlap- ping genes regulating mutation frequency as well as re- production rate, whereas organisms with low mutation rates (such as those listed in Table I) should not have overlapping reading frames in their genomes. There are observations that support this, but it is unclear whether the correlation is strong enough for this hypothesis to be valid. VIII. CONCLUSIONS In this paper we have studied the evolution of muta- tion rates in a population of multi locus genomes. The genomic mutation rate µGleading to the greatest long term growth of a strain (the optimal rate) was analyti- cally determined for reasonably general peak shifts and time-dependent replication rates σ(t) µG≈/angbracketleftk/angbracketright /angbracketleftσ/angbracketright/angbracketleftτ/angbracketright where /angbracketleftk/angbracketrightis the mean Hamming distance between suc- cessive fitness optima and /angbracketleftτ/angbracketrightis the mean time between shifts. These optimal rates were quantitatively confirmed by computational simulations of populations whose mu- tation rates were allowed to evolve. These continuous time multi-locus replicator models predict the kind of scaling of point-mutation rate with genome length that has been observed in some bac- teria and viruses/phages and puzzled over for years. When combined with the consequences of the multi- ple use/pleiotropic encoding of copying machinery these models of the evolution of mutation rate in dynamic en- vironments also suggest why lytic RNA viruses may have rates at or about the error-threshold. We would like to thank Claes Andersson and Erik van Nimwegen for useful discussions. Thanks are also due to Mats Nordahl who has given valuable comments on the manuscript. Nigel Snoad and Martin Nilsson were sup- ported by SFI core funding grants. N.S. would also like to acknowledge the support of Marc Feldman and the Cen- ter for Computational Genetics and Biological Modeling at Stanford University while preparing this manuscript. 5[1] J.W. Drake, B. Charelsworth, D. Charlesworth, and J.F. Crow. Rates of spontaneous mutation. Genetics , 148(4):1667–86, April 1998. [2] J. Maynard Smith and E. Szathm´ ary. The Major Tran- sitions in Evolution . W.H. Freeman, Oxford, 1995. [3] M Kimura. On the evolutionary adjustment of sponta- neous mutation rates. Genet. Res. , 9:23–24, 1967. [4] E.G. Leigh. Natural selection and mutability. Amer. Natur. , 104:301–305, 1970. [5] A.S. Kondrashov. Modifiers of mutation-selection bal- ance: general-approach and the evolution of mutation- rates. Genet. Res. , 66:53–69, 1995. [6] E.C. Cox and T.C. Gibson. Selection for high mutation rates in chemostats. Genetics , 77:169–184, 1974. [7] E.C. Cox. Bacterial mutator genes and the control of spontaneous mutation. Ann. Rev. Genet. , 10:135–156, 1976. [8] L. Chao, C. Vargas, B.B. Spear, and E.C. Cox. Trans- posable elements as mutator genes in evolution. Nature , 303:633–635, 1983. [9] P.D. Sniegowski, P.J. Gerrish, and R.E. Lenski. Evolu- tion of high mutation rates in experimental populations ofE. coli .Nature , 387:703–705, 1997. [10] G.T. McVean and L.D. Hurst. Evidence for a selectively favourable reduction in the mutation rate of the X chro- mosome. Nature , 386:388–392, 1997. [11] E.R. Moxon, P.B. Rainey, M.A. Nowak, and R.E. Lenski. Adaptive evolution of highly mutable loci in pathogenic bacteria. Curr. Biol. , 4:24–33, 1994. [12] E.R. Moxon and D.S. Thaler. The tinkerer’s evolving toolbox. Nature , 387:659–662, 1997. [13] M. Radman. Enzymes of evolutionary change. Nature , 401:866–869, 1999. [14] E.G. Leigh. The evolution of mutation rates. Genetics Suppl. , 73:1–18, 1973. [15] J.H. Gillespie. Mutation modification in a random envi- ronment. Evolution , 35:468–476, 1981. [16] K. Ishii, H. Matsuda, Y. Iwasa, and A. Saskai. Evolu- tionarily stable mutation rate in a periodically changing environment. Genetics , 121:163–174, January 1989. [17] J.H. Gillespie. The Causes of Molecular Evolution . Ox- ford University Press, Oxford, UK, 1991. [18] M. Eigen and P. Schuster. The hypercycle. A principle of natural self-organization. Part A: emergence of the hy- percycle. Naturwissenschaften , 64:541–565, 1977. [19] M. Nilsson and N. Snoad. Error thresholds on dynamic fitness landscapes. Phys. Rev. Lett. , 84:191–194, 2000. 6

arXiv:physics/0004043v1 [physics.atom-ph] 19 Apr 2000Characterization of a high-power tapered semiconductor am plifier system D. Voigt, E.C. Schilder, R.J.C. Spreeuw, and H.B. van Linden van den Heuvell Van der Waals-Zeeman Instituut, Universiteit van Amsterda m, Valckenierstraat 65, 1018 XE Amsterdam, the Netherlands e-mail: voigt@wins.uva.nl (February 2, 2008) We have characterized a semiconductor amplifier laser sys- tem which provides up to 200 mW output after a single-mode optical fiber at 780 nm wavelength. The system is based on a tapered semiconductor gain element, which amplifies the out - put of a narrow-linewidth diode laser. Gain and saturation are discussed as a function of operating temperature and in- jection current. The spectral properties of the amplifier ar e investigated with a grating spectrometer. Amplified sponta - neous emission (ASE) causes a spectral background with a width of 4 nm FWHM. The ASE background was suppressed to below our detection limit by a proper choice of operating current and temperature, and by sending the light through a single-mode optical fiber. The final ASE spectral density was less than 0.1 nW/MHz, i.e. less than 0.2% of the op- tical power. Related to an optical transition linewidth of Γ/2π= 6 MHz for rubidium, this gives a background sup- pression of better than −82 dB. An indication of the beam quality is provided by the fiber coupling efficiency up to 59 %. The application of the amplifier system as a laser source for atom optical experiments is discussed. 42.55.Px: Semiconductor lasers; laser diodes 42.60.Da: Resonators, cavities, amplifiers, arrays, and ri ngs 32.80.Pj: Optical cooling of atoms; trapping I. INTRODUCTION The techniques of laser cooling and trapping of neu- tral atoms [1] require stable, narrow linewidth and fre- quency tunable laser sources. Commonly used systems for the near infrared wavelengths are based on external grating diode lasers (EGDL) [2]. Optical feedback from a grating narrows the linewidth to less than 1 MHz and provides tunability. High-power single transverse mode diode lasers can provide up to 80 mW optical output at wavelengths below 800 nm. In this power range diode lasers thus provide a less costly alternative to Ti:Sapphir e lasers. If more power is required, the output of an EGDL can be amplified. Presently, there are three common techniques based on semiconductor gain elements: (i) Injection-locking of a single-mode laser diode [3] by seed- ing light from an EGDL results typically in 60-80 mW optical power at 780 nm wavelength. (ii) Amplification in a double-pass through a broad-area emitting diode laser (BAL) [4–7]. This yields an optical output of typically 150 mW after spatial filtering. A disadvantage is the rel- atively low gain of 10-15, requiring high seed input power.The BAL gain can be improved using phase conjugating mirrors in the seed incoupling setup [8]. (iii) Travelling- wave amplification in a semiconductor gain element with a tapered waveguide (TA) [5,9,10]. Compared to a BAL this yields higher gain and higher power after spatial fil- tering. This approach requires much lower input and a less complex optical setup than a BAL. However, a TA gain element is considerably more expensive. In this work we investigate a TA system that ampli- fies the narrow-linewidth seed beam of an EGDL and provides up to 200 mW optical output from a single- mode optical fiber. We operate the system on the D2 (5S1/2→5P3/2) transition of rubidium, at a wavelength of 780 nm. The input facet of the tapered gain element element has the typical width of a low power single- transverse mode diode laser ( ≈5µm). A seed beam is amplified in a single pass and expanded laterally by the taper to a width of typically 100 −200µm [9] such that the light intensity at the output facet is kept below the damage threshold and the beam remains diffraction limited. The output power can be much larger than from a single-mode waveguide. In previous work, TA’s have been used as sources for frequency-doubling and pumping solid state lasers [11]. Apart from the achievable output power, frequency tun- ability of the narrow-linewidth output [12], simultaneous multifrequency generation [13], and spatial mode prop- erties, including coupling to optical fibers [14] have been addressed. In this paper we investigated the broadband spectral properties of the TA. The background due to amplified spontaneous emission (ASE) [15] in the gain element was minimized by adjusting the amplifier’s operating condi- tions, i.e. temperature, injection current and seed input power. We also investigated the coupling efficiency of the TA output to a single-mode optical fiber, and find that the latter acts both as spatial andspectral filter. The properties of three gain elements of same type are com- pared. Atom optical applications usually require good sup- pression of spectral background. E.g. in far off-resonance optical dipole traps [16], scattering of resonant light fro m the background causes heating and atom loss. We discuss the consequences of ASE background in such schemes. 1II. AMPLIFIER SETUP Our amplifier system consists of a seed laser, the out- put beam of which is amplified in a single pass by the tapered gain element, as shown in Fig. 1. The TA output is coupled into a single-mode optical fiber [17]. The seed laser is an EGDL with a linewidth of less than 1 MHz. It contains a 60 mW single-mode laser diode ( Hitachi , HL 7851G98). From the EGDL we have 28 mW of power available to seed the amplifier at 780 nm wavelength. Coupling of the seeding beam to the amplifier is done by mode-matching the seed laser with the backwards travel- ling beam emitted by the TA. The divergence angles from seed laser emission and backward directed TA emission are similar. Hence, sufficient mode-matching is obtained using identical collimation lenses for both ( f= 4.5 mm, N.A.=0.55). Additional mode shaping, e.g. with anamor- phic prism pairs is not necessary. An optical isolator with 60 dB isolation protects the stabilized seed laser from feedback by the mode-matched beam of the amplifier. The 5 mm aperture of the isolator is sufficiently large that it does not clip the elliptical seed beam. The TA was a SDL 8630E (ser.no. TD310 [18]). Ac- cording to the manufacturer’s data sheet the output power ranges from 0 .5−0.55 W within a wavelength tun- ing range from 787 −797 nm, at an operating temper- ature of 21◦C. The beam quality parameter is specified as typically M2<1.4 [18]. The TA should be protected from any reflected light, because it will be amplified in the backward direction and may destroy the entrance facet. We used an output collimator of large numerical aperture ( f= 3.1 mm, N.A.=0.68) and sent the beam through a second 60 dB optical isolator. The plane of the tapered gain element is vertically oriented, so that diffraction yields a large horizontal divergence. This is collimated similar to the seed input, but yields a focus in the vertical plane. With a cylindrical lens ( f= 100 mm), we compensated for astigmatism of the beam, in order to couple into a single-mode optical fiber. The astigmatism correction is also shown in Fig.1. There is a considerable loss in optical power due to the isolator transmission. Taking also into account small re- flection losses of the lens surfaces, we estimate the useful output power to be 78% of the optical power emitted by the TA facet. In the remainder of this paper, all quoted powers are as measured with a power meter behind the optical isolator. The amplifier was provided as an open heatsink device. We mounted it on a water cooled base and stabilized it to the desired operating temperature within a few mK by a 40 W thermo-electric cooler. Thermal isolation from the ambient air and electromagnetic shielding were provided by a metal housing. When operating the amplifier at temperatures below the dew point, we flushed the con- tainment with dry nitrogen. It is necessary to have a compact, stable mounting of the gain element and collimators. We mounted thecollimators in a commercial xyflexure mount to allow for lateral lens adjustment. The axial zadjustment is done by two small translation stages. All adjustments except the zdirection of the output collimator are acces- sible from outside. This proved to be very convenient for mode-matching the seed beam and also for compensat- ing beam displacement of the TA output when changing temperature or current. The narrow spectral line of seed laser and amplifier output was monitored by an optical spectrum analyzer of 1 GHz free spectral range with 50 MHz resolution. The amplifier’s broad spectral background was analyzed using a grating spectrometer with a resolution of 0.27 nm. Also the output of the single-mode fiber was recorded with the spectrometer. FIG. 1. Amplifier setup with seed laser and gain element: external grating diode laser (EGDL), 60 dB optical isola- tors (OI), tapered amplifier (TA), single-mode optical fiber (OF), optical spectrum analyzer (SA), and grating spectrom - eter (GS). A top and side view of the gain element is shown with input and output collimators (IC,OC), and a cylindrica l lens (CL) compensating for astigmatism (not to scale). III. UNSEEDED OPERATION OF THE AMPLIFIER When the TA receives no seed input, it operates as a laser diode. Thus, when the injection current, ITA, is increased from zero, the optical output power shows the lasing threshold (see Fig. 3ab). Generally, both the oper- ating wavelength and the optical power of a laser diode depend on the temperature. This property is shown in Fig.2. The emission spectrum of the lasing tapered gain element is almost Gaussian shaped, with an 1 /e2width of 4 nm. It appears as a background of ASE also in the spectra when operating the gain element as an amplifier (see below). The oscillatory structures on the spectra are artifacts of the spectrometer. In the fitted spectra, 2we evaluated the center wavelength at each temperature setting. It increases with temperature with a slope of 0.28 nm/K, typical for semiconductor lasers (Fig. 2b). The temperature dependence of the output power is shown in Fig. 2c. We operated the TA within the spec- ifications of the manufacturer’s data sheet, that recom- mends to keep the optical power at the output facet below 550 mW. As the temperature increases, the conversion efficiency (mW/A) decreases and the threshold current increases. This can be seen in Fig.3ab (open symbols) where we plot the optical output power Pvs. current ITAfor two temperature settings. The threshold cur- rent of the unseeded TA increases from 0.78 A (5◦C) to 0.86 A (14◦C). From the slopes above threshold, we find that the conversion efficiency decreases from 0.7 W/A (5◦C) to 0.5 W/A (14◦C). In order to measure the un- perturbed output of the unseeded TA, one has to prevent light emitted from the entrance facet of being reflected. Even a very weak reflection would be amplified in the forward direction. For the unseeded TA, we also measured the light prop- agating backwards from the amplifier’s entrance facet. It reaches typically a power of 10 −25 mW for injection currents from 1 −1.4 A. Hence the necessity of a good isolation of the seeding laser. FIG. 2. Temperature dependence of the unseeded amplifier at 1.2 A injection current: (a) spectra, (b) center waveleng ths, (c) output power after optical isolator. Solid lines indica te linear fits. IV. AMPLIFICATION OF A SEED BEAM Amplification of a seed beam is evident in the output power of the TA. In Fig. 3ab, we have plotted the output power for distinct seed powers, Pseed, at two tempera- ture settings. For the larger seed inputs of 8.6 mW and 5.3 mW, respectively, the amplifier was well saturated. The saturation is evident from Fig. 3c where Pseedwas varied for injection currents from 0 .8−1.3 A.FIG. 3. Amplifier output vs. injection current ITAand seed power Pseed. The lasing thresholds for the unseeded amplifier are 0.86 A (14◦C) and 0.78 A (5◦C), indicated by vertical dotted lines. With Pseed≈4 mW the device appears to be sat- urated for all current settings. For seed powers be- tween 2 −4 mW, the amplification ranges from 70 −140, e.g. 320 mW output with 4 mW seed. We now discuss the spectral properties of the TA and in particular the suppression of ASE background. Fig. 4 shows the power spectral density of the TA output be- forean optical fiber for 16◦C and 5◦C operating tem- perature. In both cases the amplifier was saturated with 28 mW seed input. For comparison also the correspond- ing spectra of the unseeded amplifier are shown. In satu- ration, the broad ASE background is distinguished from a narrow peak of the amplified seed signal. The width of the peak is given by the bandwidth of the spectrometer, 0.27 nm FWHM. (By means of an optical spectrum ana- lyzer and Doppler-free spectroscopy signals of rubidium, we could verify that the amplified signal has a narrow width comparable to that of the EGDL.) The influence of the operating temperature is obvious first by the increased output power at lower tempera- ture: 323 mW (16◦C) and 410 mW (5◦C), respectively. Second, both the peak level and total amount of ASE background are better suppressed at lower temperature. We attribute this to the shift of the gain profile of the TA towards the seed wavelength of 780 nm at lower temper- ature [15]. The fraction of ASE background in the TA output is obtained by integrating the power spectral den- sities in Fig. 4, yielding 5.6% (16◦C) and 1.4% (5◦C), respectively. More than the total ASE fraction, for atom-optical ap- plications the important figure is the fraction of ASE within the natural linewidth of the atomic transition used. We define this ratio ǫby comparing the power in the peak with the ASE power in a bandwidth given by a typical atomic natural linewidth, e.g. Γ /2π= 6 MHz for rubidium. For 16◦C (5◦C), the peak value 3of +2.5 dBm/nm ( −2.0 dBm/nm) of ASE is then re- expressed as 22 nW/Γ (7.9 nW/Γ). With 323 mW (410 mW) in the narrow line, the suppression ratio ǫ is−72 dB ( −77 dB). We can thus optimize the spectral properties of the TA output by an appropriate choice of operating temperature. Even better suppression can be achieved by use of an optical fiber as spectral filter, as discussed in the following section. FIG. 4. Spectrum of the amplifier output (before the fiber). The seed power is 28 mW at 1.2 A injection current. The dashed curves are for unseeded operation. Pis the total opti- cal power, ASE is the fraction of background power, and ǫis the ASE suppression for the power spectral density in units of mW/Γ (Γ /2π= 6 MHz). FIG. 5. Transmission through a single-mode optical fiber. (a) Fiber input and output power with and without seed, (b) fiber transmission with and without seed, (c) Fiber input and output in dependence on seed power, (d) corresponding fiber transmission. The symbols: seeded with 28 mW (solid), un- seeded (open), fiber input (up triangles), fiber output (down triangles), fiber transmission (circles).V. SPATIAL AND SPECTRAL FILTERING USING AN OPTICAL FIBER For many applications, laser beam quality is an impor- tant property, e.g. for optical dipole traps. A convenient method to obtain spatial filtering is to send the light through a single-mode optical fiber. An additional ad- vantage of the fiber is a decoupling of optical alignment between different parts of the experimental setup. Here the coupling efficiency is discussed and the spectrum of the transmitted light is compared with the spectrum be- fore the fiber. We observe that spatial filtering by the fiber is accompanied by spectral filtering. Evidently, the contribution of ASE in the TA beam is spatially distin- guishable from the amplified seed signal. We find that the spatial mode properties of the satu- rated TA output are slightly different for different injec- tion currents. Fig. 5ab represents the fiber transmission vs. current, ITA. The fiber coupling was optimized for a current of 1 A and the TA was saturated. A maximum transmission of 46% is achieved. For comparison, with an EGDL, after circularizing the beam using an anamorphic prism pair, we typically obtain a fiber transmission of 75%. The slope in the transmission curve is probably due to a beam displacement caused by the current-dependent thermal load of the gain element. Such a displacement was also observed when the operating temperature was changed. With the fiber coupling thus optimized, light from the unseeded TA has less transmission than the am- plified seed signal. Fig. 5cd shows for a fixed current of 1 A, that the fiber transmission is almost independent of the seed input power, i.e. the beam shape does not change. Also the light after the fiber has been analyzed us- ing the grating spectrometer, see Fig. 6a for an operat- ing temperature of 5◦C. For the saturated amplifier a spectral ASE background cannot be distinguished after the fiber, since the peak is identical with the spectrom- eter response function. (This response function was ob- tained by recording the spectrum of the narrow-linewidth EGDL laser. A similar response was also obtained us- ing a HeNe laser.) Thus we can only give an upper limit for the ASE contribution of 0.2%. The suppres- sion ratio is ǫ <−82 dB, with an ASE level of less than −12.5 dBm/nm or 0.7 nW/Γ, respectively. This should be compared to the value of ǫ=−77 dB before the fiber, as seen in Fig. 4b for 5◦C. For comparison, at 16◦C, we found an ASE suppression of only −76 dBm afterthe fiber. The ASE background depends also on the degree of amplifier saturation, as shown in Fig. 6b. The ASE frac- tion is plotted vs. seed power for light before and after the fiber. It decreases quickly as the TA saturates. From the spectra acquired before the fiber (up triangles), it is evident that the increase of seed power into the saturated regime suppresses the ASE. Whereas mode matching of the seed input beam was not difficult for achieving max- 4imum output power, optimal ASE suppression required a more careful alignment, thus optimizing the TA satu- ration. It is also obvious from the figure, that larger gain of the TA improves the output spectrum (circles, larger op- erating current). We can summarize the results of Sec. IV and V as fol- lows: The spectral properties of the TA can be optimized by choosing an appropriate operating temperature, spec- tral filtering with an optical fiber and saturation of the gain element. FIG. 6. Spectral filtering by a single-mode optical fiber. (a) Saturated with 28 mW seed power at 1.2 A current, 130 mW power after the fiber, 410 mW in front of it. ASE background is not distinguishable from the spectrometer response func - tion after the fiber. (b) The ASE fraction depends on the saturation: fiber input (up triangles) and output (down tri- angles) at 1.2 A current. For comparison: fiber input with 1.45 A current (circles). VI. VARIATION IN THE PROPERTIES OF INDIVIDUAL GAIN ELEMENTS We compared the gain element with two other gain elements of the same type (8630E). One gain element (ser. no. TD430, 777 nm) was used in the setup de- scribed above. A second (ser. no. TD387, 790 nm) was implemented in a commercial TA system ( TUI Optics GmbH , TA100) and operated on both the D2and the D1(5S1/2→5P1/2) transition of rubidium at 780 nm and 795 nm, respectively. For the different gain elements, we found considerable differences in their beam quality and consequently their fiber coupling efficiency. Whereas TD310 and TD387 showed a dominant double-lobed mode structure in the far field and permitted only a fiber transmission of 46%, the TD 430 beam showed a less pronounced lobe struc- ture [20]. With this gain element, we could couple 59% to the fiber and obtained 200 mW of optical power after the fiber and an ASE suppression of better than −84 dBm. Also the amplification properties showed striking dif-ferences among the gain elements. TD 430 has similar saturation properties as TD310. In contrast, TD387 op- erates at maximum output power already without seed. This may be due to different antireflection coatings at the TA facets. Hence, the TD387 requires (permanent) mon- itoring by a spectrometer in order to optimize seed incou- pling and ASE suppression. The current of the TD 387 cannot be tuned continuously, because it shows discrete “locking-ranges”, resembling the injection-locking beha v- ior of single-mode diode lasers. VII. FAR OFF-RESONANCE DIPOLE POTENTIALS WITH SPECTRAL BACKGROUND In this section we present an estimate of the conse- quences of the broad spectral ASE background for light scattering in optical dipole traps. A background that cov- ers atomic resonances, leads to extra resonant scattering. Usually the detuning δfor a dipole trap is chosen as large as possible, given the available laser intensity I. The rea- son is that off-resonance scattering scales as Γ′ OR∝I/δ2 at low saturation and large detuning, whereas the dipole potential is only inversely proportional to the detuning, U ∝I/δ[21]. In the presence of a resonant background the total scat- tering rate of the atoms is Γ′= Γ′ OR+Γ′ R, where Γ′ Rrep- resents the resonant scattering. For a fixed depth of the optical dipole potential this results in a maximum useful laser detuning, δmax, at which the scattering rate of the atoms, Γ′, is minimized. With low atomic saturation by a weak spectral back- ground, we can write Γ′ R≈(Γπ/4)ǫI/I0. Here I0is the saturation intensity, i.e. 1.6 mW/cm2for the D2line of rubidium. Hence, with the requirement of a fixed potential U, the two scattering contributions scale as Γ′ OR∝1/δand Γ′ R∝δ, respectively. This results in the optimum detuning and minimum scattering rate, δmax=±Γ/√ 2πǫ, Γ′= 2√ 2πǫU/¯h. As an example we consider atoms cooled to a temper- ature of a few µK in optical molasses and require an optical potential depth of U/h≈1 MHz. If the allow- able scattering rate is, e.g. Γ max<100 s−1, this yields a required background suppression ǫ <−110 dB and optimum detuning δmax≈760 GHz. Such a small back- ground contribution is of course beyond the resolution of our spectrometric data, with which we observe at best an upper limit of IR(δ)<0.7 nW /Γ for a total opti- cal power of 200 mW. This corresponds to a background suppression of ǫ <−84 dB. With a detuning of 760 GHz, the extension of the optical potential is restricted to less than 250 µm. 5VIII. CONCLUSIONS We have investigated a tapered semiconductor ampli- fier system, that provides 150-200 mW narrow linewidth output from a single-mode optical fiber, where the fiber transmission is up to 59%, depending on the actual gain element in use. The system requires less than 5 mW seed input to saturate with an amplification up to 140 at this seed level. The output of the amplifier includes a broad spectral background of amplified spontaneous emission. We have found three means of reducing this background: (i) Choosing the operating temperature such that the gain profile of the amplifier is spectrally centered as close as possible to the amplified wavelength, (ii) spec- trally filtering the output beam with a single-mode op- tical fiber, and (iii) saturating the amplifier with suffi- cient seed input power. With those measures, the ASE background is below the resolution of our spectrometer. That is, the ASE fraction is less than 0.2% of the op- tical power in the beam and the peak level is less than 0.1 nW/MHz. Relating the power spectral density of the background to the natural transition linewidth of rubid- ium (Γ /2π= 6 MHz), the ASE suppression is better than -82 dB. We discussed the atom optical application of such an amplifier system with far off-resonance dipole potentials. A broad ASE background implies here an optimum laser detuning with which light scattering by atoms is mini- mized. A tapered amplifier system may be a lower-cost option to a Ti:Sapphire laser. The available single-transverse mode optical power and spectral properties are similar to those of broad-area semiconductor laser amplifiers. IX. ACKNOWLEDGEMENTS We wish to thank K. Dieckmann, A. G¨ orlitz, W. Kaen- ders, J. Schuster, I. Shvarchuck, B. Wolfring, A. Zach, and C. Zimmermann for helpful information. This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek van de Materie” (Foundation for the Fundamental Research on Matter) and was made possible by financial support from the “Nederlandse Or- ganisatie voor Wetenschappelijk Onderzoek” (Nether- lands Organization for the Advancement of Research). R.S. has been financially supported by the Royal Nether- lands Academy of Arts and Sciences. [1] S. Chu: Rev. Mod. Phys. 70, 685 (1998) C.N. Cohen-Tannoudji: ibid. 70, 707 (1998) W.D. Phillips: ibid. 70, 721 (1998) C.S. Adams, E. Riis: Prog. Quant. Electr. 21, 1 (1997)[2] C.E. Wieman, L. Hollberg: Rev. Sci. Instrum. 62, 1 (1991) [3] G.R. Hadley: IEEE J. Quant. Electr. QE-22 , 419 (1986) P. Spano, Salvatore Piazzolla, Mario Tamburrini: IEEE J. Quant. Electr. QE-22 , 427 (1986) [4] G.L. Abbas, S. Yang, V.W.S. Chan, J.G. Fujimoto: IEEE J. Quant. Electr. QE-24 , 609 (1988) L. Goldberg, M.K. Chun: Appl. Phys. Lett. 53, 1900 (1988) [5] L. Goldberg, D. Mehuys, M.R. 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arXiv:physics/0004044v1 [physics.bio-ph] 19 Apr 2000Contraction of a Bundle of Actin Filaments: 50 years after Szent-Gyorgyi† Ken SEKIMOTO§and Hatsumi NAKAZAWA¶ Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto, 606-01 Japan Abstract Biological systems are among the most challenging subjects for theoretical physicists, as well as experimentalists or simulationists. Physical pr inciples should have been both constraints and guide-lines for the evolution of living sys tems over billions of years. One generally aims at clarifying what physical principles, pos sibly new ones, are behind the phenomena of biological interest and at understanding how t hey work within the entire biological world. In the present talk we describe an example of such an effort. Since the discovery of ‘superprecipitation’ by Szent-Gyor gyi’s group in 1940’s, it has been a long puzzle how an assemblage of actin filaments with ra ndom orientation can contract in the presence of the two-headed myosin molecules undergoing actin-activated ATP-hydrolysis reaction. It is widely accepted that during the contraction the two-headed myosin mediates the relative sliding of two actin filaments w hose polarity directions are not parallel but rather anti-parallel. But this fact solely doe s not account for the shortening. We propose a dynamical model which, upon numerical simulati on, exhibits the shortening of an bundle of the actin filaments which are initially dirstr ibuted randomly both in space along a line and in polarity direction. In the course of short ening several clusters of actins appears along the bundle. The model also shows the sor ting of the actin filaments according to their polarity in the late stage. These findings are in accordance with the recent experiment by Takiguchi. †Invited talk presented at APCTP Inauguration Conference, J une 4-10, 1996, Seoul, Korea §To whom the correspondence should be addressed; sekimoto@y ukawa.kyoto-u.ac.jp ¶Present address: International Institute for Advanced Res earch, Central Research Laborato- ries, Matsushita Electric Industrial Co., Ltd. 11 Introduction There is much interest in biological systems from a physicis ts point of view by several reasons. First, by looking at the diversity and hierarchy of biological phenomena and at the billions of years of their evolution, it is a challenge to unveil universal phenomena or universal origins of those systems based on physical prin ciples. For instance, ATP (adenosine triphosphate) is often described in biology tex tbook as the energy source [energy donner], as the substrate of transferase [phosphate donner ], and as the substrate of allosteric enzyme [regulation factor]. We are, however, tempted to sea rch more unified view of the role of ATP since it should have appeared on the earth initially be aring a single role. Secondly, the biological systems are the very representatives of comp lex systems. By regarding them as systems of active elements we are inspired with many physi cal ideas and models. If we view the subjects of biology which have become also the s ubjects of physicists, we find that there are some frameworks elaborately introduce d so that physicists can de- velop their idea upon it. Protein folding is studied based up on the Anfinsen’s dogma that (most) natural proteins rest in their equilibrium folded st ates. Neural network, despite the prohibiting complexity in reality, is studied based upon ma thematical realization of Hebb’s hypothesis. Fluctuating membrane is studied on the basis of elasticity theory including entropic or Helfrich interaction, and molecular evolution is studied as stochastic process. Morphogenesis and pattern formation have been studied in th e framework of bifurcation theory, etc. Protein dynamics can be one of the near future ta rget of physicists, being stimulated by the recent development and need of nanoscale h andling of soft materials, though the theoretical framework for it is not yet establish ed. It is said that the biological processes which appear to be pu rely physical phenomenon, such as symmetry breakdown or instability are even sometime s coded explicitly on the DNA. It should be, however, still meaningful to ask “how did such a biological process happen to appear and become incorporated into evolutionary proces s?” Upon the appearance of a new biological function, it should have been quite primiti ve and unsophisticated, which works in barely efficient way or in a poorly organized way. Expl oring the mechanism of such primitive functions should then be a subject of physics of biological interest as well [1]. As such an example, we present in this paper how the syste m of random assemblage of myosin and actin filaments (both being the constituent prote ins of our muscle) can exhibit stochastic contraction phenomena, which is recently studi ed experimentally in detail [2]. 2(a) (b) (c) Figure 1: (a) A head of the two- headed myosin (‘seed- leaf’) translocates an actin filament (thin ar- rowed line) in the for- ward direction indi- cated by an open thick arrow. (b) and (c) The action of a myosin onto two filaments.Figure 2: Top: Uniaxial assem- blage, or bundle, of actin fila- ments, and the myosin molecules bound to them. Bottom: As the myosins translocate the filaments, overall shortening of the bundle occurs.b(a) (b)b b b Figure 3: The action of myosin can contributes to either (a) shortening or (b) to elongation of the actin bundle. In the case (b) it can happen that the elon- gation is interrupted, as ar- gued in the text. 2 System of many myosins and actin filaments: A paradox We are considering the system consisting of myosins and acti n filaments. Each myosin has two globular heads (shown by the symbols like seed-leave s in Fig. 1), and each actin filaments has its own polarity direction (indicated by the th in arrow lines in Fig. 1). If a globular head of a myosin is within the reach of an actin filame nt in the presence of ATP, the globular head consumes the hydrolysis energy of ATP to dr ive the actin filament to the forward direction (indicated by the open thick arrow in Fig. 1(a)). The net relative motion of actin filaments is brought by the action of a myosin only whe n the myosin is bound to non-parallel pair of filaments (Fig. 1(c)), but not to parall el pair (Fig. 1(b)). We focus here on the recent experiment by Takiguchi [2], He pr epared a bundle of many actin filaments which are assembled uniaxially but rand omly with respect both to their position and to their polarity direction. It has been d emonstrated [2] that the bundle 3of actin filaments undergoes longitudinal shortening in the presence of myosin and ATP molecules. We will describe the experimental procedure of [ 2] in more detail. First a long and thick bundle of many actin filaments is prepared in me thyl cellulose aqueous solution. To this bundle the two-headed myosins (so called h eavy meromyosin, or HMM) and an abundance of ATP molecules are added. The bundle then s tarts to contract slowly in length, while it thickens so as to conserve its volume (Fig . 2). This shortening often occurs in a wiggling way. After the bundle has shortened appr eciably, several needle-like subbundles appear from the main bundle. It has been shown tha t in these subbundles the polarity of the actin filaments is not random but is oriented o utward with respect to the original bundle. Experiment like this dates back to late 40’s, when Szent-Gyo rgyi’s group discovered so called superprecipitation, the phenomenon that a three-di mensional random assembly of actin filaments and myosins shrinks dramatically after the a ddition of a certain amount of ATP [3]. Such experiment has been recently also done and refin ed [4]. Takiguchi’s setup [2] may be regarded as a more idealized one to see how the contract ion occurs. Such an ideal random distribution is realized only in in vitro experiment, but the situations more or less like this have been found in nature such as in the contractile rings that appear during the mitotic period of cell division cycle [5] or in stress fibers o bserved in the locomoting cells during their contraction period [6, 7]. The experiments men tioned above, therefore, could be regarded as a simulation of in vivo systems or, at least, as a hypothetical simulation of evolutionally primitive stages of muscle contraction or cell motility. The question how this primitive system undergoes shortening has, however, n ot been studied for a long time since the discovery by Szent-Gyorgyi’s group. It is because the highly organized structure of muscle was found [8] shortly after the former discovery, a nd the main stream of muscle study has been focused towards a dynamics of single globular head of myosin and its regulation mechanism [9]. It is Hatano who seriously questioned how the actin bundle ca n shorten in the prim- itive situation like in Fig. 2, and he came across the followi ng paradox [10]: When the sliding of the oppositely oriented actin filaments starts fr om the state shown in Fig. 3(a), the overlap between the two filaments would increase, leadin g to the shortening of the bundle. On the other hand, when the sliding of the filaments st arts from the state shown in Fig. 3(b), the action of the myosin would decrease the over lap between the filaments, 4leading to the elongation of the bundle. Since the both situa tions should occur equally likely in a bundle, there would be no net shortening at all. In fact so-called bipolar kinesin, the other motor protein closely related to myosin, is discov ered to appear during the cell division process and this protein is thought to act to separate the two spindle-poles by the mechanism shown in Fig. 3(b) [11]. We would note that the abov e paradox cannot be lifted by considering the effect of simultaneous action of many myos ins to an actin filament, as it occurs experimentally as far as we assume the continuous action of myosin molecules on the actin filaments, while such model could predict the undul ational instability of filament density [12]. 3 Simple model and simulation Our simple idea to resolve Hatano’s paradox is to take into ac count the finite distance, sayb, by which a globular head of myosin can continuously drive a single actin filament (Fig. 3). The limitation of this distance may come from the dy namic fluctuation of the myosin heads as well as by the fluctuation of the lateral arran gement of actin filaments within a bundle. Our reasoning for the shortening is as follo ws: If the myosin acts in the situation of Fig. 3(a) each globular head can translocate th e respective filament fully by the distance bon the average (hereafter we assume that bis sufficiently smaller than the length of each filament, which we denote by ℓ), while in the situation of Fig. 3(b) the translocation of actions by myosin can be interrupted when o ne of its two heads meets with the rear end of an actin filament. The elongation of the bu ndle to which the filaments shown in Fig. 3(b) belong is, therefore, less extensive than the shrinking of the bundle to which the filaments shown in Fig. 3(a) belong. The interrup tion of the elongation will occur by the probability proportional to b/ℓin the approximation up to the lowest order ofb/ℓ≪1. The net shrinkage per each action of myosin will then be rou ghly scaled by ∼b·b/ℓif a single myosin acts to the pair of filaments. Actually the s hortening by this mechanism should be still less efficient due to the presence of other myosins interacting with those actin filaments. We believe, however, that the bas ic mechanism of shortening may be captured by the present simple model. We performed a numerical simulation based on the simple idea described in Fig. 3. The algorithm of the simulation is as follows: First we distribu te randomly Nactin filaments of the length ℓover an interval −L0 2≤x≤L0 2along the x−axis. We choose the parameters so 5Figure 4: Snapshots of actin distribution; the blue dots indicate the center posi- tion of actins oriented rightward, and the green ones indicate those oriented left- ward. Time proceeds from the bottom to top. For the parameter values used, see the text.2000 4000 6000 8000Time255075100125150N_s2000 4000 6000 8000Time102030405060L Figure 5: Top:Evolution of the length of whole bundle including the polar arms (the upper curve) and the length of the bundle less the polar arms (the lower curve). Bottom: The evolution of the number of actin filaments sorted out into the polar arms. The total number of actins, N, is 1000 in this calculation. thatNℓ≫L0holds to assure substantial overlapping of the filaments alo ng the x-axis. For evolution, we define the ‘unitary action’ by a two-headed myo sin:(1)We chose randomly a spatial point, say at x=xM, where the myosin translocates a pair of actins (see (2) belo w). (2)Among all the filaments that extend through the point x=xMwe then choose randomly a pair of anti-parallel actin filaments. One of the chosen fila ment is oriented toward the positive xdirection (i.e., rightward) and is centered at, say, x+, while the another chosen filament is oriented toward the negative xdirection (i.e., leftward) and is centered at, say, x−.(3)We move these two filaments by the same distance but in the oppo site direction, according to the scheme described in Fig. 3. As seen from this figure the distance of sliding is given by U(x+, x−, xM)≡Min[b, xM−x++ℓ/2, x−+ℓ/2−xM].(4)As for the rest of the filaments in the bundle, those filaments in the region of x > x c(x+, x−)≡(x++x−)/2 6are displaced by + U[−U] ifx+> x −[x+< x −], respectively. Also, those filaments in the region of x < x c(x+, x−) are displaced inversely so that Uin the last sentence is replaced by−U. The evolution of the system is obtained by applying this uni tary action from (1) to (4) repeatedly. The result of the simulation is represented by the distribut ion of filaments position (Figs. 4) and by the density of filaments along x-axis (Fig. 6). The parameters used there areN= 200, ℓ= 0.25, and L0= 50. These values are comparable to the experimental values (in unit of µmfor the lengths) [2]. The total time lapse of the evolution is such that each filament undergoes, on the average, six times the un itary action of myosin. These figures reveal the overall shortening of the assembly o f filaments, and also shows the clustering of the filaments in rather symmetric fashion with respect to their polarity. In Fig. 5 we show another run with extended time lapse. There app ear the ‘arms’ from both ends of the bundle, which consist of filaments with unique out ward polarities. The length of these arms and the number of the filaments in these arms incr ease in time. Although the simulation is restricted in one dimensional space, the char acteristic features of the evolution thus found are in agreement with experimental observation o f (i) shortening of the main bundle, (ii) inhomogenization of its thickness and (iii) ge neration of polar subbundles. [2]. 4 Construction of continuum model The above algorithm of simulation can be cast into the evolut ion equation for the densities of actin filaments, ρ+(x+) and ρ−(x−), oriented rightward and leftward, respectively. For this we needed to assume the smallness of the ratio, b/ℓ≪1, and restrict ourselves to the limit of weak spatial heterogeneity, |ρ′ ±| ≪ℓρ±. Suppose that a myosin is at x=xM and starts to exert the unitary action to a pair of anti-paral lel filaments. Let us denote by P(x+, x−;xM)dx+dx−the probability that these two filaments are centered at x=x+∼ x++dx+(the rightward filament) and at x=x−∼x−+dx−(the leftward filament), respectively. In the mean field approximation, this is given as P(x+, x−;xM) =ρ+(x+)ρ−(x−)θ(ℓ 2− |x+−xM|)θ(ℓ 2− |x−−xM|) /integraltext |x′ +−xM|<ℓ 2dx′+/integraltext |x′ −−xM|<ℓ 2dx′−ρ+(x′+)ρ−(x′−), (1) where we introduced a step function θ(z) = 1 for z >0 and θ(z) = 0 for z≤0. Us- ing this distribution we introduce the weighed average of an y function, say O(x+, x−, xM), overx+andx−withxMbeing fixed, <O>xM, as<O>xM≡/integraltextdx+/integraltextdx−P(x+, x−;xM) 7O(x+, x−, xM). Then the displacement field u(x;xM) of the actin densities caused by a unitary action of myosin at xMis given by u(x, xM) =< U(x+, x−, xM) sgn( x+−x−) sgn(x−xc(x+, x−))>xM, where sgn( z)≡2θ(z)−1. Using the gradient expansion of the den- sities of actin filaments, ρ+(xM+X+)ρ−(xM+X−)≃ρ+(xM)ρ−(xM)+ρ′ +(xM)ρ−(xM)X++ ρ+(xM)ρ′ −(xM)X−+ . . . (the prime denotes to take the spatial derivative), the weighed average can be evaluated up to the first order approximation t o give u(x, xM)≃/bracketleftBigg −b2 ℓ+bℓ 6/parenleftBiggρ′ +(xM) ρ+(xM)−ρ′ −(xM) ρ−(xM)/parenrightBigg/bracketrightBigg sgn(x−xM). (2) Here we have noted that sgn( x−xc(x+, x−)) can be safely replaced by sgn( x−xM) in the coarse grained description which deals with only the length scales larger than ℓ. The zeroth order term −b2/ℓin the angular bracket represents the tendency of shortenin g described already, and the second term represents the correction due t o the spatial inhomogeneity of the filament densities. If, for example, ρ′ +<0 and ρ′ −>0 hold at x=xM, the latter term predicts that the shortening is enhanced compared with the homogeneous case. It is understandable since ρ′ +<0 and ρ′ −>0 imply the situations like Fig. 3(a) is be more likely to be found at xMthan those like Fig. 3(b). The mean drift velocity of the bundle, ¯ v(x), is obtained as the integration of u(x, xM) with respect to xMmultiplied by the frequency factor κ(xM) with which myosins exert the unitary actions per unit time and per unit interval along thex−axis. The evolution equation for ρσ(σ=±) is then∂ ∂tρσ=−∂ ∂x[¯v(x)ρσ]. Hereafter we the simplest choice that the factor κ(xM) is an overall constant, say κ(xM) =κ0. This case is that one can solve the evolution equation analytically and, at the same t ime, that the essential feature of shortening and clustering of the bundle is preserved (see be low). Performing the integration with respect to xMthe evolution equation becomes; ∂ ∂tρσ(x, t) =−∂ ∂x/braceleftBigg κ0/parenleftBigg −2b2 ℓx+bℓ 3log/bracketleftBiggρ+(x, t) ρ−(x, t)/bracketrightBigg/parenrightBigg ρσ(x, t)/bracerightBigg , σ =±. (3) The solution of initial value problem can be given via parame tric representation as follows: ˆx(X, t) =Xe−2κ0b2 ℓt+ℓ2 6b/parenleftbigg 1−e−2κ0b2 ℓt/parenrightbigg log/bracketleftBiggρ+(X,0) ρ−(X,0)/bracketrightBigg , (4) ρσ(ˆx(X, t), t) =ρσ(X,0)/bracketleftBigg∂ˆx(X, t) ∂X/bracketrightBigg−1 , σ =±. (5) Figure 7 shows two examples of the solution of (3), the one sta rting from the actin densities with in-phase undulation (the left column) and the other one starting with anti-phase 8undulation (the right column), respectively. If we neglect ed the logarithmic correction term in (3), the solution would simply represent the affine con traction of the bundle, i.e., ρσ(x, t) =tc tc−tρσ/parenleftBig tc tc−tx,0/parenrightBig forσ=±, where tc= (2κ0b2/ℓ)−1L0is the time at which the bundle with the initial length of L0shrinks down to a point. As seen from Fig. 7 the correction term acts to promote the clustering of actin fi laments of both rightward and leftward polarity. Two remarks are in order here: We shou ld note that the simulation described in the previous section, as well as the experiment s with low myosin concentration, would correspond to the slightly different choice of κ(xM), that is, κ(xM) =κ0L0/L(t), where L(t) is the total length of the bundle at time t. This overall factor, L(t)−1would change the time scale of the evolution of Fig. 7, but does not c hange the features of the evolution of the density profiles. We would also note that the generation of the arm cannot be handled within the present approximation in which the spa tial variation of actin densities is assumed to be small. 5 Summary We have proposed a simple model for the of contraction of the r andom uniaxial assembly of actin filaments, mediated by the two headed myosin molecul es which translocates anti- parallel actin filaments. Simulation result have agreed at l east in the qualitative level with the experimental observation: the shortening of the actin b undle, the clustering of density and also the generation of polar arms. The experimental situ ation studied here may well correspond to the stage of evolution where the collective tr ansport by motor proteins had first come into existence in the biological world. More gener ally, it would be interesting to study from physicists’ viewpoint how a function, in its mo st primitive form, has been first acquired by biological systems at any level of evolutio nary history; the problem how an allosteric enzyme have acquired the function to transloc ate the other molecule is a challenging problem in this respect. Acknowledgements The authors gratefully appreciate K. Takiguchi for the kind introduction to his experiments. They also thank very much F. Oosawa, Y. Oono, K. Tawada and M. I shigami for valuable critical comments on the subject. Lastly but not least one of the author (K.S.) would like 9to acknowledge the organizers of the Inauguration Conferen ce of APCTP for the enjoyable meeting and their hospitality. References [1] The question how the systematic motion comes out of noisy environment may be explored also in this light. See, for example, D. A. McQuarri e,J. Appl. Prob. 4(1967) 413, and the references therein; see also, A. Ajdari and J. Pr ost,Comptes Rendus Acad. Sci. II 315(1992) 1635. [2] K. Takiguchi, J. Biochem. 109, (1991) 520. [3] A. Szent-Gyorgyi, Chemistry of Muscle Contraction (Academic Press, NY., 1947 & 1951). [4] S. Higashi-Fujime, J. Cell. Biol. 101(1985) 2335. [5] I. Mabuchi Int. Rev. Citology 101(1986) 175. [6] T. J. Mitchison and L. P. Cramer, Cell84(1996) 371 [Review]. [7] J. M. Sanger and J. W. Sanger, J. Cell Biol. 86(1980) 568. [8] H. Huxley and J. Hanson, Nature 173(1954) 973. [9] B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, and J. D. Watson, Molecular Biology of the Cell (3rd ed.) (1994). [10] S. Hatano, Int. Rev. Citology 156(1994) 199. [11] A. S. Kashina, R. J. Baskin, D. G. Cole, K. P. Wedaman, W. M . Saxton and J. M. Sholey, Nature 379(1996) 270. [12] H. Nakazawa, PhD thesis (in Japanese) of Nagoya Univers ity, 1996; H. Nakazawa and K. Sekimoto, J. Phys. Soc. Jpn. , in press. 10-30 -20 -10 10 20 30X -40-202040 -30 -20 -10 10 20 30X -40-202040 -30 -20 -10 10 20 30X -40-202040 -30 -20 -10 10 20 30X -40-202040 Figure 6: The snapshots of density pro- files of the actin filaments oriented right- ward (upper curve) and those oriented leftward (lower curve), taken from the data shown in Fig. 4. The time lapse is such that the average times of the uni- tary actions undergone by each filament are, respectively, 0, 1, 2, and 3 from the top to the bottom.-30 -20 -10 10 20 30X -4-224 -30 -20 -10 10 20 30X -4-224 -30 -20 -10 10 20 30X -4-224 -30 -20 -10 10 20 30X -4-224 Figure 7: Solutions of the evolution equation (3) in the text, from the two ini- tial conditions (the top raw). The time lapse of the evolved states (the bottom raw) are the same for both cases. 11

arXiv:physics/0004045v1 [physics.atom-ph] 19 Apr 2000High-harmonic generation from a confined atom Carla Figueira de Morisson Faria and Jan-Michael Rost Max-Planck-Institut f¨ ur Physik komplexer Systeme, N¨ othnitzer Str. 38, 01187 Dresden, Germany (February 9, 2014) The order of high harmonics emitted by an atom in an intense laser field is limited by the so-called cutoff fre- quency. Solving the time-dependent Schr¨ odinger equation , we show that this frequency can be increased considerably by a parabolic confining potential, if the confinement parame- ters are suitably chosen. Furthermore, due to confinement, the radiation intensity remains high throughout the extend ed emission range. All features observed can be explained with classical arguments. 32.80.Rm, 42.65.Ky, 42.50.Hz Typical features of the emission spectra of an atom in a strong laser field, known as “the plateau” and “the cutoff”, are a wide frequency region with harmonics of comparable intensities, and an abrupt intensity decrease at the high-energy-end of the plateau. For a monochro- matic driving field, the cutoff energy is given by εmax= |ε0|+ 3.17Up, where |ε0|andUpare the ionization po- tential and the ponderomotive energy, respectively. This simple cutoff law, derived by classical means only [1,2], or using more refined methods [3], corresponds to the physi- cal picture referred to as the “three-step model” [1–3]: A bound electron exposed to the laser field leaves the atom through tunneling at a time t0(step 1), propagates in the continuum, being driven back towards its parent ion at a later time t1(step 2), and finally falls back to a bound state under emission of high harmonics (step 3). This scenario describes the spectral features observed experi- mentally very well [4]. The cutoff frequency, in quanti- tative agreement with the experiment, is related to the maximum kinetic energy the electron has upon return, Ekin(t1,t0). According to this picture, in order to increase the cut- off energy, one must increase the kinetic energy of the returning electron. Indeed, the exisiting proposals to ex- tend the plateau towards higher energies reach a higher value ofEkin(t1,t0) by different means. However, this does not necessarily imply an efficient generation of high- order harmonics up to this larger cutoff energy. For instance, a rather complex situation with several “cutoffs ” [5] emerges by using bichromatic fields with driving waves of comparable intensities. An illustrative example is presented in [6], using a driving field of linearly polarized monochromatic light of frequency ωand its second harmonic. Under such conditions the monochro- matic cutoff, as a function of the field-strength ratio be- tween the two driving waves, splits into two branches.Thereby, the upper branch extends up to |ε0|+5Up. How- ever, the harmonics emerging up to the cutoff of the up- per branch are weak compared to those from the lower branch and therefore irrelevant to the emission spectrum. The reason is simple: The intensity of the harmonics is strongly influenced by step 1 which is the tunneling pro- cess out of the binding potential under the influence of the field. If the field amplitude is small at the emission timet0(which is the case for the upper branch) then the tunneling barrier is large and the generated harmonics will be weak compared to those which originate from an effective tunneling process (as it is the case for the lower branch). Another idea to increase the cutoff energy is to use a static electric field. It provides an additional force which accelerates the electron towards the atomic core resulting in a higher kinetic energy Ekin(t1,t0). Indeed, it has been demonstrated that with an electric field whose strength is only a few percent of the amplitude of the laser field one can considerably enlarge the cutoff energy [7,8]. However, the scheme suffers from two principal limitations. First, the increased kinetic energy occurs mainly for electrons with long excursion times. Due to wave packet spread- ing, those trajectories have negligible influence on the harmonic spectra. This problem has been overcome by introducing an additional magnetic field to restrict the spreading [8]. A second, more severe limitation is the pronounced bound-state depletion caused by the static electric field: the atom is irreversibly ionized within a few field cycles, such that no appreciable high-harmonic generation takes place. The bound-state depletion which prevents an effective extension of the high-harmonic frequency points to the principle dilemma easily described in the picture of the returning electron: To extend the plateau and increase the cutoff, a kinetic energy of the returning electron, as large as possible, is desirable. On the other hand, an electron with such a high energy will leave the atom and is lost for the possible generation of high harmonics in consecutive laser cycles. Hence, we need a mechanism which brings an electron back to the nucleus, despite the fact that it has a kinetic energy so high that it would be irreversibly driven away from the core. Naively, a simple wall for the electron should already do this. However, one must avoid that the abrupt reflection of the charged electron at a wall leads to Bremsstrahlung which masks the desired high- harmonic generation of the atom. In the following we will show that the idea of bringing 1back the fast electron by an additional confinement and thereby extending the cutoff for the spectrum without additional depletion does indeed work for a suitably soft confinement potential. We consider a one-dimensional situation, which is a reasonable approximation for linearly polarized light. Atomic units are used throughout. The binding of the electron is described by the potential Va(x) =−1.1 exp/parenleftbig −x2/1.21/parenrightbig , (1) which supports a single bound state |0/angbracketrightat energyε0= −0.58 a.u., corresponding to the Argon ionization poten- tial. The system is exposed to a monochromatic laser fieldE(t) =E0sinωtand the additional confining po- tential (Fig.1) Vh(x) =Ω2 h 2x2h(x), (2a) h(x) =  1,|x|<x0 cos/parenleftbigπ 2θ/parenrightbig , x0≤ |x| ≤xmax 0,|x|>xmax(2b) withθ= (|x| −x0)/(xmax−x0). The parameter x0= nE0/ω2is chosen to be a multiple of the electron excur- sion amplitude, and xmax= 2x0.Note, that for the pa- rameter range chosen, identical emission spectra are ob- tained with and without truncation of the harmonic po- tential indicating that even in the truncated potential de- pletion has negligible influence. Thus, the electron does not reach the edges of Vh(x), which indicates an effective confinement .Futhermore, this shows that the confining potential does not generate harmonics itself. Therefore, high-harmonic generation still takes place only near the atomic core, for which the coordinate xis considerably smaller than the electron excursion amplitude. FIG. 1. Schematic representation of an atom in an external confining potential Vh(x) (c.f. Eq. (2a)). The parameter x0 for which the potential is truncated and the electron excurs ion amplitude α0, for the parameters of Fig.2(a), as well as the non-truncated potential, are indicated in the figure.The evolution of the electronic wave packet is described by the time-dependent Schr¨ odinger equation id dt|ψ(t)/angbracketright=/bracketleftbiggp2 2+V(x)−p·A(t)/bracketrightbigg |ψ(t)/angbracketright, (2) withV(x) =Va(x) +Vh(x),and the emission spectra are given by σ(ω) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ 0d(t)exp[−iωt]/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 , (3) where the dipole acceleration d(t) =/angbracketleftψ(t)| −dV(x)/dx+E(t)|ψ(t)/angbracketright (4) is computed by means of Ehrenfest’s theorem [9]. We take the atom initially in the ground state |0/angbracketright. FIG. 2. Harmonic spectra calculated using the TDSE (c.f. Eq. (3)). Part (a): Field amplitude E0= 0.08 a.u.,field fre- quency ω= 0.057 a.u.,without (dashed line) and with (solid line) confinement (Ω h= 0.019 a.u).Part (b): Field strengths E0= 0.06 a.u., E0= 0.07 a.u.andE0= 0.08 a.u.,confine- ment curvature Ω h= 0.019 a.u.and the same frequency as in the previous part. The classical cutoff energies, given by th e cutoff law |ε0|+ 4.55Up, correspond to the harmonic orders n= 33, n= 41 and n= 49 and are indicated by arrows in the figure. In part (b), only the harmonic intensities are given, connected by lines. 2Furthermore, in the results to be presented we chose x0= 73.87 a.u.,which corresponds to three times the ex- cursion amplitude of an electron in a monochromatic field withE0= 0.08 a.u.andω= 0.057 a.u.. With these field parameters and a reasonable choice of Ω h,one indeed finds that the high-harmonic spectrum extends beyond the cutoff energy εmax=|ε0|+ 3.17Upwithout signifi- cant loss of intensity, see Fig. 2. More specifically, we have determined a cutoff energy of |ε0|+4.55Upwhich is a 50% increase compared to the case without trapping. The classical argument for the cutoff energy applies to the situation with confinement as well and we find very good agreement between the cutoff in the quantum spec- tra (e.g. Fig. 2) and the classical cutoff. The latter has been determined in analogy to the situation without con- finement: Starting with an electron of velocity zero, its trajectory is propagated under the influence of the laser field and the confinement potential Vh(but without the atomic potential V(x) =Va). We vary the initial time t0for which the electron leaves the atom within a field cycle, computing Ekin(t1,t0) for return times t=t1sat- isfying the condition x(t1) = 0.The local maxima in Ekin(t1,t0) yield the classical prediction for the cutoffs in the harmonic spectra. The good agreement of the classical cutoff with the one found in the quantum spectra allows us to predict, with the classical model, the behavior of the cutoff as a function of the external parameters, i.e. the confine- ment constant Ω h, the frequency and the amplitude of the external field. We find that in the parameter range of interest the cutoff law can be written in the form εmax=|ε0|+f(Ωh,ω)Up, (5) wheref(Ωh,ω) in general neither exhibits a simple func- tional form nor can be derived analytically. However, the linear dependence on the field intensity E2 0throughUpin Eq. (5) is preserved just as in the case without confine- ment, see Fig. 3. Only for large confinement constants or electron trajectories with long excursion times f(Ωh,ω) becomes slightly intensity-dependent.FIG. 3. Cutoff energies computed using the classical model, as functions of the field intensity E2 0, for confinement con- stants 0 ≤Ωh≤0.019 a.u..andω= 0.057 a.u.. The general behavior of f(Ωh,ω) is rather com- plex. Nevertheless, asymptotically a simple and famil- iar behavior is recovered: For very high frequency, the monochromatic cutoff constant is is approached, i.e., f(Ωh,ω→ ∞)→3.17, as can be seen in Fig. 4. For finite frequency ωthe cutoff energy increases with grow- ing Ω h. In fact, the lower the frequency, the more sen- sitively the cutoff law depends on Ω h. This property is the actual reason why one can obtain an increased cutoff energy with a confinement. For very low frequencies, the cutoff energy can be easily extended beyond |ε0|+9Up.In practice, however, there is a lower frequency limit to gen- erate an appreciable intensity of high harmonics in the present context. If the confinement frequency is compa- rable to the laser frequency, Ω h∼ω, the confinement potential itself starts to contribute to the harmonic gen- eration process, ceasing to be a passive element. Hence, the condition for HHG under a confinement potential can be written as Ω h/ω≪1. However, there is also the usual upper limit in frequency ωwhich comes from the requirement that the atom in the laser field must be in the tunneling regime [10]. FIG. 4. Cutoff energies computed using the classical model, as functions of the frequency ωof the driving field, for con- finement curvatures 0 ≤Ωh≤0.019 a .u..The circle in the figure corresponds to ω= 0.057 a.u.and Ω h= 0.019 a.u.,for which the spectra in Fig. 2 have been calculated. Typical frequencies used in HHG experiments, and for which a long plateau is obtained, are in the vicinity of ω= 0.057 a.u.. For this frequency a confinement indeed leads to a larger cutoff energy as demonstrated in Fig. 2. In conclusion, we have presented a new scheme for in- creasing the cutoff energy of the high-harmonic spectra of an atom under the influence of a strong laser field. Placing the atom in a confining parabolic potential, we have shown that the cutoff energy can be increased by more than fifty percent. An effective increase of the cut- 3off requires a careful choice of the confinement strength. The confinement curvature Ω hmust be strong enough for the electron to be appreciably accelerated towards the parent ion, but weak enough for it to move in a “quasi-continuum”. If Ω his too weak, the conventional cutoff law |ε0|+ 3.17Upis not altered by it. If Ω his too strong, the electron moves as a bound particle that does not generate higher harmonics. In the extreme case, one observes the dipole response of a harmonic oscillator, i.e., equally spaced resonances. A rough indication of whether the electron is in a “quasi-continuum” is given by the ratio of the energy difference between two con- secutive levels of the confinement potential, ∆ εh= Ωh, and the ionization potential of the atom in question. If Ωh/|ε0| ≪1,this condition is fulfilled. Also, as already discussed, the ratio between the frequency ωof the ex- ternal field and the confinement curvature Ω hplays an important role. If Ω h/ω∼1,the parabolic potential con- tributes too actively to the harmonic generation process, and the plateau and cutoff are not present in the spectra. The best results have been obtained for x0∼100 a.u., Ωh∼0.02 a.u.andω∼0.04 a.u.In this case, the energy difference between two consecutive levels of the confine- ment potential is still of the order of one tenth of the ionization potential |ε0|and Ω h/ω∼0.5. For this pa- rameter range, the cutoff energy can be extended until approximately |ε0|+ 6Up. On a more technical level, yet very interesting from the theoretical point of view, we have seen that the cutoff law is given by the classical picture of an electron moving under the influence of the laser field and the confinement potential. Very good agreement between the quantum- mechanical full calculation and the classical model occurs for a wide range of field strengths, frequencies around ω∼0.05 a.u.and confinement curvatures of the order of Ωh∼10−2a.u.Thereby we have found that the cutoff law strongly dependens on the confinement curvature Ω h and the frequency ωof the laser field, but only linearly on the field intensity E2 0. The proposed setup presents several advantages over the schemes using static fields. For instance, using a confining potential, one can achieve a considerable ex- tension of the cutoff energy already for the trajectories corresponding to shortelectron excursion times, whereas using static fields one mainly affects electron trajectories withlongexcursion times. Due to wave-packet spread- ing, the former trajectories are far more important for the harmonic spectra than the latter. In order to reduce the spreading one needs very strong magnetic fields [8]. Another noteworthy feature of a confinement potential is that one can obtain stronger harmonics than in the static field, or even in the monochromatic case. In fact, a serious disadvantage concerning static electric fields is an appreciable decrease in the harmonic intensities com- pared to the field free case, due to depletion, i.e. irre- versible ionization. This problem is not present in our scheme. The most serious disadvantage of our proposal is thatto date we are not aware of a direct possibility for an experimental realisation, similarly to the so far proposed extension of the cut off energy by using a combination of a static electric and magnetic fields. In the latter case the magnetic field necessary is unrealistically large for a laboratory application [11]. For our situation, a true electromagnetic trap is too macroscopic compared to the paramater range we need. On the other hand there might be exciting possibilities in the future to design a confined atom as described in a quantum-dot like device, for in- stance as an impurity. An important issue here, how- ever, is the limitation in the radiation intensity in order to avoid the damage threshold. Acknowledgements: We would like to thank K. Richter, D. B. Miloˇ sevi´ c, M. L. Du and K. Leo for useful discussions. [1] P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993). [2] M. Yu. Kuchiev, Pis’ma Zh. Eksp. Teor. Fiz. 45, 319 (1987) (JETP Lett .45(7), 404 (1987)); K. C. Kulan- der, K. J. Schafer, and J. L. Krause in: B. Piraux et al.eds.,Proceedings of the SILAP conference , (Plenum, New York, 1993). [3] M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier and P. B. Corkum, Phys. Rev. A 49, 2117 (1994); W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 41, 4112 (1990) and 50, 1540 (1994). [4] For a recent review, consult P. Sali` eres, A. L’Huillier , P. Antoine, and M. Lewenstein, Adv. At. Mol. Phys. 41, 83 (1999). [5] See, e.g., C. Figueira de Morisson Faria, M. D¨ orr, W. Becker, and W. Sandner, Phys. Rev. A 60, 1377(1999); C. Figueira de Morisson Faria, W. Becker, M. D¨ orr, and W. Sandner, Laser Phys. 9, 388 (1999). [6] C. Figueira de Morisson Faria, D.B. Miloˇ sevi´ c, and G. G . Paulus, to appear in Phys. Rev. A. [7] M.Q. Bao, and A. F. Starace, Phys. Rev. A 53, R3723 (1993); A. Lohr, W. Becker, and M. Kleber, Laser Phys. 7, 615 (1997); B.Wang, X. Li, and P. Fu, J. Phys. B 31, 1961 (1998). [8] D.B. Miloˇ sevi´ c, and A. F. Starace, Phys. Rev. A 60, 3160 (1999); D. B. Miloˇ sevi´ c, and A. F. Starace, Phys. Rev. Lett.82, 2653 (1999); D. B. Miloˇ sevi´ c, and A. F. Starace, Laser Phys. 10, 278 (2000). [9] See, e.g., K. Burnett, V.C. Reed, J. Cooper, and P. L. Knight, Phys. Rev. A 45, 3347 (1992); J.L. Krause, K. Schafer, and K. Kulander, Phys. Rev. A 45, 4998 (1992). [10] The so-called “tunneling regime” is characterized by a Keldysh parameter γ=/radicalbig |ε0|/2Upsmaller than one. [11] See, e.g., Phys. Today 51(10), 21 (1998); Yu. B. Kudasov et al, Pis’ma Zh. Eksp. Teor. Fiz. 68, 326 (1998)(JETP Lett.68, 350 (1998)). 4

arXiv:physics/0004046v1 [physics.atom-ph] 19 Apr 2000Nonlinear, ground-state, pump-probe spectroscopy P. R. Berman and B. Dubetsky Physics Department, University of Michigan, Ann Arbor, MI 4 8109-1120 (September 24, 2013) A theory of pump-probe spectroscopy is developed in which op tical fields drive two-quantum, Raman-like transitions between ground state sublevels. Th ree fields are incident on an ensemble of atoms. Two of the fields act as the pump field for the two-quantu m transitions. The absorption or gain of an additional probe field is monitored as a function of its detuning from one of the fields which constitutes the pump field. Although the probe absorption sp ectrum displays features common to those found in pump-probe spectroscopy of single-quantum t ransitions, new interference effects are found to modify the spectrum. Many of these features can be ex plained within the context of a dressed atom picture. 32.80.-t, 42.65.-k, 32.70.Jz I. INTRODUCTION Of fundamental interest in nonlinear spectroscopy is the re sponse of an atomic vapor to the simultaneous application of a pump and a probe field. A calculation of the probe field abso rption is relatively straightforward [1,2] in the weak probe field limit. Let Ω and Ω′denote the pump and probe field frequencies, ∆ = Ω −ωthe pump field detuning from atomic resonance ω, andδ1= Ω′−Ω the probe-pump detuning. For a pump field detuning |∆| ≫γe,χ, where γeis the upper state decay rate and χis a pump-field Rabi frequency, one finds the spectrum to consi st of three components. There is an absorption peak centered near δ1=−∆ (Ω′=ω),an emission peak centered near δ1= ∆ (Ω′= 2Ω−ω) and a dispersive like structure centered near δ1= 0. Experimentally, a spectrum exhibiting all these features was first obtained by Wu et al. [3]. The absorption and emission peaks can be given a simple i nterpretation in a dressed-atom picture [4], but the non-secular structur e centered at δ1= 0 is somewhat more difficult to interpret [5,6]. The width of these spectral components is on the order ofγe, neglecting any Doppler broadening. The spectral response can change dramatically when atomic r ecoil accompanying the absorption or emission of radiation becomes a factor [7], as in the case of a highly coll imated atomic beam or for atoms cooled below the recoil limit. In this limit, the absorption and emission peaks are e ach replaced by an absorption-emission doublet, and the dispersive-like structure is replaced by a pair of absorpti on-emission doublets. The spectrum can be given a simple interpretation in terms of a dressed atom theory, including quantization of the atoms’ center-of-mass motion [7]. It turns out, however, that at most one absorption-emission do ublet (one of the central ones) can be resolved unless the excited state decay rate is smaller than the recoil shift . Since this condition is violated for allowed electronic transitions, it is of some interest to look for alternative l evel schemes in which this structure can be resolved fully. If the optical transitions are replaced by two-photon, Rama n-like transitions between ground state levels, the widths of the various spectral components are determined by ground state relaxation rates, rather than excited state decay rates. As a result, the probe’s spectral response should be f ully resolvable. Raman processes have taken on added importance in sub-Doppler [8] and sub-recoil [9] cooling, a tom focusing [10], atom interferometry [11–14], and as a method for probing Bose condensates [15]. In this article we propose a scheme for pump-probe spectrosc opy of an atomic vapor using Raman transitions. This is but one of a class of interactions that can be consider ed under the general heading of nonlinear ground state spectroscopy . The spectral response is found to be similar to that of tradi tional pump-probe spectroscopy [1]; however, new interference phenomena can modify the spectrum [Sec. II I]. The interference phenomena can be interpreted in terms of a dressed atom picture [Sec. IV]. Although part of th e motivation for this work is the study of recoil effects, such effects are neglected in this article. II. EQUATIONS OF MOTION The atom field geometry is indicated schematically in Fig. 1. Three-level atoms interact with two optical fields, E1 andE2, producing strong coupling between initial and final levels 1 and 2 via an intermediate excited state level e. FieldE1couples only levels 1 and e, while field E2couples only levels 2 and e. In addition, there is a weak probe fieldEthat couples only levels 1 and e. As a consequence, fields EandE2can also drive two-photon transitions 112e E E1E2 2121 1 1 ; δ ωδ =Ω−Ω+ =Ω−Ω/G25 FIG. 1. Schematic diagram of the atom-field system. Fields E1andEdrive only the 1 −etransition and field E2only the 2−etransition. between levels 1 and 2. Levels 1 and 2 are pumped incoherently at rates Λ 1and Λ 2, respectively, and both states decay at rate Γ .The incoherent pumping and decay represent an oversimplifie d model for atoms entering and leaving the interaction volume. The incident fields are assumed to be nearly copropagating so that all two-photon Doppler shifts can be neglected. In this limit and in the limit of larg e detuning on each single photon transition, one can consider the atoms to be stationary with regards to their int eraction with the external fields. We wish to calculate the linear probe absorption spectrum. The electric field can be written as E(R,t) =1 2/bracketleftBig E1ei(k1·R−Ω1t)+E2ei(k2·R−Ω2t)+Eei(k·R−Ωt)/bracketrightBig +c.c., (1) where Ω 1,Ω2, and Ω are the field frequencies, k1,k2, andkthe field propagation vectors, and c.c.stands for complex conjugate. In an interaction representation, neglecting a ny decay or incoherent pumping of the ground state levels, the state probability amplitudes obey the equations of moti on. i˙ae=χ1e−i∆1ta1+χ2e−i∆2ta2+χe−i∆ta2−i(γe/2)ae, (2a) i˙a1=χ1ei∆1tae+χei∆tae, (2b) i˙a2=χ2ei∆2tae, (2c) whereχj=−dejEj/2¯h(j= 1,2) andχ=−de1E/2¯hare Rabi frequencies (assumed to be real and positive), dej is a dipole moment matrix element, and ∆ j= Ω j−ωejand ∆ = Ω −ωe1are atom-field detunings .Assuming that the magnitude of the detunings are much larger than γeand any Doppler shifts associated with the single photon transitions, it is possible to adiabatically eliminate the excited state amplitude to arrive at the following equation s for the ground state amplitudes: i˙a1=S1a1+S/parenleftbig eiδ1t+e−iδ1t/parenrightbig a1+ge−i˜δta2+g′e−iδ′ta2; (3a) i˙a2=S2a2+gei˜δta1+g′eiδ′ta1, (3b) where ˜δ= ∆2−∆1= Ω2−Ω1+ω21; (4a) δ′= ∆2−∆ = Ω 2−Ω +ω21; (4b) δ1= ∆−∆1= Ω−Ω1=˜δ−δ′, (4c) are detunings associated with two-quantum processes and g=χ1χ2/∆;g′=χχ2/∆;S1=χ2 1/∆;S2=χ2 2/∆;S=χχ1/∆, (5) are Rabi frequencies or Stark shifts associated with two qua ntum processes. In writing Eqs. (3), we assumed that ∆≈∆1≈∆2and|∆| ≫/vextendsingle/vextendsingle/vextendsingle˜δ/vextendsingle/vextendsingle/vextendsingle,|δ′|,|δ1|. It will prove convenient, especially when going over to a dre ssed atom picture, to introduce a representation in which 2a1=b1e−i˜δt/2e−i(S1+S2)t/2;a2=b2ei˜δt/2e−i(S1+S2)t/2. (6) Combining Eqs. (3) and (6) one finds i˙b1=−(δ/2)b1+gb2+S/parenleftbig eiδ1t+e−iδ1t/parenrightbig b1+g′eiδ1tb2; (7a) i˙b2= (δ/2)b2+gb1+g′e−iδ1tb1, (7b) where δ=˜δ−(S1−S2). (8) The corresponding equations for density matrix elements ρ11=|b1|2,ρ22=|b2|2,ρ12=b1b∗ 2=ρ∗ 21are ˙ρ11=−ig(ρ21−ρ12)−ig′eiδ1tρ21+ig′e−iδ1tρ12−Γρ11+ Λ1; (9a) ˙ρ22=ig(ρ21−ρ12) +ig′eiδ1tρ21−ig′e−iδ1tρ12−Γρ22+ Λ2; (9b) ˙ρ12=iδρ12−ig(ρ22−ρ11) +ig′eiδ1t(ρ22−ρ11)−iS/parenleftbig eiδ1t+e−iδ1t/parenrightbig ρ12−Γρ12, (9c) where the incoherent pumping and decay terms have been intro duced. It is important to note that, in this represen- tation, the frequency appearing in the g′terms isδ1=δ′−˜δ= Ω−Ω1. In other words, the effective field frequency associated with field E2in this representation is Ω 1rather than Ω 2. It follows from the Maxwell-Bloch equations that the probe a bsorption coefficient, α, and index change, ∆ n, are given by α=kNd2 1e 2¯hǫ0Im/parenleftbiggρ′ 1e χ/parenrightbigg ; (10a) ∆n=−kNd2 1e 2¯hǫ0Re/parenleftbiggρ′ 1e χ/parenrightbigg , (10b) whereNis the atomic density, ρ′ 1e≈(χ/∆)/bracketleftBig ρ(0) 11+χ1ρ+ 11+χ2ρ+ 12/bracketrightBig , (11) andρ(0) 11,ρ+ 11, andρ+ 12are coefficients that appear in the solution of Eqs. (9) (to firs t order inχ) written in the form: ρjj′=ρ(0) jj′+ρ+ jj′eiδ1t+ρ− jj′e−iδ1t;j,j′= 1,2 (12) The first and third terms in Eq. (11) are analogous to the terms that appear in conventional theories of pump-probe spectroscopy, but the second term is new and leads to qualita tively new features in the probe absorption spectrum. An expression for ρ′ 1eis given in Appendix A. The absorption coefficient is plotted i n Figs. 2(a)-(c) for several values ofδ/g, and η=/radicalbig χ1/χ2. (13) Ifη≪1, the two-quantum probe absorption spectrum has the same st ructure as the probe absorption spectrum involving single quantum transitions. The situation chang es ifη>∼1. For example, aside from an interchange of absorption and gain components as a function of δ1, the probe spectrum for single quantum transitions depends only on the magnitude of the pump field detuning. This is clearly notthe case for two-quantum transitions, as is evident from Fig. 2(a) drawn for η= 1,Γ/g= 0.1,δ/g=±1. Probe absorption and gain areinterchanged when δchanges sign, but the ratio of the amplitude of the absorption to gain peakchanges whenδchanges sign. There is another subtle difference present in these spectra. The sense of the c entral dispersive component is opposite to that for single quantum transitions. With decreasing η, the sense of the central component would reverse, as the spe ctrum reverts to the same structure found in pump-probe spectroscopy of si ngle quantum transitions. The probe response also depends on the sign of ∆ (through g=χ1χ2/∆); this feature follows from the dependence of the spectrum on the sign ofδand the relationship ρ′ 1e(−δ,−∆,−δ1) =−ρ′ 1e(δ,∆,δ1)∗, (14) which can be derived using Eqs. (A3)-(A7) of Appendix A. It is also possible for the components centered at positive or negative δ1to vanish (in the secular approximation) for certain values ofη, as can be seen in Fig. 2(b). 3-3.0 -1.5 0.0 1.5 3.0 δ1/g-4-20246Absorption (arbitrary units)δ/g=1; Γ/g=0.1; η=2 δ/g= -1; Γ/g=0.1; η=2 (a) -3.0 -1.5 0.0 1.5 3.0 δ1/g-4-20246Absorption (arbitrary units)δ/g=1; Γ/g=0.1; η=0.786 δ/g= -1; Γ/g=0.1; η=1.27 (b) -3.0 -1.5 0.0 1.5 3.0 δ1/g-2-1012Absorption (arbitrary units)δ/g=0; Γ/g=0.1; η=1/5 -3.0 -1.5 0.0 1.5 3.0 δ1/g-0.10.00.1Absorption (arbitrary units)δ/g=0; Γ/g=0.1; η=1 -3.0 -1.5 0.0 1.5 3.0 δ1/g-0.2-0.10.00.10.2Absorption (arbitrary units)δ/g=0; Γ/g=0.1; η=5 (c) FIG. 2. Probe field absorption in arbitrary units. Positive o rdinate values correspond to probe absorption and negative values to probe gain. The case of δ/g= 0 is shown in Fig. 2(c) for η= 1/5,1,5, and ∆>0.Ifη= 1/5, the spectrum is similar to that found for single quantum transitions [1]. For η= 1,the spectral component at negative δ1is found to vanish. When η>∼1, there is a dispersive-like structure centered at δ1= 0 that is not found in the pump-probe spectroscopy of single quantum transitions. Expressions for the three comp onents are given in Eqs. (A8) of Appendix A for |g| ≫Γ, Γ≪η2. 4III. DRESSED ATOM APPROACH The spectral features seen in Figs. 2 (a),(b) can be explaine d using a dressed atom approach. Semiclassical dressed states for two-quantum transitions can be introduced via th e transformation [16] /parenleftbigg |A/angbracketright |B/angbracketright/parenrightbigg =T/parenleftbigg |1/angbracketright |2/angbracketright/parenrightbigg ; (15a) T=/parenleftbigg cos(θ)−ψsin (θ) ψsin(θ) cos(θ)/parenrightbigg , (15b) where ωBA=/radicalbig δ2+ 4g2 (16) is the frequency separation of the dressed states, cos(θ) =/bracketleftbigg1 2/parenleftbigg 1 +δ ωBA/parenrightbigg/bracketrightbigg1/2 , (17) and ψ=|∆|/∆. (18) The angleθis restricted such that 0 ≤θ≤π/4 forδ>0 andπ/4≤θ≤π/2 forδ<0. Forθ∼0 (δ>0,|g/δ| ≪1), |A/angbracketright ∼ |1/angbracketright, while forθ∼π/2 (δ<0,|g/δ| ≪1),|B/angbracketright ∼ |1/angbracketright. In the secular approximation, Γ≪ωBA, (19) it follows from Eqs. (9) and (15) that, to zeroth order in the p robe field, the diagonal dressed state density matrix elements are given by ρ(0) AA= (Λ 1/Γ)cos2(θ) + (Λ 2/Γ)sin2(θ)≡ΛA/Γ; (20a) ρ(0) BB= (Λ 2/Γ)cos2(θ) + (Λ 1/Γ)sin2(θ)≡ΛB/Γ; (20b) ρ(0) AA−ρ(0) BB= (Λ A−ΛB)/Γ = [(Λ 1−Λ2)/Γ]cos(2θ);. (20c) Note that/parenleftBig ρ(0) AA−ρ(0) BB/parenrightBig has the same sign as (Λ 1−Λ2) ifδ>0 and the opposite sign if δ<0. It is now possible to use the energy level diagram (Fig. 3) to r ead directly the probe absorption spectrum. The probe field is absorbed (or amplified) via two quantum transit ions between states |A/angbracketrightand|B/angbracketright. The two quantum transitions involve one photon from the probe field and one ph oton from either fieldE1orE2, since all of these fields couple states |A/angbracketrightand|B/angbracketrightto state |e/angbracketright. It is important to remember that the effective field frequenc y of fieldE2is equal to Ω 1in this interaction representation. Fields E1andEcouple state |e/angbracketrightto the components of states |A/angbracketrightand |B/angbracketrightinvolving state |1/angbracketright, while field E2couples state |e/angbracketrightto the components of states |A/angbracketrightand|B/angbracketrightinvolving state |2/angbracketright. For example the matrix element for the two-quantum process f rom state |A/angbracketrightto|B/angbracketrightinvolving absorption of a probe photon and emission of a field E2photon is −iχ −i∆cos(θ)(−iχ2) Γ−i(δ1−ωBA)cos(θ), while that for absorption of a probe photon and emission of a fi eldE1photon is −iχ −i∆cos(θ)(−iχ1) Γ−i(δ1−ωBA)ψsin(θ). These two processes add coherently, such that probe absorpt ion via transitions from state |A/angbracketrightto|B/angbracketrightis proportional to the sum of these two matrix elements squared, multiplied b y the population difference/parenleftBig ρ(0) AA−ρ(0) BB/parenrightBig .In other words, the probe absorption at δ1=ωBAis proportional to a quantity C+given by 5cos1sin2 Aθψθ = −cos2sin1 Bθψθ = + /G4CA/G4CB /G47/G47e EEE1 E2E1E2 FIG. 3. Dressed-state energy level diagram. In the interact ion representation adopted in the text, the frequency of fiel dE2 must be set equal to Ω 1in calculating resonance conditions. For (Λ A−ΛB)>0, solid arrows correspond to probe absorption centered at δ1=ωBAand dashed arrows correspond to probe gain centered at δ1=−ωBA. C+= (g/∆Γ)[(Λ 1−Λ2)/Γ]cos(2θ)/parenleftbigg ψηsin (θ)cos(θ) +1 ηcos2(θ)/parenrightbigg2 . (21) Similarly, probe gain via transitions from state |A/angbracketrightto|B/angbracketrightatδ1=−ωBAis proportional to C−= (g/∆Γ)[(Λ 1−Λ2)/Γ]cos(2θ))/parenleftbigg ψηsin (θ) cos(θ)−1 ηsin2(θ)/parenrightbigg2 . (22) A formal derivation of these results is given in Appendix B. For the sake of definiteness, let us take (Λ 1−Λ2)>0; thenC+corresponds to absorption for δ >0 and to gain forδ <0, whileC−corresponds to gain for δ >0 and to absorption for δ <0. Note that the component centered atδ1=−ωBAvanishes if ∆ >0 and tan(θ) =η2, while that at δ1=ωBAvanishes if ∆ <0 and tan(θ) =η−2. The values ofA±=±C±/bracketleftbig Γ2∆/|g|(Λ1−Λ2)/bracketrightbig are plotted in Fig. 4 as a function of δ/gfor ∆>0 andη= 1,2. For ∆<0, one can use the relationship A±(−∆,−δ) =∓A∓(∆,δ). 0 2.5 5 7.5 - 7.5 - 5 - 2.5 0 0.5 - 0.5 1A /g=2=1 =1 =2A+ A- FIG. 4. Amplitude A+of the peak centered at δ1=ωBAand amplitude A−of the peak centered at δ1=−ωBA, for ∆ >0. Positive values of A±correspond to absorption and negative values to gain. One sees that the interference between two channels for abso rption and emission plays an important role. This interference arises only for a semiclassical description o f the pump fields. If pump fields E1andE2are quantized and in pure number states, this interference does not occur, sin ce the final states for the two channels are orthogonal. 612+ A = 122 2+ G =2 12 1e e e e(a) (b) FIG. 5. Schematic representation of the 1 →2 transition probability leading to probe absorption or pro be gain in lowest order perturbation theory in the bare basis. The thin arrow r epresents the probe field, the broad filled arrows field E1, and the broad open arrows field E2. (a) absorption, (b) gain. Terms involving the sequential a bsorption and emission of the same field have been neglected, since such terms result only in Stark sh ifts of levels 1 and 2. The diagrams are drawn for ˜δ >0; if˜δ <0, the roles of absorption and gain would be interchanged. The probe absorption vanishes in the secular approximation (19) whenδ= 0,since, in this case, θ=π 4and the populations of the dressed states are equal. The lowest orde r dressed atom approach is not useful in this limit. Typical spectra are shown in Fig. 2 (c) and were discussed in Sec. III . IV. CONCLUSION The probe absorption spectrum has been calculated for two-q uantum transitions between levels that are simulta- neously driven by a two-quantum pump field of arbitrary inten sity. In addition to features found in conventional pump-probe spectroscopy of single quantum transitions, ne w features have been found that can be identified with interference phenomena. Both Doppler and recoil effects wer e neglected in out treatment. For nearly copropagating fields, effects arising from these processes are negligible. Doppler shifts can be accounted for by the replacements δ1→δ1+ (k1−k)·v,δ1−˜δ→δ1−˜δ+ (k2−k)·v,andδ1+˜δ→δ1+˜δ+ (2k1−k2−k)·vin the equations in the Appendix. The dependence of the interference effect of the signs of ∆ and ˜δcan be understood in the bare atom picture in a perturbative limit. A schematic representation of the prob ability amplitude leading to probe absorption at δ1=˜δis shown in Fig. 5(a). Each arrow represents an interaction wit h one of the fields. The two contributions to the final state amplitude add coherently. Putting in the appropriate energy denominators, one finds that the absorption varies as A=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglei2χχ∗ 2 (γe/2−i∆)[Γ−i(∆−∆2)]+i4χχ∗ 2|χ1|2 (γe/2−i∆)[Γ−i(∆−∆1)] (γe/2−i∆)[Γ−i(∆−∆2)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (23) For/vextendsingle/vextendsingle/vextendsingle˜δ/vextendsingle/vextendsingle/vextendsingle≫Γ,and|∆| ≫γe, this equation reduces to A=/vextendsingle/vextendsingle/vextendsingle/vextendsingleχχ∗ 2 ∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle21 Γ2+/parenleftBig δ1−˜δ/parenrightBig2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 +|g|η2ψ ˜δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (24) which shows the dependence on the signs of ∆ ( ψ=|∆|/∆) and ˜δ.A similar calculation for the emission component represented schematically in Fig. 5(b) leads to G=/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ∗χ2 1χ∗ 2 ∆2˜δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle21 Γ2+/parenleftBig δ1+˜δ/parenrightBig2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−|g|η−2ψ ˜δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (25) New effects will arise if the fields are not copropagating and t he active medium is a subrecoil cooled atomic vapor, a highly collimated atomic beam, or a BEC. As for single quant um transitions [7], each component of the spectrum undergoes recoil splitting. Since the center-of-mass mome ntum states differ for two-quantum processes involving field s E1andEfrom those involving fields E2andE, one would expect a spectrum consisting of 8 absorption and 8 gain components. 7V. ACKNOWLEDGMENTS This work is supported by the U. S. Army Research Office under Gr ant No. DAAG55-97-0113 and by the National Science Foundation under Grant No. PHY-9800981. We are grat eful to the Prof. G. Raithel for fruitful discussions. APPENDIX A: BARE STATE CALCULATIONS Substituting Eqs. (12) into Eqs. (9), one finds to zeroth orde r in the probe field that w(0)=ρ(0) 22−ρ(0) 11=(Λ2−Λ1) ΓΓ2+δ2 Γ2+δ2+ 4g2; (A1a) ρ(0) 11=1 2/bracketleftbigg(Λ2+ Λ1) Γ−w(0)/bracketrightbigg ; (A1b) ρ(0) 12=−ig Γ−iδw(0)=/parenleftBig ρ(0) 21/parenrightBig∗ , (A1c) and that, to first order in the probe field, w+=ρ+ 22−ρ+ 11,ρ+ 12,ρ+ 21, andm+=ρ+ 22+ρ+ 11satisfy m+= 0; (A2a) (Γ +iδ1)w+−2igρ+ 21+ 2igρ+ 12= 2ig′ρ(0) 21; (A2b) [Γ +i(δ1−δ)]ρ+ 12+igw+=−ig′w(0)−iSρ(0) 12; (A2c) [Γ +i(δ1+δ)]ρ+ 21−igw+=iSρ(0) 21. (A2d) Equation (11) can be rewritten as ρ′ 1e≈(χ∗/∆)/bracketleftbigg ρ(0) 11−1 2y0+y12/bracketrightbigg , (A3) where y0= (χ1/χ)∗w+; (A4a) y12= (χ2/χ)∗ρ+ 12; (A4b) y21=/parenleftbiggχ2χ∗ 1 χ∗χ1/parenrightbigg ρ+ 21, (A4c) and we have allowed the Rabi frequencies to be complex. The quantities y0,y12, andy21satisfy the coupled equations: [Γ +i(δ1−δ)]y12+i˜gη−2y0=a; (A5a) (Γ +iδ1)y0−2i˜gη2y21+ 2i˜gη2y12=b; (A5b) [Γ +i(δ1+δ)]y21−i˜gη−2y0y=c, (A5c) where b=−2|g|2 Γ +iδw(0)= 2c (A6a) a=−i|g|ψη−2w(0)−|g|2 Γ−iδw(0). (A6b) andψ= (|∆|/∆). Note that the equations do not depend on the phase of the va rious Rabi frequencies, but dodepend on the sign of ∆. Explicit solutions for y0andy12are: y0=−i2a|g|ψη2(δ+δ1−iΓ) +b/bracketleftBig δ2−(δ1−iΓ)2/bracketrightBig + 2c|g|ψη2(δ−δ1+iΓ) (δ1−iΓ)/parenleftBig δ2−δ2 1+ 2iΓδ1+ Γ2+ 4|g|2/parenrightBig ; (A7a) y12=ia/bracketleftBig δ2 1+δ(δ1−iΓ)−2iΓδ1−Γ2−2|g|2/bracketrightBig −bη−2|g|ψ(δ+δ1−iΓ)−2c|g|2 (δ1−iΓ)/parenleftBig δ2−δ2 1+ 2iΓδ1+ Γ2+ 4|g|2/parenrightBig . (A7b) 8The line shape is totally non-secular when δ= 0.In the limit that ∆ >0,|g| ≫Γ, and Γ ≪η2,one finds that the absorption coefficient αforδ1≈0 is α∼ −1 4/parenleftbiggkNd2 1e 2¯hǫ0∆/parenrightbiggδ1Γ (δ2 1+ Γ2), (A8a) that forδ1≈2|g|is α∼1 8/parenleftbiggkNd2 1e 2¯hǫ0∆/parenrightbigg(δ1−2|g|)Γ/bracketleftBig (δ1−2|g|)2+ Γ2/bracketrightBig(1 +η−2), (A8b) and that for δ1≈ −2|g|is α∼1 8/parenleftbiggkNd2 1e 2¯hǫ0∆/parenrightbigg(δ1+ 2|g|)Γ/bracketleftBig (δ1+ 2|g|)2+ Γ2/bracketrightBig(1−η−2). (A8c) Note that the component at δ1=−2|g|vanishes ifη= 1. For ∆ <0, one can use Eq. (14). APPENDIX B: DRESSED-STATE CALCULATIONS Equation (7) can be written in the form i¯h˙b= (V+Vp)b, (B1) where V= ¯h/parenleftbigg −δ/2g∗ g δ/ 2/parenrightbigg , (B2) Vp= ¯h/parenleftbigg Seiδ1t+S∗e−iδ1tg′∗eiδ1t g′e−iδ1t0/parenrightbigg , (B3) g=χ1χ∗ 2 ∆;g′=χχ∗ 2 ∆;S=χ∗χ1 ∆, (B4) and we have allowed for complex Rabi frequencies, χ1=|χ1|eiφ1,χ2=|χ2|eiφ2,χ=|χ|eiφ. (B5) If one introduces semi-classical dressed states via the tra nsformation bd=Tcb, (B6) where bd=/parenleftbigg A B/parenrightbigg , (B7) Tc=/parenleftbigg cos(θ)eiφd/2−e−iφd/2sin (θ) eiφd/2sin (θ)e−iφd/2cos(θ)/parenrightbigg , (B8) and φd=φ1−φ2+π 2(1−ψ) (B9) (recall that ψ=|∆|/∆), then the dressed-state Hamiltonian is given by 9Vd= ¯h/parenleftbigg −ωBA/2 0 0ωBA/2/parenrightbigg +TcVpT† c. (B10) The dressed state density matrix, ρd=/parenleftbigg ρAAρAB ρBAρBB/parenrightbigg (B11) evolves as /parenleftbiggd dt+ Γ/parenrightbigg ρd≈1 i¯h[Vd,ρd] +/parenleftbigg ΛA0 0 Λ B/parenrightbigg , (B12) Off-diagonal terms have been neglected in the matrix represe nting the incoherent pumping, since they give rise to terms of order Γ /ωBA≪1 (secular approximation). The dressed state density matrix is expanded as ρd=ρ(0) d+ρ+ deiδ1t+ρ− de−iδ1t, (B13) and it is found from Eqs. (B1)-(B3), (B6)-(B13) that ρ+ dobeys the equation of motion /parenleftbiggd dt+ Γ/parenrightbigg ρ+ d=i/parenleftbigg 0 ( ωBA−δ1)ρAB −(ωBA+δ1)ρBA 0/parenrightbigg +1 i¯h/bracketleftBig Vpd,ρ(0) d/bracketrightBig , (B14) where Vpd= ¯h/parenleftbigg cos(θ)/bracketleftbig Scos(θ)−g′∗eiφdsin (θ)/bracketrightbig ; cos(θ)/bracketleftbig Ssin(θ) +g′∗cos(θ)eiφd/bracketrightbig sin(θ)/bracketleftbig −g′∗sin (θ)eiφd+Scos(θ)/bracketrightbig ; sin (θ)/bracketleftbig Ssin (θ) +g′∗cos(θ)eiφd/bracketrightbig/parenrightbigg (B15) In the secular approximation, the steady state solution of E q. (B14) is ρ+ d=/parenleftbigg 0ρ+ AB ρ+ BA 0/parenrightbigg , (B16) where ρ+ AB=icos(θ)/bracketleftbig Ssin (θ) +g′∗cos(θ)eiφd/bracketrightbig/parenleftBig ρ(0) AA−ρ(0) BB/parenrightBig /(Γ +i(δ1−ωBA)), (B17a) ρ+ BA=−isin(θ)/bracketleftbig −g′∗sin(θ)eiφd+Scos(θ)/bracketrightbig/parenleftBig ρ(0) AA−ρ(0) BB/parenrightBig /(Γ +i(δ1+ωBA)). (B17b) The coherence ρ′ 1eneeded in Eq. (10) for the absorption coefficient and index cha nge is given by ρ′ 1e≈(1/∆)/bracketleftBig χ∗ρ(0) 11+χ∗ 1ρ+ 11+χ∗ 2ρ+ 12/bracketrightBig . (B18) The first term can be evaluated using Eq. (A1b) for ρ(0) 11; it contributes to the index change, but not the absorption. For the remaining terms, one rewrites ρ+ 11andρ+ 12in the dressed basis using Eqs. (B6),(B8),(B11), and uses Eq . (B5) to extract all the phase factors to arrive at ρ′ 1e≈(χ∗/|∆|)/bracketleftBig ψρ(0) 11+f++f−/bracketrightBig (B19) where f+=i|g| [Γ +i(δ1−ωBA)]cos(2θ)(Λ1−Λ2) Γcos2(θ)/parenleftbigg ψηsin(θ) +1 ηcos(θ)/parenrightbigg2 , (B20a) f−=−i|g| [Γ +i(δ1+ωBA)]cos(2θ)(Λ1−Λ2) Γsin2(θ)/parenleftbigg ψηcos(θ)−1 ηsin (θ)/parenrightbigg2 . (B20b) Note that the approach and results of Sec. III are unchanged i f one uses complex dressed states defined by /parenleftbigg |A/angbracketright |B/angbracketright/parenrightbigg =T∗ c/parenleftbigg |1/angbracketright |2/angbracketright/parenrightbigg . (B21) 10[1] B. R. Mollow, Phys. Rev. A 5, 2217 (1972). [2] S. Haroche and S. Hartmann, Phys. Rev. A 6, 1280 (1972). [3] F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, Phys. Rev . Lett. 38, 1077 (1977). [4] C. Cohen-Tanoudji and S. Reynaud, J. Phys. B 10, 345 (1977). [5] G. Grynberg and C. Cohen-Tannoudji, Optics Comm. 96, 150 (1993). [6] P. R. Berman and G. Khitrova, Optics Comm. xx, xxxx (2000). [7] P. R. Berman, B. Dubetsky, and J. Guo, Phys. Rev. A 51, 3947 (1995). [8] See, for example, P. D. Lett, W. D. Phillips, S. L. Rolston , C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989); J. Dalibard and C. Cohen-Tannoudji, ibid.6, 2023 (1989); P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, ibid.6, 2058 (1989); D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, and S . Chu, ibid.6, 2072 (1989); A. Aspect, E. Arimondo, R. Kaiser, N. Vanteenkiste, and C. Cohen-Tannoud ji,ibid.6, 2112 (1989). [9] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988); M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992). [10] M. Prentiss, G. Timp, N. Bigelow, R. E. Behringer, J. E. C unningham, Appl. Phys. Lett. 60, 1027, (1992); T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, Appl. Phys. B 54, 375 (1992). [11]Atom Interferometry , edited by P.R. Berman (Academic, San Diego, 1997). [12] D. S. Weiss, B. C. Young, S. Chu, Phys. Rev. Lett. 70, 2706 (1993). [13] A. Peters, K. Y. Chung, and S. Chu, Nature, 400, 849 (1999). [14] T. L. Gustavson, P. Bouyer, and M. A. Kasevich, Phys. Rev . Lett. 78, 2046 (1997). [15] D. M. Stamper-Kurn, A. P. Chikkatur, A. G¨ orlitz, S. Inn ouye, S. Gupta, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett.83, 2876 (1999). [16] P. R. Berman, Phys. Rev. A 53, 2627 (1996). 11

Nakhmanson p. 1 of 20Informational interpretation of quantum mechanics Raoul Nakhmanson Nakhmanson@t-online.de Ascribing to inanimate matter a possibility to receive, work on and transfer information allows us to explain quantum-mechanical phenomena including"delayed-choice"- and "Einstein-Podolsky-Rosen (EPR)"-type experimentsadhering to the basis of local realism, and to suggest essentially newexperiments with microparticles and atoms in which information plays theprincipal rôle. PACS numbers: 03.65.Bz , 05.70.-a , 89.70.+c . I. Introduction The microworld has surprised the "classical" physicists with the following paradoxes: •Before quantum mechanics (QM) was created: quantization of mass, charge, energy, angular momentum; the identity of particles of the same type, wave-particleduality, quantum randomness. •In QM: statistical predictions, Heisenberg's uncertainty principle, Pauli's exclusion principle, and (implicitly but immanently) collapse of the wave function. •In standard (Copenhagen) interpretation of QM: rejection of the classical realism, a ban on speaking about non-measured parameters, trajectories, etc.; Bohr'scomplementarity principle. •In hidden-parameter interpretation: Bell's theorem and non-local instantaneous interaction between far separated microobjects. •In experiments: results in delayed-choice, EPR, Aharonov-Bohm arrangements. The Copenhagen interpretation is only a translation of the mathematical formalism of QM to the ordinary language but not an interpretation in a common sense, because itdoes not explain how, why, and in which framework this formalism works. Thecommon sense of Planck, Einstein, de Broglie, Schrödinger and many others could notaccept it: "ceterum censeo copenhaginem esse delendam" . But all attempts to find a realistic interpretation were unproductive: they did not forecast new results andintroduce doubtful peculiarities e.g. non-local interaction or multiplicating worlds.Feynman told his students that the quantum world was not like anything that we know;and although everybody knows QM, many people use it, some of them develop it, butnobody understands it. Almost equal is Gell-Mann's sentence: "QM, that mysterious,confusing discipline, which none of us really understands but which we know how touse." How must the productive common-sense interpretation of the quantum mechanics look like?Nakhmanson p. 2 of 20!It must accept the existence of real and causal world of local-interacting objects; !It must make quantum mechanics evident [1] and explain the above mentioned paradoxes as well as all "experimenta crucis" performed up to now; !It must be verifiable by means of experiments; !It must show the frame of quantum mechanics and new horizons beyond it. A version of such interpretation has been developed by the author since 1977 [2-4] and will be summarized as below. It is based on an assumption what microparticles canreceive, work on and transfer information, that means, they have some consciousness(spirit, ghost, intelligence), or they are automatons created and programmed from someoutside intelligence. This assumption is not a totally new one: just after quantumrandomness was discovered people spoke about the "free will" of electrons. Later andmore extensively this assumption was discussed e.g. in [5]. But the forerunners did notdevelop it to the level of a constructive hypothesis that could be tested experimentally. In Section II the indications for such an assumption are named, the informational interpretation of quantum mechanics is declared and applied to explain the notions ofQM and its "paradoxes". In particular, it will be shown that in the new world picture,Bell's theorem [6] is not valid anymore. Besides this there are some remarks about oneold paradox of physics, namely, the second law of thermodynamics. Section IIIconsiders the results of "experimenta crucis" such as those of "delayed choice" [7, 8] and Einstein- Podolsky-Rosen ("EPR") [9, 10] ones. Section IV presents schemes of"Gedankenexperimente" coming out of the frame of QM to have informational contactswith "inanimate" matter. The last schemes do not take into account the direct interactionbetween matter and human consciousness [11-17] which will be briefly touched inSection V. II. Foundation of hypothesis The idea of microparticle consciousness cannot avoid the following questions: What clues do we have on the existence of microparticle consciousness?Have the particles sufficient complexity for it?What conduced the evolution of microparticle consciousness? The clues supporting the assumption that "inanimate" matter has some kind of consciousness are the following: ♦the formula E=mc2 connecting mass m and energy E by virtue of the light velocity in vacuum c - the maximal velocity of the spreading of information; ♦the informational character of the wave function ψ describing probability in contrast to classical potentials describing "tangible" fields transporting energy and impulse; ♦"teleological" movement of matter seemingly in the principle of the least action;Nakhmanson p. 3 of 20♦quantum-mechanical stochastics which can be seen as optimal tactics of behavior to search for all possible alternatives. The acceptance of consciousness of microparticles assumes their complex structure. Theoretically it was shown (first perhaps by Markov [18] ) that one microparticle cancontain a whole universe ("fridmon", "baby universe"). If one does not go deeper than Planck's length ( ≈ 10 -35 m), one finds in a microobject ( ∅ ≈ 10-15 to 10-20 m ) o f t h e order of 10 45 - 10 60 "Planck cells", that is much more not only than the number of neurons in the human brain, but than the number of atoms contained in all known livingbeings too. Any detailed assumptions about structures and processes providing the workof particle consciousness are beyond the scope of this article. The basic circumstance which supports the development of any consciousness is free will. Without free will the consciousness is useless. If free will exists, humans (and notonly they) have the choice of alternatives, taking into account physical and socialconditions. The more intelligent a choice is, the higher the person's rating in evolutionis. Therefore: free will evolves intelligence. The roots of free will do not lie in the macroworld which is ruled by deterministic laws. They lie in the microworld, and quantum randomness point to it. We cannot besure that human consciousness is the only product of free will. It is possible that earlier,free will created some consciousness at the level of its roots, i.e., in the microworld.Because the time (measured not in seconds but in events) flowed there much faster, thisconsciousness had a longer evolution period. Perhaps the golden age of it is over, andnow we have to do only with a "rudimentary" consciousness (so called by Cochran, andBohm and Hiley [5] ) of automatons following the known rules of QM in the majority ofnatural and experimental situations being observed up to now. The new conception accepts reality existing beyond our sense-organs and measuring apparatuses and "exculpates" those who say and/or think like "the atom being in point A emits one photon having energy E in the direction AB at the moment t " without being sure whether all these parameters are measured. Moreover, the questions like "But howdoes your electron know, falling from the upper level, where it has to stop?" (Rutherfortto Bohr, 1911) are permitted. The QM is only a theory of measurements carried out onthe microobjects. For example, the accuracy of the simultaneous measurements ofcomplementary variables is restricted because of an interaction with a measuringapparatus that is reflected in Heisenberg's uncertainty principle. But microparticles knowtheir variables, they remember what happened and tell it to others. To do this, they musthave synchronized clocks, measuring rules, and reference points for space and time. Inthis sense it is possible to speak about special ("absolute") coordinates and time, likeGreenwich's ones. If we can communicate with particles (see Section IV), the dream ofEinstein and other realists, to know the values of all variables included in a theory, canbecome true. The Fermi-particles obey the Pauli's exclusion principle. So they can better search for all possibilities. Such a behavior is typical of scientists: each of them tries to find his owntheme. Sometimes human's behavior is like a Bose-particle's one. Phenomena such asfashion in dress or music, and applause or coughing in concert halls, are examples ofBose-condensation. The same person can manifest himself as boson or fermion. ForNakhmanson p. 4 of 20particles this was only possible in a "big bang" time. Are we now at the same beginning stage of evolution? The evolution of the quantum state pursues some purpose that can be formalized into an integral equation via the least action principle. The scientists are successfullyguessing a view of the least action function deeper than its meaning. The consciousnessof each elementary particle solves this equation taking into account all information which it has. The solution is its own wave function ψ which reflects the strategy of particle behavior. Two or more particles coming once in interaction can have some group strategy and common non-factorized ("entangled") ψ-function controlling their behavior in the following near future. Where is the ψ-function? It is not in the real 3-dimensional space. It is in imaginary configurational space, which, in its turn, is in the imagination (consciousness) of theparticle. This explains why it does not affect other particles being in the same 3d-space.When the particle receives new information (it can take place by any interaction withmicro- or macroobjects), it corrects its strategy. Thus occurs the collapse of the wave function. It occurs not in the real (infinite) space, but in the consciousness of particle.The consequent time is determined by the rapidity of this consciousness. Therefore, compared with space-time conditions of experiment, collapse is local and instantaneous.Von Neumann [11], London and Bauer [12], and Wigner [13] suggested that human consciousness provokes a collapse of ψ-function. This is not so: human consciousness collapses only the human knowledge about the ψ-function. The laws of both collapses lie beyond physics. The renovation of the knowledge’s because of interactions can gowithout alteration of mass, energy and entropy: the new state of particle can have thesame energy and entropy, but different actual information [3]. Through the development of QM the physicists came to the idea about "holism" of the material world. The new interpretation explains this holism as having an informationalorigin. The world is entangled by information, its "internet" has existed since the bigbang. The particle can know only information lying in the lower light cone, i.e., belonging to the past. To calculate its own strategy (i.e. ψ-function) successfully the consciousness of the particle must have an ability to forecast the circumstances in the future using itsknowledge about the past. It is the very natural result of development of everyconsciousness. Particularly, all human life is based on such ability. If a physicistcalculates the tomorrow behavior of an elementary particle in his laboratory, he looksinto the future, and only a part of it depends on his free will. In the new conception the paradox of "Schrödinger's cat" [19] vanishes. The cat is either alive or dead, and its state always can be seen by an observer. The mixed "alive-dead" cat exists only as a phantom in the consciousness of a radioactive atom (if it is stillpotentially active), in the consciousness of the observer (if he has not seen the dead cat),perhaps in the consciousness of the cat, etc., but not in the real space. All of themcannot forecast when the cat will be killed: it depends on the random generatorcontrolling the atom activity (about randomisation of quantum events see below).Nakhmanson p. 5 of 20The new conception is a "hidden-parameter" one, but these "parameters" are not mechanical ones. But what about Bell's theorem saying that if QM is valid no localhidden-parameter theory is possible? The proof of Bell's theorem is based on the next assertion: if P a is a probability of result a measured on the particle 1 in the point A having a condition (e.g. angle of analyzer) α , and Pb is a probability of result b measured on the particle 2 in the distant point B having a condition β , then β has no influence on the Pa , and vice versa. Here Bell and others saw the indispensable requirement of local realism and "separability".Mathematically it can be written as P ab(λ1i,λ2i,α,β) = Pa(λ1i,α)×Pb(λ2i,β) (Bell) , (1) where Pab is the probability of the join result ab , and λ1i and λ2i are hidden parameters of particles 1 and 2 in an arbitrary local-realistic theory. Under the influence of Bell's theorem and the experiments following it and showing, that forentangled particles the condition (1) is no longer valid, some "realists" reject locality. Inthis case an instantaneous action at a distance is possible, and one can write P ab(λ1i,λ2i,α,β) = Pa(λ1i,α,β)×Pb(λ2i,β,α) (non-locality) . (2) In principle such a relation permits a description of any correlation between a and b , particularly predicted by QM and observed in experiments. But in the frame of localrealism the condition (1) is not indispensable. Instead, one can write P ab(λ1i,λ2i,α,β) = Pa(λ1i,α,β´)×Pb(λ2i,β,α´) (forecast) , (3) where α´ and β´ are the conditions of measurements in points A and B , respectively, as they can be forecast by particles at the moment of their parting. If the forecast is good enough, i.e., α´ ≈ α and β´ ≈ β , then (3) practically coincides with (2) and has all its advantages plus locality. On the issue of "separability": The entangled particles have a common strategy and keep it as long as they can forecast the future at the moment of their parting. The new-coming unlooked-for circumstances allow them gradually to cut off and forget the oldpartnership. The wave-particle duality is a mind-body one. In the space there exists only the particle; the ψ-wave exists in its consciousness, as well as the reflection of the whole world. If there are many particles, their distribution in accordance with the ψ-function, e.g. interference fringes, looks like a product of real wave in real space. Because of free will the behavior of particles is not strictly determined. In situations allowing alternative outputs the theory gives only a distribution of priorities. Taking thisinto account the particle makes its choice. The optimal tactics of proportional proving ofall possibilities is randomization of this choice. It seems as if each particle has its ownrandom generator. The same seems to be valid for people: our unconsciousness oftendoes not choose the "optimal" way but dices obviously using the priorities. However,our consciousness does not yet make sufficient use of proportional randomization [20] . It is known that Einstein did not accept the fortuity implicated by QM: "God does not play dice." Bohr replied: "It is not the job of scientists to prescribe to God how HeNakhmanson p. 6 of 20should run the world." But the matter in question is not God but particles: they play dice. As was said, people also use random choice. Some of them believe that so theytransfer the choice of decision to God, i.e., God determines the decision and His rôle isexactly the one that Einstein wanted. On the other hand, Einstein believed in Spinoza'sGod, "which concerns of nature and give up people themselves". If Einstein thought themicroparticles might be intelligent, perhaps he would give up microparticles themselves,so that his God can concentrate Himself on something more important. Very likely the particles are artificial things like some small space ships. Division into different sorts or species with internal identities is typical for mass products. Itsimplifies production, usage, repairs, and replacement of such objects. Technicalobjects, plants, animals and humans illustrate this very well. In the last three cases theproduction is ruled on the genetic level. For example, people have a very narrowstatistical distribution of sizes, masses, and performance; the world records in sportdiffer from the average results not more than twice. The identity of particles of one sortin QM is analogous to the identity of vehicles of one sort with respect to traffic rules.The individual differences (registration number, color, firma-producer, sex and name ofdriver, etc.) lie beyond the rules. The individual knowledge of elementary particles liealso beyond QM. Connection between physics and information was touched in latent form in earlier discussions around the second law of thermodynamics and Maxwell's "demon" yet in19th century. Later it was manifested, e.g., in [21] . One modification of demon'smachine is shown in Fig. 1. There is a chamber having two pistons 1 and 2 and filled with gas. The pressure of gas is estimated by atmospheric one, temperature and theweight W sitting on the piston 2 . If last three parameters are constant and the chamber has no leakage, the pressure of gas and the volume V occupied by gas are constant too. FIG. 1. The chamber with two pistons 1 and 2 contains gas occupying volume V. The "demon" moves the piston 1 in time intervals between hits of molecules and lifts the weight W by virtue of thermal energy of surrounding only. The "demon" stays beyond the piston 1 and moves it, but only in time intervals between the hits of molecules. Therefore the demon does not spend any energy to do it./i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 12 VW 2Nakhmanson p. 7 of 20Because V is constant the piston 2 moves up elevating the weight W . The molecules reflecting from moving piston 2 lose their velocity i.e. are cooled, but renew their energy through contact with surrounding. Therefore the demon's machine transform100% of thermal energy coming from the surroundings into mechanical energy, realizingthe "perpetuum mobile of the second kind". If the piston 2 were fixed at the beginning, the gas would be compressed and, without performing any work on the gas, its entropywould be decreased. Both "results" contradict the second law of thermodynamics. To save the second law of thermodynamics the concept of information was called up [21]. The law was rewritten: instead of ∆S ≥ 0 there came ∆S ≥ ∆I , (4) where ∆S is the entropy produced (in our case by demon and his measuring apparatus) to have a new information ∆I (and to erase the old one). In our last case the demon spends a part of this information ∆I' ∆I' ≤ ∆I , (5) to move the piston 1 skillfully to compress the gas and change its entropy ∆S' without work. Again ∆S' ≥ ∆I' , (6) where both partners are negative. Combining (4) to (6) leads to ∆S ≥ ∆I ≥ −∆I' ≥ −∆S' , i.e. ∆S + ∆S' ≥ 0, (7) i.e. the common entropy increases (or stays constant for reversible processes) in accordance with the (corrected) second law. "We cannot get anything for nothing, noteven an observation", - Gabor concluded. Nevertheless, the second law persists to be a puzzle which cannot be deduced from mechanics with its time-symmetrical laws and, as well as the "time arrow", must beaccepted ad hoc . The informational interpretation of QM can open up a new possibility to understand the second law: states with maximum entropy can be preferred by particles for havingsome purposeful advantages. For example, molecules of gas in a chamber search thespace and walls better when they are distributing themselves stochastic-homogeneously,both as an ensemble as well as individuals averaged in time. Accordingly, beinginjected in the chamber, the molecules try to develop such a distribution faster and keepit forever. In the state with maximum entropy the molecules disperse themselvesrandomly, but if they discover a deviation from this state they act purposefully anddepress the fluctuation using control of the collision parameters (by collisions with wallsand with each other) and so producing the macroscopic asymmetry of time. In the book The Chance [22] , issued in 1913, the year of Bohr's atom model, Emil Borel, a notable pure and applied mathematician and an acquaintance of the de Brogliefamily, remarked that humans cause entropy to decrease in small volumes by means ofprocesses accompanied by increasing of entropy in big volumes, and then went on asfollows:Nakhmanson p. 8 of 20In other words, the structure of the universe is becoming more and more subtle ... , It is probable that similar phenomena are going on other scales as well, too large or too small to beaccessible to us. Thus, the evolution of the universe may be represented as a gradualcomplication of its structure, accessible to understanding and use of beings of lesser and lessersize. As there is no absolute standard of length, we may not be afraid of such a lessening ofscales; it seems to us presently that beings of molecular sizes and, all the more, beings so smallin respect to molecules as we are to sun, are objects scarcely deserving our attention; but it isquite possible that the progressing complexity of the universe will create or has already createdsome beings with an organization much more complex than ours. Following Borel's idea the second law and time arrow in our macroscopical world can also be ascribed to the intellectual activity of microparticles drawing off the"negentropy" in their world (Is part of that drawn off also? [23] ) . Therefore themicroworld must own a lot of "negentropy", e.g., information. Because information,more as energy, is the essence of life, can mankind ask microparticles to reverse thesecond law (and, on occasion, the direction of time) and to return a part of thisnegentropy to the macroscopical world? To do so, we must have an informationalcontact with them. Some ideas of such contacts will be presented in Sections IV and V. III. experimenta crucis In some experiments performed up to now, the information spreading was implied and one tried to interrupt it, however without success. In others one tried to achieve theinformation about a material object without having any material contact with it. Belowwe discuss four of the experiments from the new point of view. FIG. 2. "Delayed-choice" experiment with Mach-Zender interferometer. S is the source of photons, B1 and B2 are the beamsplitters, M1 and M2 are the mirrors, D1 and D2 are the detectors, P is the Pockels cell. The ciphers indicate probabilities to find a photon. •Delayed-choice experiment. It was suggested by Wheeler [7] and really performed in Marylend [8]. Fig. 2 shows the schema. The particles (photons) comingfrom source S enter into the Mach-Zender interferometer with beamsplitters B 1 and"S B1B2 MM1 2D1D2 11/21/2 1/21/21/2 or 00 or 1/4 1 or 1/4PNakhmanson p. 9 of 20B2 , mirrors M1 and M2 and detectors D1 and D2 . In one ("upper") arm of the interferometer there is the Pockels cell P , a very fast shutter controlled by a voltage. The interferometer is tuned so that if the Pockels cell is opened all photons are comingto the detector D 1 due to constructive interference. The flow of photons is so weak that no more than one photon in the interferometer is a rule. The Pockels cell is at all times opened (closed), except for a short time interval τ (several nanoseconds) when it is closed (opened). As a result of experiments having been performed, the interference after B2 appears only if the Pockels cell was opened during the τ at the appropriate moment when a photon can pass the Pockels cell, independent of its state at other times. This result cannot be explained on the level of classical local realism thinking that, after B1 , a photon really chooses one arm (upper or lower) of the interferometer: If the photon really traveled through the lower arm (this "must be" in 50% of the cases), it istoo far (several meters) from the Pockels cell, and at the above mentioned "appropriatemoment" is under no "influence" by the cell (action spreading with velocity u>c i s supposed to be impossible). It must be emphasized that independent of the conditionu>c the "influence" was not ascribed to some known physical mechanism and thought more as an information-like abstraction. From a new point of view all this looks different and very natural. When a photon meets the first beamsplitter B 1 it brings new information from the source S . Simultaneously it receives from B1 the fresh information about the past of the world, particularly about the interferometer and Pockels cell. The physical conditions at B1 induce a 50% choice. But the decision of photons must be at random: It is the optimal tactics for an ensemble of disconnected photons. The consciousness of the photon works to find an optimal strategy, i.e., ψ-function. In this case it is a 50% choice of arm after B1 and interference after B2 , if Pockels cell just appears opened. To find it, the photon solves a variation problem (e.g., the wave equation). Let us suggest that after B1 the photon takes the lower arm. It meets the mirror M2 and acquires some new information about the past of the world. In respect to inteferometer, it is the same information, independent of the state of the Pockels cell inthe upper arm, because in the "appropriate moment" the photon is nearer to M 2 as P or even already has left M2 . Therefore the part of the wave function of photons relating to the interferometer stays the same. Particularly, if at the beginning the Pockels cellappears to be opened, the photon keeps the idea to interfere after B 2 . Finally, the photon meets the beamsplitter B2 and must again choose its path. Simultaneously it receives from B2 the fresher information about the past of the world, including an actual history of the upper arm of the interferometer with the state ofPockels cell at the "appropriate moment". This information is brought to B 2 with velocity of light, e.g., by thermal photons emitted by Pockels cell itself: at room temperature the characteristic time τ of several nanoseconds is enough for B2 t o receive from P several thousands of thermal photons. Actually, the number of thermal photons emitted by one and arriving at the other macroobject is N ≈ 5⋅1010 S1 S2 T3 τ / L2 ,Nakhmanson p. 10 of 20where S1 and S2 are the effective areas of the first and the second macroobjects, respectively, T is the absolute temperature of S1 , τ is the characteristic time, and L is the distance between S1 and S2 . Substituting the typical values S1 = S2 = 2 cm2 , T = 293°K, τ = 10-8sec, and L = 103 cm, one finds N ≈ 5000 . Now the photon has all necessary information to make a decision, namely, to prefer a direction of constructive interference (if the Pockels cell at the "appropriate moment"was opened) or to make a 50% random choice between two possible directions (if thecell at the "appropriate moment" was closed). If the state of Pockels cell in the"appropriate moment" was different than before, a reduction of the strategy (i.e., a reduction of wave function ψ relating to the experiment) takes place in the consciousness of the photon after interaction with the beamsplitter B 2 . To have a success in the "delayed-choice" experiment (i.e., to have a result going outside the frame of QM) one can try to cut off the informational contact between thePockels cell and the beamsplitter B 2 , e.g. by introducing a deep-cooled filter which is transparent for optical photons being measured but absorbent for thermal photons. Inspite of the fact that all the atoms of the filter would know the state of the Pockels cell, ifthe filter is cooled by liquid helium, no thermal photons come from it to B 2 during the characteristic time τ . Further, one can put the Pockels cell together with its voltage source in a separate electromagnetic screen to cut off the radio channel [24]. Besides, toprevent a forecast of a state of the Pockels cell by B 2 and other "member" of the experiment, it is better to control the cell using not a regular but "good" randomgenerator. •In scheme Fig. 2 one can indicate the existence of a non-transparent object (e.g. closed Pockels cell) if the photon is detected by D2 , in spite of the fact that the photon has traveled via the lower arm and does not touch the object. The developing of thetheme is co-called "interaction-free measurements" [25]. These experiments can benaturally explained within the scope of the informational interpretation of QM: thephoton receives the actual information about the object from the beamsplitter B 2 . Moreover, because the experiments are performed in stationary conditions, the photonknows all about object just when it is born. •The situation discussed by Einstein, Podolsky, and Rosen [26], and in modern form by Bohm [27] and Bell [6], is more complex. Here two particles flying in the oppositedirections have a common non-factorizable wave function, i.e., common correlatedstrategy. As a consequence, the result of interaction of one particle with somemeasurement apparatus is strongly correlated with the result of the interaction of anotherparticle with another measurement apparatus, in spite of a large space separating of thetwo apparatuses. As it was seemed, such a correlation is possible only if the particles areconnected by instantaneous non-local interaction. The informational interpretationoffers another possibility: each apparatus and/or the particle interacting with it canforecast the state of the other apparatus at the "appropriate moment" of measuring with a good probability. It was mentioned in Section II that such a forecast is the very naturalresult of evolution of microparticle consciousness. In all EPR-experiments made up tonow, such forecasts could take place, without assuming a superluminal velocity ofinterconnection, because the states of the measurement apparatuses have been changedNakhmanson p. 11 of 20very slowly [9] or periodically [10] . To go outside the scope of QM we can try to perform EPR-experiments with a "good" random control of measurement apparatuses. It is interesting to note that the authors of cited experimental researches felt the advantage of random control (as it seems more intuitive, because they do not discuss it)and sometimes used it [8]. The peculiarity of a random signal series is the non-predictability of its next term. Therefore, these authors felt a possibility of "inanimate"matter to forecast the future, and have tried to restrict it [8, 10]. The authors of recentwork [28] said it clearly: “Selection of an analyzer direction has to be completelyunpredictable, which necessitates a physical random number generator. A pseudo-random-number generator cannot be used, since its state at any time is predetermined”(p.5039). As a “physical random number generator” they used a “light-emitting diodeilluminating a beam splitter whose outputs are monitored by photomultipliers” (p.5041). Such a point of view must be commented. It is a right prerequisite that the generator must be unpredictable for particles, but referring to above “physical” generator is notenough: If particles have “consciousness” such a generator can also be a “pseudo” one.Perhaps a good “human” pseudo-random generator is preferable because it belongs toanother civilization. •Aharonov-Bohm effect [29]. In accordance with QM the frequency of wave- function oscillations depends on the energy. If the particle has different energies indifferent arms of the interferometer, it leads to an additional phase shift and changes theinterference pattern. The experiments were performed with an electron interferometerand a magnetic vector potential and justified the predictions of QM. It is of interest thatin the experiments the electrons did not cross the magnetic field. From the old classicalpoint of view it looks like non-local action at a distance. The new interpretationexplains it very naturally: electrons know the situation from the time they are emitted.Besides, this effect emphasizes a priority of potential against field (in classical physicsthey enjoy equal rights). From the new point of view it is natural, because potential justcontributes to the action function whose minimum as a function of trajectory is wanted.It should be observed that the idea of forecasting the conditions on the trajectory is also included in the least action principle. The change from integral form to a differentialone does not remove the problem: the mathematical derivative is an abstraction, thephysical (i.e. operational) derivative is connected with two distant points, and if theparticle is in one of them, it knows only the past conditions in the second point and mustextend this into the present and future. To obtain a "non-QM" result one can try to usefast random switching of the magnetic vector potential. IV. Informational experiments If big strangers from other worlds arrived on the Earth and would like to test human intelligence, they would not, of course, do it by dipping men into a bath, throwing themfrom the tower of Pisa, or bumping them against each other in colliding beams, i.e., thestrangers would not just act in this brutal way in which physicists have treated particles,hitherto. Instead of this they would try to build an informational contact with men.Different levels of reaction on the information being proposed would be expected:Nakhmanson p. 12 of 200. No noticeable reaction. 1. A reaction showing that information is being received.2. A reaction showing that information is being deciphered correctly.3. Sending of reciprocal signals. Have physicists been searching "inanimate" matter as a possibility to build an informational contact with microparticles? The answer is "Yes". Moreover, if theparticles are intelligent, up to third reaction level and complete duplex communicationare possible. Fig. 3 introduces into this field. Fig. 3( a) shows a "black box" which is tested by the linearly polarized light beam. Inside of the box the beam meets a thick transparent glassplate fixed at the Brewster angle so that all photons pass the box. The glass plate manifest itself physically only by space shift ∆z and time delay ∆t of output photons. Further there is a movable mirror ("traffic divarication") which is controlled by theexperimenter to turn or not to turn the beam. Such a control is a brutal one like a trafficbarrier closing one of two branches of the road. FIG. 3. ( a ) - "brutal" control, ( b ) and ( c ) - informational control . In Figs. 3( b) and 3( c) the mirror is semi-transparent and immovable, and the thick glass plate is divided into eight thin plates, two of them being thicker than the remainingsix. As before, the content of the black box manifests itself physically only by the same space shift ∆z and time delay ∆t . But if the photons are intelligent and know English and Morse code, they can read the messages, namely/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 ∆z ∆t"a( ) /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0∆z ∆t"c( )/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0∆z ∆t"b( )Nakhmanson p. 13 of 20 • — • • • • — • = REF (reflect) in Fig. 3( b) , — • • • • • — • = THR (through) in Fig. 3( c) , and follow the instructions. Such a control is an informational one like traffic signs on the road. It is important to emphasize that the idea of informational experiments with particles, as it seems, has never been publicly discussed, and all experiments made with particlesup to now cannot be considered as informational ones even in retrospect, that is, therevision of their results would not enable us to make any conclusion relating to this idea. The way to communication with the particles assumes that they are interested in the information proposed (first level of reaction), that they learn to decipher the information(second level), and that they wish to communicate (third level). The interest ininformation is thought to be an inherent attribute of each consciousness. One can try touse at the beginning such an attractive "language" as music, and for advance teachingand communication special languages have developed for the project "Search forExtraTerrestrial Intelligence (SETI)" [30] . Fig. 4 shows the scheme of "binary-tree" experiment. The initial beam of microobjects (particles, atoms) enters into a system of beamsplitters (shown by circles).They can be semi-transparent mirrors for photons, crystals for electrons or neutrons,Stern-Gerlach apparatuses for atoms, etc. Fig. 4 shows only five rows of beamsplitters,but there can be as many as experimentally feasible. According to present-daytheoretical ideas and practical experience, each of the output beams has the sameintensity, namely, 1/32 of intensity of the initial beam (real beamsplitters may have, ofcourse, some absorption, but here it is not a matter of principle). Into each of the right channels of the binary tree is introduced an "informational cell" (shown by birds), which is a device leaving unchanged the intensity of the beam passingit, but offering some information to particles. For polarized photons such a cell may be aset of transparent plates fixed at the Brewster angle, and the information can be coded,say, by differences in the materials of plates, their thickness, and distances between them(see Figs. 3( b), 3(c)) . Absorption caused by the information cells may be compensated by introducing into each of the left channels a "compensating cell" bearing noinformation, more precisely, bearing less, or less significant information (e.g. the sameglass plates placed randomly or periodically). The information in each subsequent rowis a sequel of that in the preceding row. The commonly accepted point of view is that the introduction of information cells, together with compensators, will not change the uniform probability distribution ofparticles in the output beams. But if particles are intelligent, and are able to notice theinformation offered to them, they may become interested in it. After a number of rows,the particles should notice that the information is offered only in the right channels, andshould prefer the choice of right channels in passing through the followingbeamsplitters. In other words, particles could develop a "conditioned reflex", ofessentially the same kind as in behavior experiments on living beings.Nakhmanson p. 14 of 20FIG. 4. Binary-tree experiment. Circles stand for beamsplitters, birds denote informational cells, worm denotes the particle. Ciphers and apples (=0) show theprobability of detecting the particle in the case of the most rapid formation of a rigid,conservative conditioned reflex. The absorption of the particles by informationalcells is assumed to be negligible (i.e. the birds do not eat worms more). Such an inquisitiveness of particles should lead to a change of their distribution in the output beams. For example, if the conditional reflex appears immediately and theparticles are "conservative", i.e., they are no more of interest to the left channels, thedistribution of probability to find the particle in different branches of the binary tree islike the one shown in Fig. 4 by ciphers. Deviation from the uniform distribution of particles in the output beams will mean that the particles at least recognize the information offered and have an interest in it.This, however, still does not mean that the particles understand this information: peopleof modern times were interested in ancient hieroglyphic symbols long before theylearned how to interpret them. To establish a deciphering stage, one can, starting fromsome row of a binary tree, introduce some specific "requests" into the information cells.For example, one can "ask" particles to choose a left channel after the next beamsplitterrather than a right one. Because between the output branches of the binary tree and thetrajectories of the particles there is a one-to-one interrelation, the honoring of such kindof requests can easily be detected by an experimenter. However, the possibilities of anexperiment typified in Fig. 4 are not exhausted by this second level of communication.Purposefully choosing direction at each subsequent beamsplitter, the particle, in its turn,can send information to the experimenter using "right" and "left" as a binary code. For000 000321 321 321 321 161 161 161 161 161 161 81 8181 81 818181 81 41 41 41 41414141 414141 21 21 100 0 00 0 000 000 0000 0000 00 00 00 0 0(31) /i0/i0/i0/i0/i0/i0Don't sit un-der theap-ple tree withan-y-one else butme 0 000 00 00 00 00 00Nakhmanson p. 15 of 20example, extreme left and extreme right trajectories in Fig.4 present 00000 = 0 and 11111 = 31, respectively. The scheme in Fig. 5 shows another possibility. The short light impulse passes through the semi-transparent mirror E and penetrate into a volume limited by two mirrors A and B having reflection near to 100% . If the mirrors are perpendicular to the direction of light propagation, then the light impulse will undergo hundreds ofreflections before it is essentially extinguished. At each reflection of the "catched"impulse a small part of the light goes through the mirrors A or B and is registered by detectors Da or Db . A semi-transparent mirror C is placed in the middle position between A and B and parallel to them. The impulses registered by detectors follow with time intervals ∆t = AB / c . Along the optical path between A and C there is an informational cell I controlled from the source of information SI , so that information is renewed with the frequency 1/ ∆t . The synchronization is provided from the detector Da . FIG. 5. The "dynamic" informational experiment. A , B , C , and E are mirrors, Da and Db are detectors, IC is the informational cell, SI is the source of information. The Pockels cell P is closed only at the short time interval at the beginning to prevent an overload of Da by the reflection from A . Above, output impulses as functions of time are shown for the case of the first level informational contact and the most rapidformation of a rigid, undisturbable conditioned reflex. The moment the SI is switched on is denoted by arrows. The timing of SI is controlled from Da . Before the informational cell I is activated the numbers of photons in the volumes AC and CB are equal and the impulses registered by Da and Db have the same amplitudes (the absorption is assumed to be negligible). But after it's activation, andprovided the photons are able to perceive the information proposed and to be interestedin it, then, after a number of transitions in AB , they should develop a conditioned reflex and, thus, prefer to remain in the volume AC to obtain information. They are able to do so because of their liberty of choice in interacting with the semi-transparent mirror C . This means that the photons would mainly reflect from the mirror C when approaching1 21 01 41 2 1 8 1 641 161 321 21 03 47 815 1631 3263 64 1 2 t tDa Db laser AE C BPSI IC DbDa 99.9% 99.9% 50%50%Nakhmanson p. 16 of 20it from the left, and would mainly go through the mirror C when approaching it from the right. The number of photons in the volume AC and the amplitude of the output impulses beyond the mirror A would increase (twice, at most). The corresponding decrease of the number of photons and of output impulses (to zero as a limit) should beobserved for the volume CB . In Fig. 5, at the top, the graphs of intensity of output impulses are represented as functions of time, for the case of informational contact ofthe first level and of most rapid formation of a rigid, undisturbable conditioned reflex.The moment of activation of the informational cell is shown by arrows. The scheme Fig. 5 can also be used for informational contacts of second and third levels by redistribution of photons between AC (e.g. representing "0") and CB (representing "1"). FIG. 6 Informational experiment with a single atom. 1, 2, 3, and 4 are the energy levels; D23 and D24 are detectors, S is the source of light, M is the modulator, SI is the source of information . The experiments illustrated in Fig. 4 and Fig. 5 can be called "coordinate-impulse" ones to distinguish them from the "energy-time" experiment whose scheme is shown inFig. 6 . Here a four-level quantum system, e.g., an atom, with one low (1), one high (2),and two intermediate (3,4) energy levels is pumped by intensive radiation inducing the 1→2 transition, so that the atom stays not in the state 1 but immediately is translated into the state 2. From it, the atom makes transition spontaneously to the state 3 or 4, and latermakes transition to the state 1 completing the cycle. The radiation corresponding to some of transitions 2 →3, 2→4, 3→1 and 4→1 are detected (in Fig. 6 two detectors are shown). Besides, there is an informational action on the atom, e.g., by modulation oflight coming from the source S . The modulator M is controlled by the source of information SI , which, in turn, is connected with one or more detectors to close the feedback loop. The feedback works in such a way as to stimulate a channel and rate of transitions, in the case of Fig. 6, the 2 →4→1 transitions. The source SI sends a message, e.g., one line of a page or a measure of a music, only if it receives a signal from detector D 24 . Each next message continues the previous one, i.e., is the next line or the next measure.12 34«««D24D23 SI S MNakhmanson p. 17 of 20If the atom has an intelligence and is interested in the information being proposed, it develops a conditioned reflex and will prefer the 2 →4 transition to the 2 →3 one. Besides, the rates of both 2 →4 and 4 →1 transitions must increase. All this can be registered by the experimenter. To be sure that the effect is connected with information,one can make a control experiment to cut off the feedback or/and to use some "trivial"information, etc. If the detector D 24 has a small aperture, the spatial orientation of emitted photons toward to D24 may also develop itself as a part of conditioned reflex. In spite of this interesting possibility, one must prefer to use effective detection of emitted photons tofacilitate the development of a conditioned reflex. Perhaps the combination of an iontrap and resonator [32] provides a good opportunity for informational experiments withsingle atoms. A progress in semiconductor device technology possibly can also be used for such an "energy-time" experiment. In a small metal-oxide-silicon field-effect transistor(MOSFET) one can observe random telegraph signals corresponding to the charge anddischarge of a single electron trap located at the Si-SiO 2 interface [33]. As in the previous case, the new information may be sent to the MOSFET (i.e. to the trap) bymodulation of light or sound only after the next capture or/and emission of an electronby the trap. It closes the feedback loop, and the experimenter may find an increase in thecapture and emission rate. Like with the schemes of Fig. 4 and Fig. 5, in two last cases one may hope to observe not only an interest of a quantum object (atom, electron trap) to receive a newinformation, but deciphering it also, as well as the sending of messages from the objectto the experimenter being coded in states of the atom or electron trap and time intervalsbetween the states. V. Informational world of matter and human consciousness If the microparticles and atoms are really "intelligent", why they do not manifest themselves more obviously? For example, why do molecules of air in the room notcongregate themselves near radio or TV to hear music and to watch a movie? Asemantic answer could be, e.g., that they are not interested in it. Physicists can also takeinto account the large difference in spatial and temporal scales between structuresanalogous in their intelligence functions, say, something like the relation between thesize of the solar system and that of the atom (10 21). The molecules of air have millions of collisions per second, and each of them brings a lot of information. In the context theweak and slow alternations of pressure (due to music) or pixel brightness (on TV screen)are noteless. Therefore special conditions and arrangements as described in the previousSection supported by screening and cooling are needed. They can serve not only to havean informational contact with separate particles and atoms. Because it appears that thematerial world is entangled by information, the above-mentioned arrangements can alsoserve as interfaces to the "internet of matter" .Nakhmanson p. 18 of 20It is not to exclude that the evolution of plants, animals and men had found possibilities to develop the interfaces connecting them to the ghost of matter. Such aconnection would satisfy not only their inquisitiveness but would be useful also. Thosewho believe in parapsychological phenomena (among them were many notablephysicists), as well as scientists searching mind-matter connections [14-17], can find asolid base in the informational interpretation of QM. For example, information mustspread in an internet of matter without an attenuation law 1/ r 2 and pass through usual screens - the peculiarities ascribed to these phenomena. Telepathy and clairvoyance canuse the internet of matter to send and receive information, and forecasting can use inaddition the corresponding presages of particles. The known descriptions of telekinesisimplicate the appearance of mechanical energy. Such an energy can be transformedfrom thermal energy with apparent violation of the second law of thermodynamics (seeend of Section II). It is known that people have several levels of memory. The first one registers all information received by the sense-organs but keeps it only for a short time. Afterselection the essential part of it is transferred into the second level, etc. The moreimportant the information the deeper level it reaches and the longer it is conserved there.The deepest known level is the genetic one. The idea of informational contact betweenthe world of matter and living beings permits the existence of deeper levels of memorysituated in the atoms and elementary particles of these beings [34]. If this is so, thebeliefs in souls floating in air and in reincarnation have chances to be accepted byphilosophiae naturalis . VI. Conclusion In this paper the informational interpretation of quantum mechanics was presented. It is realistic with local interactions, it is in agreement with common sense, it allowsexperimental verification and explains the quantum paradoxes very naturally. It shedsnew light to problems of thermodynamical irreversibility and mind-matter interaction,and extends the field for scientific, technical, science-fiction, and religious speculations.Anyone of these perspectives is interesting enough to prove it. As it seems not only Boltzmann-Shannon but semantic information also must be introduced in physics. Was it really a Word at the beginning, if it was?Nakhmanson p. 19 of 20References [1] "... we want more than just a formula. First we have an observation, then we have numbers that we measure, then we have a law which summarizes all the numbers. But thereal glory of science is that we can find a way of thinking such that the law is evident ." The Feynman lectures on physics (Addison-Wesley, MA, 1966), p.26-3. [2] R.Nakhmanson, Preprint 38 -79 (Institute of Semiconductor Physics, Novosibirsk, 1980); see also A.Berezin and R.Nakhmanson. J. of Physics Essays 3, 331 (1990). 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[9] S.J.Freedman and J.F.Clauser, Phys.Rev.Lett. 28, 938 (1972). [10] A.Aspect, J.Dalibard, and G.Roger, Phys.Rev.Lett. 49, 1804 (1982). [11] J. von Neumann, Die mathematischen Grundlagen der Quantenmechanik (Springer, Berlin, 1932). [12] F.London and E.Bauer, La théorie de l'observation en mécanique quantique , No.755 of Actualités scientifiques et industrielles: Exposés de physique générale (Hermann et Cie, Paris, 1939), engl. transl. in Quantum Theory and Measurement , ed. by J.A.Wheeler and W.H.Zurek (Princeton University Press, Princeton, NJ, 1983). [13] E.P.Wigner, in The Scientist Speculates , ed. by I.J.Good (London, Heinemann, 1961); see also E.P.Wigner, Symmetries and Reflections (M.T.I. Press, Cambridge, MA, 1970). [14] L.Bass, Found.Phys. 5, 159 (1975). [15] J.Hall, C.Kim, B.McElroy, and A.Shimony, Found. Phys. 7, 759 (1977). [16] R.G.Jahn and B.J.Dunne, Found. Phys. 16, 721 (1986). [17] D.I.Radin and R.D.Nelson, Found. Phys. 19, 1499 (1989). [18] M.A.Markov, On the Nature of Matter , (Moscow, 1963, in Russian); id., Ann. Phys. 59, 109 (1970). [19] E.Schrödinger, Naturwiss. 23, 807 (1935).Nakhmanson p. 20 of 20[20] The roulette, lottery, Monte-Carlo method of numerical calculation etc. are without question. The election of Doge in Venice (697-1797 a.C.) was going in ten steps, five ofthem used random event generators. This experience was successful: only one of doge actedultra vires . If the democracy means the same chance for everyone to realize his program, the election must serve only to estimate the priorities ( ψ-function), not the decision, otherwise we have the dictatorship of majority. The decision must be estimated byproportional random procedure. Thereto such a procedure makes coalitions and wars for51% meaningless. If Buridan's ass ventured to use a random procedure in choosing hisbundle of hay, he would thus "quantize" himself into one of the two states, and thus escapestarvation (noted by V.G.Yerkov). [21] L.Szillard, Zs. f. Physik 53, 840 (1929); D.Gabor, MIT Lectures (Massachusetts, 1951); L.Brillouin, Science and Information Theory (Academic Press, New York, 1970); I.Prigogine and I.Stengers, Order out of Chaos (Shambhala, Boulder, 1984). It must be noted, that the matter of discussion cited is information defined by Shannon's formula(which coincides, within the sign, with Boltzman's one for entropy), not the semantics. [22] E.Borel, Le Hasard (Paris, 1913). [23] "Great fleas have little fleas, Upon their backs to bite 'em;Little fleas have lesser fleas,And so ad infinitum ." ( J. Swift ) [24] The efficiency of other fast channels (neutrinos, gravitons, ...?) can not be excluded. [25] A.C.Elitzur and L.Vaidman, Found. Phys. 23, 987 (1993); P.Kwiat, H.Weinfurter, T.Herzog, A.Zeilinger, and M.Kasevich, Ann. NY Acad. Sci. 755, 383 (1995); A.G.White, J.R.Mitchell, O.Nairz, and P.G.Kwiat, Phys. Rev. A 58, 605 (1998). [26] A.Einstein, B.Podolsky, and N.Rosen, Phys. Rev. 47, 777 (1935). [27] D.Bohm, Quantum Theory (Prentice Hall, New York, 1951). [28] G.Weihs, T.Jennewein, Ch.Simon, H.Weinfurter, and A.Zeilinger, Phys. Rev. Lett. 81, 5039 (1998). [29] Y.Aharonov and D.Bohm, Phys. Rev. 115, 485 (1959). [30] H.Freudental, LINCOS: Design of a Language for Cosmic Intercourse (North-Holland, Amsterdam, 1960). [31] L.Brown, C.Tobias, and S.H.Stept, Don't Sit Under The Apple Tree . [32] G.Rempe, W.Schleich, M.O.Scully, and H.Walther, in Proceedings of 3rd International Symposium on Foundation of Quantum Mechanics (Physical Society of Japan, Tokyo, 1989). [33] K.S.Ralls, W.J.Skocpol, L.D.Jackel, R.E.Howard, L.A.Fetter, R.W.Epworth, and D.M.Tennant, Phys. Rev. Lett. 52, 228 (1984). [34] From this point of view the inquisition had found the best way to spread heresy by burning.

arXiv:physics/0004048v1 [physics.plasm-ph] 20 Apr 2000EXCITATION OF NONLINEAR TWO-DIMENSIONAL WAKE WAVES IN RADIALLY-NONUNIFORM PLASMA Arsen G. Khachatryan Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia It is shown that an undesirable curvature of the wave front of two- dimensional nonlinear wake wave excited in uniform plasma b y a relativistic charged bunch or laser pulse may be compensated by radial cha nge of the equilibrium plasma density. PACS number(s): 52.40.Mj, 52.40.Nk, 52.75.Di, 52.35.Mw The progress in the technology of ultrahigh intensity laser s and high cur- rent relativistic charged bunch sources permits the use of l aser pulses [1] or charged bunches [2] for excitation of strong plasma waves . The excited plasma waves can be used both for acceleration of charged par ticles and focusing of bunches to get high luminosity in the linear coll iders [2]. At present the plasma-based accelerator concepts are activel y developed both theoretically and experimentally (see the overview in Ref. [3] and references therein). The amplitude of longitudinal electric field Emaxin relativistic wake-waves excited in cold plasma is limited by the relativistic wave-b reaking field [4] Erel= [2(γ−1)]1/2EWB/β, where γ= (1−β2)−1/2is a relativistic factor, β=vph/cis a dimensionless phase velocity of the wave, EWB=meωpevph/e (EWB[V/cm ]≈0.96n1/2 p[cm−3]) is the conventional nonrelativistic wave- breaking field, ωpe= (4πnpe2/me)1/2is the electron plasma frequency, npis the equilibrium density of plasma electrons, meandeare the mass and abso- lute value of the electron charge. The acceleration rate in t he wake fields can reach tens of GeV/m , that much exceeds the rates reached in conventional accelerators. The linear wake-field theory is valid when Emax≪EWB. In case of wake wave excitation by relativistic charged particle bunch [Pl asma Wakefield Ac- 1celerator (PWFA)] this corresponds to the condition α=|q|nb/upslopeenp≪1 [2], where qis the charge of bunch particles, nbis their concentration. In the scheme of wake wave excitation by a short laser pulse (of the l ength compa- rable with plasma wavelength λp= 2πvph/upslopeωpe; Laser Wakefield Accelerator (LWFA)), the linear theory is valid when a2=e2E2 0/upslope(m2 ec2ω2 0)≪1 [5], where ω0≫ωpeandE0are the frequency and amplitude of laser radiation respectively. The phase velocity of the wake wave is equal to the bunch ve- locity vbin PWFA and to the laser pulse group velocity vg≈c(1−ω2 pe/2ω2 0) ( that corresponds to γ=γg≈ω0/ωpe≫1) in LWFA. The one-dimensional nonlinear wake waves excited by wide dr ivers (when kprd≫1, where kp=ωpe/vphis the wavenumber and rdis the radius of the driver) are studied in sufficient detail both for PWFA [6] a nd for LWFA [7]. These studies testify to the feasibility of excitation of strong nonlinear plasma waves with the amplitude of up to Erelby bunches with α/greaterorsimilar0.5 and laser pulses with a2/greaterorsimilar1. The other important results of the one-dimensional nonlinear theory are the steepening of the wake wave and the i ncrease of wavelength with amplitude. The wave with the amplitude Emax≈Erelhas the wavelength nearly γ1/2times as large as the linear wavelength λp. In reality, the transverse sizes of the drivers used are ordi narily compa- rable or less than their longitudinal size. The allowance fo r finite transverse sizes of the drivers and, accordingly, the transverse motio n of plasma elec- trons complicate the treatment of the problem in the nonline ar regime. In the general case the analytical solution of this regime seem s impossible and here the use of numerical methods are usually required. The n umerical inves- tigation of nonlinear effects in two-dimensional (axially- symmetrical) wake waves exited in uniform plasma has shown that in the nonlinea r regime the wavelength becomes dependent on the transverse (relative t o the driver prop- agation direction) coordinate [3,8-10]. The change of a non linear wavelength in the transverse direction is due to the dependence of wavel ength on the amplitude, that, in its turn, is varied in the radial directi on owing to finite cross section of the driver. This leads to a curvature of the p hase front of the nonlinear wave [8-10], to steepening and ”oscillations ” of the field in the transverse direction [10,11] and eventually to the develop ment of turbulence. From the viewpoint of acceleration and focusing of charged b unches in the wave, the curvature of the nonlinear wave front is undesired as the quality (emittance, monochromaticity) of the driven bunch worsens . In the present work we show that by means of wake wave excitation in plasma, t he den- sity of which is properly varied in the transverse direction , one can eliminate 2the nonlinear change of wavelength in the transverse direct ion and the re- lated curvature of phase front. The plasma of this kind may be produced by charged beams [12] or laser pulses [13] passing through a n eutral gas or a partially ionized uniform plasma due to an additional ioniz ation. Equations for non-zero components of plasma electrons mome ntum and electromagnetic field describing the steady nonlinear wake -fields in radially- nonuniform plasma can be obtained by simple generalization of equations for uniform plasma [10,11,14,15]: β∂Pz ∂z−∂γe ∂z−β2Ez= 0, (1) β∂Pr ∂z−∂γe ∂r−β2Er= 0, (2) −∂Hθ ∂z+β∂Er ∂z+βrNe= 0, (3) ∇⊥Hθ+β∂Ez ∂z+βzNe+βα= 0, (4) β∂Hθ ∂z−∂Er ∂z+∂Ez ∂r= 0, (5) Ne=Np(r)−α− ∇⊥Er−∂Ez ∂z. (6) As usual, Eqs. (1) and (2) were derived taking into account th e conservation of generalized momentum β2H−rotP= 0, or in our case β2Hθ+∂Pz ∂r−∂Pr ∂z= 0. (7) In Eqs. (1) - (6) γe= (1 + P2 z+P2 r+a2/2)1/2,βz, r=Pz, r/γeandNe= ne/np(r= 0) are respectively a relativistic factor, dimensionless components of velocity and dimensionless density of plasma electrons, Np=np(r)/np(0), β=vph/c,z=kp(r= 0)(Z−vpht),∇⊥=∂/∂r+ 1/r.Also the following dimensionless variables have been used: the space variable s are normalized onλp(r= 0)/2π= 1/kp(r= 0), the momenta and velocities - respectively onmecand the velocity of light and the strengths of electric and ma gnetic fields - on the nonrelativistic wave-breaking field at the axi sEWB(r= 0) = meωpe(r= 0)vph/e. The field of forces acting on relativistic electrons in the excited field is F(−eEz,−e(Er−βHθ),0). In PWFA α/negationslash= 0,a2= 0 and in 3LWFA α= 0,a2/negationslash= 0 (in Eqs. (1)-(6) the linear polarization of the laser pulse field is assumed; for the circular polarization the val ue ofashould be multiplied by the factor 21/2). We have solved Eqs. (1)-(6) numerically choosing the Gaussi an profile of the driver both in longitudinal and transverse directions: A(z, r) =A0exp[−(z−z0)2/σ2 z] exp(−r2/σ2 r), (8) where A(z, r) stands for α=nb(z, r)/np(r= 0) or a2(z, r). Shown in Fig. 1 is the nonlinear 2D plasma wake wave excited in uniform plasma [ Np(r) = 1] by the relativistic electron bunch ( α0= 0.4,σz= 2,σr= 5; for example, in this casenb0= 4×1013cm−3and the characteristic longitudinal and transverse sizes of the bunch σz,r/kpcorrespondingly are 1 .06mmand 2.65mmwhen np= 1014cm−3). One can see that the wavelength changes with the radial coordinate r. This leads to curving of the phase front and to ”oscillation s” in the transverse direction (see Fig. 2, curve 1). As |z|increases, the change of phase in transverse direction (for fixed z) becomes more and more marked. The longitudinal space parameter characterizing the nonli near wave front curving is [10]: ξ=λp 2[1−λp/Λ(0)], (9) where Λ( r) is the nonlinear wavelength. At the distance |∆z|≈ξfrom the driver the oscillation phase at the axis ( r= 0) is opposite to that on the periphery ( r/greaterorsimilarσr). Thus, in 2D nonlinear regime the nonlinear wavelength changes with rdue to nonlinear increase of the wavelength with wave ampli- tude. On the other hand, the linear wavelength λp∼n−1/2 pdecreases with equilibrium density of plasma. Hence follows an opportunit y to compensate the nonlinear increase in wavelength by reducing the wavele ngth that is due to the growth of equilibrium density of plasma. Indeed, assu me that the non- linear wavelength of the two-dimensional wake wave in the un iform plasma Λ(r) is known. Then, one can roughly compensate for the radial va riation of the nonlinear wavelength by changing the equilibrium densi ty of plasma in the radial direction according to the relation Λ(0)/Λ(r) =λp(r)/λp(0) = [ np(0)/np(r)]1/2. (10) In this case the equation for equiphase surfaces is z≈const, and, therefore, the solution for fields could be written in the form f1(z)f2(r). If we put that 4the function Λ( r) is Gaussian (that is approximately the case at least for r < σ r,according to numerical data for profiles (8)), then one can ta ke the transverse profile of the equilibrium plasma density to be al so Gaussian: np(r) =np0exp(−r2/σ2 p). (11) Then follows from Eqs. (10) and (11) that σp=r/[ln(Λ(0) /Λ(r))]1/2. (12) For example, according to (12), the numerical data for Λ( r) in the nonlinear wave shown in Fig.1 give σp≈11. Thus, in the radially nonuniform plasma, the density of which is changed according to (11) and (12), on e can practically avoid undesirable curvature of the wave front of a nonlinear wave. Figs. 3 and 4 illustrate the validity of this assertion respectively fo r PWFA and LWFA (see also Fig.2, curve 2). One can see, that the nonlinear wav elength in the nonuniform plasma with proper radial profile does not practi cally change in the transverse direction. As for the case of LWFA, one has to note that as is well known, wi thout optical guiding the diffraction limits the distance of laser -plasma interaction (and, hence, the energy gain of particles accelerated by a wa ke wave) to a few Reyleigh lengths ZR=πr2 0/λ, where r0the minimum laser pulse spot size at the focal point and λis the laser wavelength. For high-intensity laser pulses the quantity ZRis usually of the order of several millimeters. One of ap- proaches in preventing of diffraction broadening of laser pu lse and increasing of laser-plasma interaction distance is the guiding of the p ulse in preformed plasma density channel [3]. Here the unperturbed plasma den sity grows from the pulse axis ( r= 0) to its periphery, in contrast with the radial profile of the plasma density that was proposed above for prevention of phase front cur- vature of a nonlinear wake wave when the plasma density at the driver axis is maximum. In case of nonlinear wake wave excitation by high power laser pulses (of power P > P c= 2c(e/re)2[ω0/ωpe(r= 0)]2≈17[ω0/ωpe(r= 0)]2 GW, where re=e2/mec2is the classical electron radius), the process of diffraction broadening of a larger part of pulse, including t he case of proposed equilibrium profile of plasma, may be prevented or essential ly retarded due to the relativistic self-focusing [3]. One can estimate the condition of rela- tivistic self-focusing in our case from the following relat ion (see, e. g. Sec. VI in Ref. [3]): P/P c>1 + (∆ n/∆nc)(rs/r0)4, where rsis the spot size, 5∆nc= 1/πrer2 0, ∆n≃np(0)−np(σr) is the radial variation of unperturbed plasma density in the laser pulse. This work has been supported by the International Science an d Technol- ogy Center under Project No. A-013. REFERENCES [1] T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267 (1979). [2] R. D. Ruth, A. W. Chao, P. L. Morton, and P. B. Wilson, Part. Accel. 17, 171 (1985); P. Chen, Part. Accel. 20, 171 (1987). [3] E. Esarey, P. Sprangle, J. Krall, and A. Ting, IEEE Trans. Plasma Sci.24, 252 (1996). [4] A. I. Akhiezer and R .V. Polovin, Zh. Eksp. Teor. Fiz. 30, 915 (1956) [Sov. Phys. JETP 3, 696 (1956)]. [5] L. M. Gorbunov and V. I. Kirsanov, Sov. JETP 66, 290 (1987). [6] A. Ts. Amatuni, E. V. Sekhpossian, and S. S. Elbakian, Fiz . Plasmy 12, 1145 (1986); J. B. Rosenzweig, Phys. Rev. Lett. 58, 555 (1987); A. Ts. Amatuni, S. S. Elbakian, A. G. Khachatryan, and E. V. Sekh possian, Journal of Contemporary Physics (Allerton Press, Inc., NY) 28, 8 (1993); A. G. Khachatryan, Phys. Plasmas 4, 4136 (1997). [7] S. V. Bulanov, V. I. Kirsanov, and A. S. Sakharov, JETP Let t.50, 198 (1989); P. Sprangle, E. Esarey, and A. Ting, Phys. Rev. A 41, 4463 (1990); V. I. Berezhiani and I. G. Murusidze, Phys. Lett. A 148, 338 (1990). [8] C. D. Decker, W. B. Mori, and T. Katsouleas, Phys. Rev. E 50, 3338 (1994). [9] S. V. Bulanov, F. Pegoraro, and A. M. Pukhov, Phys. Rev. Le tt.74, 710 (1995). [10] A. G. Khachatryan and S. S. Elbakian, Proceedings PAC’9 9, New York, 1999. [11] B. N. Breizman, T. Tajima, D. L. Fisher, and P. Z. Chebota ev, In: Research Trends in Physics: Coherent Radiation and Particl e Acceleration , edited by A. Prokhorov (American Institute of Physics, New York, 1992), pp. 263-287. [12] A. K. Berezin et al., Plasma Phys. Rep. 20, 596 (1994). [13] V. Malka et al., Phys. Rev. Lett. 79, 2979 (1997). [14] P. Mora and T. M. Antonsen Jr., Phys. Plasmas 4, 217 (1997). [15] K. V. Lotov, Phys. Plasmas 5, 785 (1998). 6FIGURE CAPTIONS Fig. 1. The two-dimensional nonlinear wake wave in uniform p lasma [Np(r) = 1]. The parameters of the bunch are: α0= 0.4,σz= 2,σr= 5, γ= 10. (a). The density of plasma electrons Neand of the bunch. 1 — the density of plasma electrons at the axis, r= 0; 2 — the same for r= 2; 3 — r= 4; 4 — r= 5; 5 — the density of bunch at the axis α(z, r= 0). (b). The longitudinal electric field for r= 0,2,4 and 5 in the order of magnitude reduction. (c). The focusing field fr=βHθ−Er. 1 — r= 2; 2 — r= 4; 3 —r= 5. All variables are normalized. Fig. 2. The radial behavior of the normalized longitudinal e lectric field strength Ez. 1 — Ez(z=−25, r) in the nonlinear wake wave excited in uniform plasma for the case given in Fig. 1 ( |∆z|≈ξ[see Eq. (9)]); 2 — Ez(z=−25, r) in nonuniform plasma for the case given in Fig. 3. Fig. 3. The two-dimensional nonlinear wake wave in nonunifo rm plasma withσp= 11. The bunch parameters are the same as in Fig. 1. 1 — the density of plasma electrons versus zforr= 0,2,4 and 5 in the order of magnitude reduction. (b). The same for the longitudinal ele ctric field. (c). The focusing field. 1 — r= 2; 2 — r= 4; 3 — r= 5. All variables are normalized. Fig. 4. The two-dimensional nonlinear wake wave excited by l aser pulse. The pulse parameters are : a2 0= 3.6,σz= 2,σr= 5,γ= 10. (a). The dimensionless accelerating field Ezexcited in uniform plasma. r= 0,2,4 and 5 in the order of magnitude reduction. (b). The same in the nonuniform plasma, σp= 12. 7Figure 1. A. G. Khachatryan, Phys. Rev. E 00.511.522.53 -25 -20 -15 -10 -5 01 2 3 4 5 zNe (a) -0.8-0.6-0.4-0.200.20.40.60.8 -25-20-15-10 -5 0Ez z (b) -0.3-0.25-0.2-0.15-0.1-0.0500.050.10.150.2 -25-20-15-10 -5 0 (c)1 2 3 zfrFigure 2. A. G. Khachatryan, Phys. Rev. E -0.8-0.6-0.4-0.200.2 0 2 4 6 8 101 2rEzFigure 3. A. G. Khachatryan, Phys. Rev. E 00.511.522.53 -25 -20 -15-10 -5 0Ne z(a) -0.8-0.6-0.4-0.200.20.40.60.8 -25 -20 -15 -10 -5 0Ez (b)z -0.2-0.15-0.1-0.0500.050.10.15 -25 -20 -15 -10 -5 0 (c)fr z 1 23Figure 4. A. G. Khachatryan, Phys. Rev. E -1-0.8-0.6-0.4-0.200.20.40.60.81 -25-20-15-10 -5 0 (a)zEz -1-0.8-0.6-0.4-0.200.20.40.60.81 -25-20-15-10 -5 0 (b)E zz

arXiv:physics/0004049v1 [physics.bio-ph] 20 Apr 2000Molecular Dynamics Studies of Sequence-directed Curvatur e in Bending Locus of Trypanosome Kinetoplast DNA Alexey K. Mazur Laboratoire de Biochimie Th´ eorique, CNRS UPR9080 Institut de Biologie Physico-Chimique 13, rue Pierre et Marie Curie, Paris,75005, France. FAX:+33[0]1.58.41.50.26. Email: alexey@ibpc.fr (December 29, 2013) The macroscopic curvature induced in the double helical B-DNA by regularly repeated adenine tracts (A-tracts) play s an exceptional role in structural studies of DNA because thi s effect presents the most well-documented example of sequenc e specific conformational modulations. Recently, a new hypot h- esis of its physical origin has been put forward, based upon t he results of molecular dynamics simulations of a 25-mer frag- ment with three A-tracts phased with the helical screw. Its sequence, however, had never been encountered in experimen - tal studies, but was constructed empirically so as to maximi ze the magnitude of bending in specific computational condi- tions. Here we report the results of a similar investigation of another 25-mer B-DNA fragment now with a natural base pair sequence found in a bent locus of a minicircle DNA. It is shown that the static curvature of a considerable magnitude and stable direction towards the minor grooves of A-tracts emerges spontaneously in conditions excluding any initial bias except the base pair sequence. Comparison of the bending dy- namics of these two DNA fragments reveals both qualitative similarities and interesting differences. The results sugg est that the A-tract induced bending obtained in simulations re - produces the natural phenomenon and validates the earlier conclusions concerning its possible mechanism. INTRODUCTION It is now generally accepted that the double helical DNA can somehow translate its base pair sequence in tertiary structural forms. The simplest such form is a bend. Large bends in natural DNA were discovered nearly twenty years ago for sequences containing regular repeats of A nTm,with n + m >3, called A-tracts1,2. Since then this intriguing phenomenon has been thor- oughly studied, with several profound reviews of the re- sults published in different years3,4,5,6,7,8. Every A-tract slightly deviates the helical axis towards its minor groove , which results in significant macroscopic curvature when they are repeated in phase with the helical screw. How- ever, in spite of considerable efforts spent in attempts to clarify the the physical origin of this phenomenon it still remains a matter of debate because all theoretical models proposed until now contradict some of the experimental results. This problem is of general importance because the accumulated large volume of apparently paradoxicalobservations on the subject points to our lack of under- standing of the fundamental principles that govern the DNA structure. A variety of theoretical methods based upon computer molecular modeling have been used in order to get in- sight in the mechanism of DNA bending9,10,11,12,13. The most valuable are unbiased simulations where sequence dependent effects result only from lower level interactions and are not affected by a priori empirical knowledge of relevant structural features14. Such calculations can re- veal the essential physical factors and eventually shed light upon the true mechanism of DNA bending. We have recently reported about the first example of stable static curvature emerging spontaneously in free molec- ular dynamics simulations a B-DNA fragment with A- tracts phased with the helical screw, and proposed a new mechanism of bending that could explain our re- sults as well as experimental data15,16. However, the sequence used in these computational experiments was artificial in the sense that it was designed empirically so as to accelerate the development and maximize the amplitude of bending. It was never studied experimen- tally, therefore, even though it was similar to canoni- cal DNA benders, one could not exclude that the static bending observed in calculations was of a different origin than that found in experiments. Here we report the re- sults of a similar study of a complementary 25-mer DNA duplex AAAATGTCAAAAAATAGGCAAATTT. This fragment is found in the center of the first bent DNA lo- cus investigated experimentally in vitro1,2. It belongs to a minicircle DNA from the kinetoplast body of Leishma- nia tarentolae and, together with several additional A- tracts, provides planar curvature that apparently serves in vivo to facilitate the loop closure. We have only re- placed the 3’-terminal A 6tract in the original fragment2 by A 3T3because our preliminary empirical observations suggested that 3’-terminal A ntracts usually need larger time to converge to characteristic conformations with a narrow minor groove. We show that two independent long MD trajectories starting from straight conformations corresponding to canonical A and B-DNA forms both converge to stati- cally bent structures with similar bending direction and magnitude, thus giving the first example of a natural DNA fragment where this phenomenon is reproduced in simulations. The results are qualitatively similar to our 1first report16as regards the kinetics of convergence and comparison with different theoretical models of bending. At the same time, along with convergence of the overall macroscopic DNA from, we find here a remarkably larger than earlier degree of convergence in local backbone and base pair conformations. These results confirm that A- tract induced DNA bending found in calculations corre- sponds to the experimental phenomenon. They provide additional information concerning its mechanism and the intriguing relationship between the base pair sequence and the DNA structure. RESULTS Two long MD trajectories have been computed for the complementary duplex AAAATGTCAAAAAATAGGCAAATTT. The first tra- jectory referred to below as TJB started from the fiber canonical B-DNA structure and continued to 10 ns. The second trajectory (TJA) started form the fiber canonical A-DNA conformation and continued to 20 ns. The longer duration of TJA was necessary to ensure structural con- vergence. The computational protocols are detailed in Methods. Figure 1 shows two series of representative structures from the two trajectories. Each structure is an average over a nanosecond interval, with these intervals equally spaced in time. In TJB the molecule was curved always in the same direction, but both the shape of the helical axis and the magnitude of bending varied. The first two struc- tures shown in Fig. 1a are locally bent between the upper two A-tracts while their lower parts are nearly straight. In contrast, in the last three structures, the planar curva- ture is smoothly distributed over the entire molecule. In TJA, distinguishable stable bending emerged only after a few nanoseconds, but after the fourth nanosecond all average conformations were smoothly bent. In contrast to TJB, however, the bending direction was not stable, and by comparing the two time series of perpendicular projections one may see that during the first 15 nanosec- onds the bending plane slowly rotated. During the final five nanoseconds the overall bending direction was stable. In the last conformation, an S-shaped profile of the heli- cal axis is found in the perpendicular projection, which indicates that there are two slightly misaligned bends lo- cated between the three A-tracts. The orientations of the helices in this figure was chosen separately for the two trajectories, and one can notice that the left projec- tion in plate (a) is close to the right one in plate (b), in other words, the final bend directions differed by ap- proximately 90◦. In TJA, the intersection of the minor groove with the bending plane occurs close to the center of the middle A-tract while in TJB this point is shifted towards its 3’ end. In both cases, however, the narrowed minor grooves of the A-tracts appear at the inside edge of the curved axis. FIG. 1. Consecutive average structures from TJB and TJA. The average conformations were taken from one nanosecond intervals equally spaced over the trajectories , namely, during the second, fourth, sixth nanosecond, and so forth. They were all superimposed and two perpendicular views are shown in the figure. In both trajectories, the view is chosen according to the final orientation of the bending plane. Namely, a straight line between the ends of the helica l axis passes through its center in the right hand projection. Residues are coded by colors, namely, A - green, T - red, G - yellow, C - blue. 20 2 4 6 8 10 Time ns123456-180-120-600601201800102030405060012345678910012345678910RMSD Direction Angle Axshift Shortening(%)TJB FIG. 2. (a) 0 5 10 15 20 Time ns1234567-180-120-60060120180010203040506001234567801234567RMSD Direction Angle Axshift Shortening(%)TJA FIG. 2. (b)FIG. 2. The time evolution of several representative struc- tural parameters in TJB (a) and TJA (b). Nonhydrogen atom rmsd is measured with respect to the fiber canonical B-DNA model constructed from the published atom coordinates17. The bending direction is characterized by the angle (given in degrees) between the plane that passes through the two ends and the middle point of the helical axis, and the xz plane of the local DNA coordinate frame at the center of the duplex. The local frame is constructed according to the Cambridge convention18, namely, its xdirection points to the major DNA groove along the short axis of the base-pair, while the local zaxis direction is adjacent to the optimal helicoidal axis. Thus, a zero angle between the two planes corresponds to the overall bend to the minor groove exactly at the central base pair. The bending angle is measured between the two ends of the helical axis. The shift parameter is the average deviation of the helical axis from the straight line between its ends. The shortening is measured as the ratio of the lengths of the curved axis to its end-to-end distance minus one. All traces have been smoothed by averaging with a window of 75 ps in (a) and 150 ps in (b). Figure 2 displays fluctuations of various parameters that characterize the overall bending of the double he- lix. In both trajectories the rmsd from the canonical B-DNA usually fluctuates between 2 and 4 ˚A and cor- relates with the three parameters shown in Fig. 2 that measure the bending magnitude. Note that in TJA there was a short period of strong initial bending which have not been detected in Fig. 1. The most significant differ- ence between the two plates in Fig. 2 is in the dynamics of the bending direction. In TJB, the final orientation of the bend was found early and remained quite stable, although the molecule sometimes straightened giving, si- multaneously, a low bending magnitude and large tempo- rary fluctuations of direction. In contrast, during the first 15 nanoseconds of TJA, the bending plane made more than a half turn with respect to the coordinate system bound to the molecule, that is a transition occurred be- tween oppositely bent conformations by passing through a continuous series of bent states. Temporary transi- tions to the straight state were short-living and always reversed, with the bending resumed in the previous direc- tion. Owing to this turn the bend orientations in TJA and TJB converged considerably although not exactly. The zero direction in Fig. 2 corresponds to bending to- wards the minor groove at the fifth AT base pair of the middle A-tract. We see that, at the end of TJA, it is shifted from zero by an angle corresponding to rotation in one base pair step, which gives a bend towards the minor groove close to the center of the middle A-tract. In TJB, the direction deviates from zero in the opposite sense by an angle corresponding to roughly two base pair steps, resulting in a residual divergence of approximately 90◦between the two trajectories. The slow kinetics of convergence exhibited in Fig. 2 indicates, however, that still better accuracy, if ever possible, would require much longer time. 3TJB AATGTCAAAAAATAGGCAAAT0 2 4 6 8ns 4681012A0 2 4 6 8ns FIG. 3. (a) TJA AATGTCAAAAAATAGGCAAAT0 4 8 12 16ns 4681012A0 4 8 12 16ns FIG. 3. (b) FIG. 3. The time evolution of the profile of the minor groove in TJB (a) and TJA (b). The surfaces are formed by time-averaged successive minor groove profiles, with tha t on the front face corresponding to the final DNA conforma- tion. The interval of averaging was 75 ps in TJB and 150 ps in TJA. The groove width is evaluated by using space traces of C5’ atoms as described elsewhere19. Its value is given in angstr¨ oms and the corresponding canonical B-DNA level of 7.7˚A is marked by the straight dotted lines on the faces of the box.AAATTTGCCTATTTTTTGACATTTT0 2 4 6 8 10Time (ns)AAAATGTCAAAAAATAGGCAAATTTTJB -222.29140.702 FIG. 4. (a) AAATTTGCCTATTTTTTGACATTTT0 4 8 12 16 20Time (ns)AAAATGTCAAAAAATAGGCAAATTTTJA -206.708149.521 FIG. 4. (b) FIG. 4. Dynamics of B I↔BIItransitions in TJB (a) and TJA (b). The B Iand B IIconformations are distinguished by the values of two consecutive backbone torsions, εandζ. In a transition they change concertedly from (t,g−) to (g−,t). The difference ζ−εis, therefore, positive in B Istate and nega- tive in B II, and it is used in as a monitoring indicator, with the corresponding gray scale levels shown on the right in eac h plate. Each base pair step is characterized by a column con- sisting of two sub-columns, with the left sub-columns refer ring to the sequence written at the top in 5’-3’ direction from lef t to right. The right sub-columns refer to the complementary sequence shown at the bottom. 4Figure 3 displays the time evolution of the profile of the minor grooves in TJB and TJA. At the end of both trajectories the minor groove width exhibits modulations phased with the helical screw. It is significantly widened between the A-tracts and narrowed within them by ap- proximately 1 ˚A with respect to the canonical level. This magnitude of narrowing corresponds well to the values observed in experimental structures of A-tract contain- ing B-DNA oligomers20,21,22,23,24,19. In TJB, the overall waving profile established early and remained more or less constant. Interestingly, during two rather long peri- ods, a secondary minimum occurred close to the 5’ end of the middle A-tract, and at the same time the main central minimum sifted towards the 3’ end of this A-tract. These motions involve almost an entire helical turn and, appar- ently, are concerted, which demonstrates the possibility of medium range structural correlations along the dou- ble helix. Comparison of Figs. 1a and 3a suggests that there is no simple one-to-one relationship between bend- ing and minor groove modulations. Notably, the right smaller and narrower widening corresponds to a stable and strong bending point of the helical axis, while the left one, which is evidently larger, gives less or no bend- ing. In TJA, the final configuration of the minor groove established only during the last few nanoseconds, but the final profile has the same number and similar positions of local maxima and minima as that in TJB. The overall minor groove dynamics in TJA looks rather complicated and its relationship with the quasi-regular rotation of the bending plane demonstrated in Figs. 1b and 2b is not readily seen. Figure 4 displays dynamics of B I↔BIIbackbone transitions in the two trajectories. A few common features can be noticed that have been encountered previously15,16. For instance, in A-tracts, the B IIcon- formers are commonly found in ApA steps and almost never in TpT steps. They tend to alternate with B Iin consecutive ApA steps. B I↔BIItransitions often occur concertedly along the same strand as well as in oppo- site strands. The B I↔BIIdynamics comprises all time scales presented in these calculations and clearly involve s slower motions as well. Note, for instance, a particularly stable B IIconformer in the only GpA step available in the two strands. On the other hand, there is some sim- ilarity in the distributions of the B IIconformers in the two trajectories, which is a new feature compared to our previous report15,16. It is seen in Fig. 4 that in total ten BIIconformers were found at the end of TJB and eight in TJA. Among them six and five, respectively, occurred in non A-tract sequences. In three base pair steps the B II conformers are found in both final conformations, with all of them in non A-tract sequences. A careful exam- ination the two plates in Fig. 4 shows that, although in A-tracts the preferred sites of B IIconformations dif- fer, in the intervening sequences their dynamics is rather similar in TJB and TJA. This trend is demonstrated in Fig. 5 where inter-trajectory correlations are examined for the specific base pair step propensities to B Iand B II-100 -50 50 100 -100-5050100 DiagonalNon A-tractApTApA FIG. 5. Correlation between populations of B Iand B IIcon- formers in different base pair steps in TJB and TJA. Each point in the plot represents a specific base pair step, with th e corresponding average ζ−εvalues in TJB and TJA used as xandycoordinates, respectively. The TpT steps in A-tracts are omitted for clarity. conformers. We have not included here the TpT steps in A-tracts because they strongly prefer the B Iconforma- tion and, therefore, are trivially correlated. It is eviden t that, except for ApA steps in A-tracts, there was certain correlation in the average populations of B IIconformers for each specific base pair step in the two trajectories. The ApA steps apparently can adopt both conformations with little effect of the sequence context and the overall structure. Figure 6 shows variation of several helicoidal parame- ters along the duplex in three representative one nanosec- ond averaged structures. Two of them were taken from TJA, namely, the 16th and 18th nanosecond averages which we refer to as TJA16 and TJA18. They illustrate the scale and the character of fluctuations of these pa- rameters in the course of the same dynamics. The third conformation is the last average of TJB (TJB10) and it illustrates convergence of helical parameters in indepen- dent trajectories. We have chosen TJA16 and TJA18 be- cause, as seen in Fig. 1, the corresponding two structures are particularly similar. They are both smoothly bent in a virtually identical direction and their rmsd is only 0.95 ˚A. All parameters shown in the figure, except the incli- nation, exhibit jumping alternations between consecutive base pair steps. Although they look chaotic, there is a considerable similarity between TJA16 and TJA18 and less significant, but still evident similarity of the two wit h TJB10. Notably, a remarkable correspondence of alter- ations of the twist is observed in the right-hand half of the sequence. At the same time, even the TJA16 and TJA18 plots sometimes diverge. Note, for instance, that the alteration in their roll traces are phased in the central A-tract, but dephased in the other two, with a particu- larly significant difference around the TpA step. These results show that, in a statically curved double helix, the 5AAATTTGCCTATTTTTTGACATTTT-8-6-4-20246810-28-23-18-13-8-32151719212325272931333537394143454749-7-5-3-11357911-16-11-6-149Inclination Propeller Twist Tilt RollAAAATGTCAAAAAATAGGCAAATTT FIG. 6. Sequence variations of some helicoidal parameters in representative one nanosecond averaged structures from TJB and TJA. The sequence of the first strand is shown on the top in 5’ – 3’ direction. The complementary sequence of the second strand is written on the bottom in the opposite direction. All parameters were evaluated with the Curves program25and are given in degrees. The color coding is: TJA 18th ns – red, TJA 16th ns – green and TJB 10th ns – blue. base pair helicoidal parameters fluctuate around certain specific values that are stable in the nanosecond time scale. There is, however, more than one combination of these parameters for the same overall bend. At the same time, the evident convergence of the corresponding dis- tributions in TJA and TJB suggests that, at least for this particular base pair sequence, the number of such redundant combinations should not be very large. Discussion The computational experiments described here give the first example of a natural DNA fragment with phased A-tracts which in free unbiased MD simulations adopts spontaneously a statically bent shape. In the analogous earlier calculations the static curvature has been demon- strated for a different A-tract containing sequence con- structed artifically and never tested in experiments15,16. The qualitative similarity between these two simulationsis evident. Trajectories starting from canonical A and B-DNA forms both travel within the B-DNA family and, in straight, states yield rmsd from B-DNA of around 2 ˚A. TJA enters the B-DNA family with a significant tem- porary bending during the first 500 ps. Later it becomes bent in an arbitrary direction and next changes the bend direction by slowly rotating the bending plane. This ro- tation slows down after 10 ns, with the final orientation much closer to that in TJB than the initial one. In both cases the residual discrepancy was in the range of 60◦– 90◦16. The final minor groove profiles are not identical, although similar for TJA and TJB, as well as the distri- butions of the B Iand B IIbackbone conformers and base pair helicoidal parameters. The present results, there- fore, suggest that the A-tract induced DNA bending ob- served in calculations here and before corresponds to the experimental phenomenon. At the same time, there are several clear differences. Notably, the preferred bending direction here is closer the centers of the minor grooves of the A-tracts, whereas the magnitude of bending is somewhat less than in the previ- ous calculations. The bending angle in the average struc- tures shown in Fig. 1 fluctuates between 12◦and 25◦in TJB and between 7◦and 28◦in TJA, with the maximal values reached at the end in both cases. The previous ar- tifical sequence was constructed to maximize the bending and it showed the corresponding values beyond 35◦15,16. According to the experimental estimates made for “good benders” in an optimal sequence context, the magnitude of bending is around 18◦per A-tract8, which in our case gives 36◦for the overall bend because the principal bend- ing elements are the two intervening zones between the A-tracts. The bends observed here are somewhat be- low this estimate. However, in experiments, the bending magnitude differs significantly between different A-tract sequences and depends upon many environmental param- eters that are not controlled in simulations. One can ex- pect to observe in calculations sequence variations of the bending magnitude that may not exactly follow those in experiments. Therefore, whatever the possible reasons of the apparent discrepancy, the overall correspondence of the computed bending magnitudes to the experimen- tal estimates should be considered as surprisingly good. Yet another difference is a larger than earlier degree of similarity in the profiles of the minor groove, in the dis- tributions of B IIconformers, and helicoidal parameters in trajectories starting from A and B-DNA forms. It was the most surprising observation of our previous report that reasonably good convergence in terms of the macro- scopic bent shape of the double helix was not accompa- nied by the parallel convergence of microscopic confor- mational parameters. Here the two trajectories manifest clear signs of convergence for base pair step parameters as well as for the backbone conformations. Additional studies are necessary in order to tell if this difference is a sequence specific property or just an occasional effect. In spite of these differences the results presented here support our previous conclusion concerning the qualita- 6tive disagreement of the computed structural dynamics of the double helix and the most popular earlier theories of bending16. In Figs. 3, 4, and 6, multiple examples are found of strong variability of local helical parame- ters in bent substates, which argues against the local interactions and the preferred base pair orientations as the cause of bending. All three A-tracts are character- ized by a narrowed minor groove and local minima in the traces of the propeller twist and the inclination of base pairs. Nevertheless, their internal structures are not ho- mogeneous and vary from one base pair step to another. Moreover, the structures of the three are not the same and present another example of an ensemble of confor- mational substates with a common overall shape. This pattern is qualitatively different from that implied by the junction models of bending26,27. At the same time, the present results are well interpreted in the framework of our model that sees the principal cause of the bending in the backbone stiffness and the postulated excess of its specific length over that in the idealized regular B-DNA double helix16. Several non trivial observations support this view. The first observation is the microheterogeneity of the ensemble of conformations that provide the same bent form of the double helix during the last nanoseconds of both trajectories. Once the backbone have found its pre- ferred waving shape on the surface of the B-DNA cylin- der it fixes the bending direction. The thermal motion of bases is allowed, but they are forced to respect this mechanical constraint, giving rise to an ensemble of con- formations with different base orientations, but the same bent form of the double helix The second observation is an always waving minor groove profile which does not change during temporary short-living straightening. The waving profile is the di- rect consequence of the postulated excess of the specific backbone length over that in the regular B-DNA with even grooves. The main immediate cause of bending is the necessity to compress the backbone in the minor groove widenings if the parallel base pair stacking is to be preserved16. The backbone stiffness tends to cause destacking from the minor groove side, which results in bending towards the major groove. Symmetrical destack- ing is also possible, however, and transitions between var- ious types of stacking perturbations makes possible time variations of the magnitude of bending with a constant backbone profile. Finally, our model explains well the persistent bends in incorrect directions and the rotation of the bending plane observed in TJA. According to this view the exces- sive backbone length and its stiffness force the backbone to wander on the surface of the B-DNA cylinder what- ever the base pair sequence is. In the dynamics start- ing from the A-DNA structure the duplex enters the B- DNA family in strongly non-equilibrium conditions, with rapidly changing different energy components. The back- bone quickly extends to its preferred specific length tak- ing some waving profile and causing bending in an ar-bitrary direction. During the subsequent slow evolution it remains always waving, and that is why there is al- ways a preferred bend direction which is not lost during occasional straightening. Methods As in the previous report16, the molecular dynamics simulations were carried out with the internal coordinate method (ICMD)28,29. The minimal model of B-DNA was used, with fixed all bond length, rigid bases, fixed va- lence angles except those centered at sugar atoms and increased effective inertia of hydrogen-only rigid bodies as well as planar sugar angles30,16. The time step was 10 fsec. AMBER9431,32force field and atom parameters were used with TIP3P water33and no cut off schemes. The heating and equilibration protocols were same as before30,34. The Berendsen algorithm35was applied dur- ing the production run, with a relaxation time of 10 ps, to keep the temperature close to 300 K. The coordinates were saved once in 2.5 ps. The initial conformation for TJB was prepared by vacuum energy minimization starting from the fiber B-DNA model constructed from the published atom coordinates17. 375 water molecules were next added by the hydration protocol designed to fill up the minor groove30. The initial conformation for TJA was prepared by hydrating the minor groove of the corresponding A- DNA model17without preliminary energy minimization. The necessary number of water molecules was added af- ter equilibration to make it equal in TJA and TJB. During the runs, after every 200 ps, water positions were checked in order to identify those penetrating into the major groove and those completely separated. These molecules, if found, were removed and next re-introduced in simulations by putting them with zero velocities at random positions around the hydrated duplex, so that they could readily re-join the core system. This time interval was chosen so as to ensure a small enough average number of repositioned molecules which was ca 1.5. 1J. C. Marini, S. D. Levene, D. M. Crothers, and P. T. Englund, Bent helical structure in kinetoplast DNA, Proc. Natl. Acad. Sci. USA 79, 7664 (1982). 2H.-M. Wu and D. M. Crothers, The locus of sequence- directed and protein-induced DNA bending, Nature 308, 509 (1984). 3S. Diekmann, in Nucleic Acids and Molecular Biology, Vol. 1, edited by F. Eckstein and D. M. J. Lilley (Springer- Veralg, Berlin Heidelberg, 1987), pp. 138–156. 4P. J. Hagerman, Sequence-directed curvature of DNA, Annu. Rev. Biochem. 59, 755 (1990). 75D. M. Crothers, T. E. Haran, and J. G. Nadeau, Intrinsi- cally bent DNA, J. Biol. Chem. 265, 7093 (1990). 6D. M. Crothers and J. 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Mazur, A minimal model of B-DNA, Preprint http: // xxx.lanl.gov/abs/ physics/9907028, (1999). 35H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and J. R. Haak, Molecular dynamics with cou- pling to an external bath, J. Chem. Phys. 81, 3684 (1984). APPENDIX This section contains comments from anonymous ref- erees of a peer-review journal where this paper was been considered for publication, but rejected (see also 16). 8A. Journal of Molecular Biology 1. First referee These companion manuscripts describe a series of molecular dynamics trajectories obtained for DNA se- quences containing arrangements of oligo dA - oligo dT motifs implicated in intrinsic DNA bending. Unlike pre- vious MD studies of intrinsically bent DNA sequences, these calculations omit explicit consideration of the role of counterions. Because recent crystallographic studies of A-tract-like DNA sequences have attributed intrinsic bending to the localization of counterions in the minor groove, a detailed understanding of the underlying ba- sis of A-tract-dependent bending and its relationship to DNA-counterion interactions would be an important con- tribution. Although the MD calculations seem to have been car- ried out with close attention to detail, both manuscripts suffer from some troubling problems, specifically: The sequence investigated here is a 25-bp segment of the well-characterized L. tarentolae kinetoplast-DNA bending locus. Two trajectories, TJA and TJB, were computed starting from canonical A-form and B-form structures, respectively. Although the author argues that greater structural convergence between TJA and TJB has taken place in these simulations, there is still a sig- nificant disparity concerning the observed bending direc- tions in these two structures. Moreover, the extent of bending in this simulated helix is significantly less than that observed in the previous study, which is unexpected because of out-of-phase placement of the third A tract in the previous sequence. This behavior is not explained and seems difficult to rationalize. 9

arXiv:physics/0004050v1 [physics.gen-ph] 20 Apr 2000A BRIEF NOTE ON THE MAGNETIC EFFECTS OF THE ELECTRON B.G. Sidharth∗ Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Hyderabad 500 063 (India) Abstract In this paper it is shown that a recent formulation of the elec tron in terms of a Kerr-Newman type metric, exhibits a short range ma gnetic effect, as indeed has been observed at Cornell, and also an Aha ronov- Bohm type of an effect. In a recent model[1, 2, 3] it was shown how an electron could be described as a Kerr-Newman type black hole with Quantum Mechanical inp uts. Such a scheme lead to a cosmology consistent with all so called lar ge number re- lations and which predicted that the universe would continu e to expand for ever[4, 5], as indeed has been subsequently observed[6, 7]. Moreover this scheme also gives a description of the quark picture includi ng such features as the characteristic fractional charge, handedness, confi nement and an order of magnitude estimate of the masses[8, 9, 10]. We would now like to point out two additional consequences of the above model, one an extra magnetic effect in the electromagnetic va cuum and the other an Aharanov-Bohm type effect[11]. We first observe that the magnetic component of the field of a st atic elec- tron as a Kerr-Newman black hole is given in the familiar sphe rical polar coordinates by (Cf.refs.[1, 2]) Bˆr=2ea r3cosΘ + 0(1 r4), BˆΘ=easinΘ r3+ 0(1 r4), Bˆφ= 0, (1) 0E-mail:birlasc@hd1.vsnl.net.in 1whereas the electrical part is given by Eˆr=e r2+ 0(1 r3), EˆΘ= 0(1 r4), Eˆφ= 0, (2) A comparison of (1) and (2) shows that there is a magnetic comp onent of shorter range apart from the dipole which is given by the first term on the right in equation (1)– infact this model also exhibits the an omalous gyro magnetic ratio g= 2 of the electron. We would like to point out that a short range force the B(3)force mediated by massive photons has indeed been ob- served at Cornell and studied over the past few years[12]. On the other hand as the Kerr-Newman charged black hole can be approx- imated by a solinoid, we have as in the Aharonov-Bohm effect, a negligible magnetic field outdside, but at the same time a real vector pot ential /vectorAwhich would contribute to a shift in phase. Infact this shift in pha se is given by ∆δˆB=e ¯h/contintegraldisplay /vectorA./vectords (3) There is also a similar effect due to the electric charge given by ∆δˆE=−e ¯h/integraldisplay A0dt (4) where A0is the usual electro static potential given in (2). In the abo ve Kerr-Newman formulation, ( /vectorA, A 0) of (3) and (4) are given by (Cf.refs.[1, 2]) Aσ=1 2(ηµvhµv), σ, (5) From (5) it can be seen that /vectorA∼1 cA0 (6) Substitution of (6) in (3) then gives us the contribution of t he shift in phase due to the magnetic field. References [1] B.G. Sidharth, Int.J.Mod.Phys.A, (1998) 13 (15), p.259 9ff. 2[2] B.G. Sidharth, Ind.J. of Pure and Applied Phys., 35, 1997 , pp456-471. [3] B.G. Sidharth, Gravitation & Cosmology, Vol.4, No.2, 19 98, pp158-162. [4] B.G. Sidharth, International Journal of Theoretical Ph ysics, 37 (4), 1998, pp1307-1312. [5] B.G. Sidharth, Chaos Solitons & Fractals, 11 (8), 2000, p p1269-1278. [6] S. Perlmutter, et. al., Nature, 391 (6662), 1998. [7] R.P. Kirshner, Proc. Natl. Acad. Sci. Vol.96, 1999, pp.4 224-4227. [8] B.G. Sidharth, Mod.Phys. Lett. A., Vol. 12 No.32, 1997, p p2469-2471. [9] B.G. Sidharth, Mod.Phys. Lett. A., Vol. 14 No. 5, 1999, pp 387-389. [10] B.G. Sidharth, in Instantaneous Action at a Distance in Modern Physics: ”Pro and Contra” , Eds., A.E. Chubykalo et. al., Nova Science Publish- ing, New York, 1999. [11] Y. Aharonov, ”Non-Local Phenomena and the Aharonov-Bo hm Effect” in Quantum Concepts in Space and Time (Eds.) R. Penrose, C.J. Isham, Clarendon Press, Oxford, 1986, pp.41ff. [12] M.W. Evans ”Origin, Observation and Consequences of th eB(3)Field” in The Present Status of the Quantum Theory of Light, S. Jeffer s et al. (eds), Kluwer Academic Publishers, Netherlands, 1997, pp. 117-125 and several other references therein. 3

arXiv:physics/0004051v1 [physics.gen-ph] 21 Apr 2000Equivalence of Descriptions of Gravity in Both Curved and Fl at Space-time Mei Xiaochun ( Institute of Theoretical Physics in Fuzhou, No.303, Build ing 2, Yinghu Garden, Xihong Road, Fuzhou, 350025, P.R.Chian, E-mail: fzbgk@pub 3.fz.fj.cn ) Abstract It is proved in the manuscript that as long as the proper coord inate transformation is intro- duced, the equations of geodetic lines described in curved s pace-time can be transformed into the dynamic equations in flat space-time. That is to say, the E instain theory of gravity and other gravitational theories based on the curved space-tim e can be identically transformed into flat space-time to describe. As an example, the Schwarzschil d solution of the spherical symmetry gravitational field is transformed into flat space-time to st udy. The results show that there exists no any singularity in the all processes and the whole space-t ime including at the point r=0. So it seems more rational to discuss the problems of gravitation i n flat space-time. PACS number 0400 Introduction The general theory of relativity based on the curved space-t ime has got great success and becomes main current theory now. However, there still exist some fou ndational problems in it just as the definition of gravitational field’s energy, the quantizatio n of gravitation and the problem of singularity and so on. So it is always an attractive idea to re-establish g ravitational theory in flat space-time. Since the 1940’s, many theories based on flat space-time were put forward. Though all of those theories are coincident with the gravitational theory of Einstain un der the conditions of weak fields, it can not be proved that they are better than the theory of Einstain by e xperiments at present. So according to the current viewpoint, the space-time of gravitation fiel d should be non-Euclidean one. The flat space-time is always regarded as the boundary condition whe re the gravitational field is far away. In the paper, the author does not try to establish an independ ent theory in flat space-time. But it can be proved that as long as the proper coordinate transform ations are introduced and, the equations of geodetic lines in curved space-time can be transformed in to the dynamic equations in flat space-time. It means that the Einstein’s theory of gravitation and other theories based on the curved space-time can also be identically transformed into the flat space-time to describe. Then, the method is used to discuss the Schwarzschild solution of the spherical symm etry gravitational field. The results show that there exists no any singularity again in the all process es and the whole space-time including the point . So it seams more rational to discuss gravitational pr oblems in the flat space-time. The paper includes three chapters. The first chapter discuss es how to transform the Schwarzschild solution of Einstein’s theory in the spherical symmetry gra vitation field into the dynamic equations in flat space-time. The second chapter provides a general pro of to transform all gravitational theories based on curved space-time into the theories based on flat spa ce-time. The third chapter discusses some problems of foundational concepts, for example, weath er space-time is curved or not when exit gravitational fields exist. 1. The Transformation of the Schwarzschild Solution According to the general theory of relativity, the Schwarzs child metric of the spherical symmetry gravitation field is ds2=c2(1−α r)dt2−(1−α r)−1dr2−r2(sin2θdϕ2+dθ2) (1) 1In the formula, we take α= 2GM/c2. Let θ=π/2 and put Eq.(1) into the geodetic line equation, according to the familiar results in the general theory of re lativity, we have the integrals c(1−α r)dt ds=ε (2) r2dϕ ds=L c(3) HereεandLare constants. From above two formulas, the linear element dscan be eliminated and we can get r2(1−α r)−1dϕ dt=L ε(4) Defining dτ= (1−α r)dt (5) and regarding τas the eigen time , tas the coordinate time and taking ε= 1,we can write Eq.(2)as cdτ=ds (6) Eq.(4) becomes r2dϕ dτ=L (7) HereLis the angel momentum of unit mass. Eq.(7) is just the conserv ation formula of angel momen- tum. Let’s first discuss the motions of particles with static mass es. By using Eq.(6), Eq.(1) can be written as (1−α r)(dt dτ)2−1 c2(1−α r)−1(dr dτ)2−r2 c2(dϕ dτ)2= 1 (8) By using Eq.(5) and (7), we get (dr dτ)2=c2α r(1−L2 αc2r+L2 c2r2) (9) Taking the differential about dτin the formula above, we get d2r dτ2−L2 r3=−c2α 2r2(1 +3L2 c2r2) (10) It should be noted that each quantity in Eq.(10) is defined in c aved space-time. In order to express the equation in flat space-time, the further transformation is needed. Let r0,ϕ0andt0represent the space-time coordinates in flat space-time, because of the in variability of the 4-diamention interval ds2 , we have ds2=c2dt2 0−dr2 0−r2 0dϕ2 0=c2(1−α r)dt2−(1−α r)−1dr2−r2dϕ2(11) Letr0=r,ϕ0=ϕwe get the transformation relation between t0andt c2dt2 0=c2(1−α r)dt2+ [1−(1−α r)−1]dr2(12) Considering Eq.(5) and (9), we get dr=c(1−α r)/radicalbigg α r(1−L2 αc2r+L2 c2r2)dt (13) Put it into Eq.12, we have dt0=/radicalbigg (1−α r)[1−α2 r2(1−L2 αc2r+L2 c2r2)]dt (14) 2Comparing it with Eq.(5), we get dτ= (1−α r)1 2[1−α2 r2(1−L2 αc2r+L2 c2r2)]−1 2dt0 (15) Combining Eq.(7)with (10) and let r0→rthe results of the Einstein’s theory can be expressed in the similar formula of the Newtonian gravitation in flat space-t ime d2/vector r dτ2=−GM(1 +3L2 c2r2)/vector r r3(16) Letu= 1/rand by considering Eq.(7), Eq.(16) can be transformed into d2u dϕ2+u=c2α 2L2(1 +3L2 c2u2) (17) The formula can describe the perihelion precession of the Me rcury. On the other hand, We have used the eigen time τin Eq.(16). It can be proved that the effect of special relativity has been considered in the formula. The s quare of a particle’s speed in the center gravitational field is V2=V2 r+V2 ϕ (18) From Eq.(7),(9) and (15), we have V2 r= (dr dt0)2= (dr dτdτ dt0)2=c2α r(1−α r)(1−L2 αc2r+L2 c2r2)[1−α2 r2(1−L2 αc2r+L2 c2r2)]−1 V2 ϕ= (rdϕ dt0)2= (rdϕ dτdτ dt0)2=L2 r2(1−α r)[1−α2 r2(1−L2 αc2r+L2 cr2)]−1(19) Therefore, we have V2=V2 r+V2 ϕ=c2α r(1−α r)(1 +L2 c2r2)[1−α2 r2(1−L2 α2c2r+L2 c2r2)]−1 1−V2 c2= (1−α r)[1−α2 r2(1−L2 α2c2r+L2 c2r2)]−1(20) Comparing Eq.(20) with Eq.(15), we obtain dτ=/radicalbigg 1−v2 c2dt0 (21) It is completely the same as the formula of time delay in the sp ecial theory of relativity. Therefore, Eq.(16) can be written as d/vector p dt=−GMm (1 +3L2 c2r2)/radicalbigg 1−V2 c2/vector r r3(22) Heremis the mass of particle, and dτis determined by Eq.(15). Because of /vectorL→/vectorV×/vector r, it can be seen that there exist the extra two items relative to V2/c2comparing with The Newtonian theory. The problem of energy conservation is discussed below. For s implification, we only discuss the situation with L= 0 . In this case, the particle moves along the radium directi on. By using Eq.(20), Eq.(22) becomes d/vector p dt=−GMm/radicalbig 1 +α/r/vector r r3(23) By producing d/vector ron the two sides of Eq.(23) and taking the integral, we have /integraldisplayd/vector p dt·d/vector r=−/integraldisplayGMm/radicalbig 1 +α/r·d/vector r (24) 3The left side of Eq.(24) can be written as /integraldisplayd/vector p dt·d/vector r=/integraldisplayd/vector p dt·d/vector r dtdt=/integraldisplay /vectorV·d/vector p=/integraldisplay /vectorV·dm/vectorV/radicalbig 1−V2/c2=mV2 /radicalbig 1−V2/c2+mc2/radicalbigg 1−V2 c2+E1(25) HereE1is a constant. So Eq.(24)can be written as mV2 /radicalbig 1−V2/c2+mc2/radicalbigg 1−V2 c2=mc2/radicalbigg 1 +α r+E (26) HereEis a constant. Supposes V→0 when r→ ∞,we have E= 0. Eq.(26) can be written as mV2 /radicalbig 1−V2/c2+mc2(/radicalbigg 1−V2 c2−1) +mc2(1−/radicalbigg 1 +α r) = 0 (27) LetTrepresents the kinetic energy of the particle, Urepresents the potential energy of the particle. We define T=mV2 /radicalbig 1−V2/c2+mc2(/radicalbigg 1−V2 c2−1) (28) U=mc2(1−/radicalbigg 1 +α r) (29) Eq.(27)is just the formula of energy conservation T+U=E= 0 . When α/r << 1 ,V << c from Eq.(27 we get the result of the Newtonian theory. mV2 2−GMm r= 0 (30) The motion equation of photon in the center gravitational fie ld is discussed as follows. For photons, ds= 0 so dscan not be used as the parameter of the equation of geodetic li ne. In this case, we take dτto replace dsand get the same results by solving the equations of gravitat ional field (1−α r)dt dτ=ε (31) r2dϕ dτ=L (32) Letε= 1we have dτ= (1−α r)dt (33) Because ds2=c2(1−α r)dt2−(1−α r)−1dr2−r2dϕ2= 0 (34) We get c2(1−α r)(dt dτ)2−(1−α r)−1(dr dτ)2−(rdϕ dτ)2= 0 (35) From the formula above we have (dr dτ)2=c2[1−(1−α r)L2 c2r2] (36) Taking the differential about dτ, we get d2r dτ2−L2 r3=−3αL2 2r4(37) 4By using Eq.(33) and (36), we get (dr dt)2=c2(1−α r)2[1−(1−α r)L2 c2r2] (38) Suppose the speed of photon in the gravitational field is V, from Eq.3233and38we have V=/radicalbigg (dr dt)2+ (rdϕ dt)2=c(1−α r)/radicalbigg 1 +αL2 c2r3(39) It is obvious that V/negationslash= constant, so the speed of light would change with rin gravitational field. Then let’s discuss how to transform the results into the flat refer ence system. For light’s motion, if we write the metric in the flat reference system as ds2=c2dt2 0−dr2 0−dr2 0−r2 0dϕ2 0= 0 (40) the result shows that light move in a uniform speed cin the gravitational field. However, this is improper for it contradicts Eq.(39). Suppose light’s speed isV0in the flat space-time, the metric should be written as ds2=u2 0dt2 0−dr2 0−r2 0dϕ2 0= 0 (41) From Eq.(34) and (41)we have V2 0dt2 0−dr2 0−r2 0dϕ2 0=c2(1−α r)dt2−(1−α r)−1dr2−r2dϕ2(42) Letr0=r,ϕ0=ϕwe get from Eq.(42) V2 0dt2 0=c2(1−α r)dt2+ [1−(1−α r)−1]dr2(43) By using Eq.(38), we have V2 0dt2 0=c2(1−α r)2(1 +αL2 c2r3)dt2(44) There exists one degree of freedom to choose the relations be tween t0andthere. If taking t0=t, we get V0=c(1−α r)/radicalbigg 1 +αL2 c2r3(45) Comparing Eq.(45) with Eq.(39) we obtain V0=V, that is to say, the speeds of lights are the same in the both curved and flat space-times. Therefore, by th e relation α=dV0/dt0=dV/dt , the accelerations and forces are the same, so that the forms of mo tion equations are also the same. So by connecting Eq.(32) with (37), we can directly write the mo tion equation of photons in the vector’s form in flat space-time as d2/vector r dτ2=−3αL2/vector r 2r5(46) Letu= 1/r, the formula can be transformed into d2u dϕ2+u=3α 2u2(47) The formula can be used to describe the deviation of light in t he solar gravitational field. As for the time delay experiments of radar waves in the solar gravitati onal field, by considering t0=tin Eq.(38) under the condition of week field, we can get cdt0= (1 +α r)(1−L2 r2)−1 2(1−αL2 c2r3)dr (48) 5Suppose radar waves just swept over the surface of the sun wit h the radium r0and the speed of radar waves is the speed of light in vacuum, we have L=cr0in the light of angular conservation. The integral of Eq.(48) is ct0=/radicalBig r2−r2 0+αln/radicalbig r2−r2 0+r r0−α/radicalbig r2−r2 0 2r(49) The same result can be reached from the formula(2). However, it can be seen from Eq.(38) that the photon would mov e in the speed over light’ speed in vacuum when L= 0 and r < α/ 2. This is unacceptable (The problem will be discussed again later.). So we re-define the transform relations between t0,tandτas dt0= (1−α2 r2)dt= (1 +α r)dτ (50) Put it into Eq.(36), we have dr dt0=c(1 +α r)−1/radicalbigg 1−(1−α r)L2 c2r2(51) From the formula, Eq.(49) can also be reached under the condi tion of weak field. It is obvious that Eq.(50) is the simplest form to obtain the formula (49).from (36). In this way, we have V0=c/radicalbigg 1 +αL2 c2r3(1 +α r)−1(52) Because /vectorL=/vectorV0×/vector r,when r→0 ,V0→0 , there is no the motion of over light’s speed again accordin g to Eq.(52). It should be noted that Eq.46is not the dynamic equation of th e photon in the gravitational field. Because when L= 0 , d2/vector r/dτ2= 0 , it seems that the photon is not acted by force. However, th e photon has acceleration in the gravitational field. So there should be a force acting on the photon. Therefore, Eq.46can only be regarded as the equation of kine matics from which the velocity and acceleration of the phone can be obtained. But it can not be re garded as the equation of dynamics of the photon from which the force can be obtained. Now let’s discuss how to obtain the dynamic equation of the ph oton based on Eq.(46). Let photon’s speed V0→V(as well as r0→r,t0→t) , when r→ ∞,V→c. When r <∞,V < c , showing that the photon is acted by repulsion and does retarded motio n. In order to obtain the dynamic equation of photon in the gravitational field, we suppose to h ave an imaginary particle with speed /vectorV′ and /vectorV′=/vector c−/vectorV (53) HereVis the speed of photon in the gravitational field, cis the speed of photon in vacuum. The directions of Vandcare supposed always the same. When photon’s initial speed V=c, the speed of imaginary particle is V′= 0 . When the photon falls down in the gravitational field with V < c , we have V′>0 . When V= 0 , we have V′=c. So it is obvious that the imaginary particle does acceleration motion in the gravitational field similar to the general particles with static masses. Therefore, the force acted on the imaginary particle in the g ravitational field of spherical symmetry can be supposed to be /vectorF′=−GMm (1 +3L2 c2r2)/radicalbigg 1−V′2 c2/vector r r3(54) Heremis the static mass of the imaginary particle. The dynamic equ ation of imaginary particle is d/vector p′ dt=−GMm (1 +3L2 c2r2)/radicalBigg 1−(/vector c−/vectorV)2 c2/vector r r3(55) 6On the other hand, similar to the particles with static masse s, the relativity momentum of a photon in the gravitational field can be defined as /vector p=m/vectorV R(56) Heremis so-called photon’s static mass and R=R(r, θ) is the function remained to be decided. Then, we define the imaginary particle’s momentum /vector p′as /vector p′=/vector pc−/vector p=m/vector c−mV R(57) Here/vector pcis photon’s momentum in vacuum. Put Eq.(57) into Eq.(55), we get the dynamic equation of the photon in the gravitational field of spherical symmetry d/vector p dt=GMm (1 +3L2 c2r2)/radicalbigg 2cV−V2 c2/vector r r3=/vectorF (58) However, it can be seen that it is actually unnecessary for us to introduce imaginary particle. In fact, we can directly suppose that the dynamic equation of ph oton in the central gravitational field is just Eq.(58), so long as from it we can reach the identical r esults comparing with the Einstein’s theory. The concrete form of function Ris discussed as follows. We have from Eq.(56) d/vector p dt=d dtm Rd/vector r dt=m Rd2/vector r dt2+m/vectorVd dt1 R(59) Put it into Eq.(58), we get photon’s acceleration in the grav itational field d2/vector r dt2=R/vectorF m−R/vectorVd dt1 R(60) On the other hand, from Eq.(46) and (50), we can get the result of the Einstein’s theory in the flat space-time d2/vector r dt2=−3αL2/vector r 2r5(1 +α r)−2+αVr/vectorV r2(1 +α r)−1(61) Comparing Eq.(60) with (61), we get R(/vectorF m−/vectorVd dt1 R) =−3αL2/vector r 22r5(1 +α r)−2+αVr/vectorV r2(1 +α r)−1(62) Decomposing the formula in the both directions of /vector erand/vector eϕ,we have R(F m−Vrd dt1 R) =−3αL2 22r4(1 +α r)−2+αV2 r r2(1 +α r)−1(63) RVϕd dt1 R=αVrVϕ r2(1 +α r)−1(64) Putting Eq.(64)into(63) and using Eq.(52), we get R=3L2 c2r2(1 +α r+3L2 r2+3αL2 C2r3)−1(1 +2α r+αL2 c2r3+2α2L2 c2r4)−1 2 (65) When L= 0 we have V=Vr,Vϕ= 0 . In this case, Eq.(64) does not exist. But from Eq.(63)dir ectly, we have dR dr=−α 2r2/radicalbigg 1 +2α r(1 +α r)R2+α r2R (66) 7This is the quasi-one order Bernoulii equationthe solution is R= [−ex/integraldisplay√ 1 + 2x(1 +x)e−xdx+C]−1(67) Herex=α/rWhen x→,R= 1the integral constant C can be determined. After the function Ris determined by the method above, Eq.(58) can be regarded as the dynamic equation of photon in the gravitational field of spherical sy mmetry in flat space-time. It coincides with the results of the Einstein’s theory and can be used to ex plain the deviation of light as well as the time delay experiments of radar waves in the solar gravit ational field. Besides, Eq.(58) can also be used to explain the gravitational red shift of spectral line . Let’s first establish the energy conservation equation of photon in the gravitational field. Similar to Eq. (24), we multiply d/vector ron the two sides of Eq.(58) and take the integral /integraldisplayd/vector p dt·d/vector r=/integraldisplay GMm (1 +3L2 c2r2)/radicalbigg 2cV−V2 c2/vector r r3·d/vector r (68) We only considering the situation with L= 0, by using Eq.(52), the two sides of Eq.(68) can be written as mc2 R(1 +α r)−2−/integraldisplaymc2α R2r(1 +α r)−3dr=/integraldisplaymc2α 2r2(1 +α r)−1/radicalbigg 1 +2α rdr (69) The integral of the right side is /integraldisplaymc2α 2r2(1 +α r)−1/radicalbigg 1 +2α rdr=−mc2(/radicalbigg 1 +2α r−αrctg/radicalbigg 1 +2α g) +C2 (70) When r→ ∞ , we have −mc2(/radicalbigg 1 +2α r−αrctg/radicalbigg 1 +2α g)r→∞=−mc2(1−π 4) (71) So when L= 0 we can define the potential energy of photon in the central g ravitational field as U(r) =mc2(/radicalbigg 1 +2α r−αrctg/radicalbigg 1 +2α g−1 +π 4) (72) When r→ ∞ we have U(r)→0. Let mc2 R(1 +α r)−2−/integraldisplaymc2α Rr2(1 +α r)−3dr=K(r) +C1 (73) Eq.(69) can be written as K(r) +mc2(1−π 4) +mc2(/radicalbigg 1 +2α r−αrctg/radicalbigg 1 +2α r−1 +π 4) =C (74) HereCis a constant. If we choose another proper constant band let V0represent the light’s frequency in vacuum, and define the total energy of photon as E=C+b=hv0=mc2, the kinetic energy of photon can be defined as T(r) =K(r) +mc2(1−π 4) +b (75) In this way, when L= 0 , the formula of energy conservation of photon in the gravi tational field of spherical symmetry can be written as T(r) +U(r) =E=hv0 (76) 8On the other hand, in the general theory of relativity, the re d shift of spectral line is considered caused by time delay of gravitational field. In the gravitati onal theory of curved space-time, the photon is actually considered to be free one and moves along t he curved geodetic line without potential energy or no force acted on it. Therefore, the total energy of a photon is equal to its kinetic energy in the gravitational field. But if the formula E=hvis considered tenable in any point in the gravitational field, because is a variable, is also a variabl e, that is to say, that the energy of photon in the gravitational field is not conservative. This is just the price we have to pay in the gravitational theory based on the curved space-time to explain the red shif t of spectral line, thought now people seem to avoid this problem. However, this is unacceptable wh en we describer gravitational force in flat space-time. In order to keep energy conservation for pho ton in the description of flat space-time, the rational way is to suppose that the frequency of photon is only relative to its kinetic energy, and has nothing to do with its potential energy with T=hv. Here vis the frequency of photon in the point in the field. So the formula of energy conservation o f photon in the gravitational field is hv+U(r) =hv0=mc2. In this way, the formula of red shift becomes △v v0=v−v0 v0=−(/radicalbigg 1 +2α r−αrctg/radicalbigg 1 +2α r−1 +π 4) (77) Under the condition of weak field with α/r << 1 by the developing of the Taylor series, we have △v v0=v−v0 v0=−(α r−α 2r) =−GM r(78) Under the same condition, the result of the general theory of relativity is △v v0=v−v0 v0=/radicalbigg 1−α r−1 =−GM r(79) The both are the same. But in the strong field, they are not the s ame, especially when α > r, Eq.(79) becomes meaningless, but Eq.(77)is still meaningful. In fa ct Eq.(77) can be used to explain the big red shifts of the quasi-stellar objects. Taking α/r= 7.5, we have the red shift value Z=−△v/v0= 2.47 . Suppose the mass of the quasi-stellar object is 1040Kg, it can be calculated that the radium of he quasi-stellar object is r= 1.98×1012mand its mean density is ρ= 3.08×102Kg/m3. It is only 0 .22 times comparing with the mean density of the sun with ρ= 1.40×103Kg/m3. Taking α/r= 12 we haveZ= 3.42 ,r= 1.24×1012m,ρ= 1.26×102Kg/m3. The even density is similar to the sun’s density. According to the general theory of relativity, if α/r > 1ris in the inside of black hole. But according to Eq.(77), only when r→0 , we have infinite red shift with Z→ ∞ . For the common stars, r/negationslash= 0 so according to Eq.(77), the black holes do not exist actua lly. What is mainly shown above is that the results are the same und er the condition of weak field when the gravitational theory is described in the both flat an d curved space-time. Besides the red shift of spectral line, the following discussion will furth er show their differences in the strong fields owing to the fact that the coordinate times are different in th e both situations. We also only discuss particle’s motions along the radial direction of the spheri cal symmetry gravitational field with L= 0 . For a particle with static mass, when L= 0 , the speed is from Eq.(9) and (5) V=dr dt=±c/radicalbiggα r(1−α r) (80) In the formula it has been supposed that V= 0 when r→ ∞ . Definite the direction of velocity is positive along the radius vector’s positive direction. The direction of particle’s velocity is negative when the particle falls down the gravitational field. The acc eleration is α=dV dt=−1 2c2α r2(1−α r)(1−3α r) (81) 9Take the integral of Eq.80and suppose r=r0when t= 0 , we have ct=±1√α[2 3(r3 2−r3 2 0) + 2α(√r−√r0) +α3 2ln(√r−√α)(√r0+√α) (√r+√α)(√r0−√α)] (82) (82) Because t >0 the negative sign is taken when a particle falls down along t he direction of radius vector. Conversely, it takes positive sign. Let’s first discuss the particle’s motion in the area r≥α. When , r > a,Vis a negative number. When r=α,V= 0 the particle arrives at the Schwarzschild event horizon. It can be known from Eq.(81) that when r= 3αandr=α, acceleration becomes zero. When r >3α,α <0 the particle is acted by gravitation and is accelerated downward. When α < r < 3α,α >0 , the particle is acted by repulsive force and is retarded Gravitation becomes repu lsion, it seems unimaginable..When r=α , the particle is not acted by force and at rest on the surface o f event horizon. It can be known from Eq.(82) that when r=α,t=∞, that is to say, it takes the particle an infinite time to reach the event horizon. Then let’s discuss the particle’s motion beneath the event h orizon r < α . It can be known from Eq.(82) that the time has no definition inside the event h orizon because the logarithm of a negative number has no definition(At present, some peasants think that it means space and time to be exchanged each other in the black holes. This is absurd. How one dimension’s time can be transformed into three dimension’s space?). So speaking st rictly, it is meaningless to talk about particle’s speed, acceleration and motion in the area r < α . If this problem is neglected temporary, we can write Eq.(80)as V=±c/radicalbiggα r(1−α r) +V0 (83) When r=α,V= 0 so V0= 0 . Therefore, the velocity and acceleration can still be ex pressed by Eq.(80) and (81). It is obvious that we have α <0 inside the event horizon, meaning that the particles only acted by gravitation. Suppose a particle at t he point rhas a velocity upward, it would be retarded until a=V= 0 when it arrives at the event horizon and stays there at last . If a particle has a velocity downward, is would be accelerated and reach li ght’s speed at a certain place. After that, the particle would move in the speed over light’s speed and reaches an infinite speed at the point r= 0. The results is the same as that analyzed in the current the ory using the method of light cone, except that the particle’s speed would be over light’s speed . As for a photon’s motion, when L= 0, from Eq.(38) we have V=dr dt=±c(1−α r) (84) α=dV dt=c2α r2(1−α r) (85) Letr=r0when t= 0 , we have integral from Eq.(84) ct=±(r−r0+αlnr−α r0−α) (86) When a photon falls down in the area r > α , because α >0 they are retarded by repulsion. When the photon arrives at the event horizon r=α, the speeds and accelerations are equal to zero and the infinite time is need. In the area r < α , time has no definition for the same reason. Despite of this problem, we have α <0 in the area r < α . Suppose a photon has a velocity upwards inside the event horizon, it would be retarded by gravitation and has α=V= 0 when it arrives at the surface of event horizon. If the photon has a velocity downwards, it w ould be accelerated. When it arrives at the point r=α/2 , its speed would reach light’s speed in vacuum again. After that time, the photon would move in the speed over light’s speed in vacuum. When the photon arrives at the point r= 0 , its speed becomes infinite(3). 10It is obvious that there exist some things irrational, espec ially particles would move in the speeds over light’s speed in vacuum. In fact in the current theory of black holes, the motions with the speeds over light’s speed can not be avoided during the processes in which material collapses toward the center singularities of gravitational fields. At present, t hose problems are attributed to the improper selections of coordinates. In order to eliminate those defe cts, people now transform the problems into other coordinate systems to discuss, for example, the E ddington and the Kruskal coordinate system. In the new coordinate system, though the singularit ies on the surfaces of event horizons can be eliminated, they can not yet be eliminated at the point r= 0. Hawking even proved that it was impossible to eliminate all singularities in the general th eory of relativity(3). Now let’s discuss the problems in flat space-time. When t0= 0 let r=r0. According to Eq.(19), when a particle falls free down the gravitational field, we ha ve V=−c/radicalbiggα r(1 +α r)−1 2 (87) α=−1 2c2α r2(1 +α r)−2(88) ct0=2 3√α[(r0+α)3 2−(r+α)3 2] (89) It is obvious that every thing is normal in the area r > α . The particle is monotonously accelerated by gravitation. There is no any singularity in the whole spac e-time and in all physical quantities. When the particle arrives at the point r= 0 , we have V=−lim x→∞c√x√1 +x→ −c(x=α r) (90) α=−lim x→∞c2x2 2α(x+ 1)2→ −c2 2α(91) It can be seen that the particle’s speed tends to light’s spee d but can not yet reach it. Besides, acceleration and time are finite. When a particle moves along the positive direction of radius vector, as long as it has a speed at the point r V≥c/radicalbiggα r(1 +α r)−1 2 (92) the particle can escape the gravitational field and has a spee dV≥0 when it reach the point r→ ∞ . As for a photon, when it falls down the gravitational field in fl at space-time, according to Eq.(51), its velocity and acceleration are V=dr dt0=−c(1 +α r)−1(93) α=dV dt0=c2α r2(1 +α r)−3(94) It can be seen that the photon is acted by repulsion and dose th e retarded motion. When t= 0 let r=r0,we have ct0=r−r0+αlnr0 r(95) There is no any singularity in the area r > α . When the photon arrives at the point r= 0, its speed V= 0 acceleration is also finite α= lim x→∞c2x2 α(1 +x)3→0 ( x=α r) (96) But it takes photons an infinite time to reach the point r= 0 . 11Therefore after the Schwazschild solution is transformed i nto flat space-time to describe, all orig- inal singularities disappear (In flat space-time, singular ity appears in the form of over light speed’s motion.). So it is obvious that the singularities in the gene ral theory of relativity are actually caused by describing the theory in the curved space-time. The gravi tational field itself has no singularities. Meanwhile, it is known a photon can escape from the gravitati onal field when it moves along the direction of radius by the action of repulsion as long as it is not at the point r= 0 . In this way, the black holes, at least the singular black holes with infinite d ensities and infinite small volumes, do not exist. If observers are in the reference system which falls free dow n the gravitational field, in this case, the time is τand the distance between the observers and the center mass is raccording to Eq.9, we have V=dr dτ=−/radicalbiggα r=−/radicalbigg 2GM r(97) α=dV dτ=dV drdr dτ=−α 2r2=−GM r2(98) τ=−/integraldisplay/radicalbiggr αdr=−2 3√α(r3 2−r3 2 0) (99) They are just the results of the Newtonian gravitational the ory. There are no singularities when r >0 . But when r < GM/c2r, the relative speed is over light’s speed. When r→0 , the relative speed becomes infinite. So the reference system falling free down t he gravitational field is not yet a good reference system for the discussion of gravitational probl ems. The reason will be discuss in the third chapter. In brief, at least for the spherical symmetry gravitational field, it is more rational to study grav- itational problems in flat space-time than in curved space-t ime. By transforming the solution of gravitational field equation to discuss in flat space-time, t he problems can become more rational and simple. Some puzzling problems just as the singularity prob lems of black holes, the flat problem of the universal early stage and so on would be expounded. So it i s necessary for us to re-examine the conclusions of the current general relativity by transform ing them into flat space-time to study in order to get more rational results. 2. The transformations in the general situations Now let’s generally prove that it is possible to transform th e gravitational theories described in the curved space-time into in the flat space-time. In the foll owing discussion, the indexes of Egyptian letters are used to represent the 4-diamention quantities a nd the indexes of Latin letters represent the 3-diamention quantities. Let xαrepresent the 4-diamention coordinates in curved space-ti me and xα 0 represent the 4-diamention coordinates in flat space-time. The 4-diamention linear elements in both space-times are individually ds2=dxα 0dxα 0= (dx0 0)2−dxi 0dxi 0 (100) ds2=gαβdxαdxβ=g00(dx0)2−2g0idx0dxi−gijdxidxj(101) In the formulas, x0andx0 0are the time components. The equation of geodetic line of a pa rticle moving in a gravitational field is d2xα ds2+ Γα βσdxβ dsdxσ ds= 0 (102) For a certain gravitational field, suppose the metric tensor gαβhas been obtained by solving the Einstein’s equation of gravitation field or other equations based on curved space-time, we can get from the integrals of Eq.(102) xi=xi(s) (103) as well as x0=x0(s) (104) 12From Eq.(104), we can obtain s=s(x0) (105) Put it into Eq.(103), we get xi(s(x0)) =xi(x0) =xi(t) (106) The particle’s velocity and acceleration are dxi(t) dt=Vi(t) (107) d2xi(t) dt2=dVi(t) dt=αi(t) (108) In addition, two independent equations can be obtained by el iminating time t in the three equations of (106) φ1(x1, x2) = 0 φ2(x1, x3) = 0 (109) Ifxiare the coordinates in the Euclidean space, φ1andφ2represent the equations of two columnar surfaces with the axial lines meeting at right angles. The in tersecting line of the two columnar surfaces determined by Eq.(109) presents the orbit of a particle movi ng in the 3-dimention Euclidean space. If xiare the non-Euclidean space coordinates, φ1andφ2represent the two 2-dimention curved surface equations in the non-Euclidean space. Their intersecting l ine also represents the orbit of a particle moving in the 3-dimention non-Euclidean space On the other hand, as we know, any point on the 2-diamention no n-Euclidean curved surface can find a one-to-one point in the 3-diamention Euclidean space, that is to say, the 2-dimention non- Euclidean curved surface can be inlaid into the 3-dimention Euclidean space. Therefore, any point at the interesting line of the two 2-dimention non-Euclidean c urved surfaces can also find a one-to-one point in the 3-diamention flat Euclidean space. So we have the transformation relation between the points of geodetic line in the non-Euclidean space and the po ints in the Euclidean space xi=xi(xj 0) or xi 0=xi 0(xj) (110) In order to get transformation relation between time tandt0, by the condition ds2=constant , we have ds2= (dx0 0)2−dxi 0dxi 0=g00(dx0)2−2goidx0dxi−gijdxidxj(111) or c2dt2 0=c2g00dt2−2cg0idtdxi−gijdxidxj+dxi 0dxi 0 (112) By considering Eq.(106) and (110), each item on the right sid e of Eq.(112) can be expressed as the function of time t, so we get transformation relation of time t0=/integraldisplay/radicalbigg g00−2g0idxi cdt−gijdxi cdtdxj cdt+∂xi 0 ∂xl∂xi 0 ∂xkdxl cdtdxk cdtdt (113) i.e., t0=t0(t) or t =t(t0) (114) After the transformation relations of space-time coordina tes are obtained, the equation (107) in the non-Euclidean space can be transformed into that in the Eucl idean space. From Eq.(107),(110) and (114), we have ∂xi ∂xj 0dxj 0 dt0=dxi dt=Vi(t) =Vi(t(t0)) =Vi(t0) (115) This is an equation set of three variables and one order about dxj 0/dt0. We can obtain the velocity and acceleration in flat space by solving the equation set dxi 0 dt0=Vi 0(t0) (116) 13d2xi 0 dt2 0=dVi 0(t0) dt0=ai 0(t0) (117) For the particle with static mass m, the momentum in flat space is /vectorP0=m/vectorV0/radicalbig 1−V2 0/c2(118) We get d/vector p0 dt=m/radicalbig 1−V2 0/c2d/vectorV0 dt+m(/vectorV0·/vector α0)/vectorV0 c2(1−V2 0/c2)3/2=/vectorF (119) Put Eq.116and117into Eq.(119), we get the dynamic equation and force of the particle in the gravi- tational field in flat space. In general, they are different fro m the Newtonian theory. As for photon, we can obtain the corresponding equations (11 6) and (117) from its geometric equation in curved space, then define photon’s momentum in th e same form of Eq.(56). After that, the dynamic equation of photon can be established in flat spac e-time. But it is unnecessary for us to discuss it nay more here. In this way, we have achieved the tra nsformation of gravitation’s descriptions from curved space-time to flat space-time. In the same, the gravitation’s descriptions can also be tran sformed from flat space-time into curved space-time. From Eq.(119) in flat space-time, we can get by so lving the equation xi 0=xi 0(t0) (120) By introducing arbitrary coordinate transformation xα 0=xα 0(xβ) (121) we have dxα 0 ds2=∂xα 0 ∂xβdxβ ds(122) d2xα o ds2=∂2xα 0 ∂xβ∂xσdxβ dsdxσ ds+∂xα 0 ∂xβd2xβ ds2(123) Using Eq.(116) and (121), we get from Eq.(101) ds= (c2−dxi 0 dt0dxi 0 dt0)1 2dt0= (c2−Vi 0Vi 0)1 2dt0=A(t0(xβ))dt0=A(xβ)dt0 (124) Put it into Eq.(122), use Eq.(116) and (121), we have dxα 0 dt0=A−1∂xα 0 ∂xβdxβ ds=Vα 0(t0(xβ)) =Vα 0(xβ) (125) HereV0 0=c. Eq.(125) is the equation set of four variables and one order about dxβ/ds. We can obtain from Eq.(125) dxα ds=Bα(xβ) (126) Put Eq.(124) into Eq.(123) and using Eq.(126), we get A−2d2xα 0 dt2 0=BβBσ∂2xα 0 ∂xβ∂xσ+∂xα 0 ∂xβd2xβ ds2−A−1dA−1 dt0dxα 0 dt0(127) By using Eq.(124)and(126) again, we have dA−1 dt0=∂A−1 ∂xβdxβ dt0=∂A−1 ∂xβdxβ dsds dt0=∂A−1 ∂xβBβA (128) 14We can write αi 0(t0) =αi 0(t0(xβ)) and have α0 0= 0 in Eq.(117). By considering Eq.(125) and (128), Eq.(127) can be written as: ∂xα 0 ∂xβd2xβ ds2= (A−2Fi 0+BβVi 0∂A−1 ∂xβ−BβBσ∂2xα 0 ∂xβ∂xσ) (129) It is an equation set of four variables and one order about d2xβ/ds2, in which all coefficients are the function of xβ, so we can obtain d2xα ds2=Kα(xβ) (130) The formula can be re-written as d2xα ds2−Kα(dxβ dsdxσ ds)−1dxβ dsdxσ ds=d2xα ds2−KαB−1 βB−1 σdxβ dsdxσ ds= 0 (131) Let Γα βσ=−KαB−1 βB−1 σ (132) Eq.(131) becomes d2xα ds2+ Γα βσdxβ dsdxσ ds= 0 (133) Regarding Γα βσas the Christoffel sign, Eq.(133) is just the geodetic line eq uation in the new reference system. From the definition Γα βσ=1 2gαβ(∂gρσ ∂xβ+∂gρβ ∂xσ−∂gβσ ∂xρ) (134) we known that the number of independent ∂gαβ/∂xσis just the same as the number of independent Γα βσ. Meanwhile, gαβcan be got from the following formula gαβ=/integraldisplay∂gαβ ∂xσdxσ(135) So the metric tensors can be determined by Eq.(134) and (135) . It is noted that if Eq.(121) is put into Eq.(101) directly, we have ds2=g′ αβdxαdxβ(136) 136 The metric tensor gαβis different from that shown in Eq.(135) in general. The metri c tensor g′ αβ shown in Eq.(136) is the Euclidean metric in essence for it ca n return to the original form Eq.(101) by an inverse transformation. But the metric tensor gαβdetermined by Eq.(134) and (135) can not in general, so they are the non-Euclidean metrics in general . Up to now we have achieved the transformation of gravitation ’s descriptions between the curved space-time and the flat space-time, and proved their equival ence. The difference is that in curved space-time particles move along the geodetic lines without forces acting on them, but in flat space- time particles move along the non-geodetic lines acted by gr avitation. What kinds of descriptive forms are taken depends on convenience in principle, but as shown a bove, the practical results should be considered. 3. Discussions on some foundational concepts 1. Is the space-time curved or flat after all when gravitation al field exists? This is first a problem of measurement. Whether can we answer t his problem by the direct measurement? The answer is negative. Even thought the space -time is curved when the gravitational field exists, we can not detect it by the direct measurement. T his is owing to the fact that before the measurement we have to define standard ruler and clock. Bu t only in flat space-time, we can do them. In curved space-time, we have no definitions of standar d ruler and clock. When the ruler and 15clock defined in flat space-time are put into gravitational fie lds, they would change or become curved synchronously with the fields, so that the measurements can n ot show the changes of curved level of space-time. The ruler and clock in the gravitational fields c an not free themselves from the effects of gravitational fields, so it is impossible to show whether spa ce-time is curved or flat when gravitational field exists by the direct measurement. What we can do is to use indirect methods, for example, to obse rve the orbits of test particles orbit or the red shifts of spectral lines in the gravitational field s to decide the curved level of space-time. However, as shown above, we can describe the orbit of test par ticle in gravitational field by either geometric equation in curved space-time, or the dynamic equ ation in flat space-time. We can also explain the red shifts of spectral lines in the gravitationa l fields as the results of time delay or the potential energy’s changes. Two methods are equal to each ot her. So it is obvious that space-time itself can not be designated as curved or flat actually. The re ality is what kind of reference systems, curved or flat, we choose to describe it. If the curved referen ce system is chosen, the space-time is curved. If the flat reference system is chosen, the pace-time becomes flat. It is meaningless to talk about space-time itself curved or flat. So we should only use t he concept of curved or flat reference system, in spite of the concept of curved or flat space-time. 2. The equivalent principle According to the weak equivalent principle, gravitational mass and inertial mass are equivalent to each other. Let mrepresent static mass, represent inertial motion mass, we h ave m1=m/radicalbig 1−V2/c2(137) LetmGrepresent gravitational mass, comparing Eq.(22) with the N ewtonian formula of gravitation and considering the relation /vectorL=/vectorV×/vector r, we get mG=m/radicalbigg 1−V2 c2[1 +3(/vectorV×/vector n)2 c2] (138) Here/vector nis the unit radius vector. It is obvious m1/negationslash=mGin general situations. Only when V= 0 , they are equal to each other. In fact, all completed Eotvos ex periments only prove that gravitational mass and inertial mass are equal to each other when the testin g bodies on the two ends of cantilever beam are at rest each other(4). It has not yet be proved that they are equivalent when there e xists relative motion between them. It should be noted that the for mula (138) is the result of Einstein’s field equation, showing that gravitational mass and inertia l mass are not equivalent actually when the factor of speed is considered. 3. The principle of general relativity The paper’s conclusions are completely based on the Einstei n equation of gravitational field, no any new hypothesis is introduced besides transforming the t heory to the flat reference system to discuss. According to the general theory of relativity, it i s equivalent to discuss physical problems in any reference system in nature. No one is more superior. Sinc e we can discuss gravitational problems in any reference system, we can also discuss them in flat refer ence system. However, the results show that flat reference system seems more superior for the discus sion of gravitational problems. The results contradict the principle of general relativity. So we have t o discuss this problem further. Einstein established special relativity that denied the ex istence of absolutely static reference sys- tem. Later, he put forwards the principle of general relativ ity trying to cancel the superior position of the inertial reference systems. If we consider the princi ple of general relativity as that the motion equations are covariant, or the basic forms of the motion equ ations are the same in any reference sys- tem, the principle is all right. However, it does not mean tha t the concrete forms of motion equations and their solutions are the same. In special relativity, the 4-diamention coordinate transformation means that the relative speed is introduced. Let pµrepresents the 4-diamention momentum, Fµrep- resents the 4-diamention force. The basic form of motion equ ation dpµ/dt=Fµis unchanged when an inertial reference system is transformed into another in ertial reference system moving in a uniform 16speed.. But the concrete forms of the 4-diamention force Fµand particle’s motion, as well as and some physical quantities would change. For example, length cons trict, time delay, moving mass increasing and the form of force changing and so on, though according to s pecial relativity, these changes only have relative meanings. In general relativity, the coordinate transformations inv olve more problems. At present, it is considered that a solution of gravitational field equation c an still represent the same field after the solution has been transformed into another new reference sy stem. This conclusion is worthy of further discussion and consideration. If the coordinate transform ation is carried out in the 3-diamention space, there is no any problem. But in the 4-diamention space-time, because time is involved, the situation is completely different. In the general theory of relativity , the 4-diamention coordinate transformation means that acceleration or non-inertial reference system i s introduced. According to the principle of equivalence, non-inertial reference system is equal to gra vitational field. The transformation from one non-inertial reference system to another means that a gravi tational field is changed to another. So the coordinate transformations would change physical process es. Speaking clearly, for a gravitational field with a determinate form, suppose we have obtained the soluti on by solving the Einstein’s field equation, if the solution is transformed into another reference syste m, the form of the solution would change, though the basic form of field equation is unchanged. In the li ght of the principle of equivalence, it means that a new gravitational field is introduced and the o riginal solution loses its meaning. Therefore, a determinate gravitational field can only corre sponds to a determinate metric, arbitrary coordinate transformation is forbidden according to the pr inciple of equivalence. Unless the same results can be reached in new reference system for all proble ms, but this is impossible in general. For example, we can not calculate the perihelion precession of M ercury and other experiments and get the same results in the Edington or Kruskal coordinate systems. This is just the reason why the energy of gravitational field can not be defined well in the current theo ry. For a definite gravitational field with a certain of symmetry, we can only define its energy in the defin ite metric with the same symmetry. But it is allowed to transform the solutions into the inertia l or flat reference systems to discuss. In this case, what is done is to transform the geodesic lines int o the dynamic equations of gravitation without any attached force being introduced. Or speaking mo re clearly, there exist no relativity and arbitrariness in the description of gravitation. A certain of absoluteness is needed for us to describe gravitation. We should establish a united standard for grav itation. Only based on the flat reference system, we can do it. It can be seen from discussions above that though the Einstei n’s theory of gravitation has obtained great succession, there still exist some problems in its the oretical and logical foundation which need to be cleared and reformed so that the theory can become more r ational. It is obvious that we need renewing some ideas about the essence of space-time and grav itation. The author will discuss them in detail later. References [1] N.Rosen, Phys.Rev. ,57 147 (1940); N.Rosen, Am.Phys., 2 2 1 1963); N.Rosen, Gen.Rel.Grav.,435 (1973); N.Rosen., Gen.Grav., 10 69 (1978). Weitou Ni, Phys. Rev.,D, 7 2880 (1973); D.L.Lee, A.P.Lightma, Weitou Ni, Phys.Rev., D, 2880 (1973). Densov V .I.,Logunov A.A.,TMP, 50 3 (1982). Wang Yongjiu, Tang Zhiming, The Theory And Effect Of Gravitat ion, 592 1990. [2] Hun Linbao, Liu Xueming, Liu Mingzheng, General Relativ ity, 120 (1995. [3] S.W.HawkingG.F.R.Ellis, The Large Scale Structure of S pace Time,(1972). [4] Liu Liao, General Relativity, 5 (1987). 17

arXiv:physics/0004052v1 [physics.data-an] 21 Apr 2000July 25, 2013 A Comment on the Roe-Woodroofe Construction of Poisson Confi dence Intervals Mark Mandelkern and Jonas Schultz Department of Physics and Astronomy University of California, Irvine, California 92697 (submitted to Phys. Rev. D on Feb. 17, 2000) Abstract We consider the Roe-Woodroofe construction of confidence in tervals for the case of a Poisson distributed variate where the mean is th e sum of a known background and an unknown non-negative signal. We poi nt out that the intervals do not have coverage in the usual sense but can b e made to have such with a modification that does not affect the believab ility and other desirable features of this attractive construction. A simi lar modification can be used to provide coverage to the construction recently pro posed by Cousins for the Gaussian-with-boundary problem. PACS number(s): 02.50.Cw, 02.50.Kd, 02.50.Ph, 06.20.Dk 1I. INTRODUCTION A problem of long-standing interest is that of setting confid ence intervals for an unknown non-negative signal µin the presence of a known mean background bwhen the measurement nis Poisson distributed as p(n;µ+b). When n < b , the usual estimate for µ, i.e. n−b, is negative, leading in most constructions to small upper li mits that imply unrealistically high confidence in small values of µ. In a recent paper, Roe and Woodroofe [1] propose a construction that produces more believable intervals and c ontains the unifying feature that one need not decide beforehand whether to set a confidence int erval or an upper confidence bound. However, since the Roe-Woodroofe confidence belt (of confidence level α) is not constructed from an unconditional probability density and does not have coverage in the usual sense (i.e. unconditional coverage), one cannot stat e that the unconditional probability of the interval enclosing the true value is at least α. Our comment is that a straightforward modification of the Roe-Woodroofe confidence belt gives it co verage, making the construction effectively an ordering principle applied to the Poisson pdf, albeit reached by circuitous means. II. ROE-WOODROOFE CONFIDENCE INTERVALS Roe and Woodroofe are motivated by the observation [1] that t he measurement n= 0 implies that zero signal (as well as zero background) is seen ; thus, the resulting estimate forµis zero, independent of b. They argue therefore that the confidence interval for µfor n=0 must be independent of b. Extending the argument, they note that for any observation n, one has observed a result nfrom the Poisson pdf p(n;µ+b)anda background of at mostn. They formulate a method of obtaining confidence intervals b ased on the conditional probability to observe ngiven a background ≤nand obtain the desired result for n= 0 2and approximately the classical confidence intervals for n > b . While they identify their method as an ordering principle, it is not one in the same sens e as Ref.s [2] and [3] which explicitly choose a confidence belt of probability αusing the Poisson pdf p(n;µ+b) and the Likelihood Ratio Construction and invert it to find confidenc e intervals. The latter methods do not obtain intervals that are independent of bforn= 0 and yield confidence intervals which are unphysically small for n < b. Although the Roe-Woodroofe construction does not have cove rage in the usual sense , it can be easily modified to obtain coverage, by retaining the left-hand boundary of the confidence belt and adjusting the right-hand boundary so tha t for all µthe horizontal inter- vals contain probability α. In Fig. 1 we show the Roe-Woodroofe 90% intervals for b= 3 along with one-sided and central confidence belts∗for the Poisson distribution without background. We note that the Roe-Woodroofe horizontal inte rvals do not coincide with the one-sided intervals shown for µ <2.44. Therefore for some values of µin this range, the confidence belt does not satisfy the coverage requirement th at≥90% of the probability is contained. Because coverage cannot be exact when the variab le is discrete, the error for the example given here is not of great numerical significance . The minimum coverage of ∗We show the confidence belt consisting of central intervals [ n1(µ0),n2(µ0)] containing at least 90% of the probability for unknown Poisson mean µ0in the absence of any known background (dotted) and the 90% one-sided belt consisting of intervals [0,nos(µ0)](dashed). There is some arbitrariness in the choice of a central interval for a discr ete variate. We choose the smallest interval such that there is ≥90% of the probability in the center and ≤5%, but as close as possible to 5%, on the right. The alternative of requiring ≤5%, but as close as possible to 5%, on the left gives slightly less symmetrical intervals. For the latter c hoice the 90% Poisson upper limit for n= 0 isµ0= 3.0 compared to µ0= 2.62 for our choice. For µ0<2.62, according to this prescription, one cannot construct an interval containing probability >90% that does not include n= 0 and we adopt 90% one-sided intervals. 3∼0.87 is obtained at µ∼0.4. Undercoverage is more severe for greater b; forb= 10.0, the minimum coverage is ∼0.78. However, it is desirable to have coverage, which we obtai n as shown in Fig. 2 where we have changed the right side of the confi dence belt so that the horizontal intervals contain probability ≥90%. We note that the confidence intervals for small n, i.e.n < b, are unchanged. Intervals for both constructions are given in Table I. It would be nice to devise an ordering principle that can be di rectly applied to the Poisson pdf p(n;µ+b) to obtain the confidence belt shown in Fig. 2, if only because the construction we have used here is aesthetically unpleasing . This method, which consists of first determining vertical intervals per Ref. [1], and then fi xing them, leaves something to be desired. However, in the end the method of construction do es not really matter. What results here is an ordering procedure that yields a confidenc e belt with coverage and produces physically sensible intervals. B. Roe has noted [4] that our modification is equally applicab le to a construction due to R. Cousins, in which the Roe-Woodroofe method of conditioni ng is applied to the Gaussian- with-boundary [5] problem. Here, for example, an interval o f confidence level αis sought for an unknown non-negative signal µand the measurements x are normally distributed as N(x;µ). As for the Roe-Woodroofe construction referred to above, the Cousins construction produces physically sensible confidence intervals for all xincluding x <0. However this construction significantly undercovers for µ <0.5 and significantly overcovers for µ∼1. In order to produce exact coverage using the Cousins constru ction, we retain the left hand (upper) curve of the confidence belt xl(µ) and recalculate the right hand (lower) curve xr(µ) so that the horizontal intervals contain probability αusing: 2α=erf(µ−xl√ 2) +erf(xr−µ√ 2). (1) III. CONCLUSION 4For the case of Poisson distributed measurements nwith a non-negative signal mean µand known mean background b, the Roe-Woodroofe construction produces well-behaved confidence intervals, particularly for n < b where other constructions yield unphysically small intervals. Since the construction is not based on inte grating probabilities that arise from an unconditional pdf, it does not produce a confidence be lt with coverage in the usual frequentist sense. We suggest a modification that provides c overage while preserving the desirable features of the construction. While the changes i ntroduced by this modification are relatively small for the example given here (they are lar ger for greater b), nevertheless the procedure corrects a formal defect in the original const ruction. A similar modification provides coverage for a construction recently discussed by R. Cousins for the Gaussian-with- boundary problem. 5REFERENCES [1] B. P. Roe and M. B. Woodroofe, Phys. Rev. D 60, 053009 (1999). [2] G. J. Feldman and R. D. Cousins, Phys. Rev. D 57, 3873 (1998). [3] C. Giunti, Phys. Rev. D 59, 053001 (1999). [4] B. P. Roe, private communication. [5] R. D. Cousins, arXiv:physics/0001031. 6TABLES Roe-Woodroofe Modified n(observed) Lower Upper Lower Upper 0 0.0 2.44 0.0 2.44 1 0.0 2.95 0.0 2.95 2 0.0 3.75 0.0 3.75 3 0.0 4.80 0.0 4.80 4 0.0 6.01 0.0 6.01 5 0.0 7.28 0.0 7.28 6 0.42 8.40 0.16 8.42 7 0.96 9.58 0.90 9.58 8 1.52 10.99 1.66 11.02 9 1.88 12.23 2.44 12.23 10 2.64 13.50 2.98 13.51 11 3.04 14.80 3.75 14.77 12 4.01 15.90 4.52 16.01 TABLE I. Comparison of confidence intervals for the Roe-Wood roofe and modified Roe-Woodroofe constructions 7FIGURES FIG. 1. 90% Poisson confidence belts for unknown non-negativ e signal µin the presence of a background with known mean b taken to be 3.0, where nis the result of a single observation. The solid belt is the Roe-Woodroofe construction, the dotte d belt the central construction and the dashed belt the one-sided construction of 90% Poisson lower limits. Here µ0=µ+bis the parameter representing the mean of signal plus background. We illustr ate confidence belts in this manner to demonstrate the absence of coverage for the Roe-Woodroofe c onstruction and to emphasize that a naive approach to setting a confidence interval for µleads to a null interval for sufficiently small n < b, in this case n= 0. The solid line Roe-Woodroofe lower limit for n≤5 is at µ= 0. 8FIG. 2. 90% Poisson confidence belts described in Fig. 1 where the solid belt is modified as described in the text to give coverage. The dotted and dashed belts are described in the Fig. 1 caption. For n= 6,7,8,9 the lower limits of the confidence intervals coincide with t he one-sided 90% Poisson lower limits. This guarantees ≥90% probability within the horizontal intervals. 9

1The Universe: Finite or Infinite ? In APEIRON Vol. 7 Nr. 1-2, on pages 126-127, Christopher John Davison argues that the universe must be completely infinite, that is infinite in duration and in size. The core of his argument is his asking "those who claim limited size and lifespan to explain how space and material came into existence from nothing, how it will disappear again, and to explain the situation beyond the edge of a finite-sized Universe". The answers are as follows. Either the universe arose from and was preceded by absolute nothing or else it, or its creator, always existed, that is had no beginning. Only those alternatives are available; the positing of anything other than nothing as a beginning immediately requires accounting for that something's existence so that the only alternative to a beginning of nothing is no beginning at all. Thinkers over at least the past 4,000 years have consistently come to that conclusion and, since the universe arising from nothing seemed impossible and ridiculous, they consistently concluded that the universe arose from something that always existed, something that so to speak was the cause of its own existence. Shakespeare has King Lear say, "Nothing can be made out of nothing." And, that so overwhelming limitation has made thinkers throughout the ages opt for the infinite. But, under closer examination the infinite that this line of reasoning requires has problems as severe as does the universe arising from nothing. Further, the universe arising from nothing is not so insuperable a problem as it has appeared. Somewhat in the manner that Alexander resolved the challenge of the Gordian Knot [he drew his sword and cut the knot in half] it can be shown not only that the universe could have arisen from nothing but that it must have, as follows. The Problem of the Infinite We can readily grasp the idea of nothing; it is easily within our ken. But, the idea of infinity is much more difficult. We use expressions such as "without limit" and "unending" to convey the idea but we do not really comprehend ["in our gut" so to speak] what "forever", "always", "without limit" really signify. To us the symbol, ∞, subtly means a specific quantity standing at the end of a long list of increasing numbers, but its true meaning is that that list of numbers goes on and on, that if we go out and "stand on" the most distant number we will see still more going on and on, forever. So, for the universe to extend in space forever and for it to have existed forever is at least as troubling as for it to have arisen from nothing. Furthermore something existing forever means that it is its own cause. But, that contradicts the essential requirement that a cause exist independently of what it causes, that the cause "precede" in the causal sense that which it causes. To account for existence it is necessary to show why it is as compared to the alternative, it not being. Thus one must begin at the beginning, it not being. The starting point is absolute nothing -- the state before there was anything, before everything. It is the only state that requires no explanation nor accounting for its existence. It is naturally what one would expect before anything started. How a Universe Could Arise from Nothing The problem with a universe [or anything as Lear said] arising from nothing is that conservation must be maintained. The inputs and outputs, the amounts at the start, any2intermediate stages, and the finish must reconcile. There can be no overall loss nor gain. But, starting from nothing while maintaining conservation would appear to preclude any progress at all. Yet, paraphrasing Descartes, "I [part of the universe] think, therefore the universe is." The resolution of the dilemma is: The primal nothing changed into something and a conservation-maintaining equal-but-opposite un-something That Our Universe Did Arise from Nothing That initial event was so unstable that it exploded too immediately for the two opposites to recombine and cancel. That explosion was an immense shower of matter particles and energy now referred to as the "big bang". But there is another difficulty in the universe so arising from nothing - the transition. How is it possible to accommodate the transition from nothing to something plus its opposite without an infinite rate of change at the beginning ? There is only one mathematical form that can so change and fit all of the circumstances and requirements of the situation: the ±[1 - Cos(2 πft)] function [the ± for the two equal but opposite components that maintain conservation]. The infinite series of derivatives of the function make for the smooth transition. The development1 from that event, a logical and mathematical derivation of all of the fundamental laws of physics (Coulomb's Law, Ampere's Law, Newton's Laws of Motion, Newton's Law of Gravitation, relativity, radiation, fields, photons, atomic structure, nuclear structure, ..., all of the physics of the contemporary universe) from the necessary conditions and nature of that origin, shows that our universe is the joint operation of the something and the un-something, which together result in the universe's fundamental particles. Thus was the origin of the universe. As for Davison's "how it will disappear again", it is in a long-term exponential decay that began with the "big bang" and never completely ends. However, just as the area under the curve ε-t = 1, so the material universe is finite. Roger Ellman 320 Gemma Circle Santa Rosa, CA 95404, USA RogerEllman@The-Origin.org Reference [1] Ellman R., The Origin and Its Meaning , The-Origin Foundation, Inc., 1996, see http://www.The-Origin.org for where, how available.

arXiv:physics/0004054v1 [physics.class-ph] 21 Apr 2000Technique for measuring the parameters of polarization of an ultrasonic wave A. M. Burkhanov, K. B. Vlasov, V. V. Gudkov, and B. V. Tarasov January 16, 2014 Among physical phenomena consisting in variation of the pol arization of a shear ulrasonic wave the acoustic analogs of the Faraday an d the Cotton- Mouton effects are investigated at present (see [1] and [2] – t he first theorec- tical papers, [3] – discovery of rotation of the polarizatio n, and [4] –[7] – some experiments). They are observed when initially linearly po larized ultrasonic wave propagates inside a bulk specimen and are due to interac tion between elastic and magnetic subsystems or conduction electrons. Q uantitative char- acteristics of the effects are polarization parameters: ε– the ellipticity which modulus is the ratio of the minor and major ellipse axes and φ– the angle of rotation of the polarization plane or, more correctly, of the major ellipse axis ifε/negationslash= 0. Most of recent experiments on polarization phenomena we re performed with the use of phase-amplitude methods. A review of them is given in Ref. [8]. Besides, a phenomenon is considered as the acoustic analog o f magneto- optic Kerr effect if variation of the polarization occurs whi le reflection of the wave from an interface between magnetic medium and isotr opic non- magnetic one. It was predicted by Vlasov and Kuleev [9] in 196 8, however, there was no papers yet about experiments in which both the pa rameters characterizing the polarization, εandφ, were measured. We have completed such an experiment and the results will be p ublished soon. While performing it we found that a very small variatio ns of a high level signal took place and came to a conclusion that amplitu de variant of a technique should be more suitable here. First amplitude technique for a precise measurement of φwas introduced by Boyd and Gavenda [10]. Its aplicability was limited to the case where ε≈0. Though, we developed an amplitude method free of this rest riction for measuring φas well asε. A description of the technique is the subject of this paper. 1The method consists of measuring the amplitude of the voltag e,V(H), on the receiving transducer at a certain B1relative to an initial B=B0using three different angles for the receiving transducer ψwith futher processing of the data with the formulas (2), (22), and (23) presented belo w. It can be used for investigating the acoustic analogs of the Faraday and th e Cotton-Mouton effects as well. A periodic motion of the volume element over an elliptic traj ectory can be repersented with the help of amplitudes, u±, and phases, ϕ±, of circular elastic vibrations. Introducing a parameter p= (u−/u+)ei(ϕ−−ϕ+), (1) expressions for εandφhave the form: ε=1− |p| 1 +|p|, φ =−1 2Im [ln(p)]. (2) Projection of the elastic vibrations to polarization direc tion of the receiv- ing transducer can be written as follows: ur(t) =u·er= Re/braceleftBig u+exp/bracketleftBig i(ωt−ϕ+−ψ)/bracketrightBig +u−exp/bracketleftBig −i(ωt−ϕ−+ψ)/bracketrightBig/bracerightBig , (3) where * designates the complex conjugate, eris unit vector of the direction of the polarization of the receiving transducer, ψis the angle between this direction and the plane of incidence, ωis frequency, and tis time.urexcite an ac voltageVcos(ωt−α) =ηur(whereη2is the coefficient of transformation of elastic vibration energy into electric field energy, and αis a phase constant). Using Eq. (3) we have V η[cosωtcosα+ sinωtsinα] =/bracketleftBig u+cos(ϕ++ψ) +u−cos(ϕ−−ψ)/bracketrightBig cosωt +/bracketleftBig u+sin(ϕ++ψ) +u−sin(ϕ−−ψ)/bracketrightBig sinωt. (4) Since Eq. (4) is valid for arbitrary t, it may be transformed into two equations: V ηcosα=u+cos(ϕ++ψ) +u−cos(ϕ−−ψ), (5) V ηsinα=u+sin(ϕ++ψ) +u−sin(ϕ−−ψ). (6) 2Multiplying Eq. (6) by iand adding the result to Eq. (5) we obtain V ηeiα=u+ei(ϕ++ψ)+u−ei(ϕ−−ψ). (7) The method suggested here for determining the polarization of the re- flected wave consists of measuring the amplitude of the signa l at a certain B1relative to an initial B=B0using three different angles for the receiv- ing transducer: ψ1,ψ2, andψ3. We assume that ε(B0) = 0 andφ(B0) = 0. Relevant equations for the two different values of Band three of ψmay be obtained by making the appropriate substitutions into Eq. ( 7). Introducing indexesj= 0,1 for the two values of Bandk= 1,2,3 for the three values ofψforVkj,αkj,u± j, andϕ± jwe have: V10 ηeiα10=u+ 0ei(ϕ+ 0+ψ1)+u− 0ei(ϕ− 0−ψ1), (8) V11 ηeiα11=u+ 1ei(ϕ+ 1+ψ1)+u− 1ei(ϕ− 1−ψ1). (9) Dividing Eq. (9) by (8), we obtain V11 V10ei(α11−α10)=F+ 1eiψ1+F− 1e−iψ1, (10) where F± 1≡u± 1eiϕ± 1 u+ 0exp[i(ϕ+ 0+ψ1)] +u− 0exp[i(ϕ− 0−ψ1)]. (11) Similar equations for ψ=ψ2have the form V21 V10ei(α21−α10)δ2eiλ2=F+ 1eiψ2+F− 1e−iψ2, (12) whereλ2andδ2describe variations in phase and amplitude of the signal, respectively, caused by differences in transducer coupling to the sample while changingψfromψ1toψ2. One more change in ψgives the following equations in addition to (10) and (12): V31 V10ei(α31−α10)δ3eiλ3=F+ 1eiψ3+F− 1e−iψ3. (13) Hereδ3andλ3have the same origin as δ2andλ2, but correspond to changing ψfromψ1toψ3. 3After multiplying the left and right sides of Eqs. (10), (12) , and (13) by their complex conjugates we obtain /parenleftbiggV11 V10/parenrightbigg2 =/vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle2+/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsingle2+ 2/vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsinglecos (∆ϕ1+ 2ψ1), (14) /parenleftBiggV21δ2 V10/parenrightBigg2 =/vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle2+/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsingle2+ 2/vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsinglecos (∆ϕ1+ 2ψ2), (15) /parenleftBiggV31δ3 V10/parenrightBigg2 =/vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle2+/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsingle2+ 2/vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsinglecos (∆ϕ1+ 2ψ3), (16) where ∆ϕ1=ϕ+(B1)−ϕ−(B1), (17) and, due to the assumption of ε(B0) = 0 andφ(B0) = 0, δi=V10cos (ψi) Vi0cos (ψ1). (18) These operations are necessary to remove the phase αkjfrom our equa- tions since amplitude is the only parameter measured in this variant of a technique. We divide both sides of Eqs. (14)–(16) by/vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsingleto obtain |p1|−1+|p1|+ 2 cos[2(φ1−ψ1)] =(V11/V10)2 /vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsingle, (19) |p1|−1+|p1|+ 2 cos[2(φ1−ψ2)] =(V21δ2/V10)2 /vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsingle, (20) |p1|−1+|p1|+ 2 cos[2(φ1−ψ3)] =(V31δ3/V10)2 /vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsingle, (21) wherep1≡p(B1) . Thus we have three equations with three unknowns, namely/vextendsingle/vextendsingle/vextendsingleF+ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF− 1/vextendsingle/vextendsingle/vextendsingle, |p1|, andφ1. The latter two are the parameters we are interested in and corresponding solutions of the system have the form φ1=1 2tan−1/braceleftBig [/parenleftBig V2 21δ2 2−V2 31δ2 3/parenrightBig cos 2ψ1 +/parenleftBig V2 31δ2 3−V2 11/parenrightBig cos 2ψ2+/parenleftBig V2 11−V2 21δ2 2/parenrightBig cos 2ψ3] ×[/parenleftBig V2 21δ2 2−V2 31δ2 3/parenrightBig sin 2ψ1+/parenleftBig V2 31δ2 3−V2 11/parenrightBig sin 2ψ2 +/parenleftBig V2 11−V2 21δ2 2/parenrightBig sin 2ψ3]−1/bracerightBig (22) 4and |p1|=a1 c1±/bracketleftBigg/parenleftbigga1 c1/parenrightbigg2 −1/bracketrightBigg1/2 , (23) where a1=V2 11sin [2(ψ2−ψ3)] +V2 21δ2 2sin [(2(ψ3−ψ1)] +V2 31δ2 3sin [2(ψ1−ψ2)]cos 2φ1, c1=/parenleftBig V2 21δ2 2−V2 31δ2 3/parenrightBig sin 2ψ1+/parenleftBig V2 31δ2 3−V2 11/parenrightBig sin 2ψ2 +/parenleftBig V2 11−V2 21δ2 2/parenrightBig sin 2ψ3. The (−) sign should be taken before the square root in Eq.(23), sinc e it alone allows |p1|= 0 and therefore ε= 1. References [1] C. Kittel, Phys. Rev. 110, 836 (1958). [2] K. B. Vlasov, Fizika Metallov i Metallovedenie 7, 447 (1959) [Phys. Met. Metallogr. (USSR) 7, 121 (1959)] . [3] R. W. Morse and J. D. Gavenda, Phys. Rev. Lett. 2, 250 (1959). [4] H. Matthews and R. C. Le Craw, Phys. Rev. Lett. 8, 397 (1962). [5] B. Luthi, Phys. Lett. 3, 285 (1963). [6] A. M. Burkhanov, K. B. Vlasov, V. V. Gudkov, and I. V. Zhevs tovskikh, Akusticheskii zhurnal 34, 991 (1988) [Sov. Phys.-Acoustics 34, 569 (1988)] [7] B. V. Tarasov A. M. Burkhanov, and K. B. Vlasov, Fiz. Tver. Tela38, 2135 (1996) [Sov. Phys.-Solid State 38, 1176 (1996)]. [8] V. V. Gudkov and B. V. Tarasov, J. Acoust. Soc. Am. 104, 2756 (1998). [9] K. B. Vlasov and V. G. Kuleev, Fiz. Tver. Tela 10, 2076 (1968) [Sov. Phys.-Solid State 10, 1627 (1969)]. [10] R. J. Boyd and J. D. Gavenda, Phys. Rev. 152, 645 (1966). 5

Inquiries into the Nature of Free Energy and Entropy In Respect to Biochemical Thermodynamics Clinton D. Stoner Department of Surgery, Ohio State University, Columbus, Ohio 43210, U.S.A. ABSTRACT Free energy and entropy are examined in detail from the standpoint of classical thermodynamics. The approach is logi- cally based on the fact that thermodynamic work is mediated by thermal energy through the tendency for nonthermal en-ergy to convert spontaneously into thermal energy and for thermal energy to distribute spontaneously and uniformly within the accessible space. The fact that free energy is a Sec- ond-Law, expendable energy that makes it possible for ther-modynamic work to be done at a finite rate is emphasized. Entropy, as originally defined, is pointed out to be the capacity factor for thermal energy that is hidden with respect to tem- perature; it serves to evaluate the practical quality of thermal energy and to account for changes in the amounts of latent thermal energies in systems maintained at constant tempera- ture. With entropy thus operationally defined, it is possible to see that T∆∆∆∆S° of the Gibbs standard free energy relation ∆∆∆∆G° = ∆∆∆∆H° −−−− T∆∆∆∆S° serves to account for differences or changes in nonthermal energies that do not contribute to ∆∆∆∆G° and that, since ∆∆∆∆H° serves to account for differences or changes in total energy, complete enthalpy-entropy ( ∆∆∆∆H°-T∆∆∆∆S°) compensation must invariably occur in isothermal processes for which T∆∆∆∆S° is finite. A major objective was to clarify the means by which free energy is transferred and conserved in sequences of bio- logical reactions coupled by freely diffusible intermediates. In achieving this objective it was found necessary to distinguish between a ‘characteristic free energy’ possessed by all First-Law energies in amounts equivalent to the amounts of the en- ergies themselves and a ‘free energy of concentration’ that is intrinsically mechanical and relatively elusive in that it can appear to be free of First-Law energy. The findings in this re-gard serve to clarify the fact that the transfer of chemical po- tential energy from one repository to another along sequences of biological reactions of the above sort occurs through trans-fer of the First-Law energy as thermal energy and transfer of the Second-Law energy as free energy of concentration. INTRODUCTION Despite the assurance of the First Law of Thermodynamics that energy is invariably conserved, we often concern ourselves with the conservation of energy. This apparent inconsistency obviously must arise from the fact that the ambient thermal energy into which high-grade thermal energy and most other forms of energy tend to degrade has relatively little ‘available work potential’. And it must be this so-called free energy rather than an actual energy that we ordinarily seek to conserve, ‘actual energy’ referring to any of the various energies recognized by the First Law. The primary objectives in pursuing this study were to achieve fully operational understandings of free energy and its close rela- tive entropy and thereby to clarify apparent inconsistencies of the above sort and to deal with a number of other problems that tend to make the laws and fundamental equations of thermodynamics gen-erally difficult to interpret and understand. Some of the problems and their persistence appear to derive from the common practice of viewing entropy as a measure of disorder. Entropy as disorder is not measurable as such and thus is not operational. Whatever the case, the findings here suggest that several of the difficulties that commonly plague the interpretation of thermodynamic phenomena can be resolved by identifying and distinguishing between the various energies involved and taking into consideration the practi-cal qualities of the actual energies in respect to free energy. The original objective was simply to clarify the means by which free energy is transferred and conserved in sequences of stationary-state biological reactions coupled by freely diffusible intermedi-ates. The reported findings in this regard serve to emphasize the need to acknowledge the existence of an intrinsically mechanical, phantom-like free energy that can disappear without a trace and thereby appear to be free of actual energy. Energy, Temperature, Work, Work Potential, and Free En- ergy. Energy is usually defined in terms of a capacity of something to do work. Actual energy of matter may be classed broadly into two interconvertible varieties: energy of motion (kinetic energy) and energy of constrained motion (potential energy). Energy of motion can be subdivided into directed and undirected varieties. Disregarding the radiant form, thermal energy is an actual energy of undirected (random) motion of the individually mobile particu- late constituents of macroscopic amounts of matter and is a me- chanical kind of energy into which all other forms of actual energy tend to convert. Temperature is a property of macroscopic amounts of matter and serves to gauge the intensity of the thermal energy. Thermal energy transfer occurs spontaneously and net transfer along a gradient of temperature is a one-way process, occurring only from higher to lower temperature. In consequence, macro-scopic amounts of matter in thermal contact with one another tend to be at the same temperature, a fact of sufficient fundamental importance to merit belated designation as the Zeroth Law of Thermodynamics. Work may be defined roughly as any activity that is energeti- cally equivalent to lifting a weight. Since it exists only at the time it is being performed, work is generally viewed both as a nonther- mal actual energy in transit between one form or repository and another and as a means of nonthermal actual energy transfer. We shall be concerned here only with ‘thermodynamic work’ ( i.e., only with work in which the transit of nonthermal actual energy between forms or repositories occurs through intermediary thermal energy). Accordingly, ‘conservative work’, such as that done in direct conversions between gravitational potential energy and di- rected motional energy of matter, shall be ignored. Transfers of actual energy in thermodynamic work processes require specific mechanisms that depend on the nature of the energy. Thermal en-ergy transfer occurs by conduction, convection, and radiation. Consistent with the reciprocity in the definitions of work and energy, work potential shall be considered here to be the total po-tential of an energy for doing work. Although in principle all actual energies are equivalent to their work potentials and thus are equivalent with respect to work potential, they differ with respect to available work potential and thus differ with respect to quality, the quality being higher the greater the availability of the work Nature of Free Energy and Entropy 2potential. The availability of the work potential of an actual energy depends on the nature of the energy and on the conditions under which work is done. Whereas the work potential of thermal energy is completely unavailable at constant temperature, that of all other forms of actual energy can in principle become available at con-stant temperature as a result of ( i) the tendency of nonthermal ac- tual energy to convert spontaneously into thermal energy of quality (temperature) exceeding that of the ambient thermal energy and of (ii) the capacity of relatively high-quality thermal energy to do mechanical work through its tendency to distribute spontaneously and uniformly within the accessible ambient space. In the course of any work thus done, actual energy and work potential are invariably conserved, whereas energy quality, or available work potential (free energy), is invariably consumed. As will be empha-sized below, the amount of free energy consumed in the course of a thermodynamic work process depends greatly on the magnitude of the differential in quality between the intermediary and ambient thermal energies and thus on how fast the work is done. Since it is possible in principle for thermodynamic work to be done under conditions of the rate and the differential in thermal energy quality being extremely small, any nonthermal actual en-ergy can be considered to possess in principle an amount of free energy equivalent to the amount of the energy itself. In conse- quence, all actual energies other than thermal energy may be viewed as being completely interconvertible through mechanical work at ordinary ambient temperatures and thus may be viewed equally as energy of high quality. However, owing to differences in barriers to conversion, to differences in complexity of required mechanisms, and to consequent unavoidable differences in losses of free energy in real conversion processes, the qualities of the various nonthermal actual energies differ from the standpoint of practical processes. Because gravitational potential energy can be converted directly into mechanical work, the energy conserved in the course of lifting a weight is generally viewed as the form of nonthermal actual energy having the highest quality. Although completely unavailable at constant temperature, the work potential of thermal energy can be realized at the expense of a decrease in the temperature (quality) of the energy, and thus for thermal energy having a particular temperature the work potential may be viewed as being potentially available. However, because complete conversion of a given amount of thermal energy into a nonthermal actual energy would require a decrease in the tempera-ture of the given amount to absolute zero, the potential availability of the work potential of thermal energy is quite limited and that of ambient thermal energy at ordinary ambient temperatures is practi- cally nil. THERMODYNAMICS OF GASES The fundamental aspects of thermodynamics are based largely on energy changes associated with changes in the state properties pressure, volume, and temperature of an ideal gas in accordance with the ideal gas equation: PV = nRT. In consequence, the nature of free energy and its relationship to entropy are best seen by ex- amining the predictions of the laws of thermodynamics in respect to compression and expansion of a gas of this kind. Such will be done here by considering an ideal gas confined within a rigid cyl-inder of the sort shown in Figure 1. The walls of the cylinder are assumed to be permeable to thermal energy and in contact with a thermal reservoir that is sufficiently large and conductive to vali- date the assumption that its temperature remains constant despite transmissions of thermal energy between it and the gas. The gas is assumed to be confined to the cylinder and separated from a vac-uum by a massless and frictionless piston by which it can be com- pressed and allowed to expand. It is important to note that, since ideal gases and solutes in ideal solution are virtually identical in respect to mechanical thermody-namic properties at normal ambient temperatures and pressures (1), the Figure 1 system is also relevant to the thermodynamics of dis- solved substances. Thus, although the kinetics would be much different, the same thermodynamic relationships would apply if ( i) the gas of Figure 1 were a solute in ideal solution, ( ii) the vacuum were replaced by the pure solvent, and ( iii) the piston were selec- tively permeable to the solvent. If such were the case, the concen- tration and osmotic pressure of the gas as a solute would corre- spond to the concentration and pressure of the gas as a gas, and the thermodynamic relationships would be largely independent of the pressure on the solvent within the normal atmospheric range of pressure. According to the ideal gas equation and the First Law of Ther- modynamics, if an amount of work W were done on an ideal gas in compressing it from the maximum volume V 1 to a volume V2, an amount of thermal energy Q equivalent to the work done would be transmitted from the gas to the reservoir in the course of the gas achieving thermal equilibrium with its surroundings during and after the compression. According to the ideal gas equation, if the compression of one mol of the gas were conducted reversibly ( i.e., sufficiently slowly that the gas would remain in virtual thermal equilibrium with the reservoir), a minimum amount of work W rev energetically equivalent to RTln(V1/V2) of gravitational potential energy would be required to compress the gas, and, according to the First Law, an equivalent amount of thermal energy Qrev would be transmitted from the gas to the reservoir during the compres-sion. In the course of this conversion of energy of the highest qual- ity into the ambient thermal energy, the gas would be endowed with a potential for conversion of ambient thermal energy into an amount of work equivalent to that done in compressing it, as is obvious from the specifications that the temperature of the reser- voir be constant and that the compression be conducted reversibly. In other words, if one mol of the gas were compressed isothermally (reversibly) from V 1 to V2, an amount RTln(V1/V2) of actual energy equivalent in quality to gravitational potential energy would be converted into an amount RTln(V1/V2) of ambient thermal energy plus an amount RTln(V1/V2) of an energy of the gas which is ordi- narily ignored when encountered in this context, but which obvi- Figure 1. Compression and expansion of an ideal gas in a rigid- cylinder, piston system. The piston is assumed to be massless and frictionless and the temperature of the thermal reservoir is assumed to be constant. W = an amount of mechanical work; Q = an amount of thermal energy; V = volume of the gas; P = pressure of the gas. Nature of Free Energy and Entropy 3ously must be viewed as being free energy. If we were to consider this free energy to be actual energy, the First Law of Thermody-namics would of course appear to be very seriously in error. That it is not in error is evident from the fact that, if the free energy were not used for reversing the degradation of the nonthermal actual energy into ambient thermal energy, it would simply disappear, never to reappear. Thus, if the compressed gas were allowed to expand from V 2 to V1 without doing work on the surroundings, as would be the case if the massless and frictionless piston holding the gas in the compressed state (Figure 1) were suddenly released, it would do so rapidly without a net change in actual energy of any kind in either the gas or its surroundings (2). If free energy is not actually energy, what, then, is it? Since in the course of the compression phase of the above cyclic process it would become dissociated from the actual energies W and Q, it appears in this case to be an available work potential that is free of actual energy. The facts ( i) that complete recovery of the reversibly imparted free energy could be achieved only if the compressed gas were allowed to expand and do work at an infinitesimal rate and (ii) that, if the gas were allowed to expand without doing work, it would do so at the maximum possible rate with complete loss of the free energy suggest that free energy expenditure is what makes things happen within finite amounts of time. Accordingly, by com- bining the well-established equations of steady-state kinetics with those of classical equilibrium thermodynamics, it is possible to show quantitatively for stationary-state (apparent-equilibrium) reactions that the faster the reaction, the greater the amount of free energy consumed (3). To see clearly the nature of the above free energy, we must con- sider the properties of an ideal gas in respect to actual energy. Gases in molar amounts at ordinary temperatures generally consist of very large numbers of very small particles that are moving ran- domly at very high speeds. They tend to occupy uniformly the entire space accessible to them and through collisions of the parti-cles exert pressure on whatever confines them, thereby tending to increase the accessible space. The particles of an ideal gas are as- sumed to collide with perfect elasticity and to have volumes that are negligibly small in relation to the space available for their translation. In addition, the particles are assumed ( i) to be devoid of attractive and repulsive forces, ( ii) to be free of influence by fields of force such as the gravitational field, and ( iii) to possess per mol a total amount of actual energy E which depends only on the temperature. According to these assumptions, the actual energy of an ideal gas consists exclusively of thermal energy and differ- ences in this energy on a per mol basis between states at the same temperature are zero. Thus, in the above considerations, ∆E for the gas was assumed to be zero. We may now inquire as to the nature and source of the thermal energy that would appear in an ideal gas as it is compressed. The thermal energy must come from kinetic energy transmitted by the piston to the gas particles as the piston is moved toward the parti-cles, this despite the fact that an amount of thermal energy Q rev would be imparted even if the piston were moved extremely (infi- nitely) slowly. The faster the piston is moved the greater the amount of energy required and thus the greater W and Q must be. This implies that the temperature would be determined exclusively by the translational kinetic energies of the particles and that in the course of thermal equilibration with the reservoir, the kinetic ener- gies on average would return to their initial values, leaving only a diminished volume and a consequent augmentation of the pressure to account for the deposition of free energy in the gas. Accord-ingly, experimental observations on real gases suggest that allow- ing an ideal gas to expand without doing work ( i.e., without the particles expending kinetic energy) would not affect its tempera- ture (2). According to the ideal gas equation, PV would not change in the course of an isothermal change in the state of such a gas on a per mol basis. This being the case, RTln(V 1/V2) for an isothermal com- pression would be equivalent to RTln(P2/P1) and thus for such a compression one could attribute the consequent increase in free energy either to the decrease in the volume or to the increase in the pressure. That it would be appropriate to attribute the increase only to the increase in pressure is evident from the facts ( i) that driving forces for change are determined only by intensive variables (4) and ( ii) that pressure is the only intensive variable of the system under consideration. It may be noted that, since there would be no difference in ac- tual energy between the states, the increase in free energy can also be regarded as being due to an increase in the density of the trans- lational kinetic energy or in the degree to which the translational motions of the gas particles are constrained, in which cases one can readily see that the free energy would possess an attribute of poten- tial energy. In addition, since for an isothermal change the pressure and concentration would vary in direct proportion to one another, the free energy can also be regarded as being an energy of concen-tration. In fact doing so is preferable for the purposes of this study because differences in this kind of free energy between the reac- tants and products of biochemical r eactions are ordinarily deter- mined according to differences in concentrations. As will be pointed out below, the nature of this ‘free energy of concentration’ is fundamentally the same for a system in which only chemical work is possible as for the system here (Figure 1) in which only mechanical work is assumed possible. Free Energy in Relation to Entropy. Since free energy is not actually energy from the standpoint of the First Law and can dis-appear without a tr ace when of the concentration variety, it is nec- essary to account for its consumption in terms of the actual ener- gies. As indicated above, the actual energies W and Q are variable quantities that depend on how fast the compression or expansion of the gas is achieved. Although W rev for an isothermal (reversible) compression or expansion can be determined from the ideal gas equation and Qrev for such a process can be determined from Wrev by invoking the First Law, neither Wrev nor Qrev is a difference in a property, or state function, of the gas. Wrev for such a process can be determined from the ideal gas equation because, for c onditions of constant T, the state variables consist only of the intensive prop- erty P and the extensive property V which correspond to one an- other such that PV defines an energy having dimensions of me- chanical work ( i.e., dimensions of force × distance). As is evident from its exclusive dependence on pressure and volume, free energy of concentration is a property of an ideal gas. However, differences in this property between states can be calcu- lated only if the states have the same temperature. Although the free energy otherwise has the advantage that it is clearly an actual property of the gas, differences in this property between states at the same temperature are ordinarily reckoned in terms of the ther- mal energy that would be transmitted between the gas and its sur-roundings if the changes giving rise to the differences were to oc-cur reversibly. The achievement of this requires an additional ex- tensive state property that corresponds to the intensive state prop- erty T in such a way as to define an energy having dimensions of thermal energy when multiplied by T. In addition, since an iso- thermal change in the free energy would not involve a net change in the actual energy of the gas and is assumed not to involve a net change in the temperature of the surroundings, this extensive state property must be limited to thermal energy that is produced or Nature of Free Energy and Entropy 4consumed at constant temperature. In other words, ( i) as for any other kind of energy, we must have a capacity factor for thermal energy that corresponds to the intensity factor for this kind of en- ergy and ( ii) the capacity factor in this particular case must be lim- ited to thermal energy that either does not contribute to tempera-ture or is otherwise hidden with respect to temperature. The problem of recognizing and meeting this need was solved early in the history of thermodynamics research by Clausius (5), who noted that, despite the fact of the amount of thermal energy Q rev being located in the surroundings rather than in the gas, Qrev/T for an isothermal change in the state of an ideal gas corresponds to a change in a property of the gas. Clausius named this property ‘entropy’ and represented it with the symbol S. With respect to gases and other chemical substances, the symbol S is now ordinar- ily used to express entropy as an intensive quantity having the same dimensions as R, the universal gas constant ( i.e., dimensions of energy per mol per degree Kelvin). As a result, nST defines an amount of energy just as does nRT of the ideal gas equation, n being the number of mols and nS the extensive component of the energy. However, whereas nRT refers to an amount of pressure- volume work energy, nST refers to an amount of thermal energy that is somehow hidden with respect to temperature. Of course if the thermal energy of which entropy is the capacity factor were not hidden with respect to temperature, the appropriate comparable capacity factor would be the molar heat capacity of the substance. The molar heat capacity of a substance refers to the amount of thermal energy required to raise the temperature of one mol of the substance by one degree Kelvin. The heat capacity of a gas can be determined unequivocally only under conditions either of constant pressure or of constant volume, in which cases the molar heat ca-pacities are denoted by C P and CV, respectively. According to the classical kinetic theory of gases, CP and CV for an ideal gas would be constants having the values 5-2 R and 3-2 R, respectively, regardless of the magnitudes of the state properties temperature, pressure, and volume at which they are determined (6). CP exceeds CV by R be- cause an amount of pressure-volume work equivalent to R would be done on the surroundings if the heat capacity were determined at constant pressure. Thus, only CV is a heat capacity that refers exclusively to energy possessed by the gas. As one might expect from its relationship to free energy of con- centration, entropy in general simply appears, never to disappear, as free energy vanishes. As a result of the fact that its total amount invariably increases, entropy is often said not to be conserved, which, of course, is true in the sense that the amount is not a con-stant. Since the increase is determined by the amount and quality of the actual energy that invariably appears as high-quality actual energy and its associated free energy disappear, it seems more appropriate and instructive to acknowledge instead that it is the free energy (quality) of actual energy, or high-quality actual energy itself, that is not conserved. Nevertheless, entropy has been univer- sally adopted as the principal index of free energy consumption, this despite the facts that the quality of an energy is determined by the magnitude of its intensive component (4) and that entropy, being, like volume, an extensive component of an energy, is con- sequently relatively incomprehensible as an indicator of energy degradation. However, as will be outlined below, entropy has an important advantage over free energy in this respect in that it can be and has been universalized such as to make possible a proper (practical) assessment of the quality of thermal energy. In this regard it is important to note that the free energy of the intermediary thermal energy of a thermodynamic work process mediated by an ideal gas would be subject to loss by two means depending on whether the thermal energy moves spontaneously down a gradient of pressure by convection, in which case the free energy that is subject to loss would be the above-described free energy of concentration, or down a gradient of temperature by conduction and/or radiation. If entropy is to be a universal capacity factor, it must be capable of serving as the capacity factor for free energy that is subject to loss by both these means. In the case of convection, the intensive and extensive components of the energy are pressure and volume and the extensive component can be read- ily expressed in terms of entropy. Thus, for an isothermal (reversi-ble) compression of an ideal gas, the increase in free energy of concentration on a per mol basis can be represented either by −RTln(V 2/V1) or by −T∆S, ∆S being equivalent to Rln(V2/V1), a negative quantity in the case of compression, indicating a decrease in the entropy of the gas. Of course the reason why the entropy of the gas would decrease in this case is that the thermal energy on which the entropy is based would be located in the surroundings rather than in the gas. Entropy is universalized by using this vol- ume-dependent entropy in a Carnot cycle to define an entropy in terms of the thermal energy possessed by the gas, in which case the free energy that is subject to loss would have temperature and heat capacity as its intensive and extensive components. Since both C V and CP would be constants, any changes in this ‘derivative’ en- tropy must necessarily be expressed in terms of changes in the temperature and thermal energy of the gas. As is evident from the above-described properties of free energy of concentration, the volume-dependent entropy of an ideal gas could change through either compression or expansion without there being a change in the actual energy of the gas and, in the case of expansion, could change in the absence of a net change in actual energy of either the gas or its surroundings. Entropy being a state property, it is also evident that the volume-dependent entropy would not depend on the rate of change. In contrast, the tempera-ture-dependent entropy, being based on thermal energy possessed by the gas, would depend greatly on the rate of change, a fact which can be readily seen by considering irreversibility in respect to an adiabatic cycle of compression and expansion of an ideal gas (see APPENDIX ). In consequence of the different locations of the actual energies on which the volume- and temperature-dependent entropies are based, one kind could increase while the other decreases if the gas were to undergo a change in temperature and thermal energy as well as in volume. In view of the marked differences between the two entropies, it seems highly desirable to distinguish between them on a regular basis more or less as done originally by Gurney (7). In making this distinction in what follows, the kind of entropy that depends only on the volume shall be referred to as entropy of concentration to correspond with free energy of concentration. That this is appropriate can be seen by noting that the concentra-tion of a given amount of an ideal gas would vary inversely as the volume regardless of the temperature and pressure and that concentration can therefore be substituted for volume in the expression Rln(V 2/V1) for a difference in the volume-dependent entropy on a per mol basis simply by changing a sign. The kind of entropy that would depend only on the temperature and thermal energy of the gas shall be referred to as characteristic entropy, and the corresponding kind of free energy shall be referred to as characteristic free energy. It is important to note that these ‘concentration’ and ‘characteristic’ kinds of entropy and free energy are very similar to but not entirely identical with the ‘cratic’ and ‘unitary’ kinds defined by Gurney. In contrast to changes in free energy of concentration, changes in characteristic free energy are invariably accompanied by equiva- lent changes in the energies recognized by the First Law. Since, as Nature of Free Energy and Entropy 5noted above, the only actual (First-Law) energy of an ideal gas would be undirected translational kinetic energy, the characteristic free energy of such a gas must necessarily be of the above- mentioned, ‘potential’ kind possessed by thermal energy ( i.e., that recoverable through work only at the expense of a decrease in the temperature of the energy as its work potential is being realized). In view of the prediction of the ideal gas equation that the size of a change in the free energy of concentration due to a compression or an expansion between two states of concentration at the same tem-perature can be determined on a per mol basis according to RTln(V 2/V1) or RTln(P2/P1) and thus would be directly propor- tional to temperature, it is evident that changes in free energy of concentration, although not involving net changes in the character-istic free energy of the gas, would require that the gas possess characteristic free energy if the changes are to be finite, the charac- teristic free energy being that associated with the thermal energy at the temperature of the two states. Of course the higher the tempera-ture, the greater the change in free energy of concentration for a given change in concentration of the particles possessing the ther- mal energy and free energy. Changes in the characteristic free energy are ordinarily reck- oned in terms of work done on or by the gas during its compres-sion or expansion under adiabatic conditions ( i.e., under conditions in which transmissions of thermal energy between the gas and its surroundings cannot occur). If this is to be achieved, one must have knowledge of the capacity factor for thermal energy that de-termines temperature and refers only to energy possessed by the gas. As noted above, C V, the molar heat capacity determined under conditions of constant volume, meets this requirement and, for an ideal gas, would not depend on the magnitudes of the state proper-ties at which it is determined. This being the case, the total energy for one mol of such a gas would be given by C VT and finite changes in total energy on a per mol basis by CV∆T, CV being a constant. Since reversible processes do not expend free energy, CV∆T for any reversible adiabatic compression or expansion would be equivalent to the change in characteristic free energy as well as to the change in total energy. Thus, if the quality of thermal energy were judged on the basis of the predicted changes in the character- istic free energy of an ideal gas, it would appear to be equivalent to that of nonthermal actual energy. However, as outlined below, a proper assessment of the quality of thermal energy in terms of changes in the properties of a gas can be achieved only by means of the Carnot cycle. The Carnot Cycle. Owing to the extensive use of gas heat engines to exploit the work potentials of natural sources of nonthermal actual energy, an important concern of thermodynamics is the availability (quality) of the work potential of thermal energy im-parted to a gas in the course of the gas undergoing a cycle of ex-pansion and compression. Since in a strictly adiabatic or strictly isothermal cyclic process, a gas cannot do more work than is done on it, net conversion of thermal energy into nonthermal actual energy is impossible in these cyclic processes. As pointed out originally by Carnot (8), such a conversion can be achieved in a cyclic process only by combining isothermal and adiabatic proc- esses in an alternating sequence in which the gas is allowed to undergo isothermal expansion with uptake of thermal energy from a reservoir at a relatively high temperature, followed by adiabatic expansion and isothermal compression with transmission of a smaller amount of thermal energy from the gas to a reservoir at a relatively low temperature. Net conversion is achieved exclusively as a result of the fact that a gas at a relatively high temperature can do more work through isothermal expansion than is required for isothermal compression by the same factor at a relatively low tem- perature. If all four steps of the Carnot cycle were conducted reversibly, none of the characteristic free energy of the thermal energy ab- sorbed isothermally by the gas at the relatively high temperature and converted through work into the kind possessed by nonthermal actual energy would be consumed. However, complete conversion of the free energy would be possible only if the gas could undergo infinite expansion in the adiabatic expansion phase of the cycle and the temperature of the relatively cold gas and reservoir were at a temperature of absolute zero. Therefore, since available machines are of limited size, and since the thermal sinks ordinarily available have temperatures much higher than absolute zero, complete or nearly complete conversion of thermal energy into energy equiva-lent in quality to gravitational potential energy is far from being practical. Of course this is the basis for the limited potential avail- ability of the work potential of thermal energy. As is evident from the Carnot cycle and the above-noted fact that the quality of an energy is determined by the magnitude of its intensive component, the practical quality of thermal energy is higher as the temperature is higher. Determination of the quality by means of the Carnot cycle requires knowledge of the relationship between temperature and volume in the adiabatic steps. The re- quired information is obtained by defining a characteristic entropy in terms of entropy of concentration, which, as noted above, would be a function only of volume and concentration for an ideal gas. Since entropy of concentration is a capacity factor for thermal energy that is hidden with respect to temperature, this characteris- tic entropy must be scaled to thermal energy transmitted to or from the gas at constant temperature, a fact which is relevant to the above-mentioned need to know the relationship between tempera- ture and volume in the adiabatic steps. The thermal energy to be evaluated as to quality is that transmitted to the gas and converted into work isothermally at the relatively high temperature. Since entropy refers to thermal energy, it is the ‘thermal fraction’ of this thermal energy ( i.e., the fraction transmitted isothermally to the surroundings at the relatively low temperature) to which changes in the characteristic entropy of the adiabatic steps must be scaled. Of course the isothermal transmissions are assumed to be possible as a result of the reservoirs being sufficiently large to be capable of yielding and accommodating thermal energy wit hout undergoing a change in temperature. Owing to the fact that it would not be possible for the tempera- ture and thermal energy of an ideal gas to change independently of one another, a change in the characteristic entropy can be expressed properly in terms of the temperature and thermal energy only by means of the differential equation dS char = dE/T or its equivalent. Since dE = CVdT and CV would be a constant, this equation can be expressed in the forms dSchar = CVdT/T = CVdlnT and integrated between two specific temperatures to yield ∆Schar = CVln(T2/T1). The comparable expression Rln(V2/V1) for a finite change in the entropy of concentration on a per mol basis is obtained essentially in the same manner. In this case, however, it is the pressure and volume that would be incapable of changing independently of one another. In consequence of this and of the fact that the volume must change if the entropy is to change, a change in Sconc can be expressed properly in terms of volume and pressure only by means of the differential equation dSconc = PdV/T, which can be modified by substituting RT/V for P to obtain the equation in the integrable form dSconc = RdlnV. Of course the scaling of ∆Schar to ∆Sconc in the adiabatic steps is accomplished through the facts ( i) that a net change in entropy would not be possible in a reversible adiabatic process and ( ii) that, in Nature of Free Energy and Entropy 6consequence, the ratios of the initial and final temperatures and volumes in the adiabatic steps of a reversible Carnot cycle would be constrained to agree with one another according to the rela- tionship: / 21 1 2/( / )VRCTT VV= . That ∆Schar for the adiabatic steps is based on the thermal energy transmitted isothermally to the surroundings at the relatively low temperature is evident from the facts ( i) that the maximum efficiency η max of a Carnot heat engine is equivalent to 1 − Tlow/Thigh and ( ii) that ∆Schar for the adiabatic steps of the reversible cycle is therefore equivalent to ±CVln(1 − ηmax), the positive and negative signs referring to the expansion and compression steps, respectively. By noting that ηmax refers to the available (work) fraction of the thermal energy absorbed at Thigh, one can readily see that ∆Schar refers to the unavailable (thermal) fraction. For any given finite value of Thigh, the unavailable fraction is a linear function of Tlow and decreases to zero as Tlow approaches zero. The Carnot cycle is very important in that it universalizes en- tropy and thereby makes it possible to evaluate the practical quality of thermal energy and to demonstrate by theoretical means for thermodynamic work processes in general that, if such a process is to occur at a finite rate, free energy must be expended and thermal energy must be produced in an amount equivalent to the amount of free energy consumed, a fact which can be readily demonstrated by considering irreversibility in respect to an adiabatic cycle of com- pression and expansion of an ideal gas (see APPENDIX ). In any particular case, the fundamental process giving rise to the finite rate would be the net conversion of relatively high-quality actual energy, both thermal and nonthermal, into ambient thermal energy, the spontaneous and unidirectional nature of which is the basis for the Second Law of Thermodynamics, which, unlike the First Law, acknowledges the existence of free energy and says in effect that if thermodynamic work is to be done at a finite rate, free energy must be expended. Also unlike the First Law, the Second Law, owing to the fact that the individually mobile particulate constituents of macroscopic amounts of matter at finite temperatures vary widely as to translational kinetic energy, is a statistical law appropriate for application only to macroscopic phenomena. This means that the Second Law is obeyed only on average over time in processes at the microscopic level and thus that conversions of ambient thermal energy into nonthermal actual energy in chemically active sub- stances can occur at the molecular level. Of significance in this regard is the fact that translational thermal energy at the molecular level is kinetic energy of the directed variety, a consequence of which is that no distinction can be made between this form of ac- tual energy and nonthermal actual energy at the molecular level. Accordingly, energy transfer at the molecular level occurs without expenditure of free energy, and irreversibility, like temperature and pressure in respect to a gas, is a concept applicable only to macro-scopic phenomena. Real Gases. As is well known, the heat capacities of real gases increase with increase of temperature and correspond closely to those predicted for an ideal gas over wide ranges of temperature and pressure only for monatomic gases (9). The temperature de-pendence is due in large part to the fact that the translational and radiant forms of thermal energy are capable of undergoing inter- conversion with actual energy associated with quantized motions within molecules. The energies of these ‘intramolecular’ motions are reckoned in terms of characteristic entropy and thus appear to be generally viewed as being of a thermal nature, this despite the facts ( i) that the energies of some of the motions undergo oscilla- tory interconversions with attractive and repulsive potential ener- gies and ( ii) that the motional energies, being intramolecular, seem best viewed as being directed (nonrandom) kinetic energies and thus of a nonthermal nature. On the other hand, since radiant en-ergy is not confined to molecules and is ordinarily unrestricted as to direction of emission in a gas, any radiant energy emitted as a result of the motions would clearly qualify as thermal energy. In accord with their being nonthermal, the energies associated with the intramolecular motions apparently do not contribute to temperature. Such is consistent with the facts ( i) that the absolute and thermodynamic scales of temperature are based on the proper-ties of an ideal gas and ( ii) that an ideal gas is assumed to possess actual energy only of the translational kinetic kind. It may be noted that, owing to the discontinuous nature of the intramolecular mo- tions in respect to change of temperature, such must be assumed for validity of the widely accepted, sweeping generalization con-cerning the reversible Carnot cycle that the ratio of the amount of thermal energy absorbed isothermally at T high to the amount re- jected isothermally at Tlow would have the same value regardless of the nature of the working substance. Despite their apparent nonthermal nature, the energies associ- ated with the intramolecular motions appear to be recoverable only as thermal energy at the temperature of their reversible production and thus appear to differ markedly from the kind of nonthermal energy that is capable of converting spontaneously into thermal energy of quality exceeding that of the ambient thermal energy. Accordingly, their elicitation is associated with diminution of mo-lecular stability (10) and might be expected thereby more likely to diminish than to enhance chemical potential energy of the kind that possesses characteristic free energy. This kind of energy shall henceforth be referred to as ‘characteristic chemical potential en-ergy’, a distinction made necessary by the fact that free energy of concentration is ordinarily treated as if it were chemical potential energy, a practice which of course is valid when properly used but which ignores the intrinsically mechanical nature of the concentra-tion-dependent free energy and tends to elicit confusion as to the nature of characteristic chemical potential energy, particularly when used in reference to an ideal gas. Although, as noted above, the energies associated with the in- tramolecular motions also differ appreciably from what is ordinar-ily considered to be thermal energy, in view of the current practice of accounting for them appropriately in terms of characteristic entropy and of entropy being the capacity factor for thermal energy that is hidden with respect to temperature, it seems appropriate and best for practical purposes to view the energies as being latent forms of thermal energy of quality determined by the temperature at which they could be reversibly produced. By making this dis- tinction we imply that thermal energy that is not latent refers to the kind that determines temperature. Since what has been referred to above as the radiant form of thermal energy is actually electro- magnetic energy having frequency ν as its intensive component and nh as its extensive component, n being any whole positive number and h Planck’s constant, and can be said to be thermal energy only in the sense that it is capable of transmitting thermal energy and to have a temperature only in virtue of the fact that it has a certain distribution as to quality and concentration of photons of energy hν as given by Planck’s Law of Heat Radiation when in equilibrium with matter at a particular temperature (11), this ‘nonlatent’ thermal energy may be considered to consist exclu-sively of the translational kinetic kind. However, since electro-magnetic energy is readily detectable and transmissible as such, when viewed as being thermal energy it must in some sense also be viewed as being a nonlatent variety, particularly in respect to transmission of thermal energy. Nature of Free Energy and Entropy 7The intramolecular motions include rotations of entire mole- cules and various rotations, librations, and vibrations of molecular constituents, all of which are known to be quantized through the occurrence of temperature-specific changes in heat capacity and in absorption and emission of characteristic radiant energy (10, 12). Of course the possibilities for these kinds of motion are greater, the greater the complexity of the molecules and the weaker and more flexible the bonds between the constituent atoms. In view of the fact that the intramolecular motions can be elicited through inelas-tic collisions between molecules, these motions must be capable of converting into the translational motions that result in the colli- sions and, under conditions of constant temperature, must tend to be at equilibrium with the translational motions; otherwise, con-trary to Planck’s Law and the Zeroth Law of Thermodynamics, there would likely be a temperature differential between the trans- lational and radiant forms of thermal energy at equilibrium. Due to the existence of net attractive forces between the indi- vidually mobile particulate constituents of macroscopic amounts of real matter, thermal energy can also disappear and appear with increase and decrease of temperature through dissociation and association reactions that increase and decrease the number of particles whose motions contribute to and thereby determine temperature. This phenomenon can be explained on the basis of the very successful prediction of the equipartition principle of the classical kinetic theory of gases that, at any particular temperature, individually mobile particles differing as to mass, composition, and other properties will possess on average the same amount of trans- lational kinetic energy, the amount being equivalent to 1-2m v- 2, m being the mass and v- the average velocity (9, 12, 13). According to this prediction, if the molecules of a gas at a particular tempera- ture were sufficiently attracted to one another that some of the molecules could expend attractive binding energy by binding to one another at that temperature, increasing the temperature would result in mechanically induced dissociations of bound molecules and in a portion of the thermal energy added to the gas for the purpose of increasing its temperature being expended to elevate the translational kinetic energies of newly formed particles to average values consistent with the existing temperature. Since the formation of the additional particles would be accompanied by the appearance of attractive forces and thus also of attractive binding energy, it seems appropriate to view the dissociations of the attractively bound particles as constituting conversions of transla- tional kinetic (mechanical) energy into attractive binding energy. Also, since it is unlikely that attractive binding energy thus generated could be recovered as an energy of quality higher than the thermal energy required for its reversible production, and since the attractive forces would extend beyond the bounds of the particles possessing them, it seems appropriate to view the attractive binding energy as being an extramolecular kind of latent thermal energy of quality determined by the temperature at which it could be reversibly produced. As will be pointed out below, one could reasonably expect the attractive binding energy to have the quality of characteristic chemical potential energy only if the gas were supercooled with respect to thermochemical equilibrium. Since associations of molecules through mutual attractions are likely to result in some hindrance of intramolecular motions, changes in latent thermal energy of the extramolecular kind at temperatures sufficiently high for elicitation of intramolecular motions are likely to be accompanied by changes in latent thermal energy of the intramolecular kind in the same direction. In contrast to the intramolecular kind, the extramolecular kind can undergo change in response to change not only of temperature, but also of pressure and thus of concentration. Both kinds can change under conditions of constant temperature and pressure in chemical reac- tions and in first-order phase transitions, and, since both are based on system energies having the quality of thermal energy, changes in their amounts under these conditions are actually and best ac- counted for in terms of characteristic entropy. As indicated above, the prediction of the classical kinetic theory of gases that C V for an ideal gas would be a true constant implies that the amount of thermal energy possessed by a given amount of such a gas would be directly proportional to temperature. This in turn implies that an ideal gas can be considered to provide linear absolute scales for actual energy and free energy as well as for temperature. Although based on predicted properties of a fictitious gas that differs appreciably from real gases in that its particulate constituents possess only translational kinetic energy, these scales are very important in that they are commonly used with remarkable success as a framework for characterization of the thermodynamic properties of all real substances, a fact which accords with the universality of the universal gas constant R. As will in effect be suggested below, the widespread success of the ideal gas model is likely due in large part to the above-noted prediction of the equi- partition principle being applicable to liquids and solids as well as to gases and to temperature in liquids and solids being determined by the average kinetic energies of individually mobile particles consisting of clusters of attractively bound molecules, the average size of which tends to increase with decrease of temperature. THERMODYNAMICS OF LIQUIDS AND SOLIDS All naturally occurring gases possess net attractive forces and in consequence undergo condensations to form liquids and solids as temperature is decreased. An important but rarely asked question is: What determines temperature in these condensed phases? This question must be asked and answered correctly if we are to under-stand how it is possible that equilibrium differences in concentra-tion between the reactants and pr oducts of a chemical reaction conducted in solution can serve to measure differences in charac- teristic chemical potential energy between the reactants and prod-ucts at various equilibrium temperatures. Since the particulate constituents of liquids can flow and thus must be sufficiently free to undergo translational motions, we can attribute temperature in this case to the same kind of motions that determine temperature in gases. Since the particulate constituents are held in the liquid state by attractive (cohesive) forces possessed by the particles, one might expect the translational motions of the particles to be hindered in respect to intensity of translational mo-tions and thus in respect to ability to contribute to and thereby determine temperature. However, in view of ( i) the likelihood that temperature is determined in both liquids and gases by the average intensity of the actual translational motions of the constituent indi- vidually mobile particles and of ( ii) the above-noted prediction of the equipartition principle that, at any particular temperature, the individually mobile particulate constituents of all gases will pos- sess on average the same amount of translational kinetic energy, it seems likely that the average translational kinetic energy possessed by the particulate constituents of any liquid would tend to be the same as that for any gas if the liquid and gas were at the same tem- perature. If such were not the case, it would be very difficult to understand, among other things, the physical bases for the latent thermal energies and for the common observation that the me-chanical thermodynamic properties of substances in dilute (ideal) solution do not differ appreciably from those expected of an ideal gas despite the individual molecules of the substances differing greatly as to such properties as chemical composition, size, net charge, and affinity for the solvent. Nature of Free Energy and Entropy 8As is well known, condensations of gases into liquids and of the liquids into solids under conditions of constant pressure can occur very nearly reversibly at constant temperature with productions of amounts of thermal energy that greatly ex ceed the am ounts of pressure-volume work done simultaneously. It is important to in-quire as to where the excess thermal energy comes from in these processes. Since the condensations occur as a result of there being attractive forces between the particulate constituents, it must nec- essarily come primarily from conversions of attractive binding energy into thermal energy as the particles associate in the course of the condensations. In view of this and of the likelihood that the average translational kinetic energies of the individually mobile particulate constituents of at least the liquid and gas phases of any substance would be the same if the phases were at the same tem-perature, it seems likely that such conversions occur through asso- ciations of particles to form larger and thus fewer particles possess- ing on average the same amount of translational kinetic energy as the particles undergoing the associations. Thus it seems appropri-ate to view the attractive binding energy that undergoes conversion into thermal energy in phase transitions of the above sort to be latent thermal energy of the above-described extramolecular kind. As is particularly well known, thermally induced transitions of water from solid to liquid and from liquid to vapor at normal (atmospheric) pressures involve large conversions of thermal energy into latent thermal energy. It is also well known that some of the latent thermal energy is of the intramolecular kind and that the conversions are capable of occurring at constant temperature with little consumption of free energy when the nonlatent thermal energy derives from ambient thermal energy (14). By linking reversible production of latent thermal energy in these processes to isothermal (reversible) compression of an ideal gas, we can readily see that both the intramolecular and the extramolecular kinds of latent thermal energy must in fact be viewed as being equivalent in quality to ambient thermal energy at the temperature of their re-versible production. Consider, for example, a Figure 1 system in which the temperature of the thermal reservoir is at the melting temperature of ice and is maintained constant solely by intercon- versions between ice and liquid water at constant external pressure. If the gas in such a system were compressed reversibly, thermal energy would be produced isothermally in the gas and consumed isothermally in the reservoir through reversible conversion of ice into liquid water. Since all of the free energy of the nonthermal actual energy imparted to the gas would be retained by the gas and could be conserved only through reversal of the above process, it would be necessary to consider any latent thermal energy produced as a result of translational thermal energy from the gas converting ice into liquid water to be equivalent in quality to ambient thermal energy at the melting temperature of the ice. The above observations being correct, one could reasonably ex- pect the number of individually mobile particles in a liquid to de-crease as the temperature is decreased. Since the decrease in num-ber of particles would necessarily occur through attractive binding interactions among particles and thus would be accompanied by conversion of attractive binding energy into thermal energy, one could reasonably expect the heat capacities of liquids to be aug-mented by conversions of thermal energy into attractive binding energy as the temperatures of the liquids are increased for the pur- pose of measuring the heat capacities. That these expectations are consistent with what is observed experimentally in this regard is particularly clear in the case of water. Owing to the molecules of water as compared to those of most other common solvents having particularly strong tendencies to associate with one another through their capacities to serve both as a double donor and a double acceptor of hydrogen bonds, the en- ergy changes in the gas ð liquid and liquid ð solid transitions of water are particularly large. The molar heat capacity of the liquid at constant atmospheric pressure is also particularly large and is close to twice that of the solid near the normal freezing point and to twice that of the vapor near the normal boiling point (15). The above-noted expectations are clearly consistent with these experi- mental findings and with numerous experimental and theoretical observations suggesting that the particulate constituents of liquid water consist largely of labile clusters of molecules, the concentra- tion and average size of which depend on temperature and pressure (15-18). For there to be consistency with the above observations, it is necessary to suppose that, for solids, temperature and to a large extent also heat capacity are determined by undirected translational (vibrational) motions of clusters of molecules (atoms in the case of atomic solids) about mean fixed positions and that the average size of these clusters tend to increase with decrease of temperature, the increase in average size being accompanied by conversion of ki- netic and potential (attractive + repulsive) vibrational energy into thermal energy. In view of the fact that the average vibrational frequency of the clusters would decrease as the average size (mass) of the clusters increases and of experimental observations suggest- ing that the heat capacities of all naturally occurring substances tend to approach zero as temperature approaches zero, in consider-ing this to be the case for a large chunk of a solid maintained close to thermochemical equilibrium, it would seem necessary to assume that the number and average vibrational frequency of the clusters would decrease to unity and zero, respectively, as the temperature of the solid is decreased to zero. This and the further assumption that the atoms or molecules of the individual clusters vibrate co- herently and consequently emit photons as clusters of indistin- guishable photons being correct, one could think of the clusters as being Bose-Einstein condensates of a sort and of the solid as being a Bose-Einstein ideal gas in which the number of particles is not conserved more or less as the Bose-Einstein quantum statistical method of deriving Planck’s Law of Heat Radiation predicts for distinguishable clusters of indistinguishable photons in equilibrium with a blackbody (see below). Doing so would be consistent with the generally accepted Bose-Einstein quantum gas, ‘quasiparticle’ (phonon) interpretation of lattice vibrations in crystalline solids (11, 19, 20) and thus with the apparent wave-particle duality of matter in this case. Doing so would also be consistent with experi- mental observations (21, 22) indicating that water ice at subfreez- ing temperatures sublimates (evaporates) in the form of clusters of molecules. In 1913 prior to the general acceptance of the quantum hypothe- sis demanded by the extremely close agreement between experi- mental observations and the equation now known as Planck’s Law of Heat Radiation, Benedicks (23) presented an ‘agglomeration hypothesis’ which appears to be generally consistent with the above notions concerning what determines temperature and heat capacity in solids. Thus, Benedicks pointed out that Planck’s Law can be derived on the basis of the assumptions ( i) that the atoms or molecules of solids coalesce through cohesive forces to form clus-ters of increasing size as temperature is decreased and ( ii) that the thermal energy partitions among the clusters in accordance with the equipartition principle. He also pointed to the possibility of explaining the observed independence of the frequency spectrum of cavity (equilibrium blackbody) radiation on the nature of the solid by taking into account the facts ( i) that the average mass of the clusters at a particular temperature would be greater as the cohesive force between the constituent atoms or molecules is Nature of Free Energy and Entropy 9greater and ( ii) that, since the average vibrational frequency of the clusters would be lower as the average mass of the clusters is greater and higher as the cohesive force between the clusters is greater, differences among solids as to cohesive force would tend to cancel in respect to the vibrational frequency spectrum of the clusters at a particular temperature and thus also in respect to the frequency spectrum of the radiant energy emitted and absorbed by the clusters at that temperature. In 1915 A. H. Compton (24) tested the agglomeration hypothe- sis as to utility in acc ounting for experimentally observed relation- ships between temperature and heat capacity in simple solids. Us- ing Maxwell’s Distribution Law in conjunction with the agglom- eration hypothesis, he came up with a very simple equation which he judged to be at least equally as accurate as the much more com-plex but now generally accepted equation developed earlier by Debye (25) on the basis of the quantum hypothesis using arbitrarily the assumption that vibrational frequencies at low temperatures are limited to the low values characteristic of large clusters of atoms. Compton (26) also compared the agglomeration and quantum hy- potheses as to utility in acc ounting for the observed inverse rela- tionship between temperature and thermal conductivity in solids and found the agglomeration hypothesis to be clearly superior in this case. Despite its remarkable successes, the agglomeration hypothesis was largely abandoned, presumably as a result of Compton turning his attention to his well-known studies on interactions between x-rays and matter, which resulted in the general acceptance of the quantum hypothesis, and to the textbook view (27) that if the ag- glomeration hypothesis were valid, one could expect solids to be incompressible at temperatures approaching absolute zero, a view which has persisted into modern times (28) despite the fact that it was based on the faulty notion that atoms are incompressible (hard) spheres (see 29). Nevertheless, although not acknowledged or recognized, the main features of the agglomeration hypothesis were used in the subsequent development of quantum mechanics, a fact which is particularly evident in the case of wave mechanics. Thus, Bose-Einstein (quantum) statistics become essentially iden- tical with Maxwell-Boltzmann (ideal-gas) statistics as temperature is increased and assumes a condensation (degeneration) of distin-guishable particles (parcels) of energy into distinguishable clusters (cells) of indistinguishable particles of energy as temperature is decreased (9). Also, de Broglie’s fundamental ideas on wave-particle duality, which, in conjunction with Bose-Einstein statistics led Schrödinger immediately to his wave mechanics (30), origi- nated with the realization that derivation of Planck’s Law by con- sidering light quanta to be an ideal gas of photons (“atoms of light”) requires the assumption that blackbody radiation other than that at the extreme high-frequency end of the observable spectrum consists of agglomerations of photons that move coherently (31, 32, 33). Since the frequency at the high-frequency end of the ob-servable spectrum increases endlessly with increase of tempera-ture, this assumption and the essentially identical one at the heart of Bose-Einstein statistics being correct, most if not all ‘conven- tional photons’ would be clusters of photons having the same fre-quency and we could imagine a rational explanation for the pecu-liar fact of quantum mechanics that what appear to be single pho- tons can appear to split and interfere with themselves in a wave- like manner. However, the explanation would not be of a sort that would seem likely to account for the considerably more peculiar, closely related fact that electrons, neutrons, and other genuine particles in what seem highly likely to be genuine (indivisible) single-particle states also can appear to split and interfere with themselves in a wave-like manner (see, e.g., 34, 35). That temperature and heat capacity in solids is actually and strongly linked to the average size of actual particles consisting of clusters of atoms or molecules is suggested by the results of nu- merous relatively recent studies on the thermodynamic properties of ‘nanosolids’ obtained by reducing normal (bulk) solids at tem-peratures far below their melting points to particles having diame-ters of a few nanometers and then lightly compacting the particles to form pellets that can be easily handled and compared with the bulk materials (for comprehensive reviews, see 36, 37). In general, the thermodynamic properties of solids thus modified have been found to differ from the normal such as to suggest that the temperature has in effect been greatly increased. Thus, melting temperatures, at least of individual particles, are greatly decreased (38), enthalpies (39), entropies (36), heat capacities (36, 39), and vapor pressures (40) are greatly increased, and heat capacities ap- pear not to approach zero as temperature approaches zero (36, 41). Most of these changes either have been or can be adequately ex-plained in terms of the associated large increases in surface area, surface attractive binding energy, and number of particles that are individually mobile under conditions of temperature where the attractive binding energy has the quality of characteristic chemical potential energy rather than of latent thermal energy (see below). The observed persistence of heat capacity in lightly compacted nanosolids at temperatures very close to zero is consistent with the theoretical findings of Jura and Pitzer (42) predicting that the heat capacities of solids consisting of unconsolidated ‘nano-sized’ par-ticles will be detectably large at temperatures closely approximat- ing absolute zero as a result of the particles being capable of un- dergoing thermally induced translational and rotational motions at such temperatures despite the particles possessing an abundance of attractive binding energy. THIRD LAW OF THERMODYNAMICS According to the first of two extant versions of the Third Law that are relevant here, the entropy change associated with any isother- mal (reversible) process will approach zero as temperature ap- proaches zero (43, 44). It is important to inquire as to the nature of the entropy to which this version refers. Since it concerns only processes occurring at particular temperatures, characteristic en- tropy of the kind that can change only as a result of a change in temperature can be immediately ruled out, thereby avoiding the uncertainty associated with the prediction of the kinetic theory of gases that a reversible decrease in the temperature of an ideal gas to zero in a Carnot cycle would be accompanied by a decrease in characteristic entropy to − ∞, a result necessitated by the fact that achievement of the condition T = 0 in the cycle would require in- creases in the volume and entropy of concentration to +∞. On the other hand, entropy of concentration can undergo change at con- stant temperature and can do so without n ecessarily requiring a change in volume, a fact which can be readily seen by considering chemical reactions of the sort: A ð P + Q (i.e., reactions in which there is a difference in mol number between the reactants and products in the stoichiometric equation). As will be outlined be- low, at least for reactions occurring in solution virtually at constant volume, a change in entropy of concentration due to a change in the number of particles in such a system maintained at constant temperature will be accompanied by a change in the characteristic entropy of the system in the same direction and the thermal energy on which the characteristic entropy is based will be indistinguish- able from the above-described extramolecular latent kind. That a change in entropy of concentration would conform to the above version of the Third Law can be seen by noting that changes in entropy of concentration at a particular temperature serve to repre-Nature of Free Energy and Entropy 10sent changes in free energy of concentration and that, since free energy of concentration depends on the existence and density of translational kinetic energy and therefore can be expected to be zero at T = 0, a change in entropy of concentration would not be possible at T = 0. The same can be said of entropy of mixing, the nature of which can readily be seen to be identical with that of entropy of concen- tration by considering a spontaneous mixing of two distinguishable ideal gases at the same temperature and initial pressure. A change in entropy of mixing in this case is given by the sum of the in- creases in the entropies of concentration of the individual gases as each diffuses (expands) into the space occupied by the other. Ac-cording to this, entropy of mixing serves to account for the free energy of concentration that could be used for conversion of ambi- ent thermal energy into nonthermal actual energy if there were a means by which the diffusion of the substances undergoing spon-taneous mixing could be harnessed individually to do work, a pos-sible means in the above case being an apparatus consisting of a cylinder having two chambers separated initially by two opposed semipermeable pistons, the piston nearest each gas in its pure form being permeable only to that gas. In the absence of such means, the mixing would occur without a net change in actual energy, just as would be the case if the individual gases were expanding into a vacuum. Presumably, the principal kind of entropy to which the above version of the Third Law is ordinarily meant to apply is the charac- teristic entropy associated with the kinds of energy referred to above as intramolecular and extramolecular latent thermal energy. As was noted, these forms of energy do not contribute to tempera-ture and can undergo change in processes occurring at constant temperature. Since, at any particular finite temperature, they tend to be at equilibrium with the transmissible (nonlatent) forms of thermal energy, they can be expected to disappear as temperature approaches zero if sufficient time is allowed for them to do so. Owing to the tendency of the latent thermal energies to equili- brate with the transmissible forms, the Third Law is widely inter-preted to mean that the characteristic entropies of individual sub-stances approach zero as temperature approaches zero. Although the version thus obtained has proved to be very useful for practical purposes and is consistent with experimental observations suggest-ing that the heat capacities of all naturally occurring substances in macroscopic amounts maintained close to thermochemical equilib-rium will decrease to zero as the temperature is decreased to zero, it is considered not to be generally correct as a result of the possi- bility for the characteristic entropy associated with the ex-tramolecular kind of latent thermal energy to become ‘frozen in’ as the temperature is decreased. As implied above, the extramolecular kind of latent thermal energy is clearly nonthermal, is possessed by all naturally occurring substances at finite temperatures, and con-sists of attractive binding energy that is subject to loss upon tem-perature reduction through stabilization of neutralizing attractive binding interactions among the particulate constituents whose un- directed motions contribute to and thereby determine temperature. In this particular case, latent thermal energy and its associated entropy can become ‘frozen in’ as a result of free molecules con- densing during temperature reduction to form stable solids in which the molecules are not arranged as well as they could be for maximum conversion of the binding energy into the radiant and translational forms of thermal energy, which, in contrast to the binding energy, are transmissible as such and subject to direct removal through reduction of the ambient temperature. For reasons of this sort, the Third Law, as it is applied to individual substances, is generally considered to be applicable in a strict sense only to pure substances in perfect crystalline states (45). THERMODYNAMICS OF BIOCHEMICAL REACTIONS We shall be concerned here primarily with the very simple, first- order, uncatalyzed reaction: 1 2 k APk→  → ²²²ı¹²²² (1) in which A and P are reactant and pr oduct in ideal solution at con- stant temperature and pressure, k1 and k2 are the forward and back- ward rate constants, and the arrows preceding the reactant and following the product are meant to indicate that a and p, the con- centrations of A and P, are maintained constant as a result of the reaction being in a long sequence of reactions of the sort that one might expect to find in living systems. Thus, Reaction (1) as pre- sented is considered to be a stationary-state reaction in an open system of more or less constant volume and to have a net velocity in the direction indicated regardless of whether the equilibrium constant is favorable or unfavorable. Although the system as pre- sented is considered to be open and therefore relevant primarily to biological systems, in most of what follows it will be considered to be closed ( i.e., capable of exchanging only thermal energy with the surroundings). Regardless of the nature of the energy from which it derives, the available work potential of a chemical substance at a particular combination of temperature and pressure is referred to either as the Gibbs energy or the Gibbs free energy. When expressed as an in- tensive property in terms of energy per mol, the Gibbs energy is referred to either as the Gibbs potential or the chemical potential. Differences in this potential between states of a substance and between substances in well-defined states are denoted by ∆G. Since it is the difference in the intensive properties of a system that determines the driving force for change, the driving force of Reac- tion (1) as presented may be considered to be this difference be-tween A and P as given by the van’t Hoff reaction isotherm: ∆G = −RTlnK + RTln(p/a) = ∆G° + RTln(p/a) (2) Here K is the equilibrium constant, ∆G° is the so-called ‘standard’ difference in characteristic Gibbs free energy between one mol of A and one mol of P, and ∆G is the amount of Gibbs free energy consumed in the net conversion of one mol of A into one mol of P under a specific set of conditions of temperature, pressure, a, and p, the temperature, pressure, and other environmental conditions being the same as those used in the determination of K and ∆G°. Since ∆G as a driving force always refers to free energy that is consumed, it is usually presented as −∆G (i.e., as a positive quan- tity) when used in this sense. The van’t Hoff reaction isotherm serves to acknowledge that a particulate substance that is capable of undergoing chemical change possesses ( i) an available work potential that is independ- ent of its concentration and is characteristic of the substance in a given environment, and ( ii) an available work potential that de- pends only on the number of individually mobile particles per unit volume ( i.e., only on particle concentration). The isotherm, as it applies to Reaction (1), can be derived from the ideal gas equation on the basis of the observation by van’t Hoff (1) that the free en-ergy of a solute in dilute (ideal) solution depends on the osmotic pressure of the solute just as the free energy of an ideal gas would depend on its pressure. Since, for an isothermal change in the state of such a gas, the pressure and concentration would vary in direct proportion to one another, the expression for a same-temperature difference in free energy of concentration in terms of pressure Nature of Free Energy and Entropy 11obtained from the ideal gas equation may be modified by substitut- ing concentration for pressure. The expression then can be used as a measuring device to determine differences in characteristic chemical potential energy between the reactants and pr oducts of chemical reactions conducted in solution at constant temperature and pressure from equilibrium differences in translational kinetic energy density ( i.e., from equilibrium differences in free energy of concentration). The means by which this is ordinarily achieved in respect to biochemical thermodynamics, in which case the solu-tions are usually considered to be ideal, is outlined immediately below. Determination of ∆∆∆∆G° by the Equilibrium Method. At a given temperature and pressure, the Gibbs potential G of a chemical substance in ideal solution is equivalent to RTlnc + C, where c is the concentration and C is an unknown constant. As is evident from the fact that it can be expressed in the form V = Ae −G/RT, where V = 1-c = molar volume and A = eC/RT, this foundational rela- tionship of the classical approach to chemical thermodynamics has much in common with those of the Maxwell-Boltzmann statistical approach. In consequence of C being unknown, G must be deter- mined in relation to a reference Gibbs potential G° and its concen- tration component RTlnc°, which by convention are subtracted from G and RTlnc. Thus, G − G° = RTlnc − RTlnc° and G = G° + RTln(c/c°). By convention, the difference in Gibbs potential be- tween a reactant and a pr oduct, such as A and P of R eaction (1), is obtained by subtracting the Gibbs potential of the reactant from that of the product. Thus, for R eaction (1): PA P A [] [ ° ] + [ l n ( / ) l n ( / ) ] GG G G R T p p R Ta a−= ° − ° − ° ° ln ( / ) ln( / ) G G RT p a RT p a∆= ∆ + − ° ° (3) As may be seen, acquisition of the reaction isotherm in the form (Equation 2) in which it is usually presented and used in respect to Reaction (1) requires that we get rid of the RTln(p°/a°) term in Equation (3). Since the logarithm of unity is zero, this can be done very easily in this particular case by assuming a° = p°. By making this assumption, we eliminate the free energy of concentration (mechanical) component of ∆G° and thereby assume ∆G° to be equivalent to the difference only of the characteristic (chemical) kind of Gibbs free energy between one mol of A and one mol of P. Since A and P are assumed to be in the same environment, the difference in characteristic free energy on a per mol basis between them in their common environment could be determined in a closed system from their relative concentrations when the reaction is at equilibrium, in which case ∆G would be zero, p/a would be equivalent to the equilibrium constant of the reaction, and ∆G° would be equivalent to −RTlnK. This assumes of course that the equilibrium constant is not so large or small as to preclude meas- urement of a or p. As may be seen by including the reference concentrations in the reaction isotherm, the equilibrium constant is invariably a dimen-sionless quantity. Such is required if the logarithm of an equilib-rium constant is to make sense (46). Inclusion of the reference concentrations in the isotherm has the additional advantage of allowing one to see clearly how to handle a reactant or pr oduct that is maintained at constant concentration or virtually so in the course of the reaction. Thus for such a reactant or pr oduct, one need only specify the constant concentration to be the reference concentra- tion, in which case the reactant or pr oduct would be eliminated from the isotherm. Of course any special condition of this nature would limit the equilibrium constant to that condition and it would thus be necessary to specify the limitation in presenting the equi- librium constant if not done so by convention. For reactions having multiple reactants and pr oducts, one’s choice of reference concentrations is arbitrary within the restriction that the sum of the reference free energies of concentration on a per mol basis be the same for the reactants as for the products. Note that, since Gibbs energies are expressed on a per mol basis for each mol of each reactant and pr oduct specified in the stoichio- metric equation for the reaction, reactions having more than or less than one mol of a particular reactant or pr oduct in their stoichio- metric equation must be treated such that ∆G° for the reactions 2A ð P and 1-2 A ð P, for example, be equivalent to G°P − 2G°A and G°P − 1-2 G°A , respectively. Note also that, regardless of the complex- ity of the reaction, ∆G° is expressed in units of energy per mol. For reactions other than the very simple Reaction (1), one might ask: energy per mol of what? By taking into account the ideal-gas and mechanical (Figure 1) origins of the above method and using reci- procal concentrations ( i.e., molar volumes) rather than concentra- tions, one can readily see that the answer must be: per mol of individually mobile particles. In other words, one can readily see that the characteristic Gibbs energy of each mol or fraction of a mol of each reactant and pr oduct specified in the stoichiometric equation is measured in terms of the density of the translational kinetic energy possessed by one mol of individually mobile parti- cles at the temperature of the reaction system. Thus, the ‘per mol’ may be interpreted to indicate simply that the stoichiometric coefficients refer to mols rather than to molecules. The conventional way of eliminating the reference term is to as- sume for each kind of reactant and pr oduct in the stoichiometric equation a hypothetical ‘standard state’ concentration of unity in whatever units of concentration are employed. This state is ‘hypo-thetical’ in that the reactants and pr oducts are assumed to have properties identical with those they would have if they were in dilute (ideal) solution, in which case the requirement that concen-trations be in terms of numbers of individually mobile particles per unit volume is likely to be met. Although this convention invaria- bly abolishes the reference concentration term and, since the loga- rithm of unity to any power is zero, is the most logical and efficient means of doing so, the fact of its use is often stated in such a way as to give the very confusing impression that the reactants and products must be at a concentration of one molar or one molal, for example, if the value of ∆G° is to be correct, implying incorrectly that its use has a purpose in addition to specification of dimen- sional units and elimination of the reference term. Nature of Characteristic Chemical Potential Energy. As im- plied above, the characteristic component of the free energy of a chemical substance consists of all forms of available work poten-tial other than the intrinsically mechanical free energy of concen- tration. From the standpoint of chemistry, the characteristic free energy of a substance at a particular temperature consists only of that associated with what is ordinarily referred to as chemical bond energy. Bond energies are usually dealt with in terms only of ther- mal energies of bond formation or dissociation. This being the case, it is necessary to inquire as to the general nature of bond energy as a nonthermal actual energy. Since, other than the ex-tremely weak gravitational force, only Coulomb (electrostatic) forces and forces deriving therefrom are known to exist between atoms (47), bond energy as a nonthermal actual energy must con-sist of these forces and must therefore correspond to what is com-monly referred to as chemical binding energy. Since atoms and molecules possess both attractive and repulsive forces, and since the formation of a kinetically stable chemical bond at a finite temperature invariably involves the establishment of a balance between these forces within an energy barrier to the making and breaking of the bond, chemical binding energy must Nature of Free Energy and Entropy 12refer to the energies of both attractive and repulsive forces. Con- sider, for example, two kinds of charged atoms A+ and A− which possess binding energy only by virtue of their net charges. Since unlike charges attract one another, binding energy for a binding interaction between A+ and A− must necessarily be viewed as be- ing attractive. On the other hand, since like charges repel, binding energy for an interaction between A+ and A+, for example, must necessarily be regarded as being repulsive. Since the attractive binding interaction would clearly involve a net consumption of binding energy, the repulsive one must by implication involve a net production of binding energy. Since both attractive and repul- sive binding energy must be forms of nonthermal actual energy, it appears from this example that the binding energies of molecules possessing particularly large amounts of bond energy are likely to be of the repulsive kind. However, as is evident from the fact that the bulk of local matter occurs naturally as liquids and solids con- sisting of attractively bound molecules, most binding interactions between atoms are net attractive. In view of this and of the large amount of energy required to force like charges into close prox- imity to one another, it seems likely that most of the molecules we ordinarily think of as possessing relatively large amounts of bond energy are molecules in which the constituent atoms are relatively unattractive to one another and/or are constrained (bonded) in such a way as to prevent optimal neutralization of their attractive forces. If we were to consider this invariably to be the case, we would be ignoring the fact that, by virtue of the existence of energy barriers to the making and breaking of chemical bonds, it is possible for kinetically stable bonds to be formed between atoms despite the interaction being net repulsive. As is evident from the fact that most chemical reactions that are thermodynamically favorable in respect to ∆G° are exothermic, translational kinetic (mechanical) energy of atoms and molecules can be constrained as bond energy. Uncatalyzed conversions of translational kinetic energy into bond energy must occur through collisions of the atoms or molecules with sufficient energy to force the electronic and other changes ( e.g., desolvations) that constitute the energy barrier. From the fact that bound atoms and molecules tend to dissociate as temperature is increased, it is evident that the collision energy can be too high as well as too low. Since the amount of attractive binding energy possessed by a substance at constant ambient pressure depends on its temperature as well as on its chemical composition, it is necessary to distin- guish between changes in attractive binding energy that do and do not constitute changes in characteristic chemical potential energy. This distinction was in effect made above by considering thermally produced attractive binding energy to be extramolecular latent thermal energy. According to the treatment given, the attractive binding energy that accumulates in an individual substance as its temperature is increased at constant pressure from absolute zero to the point at which all the molecular constituents are sufficiently free to be individually mobile is not characteristic chemical poten- tial energy. That such is the case is evident from the above consid-eration of a reversible conversion of ice into liquid water with thermal energy deriving from nonthermal actual energy used to compress an ideal gas. As was noted, the free energy of the nonthermal actual energy would remain with the gas rather than undergo transfer with the thermal energy that would become latent thermal energy of newly formed liquid. One can reasonably expect only supercooled phases of single substances to possess latent thermal energy consisting of attractive binding energy having the quality of characteristic chemical poten-tial energy. Thus, whereas a substance in a superheated phase would be out of thermochemical equilibrium as a result of possess-ing an excess of translational kinetic energy relative to attractive binding energy, a substance in a supercooled phase would be out of equilibrium as a result of possessing an excess of attractive binding energy relative to the mechanical energy. That the excess attractive binding energies of substances in supercooled states actually have the quality of characteristic chemical potential energy is particu-larly evident from studies on single solids that have been mechani- cally or otherwise reduced to nano-sized particles at temperatures far below their melting points. Single solids thus modified are in effect in extremely supercooled states and have chemical reactivi-ties and other properties, noted above, indicative of the presence of exceptionally large amounts of characteristic chemical potential energy (48, 49). For instance, graphite reduced to extremely small particles by prolonged grinding under inert-gas conditions has been observed to be sufficiently reactive to ignite spontaneously at room temperature when exposed to air (50). Determination of ∆∆∆∆G° by the Calorimetric Method. Since con- sumptions and productions of bond free energy under the constant temperature, constant pressure, and virtually constant volume con- ditions assumed here are accompanied by pr oductions and con- sumptions, respectively, of equivalent amounts of energy having the quality of thermal energy at the specified constant temperature, differences in characteristic Gibbs free energy between the reac- tants and products of chemical r eactions can be determined in closed systems not only by the equilibrium method, but also by measuring thermal energy changes occurring as a result of the reactions. However, this ‘isothermal’ calorimetric method is much more difficult and less likely to be feasible in that it requires a means of distinguishing between thermal energy changes due to bond energy changes and irrelevant ones due to interconversions between the latent and nonlatent thermal energies. An irrelevant change will occur with any net change in the heat capacity of the reaction system. Such a change could occur as a result primarily either of a change in the total intramolecular latent thermal energy of the system or of a change in the total free energy of concentra- tion ( i.e., in the total number of individually mobile particles in the system). As may be seen by comparing the reactions A ð P + Q and A 2 ð 2A, a net increase in the total free energy of concentra- tion due to a chemical change under the conditions assumed here will be accompanied by an increase in a latent thermal energy that will be indistinguishable from if not identical with the above- described extramolecular kind. That free energy of concentration should be viewed here as elsewhere as being intrinsically mechanical rather than chemical can be seen by considering a closed Reaction (1) system in which A and P possess equal amounts of bond energy on a per mol basis and the reaction does not involve a net change in the system as to latent thermal energy. In such a system, an interconversion be-tween A and P would not involve a net change in an actual (First-Law) energy of any kind of either the system or its surroundings. Nevertheless, an irreversible net conversion would occur and thus be driven with consumption of free energy if there were a differen- tial in the concentrations of A and P. Since the system is assumed to be closed, a differential between A and P as to concentration would be accompanied by differentials as to absolute am ount of both bond energy and translational kinetic energy. Since, for a given differential in concentration, the magnitude of the driving force for the reaction would be a function only of temperature, it is evident that, of these two kinds of actual energy, only the thermal energy would be relevant in respect to the driving force. However, the fact of the reaction not being accompanied by a change in ac-tual energy would rule out the thermal energy as well as any other actual energy of the system as the source of the free energy con-Nature of Free Energy and Entropy 13sumed in driving the reaction. As will be pointed out below by comparing Reaction (1) systems with the purely mechanical Figure 1 system, the source of the free energy would be the nonthermal actual energy used to generate the differential in concentration. Since only free energy from this source could be present in the closed system, it would be necessary to attribute the driving force for the reaction simply to the differential in concentration or ‘num- ber density’ between A and P. Accordingly, the reaction would be driven as a result of there being a differential between the number of A molecules converting spontaneously into P molecules and the number of P molecules converting spontaneously into A mole- cules. Consider now a ∆G° = 0 reaction which differs from the one above in that it involves a net change in latent thermal energy and thus a net change in heat capacity. Since the reaction would not involve a net change in bond energy and the change in latent ther- mal energy would occur only in the system, maintenance of the system at constant temperature would require transmission of thermal energy between the system and its surroundings. Since for a closed system such transmission could be driven only by a gradi-ent of temperature, and since by definition any latent thermal en- ergy produced or consumed at a particular temperature would have the same quality as nonlatent thermal energy at that temperature, an interconversion between these energies in such a system main-tained at constant temperature would clearly be irrelevant to the energetics of the chemical changes that constitute the reaction. In other words, as a result of the latent and nonlatent thermal energies having different locations and the same quality at any particular temperature, an interconversion between these energies in such a system could be driven only by a gradient of temperature, which, in consequence of the constant-temperature specification, would by definition be sufficiently small as not to result in a finite consump-tion of free energy. From this it is evident that, from the standpoint of theory, a change in latent thermal energy in any particular reac- tion system of the sort considered here could be determined di-rectly from a detectable change in thermal energy only if the reac-tion could be conducted reversibly. Consequently, for a r eaction involving a significant change in latent thermal energy, the only way to obtain a reasonably accurate estimate of ∆G° by the calo- rimetric method would be to determine the difference between the amount of nonlatent (detectable) thermal energy produced or con-sumed in the course of the reaction when the reaction is conducted ‘completely irreversibly’ ( i.e., under conditions of zero work being done on the surroundings) and the amount produced or consumed when the reaction is conducted very nearly reversibly ( i.e., under conditions in which very nearly all of the work potential that be-comes available in the course of the reaction is used to do work on the surroundings) ( cf. 51, 52). For a reaction system maintained at constant temperature and volume rather than at constant temperature and pressure, the de-tectable change in thermal energy occurring under conditions of complete irreversibility would correspond to the algebraic sum of the changes in the chemical and latent thermal energies of the sys-tem. A change of this sort on a per mol basis is usually denoted by ∆E° and referred to as the difference or change in the total energy of the system. The detectable change in thermal energy occurring under conditions of near reversibility would correspond closely to the change in the latent thermal energy of the system in the oppo-site direction. Changes of this sort on a per mol basis are denoted by T∆S°. The change in bond energy on a per mol basis would be given in terms of thermal energy by the difference ∆E° − T∆S° and would correspond closely to the difference between the r eactants and products in respect to characteristic Helmholtz free energy. Changes in characteristic Gibbs free energy are reckoned in the same way except that ∆E° is replaced by ∆E° + P∆V° and referred to as the difference or change in enthalpy, denoted by ∆H°. The P∆V° component of ∆H° serves to account for the pressure- volume work that would be done on the surroundings (atmosphere) by the system or by the surroundings on the system if the reaction were to result in an increase or decrease, respectively, in the vol- ume of the system. In consequence of the P∆V° component, ∆G° for a Reaction (1), for example, could correspond exactly to the difference in bond energy between one mol of A and one mol of P in their common environment only if ∆V° for the conversion should happen to be zero. Although changes in volume ordinarily occur, they are usually small for reactions involving net changes only in small molecules in solution at constant temperature and pressure. At normal (atmospheric) pressures the amounts of energy involved are usually also small and sufficiently so to justify con-sidering them to be negligible in relation to the change in bond energy. The above means of estimating and accounting for differences in the characteristic component of the Gibbs free energy is in effect a means of doing so in terms of changes in thermal energy of both the system and its surroundings in terms only of system properties. Thus: ° ° ° sys sys sysGH T S∆= ∆− ∆ (4) ° / ° / ° ° ° sys sys sys surr sysGT HT S S S−∆ = −∆ + ∆ = ∆ + ∆ ° ( ° ° ) ° sys surr sys totalGT S S T S−∆ = ∆ + ∆ = ∆ Enthalpy-Entropy Compensation. The ∆H° and T∆S° compo- nents of the above ‘Gibbs standard free energy relation’ (Equation 4) are often referred to as differences or changes in energy and entropy, respectively, and are often referred to in such a way as to give the impression that the ‘energy’ and ‘entropy’ are different forms of energy that are more or less equally capable of driving thermodynamic processes under conditions of constant temperature and pressure. That doing this is not appropriate is evident from the fact that, since ∆H° is equivalent to the algebraic sum of ∆G° and T∆S° and serves to represent the difference or change in the total energy of the system, T∆S° must represent a difference or change in an energy of the system that is independent of the difference or change in thermal energy represented by ∆G° and has the quality of thermal energy at the temperature of the system. In other words, T∆S° must serve as indicated above to represent a difference or change in the kinds of energy referred to as latent thermal energy. Accordingly, the result ∆G° = −T∆S° for an isothermal process would mean simply that the change in thermal energy due to the change in the characteristic Gibbs free energy of the system hap-pens to be accompanied by an equivalent but oppositely directed change in the latent thermal energy of the system, and the result ∆G° = ∆H° would mean simply that the process does not involve a net change in the latent thermal energy of the system. Several problems with current and past interpretations of the Gibbs standard free energy relation have been pointed out and/or discussed in recent years by a number of investigators (see, e.g., 53-60). Some of the problems derive from the fact that reactions occurring in solution are ordinarily accompanied by a ‘solvent reaction’ which is not accounted for in stoichiometric equations and which in some cases appears to make equivalent contributions of the same sign to ∆H° and T∆S° and thus appears to make little or no contribution to ∆G° (53, 61-66). For reactions occurring in aqueous solution, the energy of this so-called ‘enthalpy-entropy compensation’ can be quite large as to amount and, in such cases Nature of Free Energy and Entropy 14where the associated change in ∆G° is very small, obviously must derive from large conversions between the latent and nonlatent thermal energies somewhat as in the case of reversible conversions between the solid and liquid forms of water. In this regard, it is important to note that, since ∆H° serves to represent the change in total energy and thus includes T∆S°, enthalpy-entropy compensa- tion in a broad sense must invariably occur in processes for which T∆S° is finite. Although widely acknowledged to be a phenomenon that is not well understood, enthalpy-entropy compensation of the solvent-dependent kind is generally thought to occur as a result of differ-ences between reactants and pr oducts with respect to strength of binding interaction with the solvent and thus with respect to amount of solvent bound or otherwise immobilized. Such is consis- tent with the fact that binding interactions between solutes and solvents can reasonably be expected to follow the same rules as binding interactions between solutes. Thus, binding interactions between molecules of a solute and its solvent to form complexes can be either favorable, with net conversion of binding energy into thermal energy, or unfavorable, with net conversion of thermal energy into binding energy. Also, since particles of an undissolved solute idealy immersed in a solvent would be completely sur- rounded by the solvent, one could expect there to be a considerable mechanical (mass-action) driving force ( −∆G conc) for solvation of the solute from the high solvent activity alone. Thus, one could expect some solvation of the solute to occur even if the attractive forces between the solute and solvent molecules should be very weak. If the attractive forces between molecules of the solute and solvent should be weaker than those between molecules of the solvent, one could expect solvation of the solute to be accompa-nied by net conversion of thermal energy into attractive binding energy due to hindrance of bond formation between solvent mole- cules more or less as proposed by Hildebrand and coworkers (67, 68), resulting in a temperature-sensitive tension in the solvent at the surfaces of the solute molecules similar to that which occurs at the interface of the solvent and its vapor. As a result of this ten- sion, one could expect the existence of a force for minimization of the amount of sp ace occupied by the solute. Accordingly, surface tension in a liquid results from molecules at the surface possessing relatively large amounts of attractive binding energy which in turn results in there being a relatively large attractive (contractive) force among molecules at the surface that tends to minimize surface area (69). Consistent with these expectations, solvations of nonpolar (hy- drophobic) solutes in water tend to be thermodynamically unfavor- able with respect to ∆G° and the solubilities of such solutes in water tend to be lower as the surface areas of the molecules are larger (for recent reviews, see 70-72). In addition, water has an exceptionally high surface tension which decreases with increase of temperature (69), and solvations of nonpolar solutes in water are generally accompanied by a temperature-sensitive decrease in the amount of sp ace occupied by the solute (73-75). In contrast to ∆G°, both ∆H° and T∆S° for solvation tend to be negative, indicat- ing that the latent thermal energy of the system tends to decrease more than the bond energy increases. The large negative T∆S° associated with the above phenomena, widely referred to collec- tively as ‘the hydrophobic effect’, is usually attributed to a tem- perature-sensitive ordering of water molecules at the surfaces of the solute molecules (75-78). Accordingly, solvation is ordinarily accompanied by an increase in heat capacity that can be readily explained in terms of reversal of the solvent ordering as the tem- perature is increased for the purpose of measuring the heat capacity (70). Consistent with the above, ‘Hildebrand’ interpretation of the hydrophobic effect, the ordering appears to be limited to a single layer of water molecules (79). Contrary to the Hildebrand view, the postulated ordering of wa- ter and the observed increase in bond energy in the hydrophobic effect have generally been interpreted to mean that there is an in-crease in hydrogen bonding, either in number of bonds or in bond strength, and that the thermodynamics of the ordering process are thus anomalous (72). If we were to consider the ordering actually to involve an increase in the number of hydrogen bonds, we could expect it to be accompanied by a decrease in bond energy rather than by the observed increase and the thermodynamics would in fact appear to be anomalous, since hydrogen bonding occurs through attractive (cohesive) forces and therefore can be expected to consume binding free energy. On the other hand, if we were to consider the ordering to involve only an increase in hydrogen bond strength and were to interpret ‘increase in bond strength’ to mean ‘increase in bond energy’, we could conceive of the thermodynam-ics of the ordering process not being anomalous and of there being no actual disagreement between the conventional and Hildebrand views. Thus it is conceivable that the ordering involves only a net increase in the amount of bending and stretching of hydrogen bonds, in which case there could be an increase in attractive bind-ing energy without a net decrease in the number of bonds. This possibility is based on the fact that, by virtue of the existence of energy barriers to the making and breaking of chemical bonds, the energy of a chemical bond must be enhanced if the bond is to be broken. In the case of sustained bending or stretching without net bond breakage, ‘enhancement energy’ would be retained somewhat as in the case of the bonds in a stretched rubber band and would consist exclusively of attractive binding energy. As will be out-lined below by considering a solvent reaction in respect to Reac- tion (1), the large negative T∆S° and the associated solvent- dependent enthalpy-entropy compensation in the hydrophobic effect can be explained readily and operationally in terms of the postulated ordering of solvent molecules simply by taking into account the fact that the ordering implies immobilization. In doing this, it shall be assumed ( i) that water is the solvent, (ii) that molecules of the solvent bind less strongly to each other than to A and P, and ( iii) that the attractive binding interaction between the solvent and P is relatively strong. This being the case, we could expect a net conversion of A into P to be accompanied by a solvent reaction involving a net conversion of mobile water into an immobile, bound form and an associated net conversion of at-tractive binding energy into thermal energy. In a closed Reaction (1) system, we could expect this conversion of binding energy into thermal energy to make a negative contribution to the ∆G° for the reaction just as would any other net conversion of binding energy into thermal energy at constant temperature and pressure. There- fore, we could expect the contributions to ∆H° and T∆S° to differ accordingly. How, then, can a difference between the reactant and product with regard to solvent immobilization account for solvent- dependent enthalpy-entropy compensation? As surmised by Ives and Marsden (62), this question can be answered by taking into account changes in a specific extensive (size) variable of closed reaction systems. Enthalpy-entropy compensation of the above sort can be ade- quately explained by invoking the above-noted prediction of the equipartition principle that, at any particular temperature, the indi-vidually mobile particles of a gas will possess on average the same amount of translational kinetic energy regardless of the nature of the particles. That this prediction is generally considered to be correct and applicable to substances in ideal solution is evident from the fact that it is implicit in the concentration term of the Nature of Free Energy and Entropy 15reaction isotherm (Equation 2) through the assumption that, at any particular temperature, only the relative concentrations of A and P as particles matters in regard to the difference in free energy of concentration. According to this assumption, differences in the characteristic features of A and P, such as size (mass), net charge, amount of bond energy, amount of bound solvent, and amount of intramolecular latent thermal energy, are of no consequence what- ever as regards the contributions of A and P to free energy of con- centration. In regard to particle size, it may be noted that Einstein (80) employed the above assumption in his very successful treat-ment of Brownian movements of particles of sufficient size to be seen by light microscopy. This and the fact that one could reasona- bly expect to be capable of demonstrating the equilibrium value of RTln(p/a) for a Reaction (1) as a difference in mechanical (os- motic) work potential equivalent to the ∆G° for the reaction imply (i) that differences in free energy of concentration are differences in the density of translational kinetic energy and ( ii) that, at any particular temperature, individually mobile particles of A and P would possess on average over time equivalent amounts of transla-tional kinetic energy regardless of their characteristic features. This being the case, one could reasonably expect equipartition of the translational kinetic energy of a closed reaction system among all the particles of the system, including, of course, those of the sol-vent. Accordingly, if P of a closed Reaction (1) system were to im- mobilize more water than does A, one could expect a net conver- sion of A into P to be accompanied by ( i) a decrease in the number of particles in the system, ( ii) an equipartitioning of the total trans- lational kinetic energy of the system among the smaller number of particles, ( iii) an augmentation of the average translational kinetic energy of the particles, ( iv) an elevation of the temperature of the system, and ( v) a transmission of thermal energy from the system to the surroundings. Since the decrease in number of particles would occur regardless of the driving force for the conversion of A into P and thus regardless of whether the reaction is conducted reversibly or irreversibly, it would make a negative contribution to both ∆H° and T∆S°. Of course if A were to immobilize more water than does P, the sign of the contribution would be positive rather than negative. In either case, the contribution could be regarded as being due to a change in free energy of concentration that is not accounted for in the stoichiometric equation for the r eaction. Since the binding of water to solute and of water to water can be expected to vary with temperature, one could expect the magnitude of the contribution to be temperature dependent. For reactions involving net changes in large biological molecules, in which cases changes in latent thermal energy on a per mol basis tend to be par-ticularly large, this dependence is particularly noticeable. A ques- tion currently of interest in this regard is why applications of the integrated form of the van’t Hoff equation: ln K = −∆H°/RT + constant (5) to reactions of the above sort do not yield correct values for ∆H° despite yielding linear plots of ln K vs 1/T (54-60). This apparent discrepancy can be resolved by noting that, since ln K is also equivalent to −∆G°/RT, the integration constant of Equation (5) must be equivalent to ∆S°/R if the equation is to be consistent with Equation (4), and that, since the validity of Equation (4) is beyond dispute, the right-hand side of Equation (5) must therefore be equivalent to −∆G°/RT. This being the case, linearity of plots of lnK vs 1/T means nothing more than that ln K is directly propor- tional to −∆G°/T, the observed proportionality constant being an estimate of 1/ R, and that ∆G° is a linear function of T. Role of Free Energy of Concentration in Free Energy Transfer and Conservation. As was noted above, the nature of free energy of concentration is fundamentally the same for a system in which only chemical work is possible as for an ideal gas system in which only mechanical work is possible. Accordingly, the concentration term of the reaction isotherm ignores completely the presence of the solvent. Thus, RTln(p/a) of Equation (2) would be applicable to a closed Reaction (1) system even if A and P were gases in an ideal gas-phase reaction. This being the case, we can apply to a closed Reaction (1) system virtually everything said above con-cerning the energy and entropy changes of an ideal gas associated with compression and expansion of the gas between two states at the same temperature simply by ignoring the means by which the two states were assumed to be achieved and considering a and p to be the two states. The fact that the volume would be essentially constant for the two states in the closed Reaction (1) case would not matter because the difference in free energy of concentration between them on a per mol basis could be determined from knowl-edge of the relative values of a and p. The fact that A and P would share the same space also would not matter because A and P would be virtually independent of one another in the ideal case and thus may be thought of as being individual systems within an overall Reaction (1) system. The individual systems of a closed Reaction (1) system may be thought of as being pitted against one another in respect to com-pression and expansion. For instance, if a and p were equal and A possessed RTln10 more bond energy than P on a per mol basis, the A system, through ‘expansion’, would be capable of ‘compressing’ the P system to the extent that p exceeds a by a factor of ten, at which point ( i) the reaction would be at equilibrium, ( ii) the amount of bond energy in the individual systems would be the same on an absolute amount basis, and ( iii) the A and P systems would differ as to free energy of concentration on a per mol basis by RTln10. Since at equilibrium the absolute amount of bond energy would be the same for the A and P systems, the difference in free energy of concentration, although generated as a result of there having been a difference in bond energy on an absolute amount basis, obviously would not be associated with such a difference at equi-librium. Therefore, since any consumption of bond free energy in the course of the reaction going to equilibrium would be accompa- nied by the production of an equivalent amount of thermal energy, and since all of this thermal energy could depart the system as a result of the temperature being maintained constant if the conver- sion of A into P should happen not to result in an increase in the latent thermal energy of the system, the free energy of concentra-tion generated in the P system, like that which one could expect to be generated in an ideal gas upon its isothermal compression, could appear to be free of the actual energy that undergoes trans- formation in the course of its generation. We turn now to the question of how the difference in free en- ergy of concentration could be utilized to do chemical work ( i.e., to convert ambient thermal energy into chemical bond energy). As noted above, Reaction (1) as presented is assumed to be linked to others through A being a product of a pr eceding reaction and through P being a reactant of a succeeding one in a stationary-state, biological-type system. To rule out the possibility of differences in free energy of concentration being used to do mechanical (os-motic) work in addition to chemical work, the individual reactions of this system shall be assumed to be located in the same com- partment. If Reaction (1) were thermodynamically favorable in respect to ∆G°, a difference in free energy of concentration of the above sort could be generated and conserved as bond free energy Nature of Free Energy and Entropy 16through the differential between a and p either pulling preceding reactions or pushing succeeding ones that are thermodynamically unfavorable in respect to ∆G° and thus require thermal energy to proceed in the same direction as Reaction (1). If the individual reactions were coupled only by freely diffusible intermediates, transmissions of actual energy between the reactions would occur only through transmissions of thermal energy. Of course the role of the differential in free energy of concentration in conserving the excess bond free energy of A relative to P on a per mol basis would be to increase the number of reactant molecules converting spontaneously into product molecules relative to the number of product molecules converting spontaneously into r eactant mole- cules in the unfavorable reactions. If the stationary-state ratio of p to a were infinitesimally smaller than the equilibrium constant of Reaction (1), all of the reactions would be operating reversibly in the thermodynamic sense and all of the bond free energy would be conserved as bond energy. Although the biological system as one in which only chemical work is possible is similar to the purely mechanical (Figure 1) system in that it employs a difference in density of translational kinetic energy to mediate the transfer of free energy from one re- pository to another, it differs markedly from the purely mechanical system in that it employs the particle-number-density (concentra-tion) aspect rather than the mechanical-energy-density aspect of the translational energy density. Owing to the need for thermal energy transfer between the system and surroundings at a low tem- perature differential to conserve free energy in the purely mechani-cal case and the lack of such need to conserve free energy in the biological case, the two systems also differ greatly as to the amount of time that would be required for efficient transfer of the actual energy. Thus, in the biological case the reactions yielding and requiring thermal energy could occur simultaneously in close proximity to one another at the microscopic level, a consequence of which would be that the requisite thermal energy transfers could occur rapidly while the temperature of the system is essentially constant. Thermal energy transfer between the biological system and its surroundings would be needed only to rid the system of the thermal energy produced as a result of the need for expenditure of bond free energy to maintain an appropriate net rate of reaction. If the system should require thermal energy to maintain its tempera-ture, this expenditure of free energy would also serve to meet this need. APPENDIX According to the main body of the text, by considering irreversibil- ity in respect to an adiabatic cycle of compression and expansion of an ideal gas, one can readily see that any irreversibility in a thermodynamic work process would be reflected only in the char- acteristic entropy and would result in the consumption of an amount of characteristic free energy equivalent to an amount of ambient thermal energy produced. That these observations are correct may be seen from the following considerations in reference to an adiabatic version of the Figure 1 model of the main body of the text. If one mol of an ideal gas in the fully expanded state V 1 at a temperature T1 of the reservoir were compressed to a volume V2 adiabatically, the temperature of the gas would increase by an amount depending on the heat capacity and on the amount of energy ∆E imparted to the gas. This amount of energy would be equivalent to the amount of work W done and thus would depend on the rate of the compression. Regardless of the rate, W and ∆E would be equivalent to CV∆T. Since S, T, and V are state functions and CV would be a constant, the expression for the net change in entropy ∆Snet = ∆Schar + ∆Sconc = CVln(T2/T1) + Rln(V2/V1) in terms of these parameters would be applicable regardless of the rate of the compression. If the gas were compressed reversibly, ∆Schar would be equivalent to −∆Sconc and T2 could be determined from the relationship / 21 1 2 (/)VRCTT V V= . If the gas were com- pressed at a finite rate, T2 would be relatively high and ∆Snet for the compression would be finite and positive as a result of the increase in characteristic entropy exceeding the decrease in entropy of concentration. In this ‘irreversible’ case, T2 could be determined only by experimental means. Although calculation of ∆Snet could be easily achieved if T2 for the irreversible compression were known, doing so would not be helpful in respect to precise quantification of the amount of free energy consumed. However, since the work potential of the gas would increase by an amount equivalent to the amount of work done regardless of the rate of the compression, doing so could serve to indicate in a semiquantitative fashion that work potential has been rendered unavailable. As indicated below, if T2 were known, the precise amount rendered unavailable could be determined theoretically by considering the gas to undergo reversible expansion to its original volume. In contrast to what would be the case for an ideal gas com- pressed irreversibly between two states at the same temperature, reversible expansion of the adiabatically compressed gas would result in a fraction of the thermal energy produced as a result of the gas being compressed at a finite rate being recovered as work, the fraction being that indicated by the efficiency of a Carnot engine operating reversibly between T2 and the lower temperature that would exist upon return of the gas to its original volume V1. Since ∆Schar for the expansion would be equivalent to −∆Sconc, this lower temperature would be equivalent to / 22 1(/)VRCTV V and would of course be higher than the temperature T1 of the reservoir by an amount depending on how fast the gas was compressed. In addi- tion to the difference in temperature, there would remain differ- ences in the pressure, thermal energy, and characteristic entropy, the difference in characteristic entropy being equivalent to that existing at the end of the compression phase. Elimination of these differences would require removal of the adiabatic constraint so that an amount of thermal energy CV∆T could undergo transmis- sion to the reservoir, CV∆T in this case being equivalent to the amount of characteristic free energy consumed as a result of the gas being compressed at a finite rate. Since the amount of the gas was specified to be one mol, this amount would be equivalent to / 22 1 1[( /) ]VRC VCT VV T − . The value thus obtained divided by the temperature of the reservoir would be the net increase in entropy of the reservoir and for the overall process. As in any cyclic process, the net changes in free energy and entropy would be changes only in the characteristic kinds. 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/CQ/CP/D0/CP/D2 /CT/BA/C1/D2 /D7/CT /D8/CX/D3/D2 /BF /D8/CW/CT /D4/D6/D3 /CT/CS/D9/D6/CT /D3/CU /CJ/BD/BF ℄ /CA/D3/CQ /CT/D6/D8 /CP/D2/CS /CB/D3/D1/D1/CT/D6/CX/CP /B4/BD/BL/BL/BD/B5 /CX/D7 /CT/DC/D8/CT/D2/CS/CT/CS /D8/D3 /D8/CW/CT/D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1 /DB/CX/D8/CW /CS/CX/D7 /D9/D7/D7/CX/D3/D2 /D3/CU /D7/CX/D1/D4/D0/CX/AS/CT/CS /CP/D7/CT/D7 /CX/D2 /D7/CT /D8/CX/D3/D2 /BG/BA /CF /CT /D7/D8/CX/D0/D0 /CP/D7/D7/D9/D1/CT/D8/CW/CP/D8 /D8/CW/CT /DA /D3/D6/D8/CX /CX/D8 /DD /AS/CT/D0/CS /D6/CT/CP/D8/CT/D7 /CX/D2 /D8/D6/CX /CP/D8/CT /AS/D0/CP/D1/CT/D2 /D8/CP/D8/CX/D3/D2 /CQ/D9/D8 /D8/CW/CT /CS/CX/DA /CT/D6/CV/CT/D2 /CT /CP/D2/CS /DB /CP/D8/CT/D6 /CW/CT/CX/CV/CW /D8/B4/D7/D9/D6/CU/CP /CT /CS/CT/D2/D7/CX/D8 /DD/B5 /AS/CT/D0/CS/D7 /CP/D6/CT /D7/D8/CX/D0/D0 /D7/D1/D3 /D3/D8/CW/BA /CC/CW/CT/D7/CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2/D7 /CP/D6/CT /CY/D9/D7/D8/CX/AS/CT/CS /CU/D3/D6 /AT/D3 /DB/D7 /CS/D3/D1/CX/D2/CP/D8/CT/CS/CQ /DD /DA /D3/D6/D8/CX /CP/D0 /D1/D3/D8/CX/D3/D2 /CP/D8 /D1/D3 /CS/CT/D6/CP/D8/CT /C5/CP /CW /D2 /D9/D1 /CQ /CT/D6/D7/B8 /CU/D3/D6 /DB/CW/CX /CW /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D7/CW/D3 /CZ/D7 /CX/D7 /D2/D3/D8/CT/AR/CT /D8/CX/DA /CT/BA /CC/CW/CT /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D8/D3 /DB /CP/D6/CS /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /CX/D7 /CS/CX/D7 /D9/D7/D7/CT/CS /CX/D2 /D7/CT /D8/CX/D3/D2 /BH /D4/D6/D3 /DA/CX/CS/CX/D2/CV /D4/D6/CP /D8/CX /CP/D0/D1/CT/D8/CW/D3 /CS/D7 /CU/D3/D6 /CS/CT/D8/CT/D6/D1/CX/D2/CX/D2/CV /D8/CW/CT /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /CP/D7 /DB /CT/D0/D0 /CP/D7 /D7/D9/CQ/B9/CV/D6/CX/CS /D7 /CP/D0/CT /D1/D3 /CS/CT/D0/CX/D2/CV /D3/CU/D8/D9/D6/CQ/D9/D0/CT/D2 /CT /CX/D2 /CP /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1/BA /BY/CX/D2/CP/D0/D0/DD /D8/CW/CT /CP/D7/CT /D3/CU /D4/CP/D6/D8/CX /D9/D0/CP/D6 /CV/CT/D3/D1/CT/D8/D6/CX/CT/D7 /CX/D7 /CS/CX/D7 /D9/D7/D7/CT/CS/CX/D2 /D7/CT /D8/CX/D3/D2 /BI/BA/BE /CC/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7/BV/D3/D2/D7/CX/CS/CT/D6 /CP /AT/D9/CX/CS /D0/CP /DD /CT/D6 /DB/CX/D8/CW /D8/CW/CX /CZ/D2/CT/D7/D7h(x,y,t) /D7/D9/CQ/D1/CX/D8/D8/CT/CS /D8/D3 /CP /CV/D6/CP /DA/CX/D8 /DDg /D3/D2 /CP /D6/D3/D8/CP/D8/CX/D2/CV /D4/D0/CP/D2/CT/D8/BA/CF /CT /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8 /D8/CW/CT /D0/CP /DD /CT/D6 /CX/D7 /D8/CW/CX/D2 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /D0/CT/D2/CV/D8/CW /D7 /CP/D0/CT /D3/CU /D8/CW/CT/CW/D3/D6/CX/DE/D3/D2 /D8/CP/D0 /D1/D3/D8/CX/D3/D2/BA /C1/D2 /D8/CW/CP/D8 /CP/D7/CT/B8 /D8/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /AS/CT/D0/CSu(x,y,t) /CP/D2 /CQ /CT /CP/D7/D7/D9/D1/CT/CS /D8 /DB /D3/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0/CP/D2/CS /D8/CW/CT /DA /D3/D6/D8/CX /CX/D8 /DDω ωω=ωez=∇ ∧u /CX/D7 /CS/CX/D6/CT /D8/CT/CS /CP/D0/D3/D2/CV /D8/CW/CT /DA /CT/D6/D8/CX /CP/D0 /CP/DC/CX/D7/BA /CF /CT /D7/CW/CP/D0/D0 /CP/D7/D7/D9/D1/CT /CU/D3/D6/D7/CX/D1/D4/D0/CX /CX/D8 /DD /CP /D4/D0/CP/D2/CT /CV/CT/D3/D1/CT/D8/D6/DD /B8 /DB/CX/D8/CW /D6/D3/D8/CP/D8/CX/D3/D2 /DA /CT /D8/D3/D6 Ω ΩΩ/CS/CX/D6/CT /D8/CT/CS /CP/D0/D3/D2/CV /D8/CW/CT /DA /CT/D6/D8/CX /CP/D0/B8 /CQ/D9/D8 /CT/DC/D8/CT/D2/D7/CX/D3/D2/D8/D3 /CP /D7/D4/CW/CT/D6/CX /CP/D0 /CV/CT/D3/D1/CT/D8/D6/DD /DB /D3/D9/D0/CS /CQ /CT /D7/D8/D6/CP/CX/CV/D8/CW/CU/D3/D6/DB /CP/D6/CS /B4/DB /CT /CX/D2 /D8/D6/D3 /CS/D9 /CT /D8/CW/CT /BV/D3/D6/CX/D3/D0/CX/D7 /CT/AR/CT /D8 /CQ/D9/D8 /D2/D3 /CT/D2 /D8/D6/CX/CU/D9/CV/CP/D0 /CU/D3/D6 /CT /CP/D7 /D8/CW/CT /D0/CP/D8/D8/CT/D6 /CX/D7 /CX/D2 /D3/D6/D4 /D3/D6/CP/D8/CT /CX/D2 /D8/CW/CT /CV/D6/CP /DA/CX/D8 /DD /D3/CU /D8/CW/CT /D4/D0/CP/D2/CT/D8/B5/BA /CC/CW/CT /D8/CX/D1/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2/D3/CU /D8/CW/CT/D7/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CJ/BE/BF ℄ /B5/BM ∂h ∂t+∇ ·(hu) = 0 /B4/BD/B5 ∂u ∂t+ (ω ωω+ 2Ω ΩΩ)∧u=−∇B /B4/BE/B5/C0/CT/D6/CT /D8/CW/CT /D9/D7/D9/CP/D0 /CP/CS/DA /CT /D8/CX/DA /CT /D8/CT/D6/D1u.∇u /CW/CP/D7 /CQ /CT/CT/D2 /CT/DC/D4/D6/CT/D7/D7/CT/CS /D9/D7/CX/D2/CV /D8/CW/CT /DB /CT/D0/D0/B9/CZ/D2/D3 /DB/D2 /CX/CS/CT/D2 /D8/CX/D8 /DD /D3/CU/DA /CT /D8/D3/D6 /CP/D2/CP/D0/DD/D7/CX/D7 u.∇u=∇(u2/2) +ω ωω∧u /B8 /CP/D2/CS /D8/CW/CT /D8/CT/D6/D1u2/2 /CX/D2 /D3/D6/D4 /D3/D6/CP/D8/CT/CS /CX/D2 /D8/CW/CT /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX/BE/CU/D9/D2 /D8/CX/D3/D2 B=gh+u2 2, /B4/BF/B5/D8/D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /D8/CT/D6/D1gh /BA /C6/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1 /CP/D2 /CQ /CT /DA/CX/CT/DB /CT/CS /CP/D7 /CP/BE/BW /AT/D3 /DB /D3/CU /CP /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CV/CP/D7 /DB/CX/D8/CW /CS/CT/D2/D7/CX/D8 /DDh /CP/D2/CS /CT/D5/D9/CP/D8/CX/D3/D2 /D3/CU /D7/D8/CP/D8/CTp=gh2/2 /B8 /CP/D2/CS /D3/D9/D6 /D6/CT/D7/D9/D0/D8/D7 /D3/D9/D0/CS /CQ /CT /D6/CT/CP/CS/CX/D0/DD /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT/CS /D8/D3 /BE/BW /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CP/CS/CX/CP/CQ/CP/D8/CX /AT/D3 /DB/D7/BA /CF /CT /D7/CW/CP/D0/D0 /D3/CU/D8/CT/D2 /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT/D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1 /CP/D7 /D8/CW/CT /AG /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CP/D7/CT/AH/B8 /CQ /DD /D3/D4/D4 /D3/D7/CX/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /AG/CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT/AH /CP/D7/CT/B8 /CU/D3/D6 /DB/CW/CX /CW /B4/BD/B5 /CX/D7 /D6/CT/D4/D0/CP /CT/CS /CQ /DD∇.u= 0 /BA/C7/D2/CT /CP/D2 /CT/CP/D7/CX/D0/DD /CW/CT /CZ /D8/CW/CP/D8 /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /B4/C8/CE/B5 q=ω+ 2Ω h /B4/BG/B5/CX/D7 /D3/D2/D7/CT/D6/DA /CT/CS /CU/D3/D6 /CT/CP /CW /AT/D9/CX/CS /D4/CP/D6 /CT/D0/B8 /CX/BA/CT/BM dq dt≡∂q ∂t+u· ∇q= 0 /B4/BH/B5/BX/CP /CW /D1/CP/D7/D7 /CT/D0/CT/D1/CT/D2 /D8hd2r /CX/D7 /D3/D2/D7/CT/D6/DA /CT/CS /CS/D9/D6/CX/D2/CV /D8/CW/CT /D3/D9/D6/D7/CT /D3/CU /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2/BA /CC /D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /B4/BH/B5/B8/D8/CW/CX/D7 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CT /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /BV/CP/D7/CX/D1/CX/D6/D7 Cf=/integraldisplay f(q)hd2r /B4/BI/B5/DB/CW/CT/D6/CTf /CX/D7 /CP/D2 /DD /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /CU/D9/D2 /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /BA /C1/D2 /D4/CP/D6/D8/CX /D9/D0/CP/D6/B8 /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D7 Γn /D3/CU /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /CP/D6/CT /D3/D2/D7/CT/D6/DA /CT/CS Γn=/integraldisplay qnhd2r /B4/BJ/B5/CC/CW/CT /D1/D3/D1/CT/D2 /D8/D7n= 0,1,2 /CP/D6/CT /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /D8/CW/CT /D8/D3/D8/CP/D0 /D1/CP/D7/D7M /B8 /D8/CW/CT /CX/D6 /D9/D0/CP/D8/CX/D3/D2 Γ /CP/D2/CS /D8/CW/CT /C8/CE/CT/D2/D7/D8/D6/D3/D4/CW /DD Γ2 /BA /CC/CW/CT /CT/D2/CT/D6/CV/DD E=/integraldisplay hu2 2d2r+1 2/integraldisplay gh2d2r /B4/BK/B5/CX/D2 /DA /D3/D0/DA/CX/D2/CV /CP /CZ/CX/D2/CT/D8/CX /CP/D2/CS /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D4/CP/D6/D8/B8 /CX/D7 /CP/D0/D7/D3 /CP /D3/D2/D7/CT/D6/DA /CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /BA /C6/D3/D8/CT /AS/D2/CP/D0/D0/DD /D8/CW/CP/D8 /CU/D3/D6 /CP/D1 /D9/D0/D8/CX/D4/D0/DD /D3/D2/D2/CT /D8/CT/CS /CS/D3/D1/CP/CX/D2/B8 /D0/CX/CZ /CT /D8/CW/CT /CP/D2/D2 /D9/D0/D9/D7 /D3/D6 /CW/CP/D2/D2/CT/D0 /CS/CX/D7 /D9/D7/D7/CT/CS /CX/D2 /D7/CT /D8/CX/D3/D2 /BI/B8 /D8/CW/CT /CX/D6 /D9/D0/CP/D8/CX/D3/D2/CP/D0/D3/D2/CV /CT/CP /CW /CQ /D3/D9/D2/CS/CP/D6/DD /CX/D7 /D3/D2/D7/CT/D6/DA /CT/CS/B8 /CX/D2 /CP/CS/CS/CX/D8/CX/D3/D2 /D8/D3Γ /BA/C1/D8 /DB/CX/D0/D0 /CQ /CT /D3/D2 /DA /CT/D2/CX/CT/D2 /D8 /CX/D2 /D8/CW/CT /D7/CT/D5/D9/CT/D0 /D8/D3 /D9/D7/CT /CP /C0/CT/D0/D1/CW/D3/D0/D8/DE /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D9/D1 hu /CX/D2 /D8/D3 /CP /D4/D9/D6/CT/D0/DD /D6/D3/D8/CP/D8/CX/D3/D2/CP/D0 /CP/D2/CS /CP /D4/D9/D6/CT/D0/DD /CS/CX/DA /CT/D6/CV/CT/D2 /D8 /D4/CP/D6/D8 hu=−ez∧ ∇ψ+∇φ /B4/BL/B5/DB/CW/CT/D6/CTψ /CP/D2/CSφ /CP/D6/CT /CS/CT/AS/D2/CT/CS /CP/D7 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /C8 /D3/CX/D7/D7/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/D7 ∆ψ=−∇ ∧ (hu), ψ =const. /D3/D2 /CT/CP /CW /CQ /D3/D9/D2/CS/CP/D6/DD /B4/BD/BC/B5 ∆φ=−∇ ·(hu), ∂φ/∂ζ = 0 /D3/D2 /CT/CP /CW /CQ /D3/D9/D2/CS/CP/D6/DD /B4/BD/BD/B5/DB/CW/CT/D6/CT /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CP/D8 /D8/CW/CT /CS/D3/D1/CP/CX/D2 /CQ /D3/D9/D2/CS/CP/D6/DD /B4/DB/CX/D8/CW /D2/D3/D6/D1/CP/D0 /D3 /D3/D6/CS/CX/D2/CP/D8/CT ζ /B5 /CP/D6/CT /D8/CW/CT /D3/D2/D7/CT/B9/D5/D9/CT/D2 /CT/D7 /D3/CU /D8/CW/CT /CX/D1/D4 /CT/D6/D1/CT/CP/CQ/CX/D0/CX/D8 /DD /D3/D2/CS/CX/D8/CX/D3/D2/BA /CF /CT /CW/CT/D6/CT /D3/D2/D7/CX/CS/CT/D6 /CP /CS/D3/D1/CP/CX/D2 /DB/CX/D8/CW /CP /D7/CX/D2/CV/D0/CT /B4/D3/D9/D8/CT/D6/B5/BF/CQ /D3/D9/D2/CS/CP/D6/DD /B8 /D7/D3 /DB /CT /CP/D2 /D8/CP/CZ /CTψ= 0 /CP/D8 /D8/CW/CX/D7 /CQ /D3/D9/D2/CS/CP/D6/DD /B4/CP/D7ψ /CX/D7 /CS/CT/AS/D2/CT/CS /DB/CX/D8/CW/CX/D2 /CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /CV/CP/D9/CV/CT /D3/D2/D7/D8/CP/D2 /D8/B5/BA/BY /D3/D6 /CP /D7/D8/CT/CP/CS/DD /D7/D3/D0/D9/D8/CX/D3/D2/B8 /D8/CW/CT /D1/CP/D7/D7 /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /B4/BD/B5 /D6/CT/CS/D9 /CT/D7 /D8/D3∇ ·(hu) = 0 /B8 /D7/D3 /D8/CW/CP/D8φ= 0/CP/D2/CS hu=− /CTz∧ ∇ψ ( /D7/D8/CT/CP/CS/DD ), /B4/BD/BE/B5/CP/D2/CS /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BH/B5 /D6/CT/CS/D9 /CT/D7 /D8/D3u· ∇q= 0 /B8 /DB/CW/CX /CW /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D8q /CX/D7 /CP /CU/D9/D2 /D8/CX/D3/D2F /D3/CU /D8/CW/CT /D7/D8/D6/CT/CP/D1/CU/D9/D2 /D8/CX/D3/D2ψ q=F(ψ) /B4/BD/BF/B5/BY/CX/D2/CP/D0/D0/DD /B8 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/B5 /D6/CT/CS/D9 /CT/D7 /D8/D3 (ω ωω+ 2Ω ΩΩ)∧u=−∇B /B4/BD/BG/B5/CC /CP/CZ/CX/D2/CV /D8/CW/CT /CS/D3/D8 /D4/D6/D3 /CS/D9 /D8 /DB/CX/D8/CWu /B8 /DB /CT /D3/CQ/D8/CP/CX/D2 u· ∇B= 0 /D3/D6/B8 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/D0/DD /B8 B=B(ψ) /BA /CC/CW/CX/D7 /CX/D7/CZ/D2/D3 /DB/D2 /CP/D7 /D8/CW/CT /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /D8/CW/CT/D3/D6/CT/D1/BA /CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /CU/D3/D6 /B4/BD/BE/B5 /CX/D2 /CT/D5/D9/CP/D8/CX/D3/D2 (14) /B8 /DB /CT /D3/CQ/D8/CP/CX/D2 /CP /D7/D4 /CT /CX/AS /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DDq /CP/D2/CS /D8/CW/CT /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /CU/D9/D2 /D8/CX/D3/D2B /CX/D2 /D8/CW/CT /CU/D3/D6/D1/BM q=−dB dψ /B4/BD/BH/B5/CB/D1/CP/D0/D0 /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2/D7 /D8/D3 /CP /D7/D8/CP/D8/CT /D3/CU /D6/CT/D7/D8/B8 /DB/CX/D8/CW /D9/D2/CX/CU/D3/D6/D1 /D8/CW/CX /CZ/D2/CT/D7/D7H /B8 /D7/CP/D8/CX/D7/CU/DD /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS/CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /D8 /DB /D3 /CQ/D6/CP/D2 /CW/CT/D7 /D3/CU /D7/D3/D0/D9/D8/CX/D3/D2/D7/BA /BY /D3/D6 /D7/D1/CP/D0/D0 /D7 /CP/D0/CT/D7/B8 /D8/CW/CT/D7/CT /CP/D6/CT /D8/CW/CT /D9/D7/D9/CP/D0 /D7/D9/D6/CU/CP /CT/DB /CP /DA /CT/D7 /D3/D2 /D3/D2/CT /CW/CP/D2/CS/B8 /DB/CX/D8/CW /D4/D9/D6/CT/D0/DD /CS/CX/DA /CT/D6/CV/CT/D2 /D8 /DA /CT/D0/D3 /CX/D8 /DD /CP/D2/CS /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2 /D7/D4 /CT/CT/CSc= (gH)1/2/B8 /CP/D2/CS/D7/D8/CT/CP/CS/DD /DA /D3/D6/D8/CX /CP/D0 /CS/CX/DA /CT/D6/CV/CT/D2 /CT/D0/CT/D7/D7 /D1/D3 /CS/CT/D7 /D3/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/BA /C1/D2 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CV/CX/D1/CT/D7/B8 /D8/CW/CT/D7/CT /D8 /DB /D3/D1/D3 /CS/CT/D7 /CX/D2 /D8/CT/D6/CP /D8/BA /CE /D3/D6/D8/CX /CP/D0 /D1/D3/D8/CX/D3/D2 /DB/CX/D8/CW /D7 /CP/D0/CTl /CP/D2/CS /D8 /DD/D4/CX /CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DDω /AT/D9 /D8/D9/CP/D8/CT/D7 /D3/D2 /D8/CX/D1/CT /D7 /CP/D0/CT ω−1/B8 /D7/D3 /CX/D8 /CT/D1/CX/D8/D7 /DB /CP /DA /CT/D7 /CP/D8 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW λ∼c/ω /B8 /CW/CT/D2 /CTλ/l /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D3/CU /D8/CW/CT /C5/CP /CW /D2 /D9/D1 /CQ /CT/D6 c/u /CQ/CP/D7/CT/CS /D3/D2 /D8/CW/CT /D0/D3 /CP/D0 /DA /CT/D0/D3 /CX/D8 /DDu∼ωl /BA /CC/CW/CT/D6/CT/CU/D3/D6/CT /DB /CT /CT/DC/D4 /CT /D8 /D8/CW/CP/D8 /CU/D3/D6 /D8/CW/CT /CP/D7/CT /D3/CU /D7/D1/CP/D0/D0 /C5/CP /CW/D2 /D9/D1 /CQ /CT/D6/D7 /D8/CW/CP/D8 /DB /CT /D7/CW/CP/D0/D0 /D3/D2/D7/CX/CS/CT/D6/B8 /DA /CT/D0/D3 /CX/D8 /DD /CS/CX/DA /CT/D6/CV/CT/D2 /CT /CP/D2/CS /CU/D6/CT/CT /D7/D9/D6/CU/CP /CT /CS/CT/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/D6/CT /D1 /D9 /CW/D7/D1/D3 /D3/D8/CW/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /DA /D3/D6/D8/CX /CX/D8 /DD /AS/CT/D0/CS /B4/D8/CW/CT/CX/D6 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /CX/D7 /D1 /D9 /CW /D0/CP/D6/CV/CT/D6/B5/BA/BF /CC/CW/CT /D1/CP/DC/CX/D1 /D9/D1 /CT/D2 /D8/D6/D3/D4 /DD /D8/CW/CT/D3/D6/DD/BF/BA/BD /BZ/CT/D2/CT/D6/CP/D0 /D4/D6/CX/D2 /CX/D4/D0/CT/D7 /CP/D2/CS /D2/D3/D8/CP/D8/CX/D3/D2/D7/BY /D3/D6 /D7/D0/D3 /DB /DA /CT/D0/D3 /CX/D8/CX/CT/D7/B8 /D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1 /D6/CT/CS/D9 /CT/D7 /D8/D3 /D8/CW/CT /D5/D9/CP/D7/CX/B9/CV/CT/D3/D7/D8/D6/D3/D4/CW/CX /B4/C9/BZ/B5 /CT/D5/D9/CP/B9/D8/CX/D3/D2/D7/B8 /DB/CX/D8/CWh≃cte /B8 /D7/D9 /CW /D8/CW/CP/D8 /B4/BD/B5 /D6/CT/CS/D9 /CT/D7 /D8/D3 /D8/CW/CT /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/CX/D0/CX/D8 /DD /D3/D2/CS/CX/D8/CX/D3/D2 ∇.u= 0 /BA /CC/CW/CT/D2/D8/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /AS/CT/D0/CS /D6/CT/D1/CP/CX/D2/D7 /D6/CT/CV/D9/D0/CP/D6 /CU/D3/D6 /CP/D2 /DD /D8/CX/D1/CT/B8 /CQ/D9/D8 /CX/D8 /CV/CT/D2/CT/D6/CP/D0/D0/DD /CS/CT/DA /CT/D0/D3/D4/D7 /D3/D1/D4/D0/CT/DC /AS/D2/CT /D7 /CP/D0/CT/DA /D3/D6/D8/CX /CX/D8 /DD /AS/D0/CP/D1/CT/D2 /D8/D7 /D7/D3 /D8/CW/CP/D8 /CP /CS/CT/D8/CT/D6/D1/CX/D2/CX/D7/D8/CX /CS/CT/D7 /D6/CX/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /AT/D3 /DB /DB /D3/D9/D0/CS /D6/CT/D5/D9/CX/D6/CT /CP /D6/CP/D4/CX/CS/D0/DD/CX/D2 /D6/CT/CP/D7/CX/D2/CV /CP/D1/D3/D9/D2 /D8 /D3/CU /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/D7 /D8/CX/D1/CT /CV/D3 /CT/D7 /D3/D2/BA /CC/CW/CT /CX/CS/CT/CP /D3/CU /D8/CW/CT /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /D8/CW/CT/D3/D6/DD /CX/D7 /D8/D3/CV/CX/DA /CT /D9/D4 /D7/D9 /CW /CP /CS/CT/D8/CT/D6/D1/CX/D2/CX/D7/D8/CX /CS/CT/D7 /D6/CX/D4/D8/CX/D3/D2 /CP/D2/CS /D6/CT/CU/CT/D6 /D8/D3 /CP /D4/D6 /D3/CQ /CP/CQ/CX/D0/CX/D7/D8/CX /CS/CT/D7 /D6/CX/D4/D8/CX/D3/D2/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8/D8/CW/CT /CT/DC/CP /D8 /CZ/D2/D3 /DB/D0/CT/CS/CV/CT /D3/CU /D8/CW/CT /AG/AS/D2/CT/B9/CV/D6/CP/CX/D2/CT/CS/AH/B8 /D3/D6 /D1/CX /D6/D3/D7 /D3/D4/CX /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /AS/CT/D0/CS /CX/D7 /D6/CT/D4/D0/CP /CT/CS/CQ /DD /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CT/D2/D7/CX/D8 /DD /B4/CP/D6/CT/CP /CU/D6/CP /D8/CX/D3/D2/B5ρ(r,σ) /D3/CU /AS/D2/CS/CX/D2/CV /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /D0/CT/DA /CT/D0σ /CP/D8/D4 /D3/D7/CX/D8/CX/D3/D2 r /BA/BY /D3/D6 /D8/CW/CT /D1/D3/D6/CT /CV/CT/D2/CT/D6/CP/D0 /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1/B8 /D8/CW/CT /CX/D2 /DA/CX/D7 /CX/CS /CS/DD/D2/CP/D1/CX /D7 /CV/CT/D2/CT/D6/CP/D0/D0/DD /D0/CT/CP/CS/D7 /D8/D3/D7/CX/D2/CV/D9/D0/CP/D6/CX/D8/CX/CT/D7 /B4/D7/CW/D3 /CZ/D7/B5/B8 /DB/CX/D8/CW /CP/D7/D7/D3 /CX/CP/D8/CT/CS /CT/D2/CT/D6/CV/DD /CS/CX/D7/D7/CX/D4/CP/D8/CX/D3/D2 /B4/CT/DA /CT/D2 /CX/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /DA/CX/D7 /D3/D7/B9/CX/D8 /DD/B5/BA /CC/CW/CX/D7 /CX/D7 /CP /D7/D3/D9/D6 /CT /D3/CU /CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0 /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0 /CS/CXꜶ /D9/D0/D8 /DD /CU/D3/D6 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/BG/CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /D1/CT /CW/CP/D2/CX /D7 /CX/D2/CX/D8/CX/CP/D0/D0/DD /CS/CT/DA /CT/D0/D3/D4 /CT/CS /CU/D3/D6 /D8/CW/CT /BX/D9/D0/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D3/D6 /C9/BZ /D7/DD/D7/D8/CT/D1/BA/C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7/B8 /CU/D3/D6 /D8/CW/CT /CP/D7/CT /D3/CU /D7/D1/CP/D0/D0 /C5/CP /CW /D2 /D9/D1 /CQ /CT/D6/D7 /D8/CW/CP/D8 /DB /CT /D3/D2/D7/CX/CS/CT/D6/B8 /D7/CW/D3 /CZ/D7 /D3 /D9/D6 /D3/D2/D0/DD /CP/CU/D8/CT/D6/CP /DA /CT/D6/DD /D0/D3/D2/CV /D8/CX/D1/CT /B4/CS/D9/CT /D8/D3 /D2/D3/D2/B9/D0/CX/D2/CT/CP/D6 /D7/D8/CT/CT/D4 /CT/D2/CX/D2/CV /D3/CU /D7/D9/D6/CU/CP /CT /DB /CP /DA /CT/D7/B5/B8 /CP/D2/CS /D8/CW/CT/DD /CX/D2 /DA /D3/D0/DA /CT /CP /DB /CT/CP/CZ/CT/D2/CT/D6/CV/DD /CS/CX/D7/D7/CX/D4/CP/D8/CX/D3/D2/B8 /D7/CX/D2 /CT /D1/D3/D7/D8 /D3/CU /D8/CW/CT /CT/D2/CT/D6/CV/DD /D6/CT/D1/CP/CX/D2/D7 /CX/D2 /D8/CW/CT /DA /D3/D6/D8/CX /CP/D0 /D1/D3/D8/CX/D3/D2/BA /BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT/AS/D2/CT /D7 /CP/D0/CT /DA /D3/D6/D8/CX /CX/D8 /DD /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7 /CQ /CT/CW/CP /DA /CT /D0/CX/CZ /CT /CX/D2 /D8/CW/CT /C9/BZ /D7/DD/D7/D8/CT/D1/B8 /CP/D2/CS /D3/D2/D0/DD /CX/D2 /D8/CT/D6/CP /D8 /DB/CX/D8/CW /D7/D9/D6/CU/CP /CT/DB /CP /DA /CT/D7 /CP/D2/CS /AT/D3 /DB /CS/CX/DA /CT/D6/CV/CT/D2 /CT /CP/D8 /D1 /D9 /CW /D0/CP/D6/CV/CT/D6 /D7 /CP/D0/CT/B8 /CP/D7 /CS/CX/D7 /D9/D7/D7/CT/CS /CP/CQ /D3 /DA /CT/BA /CF /CT /D7/CW/CP/D0/D0 /D8/CW/CT/D6/CT/CU/D3/D6/CT /CP/D7/B9/D7/D9/D1/CT /D8/CW/CP/D8 /DA /D3/D6/D8/CX /CX/D8 /DD /AT/D9 /D8/D9/CP/D8/CT/D7 /CP/D8 /D7/D1/CP/D0/D0 /D7 /CP/D0/CT/B8 /CQ/D9/D8 /CS/CX/DA /CT/D6/CV/CT/D2 /CT /CX/D7 /D7/D1/D3 /D3/D8/CW /CP/D7 /DB /CT/D0/D0 /CP/D7 /D8/CW/CT /CW/CT/CX/CV/CW /D8 h /BA/CF /CT /D7/D8/CX/D0/D0 /CS/CT/D7 /D6/CX/CQ /CT /D8/CW/CT /D0/D3 /CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7 /CQ /DD /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD ρ(r,σ) /D3/CU /AS/D2/CS/CX/D2/CV /D8/CW/CT/D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /D0/CT/DA /CT/D0σ /CX/D2 /CP /D7/D1/CP/D0/D0 /D2/CT/CX/CV/CW /CQ /D3/D6/CW/D3 /D3 /CS /D3/CU /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 r /BA /CC/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D3/D2/CS/CX/D8/CX/D3/D2 /DD/CX/CT/D0/CS/D7 /CP/D8 /CT/CP /CW /D4 /D3/CX/D2 /D8/integraldisplay ρ(r,σ)dσ= 1 /B4/BD/BI/B5/CC/CW/CT /D0/D3 /CP/D0/D0/DD /CP /DA /CT/D6/CP/CV/CT/CS /AS/CT/D0/CS /D3/CU /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /CX/D7 /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD/CS/CT/D2/D7/CX/D8 /DD /CX/D2 /D8/CW/CT /CU/D3/D6/D1 q=/integraldisplay ρ(r,σ)σdσ /B4/BD/BJ/B5/CC/CW/CX/D7 /D0/D3 /CP/D0/D0/DD /CP /DA /CT/D6/CP/CV/CT/CS /AS/CT/D0/CS /CX/D7 /CP/D0/D0/CT/CS /D8/CW/CT /D1/CP /D6/D3/D7 /D3/D4/CX /B8 /D3/D6 /D3/CP/D6/D7/CT/B9/CV/D6/CP/CX/D2/CT/CS/B8 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /BA/BT /D1/CP /D6/D3/D7 /D3/D4/CX /D7/D8/CP/D8/CT /CX/D7 /CU/D9/D0/D0/DD /CS/CT/AS/D2/CT/CS /CQ /DDρ(r,σ) /B8 /D8/CW/CT /CW/CT/CX/CV/CW /D8 /AS/CT/D0/CSh(r) /CP/D2/CS /D8/CW/CT /AT/D3 /DB /CS/CX/DA /CT/D6/CV/CT/D2 /CT/B4/CP/D7/D7/D9/D1/CT/CS /DB/CX/D8/CW/D3/D9/D8 /D1/CX /D6/D3/D7 /D3/D4/CX /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/B5/BA /CC/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /AS/CT/D0/CSu(r) /CP/D2 /CQ /CT /CP/D0/D7/D3 /D3/D2/D7/CX/CS/CT/D6/CT/CS/CP/D7 /D7/D1/D3 /D3/D8/CW/B8 /CP/D7 /CX/D8 /CX/D2 /D8/CT/CV/D6/CP/D8/CT/D7 /D8/CW/CT /DA /D3/D6/D8/CX /CX/D8 /DD /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/B8 /CP/D2/CS /CP/D2 /CQ /CT /CS/CT/CS/D9 /CT/CS /CQ /DD /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2/CU/D6/D3/D1 /CX/D8/D7 /CS/CX/DA /CT/D6/CV/CT/D2 /CT /CP/D2/CS /DA /D3/D6/D8/CX /CX/D8 /DDω=qh−2Ω /BA /CC/CW/CT /CT/D2/CT/D6/CV/DD /B4/BK/B5 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2/D0/DD /D3/D2 /D8/CW/CX/D7 /D7/D1/D3 /D3/D8/CW/AS/CT/D0/CS/B8 /DB/CX/D8/CW /D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /CX/D2/AT/D9/CT/D2 /CT /D3/CU /D0/D3 /CP/D0 /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/B8 /D0/CX/CZ /CT /CX/D2 /D8/CW/CT /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CP/D7/CT /CJ/BD/BF℄/BA/CC/CW/CT /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /BV/CP/D7/CX/D1/CX/D6/D7 /B4/BI/B5 /CX/D7 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D8/D3 /D8/CW/CT /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CV/D0/D3/CQ/CP/D0/CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /B4/CX/BA/CT /D8/CW/CT /D8/D3/D8/CP/D0 /CP/D6/CT/CP /D3/CU /CT/CP /CW /D0/CT/DA /CT/D0 /D3/CU /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD/D4 /D3/D2/CS/CT/D6/CP/D8/CT/CS /CQ /DDh /B5/BM γ(σ) =/integraldisplay ρ(r,σ)h(r)d2r /B4/BD/BK/B5/CC/CW/CT /D1/CX /D6/D3/D7 /D3/D4/CX /D1/D3/D1/CT/D2 /D8/D7 /D3/CU /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2/BM Γn=/integraldisplay γ(σ)σndσ=/integraldisplay qnh(r)d2r /B4/BD/BL/B5/DB/CW/CT/D6/CT qn=/integraldisplay ρ(r,σ)σndσ /B4/BE/BC/B5/CP/D2/CS /CP/D6/CT /D3/D2/D7/CT/D6/DA /CT/CS /CS/D9/D6/CX/D2/CV /CP/D2 /CX/D2 /DA/CX/D7 /CX/CS /CT/DA /D3/D0/D9/D8/CX/D3/D2/BA /C6/D3/D8/CT /D8/CW/CP/D8 /CU/D3/D6n≥2 /B8 /D8/CW/CT /D1/CP /D6/D3/D7 /D3/D4/CX /D1/D3/D1/CT/D2 /D8/D7 /D3/CU /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD Γc.g. n=/integraltextqnh(r)d2r /CP/D6/CT /D2/D3/D8 /D3/D2/D7/CT/D6/DA /CT/CS/B8 /CP/D7 /D8/CW/CT/D6/CT /CP/D6/CT /D4/CP/D6/D8/D0/DD/D8/D6/CP/D2/D7/CU/CT/D6/CT/CS /CX/D2 /D8/D3 /AS/D2/CT/B9/CV/D6/CP/CX/D2/CT/CS /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/BA/CC/CW/CT /D1/CX/DC/CX/D2/CV /CT/D2 /D8/D6/D3/D4 /DD S=−/integraldisplay ρ(r,σ)lnρ(r,σ)h(r)d2rdσ /B4/BE/BD/B5/D1/CT/CP/D7/D9/D6/CT/D7 /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D1/CX /D6/D3/D7 /D3/D4/CX /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2/D7 /CP/D7/D7/D3 /CX/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /D7/CP/D1/CT /D1/CP /D6/D3/D7 /D3/D4/CX /AS/CT/D0/CS /D3/CU /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /BA /CC/CW/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /CX/D2ρ /CX/D7 /D8/CW/CT /D7/CP/D1/CT /CP/D7 /CU/D3/D6 /D8/CW/CT /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CP/D7/CT /CJ/BD/BF ℄/B8 /CP/D2/CS /D8/CW/CT /CU/CP /D8/D3/D6h(r) /CX/D7 /CX/D2 /D8/D6/D3 /CS/D9 /CT/CS /D8/D3 /CX/D2/D7/D9/D6/CT /D8/CW/CP/D8 /CT/D2 /D8/D6/D3/D4 /DD /CX/D7 /D3/D2/D7/CT/D6/DA /CT/CS /CQ /DD /D1/CT/D6/CT/BH/D1/CP /D6/D3/D7 /D3/D4/CX /CS/CX/D7/D4/D0/CP /CT/D1/CT/D2 /D8 /D3/CU /AT/D9/CX/CS /D4/CP/D6 /CT/D0/D7/BA /C1/D2/CS/CT/CT/CS /D8/CW/CT /D1/CP/D7/D7 /CT/D0/CT/D1/CT/D2 /D8h(r)d2r /CX/D7 /D3/D2/D7/CT/D6/DA /CT/CS/CQ /DD /AT/D9/CX/CS /D4/CP/D6/D8/CX /D0/CT /CS/CX/D7/D4/D0/CP /CT/D1/CT/D2 /D8 /CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /D7/D9/D6/CU/CP /CT /CT/D0/CT/D1/CT/D2 /D8d2r /CX/D2 /D8/CW/CT /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CP/D7/CT/BA/BT /D8 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1/B8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 /CT/DC/D4 /CT /D8/CT/CS /D8/D3 /CQ /CT /CX/D2 /D8/CW/CT /D1/D3/D7/D8 /D4/D6/D3/CQ/CP/CQ/D0/CT /B4/CX/BA/CT/BA /D1/D3/D7/D8 /D1/CX/DC/CT/CS/B5 /D7/D8/CP/D8/CT /D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D3/D2/D7/D8/D6/CP/CX/D2 /D8/D7 /D3/CU /D8/CW/CT /BX/D9/D0/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/BA /CC/CW/CT /CT/D2 /D8/D6/D3/D4 /DD /B4/BE/BD/B5 /CW/CP/D7 /CQ /CT/CT/D2 /CY/D9/D7/D8/CX/AS/CT/CS/CQ /DD /D6/CX/CV/D3/D6/D3/D9/D7 /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0 /CP/D6/CV/D9/D1/CT/D2 /D8/D7 /B4/CX/D2 /D8/CW/CT /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CP/D7/CT/B5 /CQ/D9/D8 /D3/D8/CW/CT/D6 /CU/D3/D6/D1/D7 /CW/CP /DA /CT /CQ /CT/CT/D2/D6/CT /CT/D2 /D8/D0/DD /D4/D6/D3/D4 /D3/D7/CT/CS /B4/D7/CT/CT /CS/CX/D7 /D9/D7/D7/CX/D3/D2 /CX/D2 /CJ/BD/BH℄/B5/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /DB /CT /D7/CW/CP/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /CP /CV/CT/D2/CT/D6/CP/D0 /CU/D3/D6/D1 /D3/CU/CT/D2 /D8/D6/D3/D4 /DD S=−/integraldisplay s(ρ(r,σ))h(r)d2rdσ /B4/BE/BE/B5/CP/D2/CS /AS/D2/CS /D8/CW/CP/D8 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BD/B5 /CX/D7 /D8/CW/CT /D3/D2/D0/DD /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D0/CT/CP/CS/CX/D2/CV /D8/D3 /CP /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /CT/D2 /D8/D6/D3/D4 /DD /CT/DC/D8/D6/CT/D1 /D9/D1/BA/BF/BA/BE /BY/CX/D6/D7/D8 /D3/D6/CS/CT/D6 /DA /CP/D6/CX/CP/D8/CX/D3/D2/D7/BT /D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /CS/CX/D7 /D9/D7/D7/CX/D3/D2/B8 /D8/CW/CT /D1/D3/D7/D8 /D4/D6/D3/CQ/CP/CQ/D0/CT /D1/CP /D6/D3/D7 /D3/D4/CX /D7/D8/CP/D8/CT /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD/D1/CP/DC/CX/D1/CX/DE/CX/D2/CV /D8/CW/CT /CT/D2 /D8/D6/D3/D4 /DD /B4/BE/BE/B5 /DB/CX/D8/CW /AS/DC/CT/CS /CT/D2/CT/D6/CV/DD /B4/BK/B5/B8 /CV/D0/D3/CQ/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /B4/BD/BK/B5 /CP/D2/CS/D0/D3 /CP/D0 /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 /B4/BD/BI/B5/BA /CC/CW/CX/D7 /D4/D6/D3/CQ/D0/CT/D1 /CX/D7 /D8/D6/CT/CP/D8/CT/CS /CQ /DD /CX/D2 /D8/D6/D3 /CS/D9 /CX/D2/CV /C4/CP/CV/D6/CP/D2/CV/CT /D1 /D9/D0/D8/CX/D4/D0/CX/CT/D6/D7/B8 /D7/D3/D8/CW/CP/D8 /D8/CW/CT /AS/D6/D7/D8 /DA /CP/D6/CX/CP/D8/CX/D3/D2/D7 /D7/CP/D8/CX/D7/CU/DD δS−βδE−/integraldisplay α(σ)δγ(σ)dσ−/integraldisplay ζ(r)δ/parenleftbigg/integraldisplay ρ(r,σ)dσ/parenrightbigg hd2r= 0. /B4/BE/BF/B5/CC/CW/CT /C4/CP/CV/D6/CP/D2/CV/CT /D1 /D9/D0/D8/CX/D4/D0/CX/CT/D6/D7 /CP/D6/CT /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /D8/CW/CT /AG/CX/D2 /DA /CT/D6/D7/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/AH β, /D8/CW/CT /AG /CW/CT/D1/CX /CP/D0 /D4 /D3/D8/CT/D2/B9/D8/CX/CP/D0/AHα(σ) /D3/CU /D7/D4 /CT /CX/CT/D7σ, /CP/D2/CSζ(r) /BA/CF /CT /D7/CW/CP/D0/D0 /D8/CP/CZ /CTh /B8ρh /CP/D2/CS∇ ·u /CP/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /CW/CP/D6/CP /D8/CT/D6/CX/DE/CX/D2/CV /D8/CW/CT /D1/CP /D6/D3/D7 /D3/D4/CX /D7/D8/CP/D8/CT/BA /CC/CW/CT/D2/B8 /CX/D8 /CX/D7 /CT/CP/D7/DD /D8/D3 /CT/D7/D8/CP/CQ/D0/CX/D7/CW/B8 /CQ /DD /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D8/CX/D2/CV /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /B4/BE/BD/B5/B8 /B4/BD/BK/B5 /CP/D2/CS /B4/BD/BI/B5/B8 /D8/CW/CP/D8/BM δS=/integraldisplay [ρs′(ρ)−s(ρ)]δhd2rdσ−/integraldisplay s′(ρ)δ(ρh)d2rdσ /B4/BE/BG/B5 δγ(σ) =/integraldisplay δ(ρh)d2r /B4/BE/BH/B5 hδ/parenleftbigg/integraldisplay ρ(r,σ)dσ/parenrightbigg =/integraldisplay δ(ρh)dσ−/integraldisplay ρδhdσ /B4/BE/BI/B5/CC/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CT/D2/CT/D6/CV/DD /CP/D6/CT /D3/D2 /DA /CT/D2/CX/CT/D2 /D8/D0/DD /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /CU/D9/D2 /D8/CX/D3/D2 B /B8 δE=/integraldisplay Bδhd2r+/integraldisplay hu·δud2r /B4/BE/BJ/B5/CC/CW/CT/D2/B8 /D9/D7/CX/D2/CV /D8/CW/CT /C0/CT/D0/D1/CW/D3/D0/D8/DE /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /B4/BL/B5 /CU/D3/D6 /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D9/D1 hu /B8 /D8/CW/CT /D7/CT /D3/D2/CS /CX/D2 /D8/CT/CV/D6/CP/D0 /CP/D2 /CQ /CT /D6/CT/DB/D6/CX/D8/D8/CT/D2 /integraldisplay hu·δud2r=−/integraldisplay (∇ψ∧δu)·zd2r+/integraldisplay ∇φ·δud2r /B4/BE/BK/B5/C1/D2 /D8/CT/CV/D6/CP/D8/CX/D2/CV /CQ /DD /D4/CP/D6/D8/D7 /DB/CX/D8/CW /D8/CW/CT /CX/CS/CT/D2 /D8/CX/D8/CX/CT/D7 ∇∧(ψδu) =ψ∇∧(δu) +∇ψ∧δu /CP/D2/CS∇·(φδu) = φ∇ ·(δu) +∇φ·δu /B8 /CP/D2/CS /D9/D7/CX/D2/CV /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CU/D3/D6ψ /CP/D2/CSφ /B8 /DB /CT /D3/CQ/D8/CP/CX/D2 /integraldisplay hu·δud2r=/integraldisplay ψδωd2r−/integraldisplay φ∇ ·(δu)d2r /B4/BE/BL/B5/BI/CD/D7/CX/D2/CV /B4/BD/BI/B5 /B4/BG/B5 /CP/D2/CS /B4/BD/BJ/B5/B8 /DB /CT /CW/CP /DA /CT /AS/D2/CP/D0/D0/DD δE=/integraldisplay Bρδhd2rdσ+/integraldisplay ψσδ(ρh)d2rdσ−/integraldisplay φδ(∇ ·u)d2r /B4/BF/BC/B5/CC/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /B4/BE/BF/B5 /DA /CP/D2/CX/D7/CW/CT/D7 /CU/D3/D6 /CP/D2 /DD /D7/D1/CP/D0/D0 /CW/CP/D2/CV/CT/D7 /D3/CU /D8/CW/CT /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /D3/D2/D0/DD /CX/CU /D8/CW/CT /D3 /CTꜶ /CX/CT/D2 /D8/D3/CU /CT/CP /CW /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /DA /CP/D6/CX/CP/CQ/D0/CT /DA /CP/D2/CX/D7/CW/CT/D7/BM δ(ρh) :s′(ρ) =−βσψ−α(σ)−ζ(r) /B4/BF/BD/B5 δh:s′(ρ)−s(ρ) ρ=βB−ζ(r) /B4/BF/BE/B5 δ(∇ ·u) :φ= 0 /B4/BF/BF/B5/C6/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /D6/CX/CV/CW /D8 /CW/CP/D2/CS /D7/CX/CS/CT /D3/CU /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BF/BE/B5 /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CUσ /BA /CC/CW/CX/D7 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D8 /D8/CW/CT/D8/CT/D6/D1 /D3/D2 /D8/CW/CT /D0/CT/CU/D8 /CW/CP/D2/CS /D7/CX/CS/CT /D1 /D9/D7/D8 /CQ /CT /CP /D3/D2/D7/D8/CP/D2 /D8 /B4/D8/CW/CP/D8 /DB /CT /CP/D2 /D8/CP/CZ /CT /CT/D5/D9/CP/D0 /D8/D31 /DB/CX/D8/CW/D3/D9/D8 /D0/D3/D7/D7 /D3/CU/CV/CT/D2/CT/D6/CP/D0/CX/D8 /DD/B5/BM s′(ρ)−s(ρ)/ρ= 1 /BA /CC/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /CT/CP/D7/CX/D0/DD /CX/D2 /D8/CT/CV/D6/CP/D8/CT/CS /CX/D2s(ρ) =Aρ+ρlnρ /DB/CW/CT/D6/CT A /CX/D7 /CP/D2 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /D3/D2/D7/D8/CP/D2 /D8/BA /CF/CW/CT/D2 /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CT/CS /CX/D2 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BE/B5/B8 /D9/D7/CX/D2/CV /B4/BD/BL/B5/B8 /D8/CW/CX/D7 /DD/CX/CT/D0/CS/D7 S=−/integraldisplay ρ(r,σ)lnρ(r,σ)h(r)d2rdσ−AM /B4/BF/BG/B5/DB/CW/CX /CW /CX/D7 /CY/D9/D7/D8 /D8/CW/CT /CT/D2 /D8/D6/D3/D4 /DD /B4/BE/BD/B5 /D9/D4 /D8/D3 /CP/D2 /CP/CS/CS/CX/D8/CX/DA /CT /D3/D2/D7/D8/CP/D2 /D8 /D8/CT/D6/D1AM /B4/DB/CW/CX /CW /DB /CT /CP/D2 /D8/CP/CZ /CT /CT/D5/D9/CP/D0/D8/D3 /DE/CT/D6/D3 /DB/CX/D8/CW/D3/D9/D8 /D0/D3/D7/D7 /D3/CU /CV/CT/D2/CT/D6/CP/D0/CX/D8 /DD/B5/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D8/CW/CT /CT/D2 /D8/D6/D3/D4 /DD /B4/BE/BD/B5 /CX/D7 /D8/CW/CT /D3/D2/D0/DD /CU/D9/D2 /D8/CX/D3/D2/CP/D0 /D3/CU /D8/CW/CT/CU/D3/D6/D1 /B4/BE/BE/B5 /CU/D3/D6 /DB/CW/CX /CW /D8/CW/CT /D1/CP/DC/CX/D1/CX/DE/CP/D8/CX/D3/D2 /D4/D6/D3/CQ/D0/CT/D1 /CW/CP/D7 /CP /D7/D3/D0/D9/D8/CX/D3/D2/BA /CC/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /CX/D7 /CP/D7/D8/D3/D9/D2/CS/CX/D2/CV/CQ /CT /CP/D9/D7/CT /CX/D8 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /DB/CX/D8/CW/D3/D9/D8 /CP/D2 /DD /CT/DC/D4/D0/CX /CX/D8 /D6/CT/CU/CT/D6/CT/D2 /CT /D8/D3 /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX /CP/D0 /CP/D6/CV/D9/D1/CT/D2 /D8/D7/BA/BF/BA/BF /CC/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/D7/CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /CT/DC/D4/D0/CX /CX/D8/CT/D0/DDs(ρ) =ρlnρ /CX/D2 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BF/BD/B5/B8 /D8/CW/CT /D3/D4/D8/CX/D1/CP/D0 /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CT/D2/D7/CX/D8 /DD /CP/D2/CQ /CT /CT/DC/D4/D6/CT/D7/D7/CT/CS /CP/D7 ρ(r,σ) =1 Z(ψ)g(σ)e−βσψ/B4/BF/BH/B5/DB/CW/CT/D6/CTZ(ψ)≡eζ(r)+1/CP/D2/CSg(σ)≡e−α(σ)/BA /CC/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D3/D2/CS/CX/D8/CX/D3/D2 /B4/BD/BI/B5 /D0/CT/CP/CS/D7 /D8/D3 /CP /DA /CP/D0/D9/CT/D3/CU /D8/CW/CT /D4/CP/D6/D8/CX/D8/CX/D3/D2 /CU/D9/D2 /D8/CX/D3/D2Z /D3/CU /D8/CW/CT /CU/D3/D6/D1 Z=/integraldisplay g(σ)e−βσψdσ /B4/BF/BI/B5/CP/D2/CS /D8/CW/CT /D0/D3 /CP/D0/D0/DD /CP /DA /CT/D6/CP/CV/CT/CS /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /B4/BD/BJ/B5 /CX/D7 /CT/DC/D4/D6/CT/D7/D7/CT/CS /CP/D7 /CP /CU/D9/D2 /D8/CX/D3/D2 /D3/CUψ /CP /D3/D6/CS/CX/D2/CV/D8/D3/BM q=/integraltextg(σ)σe−βσψdσ/integraltextg(σ)e−βσψdσ=F(ψ) /B4/BF/BJ/B5/CC/CW/CX/D7 /CP/D2 /CQ /CT /D6/CT/DB/D6/CX/D8/D8/CT/D2 q=−1 βdlnZ dψ /B4/BF/BK/B5/CC/CW/CX/D7 /CX/D7 /D8/CW/CT /D7/CP/D1/CT /CU/D9/D2 /D8/CX/D3/D2/CP/D0 /D6/CT/D0/CP/D8/CX/D3/D2 /CP/D7 /CX/D2 /D8/CW/CT /CP/D7/CT /D3/CU /BE/BW /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /BX/D9/D0/CT/D6 /AT/D3 /DB/D7 /CJ/BD/BF ℄/BA/BJ/BW/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D8/CX/D2/CV /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BF/BJ/B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3ψ /B8 /DB /CT /CW/CT /CZ /D8/CW/CP/D8 /D8/CW/CT /DA /CP/D6/CX/CP/D2 /CT /D3/CU /D8/CW/CT /D4 /D3/B9/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 q2≡q2−q2=−1 βF′(ψ) /B4/BF/BL/B5/D3/D6/B8 /CP/D0/D8/CT/D6/D2/CP/D8/CX/DA /CT/D0/DD /B4/D7/CT/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BF/BK/B5/B5/BM q2=1 β2d2lnZ dψ2 /B4/BG/BC/B5/CC/CW/CT/D6/CT/CU/D3/D6/CT /D8/CW/CT /D7/D0/D3/D4 /CT /D3/CU /D8/CW/CT /CU/D9/D2 /D8/CX/D3/D2q=F(ψ) /CX/D7 /CS/CX/D6/CT /D8/D0/DD /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /DA /CP/D6/CX/CP/D2 /CT /D3/CU /D8/CW/CT/DA /D3/D6/D8/CX /CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/BA /CA/CT/D0/CP/D8/CX/D3/D2 /B4/BF/BL/B5 /CW/CP/D7 /D8/CW/CT /D7/CP/D1/CT /CU/D3/D6/D1 /CP/D2/CS /D3/D6/CX/CV/CX/D2 /CP/D7 /D8/CW/CT /AG/AT/D9 /D8/D9/CP/D8/CX/D3/D2/B9/CS/CX/D7/D7/CX/D4/CP/D8/CX/D3/D2/AH /D8/CW/CT/D3/D6/CT/D1 /CX/D2 /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /AS/CT/D0/CS /D8/CW/CT/D3/D6/DD /B8 /DB/CW/CT/D6/CTdq/dψ /CX/D7 /CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS /CP/D7 /CP /D7/D9/D7 /CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /BA/CB/CX/D2 /CTq2>0 /B8 /DB /CT /AS/D2/CS /D8/CW/CP/D8 /D8/CW/CT /CU/D9/D2 /D8/CX/D3/D2q=F(ψ) /CX/D7 /D1/D3/D2/D3/D8/D3/D2/CX /BN /CX/D8 /CX/D7 /CS/CT /D6/CT/CP/D7/CX/D2/CV /CU/D3/D6β >0/CP/D2/CS /CX/D2 /D6/CT/CP/D7/CX/D2/CV /CU/D3/D6β <0 /B4/CX/D8 /CX/D7 /D3/D2/D7/D8/CP/D2 /D8 /CU/D3/D6β= 0 /B5/BA /BT/D2/D3/D8/CW/CT/D6 /D4/D6/D3 /D3/CU /D3/CU /D8/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /CX/D7 /CV/CX/DA /CT/D2 /CX/D2/CJ/BD/BF ℄/BA/CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /CT/DC/D4/D0/CX /CX/D8/CT/D0/DDs′(ρ)−s(ρ)/ρ= 1 /CX/D2 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BF/BE/B5/B8 /DB /CT /CW/CP /DA /CT B=1 βlnZ /B4/BG/BD/B5/CC/CW/CX/D7 /D6/CT/D0/CP/D8/CX/D3/D2 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /CU/D9/D2 /D8/CX/D3/D2 /D4/D0/CP /DD/D7 /D8/CW/CT /D6/D3/D0/CT /D3/CU /CP /CU/D6/CT/CT /CT/D2/CT/D6/CV/DD /CX/D2 /D8/CW/CT /D7/D8/CP/B9/D8/CX/D7/D8/CX /CP/D0 /D8/CW/CT/D3/D6/DD /BA /CF /CT /D2/D3/D8/CT /D8/CW/CP/D8 /CQ /D3/D8/CWB /CP/D2/CSq /CP/D6/CT /CU/D9/D2 /D8/CX/D3/D2/D7 /D3/CUψ /B8 /DB/CW/CX/D0/CTφ= 0 /B8 /CU/D6/D3/D1 /B4/BF/BF/B5/B8/CP/D7 /CX/D8 /D7/CW/D3/D9/D0/CS /CU/D3/D6 /D7/D8/CT/CP/CS/DD /AT/D3 /DB/D7/BA /BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT/B8 /D8/CP/CZ/CX/D2/CV /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT /D3/CU /B4/BG/BD/B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3ψ/CP/D2/CS /D9/D7/CX/D2/CV /B4/BF/BK/B5/B8 /DB /CT /CW/CT /CZ /D8/CW/CP/D8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2q=−dB/dψ /D6/CT/D5/D9/CX/D6/CT/CS /CU/D3/D6 /CP /D7/D8/CT/CP/CS/DD /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU/D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /D7/CP/D8/CX/D7/AS/CT/CS/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT /D8/CW/CT /AT/D3 /DB /D7/CT/D0/CT /D8/CT/CS /CQ /DD /D8/CW/CT /D4/D9/D6/CT/D0/DD /D7/D8/CP/D8/CX/D7/D8/CX/B9 /CP/D0 /CT/D2 /D8/D6/D3/D4 /DD /D1/CP/DC/CX/D1/CX/DE/CP/D8/CX/D3/D2 /D4/D6/D3 /CT/CS/D9/D6/CT /CS/D3 /CT/D7 /D2/D3/D8 /CT/DA /D3/D0/DA /CT /CP/D2 /DD/D1/D3/D6/CT /CQ /DD /D8/CW/CT /AT/D3 /DB /CT/DA /D3/D0/D9/D8/CX/D3/D2/B8 /D7/D3 /D8/CW/CT/D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /D8/CW/CT/D3/D6/DD /CX/D7 /CX/D2/CS/CT/CT/CS /D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /CS/DD/D2/CP/D1/CX /D7/BA/BG /C8/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /D8/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/D7 /CX/D2 /D7/D3/D1/CT /D4/CP/D6/D8/CX /D9/D0/CP/D6 /CP/D7/CT/D7/BM/BG/BA/BD /C8 /CP/D6/D8/CX /D9/D0/CP/D6 q−ψ /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/D7/BM/CC/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/D7 /CP/D6/CT /CW/CP/D6/CP /D8/CT/D6/CX/DE/CT/CS /CQ /DD /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BF/BJ/B5 /CQ /CT/D8 /DB /CT/CT/D2q /CP/D2/CSψ /B8 /DB/CW/CX /CW /CX/D7 /CP/D0/DB /CP /DD/D7/D1/D3/D2/D3/D8/D3/D2/CX /B8 /CP/D7 /D7/CW/D3 /DB/D2 /CX/D2 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CT /D8/CX/D3/D2/BA /C1/D8 /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D0/CP /DB/D7/B8/CQ/D9/D8 /D3/D2/D0/DD /CX/D2/CS/CX/D6/CT /D8/D0/DD /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /D7/CT/D8 /D3/CU /C4/CP/CV/D6/CP/D2/CV/CT /D1 /D9/D0/D8/CX/D4/D0/CX/CT/D6/D7 β /CP/D2/CSα(σ) /BA /C1/D2 /D4/D6/CP /D8/CX /CT /DB /CT/D2/CT/CT/CS /D8/D3 /CS/CX/D7 /D6/CT/D8/CX/DE/CT /D8/CW/CT /C8/CE /D0/CT/DA /CT/D0/D7/B8 /CP/D2/CS /CZ /CT/CT/D4/CX/D2/CV /D3/D2/D0/DD /D8 /DB /D3 /D0/CT/DA /CT/D0/D7/B8q=σ0 /CP/D2/CSq=σ1 /CX/D7 /D3/CU/D8/CT/D2/D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/DA /CT /D3/CU /D1/D3/D6/CT /CV/CT/D2/CT/D6/CP/D0 /CP/D7/CT/D7/BA /CC/CW/CT/D2 /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 ρ /CY/D9/D7/D8 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2/CP /D7/CX/D2/CV/D0/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD p1 /D3/CU /AS/D2/CS/CX/D2/CV /D8/CW/CT /D0/CT/DA /CT/D0σ1 /B4/CU/D3/D6 /CX/D2/D7/D8/CP/D2 /CT/B5/B8 /DB/CX/D8/CW /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD 1−p1/D3/CU /AS/D2/CS/CX/D2/CV /D8/CW/CT /D3/D1/D4/D0/CT/D1/CT/D2 /D8/CP/D6/DD /D0/CT/DA /CT/D0σ0 /B8 /CX/BA/CT/BAg(σ) /CX/D7 /D8/CW/CT /D7/D9/D1 /D3/CU /D8 /DB /D3 /BW/CX/D6/CP /CU/D9/D2 /D8/CX/D3/D2 /D8/CT/D6/D1/D7/B8 g(σ) =g1[λδ(σ−σ0) +δ(σ−σ1)] /BA /CC/CW/CX/D7 /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD p1 /CX/D7 /CS/CX/D6/CT /D8/D0/DD /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /C8/CE /CP /DA /CT/D6/CP/CV/CT/CQ /DDq=p1σ1+ (1−p1)σ0 /B8 /D3/D6 /D6/CT/DA /CT/D6/D7/CT/D0/DDp1= (q−σ0)/(σ1−σ0) /BA /CC/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BF/BJ/B5 /CU/D3/D6q/D6/CT/CS/D9 /CT/D7 /D8/D3 q=σ0+(σ1−σ0) 1 +λeβ(σ1−σ0)ψ /B4/BG/BE/B5/BK/CC/CW/CX/D7 /D6/CT/D0/CP/D8/CX/D3/D2 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /BY /CT/D6/D1/CX/B9/BW/CX/D6/CP /D7/D8/CP/D8/CX/D7/D8/CX /D7/BA /CC/CW/CT /D8 /DB /D3 /D9/D2/CZ/D2/D3 /DB/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 λ /CP/D2/CS β /CP/D6/CT /CX/D2/CS/CX/D6/CT /D8/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D3/D2/D7/CT/D6/DA /CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /CC/CW/CT /CP/D7/D7/D3 /CX/CP/D8/CT/CS /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /CU/D9/D2 /D8/CX/D3/D2/B4/BG/BD/B5 /CQ /CT /D3/D1/CT/D7 B=1 βlng1−σ0ψ+1 βln/braceleftbigg λ+eβ(σ0−σ1)ψ/bracerightbigg/B4/BG/BF/B5/CC/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /CX/D7 /CP/D0/D7/D3 /CV/D6/CT/CP/D8/D0/DD /D7/CX/D1/D4/D0/CX/AS/CT/CS /CX/D2 /D8/CW/CT /CP/D0/D8/CT/D6/D2/CP/D8/CX/DA /CT /CP/D7/CT /CU/D3/D6 /DB/CW/CX /CWg(σ) /CX/D7 /CP /BZ/CP/D9/D7/D7/CX/CP/D2/BM g(σ) =g0e−(σ−σ∗)2 2σ2. /B4/BG/BG/B5/CC/CW/CT/D2 /D8/CW/CT /D0/D3 /CP/D0 /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /B4/BF/BH/B5 /CX/D7 /CP/D0/D7/D3 /CP /BZ/CP/D9/D7/D7/CX/CP/D2/B8 /CP/D2/CS /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /CU/D9/D2 /D8/CX/D3/D2 /B4/BG/BD/B5 /CX/D7 B=1 βln[g0(2πσ2)1/2] +1 2σ2βψ2−σ∗ψ, /B4/BG/BH/B5 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /CP /D0/CX/D2/CT/CP/D6 /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/BM q=−βσ2ψ+σ∗ /B4/BG/BI/B5/BT /D3/D6/CS/CX/D2/CV /D8/D3 /B4/BF/BL/B5 /D8/CW/CT /DA /CP/D6/CX/CP/D2 /CT /D3/CU /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /CX/D7 /D8/CW/CT/D2 /D9/D2/CX/CU/D3/D6/D1/B8 /DB/CX/D8/CW /DA /CP/D0/D9/CTq2= σ2 /B4/D1/D3/D6/CT /CV/CT/D2/CT/D6/CP/D0/D0/DD /B8 /CP/D0/D0 /D8/CW/CT /CT/DA /CT/D2 /D1/D3/D1/CT/D2 /D8/CP /D3/CU /D8/CW/CT /BZ/CP/D9/D7/D7/CX/CP/D2 /CP/D6/CT /D6/CT/D0/CP/D8/CT/CS /D8/D3σ2 /CQ /DD(q−q)2n= (2n−1)!!σn 2 /CP/D2/CS /D8/CW/CT /D3 /CS/CS /D1/D3/D1/CT/D2 /D8/CP /CP/D2 /CT/D0/B5/BA/CC/CW/CX/D7 /BZ/CP/D9/D7/D7/CX/CP/D2 /D0/D3 /CP/D0 /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D1/CP/DC/CX/D1/CX/DE/CX/D2/CV /D8/CW/CT /CT/D2 /D8/D6/D3/D4 /DD /B4/BE/BD/B5/B8/D6/CT/CS/D9 /CX/D2/CV /D8/CW/CT /D3/D2/D7/D8/D6/CP/CX/D2 /D8/D7 /D3/CU /D8/CW/CT /CV/D0/D3/CQ/CP/D0 /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 γ(σ) /D8/D3 /CX/D8/D7 /AS/D6/D7/D8 /D1/D3/D1/CT/D2 /D8/D7 Γ0≡M /B8Γ1/CP/D2/CSΓ2 /BA /CC/CW/CX/D7 /DB/CX/D0/D0 /CQ /CT /D8/CW/CT /D8/D6/D9/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT /CU/D3/D6 /CP /D4/CP/D6/D8/CX /D9/D0/CP/D6 /CX/D2/CX/D8/CX/CP/D0 /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 γ(σ) /DB/CX/D8/CW/CW/CX/CV/CW/CT/D6 /D3/D6/CS/CT/D6 /D1/D3/D1/CT/D2 /D8/D7 /CT/D5/D9/CP/D0 /D8/D3 /D8/CW/CT /CV/D0/D3/CQ/CP/D0 /D1/D3/D1/CT/D2 /D8/D7 /D3/CU /D8/CW/CX/D7 /D7/CX/D1/D4/D0/CX/AS/CT/CS /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/BA /BT /D0/CX/D2/CT/CP/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4 /CQ /CT/D8 /DB /CT/CT/D2q /CP/D2/CSψ /CP/D2 /CP/D0/D7/D3 /CQ /CT /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6 /CP/D2 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 γ(σ) /CX/D2 /D8/CW/CT /D0/CX/D1/CX/D8 /D3/CU/D7/D8/D6/D3/D2/CV /D1/CX/DC/CX/D2/CV /DB/CW/CT/D6/CTβσψ≪1 /B8 /D7/D3 /D8/CW/CP/D8 /B4/BF/BJ/B5 /CP/D2 /CQ /CT /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS/B8 /CP/D7 /CS/CX/D7 /D9/D7/D7/CT/CS /CX/D2 /CJ/BE/BG ℄ /BV/CW/CP /DA /CP/D2/CX/D7/B2 /CB/D3/D1/D1/CT/D6/CX/CP /B4/BD/BL/BL/BI/B5/BA/BG/BA/BE /CD/D2/CX/CS/CX/D6/CT /D8/CX/D3/D2/CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2/D7/CF /CT /D3/D2/D7/CX/CS/CT/D6 /CW/CT/D6/CT /D9/D2/CX/CS/CX/D6/CT /D8/CX/D3/D2/CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D7/D9 /CW /D8/CW/CP/D8u=u(y)ex /BA /CC/CW/CT /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D6/CT/D0/CP/D8/CX/D3/D2 q=−dB/dψ /D8/CW/CT/D2 /DD/CX/CT/D0/CS/D7/B8 /D1 /D9/D0/D8/CX/D4/D0/DD/CX/D2/CV /CT/CP /CW /D1/CT/D1 /CQ /CT/D6 /CQ /DDhu=dψ/dy /B8 gdh dy+2Ω hdψ dy= 0. /B4/BG/BJ/B5/CC/CW/CX/D7 /D3/D2/CS/CX/D8/CX/D3/D2 /D3/CU /CV/CT/D3/D7/D8/D6/D3/D4/CW/CX /CQ/CP/D0/CP/D2 /CT /CP/D2 /CQ /CT /D6/CT/CP/CS/CX/D0/DD /CX/D2 /D8/CT/CV/D6/CP/D8/CT/CS /CX/D2 h=H/radicalBigg 1−4Ω gH2ψ /B4/BG/BK/B5/BT /D7/CT /D3/D2/CS /D6/CT/D0/CP/D8/CX/D3/D2 /CX/D7 /D4/D6/D3 /DA/CX/CS/CT/CS /CQ /DD /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CU /D8/CW/CT /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /CU/D9/D2 /D8/CX/D3/D2 /DD/CX/CT/D0/CS/CX/D2/CV 1 2h2/parenleftbiggdψ dy/parenrightbigg2 =B(ψ)−gh /B4/BG/BL/B5/BL/BV/D3/D1 /CQ/CX/D2/CT/CS /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BG/BK/B5 /D3/CUh /CP/D2/CS /D8/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BG/BD/B5 /D3/CUB(ψ) /B8/D8/CW/CX/D7 /DD/CX/CT/D0/CS/D7 /CP /AS/D6/D7/D8 /D3/D6/CS/CT/D6 /D3/D6/CS/CX/D2/CP/D6/DD /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6ψ. /CC/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D8/CW/CT /D3/D2/D7/D8/CP/D2 /D8H /CP/D2/CS /C4/CP/CV/D6/CP/D2/CV/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/B8 /CU/D3/D6 /CX/D2/D7/D8/CP/D2 /CTg1, β /CP/D2/CSλ /CX/D2 /D8/CW/CT /CP/D7/CT /DB/CX/D8/CW /D8 /DB /D3 /C8/CE/D0/CT/DA /CT/D0/D7/BA /CC/CW/CT /D3/D2/D7/D8/D6/CP/CX/D2 /D8/D7 /D3/D2 /D8/D3/D8/CP/D0 /D1/CP/D7/D7M /B8 /CV/D0/D3/CQ/CP/D0 /D1/CP/D7/D7 /D3/CU /C8/CE /D0/CT/DA /CT/D0σ1 /CP/D2/CS /D8/D3/D8/CP/D0 /CT/D2/CT/D6/CV/DDE /B8/CX/D2/CS/CX/D6/CT /D8/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA /C6/D3/D8/CT /D8/CW/CP/D8 /CX/D2 /D6/CT/CP/D0/CX/D8 /DD /D8/CW/CX/D7 /D7/D3/D0/D9/D8/CX/D3/D2 /D1 /D9/D7/D8 /CQ /CT /DA/CX/CT/DB /CT/CS/CX/D2 /CP /DC/B9/DB/CX/D7/CT /D8/D6/CP/D2/D7/D0/CP/D8/CX/D2/CV /CU/D6/CP/D1/CT /D3/CU /D6/CT/CU/CT/D6/CT/D2 /CT/B8 /CP/D7 /CS/CX/D7 /D9/D7/D7/CT/CS /CX/D2 /D7/CT /D8/CX/D3/D2 /BI/B8 /CS/D9/CT /D8/D3 /D8/CW/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D0/CP /DB /CU/D3/D6 /D1/D3/D1/CT/D2 /D8/D9/D1/BA/BG/BA/BF /BT/DC/CX/D7/DD/D1/D1/CT/D8/D6/CX /D7/D3/D0/D9/D8/CX/D3/D2/D7/BY /D3/D6 /CP/DC/CX/D7/DD/D1/D1/CT/D8/D6/CX /D7/D3/D0/D9/D8/CX/D3/D2/D7 u=uθ(r)eθ /DB/CW/CT/D6/CT (r,θ) /CP/D6/CT /D4 /D3/D0/CP/D6 /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7/B8 /CP/D2/CShuθ= −dψ/dr /BA /CC/CW/CT/D2 /BX/D5/D9/CP/D8/CX/D3/D2 /B4/BG/BJ/B5 /CX/D7 /D6/CT/D4/D0/CP /CT/CS /CQ /DD /D8/CW/CT /DD /D0/D3/D7/D8/D6/D3/D4/CW/CX /CQ/CP/D0/CP/D2 /CT ghdh dr=1 hr/parenleftbiggdψ dr/parenrightbigg2 −2Ωdψ dr /B4/BH/BC/B5/CF/CW/CT/D2huθ=−dψ/dr ≥0 /B4 /DD /D0/D3/D2/CT/B5/B8h /CX/D7 /CP/D2 /CX/D2 /D6/CT/CP/D7/CX/D2/CV /CU/D9/D2 /D8/CX/D3/D2 /D3/CUr /B8 /D7/D3 /D8/CW/CT /DA /D3/D6/D8/CT/DC /D3/D6/CT/CX/D7 /CP /D8/D6/D3/D9/CV/CW/BA /C1/D2 /D8/CW/CT /D3/D4/D4 /D3/D7/CX/D8/CT /CP/D7/CTuθ≤0 /B4/CP/D2 /D8/CX /DD /D0/D3/D2/CT/B5 /B8 /D8/CW/CT /DA /D3/D6/D8/CT/DC /D3/D6/CT /CX/D7 /CP /CQ/D9/D1/D4 /CX/D2/CV/CT/D3/D7/D8/D6/D3/D4/CW/CX /D6/CT/CV/CX/D1/CT/D7/BA /C0/D3 /DB /CT/DA /CT/D6 /CU/D3/D6 /D0/CP/D6/CV/CT /DA /CT/D0/D3 /CX/D8/CX/CT/D7 /B4/CA/D3/D7/D7/CQ /DD /D2 /D9/D1 /CQ /CT/D6 /D0/CP/D6/CV/CT/D6 /D8/CW/CP/D2 /D9/D2/CX/D8 /DD/B5/B8 /D8/CW/CT/D8/CT/D6/D1u2 θ r /CS/D3/D1/CX/D2/CP/D8/CT/D7/B8 /D0/CT/CP/CS/CX/D2/CV /CP/D0/DB /CP /DD/D7 /D8/D3 /CP /D8/D6/D3/D9/CV/CW/BA/CC/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CU /D8/CW/CT /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /D3/D2/D7/D8/CP/D2 /D8 /CV/CX/DA /CT/D7 /CP/CV/CP/CX/D2 /D8/CW/CT /CU/D3/D6/D1 /B4/BG/BL/B5/B8 /CY/D9/D7/D8 /D6/CT/D4/D0/CP /CX/D2/CVy /CQ /DD/D8/CW/CT /D6/CP/CS/CX/D9/D7r /BA /BV/D3/D1 /CQ/CX/D2/CX/D2/CV /D8/CW/CX/D7 /D6/CT/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CW /B4/BH/BC/B5/B8 /D3/D2/CT /CV/CT/D8/D7 /CP /D3/D9/D4/D0/CT /D3/CU /AS/D6/D7/D8 /D3/D6/CS/CT/D6 /D3/D6/CS/CX/D2/CP/D6/DD/CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /DA /CP/D6/CX/CP/CQ/D0/CT/D7ψ /CP/D2/CSh. /BT/D7 /CX/D2 /D8/CW/CT /D9/D2/CX/CS/CX/D6/CT /D8/CX/D3/D2/CP/D0 /CP/D7/CT/B8 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2/CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D8 /DB /D3 /D3/D2/D7/D8/CP/D2 /D8/D7 /D3/CU /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /C4/CP/CV/D6/CP/D2/CV/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/B8 /DB/CW/CX /CW /CP/D6/CTg1,β /CP/D2/CS λ /CX/D2 /D8/CW/CT /CP/D7/CT /DB/CX/D8/CW /D8 /DB /D3 /C8/CE /D0/CT/DA /CT/D0/D7/BA /CC/CW/CX/D7 /D7/D3/D0/D9/D8/CX/D3/D2 /D1 /D9/D7/D8 /CQ /CT /DA/CX/CT/DB /CT/CS /CX/D2 /CV/CT/D2/CT/D6/CP/D0 /CX/D2 /CP /D6/D3/D8/CP/D8/CX/D2/CV/CU/D6/CP/D1/CT /D3/CU /D6/CT/CU/CT/D6/CT/D2 /CT/B8 /CS/D9/CT /D8/D3 /D8/CW/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /CP/D2/CV/D9/D0/CP/D6 /D1/D3/D1/CT/D2 /D8/D9/D1 /B8 /CP/D7 /CS/CX/D7 /D9/D7/D7/CT/CS/CX/D2 /D7/CT /D8/CX/D3/D2 /BI/BA/BH /CA/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/D7/BH/BA/BD /CC/CW/CT /C5/CP/DC/CX/D1 /D9/D1 /BX/D2 /D8/D6/D3/D4 /DD /C8/D6/D3 /CS/D9 /D8/CX/D3/D2 /C8/D6/CX/D2 /CX/D4/D0/CT/CA/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D1/CT/D8/CW/D3 /CS/D7 /CP/D6/CT /D3/D2 /DA /CT/D2/CX/CT/D2 /D8 /D8/D3 /D3/D1/D4/D9/D8/CT /D8/CW/CT /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D6/D3/D1/CP/D2 /DD /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2/BA /CC/CW/CT /CP/CX/D1 /CX/D7 /D8/D3 /CX/D2 /D6/CT/CP/D7/CT /D8/CW/CT /CT/D2 /D8/D6/D3/D4 /DD /CX/D2 /D7/D9 /CT/D7/D7/CX/DA /CT /D7/D8/CT/D4/D7 /DB/CW/CX/D0/CT /CZ /CT/CT/D4/CX/D2/CV /D3/D2/D7/D8/CP/D2 /D8 /CP/D0/D0 /D8/CW/CT /D3/D2/D7/CT/D6/DA /CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /CJ/BE/BH ℄ /CC /D9/D6/CZ/CX/D2/CV/D8/D3/D2 /B2 /CF/CW/CX/D8/CP/CZ /CT/D6 /B4/BD/BL/BL/BI/B5 /CW/CP /DA /CT /CX/D1/D4/D0/CT/D1/CT/D2 /D8/CT/CS/CP /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D1/CT/D8/CW/D3 /CS /D8/D3 /CP/D0 /D9/D0/CP/D8/CT /D8/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /DB/CX/D8/CW /D8/CW/CT /BX/D9/D0/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /CJ/BE/BI ℄/CA/D3/CQ /CT/D6/D8 /CP/D2/CS /CB/D3/D1/D1/CT/D6/CX/CP /B4/BD/BL/BL/BE/B5 /CW/CP/CS /D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /D4/D6/D3/D4 /D3/D7/CT/CS /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /CU/D3/D6/D1 /D3/CU/CP /D4/CP/D6/CP/D1/CT/D8/CT/D6/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D7/D9/CQ/B9/CV/D6/CX/CS /D7 /CP/D0/CT /CT/CS/CS/CX/CT/D7 /DB/CW/CX /CW /CS/D6/CX/DA /CT/D7 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /D8/D3 /DB /CP/D6/CS /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /CT/D5/D9/CX/B9/D0/CX/CQ/D6/CX/D9/D1 /CQ /DD /CP /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /D8/CX/D1/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2/BA /CB/D9 /CW /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D2 /CQ /CT /D9/D7/CT/CS /CQ /D3/D8/CW /CP/D7 /CP/D6/CT/CP/D0/CX/D7/D8/CX /D3/CP/D6/D7/CT /D6/CT/D7/D3/D0/D9/D8/CX/D3/D2 /D1/D3 /CS/CT/D0 /D3/CU /D8/CW/CT /D8/D9/D6/CQ/D9/D0/CT/D2 /D8 /CT/DA /D3/D0/D9/D8/CX/D3/D2/B8 /CP/D2/CS /CP/D7 /CP /D1/CT/D8/CW/D3 /CS /D3/CU /CS/CT/D8/CT/D6/D1/CX/D2/CP/B9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CX/D7 /CT/DA /D3/D0/D9/D8/CX/D3/D2 /B4/D7/CT/CT /CJ/BF℄ /CU/D3/D6 /CP /D7/CW/D3/D6/D8 /D6/CT/DA/CX/CT/DB/B5/BA/CF /CT /CW/CT/D6/CT /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT /D8/CW/CX/D7 /CP/D4/D4/D6/D3/CP /CW /D8/D3 /D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1/BA/CF /CT /AS/D6/D7/D8 /CS/CT /D3/D1/D4 /D3/D7/CT /D8/CW/CT /DA /D3/D6/D8/CX /CX/D8 /DDω /CP/D2/CS /DA /CT/D0/D3 /CX/D8 /DD u /CX/D2 /D8/D3 /CP /D1/CT/CP/D2 /CP/D2/CS /AT/D9 /D8/D9/CP/D8/CX/D2/CV /D4/CP/D6/D8/B8/D2/CP/D1/CT/D0/DDω=ω+ ˜ω /B8u=u+˜u /B8 /CZ /CT/CT/D4/CX/D2/CVh /D7/D1/D3 /D3/D8/CW/BA /CC /CP/CZ/CX/D2/CV /D8/CW/CT /D0/D3 /CP/D0 /CP /DA /CT/D6/CP/CV/CT /D3/CU /D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB/DB /CP/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BD/B5/B4/BE/B5/B8 /DB /CT /CV/CT/D8 ∂h ∂t+∇ ·(hu) = 0 /B4/BH/BD/B5/BD/BC∂u ∂t+ (ω ωω+ 2Ω ΩΩ)∧u=−∇B−ez∧Jω /B4/BH/BE/B5 B=gh+u2 2 /B4/BH/BF/B5/DB/CW/CT/D6/CT /D8/CW/CT /D9/D6/D6/CT/D2 /D8 Jω=˜ω˜u /D6/CT/D4/D6/CT/D7/CT/D2 /D8/D7 /D8/CW/CT /D3/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /AS/D2/CT/B9/CV/D6/CP/CX/D2/CT/CS /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/BA/BT/D0/D8/CW/D3/D9/CV/CW /DB /CT /CW/CP /DA /CT /D2/CT/CV/D0/CT /D8/CT/CS /D8/CW/CT /AT/D9 /D8/D9/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD ˜u2/CX/D2 /CU/D6/D3/D2 /D8 /D3/CUu2/B4/CP/D7 /DB /CT/D0/D0 /CP/D7 /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/D3/CUh /B5/B8 /DB /CT /CZ /CT/CT/D4 /D8/CW/CT /D3/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7 Jω=˜ω˜u /B8 /DB/CW/CX /CW /D6/CT/D4/D6/CT/D7/CT/D2 /D8 /D8/CW/CT /C8/CE /D8/D6/CP/D2/D7/D4 /D3/D6/D8 /CQ /DD /D7/D9/CQ/B9/CV/D6/CX/CS/B9/D7 /CP/D0/CT /CT/CS/CS/CX/CT/D7/BA /CC/CW/CX/D7 /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /CX/D7 /CY/D9/D7/D8/CX/AS/CT/CS /D7/CX/D2 /CT/B8 /CS/CT/D2/D3/D8/CX/D2/CVǫ /D8/CW/CT /D8 /DD/D4/CX /CP/D0 /D7 /CP/D0/CT /D3/CU /DA /D3/D6/D8/CX /CX/D8 /DD/AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/B8 /DB /CT /CW/CP /DA /CT˜u2∼ǫ2ω2/CP/D2/CS˜ω˜u∼ǫω2≫˜u2/B4/DB/CW/CX/D0/CT ˜ω∼ω /B5/BA/CF /CT /CS/CT/CS/D9 /CT /CP/D2 /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /B4/BG/B5/B8 /D8/CP/CZ/CX/D2/CV /D8/CW/CT /D9/D6/D0/D3/CU /B4/BH/BE/B5 /CP/D2/CS /D9/D7/CX/D2/CV /B4/BH/BD/B5/B8 ∂ ∂t(hq) +∇.(hqu) =−∇.Jω /B4/BH/BG/B5/CC/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D2 /CQ /CT /DA/CX/CT/DB /CT/CS /CP/D7 /CP /D0/D3 /CP/D0 /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D0/CP /DB /CU/D3/D6 /D8/CW/CT /CX/D6 /D9/D0/CP/D8/CX/D3/D2 Γ =/integraltextqhd2r /BA/CF /CT /D7/CW/CP/D0/D0 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /D9/D2/CZ/D2/D3 /DB/D2 /D9/D6/D6/CT/D2 /D8 Jω /CQ /DD /D8/CW/CT /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX /D4/D6/CX/D2 /CX/D4/D0/CT /D3/CU /C5/CP/DC/CX/D1 /D9/D1/BX/D2 /D8/D6/D3/D4 /DD /C8/D6/D3 /CS/D9 /D8/CX/D3/D2 /B4/C5/BX/C8/B5 /CJ/BE/BI℄/BA /BY /D3/D6 /D8/CW/CP/D8 /D4/D9/D6/D4 /D3/D7/CT/B8 /DB /CT /D2/CT/CT/CS /D8/D3 /D3/D2/D7/CX/CS/CT/D6 /D2/D3/D8 /D3/D2/D0/DD /D8/CW/CT /D0/D3 /CP/D0/D0/DD/CP /DA /CT/D6/CP/CV/CT /C8/CE /AS/CT/D0/CSq /B8 /CQ/D9/D8 /D8/CW/CT /DB/CW/D3/D0/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 ρ(r,σ,t) /D2/D3 /DB /CT/DA /D3/D0/DA/CX/D2/CV /DB/CX/D8/CW /D8/CX/D1/CT t /BA /CC/CW/CT /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CV/D0/D3/CQ/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 γ(σ) =/integraltextρhd2r /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CX/D2/D8/CW/CT /D0/D3 /CP/D0 /CU/D3/D6/D1∂ ∂t(hρ) +∇.(hρu) =−∇.J /B4/BH/BH/B5/DB/CW/CT/D6/CT J(r,σ,t) /CX/D7 /D8/CW/CT /B4/D9/D2/CZ/D2/D3 /DB/D2/B5 /D9/D6/D6/CT/D2 /D8 /CP/D7/D7/D3 /CX/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /D0/CT/DA /CT/D0σ /D3/CU /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /BA/C1/D2 /D8/CT/CV/D6/CP/D8/CX/D2/CV /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BH/BH/B5 /D3 /DA /CT/D6 /CP/D0/D0 /D8/CW/CT /C8/CE /D0/CT/DA /CT/D0/D7σ /B8 /D9/D7/CX/D2/CV /B4/BD/BI/B5/B8 /CP/D2/CS /D3/D1/D4/CP/D6/CX/D2/CV /DB/CX/D8/CW /B4/BH/BD/B5/B8 /DB /CT/AS/D2/CS /D8/CW/CT /D3/D2/D7/D8/D6/CP/CX/D2 /D8 /integraldisplay J(r,σ,t)dσ= 0 /B4/BH/BI/B5/C5/D9/D0/D8/CX/D4/D0/DD/CX/D2/CV /B4/BH/BH/B5 /CQ /DDσ /B8 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D2/CV /D3 /DA /CT/D6 /CP/D0/D0 /D8/CW/CT /C8/CE /D0/CT/DA /CT/D0/D7/B8 /D9/D7/CX/D2/CV /B4/BD/BJ/B5 /CP/D2/CS /D3/D1/D4/CP/D6/CX/D2/CV /DB/CX/D8/CW/B4/BH/BG/B5/B8 /DB /CT /CV/CT/D8/integraldisplay J(r,σ,t)σdσ=Jω /B4/BH/BJ/B5/CF /CT /CP/D2 /CT/DC/D4/D6/CT/D7/D7 /D8/CW/CT /D8/CX/D1/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D2/CT/D6/CV/DD ˙E≡dE/dt /CX/D2 /D8/CT/D6/D1/D7 /D3/CUJ /B8 /D9/D7/CX/D2/CV /B4/BK/B5 /CP/D2/CS/B4/BH/BE/B5/B8 /D0/CT/CP/CS/CX/D2/CV /D8/D3 /D8/CW/CT /CT/D2/CT/D6/CV/DD /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/D2/D7/D8/D6/CP/CX/D2 /D8 ˙E=/integraldisplay Jσhu⊥d2rdσ= 0, /B4/BH/BK/B5/DB/CW/CT/D6/CTu⊥≡ez∧u /BA /CD/D7/CX/D2/CV /B4/BE/BD/B5 /CP/D2/CS /B4/BH/BH/B5/B8 /DB /CT /D7/CX/D1/CX/D0/CP/D6/D0/DD /CT/DC/D4/D6/CT/D7/D7 /D8/CW/CT /D6/CP/D8/CT /D3/CU /CT/D2 /D8/D6/D3/D4 /DD /D4/D6/D3 /CS/D9 /D8/CX/D3/D2/CP/D7 ˙S=−/integraldisplay J∇(lnρ)d2rdσ. /B4/BH/BL/B5/BT /D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /C5/CP/DC/CX/D1 /D9/D1 /BX/D2 /D8/D6/D3/D4 /DD /C8/D6/D3 /CS/D9 /D8/CX/D3/D2 /C8/D6/CX/D2 /CX/D4/D0/CT/B8 /DB /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /D9/D6/D6/CT/D2 /D8 J/DB/CW/CX /CW /D1/CP/DC/CX/D1/CX/DE/CT/D7 /D8/CW/CT /D6/CP/D8/CT /D3/CU /CT/D2 /D8/D6/D3/D4 /DD /D4/D6/D3 /CS/D9 /D8/CX/D3/D2 ˙S /D6/CT/D7/D4 /CT /D8/CX/D2/CV /D8/CW/CT /D3/D2/D7/D8/D6/CP/CX/D2 /D8/D7 ˙E= 0 /B8 /B4/BH/BI/B5/CP/D2/CS/integraltextJ2 2ρdσ≤C(r,σ) /BA /CC/CW/CT /D0/CP/D7/D8 /D3/D2/D7/D8/D6/CP/CX/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CT/D7 /CP /CQ /D3/D9/D2/CS /B4/D9/D2/CZ/D2/D3 /DB/D2/B5 /D3/D2 /D8/CW/CT /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT/CS/CX/AR/D9/D7/CX/D3/D2 /D9/D6/D6/CT/D2 /D8/BA /BV/D3/D2 /DA /CT/DC/CX/D8 /DD /CP/D6/CV/D9/D1/CT/D2 /D8/D7 /CY/D9/D7/D8/CX/CU/DD /D8/CW/CP/D8 /D8/CW/CX/D7 /CQ /D3/D9/D2/CS /CX/D7 /CP/D0/DB /CP /DD/D7 /D6/CT/CP /CW/CT/CS /D7/D3 /D8/CW/CP/D8 /D8/CW/CT/BD/BD/CX/D2/CT/D5/D9/CP/D0/CX/D8 /DD /CP/D2 /CQ /CT /D6/CT/D4/D0/CP /CT/CS /CQ /DD /CP/D2 /CT/D5/D9/CP/D0/CX/D8 /DD /BA /CC/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D3/D2/CS/CX/D8/CX/D3/D2 /D3/D2 /AS/D6/D7/D8 /DA /CP/D6/CX/CP/D8/CX/D3/D2/D7 /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D8 /CT/CP /CW /D8/CX/D1/CTt /BM δ˙S−β(t)δ˙E−/integraldisplay ζ(r,t)δ/parenleftbigg/integraldisplay Jdσ/parenrightbigg d2r−/integraldisplay D−1(r,t)δ/parenleftbigg/integraldisplayJ2 2ρ/parenrightbigg dσd2r= 0 /B4/BI/BC/B5/CP/D2/CS /D0/CT/CP/CS/D7 /D8/D3 /CP /D9/D6/D6/CT/D2 /D8 /D3/CU /D8/CW/CT /CU/D3/D6/D1 J=−D(r,t)(∇ρ+β(t)σρhu⊥−ζ(r,t)ρ) /B4/BI/BD/B5/CC/CW/CT /C4/CP/CV/D6/CP/D2/CV/CT /D1 /D9/D0/D8/CX/D4/D0/CX/CT/D6ζ(r,t) /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D3/D2/D7/D8/D6/CP/CX/D2 /D8 /B4/BH/BI/B5/B8 /DB/CW/CX /CW /D0/CT/CP/CS/D7 /D8/D3 J=−D(r,t)/bracketleftbigg ∇ρ+β(t)ρ(σ−q)hu⊥/bracketrightbigg/B4/BI/BE/B5/CC/CW/CX/D7 /D3/D4/D8/CX/D1/CP/D0 /D9/D6/D6/CT/D2 /D8 /CX/D7 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D8/CW/CT /D3/D2/CT /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D3/D6/CS/CX/D2/CP/D6/DD /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /BE/BW /D8/D9/D6/CQ/D9/D0/CT/D2 /CT/CT/DC /CT/D4/D8 /D8/CW/CP/D8 /D8/CW/CT /D8/CT/D6/D1hu⊥ /D2/D3 /DB /D6/CT/D4/D0/CP /CT/D7 ∇ψ. /CC/CW/CT /CX/D1/D4 /CT/D6/D1/CT/CP/CQ/CX/D0/CX/D8 /DD /D3/D2/CS/CX/D8/CX/D3/D2 /CX/D1/D4 /D3/D7/CT/D7 /D8/CW/CP/D8 /D8/CW/CT/D2/D3/D6/D1/CP/D0 /D3/D1/D4 /D3/D2/CT/D2 /D8 /D3/CU /D8/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /CP/D2/CS /D3/CU /D8/CW/CT /D9/D6/D6/CT/D2 /D8 /DA /CP/D2/CX/D7/CW/CT/D7 /CP/D8 /D8/CW/CT /DB /CP/D0/D0/BA /CF /CT /D8/CW/CT/D6/CT/CU/D3/D6/CT/CW/CP /DA /CT /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2/D7/D2. /D9= 0 ( /D3/D2 /CT/CP /CW /CQ /D3/D9/D2/CS/CP/D6/DD ) /B4/BI/BF/B5/D2.∇ρ=−β(t)ρ(σ−q)h /D2u⊥ ( /D3/D2 /CT/CP /CW /CQ /D3/D9/D2/CS/CP/D6/DD ) /B4/BI/BG/B5/DB/CW/CT/D6/CT /D2 /CX/D7 /CP /D9/D2/CX/D8 /DA /CT /D8/D3/D6 /D2/D3/D6/D1/CP/D0 /D8/D3 /D8/CW/CT /DB /CP/D0/D0/BA/CC/CW/CT /CS/CX/AR/D9/D7/CX/D3/D2 /D3 /CTꜶ /CX/CT/D2 /D8D /CX/D7 /D2/D3/D8 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /C5/BX/C8/C8 /CP/D7 /CX/D8 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D8/CW/CT /D9/D2/CZ/D2/D3 /DB/D2/CQ /D3/D9/D2/CSC /D3/D2 /D8/CW/CT /D9/D6/D6/CT/D2 /D8/BA /C1/D8 /CP/D2 /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D1/D3/D6/CT /D7/DD/D7/D8/CT/D1/CP/D8/CX /D4/D6/D3 /CT/CS/D9/D6/CT/D7 /CX/D2/D7/D4/CX/D6/CT/CS /CU/D6/D3/D1/CZ/CX/D2/CT/D8/CX /D8/CW/CT/D3/D6/CX/CT/D7 /D3/CU /D4/D0/CP/D7/D1/CP /D4/CW /DD/D7/CX /D7 /D0/CX/CZ /CT /CX/D2 /CJ/BE/BJ /B8 /BE/BK ℄ /CU/D3/D6 /D8/CW/CT /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CP/D7/CT/BA /BY /D3/D6 /D8/CW/CT /D4/D9/D6/D4 /D3/D7/CT/D3/CU /D6/CT/CP /CW/CX/D2/CV /D8/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/B8 /CX/D8 /CP/D2 /D7/CX/D1/D4/D0/DD /CQ /CT /CW/D3/D7/CT/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/CX/D0/DD /BA /C0/D3 /DB /CT/DA /CT/D6/B8 /DB /CT /D7/CW/CP/D0/D0 /D7/CW/D3 /DB/CQ /CT/D0/D3 /DB /D8/CW/CP/D8D /D1 /D9/D7/D8 /CQ /CT /D4 /D3/D7/CX/D8/CX/DA /CT /D7/D3 /CP/D7 /D8/D3 /CX/D2/D7/D9/D6/CT /CT/D2 /D8/D6/D3/D4 /DD /CX/D2 /D6/CT/CP/D7/CT/BA/CC/CW/CT /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /CT/D2/CT/D6/CV/DD /B4/BH/BK/B5 /CP/D8 /CP/D2 /DD /D8/CX/D1/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/D7 /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C4/CP/CV/D6/CP/D2/CV/CT/D1 /D9/D0/D8/CX/D4/D0/CX/CT/D6β(t) /CP /D3/D6/CS/CX/D2/CV /D8/D3 β(t) =−/integraltextD∇qhu⊥d2r /integraltextD(q2−q2)(hu⊥)2d2r /B4/BI/BH/B5/CF /CT /CP/D2 /D2/D3 /DB /D4/D6/D3 /DA/CX/CS/CT /CP/D2 /CT/DC/D4/D0/CX /CX/D8 /CU/D3/D6/D1 /CU/D3/D6 /D8/CW/CT /DA /D3/D6/D8/CX /CX/D8 /DD /D9/D6/D6/CT/D2 /D8 Jω /D8/D3 /CX/D2 /D8/D6/D3 /CS/D9 /CT /CQ/CP /CZ /CX/D2/D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BH/BE/B5/BA /C1/D2/CS/CT/CT/CS/B8 /D9/D7/CX/D2/CV /B4/BI/BE/B5 /CP/D2/CS /B4/BD/BJ/B5/B8 /DB /CT /AS/D2/CS Jω=−D/bracketleftbigg ∇q+β(t)(q2−q2)hu⊥/bracketrightbigg/B4/BI/BI/B5/CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /B4/BI/BI/B5 /CX/D2 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BH/BE/B5/B8 /DB /CT /D3/CQ/D8/CP/CX/D2 ∂u ∂t+ (ω ωω+ 2Ω ΩΩ)∧u=−∇B+D/bracketleftbigg ez∧ ∇q−β(t)(q2−q2)hu/bracketrightbigg/B4/BI/BJ/B5/CB/CX/D2 /CTβ(t)≤0 /CX/D2 /D6/CT/D0/CT/DA /CP/D2 /D8 /D7/CX/D8/D9/CP/D8/CX/D3/D2/D7/B8 /D8/CW/CT /D0/CP/D7/D8 /D8/CT/D6/D1 /CX/D2 /B4/BI/BJ/B5 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/D7 /CP /CU/D3/D6 /CX/D2/CV /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0/D8/D3u /DB/CW/CX /CW /D3/D1/D4 /CT/D2/D7/CP/D8/CT/D7 /D8/CW/CT /CS/CX/AR/D9/D7/CX/D3/D2 /CP/D9/D7/CT/CS /CQ /DD /D8/CW/CT /D8/CT/D6/D1ˆez∧ ∇q∼∆u /BA /CC/CW/CX/D7 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0/D8/CT/D6/D1 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D8/CW/CT /D0/D3 /CP/D0 /C8/CE /DA /CP/D6/CX/CP/D2 /CTq2−q2/B8 /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 ρ /B8 /CP/D2/CS /DB /CT /D2/CT/CT/CS /D8/D3 /CZ /CT/CT/D4 /D8/D6/CP /CZ /D3/CU /CX/D8 /CQ /DD /D7/D3/D0/DA/CX/D2/CV /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /D8/D6/CP/D2/D7/D4 /D3/D6/D8 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BH/BH/B5 /CX/D2/CP/CS/CS/CX/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /D1/D3 /CS/CX/AS/CT/CS /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1 /B4/BH/BD/B5 /CP/D2/CS /B4/BI/BJ/B5/BA /CC/CW/CX/D7 /D7/CT/D8 /D3/CU /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CX/D2 /D6/CT/CP/D7/CT/D7/D8/CW/CT /CT/D2 /D8/D6/D3/D4 /DD /B4/CP/D8 /CP/D2 /D3/D4/D8/CX/D1/CP/D0 /D6/CP/D8/CT/B5/B8 /DB/CW/CX/D0/CT /D4/D6/CT/D7/CT/D6/DA/CX/D2/CV /CP/D0/D0 /D8/CW/CT /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D0/CP /DB/D7 /D3/CU /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0/CX/D2 /DA/CX/D7 /CX/CS /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1/BA /CF /CT /D2/D3 /DB /CW/CT /CZ /D8/CW/CP/D8 /D8/CW/CT /D7/D8/CT/CP/CS/DD /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D6/CT/CP /CW/CT/CS /CQ /DD /D8/CW/CT/D7/CT/D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CX/D7 /CX/D2/CS/CT/CT/CS /D8/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/BA/BD/BE/BH/BA/BE /CA/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D8/D3 /DB /CP/D6/CS/D7 /D8/CW/CT /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1/CC/CW/CT /CT/D2 /D8/D6/D3/D4 /DD /D4/D6/D3 /CS/D9 /D8/CX/D3/D2 /B4/BH/BL/B5 /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 ˙S=−/integraldisplayJ ρ(∇ρ+βρ(σ−q)hu⊥)d2rdσ+β/integraldisplay J(σ−q)hu⊥d2rdσ /B4/BI/BK/B5/CD/D7/CX/D2/CV /B4/BH/BI/B5 /CP/D2/CS /B4/BH/BK/B5/B8 /D8/CW/CT /D7/CT /D3/D2/CS /CX/D2 /D8/CT/CV/D6/CP/D0 /CX/D7 /D7/CT/CT/D2 /D8/D3 /CP/D2 /CT/D0 /D3/D9/D8/BA /CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /CU/D3/D6 /B4/BI/BE/B5 /CX/D2 /D8/CW/CT/AS/D6/D7/D8 /CX/D2 /D8/CT/CV/D6/CP/D0/B8 /DB /CT /CW/CP /DA /CT ˙S=/integraldisplayJ2 Dρd2rdσ /B4/BI/BL/B5/DB/CW/CX /CW /CX/D7 /D4 /D3/D7/CX/D8/CX/DA /CT /D4/D6/D3 /DA/CX/CS/CT/CS /D8/CW/CP/D8D≥0 /B8 /CP/D2/CS /D8/CW/CX/D7 /CX/D7 /D0/CT/CP/D6/D0/DD /CP /D2/CT /CT/D7/D7/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2 /D8/D3 /CP/D7/D7/D9/D6/CT/CT/D2 /D8/D6/D3/D4 /DD /CX/D2 /D6/CT/CP/D7/CT /CX/D2 /CP/D0/D0 /CP/D7/CT/D7/BA /BT /D7/D8/CP/D8/CX/D3/D2/CP/D6/DD /D7/D3/D0/D9/D8/CX/D3/D2 ˙S= 0 /CX/D7 /D7/D9 /CW /D8/CW/CP/D8J= 0 /DD/CX/CT/D0/CS/CX/D2/CV/B8/D8/D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /B4/BL/B5/B8 ∇(lnρ) +β(σ−q)∇ψ= 0 /B4/BJ/BC/B5/BY /D3/D6 /CP/D2 /DD /D6/CT/CU/CT/D6/CT/D2 /CT /C8/CE /D0/CT/DA /CT/D0σ0 /B8 /CX/D8 /DB/D6/CX/D8/CT/D7 ∇(lnρ0) +β(σ0−q)∇ψ= 0 /B4/BJ/BD/B5/CB/D9/CQ/D7/D8/D6/CP /D8/CX/D2/CV /B4/BJ/BC/B5 /CP/D2/CS /B4/BJ/BD/B5/B8 /DB /CT /D3/CQ/D8/CP/CX/D2 ∇ln(ρ/ρ0)+β(σ−σ0)∇ψ= 0 /B8 /DB/CW/CX /CW /CX/D7 /CX/D1/D1/CT/CS/CX/CP/D8/CT/D0/DD/CX/D2 /D8/CT/CV/D6/CP/D8/CT/CS /CX/D2 /D8/D3 ρ(r,σ) =1 Z(r)g(σ)e−βσψ/B4/BJ/BE/B5/DB/CW/CT/D6/CTZ−1(r)≡ρ0(r)eβσ0ψ(r)/CP/D2/CSg(σ)≡eA(σ)/B8A(σ) /CQ /CT/CX/D2/CV /CP /D3/D2/D7/D8/CP/D2 /D8 /D3/CU /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2/BA/CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /CT/D2 /D8/D6/D3/D4 /DD /CX/D2 /D6/CT/CP/D7/CT/D7 /D9/D2 /D8/CX/D0 /D8/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT /B4/BF/BH/B5 /CX/D7 /D6/CT/CP /CW/CT/CS/B8 /DB/CX/D8/CWβ= limt→∞β(t) /BA/BH/BA/BF /CB/CX/D1/D4/D0/CX/AS/CT/CS /CP/D7/CT/D7/BM/C1/D2 /D8/CW/CT /CP/D7/CT /D3/CU /D8 /DB /D3 /C8/CE /D0/CT/DA /CT/D0/D7σ0 /CP/D2/CSσ1 /B8 /D8/CW/CT /D8/D6/CP/D2/D7/D4 /D3/D6/D8 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BH/BH/B5 /CU/D3/D6 /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD p1 /CX/D7 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D8/D3 /D8/CW/CT /D8/D6/CP/D2/D7/D4 /D3/D6/D8 /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6q /B4/D7/CX/D2 /CTq=σ0+p1(σ1−σ0) /B5/B8 /DB/CW/CX /CW /CX/D7/CP/D0/D6/CT/CP/CS/DD /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT /D9/D6/D0 /D3/CU /D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BI/BJ/B5/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT /D8/CW/CT /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2/CT/D5/D9/CP/D8/CX/D3/D2/D7 /D6/CT/CS/D9 /CT /D8/D3 /D8/CW/CT /D1/D3 /CS/CX/AS/CT/CS /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1 ∂h ∂t+∇ ·(hu) = 0 /B4/BJ/BF/B5 ∂u ∂t+qhez∧u=−∇(u2 2+gh) +D[ez∧ ∇q−β(t)q2hu] /B4/BJ/BG/B5 q=(∇ ∧u).ez+ 2Ω h, q2= (q−σ0)(σ1−q) /B4/BJ/BH/B5 β(t) =−/integraltextDh(ez∧u)∇qd2r/integraltextDq2(ez∧u)2h2d2r /B4/BJ/BI/B5/D2.∇q=−β(t)q2h /D2/D9⊥ ( /D3/D2 /CT/CP /CW /CQ /D3/D9/D2/CS/CP/D6/DD ) /B4/BJ/BJ/B5/BD/BF/D2. /D9= 0 ( /D3/D2 /CT/CP /CW /CQ /D3/D9/D2/CS/CP/D6/DD ) /B4/BJ/BK/B5/DB/CW/CT/D6/CT /DB /CT /CW/CP /DA /CT /D3/D1/CX/D8/D8/CT/CS /D8/CW/CT /D3 /DA /CT/D6/B9/CQ/CP/D6 /D3/D2u /B8 /CP/D2/CS /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BJ/BH/B5 /D3/CUq2=q2−q2/CX/D7/CT/CP/D7/CX/D0/DD /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6 /CP /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /DB/CX/D8/CW /D8 /DB /D3 /DA /CP/D0/D9/CT/D7σ0 /CP/D2/CSσ1 /BA /CC/CW/CT /D2 /D9/D1/CT/D6/CX /CP/D0/CX/D1/D4/D0/CT/D1/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CX/D7 /D7/DD/D7/D8/CT/D1 /DB/CX/D0/D0 /D0/CT/CP/CS /D8/D3 /D8/CW/CT /D8 /DB /D3 /C8/CE /D0/CT/DA /CT/D0/D7 /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/BA/CB/D8/CP/D8/CX/D2/CVq2=cte /CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BJ/BH/B5 /DD/CX/CT/D0/CS/D7 /D8/CW/CT /BZ/CP/D9/D7/D7/CX/CP/D2 /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT /DB/CX/D8/CW /D0/CX/D2/CT/CP/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4 /CQ /CT/D8 /DB /CT/CT/D2q /CP/D2/CSψ /BA /CC/CW/CT/D2 /D8/CW/CT /D3 /CTꜶ /CX/CT/D2 /D8q2β /D9/D7/CT/CS /CX/D2 /B4/BJ/BG/B5 /CX/D7 /CS/CX/D6/CT /D8/D0/DD /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1/B4/BJ/BI/B5/BA /CC/CW/CX/D7 /CX/D7 /D7/D9Ꜷ /CX/CT/D2 /D8 /CU/D3/D6 /D8/CW/CT /D4/D9/D6/D4 /D3/D7/CT /D3/CU /AS/D2/CS/CX/D2/CV /D8/CW/CT /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1/B8 /CQ/D9/D8 /D1/D3/D6/CT /D6/CT/AS/D2/CT/CS/D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D1/D3 /CS/CT/D0/D7 /CP/D2 /CQ /CT /D9/D7/CT/CS /CP/D7 /CS/CX/D7 /D9/D7/D7/CT/CS /CQ /DD /CJ/BD/BK℄ /C3/CP/DE/CP/D2 /D8/D7/CT/DA /CT/D8 /CP/D0/BA /B4/BD/BL/BL/BK/B5 /CX/D2 /D8/CW/CT /D3/D2 /D8/CT/DC/D8/D3/CU /C9/BZ /D1/D3 /CS/CT/D0/D7/BA/BH/BA/BG /CC/CW/CT /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /D0/CX/D1/CX/D8/BM/CC/CW/CT /CP/D7/CT /D3/CU /D3/D6/CS/CX/D2/CP/D6/DD /BE/BW /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /D8/D9/D6/CQ/D9/D0/CT/D2 /CT /CX/D7 /D6/CT /D3 /DA /CT/D6/CT/CS /CX/D2 /D8/CW/CT /D0/CX/D1/CX/D8h→1 /B8q→ω/CP/D2/CS /D9=− /CTz∧ ∇ψ /BA /CC/CW/CT /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BI/BJ/B5 /D8/CW/CT/D2 /CQ /CT /D3/D1/CT/D7 ∂u ∂t+ (u.∇)u=−1 ρ∇p+D(∆u−β(t)ω2u) /B4/BJ/BL/B5/DB/CW/CT/D6/CT /DB /CT /CW/CP /DA /CT /D9/D7/CT/CS /D8/CW/CT /DB /CT/D0/D0/B9/CZ/D2/D3 /DB/D2 /CX/CS/CT/D2 /D8/CX/D8 /DD /D3/CU /DA /CT /D8/D3/D6 /CP/D2/CP/D0/DD/D7/CX/D7 ∆u=∇(∇u)−∇ ∧ (∇ ∧u)/DB/CW/CX /CW /D6/CT/CS/D9 /CT/D7 /CU/D3/D6 /CP /D8 /DB /D3/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /AT/D3 /DB /D8/D3∆u=ez∧ ∇ω /BA /BX/D5/D9/CP/D8/CX/D3/D2/B4/BJ/BL/B5 /CX/D7 /DA /CP/D0/CX/CS /CT/DA /CT/D2 /CX/CUD /CX/D7 /D7/D4/CP /CT /CS/CT/D4 /CT/D2/CS/CP/D2 /D8 /D9/D2/D0/CX/CZ /CT /DB/CX/D8/CW /CP /D9/D7/D9/CP/D0 /DA/CX/D7 /D3/D7/CX/D8 /DD /D8/CT/D6/D1/BA /C1/D2 /D4/D6/CT/DA/CX/D3/D9/D7/D4/D9/CQ/D0/CX /CP/D8/CX/D3/D2/D7 /D8/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /DB /CP/D7 /CV/CX/DA /CT/D2 /D3/D2/D0/DD /CX/D2 /CX/D8/D7 /DA /D3/D6/D8/CX /CX/D8 /DD /CU/D3/D6/D1 ∂ω ∂t+∇(ωu) =∇/bracketleftbigg D/parenleftbigg ∇ω+β(t)ω2∇ψ/parenrightbigg/bracketrightbigg/B4/BK/BC/B5/CP/D2/CS /D8/CW/CT /CT/D5/D9/CX/DA /CP/D0/CT/D2 /CT /DB/CX/D8/CW /B4/BJ/BL/B5 /CX/D7 /D2/D3/D8 /D3/CQ /DA/CX/D3/D9/D7 /CP/D8 /AS/D6/D7/D8 /D7/CX/CV/CW /D8/D7 /DB/CW/CT/D2D /CX/D7 /D7/D4/CP /CT /CS/CT/D4 /CT/D2/CS/CT/D2 /D8/BA /BT /D8/CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1/B8 /DB /CT /CW/CP /DA /CT /CU/D6/D3/D1 /B4/BJ/BL/B5 /D8/CW/CT /CX/CS/CT/D2 /D8/CX/D8 /DD ∆u=βω2u /B4/BK/BD/B5/DB/CW/CX /CW /CP/D2 /CQ /CT /CS/CT/CS/D9 /CT/CS /CS/CX/D6/CT /D8/D0/DD /CU/D6/D3/D1 /D8/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT/BA /C1/D2/CS/CT/CT/CS /CU/D3/D6 /CP /D7/D8/CP/D8/CX/D3/D2/CP/D6/DD /D7/D3/D0/D9/D8/CX/D3/D2 ω=F(ψ) /B8 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /CX/CS/CT/D2 /D8/CX/D8 /DD ∆ /D9= /CTz∧ ∇ω /CQ /CT /D3/D1/CT/D7 ∆u=−F′(ψ)u /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D8/D3 /B4/BK/BD/B5/CU/D3/D6 /CP /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT /D8/CW/CP/D2/CZ/D7 /D8/D3 /B4/BF/BL/B5/BA/CF /CT /D2/D3 /DB /CP /D3/D9/D2 /D8 /CU/D3/D6 /CP /CS/CT/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /AT/D9/CX/CS /D0/CP /DD /CT/D6 /CQ/D9/D8 /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8 /D8/CW/CT /CT/D0/CT/DA /CP/D8/CX/D3/D2 /DB/CX/D8/CW/D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT /CP /DA /CT/D6/CP/CV/CT /D8/CW/CX /CZ/D2/CT/D7/D7H /CX/D7 /DB /CT/CP/CZ/B8 /D7/D3 /D8/CW/CP/D8 h=H(1 +η) /DB/CX/D8/CWη≪1 /B4/BK/BE/B5/CC /D3 /AS/D6/D7/D8 /D3/D6/CS/CT/D6 /D8/CW/CT /AT/D3 /DB /CX/D7 /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CP/D2/CS /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BD/B5 /D6/CT/CS/D9 /CT/D7 /D8/D3∇.u= 0 /B8 /D3/D6 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/D0/DD u=−ez∧∇ψ /B4/D8/CW/CT/D6/CT /CX/D7 /CP /CU/CP /D8/D3/D6H /DB/CX/D8/CW /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /B4/BL/B5/B5/BA /C1/D2 /D8/CW/CT /D5/D9/CP/D7/CX/B9/CV/CT/D3/D7/D8/D6/D3/D4/CW/CX /D0/CX/D1/CX/D8 /D3/CU /D7/D1/CP/D0/D0 /CA/D3/D7/D7/CQ /DD /D2 /D9/D1 /CQ /CT/D6/D7ω≪Ω /B8 /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D9/D1 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/B5 /D6/CT/CS/D9 /CT/D7 /CP/D8 /DE/CT/D6/D3 /D3/D6/CS/CT/D6/D8/D3 /D8/CW/CT /CV/CT/D3/D7/D8/D6/D3/D4/CW/CX /CQ/CP/D0/CP/D2 /CT u=gH 2Ωez∧ ∇η /D3/D6ψ=−gH2 2Ωη /B4/BK/BF/B5/BD/BG/CC/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DD /D8/CW/CT/D2 /D6/CT/CS/D9 /CT/D7 /D8/D3 ζ≡Hq−2Ω≃ω+ψ L2 R /B4/BK/BG/B5/DB/CX/D8/CW /D8/CW/CT /CA/D3/D7/D7/CQ /DD /D6/CP/CS/CX/D9/D7 /D3/CU /CS/CT/CU/D3/D6/D1/CP/D8/CX/D3/D2 LR=√gH 2Ω /B4/BK/BH/B5/CC/CW/CT /D8/CT/D6/D11 LRψ /CX/D2 /B4/BK/BG/B5 /D6/CT/CP/D8/CT/D7 /CP /D7/CW/CX/CT/D0/CS/CX/D2/CV /D3/CU /D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /DA /D3/D6/D8/CX /CT/D7 /B4/D7/CX/D1/CX/D0/CP/D6 /D8/D3/D8/CW/CT /BW/CT/CQ /DD /CT /D7/CW/CX/CT/D0/CS/CX/D2/CV /CX/D2 /D4/D0/CP/D7/D1/CP /D4/CW /DD/D7/CX /D7/B5 /D3/D2 /CP /D0/CT/D2/CV/D8/CW /D7 /CP/D0/CT∼LR /BA /C1/D2 /D8/CW/CT /D0/CX/D1/CX/D81/LR→0 /B8/DB /CT /D6/CT /D3 /DA /CT/D6 /D8/CW/CT /BE/BW /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /BY /D3/D6 /AS/D2/CX/D8/CTLR /DB /CT /CP/D2 /CT/DC/D8/CT/D2/CS /D8/CW/CT /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0/D8/CW/CT/D3/D6/DD /D3/CU /D8/CW/CT /BE/BW /BX/D9/D0/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D8/D3 /D8/CW/CT /C9/BZ /CP/D7/CT /CQ /DD /D7/CX/D1/D4/D0/DD /D6/CT/D4/D0/CP /CX/D2/CV /D8/CW/CT /DA /D3/D6/D8/CX /CX/D8 /DDω /CQ /DD /D8/CW/CT/D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /DA /D3/D6/D8/CX /CX/D8 /DDζ /CJ/BD/BI /B8 /BD/BJ/B8 /BD/BK/B8 /BD/BL ℄/BA/BI /CC/CW/CT /CP/D7/CT /D3/CU /CX/D6 /D9/D0/CP/D6 /CS/D3/D1/CP/CX/D2/D7 /D3/D6 /CW/CP/D2/D2/CT/D0/BM/BI/BA/BD /CB/D8/CP/D8/CX/D7/D8/CX /CP/D0 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1/C1/D2 /CP /CS/CX/D7/CZ/B8 /D8/CW/CT /CP/D2/CV/D9/D0/CP/D6 /D1/D3/D1/CT/D2 /D8/D9/D1 L=/integraldisplay h(r∧u)zd2r /B4/BK/BI/B5/CX/D7 /D3/D2/D7/CT/D6/DA /CT/CS/BA /CC/CW/CX/D7 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D3/D2/D7/D8/D6/CP/CX/D2 /D8 /CP/D2 /CQ /CT /CP /D3/D9/D2 /D8/CT/CS /CU/D3/D6 /CQ /DD /CP/CS/CS/CX/D2/CV /CP /D8/CT/D6/D1βλδL /CX/D2/D8/CW/CT /AS/D6/D7/D8 /D3/D6/CS/CT/D6 /DA /CP/D6/CX/CP/D8/CX/D3/D2 /B4/BE/BF/B5/BA /CF /CT /CP/D2 /DB/D6/CX/D8/CTδL=/integraltextδh(r∧u)zd2r+/integraltexth(ez∧r).δud2r /B8 /CP/D2/CS/D8/CW/CT /D7/CT /D3/D2/CS /D8/CT/D6/D1 /CP/D2 /CQ /CT /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CUδω /CP/D2/CSδ(∇.u) /CQ /DD /CP /C0/CT/D0/D1/CW/D3/D0/D8/DE /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2/D3/CUh(ez∧r) /CP/D2/CP/D0/D3/CV/D3/D9/D7 /D8/D3 /B4/BL/B5/B8 /CU/D3/D0/D0/D3 /DB /CT/CS /CQ /DD /CP/D2 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /CQ /DD /D4/CP/D6/D8/BA /CC/CW/CX/D7 /CX/D7 /CP/D2/CP/D0/D3/CV/D3/D9/D7 /D8/D3 /D8/CW/CT/CU/D3/D6/D1 /D9/D0/CP/CT /B4/BE/BK/B5/B4/BE/BL/B5 /D9/D7/CT/CS /CU/D3/D6 /CT/DC/D4/D6/CT/D7/D7/CX/D2/CV /D8/CW/CT /CT/D2/CT/D6/CV/DD /DA /CP/D6/CX/CP/D8/CX/D3/D2/BA /CF /CT /CP/D2 /D3/D1 /CQ/CX/D2/CT /D8/CW/CT /CT/D2/CT/D6/CV/DD /CP/D2/CS/D1/D3/D1/CT/D2 /D8/D9/D1 /DA /CP/D6/CX/CP/D8/CX/D3/D2/D7 /CQ /DD /CS/CT/AS/D2/CX/D2/CV h[u−λ(ez∧r)] =−ez∧ ∇ψ′+∇φ′/B4/BK/BJ/B5/CX/D2/D7/D8/CT/CP/CS /D3/CU /B4/BL/B5/BA /BT /CS/CS/CX/D2/CV /D8/CW/CT /D2/CT/DB /D8/CT/D6/D1/D7 /CX/D2 /B4/BF/BD/B5/B4/BF/BF/B5 /DD/CX/CT/D0/CS/D7 /D8/CW/CT /BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT /B4/BF/BH/B5/B4/BG/BD/B5 /CU/D3/D6 /D8/CW/CT/DA /CT/D0/D3 /CX/D8 /DD u′=u−λ(ez∧r) /D7/CT/CT/D2 /CX/D2 /D8/CW/CT /D6/CT/CU/CT/D6/CT/D2 /CT /CU/D6/CP/D1/CT /D6/D3/D8/CP/D8/CX/D2/CV /CP/D8 /CP/D2/CV/D9/D0/CP/D6 /DA /CT/D0/D3 /CX/D8 /DDλ /BA /BT /B9 /D3/D6/CS/CX/D2/CV/D0/DD /B8 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CU /D8/CW/CT /BU/CT/D6/D2/D3/D9/CX/D0/D0/CX /CU/D9/D2 /D8/CX/D3/D2 /D1 /D9/D7/D8 /CQ /CT /D1/D3 /CS/CX/AS/CT/CS /CQ /DD /CP /D8/CT/D6/D1 /D3/CU /CT/D2 /D8/D6/CX/CU/D9/CV/CP/D0/CU/D3/D6 /CT/BM /DB /CT /D1 /D9/D7/D8 /D9/D7/CTB′(ψ′) =gh+u′2 2−λ2r2/CX/D2/D7/D8/CT/CP/CS /D3/CUB(ψ) /BA /CF /CT /AS/D2/CS /D8/CW/CT/D6/CT/CU/D3/D6/CT /D8/CW/CP/D8 /D8/CW/CT/BZ/CX/CQ/CQ/D7 /D7/D8/CP/D8/CT /B4/CX/D8/D7 /D0/D3 /CP/D0/D0/DD /CP /DA /CT/D6/CP/CV/CT/CS /DA /CT/D0/D3 /CX/D8 /DD /AS/CT/D0/CS/B5 /CX/D7 /CP /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/DB/CW/CX /CW /CX/D7 /D7/D8/CT/CP/CS/DD /CX/D2 /CP /D6/CT/CU/CT/D6/CT/D2 /CT /CU/D6/CP/D1/CT /D6/D3/D8/CP/D8/CX/D2/CV /CP/D8 /CP /D1/D3 /CS/CX/AS/CT/CS /CP/D2/CV/D9/D0/CP/D6 /DA /CT/D0/D3 /CX/D8 /DD Ω +λ /BA /CC/CW/CX/D7/DA /CT/D0/D3 /CX/D8 /DD /CX/D7 /CX/D2/CS/CX/D6/CT /D8/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D3/D2/D7/D8/D6/CP/CX/D2 /D8 /D3/D2 /CP/D2/CV/D9/D0/CP/D6 /D1/D3/D1/CT/D2 /D8/D9/D1/BA /C6/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT/D6/CT/D7/D9/D0/D8 /CP/D2 /CQ /CT /D6/CT/CP/CS/CX/D0/DD /CT/DC/D8/CT/D2/CS/CT/CS /D8/D3 /D8/CW/CT /D7/CW/CP/D0/D0/D3 /DB /DB /CP/D8/CT/D6 /D7/DD/D7/D8/CT/D1 /D3/D2 /D8/CW/CT 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arXiv:physics/0004057v1 [physics.data-an] 24 Apr 2000The information bottleneck method Naftali Tishby,1,2Fernando C. Pereira,3and William Bialek1 1NEC Research Institute, 4 Independence Way Princeton, New Jersey 08540 2Institute for Computer Science, and Center for Neural Computation Hebrew University Jerusalem 91904, Israel 3AT&T Shannon Laboratory 180 Park Avenue Florham Park, New Jersey 07932 30 September 1999 We define the relevant information in a signal x∈Xas being the in- formation that this signal provides about another signal y∈Y. Examples include the information that face images provide about the n ames of the peo- ple portrayed, or the information that speech sounds provid e about the words spoken. Understanding the signal xrequires more than just predicting y, it also requires specifying which features of Xplay a role in the prediction. We formalize this problem as that of finding a short code for Xthat preserves the maximum information about Y. That is, we squeeze the information that X provides about Ythrough a ‘bottleneck’ formed by a limited set of codewords ˜X. This constrained optimization problem can be seen as a gene ralization of rate distortion theory in which the distortion measure d(x,˜x) emerges from the joint statistics of XandY. This approach yields an exact set of self consistent equations for the coding rules X→˜Xand˜X→Y. Solutions to these equations can be found by a convergent re–estimatio n method that generalizes the Blahut–Arimoto algorithm. Our variationa l principle pro- vides a surprisingly rich framework for discussing a variet y of problems in signal processing and learning, as will be described in deta il elsewhere. 11 Introduction A fundamental problem in formalizing our intuitive ideas ab out information is to provide a quantitative notion of “meaningful” or “rele vant” information. These issues were intentionally left out of information the ory in its original formulation by Shannon, who focused attention on the proble m of transmit- ting information rather than judging its value to the recipi ent. Correspond- ingly, information theory has often been viewed as being str ictly a theory of communication, and this view has become so accepted that m any people consider statistical and information theoretic principle s as almost irrelevant for the question of meaning. In contrast, we argue here that i nformation the- ory, in particular lossy source compression, provides a nat ural quantitative approach to the question of “relevant information.” Specifi cally, we formu- late a variational principle for the extraction or efficient r epresentation of relevant information. In related work [1] we argue that this single informa- tion theoretic principle contains as special cases a wide va riety of problems, including prediction, filtering, and learning in its variou s forms. The problem of extracting a relevant summary of data, a compr essed description that captures only the relevant or meaningful i nformation, is not well posed without a suitable definition of relevance. A typi cal example is that of speech compression. One can consider lossless compr ession, but in any compression beyond the entropy of speech some component s of the signal cannot be reconstructed. On the other hand, a transcript of t he spoken words has much lower entropy (by orders of magnitude) than the acou stic waveform, which means that it is possible to compress (much) further wi thout losing any information about the words and their meaning. The standard analysis of lossy source compression is “rate d istortion the- ory,” which characterizes the tradeoff between the rate, or s ignal represen- tation size, and the average distortion of the reconstructe d signal. Rate distortion theory determines the level of inevitable expec ted distortion, D, given the desired information rate, R, in terms of the rate distortion function R(D). The main problem with rate distortion theory is in the need to specify the distortion function first, which in turn determines the r elevant features of the signal. Those features, however, are often not explic itly known and 2an arbitrary choice of the distortion function is in fact an a rbitrary feature selection. In the speech example, we have at best very partial knowledge of what precise components of the signal are perceived by our (neura l) speech recog- nition system. Those relevant components depend not only on the complex structure of the auditory nervous system, but also on the sou nds and utter- ances to which we are exposed during our early life. It theref ore is virtually impossible to come up with the “correct” distortion functio n for acoustic signals. The same type of difficulty exists in many similar pro blems, such as natural language processing, bioinformatics (for examp le, what features of protein sequences determine their structure) or neural cod ing (what informa- tion is encoded by spike trains and how). This is the fundamen tal problem of feature selection in pattern recognition. Rate distorti on theory does not provide a full answer to this problem since the choice of the d istortion func- tion, which determines the relevant features, is not part of the theory. It is, however, a step in the right direction. A possible solution comes from the fact that in many interest ing cases we have access to an additional variable that determines what i s relevant. In the speech case it might be the transcription of the signal, if we are interested in the speech recognition problem, or it might be the speaker ’s identity if speaker identification is our goal. For natural language pro cessing, it might be the part of speech labels for words in grammar checking, bu t the dictionary senses of ambiguous words in information retrieval. Simila rly, for the protein folding problem we have a joint database of sequences and thr ee dimensional structures, and for neural coding a simultaneous recording of sensory stimuli and neural responses defines implicitly the relevant variab les in each domain. All of these problems have the same formal underlying struct ure: extract the information from one variable that is relevant for the predi ction of another one. The choice of additional variable determines the relev ant components or features of the signal in each case. In this short paper we formalize this intuitive idea using an informa- tion theoretic approach which extends elements of rate dist ortion theory. We derive self consistent equations and an iterative algori thm for finding representations of the signal that capture its relevant str ucture, and prove 3convergence of this algorithm. 2 Relevant quantization LetXdenote the signal (message) space with a fixed probability me asure p(x), and let ˜Xdenote its quantized codebook or compressed representatio n. For ease of exposition we assume here that both of these sets a re finite, that is, a continuous space should first be quantized. For each value x∈Xwe seek a possibly stochastic mapping to a repre- sentative, or codeword in a codebook, ˜ x∈˜X, characterized by a conditional p.d.f. p(˜x|x). The mapping p(˜x|x) induces a soft partitioning of Xin which each block is associated with one of the codebook elements ˜ x∈˜X, with probability given by p(˜x) =/summationdisplay xp(x)p(˜x|x). (1) The average volume of the elements of Xthat are mapped to the same codeword is 2H(X|˜X), where H(X|˜X) =−/summationdisplay x∈Xp(x)/summationdisplay ˜x∈˜Xp(˜x|x) logp(˜x|x) (2) is the conditional entropy of Xgiven ˜X. What determines the quality of a quantization? The first fact or is of course the rate, or the average number of bits per message nee ded to specify an element in the codebook without confusion. This number per element of Xis bounded from below by the mutual information I(X;˜X) =/summationdisplay x∈X/summationdisplay ˜x∈˜Xp(x,˜x) log/bracketleftBiggp(˜x|x) p(˜x)/bracketrightBigg , (3) since the average cardinality of the partitioning of Xis given by the ratio of the volume of Xto that of the mean partition, 2H(X)/2H(X|˜X)= 2I(X;˜X), via the standard asymptotic arguments. Notice that this quanti ty is different from the entropy of the codebook, H(˜X), and this entropy normally is not what we want to minimize. 4However, information rate alone is not enough to characteri ze good quan- tization since the rate can always be reduced by throwing awa y details of the original signal x. We need therefore some additional constraints. 2.1 Relevance through distortion: Rate distortion theory In rate distortion theory such a constraint is provided thro ugh a distortion function, d:XטX→R+, which is presumed to be small for good represen- tations. Thus the distortion function specifies implicitly what are the most relevant aspects of values in X. The partitioning of Xinduced by the mapping p(˜x|x) has an expected distortion /an}bracketle{td(x,˜x)/an}bracketri}htp(x,˜x)=/summationdisplay x∈X/summationdisplay ˜x∈˜Xp(x,˜x)d(x,˜x). (4) There is a monotonic tradeoff between the rate of the quantiza tion and the expected distortion: the larger the rate, the smaller is the achievable distor- tion. The celebrated rate distortion theorem of Shannon and Kolmo gorov (see, for example Ref. [2]) characterizes this tradeoff through th e rate distortion function, R(D), defined as the minimal achievable rate under a given con- straint on the expected distortion: R(D)≡ min {p(˜x|x):/an}bracketle{td(x,˜x)/an}bracketri}ht≤D}I(X;˜X). (5) Finding the rate distortion function is a variational probl em that can be solved by introducing a Lagrange multiplier, β, for the constrained expected distortion. One then needs to minimize the functional F[p(˜x|x)] =I(X;˜X) +β/an}bracketle{td(x,˜x)/an}bracketri}htp(x,˜x) (6) over all normalized distributions p(˜x|x). This variational formulation has the following well known consequences: 5Theorem 1 The solution of the variational problem, δF δp(˜x|x)= 0, (7) for normalized distributions p(˜x|x), is given by the exponential form p(˜x|x) =p(˜x) Z(x, β)exp [−βd(x,˜x)], (8) where Z(x, β)is a normalization (partition) function. Moreover, the La- grange multiplier β, determined by the value of the expected distortion, D, is positive and satisfies δR δD=−β . (9) Proof. Taking the derivative w.r.t. p(˜x|x), for given xand ˜x, one obtains δF δp(˜x|x)=p(x)/bracketleftBigg logp(˜x|x) p(˜x)+ 1 −1 p(˜x)/summationdisplay x′p(x′)p(˜x|x′) +βd(x,˜x) +λ(x) p(x)/bracketrightBigg ,(10) since the marginal distribution satisfies p(˜x) =/summationtext x′p(x′)p(˜x|x′).λ(x) are the normalization Lagrange multipliers for each x. Setting the derivatives to zero and writing log Z(x, β) =λ(x)/p(x), we obtain Eq. (8). When varying the normalized p(˜x|x), the variations δI(X;˜X) and δ/an}bracketle{td(x,˜x)/an}bracketri}htp(x,˜x)are linked through δF=δI(X;˜X) +βδ/an}bracketle{td(x,˜x)/an}bracketri}htp(x,˜x)= 0, (11) from which Eq. (9) follows. The positivity of βis then a consequence of the concavity of the rate distortion function (see, for exam ple, Chapter 13 of Ref. [2]). 62.2 The Blahut–Arimoto algorithm An important practical consequence of the above variationa l formulation is that it provides a converging iterative algorithm for self c onsistent determi- nation of the distributions p(˜x|x) and p(˜x). Equations (8) and (1) must be satisfied simultaneously for co nsistent probability assignment. A natural approach to solve these e quations is to alternately iterate between them until reaching convergen ce. The following lemma, due to Csisz´ ar and Tusn´ ady [3], assures global conv ergence in this case. Lemma 2 Letp(x, y) =p(x)p(y|x)be a given joint distribution. Then the distribution q(y)that minimizes the relative entropy or Kullback–Leibler di - vergence, DKL[p(x, y)|p(x)q(y)], is the marginal distribution p(y) =/summationdisplay xp(x)p(y|x). Namely, I(X;Y) =DKL[p(x, y)|p(x)p(y)] = min q(y)DKL[p(x, y)|p(x)q(y)]. Equivalently, the distribution q(y)which minimizes the expected relative en- tropy,/summationdisplay xp(x)DKL[p(y|x)|q(y)], is also the marginal distribution p(y) =/summationtext xp(x)p(y|x). The proof follows directly from the non–negativity of the re lative entropy. This lemma guarantees the marginal condition Eq. (1) throug h the same variational principle that leads to Eq. (8): Theorem 3 Equations (1) and (8) are satisfied simultaneously at the min i- mum of the functional, F=−/an}bracketle{tlogZ(x, β)/an}bracketri}htp(x)=I(X;˜X) +β/an}bracketle{td(x,˜x)/an}bracketri}htp(x,˜x), (12) where the minimization is done independently over the convex sets of the normalized distributions, {p(˜x)}and{p(˜x|x)}, min p(˜x)min p(˜x|x)F[p(˜x);p(˜x|x)]. 7These independent conditions correspond precisely to alte rnating iterations of Eq. (1) and Eq. (8). Denoting by tthe iteration step, /braceleftBiggpt+1(˜x) =/summationtext xp(x)pt(˜x|x) pt(˜x|x) =pt(˜x) Zt(x,β)exp(−βd(x,˜x))(13) where the normalization function Zt(x, β)is evaluated for every tin Eq. (13). Furthermore, these iterations converge to a unique mi nimum ofFin the convex sets of the two distributions. For the proof, see references [2, 4]. This alternating itera tion is the well known Blauht-Arimoto (BA) algorithm for calculation of the rate distortion function. It is important to notice that the BA algorithm deals only wit h the op- timal partitioning of the set Xgiven the set of representatives ˜X, and not with an optimal choice of the representation ˜X. In practice, for finite data, it is also important to find the optimal representatives whic h minimize the expected distortion, given the partitioning. This joint optimization is similar to the EM procedure in statistical estimation and does not in general have a unique solution. 3 Relevance through another variable: The Information Bottleneck Since the “right” distortion measure is rarely available, t he problem of rel- evant quantization has to be addressed directly, by preserv ing the relevant information about another variable. The relevance variable, denoted he re by Y, must not be independent from the original signal X, namely they have positive mutual information I(X;Y). It is assumed here that we have access to the joint distribution p(x, y), which is part of the setup of the problem, similarly to p(x) in the rate distortion case.1 1The problem of actually obtaining a good enough sample of thi s distribution is an interesting issue in learning theory, but is beyond the scop e of this paper. For a start on this problem see Ref. [1]. 83.1 A new variational principle As before, we would like our relevant quantization ˜Xto compress Xas much as possible. In contrast to the rate distortion problem, how ever, we now want this quantization to capture as much of the information abou tYas possible. The amount of information about Yin˜Xis given by I(˜X;Y) =/summationdisplay y/summationdisplay ˜xp(y,˜x) logp(y,˜x) p(y)p(˜x)≤I(X;Y). (14) Obviously lossy compression cannot convey more informatio n than the orig- inal data. As with rate and distortion, there is a tradeoff bet ween compress- ing the representation and preserving meaningful informat ion, and there is no single right solution for the tradeoff. The assignment we a re looking for is the one that keeps a fixed amount of meaningful information ab out the rel- evant signal Ywhile minimizing the number of bits from the original signal X(maximizing the compression).2In effect we pass the information that X provides about Ythrough a “bottleneck” formed by the compact summaries in˜X. We can find the optimal assignment by minimizing the function al L[p(˜x|x)] =I(˜X;X)−βI(˜X;Y), (15) where βis the Lagrange multiplier attached to the constrained mean ingful information, while maintaining the normalization of the ma pping p(˜x|x) for every x. Atβ= 0 our quantization is the most sketchy possible—everythin g is assigned to a single point—while as β→∞ we are pushed toward arbitrar- ily detailed quantization. By varying the (only) parameter βone can explore the tradeoff between the preserved meaningful information a nd compression at various resolutions. As we show elsewhere [1, 5], for inte resting special cases (where there exist sufficient statistics) it is possibl e to preserve almost all the meaningful information at finite βwith a significant compression of the original data. 2It is completely equivalent to maximize the meaningful info rmation for a fixed com- pression of the original variable. 93.2 Self-consistent equations Unlike the case of rate distortion theory, here the constrai nt on the meaning- ful information is nonlinear in the desired mapping p(˜x|x) and this is a much harder variational problem. Perhaps surprisingly, this ge neral problem of extracting the meaningful information—minimizing the fun ctionalL[p(˜x|x)] in Eq. (15)—can be given an exact formal solution. Theorem 4 The optimal assignment, that minimizes Eq. (15), satisfies t he equation p(˜x|x) =p(˜x) Z(x, β)exp/bracketleftBigg −β/summationdisplay yp(y|x) logp(y|x) p(y|˜x)/bracketrightBigg , (16) where the distribution p(y|˜x)in the exponent is given via Bayes’ rule and the Markov chain condition ˜X←X←Y, as, p(y|˜x) =1 p(˜x)/summationdisplay xp(y|x)p(˜x|x)p(x). (17) This solution has a number of interesting features, but we mu st emphasize that it is a formal solution since p(y|˜x) in the exponential is defined implicitly in terms of the assignment mapping p(˜x|x). Just as before, the marginal distribution p(˜x) must satisfy the marginal condition Eq. (1) for consistenc y. Proof. First we note that the conditional distribution of yon ˜x p(y|˜x) =/summationdisplay x∈Xp(y|x)p(x|˜x), (18) follows from the Markov chain condition Y←X←˜X.3The only varia- tional variables in this scheme are the conditional distrib utions, p(˜x|x), since other unknown distributions are determined from it through Bayes’ rule and consistency. Thus we have p(˜x) =/summationdisplay xp(˜x|x)p(x), (19) 3It is important to notice that this not a modeling assumption and the quantization ˜X isnotused as a hidden variable in a model of the data. In the latter, the Markov condition would have been different: Y←˜X←X. 10and p(˜x|y) =/summationdisplay xp(˜x|x)p(x|y). (20) The above equations imply the following derivatives w.r.t. p(˜x|x), δp(˜x) δp(˜x|x)=p(x) (21) and δp(˜x|y) δp(˜x|x)=p(x|y). (22) Introducing Lagrange multipliers, βfor the information constraint and λ(x) for the normalization of the conditional distributions p(˜x|x) at each x, the Lagrangian, Eq. (15), becomes L=I(X,˜X)−βI(˜X, Y)−/summationdisplay x,˜xλ(x)p(˜x|x) (23) =/summationdisplay x,˜xp(˜x|x)p(x) log/bracketleftBiggp(˜x|x) p(˜x)/bracketrightBigg −β/summationdisplay ˜x,yp(˜x, y) log/bracketleftBiggp(˜x|y) p(˜x)/bracketrightBigg −/summationdisplay x,˜xλ(x)p(˜x|x). (24) Taking derivatives with respect to p(˜x|x) for given xand ˜x, one obtains δL δp(˜x|x)=p(x) [1 + log p(˜x|x)]−δp(˜x) δp(˜x|x)[1 + log p(˜x)] −β/summationdisplay yδp(˜x|y) δp(˜x|x)p(y)[1 + log p(˜x|y)] −βδp(˜x) δp(˜x|x)[1 + log p(˜x)]−λ(x). (25) Substituting the derivatives from Eq’s. (21) and (22) and re arranging, δL δp(˜x|x)=p(x)/braceleftBigg log/bracketleftBiggp(˜x|x) p(˜x)/bracketrightBigg −β/summationdisplay yp(y|x) log/bracketleftBiggp(y|˜x) p(y)/bracketrightBigg −λ(x) p(x)/bracerightBigg .(26) 11Notice that/summationtext yp(y|x) logp(y|x) p(y)=I(x, Y) is a function of xonly (independent of ˜x) and thus can be absorbed into the multiplier λ(x). Introducing ˜λ(x) =λ(x) p(x)−β/summationdisplay yp(y|x) log/bracketleftBiggp(y|x) p(y)/bracketrightBigg , we finally obtain the variational condition: δL δp(˜x|x)=p(x)/bracketleftBigg logp(˜x|x) p(˜x)+β/summationdisplay yp(y|x) logp(y|x) p(y|˜x)−˜λ(x)/bracketrightBigg = 0,(27) which is equivalent to equation (16) for p(˜x|x), p(˜x|x) =p(˜x) Z(x, β)exp (−βDKL[p(y|x)|p(y|˜x)]), (28) with Z(x, β) = exp[ β˜λ(x)] =/summationdisplay ˜xp(˜x) exp (−βDKL[p(y|x)|p(y|˜x)]), the normalization (partition) function. Comments: 1. The Kullback–Leibler divergence, DKL[p(y|x)|p(y|˜x)],emerged as the relevant “effective distortion measure” from our variation al principle but is not assumed otherwise anywhere! It is therefore natur al to con- sider it as the “correct” distortion d(x,˜x) =DKL[p(y|x)|p(y|˜x)] for quantization in the information bottleneck setting. 2. Equation (28), together with equations (18) and (19), det ermine self consistently the desired conditional distributions p(˜x|x) and p(˜x). The crucial quantization is here performed through the conditi onal distri- butions p(y|˜x), and the self consistent equations include also the opti- mization over the representatives, in contrast to rate dist ortion theory, where the selection of representatives is a separate proble m. 123.3 The information bottleneck iterative algorithm As for the BA algorithm, the self consistent equations (16) a nd (17) suggest a natural method for finding the unknown distributions, at ev ery value of β. Indeed, these equations can be turned into converging, alte rnating iterations among the three convex distribution sets, {p(˜x|x)},{p(˜x)}, and{p(y|˜x)}, as stated in the following theorem. Theorem 5 The self consistent equations (18), (19), and (28), are sati sfied simultaneously at the minima of the functional, F[p(˜x|x);p(˜x);p(y|˜x)] =−/an}bracketle{tlogZ(x, β)/an}bracketri}htp(x) (29) =I(X;˜X) +β/an}bracketle{tDKL[p(y|x)|p(y|˜x)]/an}bracketri}htp(x,˜x),(30) where the minimization is done independently over the convex sets of the normalized distributions, {p(˜x)}and{p(˜x|x)}and{p(y|˜x)}. Namely, min p(y|˜x)min p(˜x)min p(˜x|x)F[p(˜x|x);p(˜x);p(y|˜x)]. This minimization is performed by the converging alternati ng iterations. De- noting by tthe iteration step,   pt(˜x|x) =pt(˜x) Zt(x,β)exp(−βd(x,˜x)) pt+1(˜x) =/summationtext xp(x)pt(˜x|x) pt+1(y|˜x) =/summationtext yp(y|x)pt(x|˜x)(31) and the normalization (partition function) Zt(β,˜x)is evaluated for every t in Eq. (31). Proof. For lack of space we can only outline the proof. First we show t hat the equations indeed are satisfied at the minima of the functi onalF(known for physicists as the “free energy”). This follows from lemm a (2) when applied toI(X;˜X) with the convex sets of p(˜x) and p(˜x|x), as for the BA algorithm. Then the second part of the lemma is applied to /an}bracketle{tDKL[p(y|x)|p(y|˜x)]/an}bracketri}htp(x,˜x) which is an expected relative entropy. Equation (28) minimi zes the expected relative entropy w.r.t. to variations in the convex set of th e normalized 13{p(y|˜x)}. Denoting by d(x,˜x) =DKL[p(y|x)|p(y|˜x)] and by λ(˜x) the normal- ization Lagrange multipliers, we obtain δd(x,˜x) = δ/parenleftBigg −/summationdisplay yp(y|x) logp(y|˜x) +λ(˜x)(/summationdisplay yp(y|˜x)−1)/parenrightBigg (32) =/summationdisplay y/parenleftBigg −p(y|x) p(y|˜x)+λ(˜x)/parenrightBigg δp(y|˜x). (33) The expected relative entropy becomes, /summationdisplay x/summationdisplay y/parenleftBigg −p(y|x)p(x|˜x) p(y|˜x)+λ(˜x)/parenrightBigg δp(y|˜x) = 0 , (34) which gives Eq. (28), since δp(y|˜x) are independent for each ˜ x. Equation (28) also have the interpretation of a weighted average of th e data conditional distributions that contribute to the representative ˜ x. To prove the convergence of the iterations it is enough to ver ify that each of the iteration steps minimizes the same functional, i ndependently, and that this functional is bounded from below as a sum of two n on–negative terms. The only point to notice is that when p(y|˜x) is fixed we are back to the rate distortion case with fixed distortion matrix d(x,˜x). The argument in [3] for the BA algorithm applies here as well. On the other h and we have just shown that the third equation minimizes the expect ed relative entropy without affecting the mutual information I(X;˜X). This proves the convergence of the alternating iterations. However, the si tuation here is similar to the EM algorithm and the functional F[p(˜x|x);p(˜x);p(y|˜x)] is convex in each of the distribution independently but notin the product space of these distributions. Thus our convergence proof does not imply uniqueness of the solution. 3.4 The structure of the solutions The formal solution of the self consistent equations, descr ibed above, still requires a specification of the structure and cardinality of ˜X, as in rate distortion theory. For every value of the Lagrange multipli erβthere are corresponding values of the mutual information IX≡I(X,˜X), and IY≡ 14I(˜X, Y) for every choice of the cardinality of ˜X. The variational principle implies that δI(˜X, Y) δI(X,˜X)=β−1>0, (35) which suggests a deterministic annealing approach. By increasing the value ofβone can move along convex curves in the “information plane” ( IX, IY). These curves, analogous to the rate distortion curves, exis ts for every choice of the cardinality of ˜X. The solutions of the self consistent equations thus correspond to a family of such annealing curves, all startin g from the (trivial) point (0 ,0) in the information plane with infinite slope and parameter ized by β. Interestingly, every two curves in this family separate (b ifurcate) at some finite (critical) βthrough a second order phase transition. These transitions form a hierarchy of relevant quantizations for different car dinalities of ˜X, as described in [1, 5, 6]. Further work The most fascinating aspect of the information bottleneck p rinciple is that it provides a unified framework for different information proce ssing problems, including prediction, filtering and learning [1]. There are already several successful applications of this method to various “real” pr oblems, such as semantic clustering of English words [6], document classifi cation [5], neural coding, and spectral analysis. Acknowledgements Helpful discussions and insights on rate distortion theory with Joachim Buh- mann and Shai Fine are greatly appreciated. Our collaborati on was facili- tated in part by a grant from the US–Israel Binational Scienc e Foundation (BSF). References [1] W. Bialek and N. Tishby, “Extracting relevant informati on,” in prepara- tion. 15[2] T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991). [3] I. Csisz´ ar and G. Tusn´ ady, “Information geometry and a lternating mini- mization procedures,” Statistics and Decisions Suppl. 1, 205–237 (1984). [4] R. E. Blahut, “Computation of channel capacity and rate d istortion func- tion,” IEEE Trans. Inform. Theory IT-18, 460–473 (1972). [5] N. Slonim and N. Tishby, “Agglomerative information bot tleneck,” To appear in Advances in Neural Information Processing systems (NIPS-1 2) 1999. [6] F. C. Pereira, N. Tishby, and L. Lee, “Distributional clu stering of En- glish words,” in 30th Annual Mtg. of the Association for Computational Linguistics , pp. 183–190 (1993). 16

arXiv:physics/0004058v1 [physics.atom-ph] 24 Apr 2000Precision measurements of sodium - sodium and sodium - noble gas molecular absorption M. Shurgalin, W.H. Parkinson, K. Yoshino, C. Schoene* and W. P. Lapatovich* Harvard-Smithsonian Center for Astrophysics, 60 Garden St , MS 14, Cambridge MA 02138 USA ∗OSRAM SYLVANIA Lighting Research, 71 Cherry Hill Dr, Beverl y, MA 01915 USA Precision measurements of molecular absorption PACS numbers: 39.30.+w 07.60.Rd 33.20.Kf Submitted to Measurement Science and Technology, January 2 000 Abstract. Experimental apparatus and measurement technique are desc ribed for precision absorption measurements in sodium - noble gas mix tures. The ab- solute absorption coefficient is measured in the wavelength r ange from 425 nm to 760 nm with ±2% uncertainty and spectral resolution of 0.02 nm. The precision is achieved by using a specially designed abso rption cell with an accurately defined absorption path length, low noise CCD d etector and double-beam absorption measurement scheme. The experimen tal set-up and the cell design details are given. Measurements of sodium at omic number density with±5% uncertainty complement absorption coefficient measure- ments and allow derivation of the reduced absorption coeffici ents for certain spectral features. The sodium atomic number density is meas ured using the anomalous dispersion method. The accuracy of this method is improved by employing a least-squares fit to the interference image reco rded with CCD detector and the details of this technique are given. The mea surements are 1aimed at stringent testing of theoretical calculations and improving the val- ues of molecular parameters used in calculations. Keywords: absorption cell, molecular absorption, anomalous dispers ion method 1. Introduction Atomic collision processes significantly influence the abso rption and emis- sion of light by atomic vapors at high pressures. As a result t he absorption and emission spectra reveal not only atomic line broadening but also very broad, essentially molecular features with rich rotationa l-vibrational struc- ture and satellite peaks due to formation of molecules and qu asi-molecules. Since pioneering work by Hedges et al. [1], such spectra have been a subject of extensive studies, both theoretical and experimental, and proved to be a rich source of information about the interaction potentials, co llision dynamics and transition dipole moments [2-12]. The experimental approa ches employed in- clude absorption measurements [4,5,8,9,12], laser-induc ed fluorescence [3,6,9] and thermal emission spectra [7]. While laser-induced fluor escence and emis- sion spectra provide the shapes and positions of many molecu lar bands, the measurements of absorption coefficient spectra also give abs orption coeffi- cients over a large spectral range. Absolute measurements o f the absorption spectra may allow more comprehensive tests of theoretical c alculations. As a result, better differentiation between different theoreti cal approaches and improved values for various molecular parameters and poten tials can be ob- tained. However, in many cases absorption spectra are obtai ned on a rel- ative scale or only the absorption coefficient or optical dept h is measured accurately. Extraction of absolute cross-sections (or red uced absorption co- efficients) from traditional measurements of the optical dep th as well as any quantitative comparisons of absorption spectra with theor etical calculations require accurate knowledge of the absorption path length an d the absorbing species concentrations. Most of absorption spectroscopy experiments with hot and co rrosive va- pors such as sodium have been performed using heat pipes [8,9 ,13]. In a heat pipe the alkali vapor is contained in the hot middle of th e furnace be- tween cold zones where windows are located. Buffer noble gas f acilitates the alkali containment and protects cold windows from alkali de posits. In this 2type of absorption cell the vapor density is not uniform at th e ends of the absorption path and the path length is not accurately defined . In addition, the temperature of the vapor-gas mixture is not uniform and a t higher alkali vapor densities formation of fog at the alkali - buffer gas int erface seriously affects the optical absorption coefficient measurements [13, 14]. Absorption cells have been designed, where heated windows, placed in th e cell hot zone, define the absorption length with good precision [14-16]. Th e absorption cell described in [15] is suitable for hot sodium vapors up to 1000 K but it is difficult to make it with a long absorption path. The cell descr ibed in [16] is not suitable for corrosive vapors and may still have problem s with window transmission due to metal deposits [16]. Schlejen et. al. [14] designed a cell specifically for spectroscopy of sodium dimers at high tempe ratures. Their cell allowed uniform concentration of absorbers and unifor m temperature up to 1450K with an absorption length defined accurately by hot s apphire win- dows. However, the cell design is not suitable for spectrosc opy of gas - vapor mixtures because it was completely sealed and did not easily enable changing the mixtures. As well as defining the absorption length accurately, an equa lly important aspect is measuring the alkali vapor density. While the nobl e gas density can be calculated reasonably well from the measurements of pres sure and temper- ature using the ideal gas law, it is difficult to establish the d ensity of alkali atoms. In the majority of reported experiments alkali conce ntration was determined from the temperature and published saturated va por pressure curves but this approach can introduce significant uncertai nties. For exam- ple, in measurements of oscillator strengths or f-values significant discrepan- cies were often obtained between oscillator strengths meas ured by methods involving knowledge of the number density and by methods not requiring it [17]. Even if the vapor pressure curve is well known for pure s aturated vapor, introducing buffer gas or using unsaturated vapors prohibit accurate knowl- edge of the vapor density along the absorption path. To achie ve a higher precision in determination of absolute cross-sections or r educed absorption coefficients it is necessary to measure the alkali vapor densi ty directly. In this paper we describe experimental apparatus and techni que used for precision measurements on an absolute scale of molecula r absorption co- efficients in sodium vapor + noble gas mixtures. To overcome th e above- mentioned difficulties with definition of absorption length w e have designed a special absorption cell. In our cell, heated sapphire wind ows, resistant to 3hot sodium vapor, are used to define the absorption path. A hig h temper- ature valve, kept at the same temperature as the cell itself, is utilized to introduce different noble gases. A separate sodium reservoi r, maintained at a lower temperature, is used to control the sodium vapor pres sure indepen- dently of the cell temperature. The cell can be operated at te mperatures up to 900K. During the spectral measurements we measure and m onitor the sodium number density at different temperatures and pres sures using the anomalous dispersion or ’hook’ method [8,9,18,19]. The ’hook’ method allows accurate measurement of nflvalue where nis the atomic number den- sity,fis the atomic line oscillator strength and lis the absorption length. If the absorption length and f-value for sodium D lines are known, the sodium number density is accurately obtained. The next section con centrates on the details of the experiment and the absorption cell design. 2. Experiment 2.1 Experimental set-up Fig. 1 shows schematically the experimental set-up that is u sed for our absorption measurements. The light source is a 100W halogen lamp pow- ered by a voltage-stabilized DC power supply. A well-collim ated beam of white light is produced with the help of a condenser lens, an a chromat lens, a pinhole aperture 0.4 mm diameter and another achromat lens of shorter focal length. The light beam is sent through a Mach-Zender in terferometer and focused on the entrance slit of 3m Czerny-Turner spectro graph (McPher- son model 2163) with a combination of spherical and cylindri cal lenses. An absorption cell is placed in the test-beam arm of the Mach-Ze nder interfer- ometer. Beam blocks are used in both arms to switch the beams o r block them altogether. The light beam through the reference arm of the Mach- Zender interferometer is used as a reference for the absorpt ion in the usual manner of double-beam absorption spectroscopy [9]. The spectra are recorded with a two-dimensional CCD detecto r (Andor Technology model V420-0E) placed in the focal plane of the sp ectrograph. This detector has 1024 pixels horizontally and 256 pixels ve rtically with pixel size of 26x26 m. For spectral measurements the detector is us ed in the ver- tical bin mode, that is, as a one-dimensional array detector . The stigmatic spectrograph has a plane diffraction grating with 2400 groov es/mm and the- oretical resolution of ∼0.005 nm at 500 nm wavelength. We used 150 m 4entrance slit width which gives actual resolution of 0.02 nm . At least 5 pix- els of the array detector are used over 0.02 nm wavelength int erval and as a result smoother spectral data are obtained from the array de tector. The overall spectral range determined by the diffraction gra ting and the detector sensitivity is 425 nm to 760 nm. To record spectra th rough this spec- tral range the diffraction grating is rotated through 160 diff erent positions by a stepper motor. Backlash is avoided by rotating the grating in one direction from its calibration position at 425 nm, which in turn is set b y rotating the grating beyond the calibration point. The calibration poin t is located by a rotation photosensor placed on the worm screw of the spectro graph sine-bar mechanism. The photosensor signal is sent to a programmable stepper motor controller (New England Affiliated Technology NEAT-310) whi ch drives the stepper motor and allows the grating to be set at the calibrat ion position automatically. Each position of the grating permits recording consecutive ly spectral in- tervals ranging from about 3 nm at 430 nm to 1.1 nm at 760 nm, whi ch are determined by the linear reciprocal dispersion of the sp ectrograph at a given wavelength and the overall length of the array detecto r. All grating positions are wavelength-calibrated using a large number o f different atomic lines obtained from a number of different hollow cathode spec tral lamps. The wavelength calibration enables identifying the wavelengt h of any pixel of the CCD detector within ±0.05 nm in the range 425 to 760 nm. To measure the sodium atomic density using the anomalous dis persion or ’hook’ method, both beams through the Mach-Zender interf erometer are unblocked and interfere to produce a spectrally dispersed t wo-dimensional interference pattern in the focal plane of the spectrograph . The Mach-Zender interferometer is adjusted to localize the interference fr inges at infinity. This insures that the integral sodium number density along the ab sorption path is measured. The CCD detector is used in its normal two-dimen sional array mode to record the interference pattern. From the analysis o f the interference pattern recorded in the vicinity of sodium D lines, sodium nu mber density is derived. The general description of the ’hook’ method is giv en in [17,18,19] and the details of analyzing the interference pattern recor ded with CCD detector are given in the next section. A glass plate and a sta ck of windows, identical to those used in the absorption cell, are placed in the reference arm of the Mach-Zender interferometer [17]. These compensatin g optics remain in the reference beam during spectral absorption measureme nts as well and 5have no effect on the spectral measurements due to the nature o f the dual- beam absorption technique. A simple vacuum system consisting of a turbomolecular pump ( Sargent Welch model 3106S) backed by a rotary vane pump (Sargent Welc h model 1402) is used to evacuate the absorption cell. The turbomole cular pump can handle short bursts of increased pressure and gas flow load an d therefore it is utilized also to pump gases from the cell. A liquid nitrogen t rap is placed in between the cell and the pump to trap sodium vapor. A precisio n pressure gauge (Omega Engineering model PGT-45) is used to measure ac curately the pressure of noble gases when filling the cell. An experiment control and data acquisition computer (Penti um PC) controls the CCD detector, spectral data acquisition and th e spectrograph diffraction grating via the stepper motor controller connec ted to the serial port. The absorption cell temperature is monitored constan tly through a number of thermocouples connected via commercial plug-in d ata acquisi- tion board (American Data Acquisition Corporation) and the cell heaters are controlled via output channels of the data acquisition b oard and solid state relays. Andor Technology CCD software and custom ’in- house’ written software are used to perform all these tasks. 2.2 Absorption cell Fig. 2 shows the schematic diagram of the absorption cell. Th e cell body is made of stainless steel (SS) 316 and is approximately 470 m m in length and 8 mm internal diameter. A vertical extension is welded to the middle of the cell body, 70 mm in length and 11 mm internal diameter. A sodiu m reservoir is located at the end of this extension. It is made of SS 316 wit h internal diameter 5.5 mm and 70 mm length and it is connected using a Swa gelok fitting which enables disconnection for loading sodium. The sodium reservoir is heated with a separate heater to introduce sodium vapor in to the cell or it can be cooled with a circulating water cooler to reduce the alkali number density. Both the heater and cooler are made to slide on and off the sodium reservoir. A valve is connected to the vertical extension th rough which the cell is evacuated and noble gases can be admitted. This valve is a special bellows-sealed high-temperature valve (Nupro Company) ra ted to work at temperatures up to 920K. The valve is heated to the same tempe rature or 5 to 10 K higher than the cell itself to prevent sodium from cond ensing in the 6valve. The major problem one faces when designing an absorption cel l with heated windows is making good vacuum seals for the windows. I n case of sodium, sapphire proved to be material of choice for the wind ows because of its excellent resistance to hot sodium [14]. However, it is d ifficult to make a reliable sapphire to metal seal that withstands repeated he ating cycles up to 900K. In our design (Fig. 2) the sapphire windows are sealed i nto polycrys- talline alumina (PCA) tubes with special sealing ceramics ( frit) used in the construction of commercial high-pressure sodium (HPS) lam ps. The sealing technique used was similar to the one described by Schlejen et. al. [14]. The PCA tubes have 10.2 mm outside diameter and 110 mm length. The y are made from commercial OSRAM SYLVANIA standard HPS lamps ULX8 80 by cutting off slightly larger diameter portions. Since PCA a nd sapphire have similar thermal expansion coefficients, such window-tube as sembly retains its integrity over a wide range of temperatures. The tubes are in serted into the heated cell body so that the windows are located in the hot zon e while the tube ends extend to cooler cell ends where Viton O-rings are u sed for vacuum seals. Additional external windows made of fused silica are used with O-ring seals to create an evacuated transition zone from the heated middle of the cell to the cooler ends. These silica windows are not heated. Our cell design allows sodium to condense along the PCA tube t owards the colder zone where O-ring seals are located. To reduce the amount of sodium condensed there the PCA tubes were carefully selecte d in outside diameter tolerance and straightness to match closely the in ternal diameter of the SS cell body at room temperature. When the cell is heate d the SS expands more than PCA thus creating some space for sodium to c ondense. Once the sodium build-up reaches the hot zone, no more sodium is lost into the void along the PCA tubes. The windowed ends of the PCA tubes rest against the stepped pr ofiles inside the SS cell body as shown in Fig. 2. These stepped profil es determine the positions of the heated windows and thus the absorption l ength. To ensure that the PCA tubes are always firmly pressed against th ese stepped profiles regardless of the thermal expansion differences bet ween PCA and SS, compression springs are used to push the PCA tubes via stacks of spacers, made of SS 316, and the external windows. Cap nuts complete th e assembly of the windows, PCA tubes, O-ring seals and spacers as Fig. 2 i llustrates. These caps allow easy removal of all windows for cleaning if n eeded as well as 7adjustment of the spring compression. The compression spri ngs are chosen to produce about 12 N force, equivalent to about 1.5 atm on the surface area of the heated windows. The absorption length at room temperature is measured using a special tool made of two rods about 4 mm in diameter inserted into a tub e of 6 mm outside diameter. One rod is permanently fixed while the othe r can slide in and out with friction, thus allowing change in the overall le ngth of the tool. The ends of the tool are rounded and polished. With one sapphi re window completely in place at one end of the cell, the tool is inserte d into the cell and the second PCA tube with sapphire window is put in place. T he tool adjusts its length precisely to the distance between two sap phire windows. Then the PCA tube and the tool are carefully taken out and the l ength of the tool is measured with a micrometer. In our cell the absorp tion length at room temperature was measured 190.03 ±0.025 mm. The absorption length at operating temperature is calculated from the temperatur e of the cell and the thermal expansion coefficient for SS of 18 ±2.2 x 10−6K−1[20]. Since the change in the absorption path length due to thermal expan sion is a small percentage of the overall length, large uncertainties in th e thermal expansion calculation do not lead to a large uncertainty in the resulti ng absorption length at a given temperature. The whole absorption cell including the valve is heated by se ts of heaters made of Nickel-Chromium wire. Separate sets of heaters are u sed to heat the cell and the valve. Each heater set consists of two separate h eaters. One is switched on constantly while the other one is used in on-off mo de, controlled from the experiment control computer, to maintain average c ell and valve temperatures constant. Six type K thermocouples are used to measure the temperatures at different points. Three thermocouples are p laced in contact with the main cell body, one of them in the middle and the other two at the locations of heated windows. Another thermocouple meas ures the valve temperature. Two thermocouples are used to measure the temp eratures at the bottom and at the top of the sodium reservoir. The heaters are isolated from each other and the SS parts of the cell by embedding them i nto insula- tion made of moldable Alumina-Silica blankets (Zircar Prod ucts). All heated parts are wrapped into thermal insulation made of Alumina bl ankets (Zircar Products). The positions of the heaters are chosen as shown s chematically in Fig. 2. The middle part of the cell of ∼60 mm length does not have heaters but is nevertheless heated sufficiently by thermal co nductance. Also 8the thermal insulation is adjusted in such a way that the temp erature, mea- sured with thermocouple in the middle of the cell, is 5 to 10 K l ower than the temperature at the points where the heated windows are locat ed. Heating sapphire windows to a slightly higher temperature ensures t hey remain free from any deposits during the operation of the cell. The cell body thermocouple and the valve thermocouple readi ngs give an average temperature of the cell body. The gas mixture temp erature is assumed to be equal to the average temperature of the cell bod y. Since the thermocouples are located between the SS cell body and the he aters, they may give readings of slightly higher temperature than the ac tual temperature of the cell and the gas inside it. Given this fact and the tempe rature reading differences between the thermocouples, the uncertainty in t he gas mixture temperature is estimated to be + 10 K - 50 K. 2.3 Measurement technique Absorption spectroscopy is based on Beer’s law describing a bsorption of light in homogeneous absorbing media I1(λ) =I0(λ)exp(−k(λ)l) (1) where I1is the transmitted intensity of light, I0is the incident intensity of light, kis absorption coefficient and lis the absorption length. In real experimental measurements the transmission through optic s, absorption cell windows and spectrograph, detector sensitivity and light s ource spectral char- acteristics all have to be taken into account. In the dual bea m arrangement for absorption spectroscopy the test St(λ) and reference Sr(λ) beam spectra, obtained from the detector, are St(λ) =k0 t(λ)I0(λ) exp (−τt(λ)−k(λ)l) (2) Sr(λ) =k0 r(λ)I0(λ)exp(−τr(λ)) (3) where I0is the intensity of the light source, k(λ)is the absorption coeffi- cient to be measured, lis the absorption length and k0 t,τtandk0 r,τrare the coefficients that take into account the detector efficiency, ab sorption of all optics elements such as windows and lenses and spectrograph transmission. To eliminate all unknown parameters represented in (2) and ( 3) by k0 t,τt, 9k0 randτr, we measure first the reference spectra (the spectra obtaine d with- out sodium in the absorption path and thus without atomic and molecular absorption we are interested in). Sodium concentration in t he absorption path is reduced to less than 1014cm−3by cooling the sodium reservoir down to between +5 to + 10 C using the circulating water cooler arou nd it. At densities below 1014cm−3the molecular absorption of both sodium-sodium and sodium-noble gas is negligible and k(λ) = 0. Both test and reference beam spectra are taken at each grating position and their rat io S0 t(λ) S0 r(λ)=k0 t k0 rexp(τr(λ)−τt(λ)) (4) is calculated. Thus obtained reference spectra contain inf ormation about all unknown parameters. To reduce statistical error a number of measurements are performed and the average is calculated. To measure the absorption spectra of sodium-sodium and sodi um-noble gas molecules, the sodium vapor is introduced in the absorpt ion path by heating the sodium reservoir. Both test and reference beam s pectra are taken at each diffraction grating position and their ratio is calculated: SNa t(λ) SNar(λ)=k0 t k0rexp(τr(λ)−τt(λ)−k(λ)l) (5) Once again to reduce statistical error a number of measureme nts are performed and averaged. From (4) and (5) it follows that the a bsorption coefficient k(λ) is obtained from measurements of absorption and reference spectra with all unknown parameters eliminated: k(λ) =−1 lln/parenleftBigg SNa t SNar/slashBigg S0 t S0r/parenrightBigg (6) Using the procedure described above we are able to measure th e absolute absorption coefficient with as small as ±1 % statistical error in the range 425 - 760 nm with spectral resolution ∼0.02 nm. Derivation of the reduced absorption coefficient for sodium- sodium and sodium-noble gas quasi-molecules requires accurate knowl edge of atomic num- ber densities. The atomic density for noble gas is calculate d from pressure and temperature using the ideal gas relationship and the sod ium density is measured by the ’hook’ method [17-19]. Fig. 3 shows the ’hook ’ interference 10pattern recorded with CCD detector in the focal plane of the s pectrograph. The analysis of this pattern and extraction of the sodium ato mic number density is performed by a least-squares fit of the interferen ce fringe model to the recorded pattern using software specifically written for this purpose. The following equation can be used to describe the positions ykof interference fringes of maximum intensity in the focal plane of the spectr ograph [19]: yk=a/parenleftBig kλ−/parenleftBig A1 λ1−λ−A2 λ2−λ/parenrightBig N+ ∆nl/parenrightBig , A1=r0g1f1lλ3 1 4πandA2=r0g2f2lλ3 2 4π where r0is the classical electron radius, g1, f1, λ1are respectively the g-factor, the oscillator strength and the wavelength of sodium D1 line ,g2, f2, λ2are re- spectively the g-factor, the oscillator strength and the wavelength of sodi um D2 line, lis the absorption path length, ∆ nis the coefficient accounting for optical path length difference between test and reference be ams of the Mach- Zender interferometer, ais the scaling factor accounting for imaging proper- ties of the optical set-up, kis the fringe order and Nis the sodium number density. The above equation is valid at wavelengths separat ed from the atomic line core by more than the FWHM of the broadened line, λ−λi>>∆λ[17]. Our model calculations of the ’hook’ interference pattern, which included the atomic line broadening, showed that the error introduce d by neglecting the atomic line broadening in the above equation, is negligi ble when atomic number density of sodium is above 5x1014cm−3and noble gas pressure is be- low 500 Torr. These conditions are always met in our measurem ents. After some algebraic manipulations the following fringe model eq uation is obtained which gives positions yiof a number of fringes in terms of 2D CCD detector coordinates and fit parameters: yi=a3+a2λ+a1iλ+a4/parenleftbiggA1 λ1−λ+A2 λ2−λ/parenrightbigg (7) where yiis the vertical fringe coordinate at a given wavelength λ,i= 0,1,2,3,4,5 denotes adjacent fringes seen by the detector a nda1,a2,a3and a4are the fit parameters. The sodium number density is calculat ed from fit parameters a1anda4: N=a4 a1(8) From the recorded interference pattern three to five interfe rence fringes, defined at maximum intensity, are extracted at each side of th e sodium dou- 11blet and the ( y,λ) coordinates for each fringe are calculated from the CCD pixel coordinates and wavelength calibration to provide th e data set for the least-squares fit. Fig. 4a shows the interference fringes ob tained from the image presented in Fig. 3 and Fig 4b shows the fitted model curv es. Since a large number of points are used to locate the fringe position s, higher accuracy can be achieved in comparison with traditional methods of ex tracting atomic number density from measuring the location of the ’hook’ max ima of a single fringe [17]. The main limitation on the accuracy of this new t echnique is im- posed by wavelength calibration, especially at lower atomi c densities. Using the system described above, the sodium density is routinely measured with± 2 % uncertainty, given the wavelength uncertainty ±0.03 nm in the vicinity of the sodium D-lines and the uncertainty in the interferenc e fringe position of±5 pixels. During consecutive spectral measurements used in calculating the resultant average spectra, the sodium number density wa s measured at the beginning of each measurement and was found to remain con stant within ±4% to±5%. 3. Measurement results Fig. 5 presents a spectrum of the absolute absorption coeffici ent of a sodium - xenon mixture measured at 900 K+10K −50Kcell temperature. The Xe pressure is 400±0.5 Torr, which gives xenon density 4.29x1018cm−3at 900 K temperature. Sodium density is measured as 2.05 ±0.06 x 1016cm−3. The absorption coefficient in the 425 nm to 760 nm range consists of contributions from the broadened sodium atomic lines around 589 nm, the sod ium - noble gas and the sodium - sodium molecular spectra. From 460 nm to a bout 550 nm a blue wing of the sodium dimer absorption is apparent [5]. At 560 nm there appears a sodium-xenon blue wing satellite feature [6 ] and towards the longer wavelength of the significantly broadened sodium D-l ines there are red wings of the sodium dimer [5] and the sodium-xenon molecu les [6]. Fig 6 shows a spectrum of the absolute absorption coefficient of a so dium - argon mixture measured at 900 K+10K −50Kcell temperature. The Ar pressure is 401 ±0.5 Torr, which gives argon density 4.3x1018cm−3at 900 K temperature. Sodium density is measured as 1.00 ±0.04 x 1016cm−3. This spectrum is similar to the sodium - xenon spectrum shown in Fig. 5 excep t that the sodium - argon blue wing satellite is located at a slightly sh orter wavelength of 554.5 nm and the sodium - argon red wing extends further from t he sodium 12atomic line core. The magnitude of the absorption coefficient is lower in proportion to the lower sodium density. Fig. 7 illustrates rotational-vibrational features of sod ium dimer absorp- tion at a 0.02 nm resolution in the vicinity of the 520 nm band. The features are a complicated superposition of many rotational-vibrat ional bands of the sodium dimer and identification of these bands has not yet bee n attempted. The statistical uncertainty in the absorption coefficient ma gnitude is indi- cated. This uncertainty includes both detector statistica l errors and the uncertainty in the absorption path length. At any wavelengt h in the 425 nm to 760 nm range the uncertainty in the absorption coefficien t does not exceed±2% where absorption coefficient values are larger than 0.008 c m−1. The measured spectra can serve for stringent quantitative t ests of theoretical calculations [21]. Preliminary comparisons showed good ov erall agreement between the measurements and theoretical calculations at a temperature of 870 K, which is within our experimental temperature uncerta inty [22]. Full details of the calculations and comparisons with experimen t will be presented in the forthcoming publication [23]. A reduced absorption coefficient is calculated for the blue wi ng of sodium dimer absorption, which is well separated from the rest of th e spectrum, using the measured absorption coefficient and sodium atomic number density and it is presented in Fig. 8. Apart from the broad and strong molecu lar absorption arising mostly from transitions from bound to bound states b etween X1Σ+ g andB1Πumolecular singlet states of the sodium dimer, there are also features from the triplet transitions 23Πg←a3Σ+ uandc3Πg←a3Σ+ u [5,24]. Since the sodium-xenon molecular absorption bands are very close to the sodium D-lines, it is difficult to separate them completely fr om atomic lines and to derive the reduced absorption coefficient. Fig.9 prese nts the absolute absorption coefficient in the vicinity of sodium-xenon blue s atellite features at different xenon densities and sodium density of 7.7x1015cm−3and at 900 K temperature. There are two satellite features present at 5 60 nm and 564 nm. The positions and relative magnitude of these features c an provide some insights into potentials of the sodium-xenon molecule as we ll as transition dipole moments [21,23]. 4. Conclusion 13Details of precision absorption measurements in sodium - no ble gas mix- tures at high spectral resolution have been presented. To pe rform more stringent tests of theoretical calculations and molecular parameters used in the calculations the goal was to obtain the absorption coeffic ient spectra on an absolute scale with better than ±2% uncertainty at near UV and visible wavelengths. To achieve such precision an absorption cell f or containment of sodium vapor with accurately defined absorption path was con structed. The measurements were performed using double-beam absorption measurement scheme eliminating all unknown parameters such as detector sensitivity and optics transmission. A low noise CCD detector was used to rec ord the spectra and a number of separate measurements were averaged to reduc e statistical error. To measure accurately the alkali number density the a nomalous dis- persion or ’hook’ method was employed. The accuracy of the ’h ook’ method was improved by means of least-squares fit to the interferenc e fringes image recorded using 2D CCD detector in the focal plane of the spect rograph. The measurements obtained with the apparatus and technique des cribed extend the available data on the sodium - sodium and sodium - rare gas absorption to different temperatures and higher precision and spectral resolution. 5. Acknowledgements This work is supported in part by National Science Foundatio n under grant No PHY97-24713. The authors would like to acknowledge useful dis- cussions with J.F. Babb, H. Adler and G. Lister and generous e quipment and materials support from OSRAM SYLVANIA. References [1] R.E. Hedges, D.L. Drummond and A. Gallagher 1972 Phys. Rev. A 6, 1519 [2] D.L. Drummond and A. Gallagher 1974 J. Chem. Phys. 60, 3246 [3] W. Demtrder and M. Stock 1975 J. Mol. Spectr. 55, 476 [4] J.F. Kielkopf, and N.F. Allard 1980 J. Phys. B 13, 709 [5] J. Schlejen, C.J. Jalink, J. Korving, J.P. Woerdman and W . Mller 1987J. Phys. B 20, 2691 [6] K.J. Nieuwesteeg, Tj. Hollander and C.Th. J. Alkemade 19 87J. Phys. B20, 515 14[7] J. Schlejen, J.P. Woerdman and G. Pichler 1988 J. Mol. Spectr. 128, 1 [8] K. Ueda, H. Sotome and Y. Sato 1990 J. Chem. Phys. 94, 1903 [9] K. Ueda, O. Sonobe, H. Chiba and Y. Sato 1991 J. Chem. Phys. 95, 8083 [10] D. Gruber, U. Domiaty, X. Li, L. Windholz, M. Gleichmann and B. A. He 1994 J. Chem. Phys. 102, 5174 [11] J. Szudy and W.E. Baylis 1996 Phys. Rep. 266, 127 [12] P.S. Erdman, K.M. Sando, W.C. Stwally, C.W. Larson, M.E . Fajardo 1996Chem. Phys, Lett. 252, 248 [13] A. Vasilakis, N.D. Bhaskar and W Happer 1980 J. Chem. Phys. 73, 1490 [14] J. Schlejen, J. Post, J. Korving and J.P. Woerdman 1987 Rev. Sci. Instrum. 58, 768 [15] A.G. Zajonc 1980 Rev. Sci. Instrum. 51, 1682 [16] Y. Tamir and R. Shuker 1992 Rev. Sci. Instrum. 63, 1834 [17] W.H. Parkinson 1987 Spectroscopy of Astrophysical Plasmas (Cam- bridge University Press) [18] D. Roschestwensky 1912 Ann. Physik, 39, 307 [19] M.C.E. Huber and R.J. Sandeman 1986 Rep. Prog. Phys. 49397 [20] American Institute of Physics Handbook 1972 4-138 (McGraw Hill Book Company) [21] H-K. Chung, M. Shurgalin and J.F. Babb 1999 52ndGEC, Bull. APS , 44, 31 [22] H-K. Chung and J.F. Babb, private communication [23] H-K. Chung, K. Kirby, J.F. Babb and M. Shurgalin, 2000, in prepa- ration [24] D.Veza, J. Rukavina, M. Movre, V. Vujnovic and G. Pilche r 1980 Optics Comm. 3477 Figure 1. Schematic diagram of the experimental set-up. Figure 2. Schematic diagram of the absorption cell. Figure 3. Image of a ’hook’ pattern obtained with a two-dimensional CCD detector. 15Figure 4. Interference fringes extracted from ’hook’ pattern image ( a) and fitted model curves (b). Figure 5. Absorption coefficient of sodium - xenon mixture at 900 K. Figure 6. Absorption coefficient of sodium - argon mixture at 900 K. Figure 7. Rotational-vibrational features of sodium dimer absorpti on spectra at 0.02 nm resolution. Figure 8. Reduced absorption coefficient for the blue wing of sodium dimer molecular absorption. Figure 9. Blue wing of sodium-xenon molecular absorption. 16Cylindrical lens CCD detector 256x1024Stepper motor controller3m Czerny-Turner spectrograph Experiment control and data acquisition computerAbsorption cell Beam bl Pinho Cell window compensation HBeam f optiCompensation plateMach-Zender interferometerPressure gvacuum vacuumnoble gasSodium reservoir heValve heaterSodium reservoir ValveQuartz windows Cap nutSapphire windows Cell bodyPCA t Compression spring Cell heatersThermal insulationa bwavelength, nm450500550600650700750k ( cm-1 ) 0.000.020.040.060.080.10wavelength, nm450500550600650700750k ( cm-1 ) 0.0000.0050.0100.0150.0200.025wavelength, nm520.0520.4520.8521.2521.6522.0k ( cm-1 ) 0.0000.0050.0100.0150.0200.0250.0300.0350.040 uncertaintywavelength, nm425450475500525550k/n2 x 10-18 ( cm2 ) 0.000.250.500.751.001.251.501.752.002.25 c 3Πg← a 3Σu+ Β 1Πu ← X 1Σg+2 3Πg ← a 3Σu+wavelength, nm550555560565570575580585k ( cm-1 ) 0.00000.00250.00500.00750.01000.01250.0150 Xe: 10 TorrXe: 100 TorrXe: 200 TorrXe: 400 Torr

arXiv:physics/0004060v1 [physics.bio-ph] 25 Apr 2000Real time encoding of motion: Answerable questions and questionable answers from the fly’s visual system Rob de Ruyter van Steveninck,1Alexander Borst,2and William Bialek1 1NEC Research Institute 4 Independence Way Princeton, New Jersey 08540 USA 2ESPM—Division of Insect Biology University of California at Berkeley 201 Wellman Hall Berkeley, California 94720, USA In the past decade, a small corner of the fly’s visual system ha s become an im- portant testing ground for ideas about coding and computati on in the nervous system. A number of results demonstrate that this system ope rates with a preci- sion and efficiency near the limits imposed by physics, and mor e generally these results point to the reliability and efficiency of the strateg ies that nature has se- lected for representing and processing visual signals. A re cent series of papers by Egelhaaf and coworkers, however, suggests that almost all t hese conclusions are incorrect. In this contribution we place these controversi es in a larger context, emphasizing that the arguments are not just about flies, but r ather about how we should quantify the neural response to complex, naturali stic inputs. As an example, Egelhaaf et al. (and many others) compute certain c orrelation func- tions and use the apparent correlation times as a measure of t emporal precision in the neural response. This analysis neglects the structur e of the correlation function at short times, and we show how to analyze this struc ture to reveal a temporal precision 30 times better than suggested by the cor relation time; this precision is confirmed by a much more detailed information th eoretic analysis. In reviewing other aspects of the controversy, we find that th e analysis methods used by Egelhaaf et al. suffer from some mathematical inconsi stencies, and that in some cases we are unable to reproduce their experimental r esults. Finally, we present results from new experiments that probe the neural r esponse to inputs that approach more closely the natural context for freely fly ing flies. These new experiments demonstrate that the fly’s visual system is even more precise and efficient under natural conditions than had been inferred fro m our earlier work. 11 Introduction Much of what we know about the neural processing of sensory in formation has been learned by studying the responses of single neurons to r ather simplified stimuli. The ethologists, however, have argued that we can r eveal the full rich- ness of the nervous system only when we study the way in which t he brain deals with the more complex stimuli that occur in nature. On t he other hand it is possible that the processing of natural signals is deco mposable into steps that can be understood from the analysis of simpler signals. But even then, to prove that this is the case one must do the experiment and use c omplex natural stimuli. In the past decade there has been renewed interest i n moving beyond the simple sensory inputs that have been the workhorse of neu rophysiology, and a key step in this program has been the development of more pow erful tools for the analysis of neural responses to complex dynamic inputs. The motion sensi- tive neurons of the fly visual system have been an important te sting ground for these ideas, and there have been several key results from thi s work: 1. The sequence of spikes from a motion sensitive neuron can b e decoded to recover a continuous estimate of the dynamic velocity traje ctory (Bialek et al. 1991; Haag and Borst 1997). In this decoding, individu al spikes contribute significantly to the estimate of velocity at each point in time. 2. The precision of velocity estimates approaches the physi cal limits imposed by diffraction and noise in the photoreceptor array (Bialek e t al. 1991). 3. One or two spikes are sufficient to discriminate between mot ions which differ by displacements in the ‘hyperacuity’ range, an order of magnitude smaller than the spacing between photoreceptors in the reti na (de Ruyter van Steveninck and Bialek 1995). Again this performance app roaches the limits set by diffraction and receptor noise. 4. Patterns of spikes which differ by millisecond shifts of th e individual spikes can stand for distinguishable velocity waveforms (d e Ruyter van Steveninck and Bialek 1988), and these patterns can carry mu ch more information than expected by adding up the contributions of individual spikes (de Ruyter van Steveninck and Bialek 1988, Brenner et al. in press). 5. The total information that we (or the fly) can extract from t he spike train continues to increase as we observe the spikes with gre ater temporal resolution, down to millisecond precision (de Ruyter van St eveninck et al. 1997, Strong et al. 1998). 6. These facts about the encoding of naturalistic, dynamic s timuli cannot be extrapolated simply from studies of the neural response t o simpler signals. The system exhibits profound adaptation (Maddess and Laughlin 1985, de Ruyter van Steveninck et al. 1986, Borst and Egelhaa f 1987, de Ruyter van Steveninck et al. 1996, Brenner et al. submitted) , so that the encoding of signals depends strongly on context, and the statistical 2structure of responses to dynamic stimuli can be very differe nt from that found with simpler static or steady state stimuli (de Ruyter van Steveninck et al. 1997). We emphasize that many of these results from the fly’s visual s ystem have direct analogs in other systems, from insects to amphibians to prim ates (Rieke et al. 1997). In a series of recent papers, Egelhaaf and coworkers have cal led these results into question (Warzecha and Egelhaaf 1997, 1998, 1999; Warz echa et al. 1998). Several of these papers are built around a choice of a stimulu s very different from that used in previous work. Rather than synthesize a sti mulus with known statistical properties, they sample the time dependent mot ion signals generated by a fly tethered in a flight simulator. The simulator is operat ed in closed loop so that the fly, by producing a yaw torque which is measured ele ctronically, moves a pattern on a CRT monitor, while the animal itself stay s stationary. For experiments on the responses of the motion sensitive neu rons these patterns and motions are replayed to another fly, again through a monit or. In their judgement these stimuli “are characteristic of a normal beh avioral situation in which the actions and reactions of the animal directly affect its visual input” (Warzecha and Egelhaaf 1998). For these stimuli, Warzecha and Egelhaaf claim that the timi ng of individ- ual spikes has no significance in representing motion signal s in the fly’s motion sensitive neurons. Instead they suggest that the neuron’s r esponse should be averaged over time scales of order 40 to 100 ms to recover the e ssential infor- mation, and that timing of spikes within this averaging wind ow is irrelevant. These claims are in conflict with points [1], [4], and [5] abov e. As part of their discussion of these points Warzecha and Egelhaaf make repea ted references to the noisiness of the neural response, in apparent contradic tion of points [2] and [3], although they do not address specifically the quantitat ive results of the ear- lier work. Finally, they suggest that there is no difference b etween the statistics of spike trains in response to steady state vs. dynamic stimu li, in contradiction of point [6]. Obviously the recent work of Egelhaaf and coworkers raises m any different issues. In this contribution we try to focus on three problem s of general interest. First, how do we define a meaningful “naturalistic stimulus, ” and does their “behaviourally generated” stimulus fall into this categor y? In particular, how do we reach an effective compromise between stimuli that occu r in nature and stimuli that we can control and reproduce reliably in the lab oratory? Second, how do we characterize the neural response to complex dynami c inputs? In particular, how do we evaluate all the relevant time scales i n the sensory signal itself and in the spike train? Again, these are issues that we must face in the analysis of any neural system for processing of sensory info rmation; indeed there are even analogous issues in motor systems. Thus the fly’s vis ual system serves here as an example, rather than as an end in itself. Before we begin our discussion of these two points, we must be clear that the first question—what is a natural stimulus?—is a question abo ut the biology and 3ecology of the animal we are studying, as well as a question ab out the design and constraints of a particular experimental setup. One might w ell disagree about the best strategy for generating naturalistic stimuli in th e lab. On the other hand, our second question—how do we characterize the respon se to complex signals?—is a theoretical issue which is not tied to the part iculars of biology. On this issue there are precise mathematical statements to b e made, and we hope to make clear how these mathematical results can be used as a rigorous guide to the analysis of experiments. The third and final question we address concerns the comparis on between static and dynamic stimuli. Although we believe that the mos t interesting problems concern the way in which the brain deals with the com plex, dynamic stimuli that occur in nature, much has been learned from simp ler static stimuli and there are nagging questions about whether it really is ‘n ecessary’ to design new experiments that need more sophisticated methods of ana lysis. For reasons that will become clear below, the comparison of static and dy namic stimuli also is crucial for understanding whether many of the lesson s learnt from the analysis of the fly’s motion sensitive neurons will be applic able to other systems, especially the mammalian cortex. 2 What is a natural stimulus? The fly’s motion sensitive neuron H1 offers a relatively simpl e testing ground for ideas about the neural representation of natural signals. T his cell is a wide field neuron, so rather than coding the motion of small objects or a component of the local velocity flow field, H1 is responsible primarily for cod ing the rigid body horizontal (yaw) motion of the fly relative to the rest of the w orld. Thus there is a limit in which we can think of “the stimulus” as being a sin gle function of time, ν(t), which describes this angular velocity trajectory. It sho uld be clear that this description is incomplete: the neural response is affected also by the mean light intensity, the spatial structure of the visual st imulus, and the area of the compound eye that is stimulated. Further, the system is h ighly adaptive, so that the encoding of a short segment of the trajectory ν(t) will depend strongly on the statistics of this trajectory over the past several se conds. Traditional experiments on motion sensitive neurons (as on other sensory cells) have used constant stimuli (motion at fixed velocity) , pulsed stimuli (step- wise motion), or have analysed the steady state behaviour in response to sinu- soidal motion at different frequencies. In nature, trajecto ries are not so simple. Instead one can think of trajectories as being drawn from a di stribution P[ν(t)] or “stimulus ensemble.” A widely used example of stimulus en sembles is the Gaussian ensemble, in which the distribution of trajectori es is described com- pletely by the spectrum or correlation function. We can cons truct spectra and correlation functions so that there is a single characteris tic stimulus amplitude— the dynamic range νrmsof velocity signals—and a single characteristic time τc in the dynamics of these signals. A reasonable approach to th e study of natural- istic stimuli might then be to explore the coding of signals i n H1 using stimulus 4ensembles parametrized by νrmsandτc. Most of the results enumerated above have been obtained in this way. In their recent papers (Warzecha and Egelhaaf 1997, Warzech a et al. 1998), as well as in their contribution to this volume, Warzecha and Egelhaaf argue that the stimulus ensembles used in experiments on H1 have be en restricted unfairly to short correlation times. Put another way, the st imuli used in these experiments have included high temporal frequency compone nts. Warzecha and Egelhaaf suggest that these high frequency components bias the response of the motion sensitive cells to artificially high temporal precis ion which is not relevant for the behaviourally generated stimuli that they use.1The question of whether timing precision is important under truly natural conditio ns is left open. Independent of what is truly natural, one can argue that expe riments with short correlation times have provided evidence on what the fl y’s visual system can do. Although we seldom sit in dark rooms and wait for dim fla shes of light, such experiments led to the demonstration that the human vis ual system can count single photons (Hecht et al. 1942). In this spirit, stu dies of H1 using stimuli with short correlation times have revealed that the fly’s nervous system can estimate velocity with a precision limited by noise in th e photoreceptor array and that timing relations between neural responses an d stimulus events can be preserved with millisecond precision, even as the sig nals pass through four stages of neural circuitry. It would seem strange that s uch impressive performance would evolve if it were irrelevant for fly behavi our. Instead of choosing trajectories ν(t) from a known probability distribution, we could try to sample the trajectories that actually occur i n nature. Here we have to make choices, and these will always be somewhat subje ctive: Dethier (1976) reports that female flies spend 12.7%, and male flies 24 .3% of their time walking or flying. The other activities on Dethier’s list are feeding, regurgitating, grooming and resting, during which information from the fly’ s motion sensitive cells presumably is not too relevant. So it seems the fly could live quite happily without its tangential cells most of its time. On the other ha nd, during periods of flight, the responses of its motion sensitive cells are str ongly modulated. On top of that, the depth and speed of modulation may vary as the fl y switches from periods of relatively quiet cruising to episodes of fast and acrobatic pursuit or escape, and back (Land and Collett 1974). Although it is not c lear at the outset what portion of the total behavioural repertoire we should a nalyse, the thing that presumably tells us most about the “design” of the fly is t he dynamics of neural signal processing during top performance. Correspo ndingly, Warzecha 1In fact Warzecha and Egelhaaf make two different arguments ab out high frequency stimuli. They make repeated references to the integration times and n oise in the fly’s visual system, all of which limit the reliability of responses to high frequ ency components in the input. These arguments generally are presented in qualitative ter ms, but Warzecha and Egelhaaf (1999) state explicitly that signals above 30 Hz are undetec table above the noise and hence can have no impact on the statistics of the spike train. On the other hand, Warzecha and Egelhaaf (1997) argue that the inclusion of high frequency c omponents in the input causes an unnaturally tight locking of spikes to stimulus events, c ausing us to overestimate the significance of spike timing for the coding of behaviorally r elevant stimuli. It should be clear that these two arguments cannot both be correct. 5and Egelhaaf propose to use stimuli that are representative of the trajectories experienced by a fly in flight, and we agree that this is an excel lent choice. There are still some difficulties, however. Warzecha and Egel haaf propose that meaningful data can be obtained from “behaviourally genera ted” trajectories ν(t) recorded from flies that are tethered in a flight simulator ap paratus in which the fly’s measured torque is to move a pattern on a CRT monitor i n the visual field of the fly. The combination of fly, torque meter, and movin g pattern thus acts as a closed loop feedback system whose dynamical proper ties are determined both by the fly and by the gain and bandwidth of the mechanical a nd electronic components involved. The data presented by Warzecha and Ege lhaaf (1997, 1998, and this volume) strongly suggest that the dynamics of the feedback system are dominated by the electromechanical properties o f their setup, and not by the fly itself. This is most clearly seen from direct com parisons between the trajectories in the flight simulator and those observed i n nature. Trajectories during free flight were recorded in the classic work of Land and Collett (1974), who studied chasing behaviour in Fannia canicularis and found turning speeds of several thousand degrees per second . Wehrhahn (1979), Wehrhahn et al. (1982) and Wagner (1986a,b,c) report very si milar results for the housefly Musca, and recent publications (Schilstra and v an Hateren 1998, van Hateren and Schilstra 1999) report flight measurements a t high temporal and spatial resolution, from Calliphora flying almost free. In their published dataset flies made about 10 turns per second, during which hea d velocities easily exceeded 1000◦/s, while maximum head turning velocities were well over 3000◦/s. If we compare the results of these studies to the motion tr aces used in the experiments by Warzecha and Egelhaaf (1997, 1998) we see that their traces are considerably smoother, and do not go beyond 100◦/s. These differences are illustrated in Fig. 1, where we make an explicit comparison b etween free flight data obtained by Land and Collett (1974) and the motion trace s data presented in Fig. 1 of Warzecha and Egelhaaf (1997). It is clear that the re are dramatic differences in the frequency of alternation and, especially , in the amplitude of the motion signals. We are not sure how Egelhaaf and Warzecha can maintain their claim that “there are likely to be few instances in the normal world where visual motion encompasses a wider dynamic range than that which cou ld be tested here” (Warzecha et al. 1998, p. 362). Simple theoretical arg uments suggest that these differences between the flight simulator trajecto ries and true natural trajectories will have enormous consequences for the relia bility of responses in the motion sensitive neurons. Warzecha and Egelhaaf (Fig 6 o f their keynote paper in this volume) report estimates of the signal and nois e power spectra in the graded voltage response of a motion sensitive cell. If we scale the signal to noise power ratio they present in proportion to the ratio b etween the power spectrum of natural motion and the velocity power spectrum t hey used, then the signal to noise ratio will increase so much that the natur al trajectories will produce signal resolvable against the noise at frequen cies well above 200 Hz. This would mean that events in natural stimuli will be loc alizable with millisecond precision. There are other differences between the stimulus conditions studied by War- 6zecha and Egelhaaf and the natural conditions of free flight. Outdoors, in the middle of the afternoon, light intensities typically are tw o orders of magnitude larger than are generated with standard laboratory display s (Land 1981). Fur- ther, the wide field motion sensitive cells gather inputs fro m large portions of the compound eye (Gauck and Borst 1999), which extends backw ard around the head to cover a large fraction of the available solid angl e; rotation of the fly produces coherent signals across this whole area, and it i s very difficult to reproduce this “full vision” in the lab with CRT displays. Wh ile it is difficult to predict quantitatively the consequences of these differe nces, the qualitative effect is clear: natural signals are much more powerful and “c leaner” than the stimuli which Warzecha and Egelhaaf have used. We can take a substantial step toward natural stimulus condi tions by record- ing from a fly that itself rotates in a natural environment alo ng a trajectory representative of free flight. Preliminary results from suc h experiments will be analysed in more detail below, and a detailed account is fort hcoming (Lewen et al. in preparation). A female wild fly (Calliphora), caugh t outdoors, was placed in a plastic tube and immobilized with wax. A small inc ision was made in the back of the head, through which a microelectrode could be advanced to the lobula plate to record from H1. The fly holder, electrode h older and manip- ulator were assembled to be as light and compact, yet rigid, a s possible. In this way the fly and the recording setup could be mounted on the axle of a stepper motor (Berger-Lahr, RDM 564/50, driven by a Divi-Step D331. 1 interface with 10,000 steps/revolution) and rotated at speeds of up to seve ral thousand de- grees per second. The motor speed was controlled through the parallel port of a laptop computer by means of custom designed electronics, an d was played out at 2ms intervals. The data presented here are from an experim ent in which the setup was placed outside on a sunny day, in a wooded environme nt not far from where the fly was caught. A simple, but crucial, control is nec essary: H1 does not respond if the fly is rotated in the dark, or if the visual sc ene surrounding it rotates together with the fly. We can thus be confident that H1 i s stimulated by visual input alone, and not by other sensory modalities, and also that electronic crosstalk between the motor and the neural recording is negl igible. The motion trace ν(t) was derived from a concatenation of body angle readings ove r the course of the flight paths of a leading and a chasing fly as depic ted in Fig. 4 of Land and Collett (1974). For technical reasons we had to li mit the velocity values to half those derived from that figure, but we have no re ason to believe that this will affect the main result very much. Translationa l motion compo- nents were not present, representing a situation with objec ts only at infinity. Padded with a few zero velocity samples, this trace was 2.5 se conds long. That sequence was repeated with the sign of all velocity values ch anged, to get a full 5 second long sequence. This full sequence was played 200 tim es in succession while spikes from the axon terminals of H1 were recorded as an analog waveform at 10 kHz sampling rate. In off line analysis spike occurrence times were derived by matched filtering and thresholding. Before looking at the responses of H1, we emphasize several a spects of the stimulus conditions: 7•The motion stimulus is obtained from direct measurement of fl ies in free flight, not from a torque measurement of a tethered fly watchin g a CRT monitor. As argued above, the electromechanical propertie s of the setup used by Warzecha and Egelhaaf are likely to have drastic effec ts on the frequency and amplitude characteristics of the motion. •The field of view experienced by the fly in our setup is almost as large as that for a free flying fly. Most of the visual field is exposed to m ovement, with the exception of a few elements (e.g. the preamplifier) t hat rotate with the fly, and occupy just a small portion of the visual field . •The experiment is done outside, in an environment close to wh ere our experimental flies are caught, so that almost by definition we stimulate the fly with natural scenes. •The experiment is performed in the afternoon on a bright day. ¿From dim to bright patches of the visual scene the effective estimated photon flux for fly photoreceptors under these conditions varies from 5 ×105to 5×106 photons per second per receptor. Warzecha and Egelhaaf’s ex periments (as many experiments of ours) were done with a fly watching a Te ktronix 608 cathode ray tube, which has an estimated maximum photon fl ux of about 105photons per second per receptor. Figure 2 shows the spike trains generated by H1 in the “outdoo r” experiment, focusing on a short segment of the experiment just to illustr ate some qualitative points. The top trace shows the velocity waveform ν(t), and subsequent panels show the spikes generated by H1 in response to this trajector y (H1+) or its sign reverse (H1 −). Visual inspection reveals that some aspects of the respon se are very reproducible, and further that particular events i n the stimulus can be associated reliably with small numbers of spikes. The firs t stimulus zero crossing at about 1730 ms is marked by a rather sharp drop in th e activity of H1+, with a sharp rise for H1 −. This sharp switching of spike activity is not just a feature of this particular zero crossing, but occurs i n other instances as well. Further, the small hump in velocity at about 2080 ms las ts only about 10 ms, but induces a reliable spike pair in H1+ together with a sh ort pause in the activity of H1 −. The first spike in H1 −after this pause (Fig. 2c) is timed quite well; its probability distribution (Fig. 2e) has a standard deviation of 0.73 ms. Thus, under natural stimulus conditions individual spikes can be locked to the stimulus with millisecond precision. In fact the first few spikes after the pause in H1 −have even greater internal or relative temporal precision. The raster in Fig. 2c shows t hat the first spike meanders, in the sense that the fluctuation in timing from tri al to trial seems to be slow. This suggests that much of the uncertainty in the tim ing of this spike is due to a rather slow process, perhaps metabolic drift. To out side observers, like us, these fluctuations just add to the spike timing uncertain ty, which even then is still submillisecond. Note, however, that to some extent the fly may be able to compensate for that drift. If the effect is metabolic, then di fferent neurons might 8drift more or less together, and the time interval between sp ikes from different cells could be preserved quite well in spite of temporal drif t of individual spikes. Similarly, within one cell, spikes could drift together (Br enner et al., 2000), and this indeed is the case here. As a result the interval between the first spike and the next is much more precise, with a 0.18 ms standard deviato n, and it does not seem to suffer from these slow fluctuations (Fig. 2d). The timi ng accuracy of ensuing intervals from the first spike to the third and fourth , although becoming gradually less well defined, is still submillisecond (Fig. 2 f). So it is clear that some identifiable patterns of spikes are generated with a tim ing precision of the order of a millisecond or even quite a bit better. Although we have emphasized the reproducibility of the resp onses to natural stimuli, there also is a more qualitative point to be made. Al l attempts to characterize the input/output relation of H1 under laborat ory conditions have indicated that the maximum spike rate should occur in respon se to velocities below about 100◦/s, far below the typical velocities used in our experiments . Indeed, many such experiments suggest that H1 should shut do wn and not spike at all in response to these extremely high velocities. In par ticular, Warzecha and Egelhaaf (1998) claim that spike rates in H1 are essentia lly zero above 250◦/s, that this lack of sensitivity to high speeds is an essenti al result of the computational strategy used by the fly in computing motion, a nd further that this behaviour can be used to advantage in optomotor course c ontrol. The outdoor experiment demonstrates that none of these conclus ions are relevant to more natural conditions, where H1’s response peaks at about 1000◦/s (Lewen et al. in prep.) and responds robustly and reliably to angula r velocities of over 2000◦/s. The arguments presented here rested chiefly on visual inspec tion of the spike trains, and this has obvious limitations. Our eyes are drawn to reliable features in the response, and one may object that these cases could be a ccurate but rare, so that the bulk or average behaviour of the spike train is muc h sloppier. To proceed we must turn to a more quantitative approach. 3 How do we analyse the responses to natural stimuli? When we deliver simple sensory stimuli it is relatively easy to analyse some measures of neural response as a function of the parameters t hat describe the stimulus. Faced with the responses of a neuron to the complex , dynamic signals that occur in nature–as in Fig. 1–what should we measure? How do we quantify the response and its relation to the different features of the stimulus? The sequence of spikes from a motion sensitive neuron constitut es an encoding of the trajectory ν(t). Of course, this encoding is not perfect: there is noise in t he spatiotemporal pattern of the photon flux from which motion i s computed, the visual system has limited spatial and temporal resolution, and inevitably there is internal noise in any physical or physiological system. Thi s may cause identical 9stimuli to generate different responses. The code also may be ambiguous in the sense that, even if noise were absent, the same response c an be induced by very different stimuli. Conceptually, there are two very diff erent questions we can ask about the structure of this code. First, we can ask abo ut the features of the spike train that are relevant for the code: Is the timin g of individual spikes important, or does it suffice to count spikes in relativ ely large windows of time? Are particular temporal patterns of spikes especiall y significant? Second, if we can identify the relevant features of the spike train th en we can ask about the mapping between these features of the response and the st ructure of the stimulus: What aspects of the stimulus influence the probabi lity of a spike? How can we (or the fly) decode the spike train to estimate the stimu lus trajectory, and how precisely can this be done? There are two general approaches to these problems. One is to compute correlation functions. A classic example is the method of “r everse correlation” in which we correlate the spike train with the time varying in put signal (see Section 2.1 in Rieke et al. 1997). This is equivalent to compu ting the aver- age stimulus trajectory in the neighbourhood of a spike. Oth er possibilities include correlating spike trains with themselves or with th e spike trains of other neurons. A more subtle possibility is to correlate spike tra ins that occur on different presentations of the same time dependent signal, o r the related idea of computing the coherence among responses on different presen tations (Haag and Borst 1997). All of these methods have the advantage that sim ple correlation functions can be estimated reliably even from relatively sm all data sets. On the other hand, there are an infinite number of possible corre lation functions that one could compute, and by looking only at the simpler one s we may miss important structures in the data. An alternative to computing correlation functions is to tak e an explicitly probabilistic point of view. As an example, rather than comp uting the average stimulus trajectory in the neighbourhood of a spike, as in re verse correlation, we can try to characterize the whole distribution of stimuli in the neighbourhood of a spike (de Ruyter van Steveninck and Bialek 1988). Simila rly, rather than computing correlations among spike trains in different pres entations of the same stimulus, we can try to characterize the whole distribution of spike sequences that occur across multiple presentations (de Ruyter van Ste veninck et al. 1997, Strong et al. 1998). The probability distributions themsel ves can be difficult to visualize, and we often want to reduce these rather comple x objects to a few sensible numbers, but we must be sure to do this in a way that do es not introduce unwarranted assumptions about what is or is not important in the stimulus and response. Shannon (1948) showed that there is a unique way of doing this, and this is to use the entropy or information associated with the probability distributions. Even if we compute correlation functions, i t is useful to translate these correlation functions into bounds on the entropy or in formation, as is done in the stimulus reconstruction method (Bialek et al. 1991, R ieke et al. 1997, Haag and Borst 1997, Borst and Theunissen 1999). Although th e idea of using information theory to discuss the neural code dates back nea rly to the inception of the theory (MacKay and McCulloch 1952), it is only in the la st ten years 10that we have seen these mathematical tools used widely for th e characterization of real neurons, as opposed to models. 3.1 Correlation functions Although we believe that the best approach to analyzing the n eural response to natural stimuli is grounded in information theory, we follo w Warzecha and Egel- haaf and begin by using correlation functions. From an exper iment analogous to the one in our Fig. 2, Warzecha et al. (1998) compute the corre lation function of the spike trains of simultaneously recorded H1 and H2 cells, ΦspikeH1 −spikeH2 (τ), and also the average crosscorrelation function among spike trains from different presentations (trials) of the same stimulus traje ctory, ΦcrosstrialH1 −H2(τ). If the spike trains were reproduced perfectly from trial t o trial, these two correlation functions would be identical; of course this is not the case. Warzecha and Egelhaaf conclude from the difference between the two correlation functions that the spikes are not “precisely ti me coupled” to the stimulus, and they argue further that the scale which charac terizes the preci- sion (or imprecision) of spike timing can be determined from the width of the crosstrial correlation function Φ crosstrialH1 −H2(τ). This is one of their arguments in support of the notion that the time resolution of the spike train under nat- ural conditions is in the range of 40 to 100 milliseconds, one or two orders of magnitude less precision than was found in previous work. The crosstrial correlation function obviously contains in formation about the precision of the neural response, but there is no necessary m athematical re- lation between the temporal precision and the width of the co rrelation func- tion. To make the discussion concrete, we show in Fig. 3a the a utocorrelation Φspike−spike(τ) and in Fig. 3b the crosstrial correlation function Φ crosstrial (τ) computed for the outdoor experiment. We see that Φ crosstrial (τ) is very broad, while Φ spike−spike(τ) has structure on much shorter time scales, as found also by Warzecha and Egelhaaf. But the characterization of the cros strial correlation function as broad does not capture all of its structure: rath er than having a smooth peak at τ= 0, there seems to be a rather sharp change of slope or cusp, and again this is seen in the data presented by Warzecha and Eg elhaaf, even though the stimulus conditions are very different. This cusp is a hint that the width of the correlation function is hiding structure on muc h finer time scales. Before analyzing the correlation functions further, we not e some connections to earlier work. Intuitively it might seem that by correlati ng the responses from different trials we are probing the reproducibility of spike timing in some detail. But because Φ crosstrial (τ) is an average over pairs of spikes (one from each trial), this function is not sensitive to reproducible patterns of s pikes such as those we have seen in Fig. 2. In fact, the crosstrial correlation func tion is equal (with suitable normalization) to the autocorrelation function o f the time dependent rater(t) that we obtain by averaging the spike train across trials. T hus the crosstrial correlation does not contain information beyon d the usual poststim- ulus time histogram or PSTH, and the time scales in the correl ation function just measure how rapidly the firing rate can be modulated; aga in, there is no 11sensitivity to spike timing beyond the rate, and hence no sen sitivity to spike patterns. Since the crosstrial correlation function is equ al to the autocorrela- tion of the rate, the Fourier transform of Φ crosstrial (τ) is equal to the power spectrum of the rate, which has been used by Bair and Koch (199 6) to discuss the reproducibility of responses in the motion sensitive ne urons of monkey vi- sual cortex. If we Fourier transform both the crosstrial cor relation function and the spike-spike correlation, their ratio is proportional t o the crosstrial coherence considered by Haag and Borst (1997) in their analysis of H1. Even granting the limitations of the correlation function a s a probe of spike timing, we would like to reveal the finer time scale structure that seems to be hiding near τ= 0. To do this we consider a simple model that can be generaliz ed without changing the basic conclusions. Imagine that each s pike has an “ideal” time/angbracketleftti/angbracketrightrelative to the stimulus, and that from trial to trial the act ual arrival time of the ith spike fluctuates as ti=/angbracketleftti/angbracketright+δti. The meandering of spikes from trial to trial in Fig. 2c suggests that the δtiandδtjof nearby spikes i and j are correlated, and if these correlations extend over a sufficien tly long time (roughly 10 ms is sufficient) then there is a simple approximate equatio n relating the crosscorrelation among trials to the autocorrelation and t he distribution of time jitter, P(δti): Φcrosstrial (τ) = Φ PP(τ)⊗Φspike−spike(τ), where ⊗denotes convolution and Φ PP(τ) is the autocorrelation of the distri- bution P(δti). Thus Φ PP(τ) can be computed from the measured correlation functions by deconvolution. For our outdoor experiment we fi nd that Φ PP(τ) has a width of 3.1 ms (Fig. 3c), so that a reasonable estimate f or the width of the underlying jitter distribution is δtrms= 3.1/√ 2≈2.2 ms. This analysis shows that the difference between the crosstrial and the spik e-spike correlation functions is consistent with jitter in the range of a few mill iseconds, not the many tens of milliseconds claimed by Warzecha and Egelhaaf.2 Because the interpretation of correlation functions is a cr ucial issue, let us give an example from spatial vision, where it is clear that th e width of the correlation function (correlation length) is not a good ind icator of the precision required to read out a signal. It is well documented that natu ral scenes typically have broad spatial correlations, often associated with 1/f -like power density spectra (Srinivasan et al. 1982, Field 1987, Ruderman and Bi alek 1994). Using the same reasoning that Warzecha and Egelhaaf apply to spike trains, one would conclude that the visual system should not bother to use high spatial resolution. 2There are further difficulties in the interpretation of corre lation functions offered by War- zecha and Egelhaaf. One of their arguments for the irrelevan ce of high frequency stimuli is based on a comparison of the velocity spectrum with the spect rum of fluctuations in the time dependent rate (Fig. 2 of Warzecha and Egelhaaf 1997); the sp ectrum of the time dependent rate should be the Fourier transform of the crosstrial corre lation function, as noted above. On the down going slope, across a decade of frequency the decl ine in the response spectrum is slower than the decline in the stimulus spectrum. If we defi ne a transfer function by taking the ratio of the response and stimulus spectra, then the cell is amplifying the higher frequency components, not attentuating them as Warzecha and Egelhaaf claim. This is consistent with the experiments of Haag and Borst (1998) demonstrating that the motion sensitive neurons have active membrane mechanisms to achieve such amplificati on. 12This would be true for environments with Gaussian statistic s, where second order descriptions–the simplest correlation functions–a re sufficient. But the world we live in definitely is not Gaussian. It is made out of ob jects that typically have well defined edges, and these edges are import ant to us, not least because they are often associated with rigid objects. The width of the spatial correlation function is defined, very roughly, by th e apparent size of the objects in our visual field. But this width has nothing to do wi th the precision with which we can estimate the position of edges and hence the location of object boundaries. Just as for spatial edges, the location o f temporal edges may also be important, and we can look at horse racing for an ex ample: In Warzecha and Egelhaaf’s interpretation we would not need to time horses any more precisely than the width of the “horse density” correla tion function, which corresponds roughly to the time required for the entire hors e to cross the finish line. Yet fortunes are won and lost over differences correspo nding to a fraction of a horse’s nose. What matters here is that we attach importa nce to features that are defined very sharply in time, and this temporal preci sion cannot be measured from the width of one simple correlation function. For precisely the same reason one cannot equate the relevant time scale of reti nal image motion to spike timing precision, as Warzecha and Egelhaaf argue in this volume (Sect. 6). Let us then turn to an information theoretic approach. 3.2 Information Looking at the responses to repeated presentations of a natu ral complex dynamic stimulus, as in Fig. 2, we see many different features, some of which have been noted above: there are individual spikes which are reproduc ed from trial to trial with considerable accuracy; there are patterns of spikes in which the intervals between spikes are reproduced more accurately than the abso lute spike times, so that the patterns appear to ‘meander’ from trial to trial; there are trials in which spikes are deleted, apparently at random, and trial s in which extra spikes appear. How are we to make sense out of this variety of p henomena? Specifically, we want to know whether the detailed timing of s pikes is important for the encoding of naturalistic stimuli. How can we analyse data of this sort to give us a direct answer to this question about the structure o f the neural code? Intuitively, the sequence of action potentials generated b y H1 “provides in- formation” about the motion trajectory. If the response of H 1 were always the same, independent of the trajectory, of course no informati on would be pro- vided. Generally, then, the greater the range of possible re sponses the greater is the capacity of the cell to provide information: if we thin k of segments of the neural response as being like words in a language, then the ab ility of the neuron to ‘describe’ the input is enhanced if it has a larger vocabul ary. On the other hand, it clearly is not useful to generate words at random, no matter how large our vocabulary, and so there must be a reproducible relation ship between the choice of words and the form of the motion trajectory. These i ntuitive ideas have a precise formulation in Shannon’s information theory (Sha nnon 1948): the size of the neuron’s ‘vocabulary’ is measured by the entropy of th e distribution of 13responses, the (ir)reproducibility of the relation betwee n stimulus and response is related to the conditional or noise entropy computed from the distribution of responses seen in multiple trials, and the information th at the response con- veys about the stimulus is the difference between the entropy and the noise entropy (see also de Ruyter van Steveninck et al. 1997). Thes e measures from information theory are not just one of many possible ways of q uantifying the neural response; Shannon proved that these are the only meas ures of variabil- ity, reproducibility and information that are consistent w ith certain simple and intuitively plausible constraints. If we believe that the neural code makes use of a time resoluti on ∆t, then we can describe the neural response in discrete time bins of t his size. If ∆ tis very large this amounts to counting the number of spikes in ea ch bin, while as ∆tbecomes small this description becomes a binary string in wh ich we record the presence or absence of individual spikes in each bin. As o ur time resolution improves (smaller ∆ t) the size of the response ‘vocabulary’ increases because we are distinguishing as different responses that were, at larg er ∆t, lumped together as being the same. Quantitatively, the entropy of the respon ses is a function of time resolution, so that the capacity of the neuron to conv ey information is greater at smaller ∆ t, as first emphasized by MacKay and McCulloch (1952). The question of whether spike timing is important to the neur al code is then whether neurons make efficient use of this extra capacity (Rie ke et al. 1993, 1997). In the next section we address precisely this questio n in the context of the ‘outdoor’ experiment on H1, reaching conclusions that p arallel closely those from our earlier work (de Ruyter van Steveninck et al. 1997, S trong et al. 1998). First we consider the results of Egelhaaf and coworkers, who have drawn nearly opposite conclusions. Warzecha and Egelhaaf (1997) and Egelhaaf and Warzecha (199 9) set out to study the dependence of information transmission on time re solution, along the lines indicated above. Specifically, they count spikes in bi ns of size ∆ tand then ask how much information this spike count on a single trial pr ovides about the local firing rate, or “Stimulus Induced Response Component” (SIRC) computed as an average over many trials. Their information measure sh ows a peak for a window width of ∆ t= 80 ms (Warzecha and Egelhaaf 1997, Fig. 3), from which they conclude that this is the time resolution at which signals are best represented by H1. It is not clear what measure of informatio n Warzecha and Egelhaaf (1997) are using to find the optimum: the rate at whic h the spike train provides information about the stimulus must be a monotonic function of the time resolution. By marking spike arrival times more accura tely we can only gain, and never lose, information. Thus a proper measure of i nformation rate vs. time resolution cannot show the behavior reported by Warzec ha and Egelhaaf. In the present volume they substitute the information theor etical analysis by one in which they quantify the same difference (that is, betwe en the SIRC and a running window average count of the single trial spike trai n) by a standard deviation. This standard deviation reaches a minimum for a w indow width ∆ t of about 50ms. This analysis, as their correlation function analysis, is based on a consideration of second order statistics, and is therefor e subject to the same 14shortcomings discussed before. Both these approaches suffer from the same fundamental probl em: Warz- echa and Egelhaaf do not quantify the relation between the ne ural response and the stimulus, but instead between spike counts and the SIRC. Implicitly, then, they postulate that the stimulus is encoded exclusively in t he time dependent firing rate, or the SIRC as they prefer to call it, and further t hat all information about the local rate can be “read out” by counting spikes.3As in the analysis of crosstrial correlation functions, this ignores by const ruction the possibility that temporal patterns of spikes may play a special role in th e code, and their reasoning is therefore circular. For many investigators th is issue of whether pat- terns are important is the question about the structure of th e neural code, and in the case of H1 it is now more than a decade since de Ruyter van Steveninck and Bialek (1988) reported that patterns with short intersp ike intervals carry a considerable excess of information about the stimulus (see also Rieke et al. 1997 and Brenner et al. in press). The approach taken by de Ruyter v an Steveninck et al. (1997) and by Strong et al. (1998) describes the neural response at fine time resolution as a binary string, marking the presence or a bsence of spikes in each small time bin, and hence all patterns of spikes are incl uded automatically. This is the approach that we will use below for the analysis of the outdoor experiment. In principle, the methods used by de Ruyter van Steveninck et al. (1997) and by Strong et al. (1998) are independent of any model for th e structure of the neural code: we do not need to assume that we know which fea tures of the neural response are relevant, nor do we need to assume which f eatures of the stimulus are most important for the neuron. A number of resul ts on information transmission by H1 have been obtained with a less direct meth od, in which we use the spike train to reconstruct the stimulus and then meas ure the mutual in- formation between the stimulus and the reconstruction (Bia lek et al. 1991, Haag and Borst 1997, 1998). Warzecha and Egelhaaf emphasize that errors in the re- construction result only in part from noise, and they claim t hat one therefore cannot conclude anything about the reliability of neurons f rom the quality of reconstructions (Warzecha and Egelhaaf 1997; see also thei r contribution to this volume). The thrust of their argument is that there need be no conflict between their claim of imprecision in the coding of behaviourally re levant stimuli and previous work demonstrating precise reconstruction of the velocity waveforms, because the reconstruction doesn’t really measure the prec ision of the neural system. But this discussion ignores the fact that the recons truction method pro- vides a lower bound on the performance of the neuron (Rieke et al. 1997, Borst and Theunissen 1999). Thus it is possible that reconstructi on experiments un- derestimate the precision of neural coding and computation , but properly done the reconstruction method cannot overestimate neural perf ormance. 3Even if the changing stimulus serves only to modulate the spi ke rate, it might be that different rates can be distinguished more easily because, fo r example, the shape of the inter- spike interval distributon changes as function of rate. Thi s is known to occur in many cells. Mathematically, counting spikes is the optimal way of recov ering rate information only if the spike train is a modulated Poisson process. 15Since the reconstruction procedure is a bound on performanc e and not a di- rect measurement, it is reasonable to ask how tight this boun d will be. Warzecha and Egelhaaf state that the reconstruction of velocity sign als would underesti- mate the performance of the neuron if the cell is sensitive to derivatives of the velocity; specifically they claim that the coherence betwee n the stimulus and reconstruction would be reduced if the neuron were sensitiv e to derivatives. In fact, the particular reconstruction procedure of Bialek et al. (1991) is invariant to linear transformations of the signal such as differentiat ion and integration, and the computation of coherence always is invariant to thes e transformations (Lighthill, 1958). Is there any independent way to assess th e efficacy of the re- construction method? One approach is to try different recons truction algorithms (Warland et al. 1997). Another is to check for consistency am ong different mea- sures of coherence (Haag and Borst 1997). Finally, we can com pare the noise levels in the reconstructions with the noise levels that wou ld be generated by an ideal observer who is limited only by noise in the photorec eptors. In the high frequency limit (of order 30 Hz), where the ideal observ er’s performance can be calculated from photoreceptor measurements, the ide al observer does not perform substantially better than the reconstruction (Bia lek et al. 1991), which demonstrates that H1’s response approaches ideal observer performance. This can only be true if the fly’s visual brain makes efficient use of t he information present in the array of photoreceptors, and does not add a sub stantial amount of noise to the computation of motion. This finding is confirme d by measure- ments of neural performance that do not depend on reconstruc tions (de Ruyter van Steveninck and Bialek, 1995). Of course, the accuracy an d efficiency of the reconstructions also imply the functional correctness of the reconstruction algorithm. Criticism of the reconstruction algorithm itse lf cannot invalidate the demonstration of accurate reconstructions. 4 Information transmission with natural stimuli In the following we use methods described in detail by Strong et al. (1998) to quantify information transmission in our natural motion ex periment. Briefly, we analyse the statistics of firing patterns that H1 produces in response to the stimulus used in our experiment, and consider segments of th e spike train with length Tdivided in a number of bins of width ∆ t, where ∆ twill range from very small (order of a millisecond) up to ∆ t=T. Each such bin may hold a number of spikes, and within a bin no distinction is made on wh ere the spikes appear. However, two windows of length Tthat have different combinations of filled bins are considered to be different firing patterns, a nd are therefore distinct. From an experiment in which we repeat a reasonably long natural stimulus a number of times (here 200 repetitions of a 5 second s long sequence) we get a large number of these firing patterns, and from that se t we compute two entropies: 1. The total entropy, which characterizes the probability d istribution of all spike firing patterns of length Tthat consist of nadjacent bins each ∆ t 16wide (that is, T=n∆t). This entropy measures the richness of the ‘vo- cabulary’ used by H1 under these experimental conditions, h ence the time of occurrence of the pattern within the experiment is irrele vant. 2. The noise entropy, which gives us an estimate of how variab le the response to identical stimuli can be. We first accumulate, for each poi nt in time in the stimulus sequence, the distribution across all trials o f firing patterns that begin at that point. The entropy of this distribution me asures the (ir)reproducibility of the response at each instant. Calcu lating this for each point in time and averaging all these values we obtain th e average noise entropy. The information contained in firing patterns of length Tand resolution ∆ tis the total entropy minus the average noise entropy (Shannon 1948 ). One interesting measure is to estimate this information as we let Tbecome very long, and ∆tvery short. This limit is the average rate of information tra nsmission, as discussed by Strong et al. (1998). Here, instead, we will jus t calculate the information transmitted in constant time windows, T= 30 ms, as a function of ∆t. We choose T= 30 ms because that amounts to the delay time with which a chasing fly follows turns of a leading fly during a chase (Land and Collett, 1974); the end result, namely the dependence of information transmission on ∆t, does not depend critically on the choice of T. The data in Fig. 4a show that the information contained in a 30 ms window depends strongly on ∆ t, increasing from about 2 bits to about 5 bits when the resolution increases from ∆ t= 30 ms to ∆ t= 1 ms. Although in the limit of arbitrarily fine time resolution (∆ t→0), the information must reach a finite limit, we see no evidence for a plateau at ∆ t= 1 ms. For shorter time windows (T= 12 ms) we find that the information keeps on increasing up to ∆ t= 0.25 ms. This lack of a clear plateau makes sense: the motion sti muli themselves have a distribution of temporal features so it is not surpris ing that there is not a sharply defined single timescale in the response. We also not e that, as in earlier work with less natural stimuli (Strong et al. 1998), the info rmation rate is a bit more than half the total entropy, even at millisecond resolu tion (see Fig. 4b), so the neuron utilizes a significant fraction of its coding ca pacity even on this fine time scale. The question of whether spike timing is important in the neur al code has been debated for decades, and our present experiment addres ses the importance of millisecond resolution in information transmission by a single cell. Ultimately one would like to connect the responses of neurons to animal b ehaviour. Thus, one way to demonstrate the importance of spike timing would b e to search for experimental conditions in which the timing of just a few spikes would be correlated with a behavioral decision, in the spirit of the w ork by Newsome and colleagues (Newsome et al. 1995). Another approach is to look for other neurons that can “read” the temporal structure, for example along the lines of recent work from Usrey et al. (1998). Here we focus on the resp onse of a single neuron, and ask if the precise timing of spikes carries infor mation under natural stimulus conditions. The answer is yes. 175 Responses to static and dynamic stimuli The measured precision of responses in H1 to dynamic stimuli seems to suggest that the behavior of the fly visual system might be very differe nt from other sys- tems, especially the mammalian cortex. Neurons in visual co rtex, for example, commonly show a large variance in the responses across repea ted presentations of the same visual stimulus (Tolhurst et al. 1983). To quanti fy this observa- tion several groups have studied the variance in the number o f spikes that are counted in a window of fixed size, and then manipulated the sti mulus condi- tions to find the relation between the variance of the respons e and its mean. Typically, the variance in spike count is found to be close to or somewhat larger than the mean over a wide range of conditions; there is a tende ncy for the ratio variance/mean (the Fano factor) to be larger in larger time w indows. More recently several groups are investigating to what exte nt accurate spike timing, such as observed in H1, can be consistent with the var iability of neu- ral responses observed in cortex. Almost all experiments on the variability of responses in visual cortex had been done with static or slowl y varying stimuli, while all the work indicating precise responses and the impo rtance of spike tim- ing in H1 had been done using complex, dynamic inputs. Newsom e and collab- orators studied the responses of motion sensitive neurons i n the monkey visual cortical area MT using dynamic random dot stimuli, but their work focused on the connection of neural responses to the monkeys percept ion of coherent motion in the entire display (Newsome et al. 1995). Bair and K och (1996) reanalysed some of these data to show that when the monkey saw exactly the same dynamic dot movies the neural response showed significa nt modulations on a time scale of 30 ms or less. Strong analysed the same data t o show that the spike train of a cortical neuron could provide information a bout the movie at a rate of ∼2 bits/spike, comparable to the results in H1 (see note 19 in S trong et al. 1998), and in unpublished analyses he found that the vari ance of the spike count in windows of 30 ms or less could be significantly less th an the mean. Mainen and Sejnowski (1995) found that they could produce ir regular spike trains in a slice of cortex if they injected constant current into a neuron: after some time the cell ‘forgets’ the time at which the current was turned on and the spikes drift relative to the stimulus. With dynamic current s, however, there can be precise temporal locking of spikes to particular events i n the input signal. Berry et al. (1997) found that ganglion cells in the vertebra te retina—which are known to generate irregular and highly variable spike tr ains in response to static or slowly varying images—generate highly reproduci ble spike trains in response to more dynamic movies. A hint in the same direction had been found earlier by Miller and Mark (1992), who showed that primary au ditory neurons in the cat give less variable responses to complex speech sti muli than to pure tones. Finally, de Ruyter van Steveninck et al. (1997) showe d explicitly that the low variance, reproducible response of H1 to dynamic sti muli coexists with a much more variable response to constant velocity inputs: s tudying a range of constant velocities that drove H1 to average firing rates up t o about 70 spikes per second (which corresponds to the time average rate elici ted by dynamic 18stimuli in comparable stimulus conditions), mean counts an d variances in 100 ms windows straddled the line at which variance is equal to me an, and fell well within a cloud of points obtained from experiments in visual cortex. Taken together, all of these different results point to the co nclusion that the statistical structure of the neural response to static stim uli may be very different from that in response to dynamic or naturalistic stimuli. Th e crucial conclusion is that we cannot extrapolate from the observation of highly variable responses under one set of conditions to reach conclusions about the st ructure of the neu- ral code under more natural conditions. This fits very well wi th the ethological perspective that we introduced at the beginning of this cont ribution, and in- deed many of the analysis methods that we have discussed here were developed to meet the challenges of quantifying the neural response to more naturalistic stimuli. From a more mechanistic point of view there is now co nsiderable inter- est in understanding why neurons seem to respond so different ly to static and dynamic inputs (Schneidman et al. 1999, Jensen 1998). Against this background it came as a surprise when Warzecha a nd Egelhaaf (1999) claimed that the variance of H1’s response to constan t velocity is no dif- ferent from that in response to dynamic stimuli. It would app ear that they have done an experiment very similar to that described by de Ruyte r van Steveninck et al. (1997) but reached the opposite conclusion: while War zecha and Egelhaaf confirm the highly reproducible, low variance response to dy namic stimuli, they find similarly reproducible responses to constant velociti es. There are many issues here, but we focus first on the explicit disagreement r egarding the vari- ability of responses to constant velocity. Warzecha and Ege lhaaf themselves offer several possible explanations for the discrepancy, bu t they do not draw attention to the fact that the stimuli used in the two sets of e xperiments differ substantially; these differences exist along every stimulu s dimension known to affect the response of H1—velocity, image contrast, spatial pattern, and size of the visual field. Further, in the crucial comparison of stati c to dynamic stimuli, it is not clear what is being held constant in the Warzecha and Egelhaaf exper- iments. In the dynamic experiments changes in spike rate are of course driven by variations in angular velocity, but in their static exper iments they hold the velocity fixed and vary the image size. At best these experime nts show that H1 responds with different statistics under different conditio ns, but we still find the discrepancies disturbing. In an attempt to resolve the issue, we have gone back over seve ral years of experiments to collect all the data which may be relevant to r elation between variance and mean in static experiments, we have done new exp eriments that come close to the conditions of the Warzecha and Egelhaaf wor k, and we have designed new stimuli that highlight the differences between static and dynamic responses in a single experiment. In brief, studying 20 flies under a wide variety of static stimulus conditions, we find a broad distribution o f variances at each value of the mean, but up to rates of about 100 spikes/s there i s no overlap with the results of Warzecha and Egelhaaf (1999). Further, w hen we match the conditions of their experiments we cannot reproduce eve n the mean spike counts, let alone the variances. For example, their Fig. 4A i n this volume shows 19a mean spike count of about 6.5 in 100 ms windows for a high cont rast large field (91◦×7.5◦) pattern moving at about 36◦/s, and for the same experimental conditions Warzecha and Egelhaaf (1999) report a mean count of about 4. These values correspond to mean firing rates of 65 and 40 spikes/s re spectively. In 8 flies tested under comparable conditions (contrast, veloci ty and stimulated area) we never get rates below 120 spikes/s, consistent with the fin dings of Lenting et al. (1984). In Fig. 5 we show the response of H1 to a slowly varying velocit y ramp, and contrast this response to that obtained with dynamic veloci ties. Computing the mean spike counts in 100 ms windows across 50 trials, we see th at the static and dynamic stimuli give the same range of mean responses, yet wh en we compute the variances there are huge differences that are obvious to t he eye. The count variance during quasistatic stimulation peaks for mean cou nts that are about equal to the average count during dynamic stimulation (that is, in the two places where the dashed line in Fig. 5b intersects the smooth curve) . This is not just a coincidence of our choice of standard deviation of the dyna mic stimulus; it turns out that the fly’s visual system adapts such that the mea n firing rate during dynamic stimulation is rather insensitive to the sta ndard deviation of the dynamic stimulus (Brenner et al., submitted). For highe r values of the mean count the variance decreases strongly, due to the effect s of refractoriness (Hagiwara, 1954). So H1 has relatively low count variance bo th for low and high rates, but its count variance is high for intermediate rates . Loosely, one may think of the dynamic stimulus as switching the cell rapidly b ack and forth from a state of low rate and low variance to a state of high rate and l ow variance. By switching fast, the cell effectively bypasses the intermedi ate condition of high variance, so that its count variance for dynamic stimuli alw ays remains low, as can be seen directly from Fig. 5c. Thus, if we match windows wi th the same mean count, up to a count of about 10 for 100 ms windows, we find t hat H1’s count variance is lower in response to dynamic than to static stimuli, which was precisely the point of the original work by de Ruyter van S teveninck et al. (1997). 6 Conclusion Most of what we know about the nervous system has been learned in experiments that do not even approach the natural conditions under which animals normally operate. Much of our recent work, and the core of the debate be tween our groups and Warzecha and Egelhaaf, concerns the structure of the neu ral code under natural conditions. We emphasize that this is not an easy pro blem, and by no means are the issues specific to flies or even the visual system ; in many different sensory and motor systems we would like to design and analyse experiments on the coding and processing of more natural signals. Our approach has been to break this large question into (hope fully) man- ageable pieces, and then to use information theory as a frame work to pose these questions in a form such that suitable experiments should yi eld precise quan- 20titative answers. In particular, we endeavour to make state ments that do not depend on multiple prior assumptions, and to develop method s which can be used in analyzing many different kinds of experiments. Thus, we have used stimulus reconstruction techniques to give lower bounds on the performance of fly motion sensitive neurons, and we have been able to measure the average in- formation carried by single spikes, patterns of spikes, and continuous segments of the spike train, all without assumptions regarding the “i mportant” features of the stimulus or neural response. Many of the results obtaine d in this way point clearly toward a picture of the fly’s visual system as close to optimal in extract- ing motion information from the photoreceptor cell array, a nd then encoding this signal efficiently in the timing of action potentials of motio n sensitive neurons. In contrast to the view developed over the past decade, the re cent papers from Egelhaaf and coworkers, including their contribution to th is volume, make the explicit claim that the system is very noisy and that meaning ful information is contained only in averages over time windows containing man y spikes. In many cases these claims are introduced with plausible qualitati ve arguments. As em- phasized long ago by Bullock (1970), however, the challenge is to quantify the degree of noisiness or precision in the nervous system, and t here is a danger that a neuron may appear noisy because we have an incomplete under standing of its function. Thus we have grown skeptical about qualitative or even semiquanti- tative arguments for the imprecision of neural responses. T he interpretation of correlation functions, discussed above, provides a good ex ample: Although there may be an obvious “correlation time” in one correlation func tion, the hint that other time scales are relevant is hidden in the cusp of the cor relation function at short times. More detailed analysis shows that the relation s among correlation functions are consistent with temporal precision on scales a factor of 30 smaller than the nominal correlation time. Further, this measure of temporal precision is consistent with the results of a rigorous information the oretic analysis. Because this paper is intended (by the editors) as a response to the contri- bution of Warzecha and Egelhaaf, we have tried to understand how they have reached conclusions so nearly opposite from our own. As emph asized at the out- set, there are two different questions. First there is the pro blem of constructing an approximately natural stimulus, and then there is the pro blem of analyzing the response to such a complex signal. Although there is a who le generation of quantitative observations on insect flight trajectories, W arzecha and Egelhaaf present as the stimulus they analyse a signal that is substan tially impoverished both in amplitude and in frequency content, as is clear from F ig. 1. Further, their visual stimulus is very dim compared to daytime natura l conditions, and has a visible area much smaller than what is experienced by a f ree flying an- imal. They repeatedly stress that motion induced responses depend on many stimulus variables in addition to velocity, but never discu ss critically the ex- trapolation from their experiments to natural behaviour, i n spite of the large differences in many of the crucial variables. Similarly, alt hough there is now a decade of papers concerning the quantitative information t heoretic analyses of neural spike trains, and of the cell H1 in particular, Warzec ha and Egelhaaf do not present their results on information transmission in ab solute units (bits); 21closer examination suggests that there are more basic mathe matical problems in their approach, as outlined above. In this paper we have presented the results from a new experim ent (Lewen et al. in preparation) which brings us much closer to the natura l conditions of fly vision. Visual inspection of the responses of H1 under these conditions indicates that individual spikes are reproducible on a millisecond ti me scale, and aspects of temporal pattern in the spike train can be reproducible on a substantially submillisecond time scale. This impression is borne out by t he quantitative demonstration that the spike train conveys information wit h nearly constant efficiency down to millisecond time resolution; indeed, the i nformation provided by the spike train shows no sign of saturation as we approach m illisecond res- olution. These responses to natural stimuli thus are even mo re precise than suggested by our earlier work. In the early 1960’s Reichardt and his coworkers started work ing on flies, with a special emphasis on motion detection (Reichardt, 196 1). One of their motivations was that motion detection in flies represents a g ood compromise between a reasonable complexity of information processing properties and an amenability to quantitative analysis (Reichardt and Poggi o 1976). Over the years this intuition has proved to be very fruitful, and the fl y has turned out to be a system in which many issues could be studied, often with u nprecedented quantitative detail. 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Comparison of the rotational velocity traces reported from free flying and tethered flies. a: Rotation velocity of a fly (Fannia canic ularis) in free flight, derived from video recordings by Land and Collett (1974). b: Rotation velocity of a pattern in a flight simulator, derived from torque signal s measured from a tethered fly, as reported by Warzecha and Egelhaaf (1997). c: The data from a and b plotted on the same scale. Figure 2. Direct observations of H1 spike timing statistics in respon se to rotational motion derived from Land and Collett’s (1974) fr ee flight data (see Fig. 1a). a: A 500 ms segment of the motion trace. b: Top: raste r plot with 25 traces representing spike occurrences measured from H1. Bo ttom: raster plot of 25 traces of spike occurrences from the same cell, but in resp onse to a velocity trace that was the negative of the one shown in a. For ease of re ference we call these traces H1+ and H1 −respectively. c: Raster plot of 25 samples of the occurrence time of the first spike fired by H1 −after time t= 2080 ms in the stimulus sequence (indicated by the dashed line connect ing the axis of b to panel c). d: Raster plots of 25 samples of the interval from th e spike shown in c to the first (filled circles), second (open circles), third ( filled triangles), and fourth (open triangles) spike following the spike shown in c . Note the time axes: The rasters in c and d are plotted at much higher time resoluti on than those in b. e: Probability density for the timing of the spike shown in c. The spread is characterized by σ= 0.73 ms, where σis defined as half the width of the peak containing the central 68.3% of the total probability. If th e distribution were Gaussian, then this would be equivalent to the standard devi ation. Here we prefer this definition instead of one based on computing seco nd moments. The motivation is that there can be an occasional extra spike, or a skipped spike, giving a large outlier which has a disproportionate effect on the width if it is calculated from the second moment. Filled squares represen t the experimental histogram, based on 200 observations, while the solid line i s a Gaussian fit. f: Probability densities for the same interspike interval sho wn in d. The definition ofσis the same as the one in e. Figure 3. Correlation functions for H1 during stimulation with natur al mo- tion, all computed at 0.2 ms resolution. a: The spike–spike a utocorrelation Φspike−spike(t), normalized as a conditional rate. There are strong oscill ations in the conditional rate, due to neural refractoriness. b: The c ross-trial correlation function Φ crosstrial (t), computed as the correlation function of the estimated time dependent rate minus a contribution from Φ spike−spike(t) scaled by 1 /N (Nis the number of trials) to correct for intratrial correlati ons. c: Autocorre- lation of the assumed underlying distribution of spike jitt er times, computed by deconvolving the data in b by those in a. See text for further e xplanation. Figure 4. Information and coding efficiency in firing patterns for natur alistic motion stimuli (see legend for Fig. 2). a: Total entropy, noi se entropy and infor- 27mation in an observation window of T= 30 ms, as a function of time resolution, ∆t. From the trends observed in partitioning the finite dataset we estimate first and second order extrapolations to the entropies for an infinite dataset. Filled symbols are first order, open symbols are second order extrapolations. The deviation between first and second order extrapolations is small, indicating that systematic errors in the entropy estimates are small (f or details see Strong et al. 1998). Statistical errors were estimated from the spr ead in the differ- ent partitions of the original dataset. These errors are sma ller than the size of the symbols. b: Coding efficieny (information divided by tota l entropy) as a function of ∆ t. Figure 5. Mean count and variance compared for quasistatic and dynami c velocity stimuli. a: Stimulus. for the first 48 s the velocity is slowly ramped up and down. From 50 to 72 seconds the stimulus is dynamic with a s tandard devi- ation of 100◦/s, and a cutoff frequency of 250 Hz. Note that the vertical sca les (left for the quasitationary and right for the dynamic stimu lus) are different. The peak at 50 s is a reset phase in which the pattern is moved at maximum speed so as to bring the stimulus pattern into exact register on every trial. b: Trial average spike count in 100 ms windows, as a function of t ime. The dashed line represents the time averaged count in response to the dy namic stimulus. c: Spike count variance in 100 ms windows, as a function of time. 28de Ruyter et al. FIG 1 angular velocity ( °/s) -100-50050-200002000a b c time (s)0.0 0.5 1.0 1.5 2.0-200002000de Ruyter et al. FIG 2.time (ms)1700 1800 1900 2000 2100 2200velocity (°/s) -2000-1000010002000 interval from spike 0 (ms)0 5 10 15010002000a b c d location of spike 0 (ms)2085 2090prob. density (1/s) 010002000 σ=0.73 msσ=0.18 ms σ=0.46 ms σ=0.77 msσ=0.84 msH1+ H1- H1- e f2000conditional rate (1/s) 010020030040050060049005000conditional rate (1/s)0100200300400 time (ms)-50 -40 -30 -20 -10 0 10 20 30 40 50jitter autocorrelation (1/s) 050100150a b c de Ruyter et al. FIG 3.T = 30 ms time resolution (ms)1 10entropy (bits) 0246810 total entropy, 1st order total entropy, 2nd order noise entropy, 1st order noise entropy, 2nd order information, 1st order information, 2nd orderT = 30 mscoding efficiency 0.00.20.40.60.81.0 de Ruyter et al. Fig. 4abvelocity (/s ) -20-1001020 velocity (°/s) -400-2000200400mean count in 100 ms 01020 time (s)0 10 20 30 40 50 60 70count variance in 100 ms 0246 de Ruyter et al. FIG 5a b c

arXiv:physics/0004061v1 [physics.ins-det] 25 Apr 2000STUDY OF NUCLEATING EFFICIENCY OF SUPERHEATED DROPLETS BY NEUTRONS B.ROY, B.K. CHATTERJEE, M. DAS and S.C. ROY Department of Physics, Bose Institute, Calcutta 700009, In dia Superheated droplets have proven to be excellent detectors for neutrons and could be used as a neutron dosimeter. To detect accurately th e volume of superheated droplets nucleated, an air displacement system has been dev eloped. Here the air expelled due to volume change upon nucleation displaces a co lumn of water through a narrow horizontal glass tube, and, the displacement of wat er is linearly related to the nucleated volume and has the added advantage of being lea k free. In presence of neutrons, the rate of nucleation (rate of decr ease in the volume of superheated droplets) is proportional to the residual volu me of superheated droplets and the neutron flux ( φ). Hence the volume of accumulated vapour (or the volume of the displaced air) is given as: V=V0/parenleftBig 1−e−t/τ/parenrightBig (1) where τ= M/( φρηN Ad/summationtextniσi), M is the molecular weight and ρis the density of the superheated liquid, NAis the Avogadro number, niis the abundance of the ithspecies of nucleon (whose neutron elastic scattering cross section isσi) in the molecule, ηis the efficiency of nucleation of the droplet by a recoil nucleon , d is the average droplet volume and V0is the volume of vapour of the entire superheated liquid. By l east squares fitting of the volume of displaced air (V) as a functio n of time (t), V0andτ are obtained. From τone may obtain ηif the other parameters are known. Results of an experiment performed with Freon- 12 samples using an Am -Be neutron source are presented in Fig. 1. Figure Captions Fig.1. Variation of volume (scaled by the mass of sample and the area of cross section of the tube) of displaced air as a function of time for Freon-12; solid curve is the least-squares-fit to experimental data. 1

arXiv:physics/0004062v1 [physics.atom-ph] 25 Apr 2000Testing cosmological variability of fundamental constants D. A. Varshalovich, A. Y. Potekhin, and A. V. Ivanchik Ioffe Physico-Technical Institute, 194021 St.Petersburg, Russia Abstract. One of the topical problems of contemporary physics is a poss ible variability of the fundamental constants. Here we consider possible var iability of two dimensionless constants which are most important for calculation of atomi c and molecular spectra (in particular, the X-ray ones): the fine-structure constant α=e2/¯hcand the proton-to- electron mass ratio µ=mp/me. Values of the physical constants in the early epochs are estimated directly from observations of quasars – the mo st powerful sources of radiation, whose spectra were formed when the Universe was s everal times younger than now. A critical analysis of the available results leads to the conclusion that present-day data do not reveal any statistically significan t evidence for variations of the fundamental constants under study. The most reliable up per limits to possible variation rates at the 95% confidence level, obtained in our w ork, read: |˙α/α|<1.4×10−14yr−1,|˙µ/µ|<1.5×10−14yr−1 on the average over the last 1010yr. INTRODUCTION Contemporary theories (SUSY GUT, superstring and others) n ot only predict the dependence of fundamental physical constants on energy1, but also have cos- mological solutions in which low-energy values of these con stants vary with the cosmological time. The predicted variation at the present e poch is small but non- zero, and it depends on theoretical model. In particular, Da mour and Polyakov [1] have developed a modern version of the string theory, who se parameters could be determined from cosmological variations of the coupling constants and hadron- to-electron mass ratios. Clearly, a discovery of these vari ations would be a great step in our understanding of Nature. Even a reliable upper bo und on a possible variation rate of a fundamental constant presents a valuabl e tool for selecting viable theoretical models. 1)The prediction of the theory that the fundamental constants depend on the energy of interac- tion has been confirmed in experiment. In this paper, we consi der only the space-time variability of their low-energy limits.Historically, a hypothesis that the fundamental constants may depend on the cosmological time t(that is the age of the Universe) was first discussed by Milne [ 2] and Dirac [3]. The latter author proposed his famous “large- number hypothesis” and suggested that the gravitational constant was directly proportional to t. Later the variability of fundamental constants was analyzed, usi ng different arguments, by Gamow [4], Dyson [5], and others. The interest in the probl em has been revived due to recent major achievements in GUT and Superstring mode ls (e.g., [1]). Presently, the fundamental constants are being measured wi th a relative error of∼10−8. These measurements obviously rule out considerable varia tions of the constants on a short time scale, but do not exclude their chan ges over the lifetime of the Universe, ∼1.5×1010years. Moreover, one cannot rule out the possibility that the constants differ in widely separated regions of the U niverse; this could be disproved only by astrophysical observations and different kinds of experiments. Laboratory experiments cannot trace possible variation of a fundamental con- stant during the entire history of the Universe. Fortunatel y, Nature has provided us with a tool for direct measuring the physical constants in the early epochs. This tool is based on observations of quasars, the most powerful s ources of radiation. Many quasars belong to most distant objects we can observe. L ight from the dis- tant quasars travels to us about 1010years. This means that the quasar spectra registered now were formed ∼1010years ago. The wavelengths of the lines ob- served in these spectra ( λobs) increase compared to their laboratory values ( λlab) in proportion λobs=λlab(1 +z), where the cosmological redshift zcan be used to determine the age of the Universe at the line-formation epoc h. In some cases, the redshift is as high as z∼3−5, so that the intrinsically far-ultraviolet lines are reg- istered in the visible range. The examples are demonstrated in Fig. 1. Analysing these spectra we may study the epoch when the Universe was sev eral times younger than now. Here we review briefly the studies of the space-time variabil ity of the fine- structure constant αand the proton-to-electron mass ratio µ. FINE-STRUCTURE CONSTANT Various tests of the fundamental constant variability diffe r in space-time regions of the Universe which they cover. Local tests relate to the values of constants on the Earth and in the Solar system. In particular, laboratory tests infer the possible variation of certain combinations of constants “here and no w” from comparison of different frequency standards. Geophysical tests impose constraints on combina- tions of fundamental constants over the past history of the S olar system, although most of these constraints are very indirect. In contrast, astrophysical tests allows one to “measure” the values of fundamental constants in dist ant areas of the early Universe./G34 /G30 /G30 /G30 /G34 /G30 /G31 /G30/G7a /G20 /G20 /G3d /G20 /G32 /G2e /G38 /G31 /G30 /G38 /G30 /G50 /G4b /G53 /G20 /G30 /G35 /G32 /G38 /G2d /G32 /G35/G61 λ /G6f /G62 /G73/G31 /G30 /G35 /G30 /G31 /G30 /G35 /G33 λ /G6c /G61 /G62 /G48/G32 /G52 /G28 /G30 /G29 /G20 /G20 /G20 /G20 /G52 /G28 /G31 /G29 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G50 /G28 /G31 /G29 /G20 /G20 /G52 /G28 /G32 /G29 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G20 /G50 /G28 /G32 /G29 /G20 /G20 /G20 /G20 /G20 /G52 /G28 /G33 /G29 /G35 /G33 /G36 /G30 /G35 /G34 /G30 /G30/G31 /G33 /G39 /G30 /G31 /G34 /G30 /G30 /G7a /G20 /G20 /G3d /G20 /G32 /G2e /G38 /G34 /G33 /G33 /G35 /G48 /G53 /G20 /G31 /G39 /G34 /G36 /G2b /G37 /G36/G61/G53 /G69 /G20 /G49 /G56 /G4c /G20 /G20 /G34 /G20 /G2d /G20 /G30 FIGURE 1. Portions of quasar spectra which show the absorption lines o f H2and Si ivwith large redshifts, z= (λobs−λlab)/λlab≈2.8. The lower horizontal axis gives the wavelengths in the observer’s frame ( λobs) and the upper axis gives the wavelengths in the quasar’s fra meλlab(in ˚A). (a) The spectrum (thin line) of the quasar PKS 0528–250, o btained with the 4-meter CTIO telescope (Chile), containing H 2lines which belong to the L 4–0 branch of the spectrum; thick line plots the spectral fit. (b) The spectrum of the quasar HS 1 946+76, obtained with the 6-meter SAO telescope (Russia), containing Si ivdoublet lines which correspond to the2S1/2→2P3/2and 2S1/2→2P1/2transitions. Local tests Laboratory experiments There were a number of laboratory experimets aimed at detect ion of trends of the fundamental constants with time by comparison of freque ncy standards which have different dependences on the constants. We mention only two of the published experiments. Comparison of H-masers with Cs-clocks during 427 days revea led a relative (H– Cs) frequency drift with a rate 1 .5×10−16per day, while the rates of (H-H) and (Cs–Cs) drifts (i.e., the drifts between identical standar ds, used to control their stability) were less than 1 ×10−16per day [6]. A similar result was found in comparison of a Hg+-clock with a H-maser during 140 days [7]: the rate of the relative frequency drift was less than (2 ±1)×10−16per day. Such a drift is treated as a consequence of a difference in the l ong-term stabilityof different atomic clocks. In principle, however, it may be c aused by variation of α. That is why it gives an upper limit to the αvariation [7]: |˙α/α| ≤3.7×10−14yr−1. Geophysical tests The strongest bound on the possible time-variation rate of αwas derived in 1976 by Shlyakhter [8], and recently, from a more detailed an alysis, by Damour and Dyson [9], who obtained |˙α/α|<0.7×10−16yr−1, The analysis was based on measurements of isotope ratios in the Oklo site in Gabon, w here a unique natural uranium nuclear fission reactor had operated 1.8 bil lion years ago. The isotope ratios of samarium produced in this reactor by the ne utron capture reaction 149Sm+n→150Sm+γwould be completely different, if the energy of the nuclear resonance responsible for this capture were shifted at leas t by 0.1 eV. Another strong bound, |˙α/α|<5×10−15yr−1, was obtained by Dyson [5] from an isotopic analysis of natural radioactive decay products in meteorites. A weak point of these tests is their dependence on the model of the phenomenon, fairly complex, involving many physical effects. For instan ce, Damour and Dyson [9] estimated possible shift of the above-mentioned resona nce due to the αvariation, assuming that the Coulomb energy of the excited state of150Sm∗, responsible for the resonance, is not less than the Coulomb energy of the ground state of150Sm. In absence of experimental data on the nuclear state in quest ion, this assumption is not justified, since heavy excited nuclei often have Coulo mb energies smaller than those for their ground states [10]. Furthermore, a corr elation between the constants of strong and electroweak interactions (which is likely in the frame of modern theory) might lead to further softening of the mentio ned bounds by 100- fold, to |˙α/α|<5×10−15yr−1, as noted by Sisterna and Vucetich [11]. In addition, the local tests cannot be extended to distant sp ace regions and to the early Universe, since the law of possible space-time var iation of αis unknown a priory . It is the extragalactic astronomy that allows us to study th ese remote regions of spacetime, in particular the regions which were c ausally disconnected at the epoch of formation of the observed absorption spectra. Astrophysical tests To find out whether αchanged over the cosmological time, we have studied the fine splitting of the doublet lines of Si iv, Civ, Mgiiand other ions, observed in the spectra of distant quasars. According to quantum electr odynamics, the relative splitting of these lines δλ/λis proportional to α2(neglecting very small corrections). Consequently, if αchanged with time, then δλ/λwould depend on the cosmological redshift z. This method of measuring αin distant regions of the Universe had been first suggested by Savedoff [12] and was used later by other aut hors. For instance, Wolfe et al. [13] derived an estimate |˙α/α|<4×10−12yr−1from an observation of the Mg iiabsorption doublet at z= 0.524.TABLE 1. Variation of αvalue estimated from redshifted Si ivfine-splitting doublets. Quasar z ∆α/α Ref. HS 1946+76 3.050079 1.58 [16] HS 1946+76 3.049312 0.34 [16] HS 1946+76 2.843357 0.59 [16] S4 0636+76 2.904528 1.37 [16] S5 0014+81 2.801356 -1.80 [16] S5 0014+81 2.800840 -1.70 [16] S5 0014+81 2.800030 1.11 [16] PKS 0424 −13 2.100027 -4.51 [15] Q 0450 −13 2.230199 -1.48 [15] Q 0450 −13 2.104986 0.02 [15] Q 0450 −13 2.066646 1.03 [15] J 2233 −60 1.867484 -1.92 [17] J 2233 −60 1.869756 -2.21 [17] J 2233 −60 1.871074 -1.41 [17] J 2233 −60 1.925971 1.11 [17] J 2233 −60 1.941979 0.48 [17] An approximate formula which relates a deviation of αat redshift zfrom its cur- rent value, ∆ αz, with measured δλ/λin the extragalactic spectra and in laboratory reads ∆αz≈cr 2/bracketleftBigg(δλ/λ)z (δλ/λ)0−1/bracketrightBigg , (1) where cr∼1 takes into account radiation corrections [14]: for instan ce, for Si iv cr≈0.9. Many high-quality quasar spectra measured in the last decad e have enabled us to significantly increase the accuracy of determination of δλ/λat large z. An example of the spectra observed is shown in Fig. 1. For the present rep ort, we have selected the results of high-resolution observations [15–17], most suitable for an analysis of αvariation. The values of ∆ α/αcalculated from these data according to Eq. (1) are given in Table 1. As a result, we obtain a new estimate of the possible deviatio n of the fine- structure constant at z= 2–4 from its present ( z= 0) value: ∆α/α= (−4.6±4.3 [stat] ±1.4 [syst]) ×10−5, (2) where the statistical error is obtained from the scatter of a stronomical data (at large z) and the systematic one is estimated from the uncertainty of the fine splitting measurement in the laboratory [18,19] (at z= 0, which serves as the reference point for the estimation of ∆ α). The corresponding upper limit of the αvariation rate averaged over ∼1010yr is|˙α/α|<1.4×10−14yr−1(3) (at the 95% confidence level). This constraint is much more st ringent than those obtained from all but one previous astronomical observatio ns. The notable ex- ception is presented by Webb et al. [20], who have analysed sp ectroscopic data of similar quality, but estimated αfrom comparison of Fe iiand Mg iifine-splitted walelengths in extragalactic spectra and in the laboratory . Their result indicates a tentative time-variation of α: ∆α/α= (−1.9±0.5)×10−5atz= 1.0–1.6. Note, however, two important sources of a possible systematic err or which could mimic the effect: (a) Fe ,iiand Mg iilines used are situated in different orders of the echelle-spectra, so relative shifts in calibration of the d ifferent orders can simu- late the effect of α-variation, and (b) were the relative abundances of Mg isoto pes changing during the cosmological evolution, the Mg iilines would be subjected to an additional z-dependent shift relative to the Fe iilines, quite sufficient to sim- ulate the variation of α(this shift can be easily estimated from recent laboratory measurements [21]). In contrast, the method based on the fine splitting of a line of the same ion species (Si ivin the above example) is not affected by these two uncertainty sources. Thus we believe that the restriction ( 3) is the most reliable at present for the long-term history of the Universe. According to our analysis, some theoretical models are inco nsistent with observa- tions. For example, power laws α∝tnwithn= 1,−1/4, and −4/3, published by various authors in 1980s, are excluded. Moreover, the Telle r–Dyson’s hypothesis on the logarithmic dependence of αont[22,5] has also been shown to be inconsistent with observations. Many regions of formation of the spectral lines, observed at large redshifts in different directions in the sky, had been causally disconnec ted at the epochs of line formation. Thus, no information could have been exchanged b etween these regions of the Universe and, in principle, the fundamental constant s could be different there. However, a separate analysis [23] has shown that αvalue is the same in different directions in the sky within the 3 σrelative error |∆α/α|<3×10−4. PROTON-TO-ELECTRON MASS RATIO The dimensionless constant µ=mp/meapproximately equals the ratio of the constants of strong interaction g2/(¯hc)∼14 and electromagnetic interaction α≈ 1/137.036, where gis the effective coupling constant calculated from the ampli tude of nucleon– π-meson scattering at low energy. In order to check the cosmological variability of µwe have used high-redshift absorption lines of molecular hydrogen H 2in the spectrum of the quasar PKS 0528–250. This is the first (and, in a sense, unique) high-red shift system of H 2 absorption lines discovered in 1985 [24]. A study of these ob jects yields information of paramount importance on the physical conditions ∼1010years ago. A possibility of distinguishing between the cosmological r edshift of spectral wave- lengths and shifts due to a variation of µarises from the fact that the electronic,TABLE 2. Comparison of wavelengths of electron-vibro-rotational lines for H 2, D2, and T2. i λ i(H2) λi(D2) λi(T2) Ki L 0–0 R(1) 1108.633 1103.351 1101.021 −8.18×10−3 L 0–2 R(1) 1077.697 1081.153 1082.760 +5 .35×10−3 L 0–9 R(1) 992.013 1015.610 1027.218 +3 .80×10−2 vibrational, and rotational energies of H 2each undergo a different dependence on the reduced mass of the molecule. Hence comparing ratios of w avelengths λiof various H 2electron-vibration-rotational lines in a quasar spectrum at some red- shiftzand in laboratory (at z= 0), we can trace variation of µ. The method had been used previously by Foltz et al. [25], whose analysis was corrected later in our papers [26,23,27]. In the latter papers, we calculated the s ensitivity coefficients Ki of the wavelengths λiwith respect to possible variation of µand applied a linear regression analysis to the measured redshifts of individua l lines zias function of Ki. An illustration of the wavelength dependences on the mass o f the nucleus is given in Table 2, where a few resonance wavelengths of hydrog en, deuterium, and tritium molecules are listed. One can see that, as the nuclea r mass increases, dif- ferent wavelengths shift in different directions. More comp lete tables, as well as two algorithms of Kicalculation, are given in Refs. [23,28]. Thus, if the proton mass in the epoch of line formation were di fferent from the present value, the measured ziandKivalues would correlate: zi zk=(λi/λk)z (λi/λk)0≃1 + (Ki−Kk)/parenleftBigg∆µ µ/parenrightBigg . (4) We have performed a z-to-Kregression analysis using a modern high-resolution spectrum of PKS 0528 −250 [28]. Several tens of the H 2lines have been identified; a portion of the spectrum which reveales some of the lines is s hown in Fig. 1. The redshift estimates for individual absorption lines with th eir individual errorbars are plotted in Fig. 2 against their sensitivity coefficients. The resulting parameter estimate and 1 σuncertainty is ∆µ/µ= (−11.5±7.6 [stat] ±1.9 [syst]) ×10−5. (5) The 2 σconfidence interval to ∆ µ/µis |∆µ/µ|<2.0×10−4. (6) Assuming that the age of the Universe is ∼15 Gyr the redshift of the H 2absorption system z= 2.81080 corresponds to the elapsed time of 13 Gyr (in the standa rd cosmological model). Therefore we arrive at the restrictio n |˙µ/µ|<1.5×10−14yr−1(7) on the variation rate of µ, averaged over 90% of the lifetime of the Universe.FIGURE 2. Redshift values inferred from an analysis of separate spect ral features in an H 2 absorption system in the spectrum of the quasar PKS 0528 −250, plotted vs. λi(µ)-sensitivity coefficients Ki. The slanted solid line shows the most probable regression a nd the dashed ones corespond to ±1σdeviations of the slope. CONCLUSIONS Despite the theoretical prediction of the time-dependence s of fundamental con- stants, a statistically significant variation of any of the c onstants have not been reliably detected up to date, according to our point of view s ubstantiated above. The upper limits obtained indicate that the constants of ele ctroweak and strong interactions did not significantly change over the last 90% o f the history of the Universe. This shows that more precise measurements and obs ervations and their accurate statistical analyses are required in order to dete ct the expected variations of the fundamental constants. Acknowledgements. This work was performed in frames of the Project 1.25 of the Russian State Program “Fundamental Metrology” and supp orted by the grant RFBR 99-02-18232.REFERENCES 1. Damour, T., and Polyakov, A.M., Nucl. Phys. B 423 , 532 (1994). 2. Milne, E., Proc. R. Soc. A 158 , 324 (1937). 3. Dirac, P.A.M., Nature 139, 323 (1937). 4. Gamow, G., Phys. Rev. Lett. 19, 759 (1967). 5. Dyson, F.J., in Aspects of Quantum Theory, edited by A. Salam and E. P. Wigner, , p. 213 (Cambridge Univ., Cambridge, 1972). 6. Demidov, N.A., Ezhov, E.M., Sakharov, B.A., et al., in Proc. of 6th European Fre- quency and Time Forum , p. 409 (1992). 7. Prestage, J.D., Tjoelker, R.L., and Maleki, L., Phys. Rev. Lett. 14, 3511 (1995). 8. Shlyakhter, A.I., Nature 25, 340 (1976). 9. Damour, T., and Dyson, F.J., Nucl. Phys. B 480 , 37 (1996). 10. Kalvius, G.M., and Shenoy, G.K., Atomic and Nuclear Data Tables 14, 639 (1974). 11. Sisterna, P.D., and Vucetich, H., Phys. Rev. D 41, 1034 (1990). 12. Savedoff, M.P., Nature 264, 340 (1956). 13. Wolfe, A.M., Brown, R.L., and Roberts, M.S., Phys. Rev. Lett 37, 179 (1976). 14. Dzuba, V.A., Flambaum, V.V., and Webb, J.K., Phys. Rev. A , submitted (e-print: physics/9808021). 15. Petitjean, P., Rauch, M., and Carswell, R.F., Astron. Astrohys. 91, 29 (1994). 16. Varshalovich, D.A., Panchuk, V.E., and Ivanchik, A.V., Pis’ma Astron. Zh. (Engl. transl.: Astron. Lett.) 22, 8 (1996). 17. Outram, P.J., Boyle, B.J., Carswell, R.F., Hewett, P.C. , and Williams, R.E., Mon. Not. Roy. Astron. Soc. 305, 685 (1999). 18. Morton, D.C., Astrophys. J. Suppl. 77, 119 (1991).; (E) 81, 883 (1992) 19. Kelly, R.L., J. Phys. Chem. Ref. Data NBS 16, Suppl.1 (1987). 20. Webb, J.K., Flambaum, V.V., Churchill, C.W., Drinkwate r, M.J., and Barrow, J.D., Phys. Rev. Lett. 82, 884 (1999). 21. Pickering, J.C., Thorne, A.P., and Webb, J.K, Monthly Not. R. Astron. Soc. 300, 131 (1998). 22. Teller, E., Phys. Rev. 73, 801 (1948). 23. Varshalovich, D.A. and Potekhin, A.Y., Space Sci. Rev. 74, 259 (1995). 24. Levshakov, S.A., and Varshalovich, D.A., Monthly Not. R. Astron. Soc. 212, 517 (1985). 25. Foltz, C.B., Chaffee, F.H., and Black, J.H., Astrophys. J. 324, 267 (1988). 26. Varshalovich, D.A. and Levshakov, S.A., JETP Lett. 58, 237 (1993). 27. Varshalovich, D.A., and Potekhin, A.Y., Pis’ma Astron. Zh. (Engl. transl.: Astron. Lett.)22, 3 (1996). 28. Potekhin, A.Y., Ivanchik, A.V., Varshalovich, D.A., La nzetta, K.M., et al., Astro- phys. J. 505, 523 (1998).

arXiv:physics/0004063v1 [physics.acc-ph] 25 Apr 2000DESY 00-065 physics/0004063 April 2000 Test Results on the Silicon Pixel Detector for the TTF-FEL Beam Trajectory Monitor S. Hillerta, R. Ischebeckb, U. C. M¨ ullerb, S. Rothb, K. Hansenb, P. Hollc, S. Karstensenb, J. Kemmerc, R. Klannera,b, P. Lechnerc, M. Leenenb, J. S. T. Ngb,1, P. Schm¨ usera, L. Str¨ uderd aII. Institut f¨ ur Experimentalphysik, Universit¨ at Hambu rg, Luruper Chausee 149, D-22761 Hamburg, Germany bDeutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22603 Hamburg, Germany cKetek GmbH, Am Isarbach 30, D-85764 Oberschleißheim, Germany dMax-Planck-Institut f¨ ur extraterrestrische Physik, Gie ssenbachstraße, D-85740 Garching, Germany Abstract Test measurements on the silicon pixel detector for the beam trajectory monitor at the free electron laser of the TESLA test facility are pres ented. To determine the electronic noise of detector and read-out and to calibra te the signal amplitude of different pixels the 6 keV photons of the manganese Kα/Kβline are used. Two different methods determine the spatial accuracy of the dete ctor: In one setup a laser beam is focused to a straight line and moved across the p ixel structure. In the other the detector is scanned using a low-intensity elec tron beam of an electron microscope. Both methods show that the symmetry axis of the d etector defines a straight line within 0.4 µm. The sensitivity of the detector to low energy X- rays is measured using a vacuum ultraviolet beam at the synch rotron light source HASYLAB. Additionally, the electron microscope is used to s tudy the radiation hardness of the detector. Key words: Beam monitor, X-ray detector, solid-state detector, imagi ng sensor. PACS: 41.85.Qg, 07.85.Fv, 29.40.Wk, 42.79.Pw. 1now at SLAC, Stanford, CA 94309, USA Preprint submitted to Nuclear Instruments and Methods A 24 N ovember 2013sensitive area of /0/0/0/0/0/0 /1/1/1/1/1/1 silicon pixel detector measured signalsimage plane photons silicon pixel detectorsBeam Trajectory Monitor undulator 12 measured points electron trajectory pinhole planeimaged photon flux0.74mm 0.5mm Fig. 1. Measurement principle: An image of the electron beam is projected through a set of pinholes onto pixel detectors. 1 Introduction It is a widespread opinion that the fourth generation synchr otron light source will be a X-ray free electron laser (FEL). It will consist of a sing le-pass FEL relying on the self-amplified spontaneous emission (SASE) mechanism [ 1] and deliver coherent radiation in the X-ray range with unprecedented brilliance . In such SASE FEL a tightly collimated electron beam of high charge density is sent through a long undulator. The SASE effect results from the interaction of th e electron bunch with its own radiation field created by the undulation motion. Thi s interaction can only take place if the electron and the photon beams overlap. At the free electron laser of the TESLA Test Facility (TTF-FE L) [2] the electron beam position must be controlled to better than 10 µm over the entire 15 m long undulator. With the beam trajectory monitor (BTM) [3] the off -axis spontaneous undulator radiation is imaged through a set of pinholes of 80 µm diameter (see Fig. 1). In order to reduce the effect of diffraction, only the h igher harmonics of the spontaneous undulator radiation will be used for BTM measur ements. A 120 nm thick silver foil across each pinhole absorbs all low energy photons and restricts the spectral range of the detected radiation to energies above 1 00 eV. From a simulation using the expected undulator spectrum the Gaussian width of the photon spot at the position of the detector 0.5 m behind the pinholes is esti mated to 30 µm. To achieve the required resolution of the BTM, the center of the photon spot will be measured with a precision of better than 1 µm using a high resolution silicon pixel detector. It delivers 12 points of the transverse beam posit ion with an accuracy of better than 10 µm over a length of 5 m using a single setup. The performance of the silicon detector with respect to noise, spatial precisi on, radiation hardness and quantum efficiency is presented in this paper. 2Fig. 2. Anode structure of the silicon pixel detector: Two pi xel rows with a charge injection line across each row and each pixel connected to th e on-chip JFETs 2 The Silicon Pixel Detector A silicon pixel detector with an active area of 0.5 mm ×0.74 mm and a total of 24 channels was designed and fabricated at the MPI Halbleite rlabor. The sensitive area of the detector consists of two rows of each 12 active pix els as shown in Fig. 2. Each pixel anode is directly connected to an on-chip JFET for low noise read-out. The pixels are 250 µm wide, with heights varying from 25 µm (nearest to beam) to 100 µm to give roughly equidistant measuring points in the projec tion along the undulator axis. High quantum efficiency is achieved using a th in entrance window technology [4]. The concept of a backside illuminated pn-junction detector has been chosen, which shows not only a high quantum efficiency for the desired photon energies, but in addition an excellent spatial homogeneity. It consists of a fully depleted n-type bulk and a non structured p+-rear contact, acting as radiation entrance window. At photon energies of about 150 eV the absorption length in sili con drops to 40 nm, which leads to signal loss in the almost field free, highly dop ed region underneath thep+contact. To reduce the width of this insensitive region the i mplantation of boron ions was done through a SiO 2layer, which has been removed afterwards. One achieves a shallow doping profile with the pn-junction placed at a depth of only 30 nm below the detector surface. Ionizing radiation wh ich penetrates through the dead layer generates electron hole pairs in the sensitiv e volume of the detector. Applying a negative voltage of about 120 V to the rear contact totally depletes the detector and causes the signal electrons to drift toward s the collecting n+pixel anodes (see Fig. 3). 3µm-20 VRFDR 0 VFFDR +8 V GRARFGA -3 ... -10 V pulsed Signal280 electrons drift pathanode0 V p-n junctionFFDR Signal Left Right Left-Right symmetry line p+First FET Pixel Sensitive Area Reset FET Sin-bulk fully depleteddeep-n deep-pn+ Radiation photon fluxn+n+p+ effective dead layer RK (-120 V)-8 V const. or (Typical voltages shown) Fig. 3. Cross section of the pixel detector. The pixels are formed by n+-implants and are isolated from each other by a 5 µm widep+grid. Each pixel anode is connected to an amplifying JFET whi ch is in- tegrated on the detector chip, thus minimizing stray capaci tances. The JFETs are operated as source followers with a given constant current o f about 200 µA from source to drain. The collected signal charge is transferred to the gate and modulates the voltage drop across the JFET. A second JFET (Reset FET) al lows to discharge the pixel anodes after each read-out cycle. The 4 mm ×2.5 mm large detector chips are mounted onto a ceramic hybrid board. Each detector pixel is connected to one channel of the CAMEX64B [5] read- out chip. It provides signal amplification, base line subtra ction, and multiplexed output. The digital steering chip TIMEX generates the neces sary control signals. Signal integration times between 2 µs and the full TTF pulse length (800 µs) can be programmed. 3 Calibration and Noise Determination An absolute energy calibration of each detector pixel is obt ained using mono- energetic photons emitted from a55Fe source at 5.90 keV (Mn K α-line, 89.5%) and 6.49 keV (Mn K β-line, 10.5%). The X-ray photons enter the detector through the back entrance window on the side opposite to the anodes. P hotons at an energy of 6 keV have an attenuation length of 30 µm in silicon and are therefore absorbed close to the surface of the detector chip. On average each pho ton at this energy produces about 1600 electron-hole pairs. The electrons dri ft to the anodes, where the charge is collected. During the drift time of 7 ns the late ral extent of the electron cloud increases to about 8 µm due to diffusion. The detector is operated at room temperature and read out wit h a rate of 5 kHz. For the given activity of the source (106Bq) and the integration time (15 µs) a 4E [keV]-2 0 2 4 6 8 10no. of events 110102103104105106 data fit K Kpedestal α β Fig. 4. Measured energy spectrum for pixel 8 (60 µm×250µm). photon is registered by the detector in 10% of the read-outs. The measured energy spectrum for one of the detector pixels is shown in Fig. 4. It c an be separated into three parts: The pedestal peak, which dominates the distrib ution, the signal peak, which consists of the K αand K βcomponents, and the region in between, which is caused by charge sharing between adjacent pixels. The energ y scale, the noise and the diffusion width are determined with a simultaneous fit to t he whole spectrum of Fig. 4 based on a model describing the two-dimensional pix el structure. The location of the pedestal defines the zero-signal level fo r each pixel. The pedestal subtraction has already been applied to the data shown in Fig . 4. The difference between the signal and pedestal peak gives an energy calibra tion for each pixel. The resulting calibration constants differ by at most 10% for neighbouring left and right pixels. For a 30 µm photon spot this corresponds to an error in position reconstruction of at most 0.8 µm if the signal is not corrected according to the different calibration constants. The Gaussian width of the pedestal peak is mainly caused by on e source of noise, the leakage current. Using the calibration one calculates e nergy resolutions between 222 eV and 391 eV, which can be translated into an equivalent n oise charge (ENC) between 60 and 106 electrons. The variation of the noise valu es are caused by the different pixel sizes which lead to variable leakage current and capacitance. Due to the dominant role of the leakage current the energy resoluti on strongly depends on integration time and temperature. Our measurements show th at the noise grows proportionally to the square root of the integration time an d decreases by a factor of two when cooling the detector by 16 K. The number of events with charge division between pixels com pared to the number of events with the photon signal fully recorded by one pixel i s directly related to the geometry of the individual pixels and the diffusion width of t he charge cloud. This 5is taken into account by the fit and leads to a Gaussian width of the charge cloud at the anode plane of about 8 µm, consistent with our estimations from diffusion. The common mode noise is defined as the variation of the zero-s ignal level com- mon to all pixels. For each event the median of the 24 pixel sig nals is taken as an estimate for the common mode value. It has been found that the common mode shows no systematic drift and varies only within its standar d deviation of 30 elec- trons. Electronic cross talk is seen by the pedestal shift of a pixel adjacent to a pixel which registered one photon totally. The cross talk am ounts to 3% of the full signal at most. This corresponds to a reduction in spatial re solution by 6% which is acceptable for our application. 4 Measurement of Spatial Accuracy using a Laser Light Spot The spatial accuracy of the pixel detector is measured in a la ser test stand [6] by projecting a laser line-focus onto the pixel structure. The light emitted by a pulsed laser diode ( λ= 670 nm) is focused using a micro-focus lens and then defocus ed in one plane using a cylindrical lens. This setup produces a s traight line-focus of about 30 µm full width on the detector surface. The line-shaped light s pot is oriented parallel to the separation line of the two pixel rows. Using a stepping device the light spot is moved across the two pixel rows with 0.068 µm per step to determine its left-right symmetry line. In addition to the pedestal correction we subtract a constan t signal offset propor- tional to the pixel size to correct for stray light falling on to the detector. For each pixel pair the relative difference between the signals of the right and the left pixel, η= (SR−SL)/(SR+SL), is calculated. The result is shown in Fig. 5 versus the m]µ laser position [-50 -40 -30 -20 -10 0 10 20 30 40 50η asymmetry -1-0.500.51 Fig. 5. Asymmetry, η, versus the position of the laser light spot. The results of a ll 12 pixel pairs are overlayed. 6 m]µ center of pixel pair [0 100 200 300 400 500 600 m]µ [η zero crossing of -2-1.5-1-0.500.511.52 Fig. 6. Zero crossing of ηversus pixel-pair position. position of the laser light spot. All 12 pixel pairs show the z ero crossing of ηat the same laser position within ±1µm. The position of the zero crossing of ηcan be extracted for each pixel pair from a straight line fit to the central data points. In Fig. 6 the resu lting zero crossing is plotted versus the center position of the corresponding pix el pair. Obviously, the laser line-focus was tilted by about 1 mrad with respect to th e center line of the pixel array. Fitting a straight line to the 12 data points giv es us the location of the laser spot. The individual measurements scatter with a s tandard deviation of 0.37µm around the reconstructed line. For measurements very close to the zero crossing one expects a linear dependence of ηon the laser position. Fig. 7 shows the behaviour of the measu redηin this region. m]µ laser position [-1.5 -1 -0.5 0 0.5 1 1.5η asymmetry -0.2-0.15-0.1-0.0500.050.10.150.2 Fig. 7. Closer look at the left-right asymmetry, η, around zero crossing. 7All 12 pixel pairs show the same dependence with the exceptio n of different offsets. We observe a periodic oscillation of 0.5 µm length which is caused by the inaccuracy of the stepping device. As these oscillations are fully corr elated one can correct for the effect and is left with a relative point-to-point reso lution of approximately 0.1µm. 5 Measurement of Spatial Accuracy using a Scanning Electron Mi- croscope The detector is installed into the focal plane of a scanning e lectron microscope (SEM). The SEM produces an electron beam with an energy of 10 k eV focused to a spot smaller than 1 µm on the surface of the detector. The SEM beam current can be adjusted up to 100 µA at the filament cathode. It is significantly reduced by several apertures in the optical system, yielding curren ts below 1 pA at the beam focus. Secondary emitted electrons from the detector s urface are collected and amplified by an open multiplier tube. Its signal is used to display a picture of the detector on a view screen. The detector hybrid is placed o nto a copper plate to remove the heat produced by the read-out electronics. Howev er, the temperature of the detector chips increases by about 15 K while the chamber i s under vacuum. The electron beam is scanned across the two pixel rows in para llel to their separation line. After each scanning line, the electron beam is displac ed by a fixed amount. The detector read-out is synchronized to the scanning frequ ency of the SEM, so that data are taken after each scanned line. Measurements ar e made with 618 to 2252 lines per mm. −50−40−30−20−10 01020304050−1.5−1−0.500.511.5 beam position [ µm]asymmetry η Fig. 8. Asymmetry, η, versus position of electron beam. The results of all 12 pixe l pairs are overlayed. 80 100 200 300 400 500−4−3−2−1012345 center of pixel pair [ µm]zero crossing of η [µm]measurement 1 measurement 2 measurement 3 fit Fig. 9. Zero crossing of ηversus pixel-pair position Analogous to the previous measurement the relative asymmet ryηbetween right and left pixels is calculated (see Fig. 8) and the zero crossings ofηare extracted. Fitting a straight line to the central data points gives again the zero crossings for the different pixel pairs. The results of three different scans are shown in Fig. 9. The scanning line of the electron beam was tilted with respect to the symme try line of the pixel detector by 10 mrad. The standard deviation of the measured z ero crossings from the reconstructed scanning line amounts to 0.47 µm. The reconstructed scanning lines from three different measurements show the same struct ure and are only shifted with respect to each other. One concludes that the deviation s of the measured zero crossings from a straight line and therefore the limitation in spatial accuracy of the detector is due to a systematic effect. The reconstructed sym metry line cannot be directly compared to the laser measurement of Section 4, bec ause the penetration depth of the electrons is much smaller than for laser light. A dditionally, the detector chip was different from the one used in the laser test. 6 Sensitivity to Vacuum Ultraviolet Radiation We measured the sensitivity of the detector to vacuum ultrav iolet (VUV) at the synchrotron radiation facility HASYLAB. For this purpose t he detector is illumi- nated with VUV radiation in the energy range between 50 eV and 1000 eV which is produced by a bending magnet of the electron storage ring D ORIS. The hybrid containing the silicon pixel detector and its read-out elec tronics is placed into the vacuum chamber of a reflectometer, where ultra high vacuum (1 0−9mbar) had to be established. The mono-energetic photon beam coming from the monochromator is focused onto the center of the pixel detector. The synchro tron light is pulsed with the 5 MHz bunch frequency of DORIS. The separation of 200 ns be tween two light 9photon energy [keV]10-11quantum efficiency 00.20.40.60.81 data parametrization SiO C Fig. 10. Quantum efficiency of the detector for VUV radiation. pulses is much shorter than the integration time of the detec tor read-out. Therefore the pixel anodes are discharged at the beginning of each inte gration period which is extended over several synchrotron light pulses. The silicon detector response was corrected to the photo-el ectron emission of one of the focusing mirrors and to a GaAs photo diode as a reference. Using the normalized signals, the quantum efficiency can be estimated from the meas ured absorption edges of the relevant elements. Detailed measurements were done at photon energies in the vicinity of the absorption edges of silicon (100 eV), o xygen (543 eV), and carbon (284 eV). For the parameterization of the absorption edges the compilation of photon absorption lengths for different elements in Ref. [7] is used. For simplicity we assume that a photon absorbed in the dead layer of the detecto r does not contribute to the signal and all other photons are fully registered. Usi ng the measured heights of the absorption edges, this model gives the thicknesses of the photon absorbing layers. These are used to calculate the quantum efficiency in t he whole spectral range from 50 eV to 1000 eV. The data points are normalized to t his result and both are presented in Fig. 10. The observed quantum efficiency is explained by the following effects: The electrical field of the detector diode does not extend up to the cathode pl ane, but leaves space for a dead layer with a thickness of the order of 30 nm, wh ich has to be penetrated by the photons before they enter the sensitive re gion of the detector. A 50 nm thin passivation layer of silicon oxide on top of the bac k entrance window leads to further absorption of photons. From a detailed inve stigation of the silicon absorption edge one can see the effect of the covalent Si-O bon d which results in a deviation from the absorption edge of pure silicon (see Fig . 11). The origin of an additional carbon contamination which leads to the appea rance of the carbon absorption edge in Fig. 10 is not yet understood. 10photon energy [eV]85 90 95 100 105 110 115 120 125 130 135quantum efficiency 00.10.20.30.40.50.60.70.8 data parametrization Fig. 11. Quantum efficiency at silicon absorption edge. The quantum efficiency of the detector is larger than 20% for ph otons in the energy range above 100 eV which will be used by the BTM. Absolute meas urements with a similar type of detector have been done using a reference dio de with known quantum efficiency [4]. The measured quantum efficiencies are comparab le with our results, but did not show the problem of the carbon absorption edge. 7 Study of Radiation Hardness In the BTM the silicon detector will be operated at a distance of only 5 mm from the electron beam of the TTF linac. It can suffer from radiatio n damages caused by a beam halo or by scraping a misaligned beam. One expects th at the radiation damages of the silicon detectors are dominated by surface eff ects. Placing the silicon detector inside an electron microscope not only allows to deter- mine its spatial accuracy, but also gives the opportunity to study radiation hardness against surface damages. The detector side containing the p ixel anodes and the am- plifier JFETs should be more sensitive to surface damages tha n the back entrance window. Therefore we place the detector in such way that the e lectrons enter the detector opposite to the entrance window. One of the two pixe l rows, including its JFETs, is irradiated using the 10 kV electron beam with be am currents of the order of several tens of pA. Irradiation takes place with all operating voltages on, including the bias voltage. Before irradiation the detector had been calibrated using t he 59.5 keV photons of an241Am source. During the irradiation procedure, the beam scan i s extended to all pixels and the detector is read out. This is done to determ ine the number of incident beam electrons from the measured signal of the unda maged pixels. From 111234567101112131415161718192021222324−10123456 pixel numbersignal [a.u.]9 Gy 29 Gy 70 Gy 119 Gy 299 Gy 824 Gy Fig. 12. Signal height of the pixels as function of the radiat ion dose; pixels 13–24 are irradiated. the mean energy loss of a 10 keV electron in silicon oxide (4.1 3 keV/ µm) one can then calculate the radiation dose. Typical dose rates between 0. 1 Gy/min and 10 Gy/min have been achieved with this method. The measurement of the b eam current was cross checked using a faraday cup which delivered consisten t results within 20%. Several data sets are taken in between the irradiation steps to determine the change of signal height and noise level. For this purpose a LED is ins talled inside the electron microscope shining onto the back entrance window o f the silicon detector. Fig. 12 gives the measured signal height versus the pixel num ber, starting from the small left pixels (1–4), going to the large pixels (12,13) an d ending with the small right pixels (21–24). The irradiated pixels on the right sid e show a decreasing signal height with increasing radiation dose. Above radiation dos es of about 300 Gy some of the irradiated pixels cease to deliver any signal at all. In Fig. 13 the dependence of noise on the radiation dose is sho wn. One can clearly distinguish between the non-irradiated pixels and the irra diated ones. Whereas the first stayed at the same noise level the equivalent noise char ge of the latter increased by a factor of three after a total radiation dose of 120 Gy. Surface damages include both the creation of oxide charges i n the passivation layer and the generation of inter-band energy levels at the interf ace between the silicon bulk and the oxide layer, the so-called interface states. Th e latter inject additional charges and therefore contribute to leakage current and noi se. The oxide charges lead to a charge up of the SiO 2layer and therefore influence the operating voltages of the integrated JFETs. The signal loss with increasing rad iation dose might be caused by a change of the operation point of the amplifying JF ET which results in a lower gain. In addition to the amplifying JFET each pixel is connected to a 120 20 40 60 80 100 12020040060080010001200140016001800 dose [Gy]ENC [e]irradiated pixels other pixels Fig. 13. Eqivalent noise charge as function of the total radi ation dose reset JFET which allows discharging the pixel anode between detector read-outs. By recording the dependence of the amplified signal on the app lied gate potential, we measured the gate potential which was necessary to close t he reset FET. This voltage had to be increased from −4 V to −6 V after a irradiation of 120 Gy. Further irradiations up to 0.8 kGy were performed, which cau sed the loss of signal in all irradiated pixels. Recovery of the pixels took place w ithin one week (165 h). Then the total radiation dose was increased to 4 kGy. This tim e the detector could not recover within the following week. After an in-situ heat ing of the detector and the read-out electronics to 130◦C for 30 minutes all pixels worked again. The effect can be explained by the removal of the oxide charges due to the heating process. Because the interface states cannot be removed by heating th e noise stayed at a high level, a factor of two above the noise of the non-irradia ted pixels. 8 Conclusions Noise, spatial accuracy and quantum efficiency of the silicon pixel detector which will be used in the beam trajectory monitor at TTF-FEL were in vestigated. The measured noise values are in the specified range and are domin ated by leakage current. The systematics of the position measurement was st udied using a laser line- focus and a scanning electron microscope. The spatial accur acy is of the order of 0.4µm, well below the required 1 µm for the operation within the beam trajectory monitor. The sensitivity to vacuum ultraviolet radiation h as been measured in a synchrotron beam line. From the observed absorption edges a quantum efficiency above 20 % is estimated at photon energies used for the BTM. Th e detector can cope with radiation doses up to 100 Gy. At the position of the B TM in TTF a radiation dose of the order of 1 Gy per week is expected. 13Acknowledgements We are grateful to D. Vogt for giving us the opportunity to ope rate our detector inside an electron microscope. For his help during the measu rements at HASYLAB we would like to thank M.-A. Schr¨ oder. We thank C. Coldewey, E. Fretwurst and M. Kuhnke for fruitful discussions about radiation hardnes s of silicon detectors. References [1] A. M. Kondratenko, E. L. Saldin, Part. Accelerators 10(1980) 207; R. Bonifacio, C. Pellegrini, L. M. Narducci, Opt. Commun. 53(1985) 197. [2] TTF-FEL Conceptual Design Report, TESLA-FEL 95-03, DES Y, June 1995; J. Rossbach et al., Nucl. Instr. and Meth. A375 (1996) 269. [3] J. S. T. Ng, TESLA-FEL 96-16, DESY, 1996; AIP Conf. Proc. #413, eds R. Bonifacio and W. A. Barletta, 199 7; J. S. T. Ng et al., Nucl. Instr. and Meth. A439 (2000) 601. [4] P. Lechner, L. Str¨ uder, Nucl. Instr. and Meth. A354 (1995) 464; R. Hartmann et al., Nucl. Instr. and Meth. A377 (1996) 191; H. Soltau et al., Nucl. Instr. and Meth. A377 (1996) 340. [5] W. Buttler et al., Nucl. Instr. and Meth. A288 (1990) 140. [6] S. Hillert, Diploma-Thesis Univ. of Hamburg, TESLA 2000 -04, DESY, Jan. 2000. [7] B. L. Henke, E. M. Gullikson, J. C. Davis, Atomic Data and N uclear Data Tables 54(1993) 181 and http://cindy.lbl.gov/optical constants . 14

arXiv:physics/0004064v1 [physics.ed-ph] 25 Apr 2000Matching Conditions on Capillary Ripples: Polarization Estudio de las condiciones de empalme para las oscilaciones de intercara entre dos fluidos. Polarizaci´ on. Arezky H. Rodr´ ıguez∗, J. Mar´ ın-Antu˜ na and H. Rodr´ ıguez-Coppola Dpto. de F´ ısica Te´ orica, Fac. de F´ ısica, Universidad de la Habana, C. de la Habana 10400, Cuba (February 2, 2008) The matching conditions at the interface between two non-mi xed fluids at rest are obtained directly using the equation of movement of the whole media. T his is a non-usual point of view in hydrodynamics courses and our aim is to fix ideas about the i ntrinsic information contained in the matching conditions, on fluids in this case. Afterward , it is analyzed the polarization of the normal modes at the interface and it is shown that this inf ormation can be achieved through a physical analysis and reinforced later by the matching con ditions. A detailed analysis of the matching conditions is given to understand the role that pla ys the continuity of the stress tensor through the interface on the physics of the surface particle movement. The main importance of the viscosity of each medium is deduced. En el presente trabajo se utilizan las ecuaciones de movimie nto de todo el sistema compuesto por dos fluidos no miscibles en reposo relativo para obtener l as condiciones de empalme en la intercara. Este procedimiento es inusual y el objetivo de ha cerlo en esa forma es fijar ideas sobre la informaci´ on que est´ a contenida en dichas condiciones de e mpalme, en este caso en fluidos. Posterior- mente es analizada la polarizaci´ on de los modos normales en la interface y se demuestra que estas caracter´ ısticas pueden ser obtenidas directamente de an´ alisis f´ ısico de las condiciones de empalme. 47.10.+g, 47.17.+e I. INTRODUCTION The boundary conditions constitute the key feature in any theory of surface waves. It is through them that one introduces in the analysis the physical consequences of specific surface effects, such as local changes of mass or stresses, thus going beyond the simple case in which one merely matches two semi-infinite bulk media at an interface (which, in particular, can be a free surface). In the last years, increased attention has been paid to the properties of capillary waves by physicists and chemists [1–8]. Ripples represent one of the few cases in which the relation between the dynamical properties of a surface and liquid flows can be predicted completely. The study of capillary ripples has clarified which proper- ties of a liquid surface determine the surface’s resistance against deformation. The boundary conditions for the stress at the interface are derived from the principle that the forces acting upon such an “interfacial” element result not only from viscous stresses in the liquid but also from stresses existing in thedeformed interface. The cause of the difference is that an interface, unlike a three dimensional liquid, can not enjoy the property of incompressibility [1]. Work done on an element of liquid is partially degrade into heat by viscous friction and partially transformed into kinetic en - ergy which is transmitted to adjoining elements. Work done on an interfacial element leads, at least partially, to an increment of the surface potential energy. It is this potential energy of the deformed interface which enables the whole system, including the interface, to carry out an oscillatory motion. In this article, first, it will be shown a non-traditional way to obtain the matching conditions at the interface be- tween two non-mixed fluid at rest, considering the equa- tion of motion of the whole media directly. Usually, uni- versity courses do not use this approach and state the boundary conditions from outside the constitutive equa- tions governing the studied problem. It is important from a pedagogical point of view to evidence that in fact, the matching conditions at the interfaces are contained, al- most in all cases, in the equation of movement for the whole system taken as the composition of all media. The ∗arezky@ff.oc.uh.cu 1other cases arise when the fluid interfaces have intrinsic properties not included on the equations of the media in that cases mentioned before. This point of view was already used in [9–13] to elab- orate a formalism, based on Surface Green Functions, which establish an isomorphism between solids and flu- ids related to the interface normal modes. The aim of this paper is to emphasize the fact that this way to establish the matching conditions allows us to introduce the phys- ical characteristics related with the boundary conditions in a firmer floating. II. EQUATIONS FOR THE BULK IN A VISCOUS INCOMPRESSIBLE FLUID. To achieve a system of equations which describes the oscillatory motion of a viscous incompressible fluid, the starting point is the linearized Navier-Stokes equation fo r a viscous fluid [14], which reads: ρ∂Vi ∂t=∂τij ∂xj+ρFi,ext (1) whereρis the density of equilibrium, Vithe components of the fluid particle, Fi,extare the external forces and τij is the stress tensor which, for a viscous incompressible fluid, has the following form [14] τij=−pδij+η/parenleftbigg∂Vi ∂xj+∂Vj ∂xi/parenrightbigg (2) Herepare the small variations of pressure, ηis the viscosity parameter and δijis the unit matrix elements. The subscripts iandjtake the values yandzidentically. In (1) the sum over repeated subscripts is understood. It had been assumed that the system is symmetric with re- spect to the xdirection. Putting (2) in (1) it is obtained −ρ∂Vy ∂t+∂ ∂y/parenleftbigg −p+ 2η∂Vy ∂y/parenrightbigg + +∂ ∂z/bracketleftbigg η/parenleftbigg∂Vy ∂z+∂Vz ∂y/parenrightbigg/bracketrightbigg = 0 (3) −ρ∂Vz ∂t+∂ ∂y/bracketleftbigg η/parenleftbigg∂Vz ∂y+∂Vy ∂z/parenrightbigg/bracketrightbigg + +∂ ∂z/parenleftbigg −p+ 2η∂Vz ∂z/parenrightbigg = 0 (4) where we have neglected the external forces. Also the continuity equation is needed, which expresses that the volume of an element of the incompressible fluid does not change during the motion. It has the form [14] ∂Vy ∂y+∂Vz ∂z= 0 (5) The first step is to obtain the equation of motion and its solution of the each medium taken as infinite. So,the parameters as density and viscosity are considered constants in the whole medium and it leads to transform equations (3) and (4) to: −ρ∂Vy ∂t−∂p ∂y+η∇2Vy= 0 (6) −ρ∂Vz ∂t−∂p ∂z+η∇2Vz= 0 (7) where∇2= (∂2/∂y2+∂2/∂z2). The solution of the system (5), (6) and (7) written as a vector field velocity, can be putted as the sum of an irro- tational field (related with the longitudinal mode) and a divergence free field (related with the transverse modes) [1], i.e.: V=V1+V2 (8) which satisfy: ∇ ×V1= 0 (9) ∇·V2= 0 (10) Any irrotational field is characterized by a scalar func- tion, the “potential” function ϕ(y,z,t), such that V1=−∇ϕ (11) and the divergence free field can be described by a vector function [14], the “ stream ” or vorticity function ψ(y,z,t), such that V2=/parenleftbigg −∂ψ ∂z,∂ψ ∂y/parenrightbigg (12) The velocity components can thus be written in terms of the potential and the stream functions, as: Vy=−∂ϕ ∂y−∂ψ ∂z(13) Vz=−∂ϕ ∂z+∂ψ ∂y(14) Substitution of eqns. (13) and (14) in the continuity condition (5) gives ∇2ϕ= 0 (15) and substitution into eqns. (6) and (7) leads to ∂ ∂y/braceleftbigg ρ∂ϕ ∂t−p/bracerightbigg +∂ ∂z/braceleftbigg ρ∂ψ ∂t−η∇2ψ/bracerightbigg = 0 (16) ∂ ∂z/braceleftbigg ρ∂ϕ ∂t−p/bracerightbigg −∂ ∂y/braceleftbigg ρ∂ψ ∂t−η∇2ψ/bracerightbigg = 0 (17) Equations (16) and (17) are simultaneously satisfied if one considers: 2ρ∂ϕ ∂t−p=C1 (18) ρ∂ψ ∂t−η∇2ψ=C2 (19) The constants C1andC2are obtained from the condi- tion at zero flow, and it gives raise to C1=poandC2= 0 wherepois the reference atmospheric pressure. The solution of equations (15), (18) and (19) are look- ing for as the following ϕ(y,z,t) = Φ(z)ei(κy−ωt)(20) ψ(y,z,t) = Ψ(z)ei(κy−ωt)(21) where the parameters κandωare the wavevector and frequency of the wave respectively. Substituting eqns. (20) and (21) in (15) and (19) gives thez-dependence of the functions ϕandψ, which satisfy: d2Φ(z) dz2−κ2Φ(z) = 0 (22) d2Ψ(z) dz2−q2tΨ(z) = 0 (23) with q2t=κ2−iρω/η Equations (22) and (23) lead to the solutions: Φ(z) =C1eκz+C2e−κz(24) Ψ(z) =C3eqtz+C4e−qtz(25) and in combination with eqns. (20) and (21) they give a solution of the form: ϕ(y,z,t) =/parenleftBig C1eκz+C2e−κz/parenrightBigei(κy−ωt)(26) ψ(y,z,t) =/parenleftBig C3eqtz+C4e−qtz/parenrightBigei(κy−ωt)(27) whereC1,C2,C3andC4are constants to be determined by boundary and matching conditions. Then, the expressions for the varying velocity com- ponents and the pressure are finally obtained by substi- tution of eqns. (26) and (27) into (13), (14) and (18), respectively. III. INTERFACE PROBLEM: MATCHING CONDITIONS. Now we will match the two media. We consider a sur- face which at rest coincides with the plane z= 0 and it separates medium M1atz <0 from medium M2at z>0. Each one is viscous and incompressible. First of all, any solution has to fulfill the continuity of the velocity field across the surface according to V(1)y=V(2)y atz= 0 (28) V(1)z=V(2)z atz= 0 (29)Superscript 1 denotes medium M1and superscript 2 denotes medium M2. The rest of the matching conditions at the interface are derived from the system of equations which govern the whole system. This is no usually done in the normal program courses at the Universities and we consider that it is important to state that the matching conditions are, almost in all of the cases, contained in the equation of movement for the whole system taken as the composition of each one. These are eqns. (3) and (4) where the parameters ηandρare constant, but taking different values on each medium. Integrating these equa- tions through the surface about z= 0 from −ǫto +ǫand later taking ǫ→0, it is obtained from eq. (3) /bracketleftBigg η1/parenleftBigg ∂V(1)y ∂z+∂V(1)z ∂y/parenrightBigg/bracketrightBigg z= 0−= =/bracketleftBigg η2/parenleftBigg ∂V(2)y ∂z+∂V(2)z ∂y/parenrightBigg/bracketrightBigg z= 0+(30) and from eq. (4) /bracketleftBigg −p1+ 2η1∂V(1)z ∂z/bracketrightBigg z= 0−= =/bracketleftBigg −p2+ 2η2∂V(2)z ∂z/bracketrightBigg z= 0+−pγ(31) wherepγis the jump due to the surface tension according with the Laplace Law [14]. The other terms in eqns. (3) and (4) vanish when ǫ→0 because they are continuous or have a finite jump at z= 0. It is easily seen according to eq. (2) that eqns. (30) and (31) are the conditions of the continuity of the stress tensor components through the interface, as expected. On the other hand, the boundary conditions of the problem are regularity at z→ ±∞ , which leads to elim- inate 4 of the 8 constants appearing in (26) and (27) for the two media. Using the resulting functions ϕandψfor each medium in conditions (28)-(31) the following system is met: /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble−iκ iκ −qt1−qt2 −1 −1i −i −2iκ2η1−2iκ2η2−η1Qt1η2Qt2 d41η2Qt2d432iκη2qt2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC1 C2 C3 C4/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble= 0 (32) to determine the remaining constants, with the defini- tionsd41=−η1Qt1−iΥ,d43= 2iκη1qt1−Υ, Υ =γκ3/ω, Qt1= 2κ2−iρ1ω/η1andQt2= 2κ2−iρ2ω/η2. In deducing the expression in system (32) which comes from (31) it was considered that pγ=−γ∂2z ∂y2(33) 3on the surface, but as we are dealing with the velocity field, then it is obtained that: ∂pγ ∂t=−γ∂2Vz ∂y2(34) which finally leads to: pγ=−γκ2 iω(−κC1+iκC3)ei(κy−ωt)(35) according to the substitution ∂/∂t→ −iωand∂/∂y→ −κ2 The vanishing of the determinant of (32) leads to the equation for the dispersion relation of the existing surfac e modes. It can be written as: ω2/bracketleftbig (ρ1+ρ2)(ρ1qt2+ρ2qt1)−κ(ρ1−ρ2)2/bracketrightbig + +γκ3[ρ1(κ−qt2) +ρ2(κ−qt1)] + + 4κ3(η2−η1)2(κ−qt1)(κ−qt2) + + 4iκ2ω(η2−η1)(ρ1κ−ρ2κ−ρ1qt2+ρ2qt1) = 0 (36) This is the dispersion relation for capillary waves for the interface of two viscous incompressible fluids. In order to obtain the constants C1,C2,C3andC4, an initial stimulus is needed according to an initial value problem [15] but this method, simple at the beginning, becomes rather complicated later and it is not good for a quite general study. As was said on the introduction, the aim of this paper is not to get inside the dispersion relation of the nor- mal modes at the interface of two viscous fluids at rest. For a better study of this subject we recommend paper [16]. We get here to show a way of solution also different from the usually taken as only the velocity vector as a function of an ” stream function ”. From now on, we will put our attention on the matching conditions and we will show that more that a mathematical information of the matching can be found on it, but also the physics of the polarization can be deduced and how the interface moves in its oscillation. From eqns. (30) and (31) it can be obtained more information about the polarization of the modes on the interface. This will be done in the next section. IV. MATCHING CONDITIONS AND POLARIZATION. There are two possible modes on the fluid: one in which the fluid particle moves in the direction of the wave prop- agation called longitudinal with notation L( Vy,0) and an- othertransverse to the direction of the wave propagation andnormal to the interface with notation TN(0, Vz). The longitudinal mode is related to the fluid compress- ibility because this motion of the fluid particles is only possible when its volume changes [14]. This analysis also holds when the mode is on the interface, but this does notmean that there are no longitudinal modes of oscillation on the interface when the fluids involved are incompress- ible. It had been shown in [1] that the interface, when oscillating, must be considered as a compressible one, be- cause precisely its change in area is responsable for the increasing of its potential energy and therefore, for its oscillation. It is important to state that this is a fun- damental argument to understand the movement of any interface in hydrodynamics. Nevertheless, now it can be shown that in spite of the compressibility of the interface, there does not exist pure longitudinal mode if we are dealing with incompressible media. Let us demonstrate this. If a pointyoon the interface is considered moving with velocity, say VSyoin they-axis direction, then, according to the continuity of velocity, the point ( yo,−ǫ) inM1and the point (yo,+ǫ) inM2must have the same velocity VSyo ifǫis small enough. As the interface is compressible, at the point y1near enough yothe velocity can be, for instance,VSy1different in general from VSyobut as the me- dia are incompressible, at the point ( y1,−ǫ) and (y1,+ǫ) the velocity must be VSyo. See Fig. 1. This is not in conformity with the continuity of the velocity through the surface and hence the pure longitudinal mode is not possible and only the TN mode seems to be valid when the media are incompressible. After these considerations during the above demon- stration, the student can keep the idea that the interface oscillations can only occur in the z-axis. This is the ac- curate moment to show to the student that things not always are as they apparently seem to be, because that assumption does not take into account the different prop- erties of each medium, whose response depends on its fundamental parameters, as density and viscosity, which are different for each medium. Hence, it is evident that it must be analyzed, precisely, the interface matching con- ditions. It is useful to compare and to support the previous qualitative analysis with a quantitative and more pro- found one regarding the interface matching conditions. Recalling carefully eqns. (30) and (31) and supposing that such a wave propagates in ydirection with move- ment only in zdirection (TN mode), then V(1)y=V(2)y= 0 and eq. (30) becomes η1∂V(1)z ∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglez= 0−=η2∂V(2)z ∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglez= 0+(37) It is known that Vzis continuous along the interface for all points. Then, the derivative with respect to yis also the same in both hands of (37) and this expression only holds ifη1=η2, i.e., if the media have the same viscosity. It does not mean for the interface to disappear because the density of each medium can be different. Then, if the viscosities of the media have not the same value, the velocity component Vyalong the interface must be differ- ent from zero to compensate the inequality (37) and to 4fulfill the continuity of the stress tensor in the ydirection given by eq. (30) yielding to a component of movement alongydirection. This mode will be called Sagittal mode or S(Vy,Vz). The above analysis was done for the general case. Now we are able to take the particular case in which one of the media is vacuum, for instance, M2, withη2= 0. Then, condition (30) becomes /bracketleftBigg η1/parenleftBigg ∂V(1)y ∂z+∂V(1)z ∂y/parenrightBigg/bracketrightBigg z= 0−= 0 (38) and it can be seen that also Vymust be non zero on the surface to hold eq. (38) with the corresponding Sagittal polarization movement. With this analysis on the conditions of stress compo- nent continuity in ydirection along the interface, it can be seen that if the two media are viscous (at least one of them), the fluid particle of the interface moves in a Sagit- talmode which combines movement in both directions: along the wave propagation in ydirection, and normal to the interface in zdirection. Then, it is qualitatively clear that the viscosity of each media plays a fundamental role in the coupling of modes even for incompressible fluids. In spite of that, it could be a mistake to say that the modes decouple if the viscosities are equal. It should not be forgotten that eq. (31) is also important in the characterization of the interface particl e behaviour and it includes the pressure on each side of the surface. According to eq. (18), the pressure is associated with the longitudinal mode and the inertial effect of the fluid particle according with the density of the media. Then, it contains the information of each components of the velocity and also of the density and according to eq. (26) the pressure has a jump through the interface. This result, in combination with the analysis of eq. (30) make difficult to understand the role played by the densities of each media on the surface polarization movement, and it can not be reached from this only analysis. This point is still a matter of investigation. V. CONCLUSIONS The present work is an attempt to give an example, us- ing the hydrodynamics, of how the study of the interface matching conditions allows us to make a plentiful and rich in details discussion. Moreover, of how the inter- face matching conditions content a sufficient information to conclude that the interface oscillation must be withaSagittal mode and not with neither a pure longitudi- nal, nor a pure transversal one. This movement has been shown to be close related to the physical properties of the media such as viscosities and that fact allows us to establish rigorously that those are the parameter which characterize the interface movement and the response of each medium to an stimulus coming from the other one. It was seen that viscosity is the main parameter in the coupling of the two modes to achieve a Sagittal one, nevertheless within the framework of this formalism it is difficult to determine the role of viscosity and of the den- sity ratios in the coupling of modes. This aspect needs further investigation. [1] E. H. Lucassen-Reynders and J. Lucassen, Advan. Col- loid Interface Sci. 2, 347 (1969). [2] J. Lucassen, Trans. Faraday Soc. 64, 2220 (1968). [3] M. van den Tempel and R. P. van de Riet, J. Chem. Phys. 42, 2769 (1965) [4] R. S. Hansen and J. A. Mann, J. Appl. Phys. 35, 152 (1964) [5] C-S. Yih, J. Fluid Mech. 27, 337 (1967) [6] A. P. Hooper and W. G. C. Boyd, J. Fluid Mech. 128, 507 (1983) [7] X. Li and R. S. Tankin, J. Fluid Mech. 226, 425 (1991) [8] J. Cousin and C. Dumouchel, Atomization and Sprays 6, (1996) [9] Garc´ ıa-Moliner, F., Ann. Physique 2, 179 (1977). [10] Velasco, V. R. and Garc´ ıa-Moliner, F., Surface Sci. 67, 555 (1977). [11] Velasco, V. R. and Garc´ ıa-Moliner, F., Physica Script a 20, 111 (1979). [12] Platero, G., Velasco, V. R. and Garc´ ıa-Moliner, F., Ph ys- ica Scripta 23, 1108 (1981). [13] Garc´ ıa-Moliner, F. and Velasco, V. R., Theory of Sin- gle and Multiple Interfaces, World Scientific, Singapore, (1992). [14] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Butterworth-Heinemann, 1987). [15] P. G. Drazin and W. H. Reid, Hydrodynamics Stability (Cambridge University Press, 1981). [16] A. Hern´ andez Rodr´ ıguez, J. Mar´ ın Antu˜ na, H. Rodr´ ı guez Coppola and C. Dopazo, Phys. Scripta 59, 313-318 (1999). FIG. 1. Relation of velocities on the interface between two viscous fluids. 5vy1(1)vy1(2) vy1S vyo(1)vyo(2) vyoS y1yo -ε+ε yz

arXiv:physics/0004065v1 [physics.flu-dyn] 25 Apr 2000Surface waves at the interface between two viscous fluids Arezky H. Rodr´ ıguez∗, J. Mar´ ın-Antu˜ na, H. Rodr´ ıguez-Coppola Dpto. de F´ ısica Te´ orica, Fac. de F´ ısica, Universidad de la Habana, C. de la Habana, Cuba C. Dopazo Dpto. de Fluidos, C.P.S, Universidad de Zaragoza, Espa˜ na (January 12, 2014) The Surface Green Function Matching analysis (SGFM) is used to study the normal modes of the interface oscillations between two non-mixed fluids by cons idering the difference in their densities and viscosities. The limiting case of viscous-inviscid sys tem is used for comparison. The role of the viscosity and the density ratios on the momentum exchange an d on the polarization of the surface modes is analyzed. 68.10.-m; 68.10.Cr; 68.10.Et I. INTRODUCTION The theory of surface waves in fluids is usually treated using the Orr-Sommerfeld equation obtained from potential method [1]. This procedure is useful to find the characterist ics of the wave, such as dispersion relation and damping but becomes rather complicated when some other features are needed such as polarization and density of modes. On the other hand, the inclusion of the viscosities of all of t he media make difficult to understand the physics of the interface. Broadly speaking, the dual effects of viscosity i s well known [1]: to dissipate the energy of any disturbance, but also it has the more complicated effect of diffusing momentum. At present, the theory for viscous cases is not nearly as complete or general as for inviscid cases and it pro vides only a partial understanding of the role of viscosity in such systems. A suitable formalism for including all the viscosities with great ease in non-homogeneous systems studying the response function has been developed elsewhere [2–5]. This formalism, the method of Surface Green Function Matching (SGFM), has been extensively used to study various inhomoge neous problems involving surface waves at solid surfaces, both free solid surface (interface between vacuum and solid ) and solid-solid interfaces [6]. It has also been used in interface involving fluids [2,4] as is the case of solid-fluid interface and even fluid-fluid interface (this last case anal yzed to give an unified treatment of waves in solids and fluids which seem to be apparently unconnected problems). As far as we know, there are no previous works where the SGFM have been applied to the hydrodynamics problems as these authors suggested. The aim of this paper is to apply the SGFM to the study of the phy sical characteristics (dispersion relation, damping and polarization) on the interface normal modes of two fluids at rest giving insight of the mechanisms of momentum exchange through the interface for different ratios between the viscosity and density of the two media. In the next section a brief outline of the main points of the SG FM is given for the fluid-fluid system at rest, highlighting the considerations made in the solution of the problem. Section III is devoted to the physical analysis of the polarization of the modes and the momentum exchange ac ross the interface. In section IV it is carried out a numerical evaluation considering the physical interpreta tion of the terms and the results for pair of fluids which are analyzed as illustration. Finally some conclusions are out lined. II. SGFM FOR TWO VISCOUS FLUIDS INCLUDING SURFACE EFFECTS The formal development of the SGFM method has been fully expl ained elsewhere [2,5] and in particular the treatment of matching with discontinuities [3], suitable f or the case of two non mixed fluids where the interface has special effects not seen in the liquid bulk. Mathematical and formal details can be found elsewhere [2] and need not ∗arezky@ff.oc.uh.cu 1be repeated here. It is only necessary to add that in fluid-flui d interfaces it is better to work with the velocity of the fluid particle in agreement with the Navier-Stokes equation , instead of the fluid particle deformation, suitable when solids are present. Consider a system formed by a fluid M1forz <0 and a fluid M2forz >0, both of them at rest. It has a planar interface at z= 0. Analysing first each bulk media individually to prepare i ts description in a suitable way for the eventual matching at the interface, the coordinate system w ill be choosen considering the planes z=const as those of interest. The notation will be for coordinates r= (ρ, z),k= (κ, q) where ρandκare 2D vectors. As explained in [5], the SGFM start with the knowledge of the G reen function G.F of the excitation studied in each bulk material constituent. Then, it is needed to analyz e the physical model for the excitation to perform later matching at the interface. Now, to know the G.F of each bulk media, the 3D differential equ ations of hydrodynamics are the starting point. The fluids are usually treated as incompressible and describ ed with the Navier-Stokes equation. However, as explained in [3], it proves convenient here to give the theory for compressible fluids, even if compressibility effects are ultimately neglected. Then, the equation of mass for isoentropic proce sses, and the momentum conservation equation that govern the fluid motion are linearized by neglecting all nonl inear terms in disturbance quantities. They may be written, respectively, as 1 c2∂ ∂tp(r, t) +ρ∇ ·V(r, t) = 0 (1) ρ∂ ∂tV(r, t) =−∇p(r, t) +η∇2V(r, t) +/parenleftBig η′+η 3/parenrightBig ∇∇ ·V(r, t) (2) where c,p,ρ,ηandη′are the velocity of sound, dynamical pressure, equillibriu m density, shear and bulk viscosities respectively, all of them considered as constants in each me dium. V(r, t) is the velocity of the fluid. We neglected the external forces and supposed that the perturbation is sm all enough to neglect the convective term for pressure in (1). All space and time dependent quantities will be Fourier tran sformed according to exp[i(κ·ρ−ωt)] where ωis a frequency. Then, for surface wave propagation, the amplit udes are functions of ( κ, ω) on one hand and of zon the other. This zdependence is due to the fact that there is no spatial invaria nce in this direction and the Fourier transform can not be accomplished. Green functions, includ ing the ones for the bulk material constituents, are then conveniently expressed as G(κ, ω;z, z′) or, simply, as G(z, z′), with ( κ, ω) understood everywhere. Time Fourier transform will be implied now on. From eq. (1) it is obtained p(r, ω) = (ρc2/iω)∇ ·V(r, ω), which putted in eq. (2) gives rise to iρωV i(r, ω) + (¯Γ−η)∂ ∂xi∇ ·V(r, ω) +η∇2Vi(r, ω) = 0 (3) withi=x, y, z and ¯Γ =−ρc2 iω+/parenleftbigg η′+4 3η/parenrightbigg (4) as the system of equations which couples the velocity compon ents. This system must be solved as a whole as it can not be decoupled in the general case. The actual G(z, z′) of each bulk media considered separately as infinity can be o btained in different ways but using, for instance, the Fourier transform 3D, it yields for the G.F [5]: G(k, ω) =1 iρω−ηk2/bracketleftbigg I+(¯Γ−η)kk iρω−¯Γk2/bracketrightbigg (5) where Iis the unit matrix and kkis a diadic product of the wave vector. Its poles ql=/parenleftbiggiρω ¯Γ−κ2/parenrightbigg1/2 qt=/parenleftbiggiρω η−κ2/parenrightbigg1/2 (6) describe the transverse and longitudinal modes of the infini te medium. In (5) the incompressible fluid can be considered taking ( ¯Γ→ ∞) and the proper limit is achieved. 2There is no physical reason for the preference of a particula r direction in the xy-plane. This spatial symmetry of the system allows us to define, for instance, k= (0, κ, q) without loosing generality but getting simplification of t he calculations. Note that ql→i|κ|if the compressibility is neglected, see eq. (4), given rise to a vanishing longitudinal mode. So, the qlpole describes the longitudinal mode due to the compressibi lity of the media. LetGSbe the Green function (G.F) of the surface system just defined andGSits surface projection. Let G−1 S be the reciprocal of GSin the two-dimensional ρorκspace. This is the central object in the SGFM analysis. In particular, knowing G−1 Sit is possible to find the surface mode dispersion relation (S MDR) and the density of modes of the surface system [2]. It is important to stress that the s ecular equation for the SMDR, namely detG−1 S= 0 (7) expresses the continuity of the velocity and the stress comp onents transmitted across z= 0. This is where the physics of the surface effects comes into the picture. These effects in troduce changes in the stress components transmitted across the interface and are ultimately measured by some sur face tensor mSwhose physical meaning is that mS, acting on the velocity field V, yields the extra forces per unit area transmitted across th e interface. Let us call G−1 SOtoG−1 Sin the absence of such surface effects, then one finds [3] G−1 S=G−1 SO+mS (8) Thus the problem is to find mSfor the surface effects one wishes to study. It will be include d in this case only the surface tension γaccording to Laplace’s Law. It can be deduced [3] that mS=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble0 0 0 0 0 0 0 0−γκ2 i ω/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble(9) There is a little difference between the former expression an d the expression obtained in [3] according to the fact that here the velocity of the fluid particle is considered ins tead the fluid particle deformation. Then, eq. (8) expresses the continuity of the velocities and the stress components transmitted across the interface at z= 0. Knowing G−1 Sone can find the dispersion relation of the surface modes (SMD R) through the secular equation (7) III. PHYSICS AND POLARIZATION OF THE SURFACE MODES The construction of G−1 SOis explained in [2]. The result, after adding (9), is G−1 S=/vextenddouble/vextenddouble/vextenddouble/vextenddoubleη1qt1+η2qt20 0 /ba∇dblg−1 S/ba∇dbl/vextenddouble/vextenddouble/vextenddouble/vextenddouble(10) where g−1 Sis a 2×2 matrix and 0is the null vector 1 ×2.g−1 Shas components (g−1 S)11=ρ1ωql1 κ2+ ql1qt1+ρ2ωql2 κ2+ ql2qt2(11) (g−1 S)22=ρ1ωqt1 κ2+ ql1qt1+ρ2ωqt2 κ2+ ql2qt2−γκ2 i ω(12) (g−1 S)12=−(g−1 S)21=/parenleftbiggρ1ωκ κ2+ ql1qt1+ 2iκη1/parenrightbigg −/parenleftbiggρ2ωκ κ2+ ql2qt2+ 2iκη2/parenrightbigg (13) We shall refer to the modes as sagittal or S polarized with (0 , Vy, Vz),transverse tangent or TT( Vx,0,0),longitudinal or L(0 , Vy,0) and transverse normal or TN(0 ,0, Vz) modes, according to the component of the velocity they have . Now, on using (10) in (7) the factorisation of the ( G−1 S)11matrix element yields a TT mode which does not interact with the others, whose dispersion relation is η1qt1+η2qt2= 0 (14) and has x-axis polarization. 3It is easily seen according to (6) that the TT mode has no solut ion but as stressed in [4], it does contribute to the density of modes and therefore plays a non trivial role in the physical properties of the interface. This mode exists but it is not a stationary one if there is other surface effects considered [4]. The rest of (10) yields the secular equation detg−1 S= 0 (15) It gives a sagittal mode with polarization S(0 , Vy, Vz) and surface tension included. It will be analyzed in the fol lowing. The factor ( g−1 S)11, see eq. (11), represents the surface movement component in ydirection due to compressibility of the media while ( g−1 S)22is azdirection surface movement. The factor ( g−1 S)12represents a coupling between y andzmovements giving rise to an S polarization mode. It means tha t the surface has both horizontal and vertical movements. In other words, the surface particles move in a ki nd of circular orbits depending of its phase difference. On the other hand, there are no important velocities in our sy stem, then compressibility can be neglected as described in [2] and we will discuss whether the S polarizati on remains or not. Putting ql=i|κ|in (11), (12) and (13) it is obtained (g−1 S)11=ρ1ωi|κ| κ2+i|κ|qt1+ρ2ωi|κ| κ2+i|κ|qt2(16) (g−1 S)22=ρ1ωqt1 κ2+i|κ|qt1+ρ2ωqt2 κ2+i|κ|qt2−γκ2 i ω(17) (g−1 S)12=−(g−1 S)21=/parenleftbiggρ1ωκ κ2+ i|κ|qt1+ 2iκη1/parenrightbigg −/parenleftbiggρ2ωκ κ2+ i|κ|qt2+ 2iκη2/parenrightbigg (18) First of all let us consider the special case where the viscos ity of M1is neglected. If we put η1= 0 in (16)-(18) it is obtained qt1→ ∞ and hence g−1 S=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρ2ωi|κ| κ2+i|κ|qt2−/parenleftbigg ρ2ωκ κ2+i|κ|qt2+ 2iκη2/parenrightbigg /parenleftbigg ρ2ωκ κ2+i|κ|qt2+ 2iκη2/parenrightbigg ρ1ω i|κ|+ρ2ωqt2 κ2+i|κ|qt2−γκ2 i ω/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble(19) Note that even though the viscosity of one of the constitutie nt media was neglected, the coupled factor remains due to the nonzero viscosity of the other fluid. So, in this lim it this mode remains as sagittal S(0 , Vy, Vz) exhibiting movements in y- and z-axis for the surface particles. The fluid was taken as incomp ressible but there is still a component of velocity on y-direction. As far as we know, nobody has ever pointed out thi s fact clear, except Lucassen in his works [7–10], who considered incompressible fluids, b ut the movement in y-axis was due to active materials on the surface, no as an effect of viscosity. This coupling of mov ements could be responsible for a less wavelength and a bigger dissipation as it will be seen later. It is in this dire ction where the viscosity plays an important role. There is more information in eqs. (16)-(18). If viscosities are neglected ( η1=η2= 0) in ( g−1 S)11and (g−1 S)12these expressions vanish because q t1→ ∞, qt2→ ∞, but doing the same in ( g−1 S)22does not yield a vanishing result. This leads to: g−1 S=/vextenddouble/vextenddouble/vextenddouble/vextenddouble0 0 0ρ1ω2+ρ2ω2−γκ2|κ|/vextenddouble/vextenddouble/vextenddouble/vextenddouble(20) The non zero component of Eq. (20) is a generalization of the K elvin dispersion relation when the density of the upper medium is included. Then, it is concluded that when there is no viscosities the S p olarized mode becomes TN mode because the coupled factor disappears and only remains ( g−1 S)22. Indeed, if at least one of the viscosities is considered the coupled factor appears giving rise to the S mode, (see eq. (19)). Hence, the v iscosity is the fundamental force which couples different modes among them. Furthermore, the viscosity is the main cause of momentum exc hange between the two media through the surface on the y-direction movement. Note that the longitudinal component movement disappears according to ( g−1 S)11→0 when η1andη2are neglected. On the other hand, in (15) the transverse norm al movement described by ( g−1 S)22 exists because of the densities and viscosities of the media , (see eq. (17)). When the viscosities are neglected as in (20), the normal component movement still remains because o f the densities of the fluids. Hence, in the case of the z-axis movement both the viscosities and densities are impor tant for the exchange of momentum. 4These results are in agreement to the fact that when the inter face particle moves according to the longitudinal mode it remains on the plane z= 0 and the viscosities are the only way for the two media to int eract, but when the interface particle moves according to the transverse norma l mode it goes into each medium sometimes at z >0 and other at z <0 and then the inertial effects of the media become important a ccording to their densities. Expressions (15)-(18) also recover the Kelvin equation for an ideal fluid with free surface, (see references in [9,1]). Neglecting the viscosities, and setting ρ1= 0 it is obtained g−1 S=/vextenddouble/vextenddouble/vextenddouble/vextenddouble0 0 0ρ2ω2−γκ2|κ|/vextenddouble/vextenddouble/vextenddouble/vextenddouble(21) It can be seen that our formalism not only recovers the expres sion for the Kelvin equation, but also recovers the z-polarization of that mode. After this analysis one can return back to the problem for bot h viscous fluids. From (15)-(18) it is obtained the secular equation for the SMDR ω2/bracketleftbig (ρ1+ρ2)(ρ1qt2+ρ2qt1)− |κ|(ρ1−ρ2)2/bracketrightbig +γκ2|κ|[ρ1(|κ| −qt2) +ρ2(|κ| −qt1)] + + 4κ2|κ|(η2−η1)2(|κ| −qt1)(|κ| −qt2) + 4iκ2ω(η2−η1)(ρ1|κ| −ρ2|κ| −ρ1qt2+ρ2qt1) = 0 (22) with the following new definition of q t= (κ2−iρω/η )1/2. This expression, which we recall corresponds to two viscous non mixed incompressible fluids, can also be accompl ished applying the potential method, although using that formalism it is rather difficult to obtain the polarizati on of the modes. This is the equation to be used to study the modes if one includ es both viscosities and surface tension effects for incompressible fluids. Expression (22) was reported in [4] t o study the surface waves at the interface between a solid and a fluid. They neglected the surface tension. One of the aim of this paper is to compare this theory with the theory which just take into account only one of the viscositi es. From expression (19) it is not difficult to achive the SMDR for the viscous-inviscid fluid interface −ω2ρ2(ρ1+ρ2) +γκ2|κ|ρ2+ 4κ2|κ|η2 2(|κ| −qt2)−4iρ2ωκ2η2= 0 (23) which reduces to equation (2.5) of [4] when γ= 0 and will be evaluated in the next section along with (22) fo r the viscous case. IV. RESULTS OF THE NUMERICAL EVALUATION In order to make a numerical study the following quantities o f length and time for nondimensionalization will be taken: time by TO=η3 2 ρ2γ2 length by LO=η2 2 ρ2γ(24) The dispersion relation (22) becomes ω2/bracketleftbig (1 +Q)( ¯ qt1+Q¯ qt2)− |κ|(1−Q)2/bracketrightbig +κ2|κ|[|κ|(1 +Q)−¯ qt1−Q¯ qt2] + +4κ2|κ|(1−N)2(|κ| −¯ qt1)(|κ| −¯ qt2) + 4iκ2ω(1−N)[−|κ|(1−Q) + ¯ qt1−Q¯ qt2] = 0 (25) for viscous fluids and eq. (23) gives rise to −ω2(1 +Q) +κ2|κ|+ 4κ2|κ|(|κ| −¯ qt2)−4iωκ2= 0 (26) for viscous-inviscid case, where Q=ρ1/ρ2,N=η1/η2and ¯ qt1=/parenleftbigg κ2−iωQ N/parenrightbigg1/2 (27) ¯ qt2=/parenleftbig κ2−iω/parenrightbig1/2(28) 5Then the characteristics of the system will be studied by its SMDR with real values of the frequency ω. Let us allow κto be complex, its real part is 2 πtimes the inverse of the wavelength and the imaginary part is the distance damping coefficient βrelated with the viscosities of the media. The dimensionles s parameters are κo= 2π/L oand ωo= 2π/To. Fig. 1 shows the SMDR for Q= 0.8. There is one mode which decreases its wavelength λand increases its distance damping coefficient βwith increasing frequency at a fixed value of the parameter N. It is also shown that when the viscosity ratio Nis increased the wavelength lightly decreases at any freque ncy. The curves split bigger at higher frequencies. On the other hand βincreases with increasing N. Also it was plotted the curves obtained with N= 0 from eq. (26) which means zero viscosity of the medium M1. It can be seen that the theory which includes all the viscosities predicts small λand bigger βfor a fixed ωwith respect to the N= 0 case. Fig. 2 shows the dependence of κandβwith respect to the variation of the density ratio Qat a fixed value of the frequency and viscosity ratio. It can be seen that λdecreases when Qincreases at a fixed N. This was deduced by Taylor in his study of the ripple formation on an infinitely th ick viscous circular jet but neglecting the air viscosity. References are given in [11]. We now prove that this is also tr ue when both viscosities are considered. Also λdecreases at a fixed Qwhen the viscosity ratio Ntakes higher values. So, the effect of the viscosity of medium M1reinforces the effect produced by the density and it can be stated that the smaller wavelength will be obtained when QandN are both bigger. It is also plotted the curve with N= 0 corresponding to the viscous-inviscid case. It can be see n that the wavelength is always smaller in the case N/negationslash= 0 (viscous-viscous case). The curves split bigger as Qincreases and βgrows rapidly at low values of Qfor a fixed value of Nand tends to saturation for higher values of Q. This small variation of βwith the variation of Qeven at a fixed value of the viscosity ratio Nreinforces the idea of the density as another mechanism of momentum exchange between the two me dia through the interface. It not only produces smaller wavelengths, but also produces lightly bigger dist ance damping coefficients β. On the other hand, the distance damping coefficient βalso increases at higher values of the viscosity ratio Nat a fixed Q. It was also plotted the curves at N= 0. It is interesting to note that the theory of viscous-invi scid case predicts a small decrease of the distance damping coefficient with increasing density ratio Q. This is in accordance to the fact that setting N= 0 means to neglect the momentum exchange through the interf ace by the viscosity and raising Qrepresents to increase the dynamic properties of the surfac e given rise to a bigger distance for the wave to travel before vanishing. Fig. 3 shows the variation of λandβwith respect to the viscosity ratio for a fixed value of ωat three values of Q. It shows that as the viscosity ratio increases, the waveleng th reduces rapidly first and tends to a limiting value for N≥1. The curves start in the value of λcorresponding to the viscous-inviscid case. Also, for a fixe d value of Nthe wavelength decreases as Qincreases, in correspondence with Fig. 2. For the coefficient βit is seen that it raises for increasing N. To illustrate this theory for real fluid combinations there w ill be used three pairs of fluids: air/water, water/aniline and water/mercury. The parameters of these fluids at room tem perature are: Element Density Viscosity Surface Tension (kg/m3) (mPa s) ( mN/m) air 1 .21 0.018 - water 998 0.890 71.99 mercury 13500 1.526 485.48 aniline 1022 3.847 42.12 Then, for the system air/water it is Q= 0.0012 and N= 0.0202. In this case the SMDR is plotted in Fig. 4. It can be seen that there is no difference of the wavelength repor ted by viscous-viscous and viscous-inviscid cases due to the small values of the density and viscosity ratios but th ere is a small increase of βfor all frequencies when the air viscosity is considered. However, the operating conditions in many gas turbine combu stors and liquid-propellant rocket engines are such that the density and viscosity ratios are higher. It could be so also in water-oil emulsions and other problems where the interface between two fluids plays an important role. The n, the SMDR for water/aniline ( Q= 0.977,N= 0.231) and water/mercury ( Q= 0.074,N= 0.583) systems were also plotted. In the first case (water/anil ine) the densities are very similar but the viscosity of the aniline is much bigg er than the water viscosity. In the case of water/mercury the viscosities are near one half one another, but the densit y of the mercury is much bigger the density of the water. The SMDR has been plotted in Fig. 5 and 6. Note than in both case s the wavelength decreases and the distance damping coefficient increases in a visible way when the viscos ity of the medium M1(water in both cases) is considered. The difference is bigger at high values of the frequency where the viscosity effects become important. Then, one can conclude that the inclusion of the viscosities of all media p roduces a substantial decrease of the wavelength if the viscosity ratio are big enough. It gives rise also to a bigger distance damping coefficient for the wave. 6V. CONCLUSIONS In the present paper the close relationship between the prop erties of low amplitude surface waves propagation with the viscosity and density ratios, in a system of two non-mixe d incompressible fluids at rest has been set out. The SGFM method was used to accomplish the dispersion relation a nd the full study of wave propagation by varying different parameters of the media. It was shown that the viscosity is a fundamental parameter fo r the coupling of different modes. It gives rise to an S polarization mode with yandzcomponents of the movement of the particles on the surface. A lso it was seen that the viscosity is the main force in producing momentum exchan ge in the longitudinal mode, but for the transverse normal mode both the viscosity and the density ratios are imp ortant to the momentum exchange. When considering surface modes, it was shown that only one of them is allowed and its wavelength is smaller when considering the viscosity of both media for fixed values of th e density ratio. Also it was seen a characteristic variation of the distance damping coefficient when the viscosity of all m edia are included. On the other hand the increasing of the density ratio also reduces the wavelength and produce s a lightly increase of the distance damping coefficient, then this factor is also important in reducing the wavelengt h of the surface waves. In order to see more real situations, three pair of fluids were analyzed and the importance of taking in consideration all the viscosities was shown. ACKNOWLEDGEMENTS We are indebted to Professors Federico Garc´ ıa-Moliner and V. R. Velasco for advices and clever discussions. This work was partially supported by an Alma Mater grant, Uni versity of Havana. [1] Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, Butte rworth-Heinemann (1987). [2] Garc´ ıa-Moliner, F., Ann. Physique 2, 179 (1977). [3] Velasco, V. R. and Garc´ ıa-Moliner, F., Physica Scripta 20, 111 (1979). [4] Platero, G., Velasco, V. R. and Garc´ ıa-Moliner, F., Phy sica Scripta 23, 1108 (1981). [5] Garc´ ıa-Moliner, F. and Velasco, V. R., Theory of Single and Multiple Interfaces, World Scientific, Singapore, (199 2). [6] Velasco, V. R. and Garc´ ıa-Moliner, F., Surface Sci. 67, 555 (1977). [7] Lucassen, J. and Lucassen-Reynders, E. H., J. Colloid In terface Sci. 25, 496 (1967). [8] Lucassen, J., Trans. Faraday Soc. 64, 2220 (1968). [9] Lucassen-Reynders, E. H. and Lucassen, J., Advan. Collo id Interface Sci. 2, 347 (1969). [10] Lucassen, J. and van der Tempel, M., Chem. Engin. Sci. 27, 1283 (1972). [11] Lin, S. P., Lian, Z. W. and Creighton, B. J., J. Fluid Mech .220, 673 (1990). FIG. 1. Dispersion relation of the surface mode for Q= 0.8. The upper part gives the wavelength κ/κ oand the lower part the distance damping coefficient β/κ o FIG. 2. Relation between wavelength and distance coefficient with respect to the density ratio at a fixed frequency for different values of viscosity ratios. The case N= 0 is the viscous-inviscid case FIG. 3. Relation between wavelength and distance coefficient with respect to the viscosity ratio at a fixed frequency for different values of density ratios FIG. 4. Dispersion relation of the surface mode for the air/w ater system. In the legend it is especified the M1as left and M2as right in the combination M1/M2, i. e., air/water in this figure. FIG. 5. Dispersion relation of the surface mode for the water /aniline system 7FIG. 6. Dispersion relation of the surface mode for the water /mercury system 85101520 ω/ωο (x 10-5)κ/κο (x 10-4) 0 2 4 6 8 1004812 non-viscous airN=0.0202air/water Q =0.0012β/κο (x 10-5)051015202530 N =0.231 non-viscous waterwater/aniline Q =0.977 ω/ωο (x 10-5)κ/κο (x 10-4) 0 2 4 6 8 100481216 non-viscous waterN =0.231β/κο (x 10-5)0,0 0,2 0,4 0,6 0,8 1,01,21,41,61,82,02,2 N =0N =0.8 N =0.5β/κο (x 10-4) N =0.1 Q2,22,32,42,52,62,72,8 N =0N =0.8N =0.5 N =0.1ω/ωο= 10-4κ/κο (x 10-3)2,32,42,52,62,72,8 Q =0.8 Q =0.5 Q =0.1ω/ωο=10-4κ/κο (x 10-3) 0,00,20,40,60,81,01,21,41,61,21,62,02,42,8 Q =0.8 Q =0.5 Q =0.1β/κο (x 10-4) N510152025 ω/ωο (x 10-5)N =0N =0.8κ/κο (x 10-4) 0 2 4 6 8 10048121620 N =0N =0.8 N =0.5 N =0.1Q=0.8β/κο (x 10-5)05101520non-viscous waterN =0.583water/mercury Q =0.074 ω/ωο (x 10-5)κ/κο (x 10-4) 0 2 4 6 8 100481216N =0.583 non-viscous waterβ/κο (x 10-5)

arXiv:physics/0004066v1 [physics.atm-clus] 25 Apr 2000A transferable nonorthogonal tight-binding model of germa nium: application to small clusters Jijun Zhao∗ Department of Physics and Astronomy, University of North Ca rolina at Chapel Hill, Chapel Hill, NC 27599, USA. International Centre for Theoretical Physics, P.O.Box 586 , Trieste 34100, Italy Jinlan Wang, Guanghou Wang National Laboratory of Solid State Microstructures, Nanji ng University, Nanjing 210093, P.R. China (February 2, 2008) Abstract We have developed a transferable nonorthogonal tight-bind ing total en- ergy model for germanium and use it to study small clusters. T he cohesive energy, bulk modulus, elastic constants of bulk germanium c an be described by this model to considerably good extent. The calculated bu lk phase di- agram for germanium agrees well with LDA results. The geomet ries and binding energies found for small Ge nclusters with n= 3−10 are very close to previous ab initio calculations and experiments. All these results suggest that this model can be further applied to the simulation of ge rmanium cluster of larger size or with longer time scale, for which ab initio methods is much more computational expensive. 36.40.Mr, 61.46.+w, 71.15.Fv, 31.15.Rh Typeset using REVT EX 1In the past decade, tight-binding molecular dynamics (TBMD ) has evolved into a power- ful approach in the simulation of semiconductor materials1–3. In the tight-binding scheme, although the system is still described in a quantum-mechani cal manner, the computational cost is significantly reduced due to the parameterization of Hamiltonian matrix elements. In many cases of material simulations, it might offer a satisf actory compromise between empirical4and first principle5,6methods for modeling the interatomic interaction. As an alternative of the accurate but costly ab initio molecular dynamics, TBMD can handle more complicated systems with acceptable accuracy1–3. For carbon and silicon, there are several well established o rthogonal7–9and nonorthogonal10–12tight-binding models. Although the orthogonal models work s well for various bulk systems1–3, Menon found that the inclusion of the nonorthogonality of t ight- binding basis is essential for describing the geometries an d binding energies of small silicon clusters11,12. Compared to carbon and silicon, there is much fewer tight-b inding models de- veloped for germanium. Recently, M.Menon has extended the n onorthogonal tight-binding (NTB) scheme to germanium and calculated the structures and cohesive energies of small Genclusters13. Although the cluster geometries obtained in Ref.[13] are g enerally consistent withab initio results, the binding energies are overestimated. In this wo rk, we perform an independent fitting of NTB parameters for germanium, which d escribes binding energies of germanium clusters better than that in Ref.[13]. This model is employed to study some bulk properties and considerably good results are obtained. In Menon’s NTB model11–13, the total binding energy Ebof a system with Naatoms can be written as a sum Eb=Eel+Erep+NaE0 (1) Eelis the electronic band energy, defined as the sum of one-elect ron energies ǫkfor the occupied states: Eel=/summationtextocc kǫk. In Eq.(1), a constant energy correction term NaE0and a repulsive interaction Erepare also included. On nonorthogonal basis set, the eigenvalues ǫkof system are determined from the secular 2equation: det|Hij−ǫSij|= 0. (2) Here the overlap matrix elements Sijare constructed in the spirit of extended H¨ uckel theory14, Sij=2Vij K(ǫi+ǫj)(3) and the nonorthogonal Hamiltonian matrix elements by Hij=Vij[1 +1 K−S2 2] (4) where S2=(Sssσ−2√ 3Sspσ−3Sppσ+ 3Sppπ) 4(5) is the nonorthogonality between two sp3bonding orbitals and Kis a environment dependent empirical parameter11. TheHijandSijdepend on the interatomic distance through the universal pa rameters Vij, which are calculated within Slater-Koster’s scheme15. The scaling of the Slater-Koster parameters Vλλ′µis taken to be exponential with the interatomic distance r Vλλ′µ(r) =Vλλ′µ(d0)e−α(r−d0)(6) where d0= 2.45˚Ais the bond length for germanium crystal in the diamond struc ture16. The repulsive energy Erepin Eq.(1) is given by the summation of pairwise potential function χ(r): Erep=/summationdisplay i/summationdisplay j>iχ(rij) =/summationdisplay i/summationdisplay j>iχ0e−4α(rij−d0)(7) where rijis the separation between atom iandj. In practice, we adopt the Slater-Koster hopping integrals Vλλ′µ(d0) fitted from the band structure of bulk germanium17. The on-site orbital energies ǫs,ǫpare taken from atomic 3calculations18. The only four adjustable parameters α,K,χ0,E0are fitted to reproduce the fundamental properties of germanium bulk and dimer. The inp ut properties include: the experimental values of bulk interatomic distance 2.45 ˚A16, dissociation energy 2.65 eV19and vibrational frequency (286 ±5 cm−1)20of Ge 2dimer, as well as theoretical bond length of Ge2(2.375 ˚A) from accurate quantum chemistry calculation at G2(MP2) l evel21. The fitted parameters are given in Table I. 4TABLES Table I. Parameters in the NTB mode developed for germanium i n this work. See text for detailed descriptions. ǫs ǫp Vssσ Vspσ Vppσ Vppπ -14.38 eV -6.36 eV -1.86 eV 1.90 eV 2.79 eV -0.93 eV d0 K χ 0 α E 0 2.45˚A 1.42 0.025 eV 1.748 ˚A−10.79 eV We can first check the validity of the present NTB scheme by stu dying the fundamen- tal properties of germanium solid in diamond phase. The obta ined cohesive energy 3.58 eV/atom is very close to experimental value 3.85 eV/atom16. Furthermore, we have calcu- lated the bulk modulus Band elastic constants C11, C12, C44of germanium and compared with experimental values22in Table II. Most of the bulk elastic properties such as B,C11, C12are well reproduced except that the C 44is overestimated by 0.35 Mbar in our model. Table II. Bulk modulus and elastic constants (in units of Mba r) of bulk germanium in diamond structure. NTB are the theoretical results from pre sent NTB model; Exper. denote the experimental values taken from Ref.[22]. C11 C12 C44 B NTB 1.125 0.545 1.019 0.738 Exper. 1.288 0.483 0.671 0.751 5FIGURES 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3-0.20.00.20.40.60.81.01.2 fccNTB LDA Clathrate IDiamondscCohesive Energy (eV) Relative volume FIG. 1. Cohesive energies vs. relative volume for bulk germa nium in simple cubic (sc), diamond and clathrate I phase from NTB (solid line) and LDA (open circ le) calculations23. By using the NTB scheme, we have also calculated the equation of states of germanium in different phases. In Fig.1, we present the zero-temperatu re phase diagram of the fcc, sc, diamond and type I clathrate obtained from NTB model, along w ith recent LDA plane- wave pseudopotential calculations23. It is worthy to noted that our NTB model is able to described the energy and atomic volume of clathrate phase. T he energy of clathrate I is 0.06 eV/atom higher than that of diamond phase and its relative vo lume is about 15% larger than diamond phase. These results are consistent with the 0.08 eV energy difference and 15% volume change from LDA calculation23. The success in clathrate phase is important since the clathrate is also four-coordinated structure23. On the other hand, it is natural to find that the agreement between LDA and NTB scheme become worse in the h igh-coordinated phases like fcc since the present model is fitted from the diamond pha se and dimer. However, the relative poor description of high coordinate phase will not influence the study on germanium 6clusters since such high coordination ( ∼12) does not exist in the geometries of germanium clusters. Considering its extreme simplicity and small num ber of adjustable parameters, the current NTB scheme gives a sufficient satisfactory overall de scription of bulk germanium properties. Therefore, one can expect that the model to give a reasonable description on the germanium clusters. In this paper, we determine the lowest energy structures of t he Ge nclusters with n= 3−10 by using TBMD full relaxation. The ground state structure s of Ge n(n= 5−10) are presented in Fig.2 and the geometrical parameters of sma ll Ge n(n= 3−7) clusters are compared with previous ab initio calculations24–27in Table III. In general, both the lowest energy structures and their characteristic bond length agr ee well with ab initio results. A brief description is given in the following. FIG. 2. Lowest energy structures of Ge n(n= 5−10) clusters. Table III. Lowest energy geometries (with characteristic b ond length parameters) of small Genclusters. Tight-binding calculation (NTB) are compared wi th previous ab initio results such as: MRSDCI24, B3LYP25, LDA26,27. The label of atom and bond for Ge nare taken from Ref.29. n Sym. Bond Bond length ( ˚A) MRSDCI24B3LYP25LDA26LDA27NTB 3 C2v 1-2 3.084 3.070 3.20 2.91 2.71 71-3 2.320 2.312 2.26 2.21 2.38 4 D2h 1-2 2.477 2.475 2.40 2.35 2.44 1-3 2.622 2.619 2.53 2.44 2.57 5 D3h 1-2 3.277 3.135 3.19 3.10 2.87 1-3 2.456 2.476 2.39 2.34 2.44 3-4 2.456 3.320 3.19 3.10 3.40 6 Oh 1-2 – 2.553 2.47 2.40 2.47 2-3 – 2.941 2.85 2.78 2.70 7 D5h 1-2 – – 2.65 2.56 2.83 1-3 – – 2.57 2.49 2.57 3-4 – – 2.59 2.51 2.53 8The minimum energy structure found for Ge 3is an isosceles triangle (C 2v) with bond length 2.38 ˚A and apex angle θ= 69.5◦, in agreement with ab initio calculations (see Table III for comparison). The linear chain has higher total energ y of about 0.95 eV. The ground state structure of germanium tetramer is a planar rhombus (D 2h) with side length 2.44 ˚A and minor diagonal length 2.57 ˚A. This structure has been predicted as ground state in all ab initio calculations21,24–28and the tight-binding bond length are close toab initio results. For the Ge 5, the lowest energy configuration is obtained as a strongly co mpressed trigonal bipyramid (D 3h). The energy of structure is lower than the perfect trigonal bipyramid by 0.62 eV and the planar edge capped rhombus by 0.35 eV. The trig onal bipyramid structure has been considered in all of the previous ab initio studies21,24–28. In those LDA based simulation without symmetry constraint25,26,28, the trigonal bipyramid is found to undergo severe compression and relax to the structure in Fig.2. A distorted octahedron (D 4h) is obtained for Ge 6as lowest energy structure. This structure is found to be energetically degenerated with a ed ge-capped trigonal bipyramid (∆E= 0.018 eV). This result agree well with recent B3LYP and Car-Par rinello calculation of Ge 627,28. In the case of Ge 7, we find a compressed pentagonal bipyramid with D 5hsymmetry as ground state and energetically lower than the face capped octahedron by 0.63 eV. The pentagonal bipyramid structure has also been obtained from LDA based simulations25,26,28. Table IV. Binding energy per atom Eb/n(eV) of Ge nclusters obtained within the present NTB model, compared to experimental values19,30,ab initio results based on G2(MP2) level21 or LDA plane-wave pseudopential26,28, as well as nonorthogonal tight-binding13calculations. n Exper.19,30G2(MP2)21LDA26LDA27NTB13NTB(present) 2 1.32 1.25 1.89 – 1.31 1.32 3 2.24 2.02 2.78 2.66 2.11 2.06 94 2.60 2.49 3.32 3.19 2.66 2.62 5 2.79 2.68 3.58 3.45 2.85 2.73 6 2.98 – 3.76 3.63 3.05 2.95 7 3.03 – 3.90 3.77 3.19 3.09 8 3.04 – 3.82 3.69 3.17 3.05 9 3.04 – 3.93 3.79 3.25 3.12 10 3.13 – 4.04 3.91 3.32 3.17 10An additional atom capped to pentagonal bipyramid of Ge 7yields the lowest energy structure for Ge 8. This structure is more stable over the bicapped octahedron by 0.08 eV. Both of these two structures are found for Ge 8in Car-Parrinello simulation, while bicapped octahedron is lower in energy by 0.03 eV28. A bicapped pentagonal bipyramid is found for Ge 9. It is more stable than a capped distorted square antiprism by 0.06 eV. The current ground st ate structure has been found in Car-Parrinello simulation for Ge 928but it is 0.08 eV higher than the capped square antiprism structure. For Ge 10, the tetracapped trigonal prism (C 3v) is found to be most stable and 0.16 eV lower than the bicapped square antiprism (D 4d). This ground state structure is consistent with previous LDA results26,28. In Table IV, we compare the binding energy per atom Eb/nfor Ge n(n= 2−10) with the other theoretical and available experimental results. Due to the local density approximation, LDA calculation26,28has systematically overestimated the cluster binding ener gies. The more accurate binding energies for small germanium clusters up t o five atoms has been provided by a more sophisticated G2(MP2) computation21. Although all the empirical parameters in our NTB model are fitted from dimer and bulk solid and there is n o bond counting correction included, the experimental cohesive energies are fairly we ll reproduced by our calculation. Typical discrepancy between our calculation and experimen t is less than 0.1 eV for those clusters. The successful description of binding energy wit hin the present size range further demonstrates the transferability of the nonorthogonal tig ht-binding approach. In Table IV, we have also included the binding energies from Menon’s NTB m odel for Ge nclusters13. Although the geometries of Ge nfound in their work is almost the same as our results, the binding energies of Ge nstarting from Ge 5in Ref.13 are about 0.10 ∼0.2 eV higher than our results and experimental values. In summary, a nonorthogonal tight-binding model for german ium has been developed in this work. The transferability of the model is tested by va rious of bulk phases. The agreements between NTB model and ab initio results for bulk solids and small clusters are 11satisfactory. For most Ge ncluster with n= 3 to 10, the ground state geometries from tight- binding model coincide with those from ab initio calculation. The only exceptional cases are Ge8and Ge 9, in which the ab initio metastable isomers are predicted as ground states by NTB scheme. However, the energy difference between the groun d state configuration and the isomer is less than 0.01 eV/atom and within the accuracy o f tight-binding approach. Therefore, the NTB model developed in this work can be applie d to explore the configuration space of larger germanium clusters with n >10, for which a global minimization at the ab initio level is significantly more expensive. Our further studies s hall include a genetic algorithm for sampling the phase space and finding possible l ow energy structural isomers of germanium clusters. Thus, first principle structural optim ization can be performed on these local minima structures. On the other hand, this model will b e also employed to simulate the thermodynamic properties such as melting and growth pro cess of germanium cluster, which require a long time scale in TBMD simulation. This work is partially supported by the U.S. Army Research Offi ce (Grant DAAG55- 98-1-0298) and the National Natural Science Foundation of C hina. The author (J.Zhao) are deeply grateful to Prof.E.Tosatti, Dr.J.Kohanoff, Dr.A .Buldum, and Prof.J.P.Lu for discussions. ∗Corresponding author: zhaoj@physics.unc.edu 12REFERENCES 1C.Z.Wang, K.M.Ho, in Advances in Chemical Physics , Vol.XCIII, p.651, Edited by I.Prigogine, S.A.Rice, (John Wiley & Sones, Inc., New York, 1996). 2C.M.Goringe, D.R.Bowler, E.Herhandez, Rep.Prog.Phys. 60, 1447(1997). 3Computational Material Science, Vol 12, No.3 (1998): speci al issue on tight-binding molec- ular dynamics, Edited by L. 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arXiv:physics/0004067v1 [physics.gen-ph] 26 Apr 2000Imprints of Discrete Space Time - A Brief Note B.G. Sidharth Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Hyderabad 500 063 Abstract We point out that the observed decay mode of the pion and the Kaon decay puzzle are really imprints of discrete micro spac e-time. In recent years ideas of discrete space time have been revive d through the work of several scholars and by the author within the context of Kerr- Newman Black Hole type formulation of the electron[1]-[5]. Further, even more recently this has been considered in the context of a sto chastic underpinning[6, 7]. Let us now consider two of the imprints that such discrete space time would have. First we consider the case of the neutral pion. Within the fra mework of the Kerr-Newman metric type formulation referred to above, it i s possible to recover the usual picture of a pion as a quark-anti quark boun d state [8, 9], though equally well we could think of it as an electron-posit ron bound state also[4, 10]. In this case we have, mv2 r=e2 r2(1) Consistently with the above formulation, if we take v=cfrom (1) we get the correct Compton wavelength lπ=rof the pion. However this appears to go against the fact that there would b e pair annihi- lation with the release of two photons. However if we conside r discrete space time, the situation would be different. In this case the Schro dinger equation Hψ=Eψ (2) 1whereHcontains the above Coulumb interaction could be written, in terms of the space and time separated wave function components as ( Cf. also ref.[2]), Hψ=EφT=φı¯h[T(t−τ)−T τ] (3) whereτis the minimum time cut off which in the above work has been take n to be the Compton time (Cf.refs.[4] and [5]). If, as usual we l etT=exp(irt) we get E=−2¯h τsinτr 2(4) (4) shows that if, |E|<2¯h τ(5) holds then there are stable bound states. Indeed inequality (5) holds good whenτis the Compton time and Eis the total energy mc2. Even if inequality (5) is reversed, there are decaying states which are relativ ely stable around the cut off energy2¯h τ. This is the explanation for treating the pion as a bound state of an electron and a positron, as indeed is borne out by its decay mode. The si tuation is similar to the case of Bohr orbits– there also the electrons w ould according to classical ideas have collapsed into the nucleus and the at oms would have disappeared. In this case it is the discrete nature of space t ime which enables the pion to be a bound state as described by (1). Another imprint of discrete space time can be found in the Kao n decay puzzle, as pointed out by the author[11]. There also we have e quations like (2) and (3) above, with the energy term being given by E(1 +i), due to the fact that space time is quantized. Not only is the fact that th e imaginary and real parts of the energy are of the same order is borne out but a s pointed out in[11] this also explains the recently observed[12] decay a nd violation of the time reversal symmetry which in the words of Penrose[13], ”t he tiny fact of an almost completely hidden time-asymmetry seems genuinel y to be present in theK0-decay. It is hard to believe that nature is not, so to speak, t rying to tell something through the results of this delicate and beau tiful experiment.” From an intuitive point of view, the above should not be surpr ising because time reversal symmetry is based on a space time continuum and is no longer obvious if space time were discrete. 2References [1] Bombelli, L., Lee, J., Meyer, D., Sorkin, R.D., Phys. Rev . Lett. 59, 1987, 521. [2] Wolf C., Nuovo. Cim. B 109 (3), 1994, 213. [3] Lee, T.D., Phys. Lett. 12 (2B), 1983, 217. [4] Sidharth, B.G., Ind. J. Pure and Applied Phys., 35 (7), 19 97, 456. [5] Sidharth, B.G., IJMPA, 13 (15), 1998, 2599. Also xxx.lan l.gov quant-ph 9808031. [6] Sidharth, B.G., Chaos, Solitons and Fractals, 11 (8), 20 00, 1269-1278. [7] Sidharth, B.G., Chaos, Solitons and Fractals, 11 (8), 20 00, 1171-1174. [8] Sidharth, B.G., and Lobanov, Yu Yu, Proceedings of Front iers of Fun- damental Physics, (Eds.)B.G. Sidharth and A. Burinskii, Un iversities Press, Hyderabad, 1999. [9] B.G. Sidharth, Mod.Phys. Lett. A., Vol. 12 No.32, 1997, p p2469-2471. [10] B.G. Sidharth, Mod.Phys. Lett. A., Vol. 14 No. 5, 1999, p p387-389. [11] Sidharth, B.G., Chaos, Solitons and Fractals, 11, 2000 , 1045-1046. [12] Angelopoulos, A., et al., Phys. Lett. B 444, 1998, 43. [13] Hawking, S., Israel, W., General Relativity: An Einste in Centenary Survey, Cambridge University Press, Cambridge, 1979. 3

arXiv:physics/0004068v1 [physics.atom-ph] 26 Apr 2000PUZZLE OF THE CONSTANCY OF FUNDAMENTAL CONSTANTS D. A. Varshalovich, A. Y. Potekhin, and A. V. Ivanchik Ioffe Physical-Technical Institute, 194021 St.Petersburg , Russia Abstract We discuss experiments and observations aimed at testing th e possible space-time variability of fundamental physical constants, predicted by the modern th eory. Specifically, we consider two of the dimensionless physical parameters which are important for atomic and molecular physics: the fine-structure constant and the electron-to-proton mass ra tio. We review the current status of such experiments and critically analyze recent claims of a detec tion of the variability of the fine-structure constant on the cosmological time scale. We stress that such a detection remains to be checked by future experiments and observations. The tightest of the fir mly established upper limits read that the considered constants could not vary by more than 0.015% o n the scale ∼1010years. Key words : fundamental constants; cosmology PACS numbers : 06.20.Jr, 06.30.-k, 98.80.Es 1 Introduction Possible variability of fundamental physical constants is one of the topical problems of contemporary physics. The modern theory (Supersymmetric Grand Unificati on Theory – SUSY GUT, Superstring models, etc.) has established that the coupling constant va lues which characterize different kinds of interactions (i) are “running” with the energy transfer and (ii) may be different in different regions of the Universe and vary in the course of cosmological evolution (e .g., Ref. [1]). The energy dependence of the coupling parameters has been reliably confirmed by high-ene rgy experiments (see, e.g., Ref. [2]), whereas the space-time variability of their low-energy limits so fa r escapes detection. Note that a numerical value of any dimensional physical para meter depends on arbitrary choice of physical units. In turn, there is no way to determine the unit s in a remote space-time region other than through the fundamental constants. Therefore it is meaning less to speak of a variation of a dimensional physical constant without specifying which of the other phy sical parameters are defined to be invariable. Usually, while speaking of variability of a dimensional phy sical parameter, one implies thatallthe other fundamental constants are fixed. So did Milne [3] and Dirac [4 ] in their pioneering papers devoted to a possible change of the gravitational constant G. More recently, a number of authors considered cosmological theories with a time varying speed of light c(e.g., Ref. [5]). However, if we adopt the standard definition of meter [6] as the length of path travele d by light in vacuum in 1/299792458 s, thenc= 2.997 924 58 ×1010cms−1identically. Similarly, one cannot speak of variability of the electron massmeor charge ewhile using the Hartree units ( /planckover2pi1=e=me= 1), most natural in atomic physics. Thus, only dimensionless combinations of the physical parameters are truly fundamen tal, and only such combinations will be considered hereafter. At present, the most promising candidate for the theory whic h is able to unify gravity with all other interactions is the Superstring theory, which treats gravi ty in a way consistent with quantum mechanics. All versions of the theory predict existence of the dilaton – a scalar partner to the tensorial graviton. Since the dilaton field φis generally not constant, the coupling constants and masse s of elementary particles, being dependent on φ, should vary in space and time. Thus, the existence of a weakl y coupled massless 1Comments on Atomic and Molecular Physics, 2000 2 dilaton entails small, but non-zero, observable consequen ces such as Jordan–Brans–Dicke-type deviations from General Relativity and cosmological variations of the gauge coupling constants [7]. These variations depend on cosmological evolution of the dilaton field and may be non-monotonous as well as different in different space-time regions. In this paper, we focus on the space-time variability of the l ow-energy limits of two fundamental constants which are of paramount importance for atomic and m olecular spectroscopy: (i) the fine-structure constant α=e2//planckover2pi1c(Sommerfeld parameter), (ii) the electron-to-proton mass ratio µ=me/mp(Born–Oppenheimer parameter). The next section presents a compendium of the basic methods a llowing one to obtain restrictions on possible variations of fundamental constants. In Sects. 3 a nd 4, we consider recent estimates of the values ofαandµ, respectively, at cosmological redshifts z= 1–4 which correspond to epochs ∼7–13 billion years ago. Conclusions are given in Sect. 5. 2 Tests of variability of fundamental constants Techniques used to investigate time variation of the fundam ental constants may be divided into extra- galactic and local methods. The latter ones include astrono mical methods related to the Galaxy and the Solar system, geophysical methods, and laboratory measure ments. 2.1 Local tests 2.1.1 Laboratory measurements Laboratory tests are based on comparison of different freque ncy standards, depending on different com- binations of the fundamental constants. Were these combina tions changing differently, the frequency standards would eventually discord with each other. An inte rest in this possibility has been repeatedly excited since relative frequency drift was observed by seve ral research groups using long term compar- isons of different frequency standards. For instance, a comp arison of frequencies of He-Ne/CH 4lasers, H masers, and Hg+clocks with a Cs standard [8, 9, 10, 11] has revealed relative drifts. Since the con- sidered frequency standards have a different dependence on αvia relativistic contributions of order α2, the observed drift might be attributed to changing of the fine -structure constant. However, the more modern was the experiment, the smaller was the drift. Taking into account that the drift may be also related to some aging processes in experimental equipment, Prestage et al. [11] concluded that the current laboratory data provide only an upper limit |˙α/α| ≤3.7×10−14yr−1. 2.1.2 Analysis of the Oklo phenomenon The most stringent limits to variation of the fine-structure constant αand the coupling constant of the strong interaction αshave been originally inferred by Shlyakhter [12] from resul ts of an analysis of the isotope ratio149Sm/147Sm in the ore body of the Oklo site in Gabon, West Africa. This r atio turned out to be considerably lower than the standard one, which is beli eved to have occurred due to operation of the natural uranium fission reactor about 2 ×109yr ago in those ores. One of the nuclear reactions in the fission chain was the resonance capture of neutrons by149Sm nuclei. Actually, the rate of the neutron capture reaction is sensitive to the energy of the relevant n uclear resonance level Er, which depends on the strong and electromagnetic interaction. Since the ca pture has been efficient 2 ×109yr ago, in means that the position of the resonance has not shifted by mo re than it width (very narrow) during the elapsed time. At variable αand invariable αs(which is just a model assumption), the shift of the resonance level would be determined by changing the differen ce between the Coulomb energies of the ground-state nucleus149Sm and the nucleus150Sm∗excited to the level Er. Unfortunately, there is no experimental data for the Coulomb energy of the excited150Sm∗in question. Using order-of-magnitude estimates, Shlyakhter [12] concluded that |˙α/α|/lessorsimilar10−17yr−1. From an opposite model assumption that αsis changing whereas α=constant, he derived a bound |˙αs/αs|/lessorsimilar10−19yr−1.Comments on Atomic and Molecular Physics, 2000 3 Damour and Dyson [13] performed a more careful analysis, whi ch resulted in the upper bound |˙α/α|/lessorsimilar 7×10−17yr−1. They have assumed that the Coulomb energy difference betwee n the nuclear states of 149Sm and150Sm∗in question is not less than that between the ground states of149Sm and150Sm. The latter energy difference has been estimated from isotope shi fts and equals ≈1 MeV. However, it looks unnatural that a weakly bound neutron ( ≈0.1 eV), captured by a149Sm nucleus to form the highly excited state150Sm∗, can so strongly affect the Coulomb energy. Moreover, heavy e xcited nuclei often have Coulomb energies smaller than those for their ground st ates (e.g., Ref. [14]). This indicates the possibility of violation of the basic assumption involved i n Ref. [13], and therefore this method may possess a lower actual sensitivity. Furthermore, a correla tion between αandαs(which is likely in the frame of modern theory) might lead to considerable softenin g of the above-mentioned bound, as estimated by Sisterna and Vucetich [15]. 2.1.3 Some other local tests Geophysical, geochemical, and paleontological data impos e constraints on a possible changing of various combinations of fundamental constants over the past histor y of the Solar system, however most of these constraints are very indirect. A number of other methods are based on stellar and planetary models. The radii of the planets and stars and the reaction rates in them a re influenced by values of the fundamental constants, which offers a possibility to check variability o f the constants by studying, for example, lunar and Earth’s secular accelerations. This was done using sate llite data, tidal records, and ancient eclipses. Another possibility is offered by analyzing the data on binar y pulsars and the luminosity of faint stars. Most of these have relatively low sensitivity. Their common weak point is the dependence on a model of a fairly complex phenomenon, involving many physical effect s. An analysis of natural long-lived α- and β-decayers in geological minerals and meteorites is much more sensitive. For instance, a strong bound, |˙α/α|<5×10−15yr−1, was obtained by Dyson [16] from an isotopic analysis of natural α- and β-decay products in Earth’s ores and meteorites. Having critically reviewed the wealth of the local tests, ta king into account possible correlated syn- chronous changes of different physical constants, Sisterna and Vucetich [15] derived restrictions on possible variation rates of individual physical constants for ages tless than a few billion years ago, which correspond to cosmological redshifts z/lessorsimilar0.2. In particular, they have arrived at the estimate ˙α/α= (−1.3±6.5)×10−16yr−1. All the local methods listed above give estimates for only a n arrow space-time region around the Solar system. For example, the epoch of the Oklo reactor (1 .8×109years ago) corresponds to the cosmological redshift z≈0.1. These tests cannot be extended to earlier evolutionary st ages of the Universe, because the possible variation of the fundamental constants is, in g eneral, unknown and may be oscillating [17, 7]. Another investigation is needed for higher cosmological re dshifts. 2.2 Extragalactic tests Extragalactic tests, in contrast to the local ones, concern values of the fundamental constants in distant areas of the early Universe. A test which relates to the earli est epoch is based on the standard model of the primordial nucleosynthesis. The amount of4He produced in the Big Bang is mainly determined by the neutron-to-proton number ratio at the freezing-out of n ↔p reactions. The freezing-out temperature Tfis determined by the competition between the expansion rate of the Universe and the β-decay rate. A comparison of the observed primordial helium mass fractio n,Yp= 0.24±0.01, with a theoretical value allows one to obtain restrictions on the difference bet ween the neutron and proton masses at the epoch of the nucleosynthesis and, through it, to estimate re lative variation of the curvature radius R of extra dimensions in multidimensional Kaluza–Klein-lik e theories which in turn is related to the α value [18, 19]. However, as noted above, different coupling c onstants might change simultaneously. For example, increasing the constant of the weak interactions GFwould cause a weak freezing-out at a lower temperature, hence a decrease in the primordial4He abundance. This process would compete with the one described above, therefore, it reduces sensitivity of t he estimates. Finally, the restrictions wouldComments on Atomic and Molecular Physics, 2000 4 be different for different cosmological models since the expa nsion rate of the Universe depends on the cosmological constant Λ. The most unambiguous estimation of the atomic and molecular constants at early epochs and in distant regions of the Universe can be performed using the extragala ctic spectroscopy. Accurate measurements of the wavelengths in spectra of distant objects provide qua ntitative constraints on the variation rates of the physical constants. This opportunity has been first note d and used by Savedoff [20], and in recent years exploited by many researchers (see, e.g., Refs. [21, 2 2] and references therein). At present, the extragalactic spectroscopy enables one to probe the physic al conditions in the Universe up to cosmological redshifts z/lessorsimilar4, which correspond, by order of magnitude, to the scales ∼1010yr in time and ∼109 parsec in space. In the following sections, we review briefly the studies of the space-time variability of the fine-structure constant αand the electron-to-proton mass ratio µ, based on the latter method. 3 Non-variability of α We have already mentioned in Sect. 2.1.1 that several labora tory tests hinted at a tentative time variation ofα, but were later refuted by measurements at a higher level of a ccuracy. A similar situation has occurred for extragalactic tests at larger space-time scales. Bahcall and Schmidt [23] were the first to use spectral observ ations of distant quasars to set a bound on the variability of the fine-structure constant. They have obtained an estimate ∆ α/α= (−2±5)×10−2 atz= 1.95. Later statistical analyses [24, 25] of fine-structure do ublet lines in quasar spectra appeared to indicate a tentative variation of α(of the order of ∼0.3% at the cosmological redshift z∼2). However, this tentative variation has been shown to result from a stat istical bias [26]. Another statistical examination of the fine-doublet wavele ngths of absorption lines in quasar spectra [21] indicated a tentative (at the 2–3 σlevel) variability of αvalues by ∼0.1% over the celestial sphere (as function of angle) at redshifts z∼2–3. However, this result has not been confirmed by a later ana lysis [22], which was based on higher-quality spectra and yielded an order of magnitude higher precision. Quite recently, Webb et al. [27] have estimated αby comparing wavelengths of Fe iiand Mg iifine- splitted spectral lines in extragalactic spectra and in the laboratory. Their result suggests a time-variation ofαat the incredibly high accuracy level of ∼10−3%: the authors’ estimate reads ∆ α/α= (−1.9± 0.5)×10−5atz= 1.0–1.6. Note, however, two important sources of a possible sy stematic error which could mimic the effect: (a) Fe iiand Mg iilines used are situated in different orders of the echelle-sp ectra, so relative shifts in calibration of the different orders can affect the result of comparison, and (b) if the isotopic composition varies during the evolution of the Uni verse, then the average doublet separations should vary due to the isotopic shifts. Were the relative abu ndances of Mg isotopes changing during the cosmological evolution, the Mg iilines would be subject to an additional z-dependent shift relative to the Feiilines, quite sufficient to simulate the variation of α(this shift can be easily estimated from recent laboratory measurements [28]). The method based on the fine splitting of a line of the same ion s pecies is not affected by these two uncertainty sources. We have studied the fine splitting of th e doublet lines of Si iv, Civ, Mgiiand other ions, observed in spectra of distant quasars. According to q uantum electrodynamics, the relative splitting of these lines δλ/λ is proportional to α2(neglecting small relativistic corrections, recently est imated by Dzuba et al. [29]). We have selected the results of high-reso lution observations [30, 31, 32], most suitable for an analysis of the variation of α. According to our analysis, presented elsewhere [33], the m ost reliable estimate of the possible deviation of the fine-structure con stant at z= 2–4 from its present ( z= 0) value: ∆α/α= (−4.6±4.3 [stat] ±1.4 [syst]) ×10−5. (1) Thus, only an upper bound can be derived at present for the lon g-term variability of α: |˙α/α|<1.4×10−14yr−1(2) (at the 95% confidence level).Comments on Atomic and Molecular Physics, 2000 5 4 Non-variability of µ The dimensionless Born–Oppenheimer constant µ=me/mpapproximately equals the ratio of the con- stant of electromagnetic interaction α=e2//planckover2pi1c≈1/137 to the constant of strong interaction αs= g2//planckover2pi1c∼14, where gis the effective coupling constant calculated from the ampli tude of π-meson–nucleon scattering at low energy. An early limit on the possible variation of this constant, |˙µ/µ|<1.2×10−10yr−1, has been derived from the concordance of K–Ar and Rb–Sr geochemical ages [34] . The first astrophysical bound [35], based on the agreement between redshifts of atomic hydrogen and ot her lines in quasar absorption spectra, turned out to be twice stronger. Orders-of-magnitude more precise analysis has become poss ible due to discovery [36] of a system of H2absorption lines in the spectrum of quasar PKS 0528 −250 at z= 2.811. A study of this system yields information about physical conditions and, in particular, the value of µat this redshift (corresponding to the epoch when the Universe was several times younger than now). A possibility of distinguishing between the cosmological redshift of spectral wavelengths and shifts due to a variation of µarises from the fact that the electronic, vibrational, and rotational ener gies of H 2each undergo a different dependence on the reduced mass of the molecule. Hence comparing ratios o f wavelengths λiof various H 2electron- vibration-rotational lines in a quasar spectrum at some red shiftzand in laboratory (at z= 0), we can trace variation of µ. We have calculated [21, 37] sensitivity coefficients Kiof the wavelengths λiwith respect to possible variation of µand applied a linear regression analysis to the measured red shifts of individual lines zias function of Ki. If the proton mass in the epoch of line formation were differe nt from the present value, the measured ziandKivalues would correlate: zi zk=(λi/λk)z (λi/λk)0≃1 + (Ki−Kk)/parenleftbigg∆µ µ/parenrightbigg . (3) We have performed a z-to-Kregression analysis using a modern high-resolution spectr um of PKS 0528−250. Eighty-two of the H 2lines have been identified. The resulting parameter estimat e and 1 σ uncertainty is ∆µ/µ= (−11.5±7.6 [stat] ±1.9 [syst]) ×10−5. (4) The 2 σconfidence bound on ∆ µ/µreads |∆µ/µ|<2.0×10−4. (5) Assuming that the age of the Universe is ∼1.5×1010yr the redshift of the H 2absorption system z= 2.81080 corresponds to the elapsed time ≈1.3×1010yr (in the standard cosmological model). Therefore we arrive at the restriction |˙µ/µ|<1.5×10−14yr−1(6) on the variation rate of µ, averaged over 90% of the lifetime of the Universe. 5 Conclusions Despite the theoretical prediction that fundamental const ants of Nature should vary, no statistically significant variation of any of the constants has been reliab ly detected up to date, according to our point of view substantiated above. The upper limits obtaine d indicate that the constants of electroweak and strong interactions did not significantly change over th e last 90% of the history of the Universe. The striking tightness of these limits is really astonishin g and has already ruled out some theoretical models (see Refs. [21, 22, 35]). A more elaborated theory (e. g., Ref. [7]) cannot be ruled out yet, but its parameters can be severely restricted (e.g., see Ref. [22]) . 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arXiv:physics/0004069v1 [physics.atom-ph] 26 Apr 2000The propensity of molecules to spatially align in intense li ght fields S. Banerjee, D. Mathur, and G. Ravindra Kumar Tata Institute of Fundamental Research, Homi Bhabha Road, M umbai 400 005, India (December 21, 2012) Abstract The propensity of molecules to spatially align along the pol arization vec- tor of intense, pulsed light fields is related to readily-acc essible parameters (molecular polarizabilities, moment of inertia, peak inte nsity of the light and its pulse duration). Predictions can now be made of which mol ecules can be spatially aligned, and under what circumstances, upon ir radiation by in- tense light. Accounting for both enhanced ionization and hy perpolarizability, it is shown that allmolecules can be aligned, even those with the smallest static polarizability, when subjected to the shortest avai lable laser pulses (of sufficient intensity). Typeset using REVT EX 1Studies of the response of matter to very intense fluxes of ele ctromagnetic radiation ad- dress fundamental physics issues of systems driven strongl y away from equilibrium. Matter is inherently unstable when subjected to strong electric fie lds of the type that can be gen- erated in intense laser light. In the case of molecules subje cted to laser light of intensity in excess of ∼1012W cm−2, distortions of potential energy surfaces, with concomita nt al- terations in electron density distributions, lead to ioniz ation, dissociation, and formation of strong dipole moments ( µ). With linearly polarized Efields of magnitudes that equal, or exceed, interatomic binding fields, the induced dipole mome nts exert torques on the molec- ular axes, µ×E, that can be large enough to spatially reorientate molecule s and their ions such that the most polarizable molecular axis points along t he light field vector. In early experiments, anisotropic angular distribution of fragmen t ions were obtained when the light polarization vector was rotated relative to the detector ax is, and these were taken to be unambiguous signatures of spatial orientation [1,2]. In re cent studies, it has been recognized that the molecular ionization rate depends on the angle that the internuclear axis makes with the light field vector, and that this also leads to anisot ropic angular distributions [3]. Indeed, the ionization rate has to be computed using the field ionization Coulomb explosion model [4], whose angular dependence arises from the fact tha t the barrier suppression is given by E·r, where rdenotes the molecular axis. Spatial alignment of isolated molecules is a subset of one of the central endeavors of physicists and chemists, namely to control the external deg rees of freedom of atomic and molecular species at the microscopic level. The polarizabi lity interaction of an intense, linearly-polarized light field with the induced dipole mome nt of molecules gives rise to a double-well potential; the resulting angular reorientati on of molecular axes is akin to the interconversion of left- and right-handed enantiomers tha t was considered by Hund over 3 decades ago in terms of similar potentials [5]. Spatial alig nment of individual molecules can also justifiably be considered a special facet of the opti cal Kerr effect [6]. On a more practical level, interest in studies of spatial alignment o f molecules has been generated be- cause of tantalizing possibilities of entirely new studies on pendular-state spectroscopy [7], coherent control experiments [8], and molecular trapping a nd focusing [9]. The crucial role of polarization in choosing or altering dissociation pathw ays has also been experimentally established [10]. It is clearly very important, therefore, to establish, both on the basis of the properties of the molecule under investigation and the char acteristics of the laser light that is used, the extent of spatial alignment that occurs. Specifi c insight is also needed on the relative importance of angle dependent ionization on the on e hand and molecular reorienta- tion on the other in making sense of measured anisotropies in fragment ion distributions. In this Letter we present results of a comprehensive study that enables predictions to be made of the propensity of molecules to spatially align in intense , pulsed, polarized light on the basis of parameters that are readily accessible. We show tha t existing experimental data, spanning work done over the past decade by several groups inc luding our own is explained by our model. We also show that it is possible to align molecul es even with extremely short light pulses irrespective of the polarizability. In any given analysis of spatial alignment three factors pla y a crucial role: (i) the peak intensity of the laser pulse, (ii) its temporal duration, an d (iii) the ratio of the molecular polarizability (ground or excited state) to the moment of in ertia ( R=α I). At high enough intensities, molecular hyperpolarizabilites may also be s ignificant although their role in the 2alignment dynamics has not hitherto been explicitly consid ered. It is also well established that field ionization of molecules is ubiquitous with short p ulse lasers. The important con- clusion of the field ionization model relevant to alignment i s the breakup of the molecule at a critical distance R cthat is larger than the equilibrium internuclear separatio n Re[12]. The stretching of the internuclear axis increases the moment of inertia and leads to a slowing down of reorientation; dissociation at R cimplies that the molecule will dissociate before the peak intensity is reached, except for ultrashort light puls es. The process of alignment is modeled by considering a rotor in a time dependent E field [13]. In this case the interaction Hamiltonian is given by H I= -µ.Ewhere µ= µ0+1 2αE+1 24γEE. The Lagrangian is: L=I 2/bracketleftbigg ˙θ2+sin2θ˙φ2/bracketrightbigg +µ.E. (1) The equation of motion is then given by d2θ dt2=−αeff 2IE(t)2sin2θ−2˙ r r˙θ, (2) where αeffis the effective polarizability. In the low-field limit this i s the linear polarizability. The modification of this in high fields is discussed later. To k eep our calculations as realistic as possible we have used a Gaussian laser pulse, E(t) =E0e−t2 2τ2cosωt. Spatial variations within the laser beam are not taken into account since it has b een shown recently that intensity selective experiments minimize focal volume effe cts [11]. In Eq. (1), the first term causes reorientation while the second term, the so-called d amping term, impedes the motion of the molecular axis towards the light field vector. In the fie ld ionization Coulomb explosion model [4], the damping term arises due to the elongation of th e molecular axis from Reto Rcafter the removal of one or two electrons by tunnel/over the b arrier ionization. We present our results as follows. We first consider the simpl est case where the polar- izability is linear and no damping term is present. It is show n that even such a picture is of wide-ranging validity and utility. We then discuss resul ts of calculations that take into account both damping and the hyperpolarizability, and we ex amine the extent to which our results are modified. Our calculations are compared with exp erimental observations. Eq. (2) is solved used a 4th-order Runge-Kutta algorithm for a range of light field parameters. We consider the laser intensity range 1012- 1015W cm−2, and pulse durations from 40 fs to 2 ps. In the linear case, molecule-specific calcu lations have not been done. Instead the A-parameter has been taken to lie in the interval 2 ×104- 6×107(covering a wide range of light and heavy species, amongst them H 2, N2, CS 2and I 2). The initial direction of the molecular axis, θ0, is taken to be random in space. After the light pulse is switc hed on, the angular position is calculated as a function of time f or various values of θ0as shown in figure 1(a). From this one obtains a plot of θfvs.θiwhere θiis the initial angular position and θfis the angular orientation of the molecular axis at a particu lar instant. This is shown in figure 1(b). It is easy to show that the angular dist ribution of the molecular axis is proportional to (dθf dθi)−1. This procedure can no longer be used when the applied field is so strong that the molecular axis crosses θ=0. In such cases we have used a counting method by interpolating the relation between θfandθito obtain the angular distribution. 3The trajectories shown in figure 1 (a) correspond to H 2exposed to 20 fs pulses at 1015 W cm−2. As can be seen from the slope in figure 1(b) the extent of reori entation in H 2 is negligible under these conditions. Similar calculation s were carried out for a range of parameters as specified above, and results are shown in figure 2. The surface demarcates the regions where spatial alignment is significant from thos e where no significant reorien- tation occurs. All points above the surface, i.e. in regions of larger intensity, polarizab ility and pulse duration correspond to molecules that are signific antly aligned, while the oppo- site holds for points that lie below the surface. Note that we are dealing here only with linear polarizabilities. The nonlinear polarizability co mponents serve only to strengthen the alignment. Thus, the demarcation based on αalone is very rigorous. It is to be noted that the above calculations pertain to the po sition of the molecular axis. However alignment is deduced from the anisotropy of fragmen t ions. To make the connection with experimental data it should be noted that the angular di stributions that are shown are those that would practically be measured using a spectromet er with a small acceptance angle. Thus, our calculations are specially relevant to the angular distributions of highly charged ions (that possess large kinetic energies). This is also important in the context of the residual angular momentum as the molecule rotates. This is sufficiently large to cause significant rotation on the time scale of the laser pulse but n egligible compared to the energy of the fragment ions which are typically in excess of 1 eV. Ill ustrative experimental data for various molecules are also shown in the figure and we note the e xcellent agreement that is obtained between our model and measured data. We now consider the damping term. Ignoring this term essenti ally assumes that the molecule is a rigid rotor in the intense field. However, as alr eady noted, when the field is sufficiently large, the ionization dynamics occur through an enhanced ionization (EI) mechanism wherein one or two electrons are removed at the firs t ionization step. Subsequent to the first ionization step that occurs at the equilibrium in ternuclear separation, the two residual atomic ions mutually repel each other, leading to a n increase in the bond length. This results in one or more Stark-shifted electronic levels rising above the potential barrier that separates the atomic cores, at which point multiple ele ctron ejection occurs, leading to molecular fragmentation. EI can modify the reorientatio n rate in two ways. Firstly, as the moment of inertia increases, the magnitude of the first te rm in Eq. (1) will reduce. In addition, the damping term will come into play, leading to a f urther decrease in the rate at which the molecule rotates towards the light field vector. It is important to investigate the extent EI might modify our first-order calculations. Since E I parameters are available only for a few molecules, we have carried out these calculations f or some standard cases. These can be extended to any other molecule once the relevant param eters are known either by calculation or experiment. Fig. 3 (a) shows the angular distribution for H 2for a pulse duration of 40 fs at a peak laser intensity of 1015W cm−2, with and without the damping term. It is clear that the reorientation of H 2is not significantly affected when the damping term is include d. There are two major reasons for this. Firstly, Ris extremely large and the torque experienced by H2is sufficient to induce reorientation despite the presence of an opposing force. Secondly, the fact that the ionization energy (and hence, the appearan ce intensity) of H 2is quite high, the damping force only comes into play close to the peak of the laser pulse, by which time the molecular axis is already aligned with the light polariz ation vector. Interestingly, the 4width of the angular distribution with damping included is a ctually smaller than when no damping is present. This arises due to the fact that the angul ar velocity without damping is larger, causing the molecular axis to execute large amplitu de oscillations about θ=0; hence, there will exist instants at which the peak of the angular dis tribution will shift away from zero. A contrary situation is depicted in Fig. 3(b) when linear CS 2molecules are exposed to 100 fs light fields. In this case, the lower ionization energy of the molecule coupled with the relatively small value of A, leads to a situation where there is virtually no reorientat ion of the S-C-S axes with the direction of the Efield at the point at which dissociation occurs. Strong alignment can be expected if it is assumed that the mol ecule survives undissociated till the peak of the laser pulse, a fact contrary to experimen tal observation, and illustrates the essentiality of EI in any such model. Similar calculatio ns have been carried out for other molecules like N 2and I 2. The situation for N 2is similar to H 2because of the similarity in the relevant parameters. In the case of I 2, reorientation of the molecule is not significant even without the damping term. Once the damping term is inclu ded there is only a small deflection of the molecular axis. Hitherto, only the polarization response that arises from t he linear term has been con- sidered. To what extent is this justified, especially at inte nsities in the range of 1012- 1015 W cm−2? It is important to note that hyperpolarizabilites are sign ificant only at the highest intensities. This is obvious when we compare the integrals w hich define the work done by the field on the molecule by each order of the hyperpolarizabi lity. It can be expected that for longer pulses, a model based on linear polarizability is sufficient since the dissociation of the molecule occurs on the rising edge of the pulse. However, as the pulse duration becomes shorter ( <50 fs), the molecule will survive till the maximum intensity is reached, and the reorientation due to the higher order terms will become comp arable to that due to the linear term, and may even exceed it! To incorporate hyperpolarizability in our calculations, c ertain approximations need to be made. Firstly, the magnitude of the second- and higher-orde r susceptibilities is not known in most cases [14]. Moreover, these quantities are tensors w ith numerous components and it is difficult to consider all the components in an exact way. We c onsider here the case for H 2 taking into account the third order term due to the electroni c response γe. The equation of motion is modified as follows: αsin2θE2→αsin2θE2+γesin4θE4+. . .. Fig. 4 shows the effect of nonlinearity on the reorientation o f the molecular axis for H 2 acted on by a 20 fs pulse.The inclusion of only the third order term [14] leads to a significantly larger reorientation of the H-H axis as compared to the case w hen only the linear term is considered. As noted previously, such effects will be signifi cant only for very short pulses. This is shown in figure 3(a) for the case of a 40 fs pulse wherein the effect of the third order term is much smaller than that for the 20 fs case. We have verifi ed that for longer pulses such higher order terms need no longer be taken into account. Of course, it is obvious that as the laser pulses get shorter even higher order terms will beg in to play a significant role. One can speculate that heavy molecules, like I 2, may align with sufficiently short pulses because of the contribution from higher order terms. It is clearly ne cessary to test this conjecture experimentally since very little is known about the high ord er polarizabilities of almost all molecules. In summary, we have considered molecular reorientation usi ng a classical model. The 5justification for using a classical model is two-fold. In int ense field-molecule interactions, classical analysis have been shown to be extremely fruitful in explaining much of the ex- perimental data. Also, exact time-dependent analysis of th e molecular response to a high- intensity pulsed light field is presently not feasible. We sh ow that, despite the obvious limitations of classical models, the results are considera ble utility in understanding a large body of experimental work on alignment, and for enabling pre dictions to be made on whether or not, and under what circumstances, molecules will be spat ially aligned when subjected to intense, polarized, short-duration light fields. The rig or of the model is demonstrated. We have also incorporated, for the first time, (i) the role of e nhanced ionization in the reori- entation of molecules and (ii) the role of hyperpolarizabil ity. It is shown that higher order contributions to the dipole moment are very important for ex tremely short light pulses . It is predicted that this will lead to alignment of molecules ev en for sub-50 fs pulses that are becoming increasingly accessible to experimentalists. The TIFR high energy, femtosecond laser facility has been se t up with substantial funding from the Department of Science and Technology, Government o f India. 6REFERENCES [1] D. Normand et al., J. Phys. B 25, L497 (1992), P. Dietrich et al., Phys. Rev. A 47, 2305 (1993), J. H. Sanderson et. al. , J. Phys. B 31, L599 (1998). [2] G. Ravindra Kumar et al., J. Phys. B 29, L95 (1996), V. R. Bhardwaj et al., Phys. Rev. A56, 2455 (1997), and references therein. [3] J. H. Posthumus et al., J. Phys. B 31, L553 (1998), Ch. Ellert and P. B. Corkum, Phys. Rev. A 59, R3170 (1999). [4] L. J. Fraskinski et al., Phys. Rev. Lett. 58, 2424 (1987). [5] F. Hund, Z. Phys. 43, 805 (1927), see also B. Friedrich and D. Herschbach, Z. Phys . D 36, 221 (1996). [6] J. H. Williams, Adv. Chem. Phys. 85, 361 (1993). [7] B. Friedrich and D. Herschbach, Phys. Rev. Lett. 74, 4623 (1995), G. R. Kumar et al., Phys. Rev. A A53, 3098 (1996), W. Kim and P. M. Felker, J. Chem. Phys. 108, 6763 (1998), J. Ortigoso et al., J. Chem. Phys. 110, 3870 (1999). [8] E. Charron, A. Giusti-Suzor, and F. H. Mies, Phys. Rev. A 49, R641 (1994). [9] B. Friedrich and D. R. Herschbach, Phys. Rev. Lett. 74, 4623 (1995), J. D. Weinstein et al., Nature 395, 148 (1998), H. Stapelfeldt et al., Phys. Rev. Lett. 79, 2787 (1997). [10] D. Mathur et al., Phys. Rev. A 50, R7 (1994), J. Phys. B 27, L603 (1994). [11] S. Banerjee, G. Ravindra Kumar, and D. Mathur, J. Phys. B 32, 4277 (1999), Phys. Rev. A 60, R3369 (1999). [12] T. Zuo and A. D. Bandrauk, Phys. Rev. A 52, R2511 (1995), T. Seideman et al., Phys. Rev. Lett. 75, 2819 (1995), J. H. Posthumus et al., J. Phys. B 29, L525 (1996), K. C. Kulander et al., Phys. Rev. A 53, 2562 (1996). [13] C. M. Dion et. al. , Phys. Rev. A 59, 1382 (1999) [14] C. C. Wang, Phys. Rev. B 2, 2045 (1970), D. P. Shelton, Mol. Phys. 60, 65 (1987). 7FIGURES FIG. 1. (a) Time evolution of the molecular alignment for var ious initial orientations at a peak intensity of 1015W cm−2and temporal width of 20 fs. (b) Alignment of the molecular ax is at the point of breakup predicted by the EI model as a function of ini tial orientation. FIG. 2. (a) Alignment of molecules for various conditions of peak intensity, pulse duration and R (see text). Points lying below the surface correspond t o the case of no alignment while all points lying on the surface, and above it, lead to the molecul ar axis being aligned along the light polarization vector. (b) Experimental data for some typica l molecules: CO [1], CO 2[1], H 2, N2[3], I2[3], CS 2(picosecond data [2], femtosecond data [11]). The axes rang es are the same as in (a). FIG. 3. Alignment dynamics calculated with enhanced ioniza tion taken into account (a) H 2(b) CS2. L≡linear polarizability, ND ≡no damping, D ≡damped rotation, NL3 ≡linear and third order polarizability. FIG. 4. Alignment of molecular axes for H 2assuming nonlinear contributions to the polar- izability at a peak intensity of 1015W cm−2and temporal duration of 20 fs. The symbols are explained in fig.3 80 15 30 45 60-60-300306090 (b)(a) H2 1x1015 W cm-2, 20 fsOrientation angle (degrees) Time (fs) 0 30 60 90020406080θθf (degrees) θθi (degrees)101102103104 10110-11012 1014 1016ττ (fs) R (x106A s/kg V)Intensity (W cm-2)(a)ττ R Intensity(b) H2/G31/G15 /G26/G36/G15 /G2C/G15CO2CO0 30 60 900.000.250.500.751.00Counts (arb.units)(b)(a) H2 1x1015 W cm-2, 40 fs L, ND L,D NL3, DCounts (arb.units) Angle (degrees) 0 30 60 900.000.250.500.751.00 CS2 1x1015 W cm-2, 100 fs L, ND, R L, D, R L, D, P Angle (degrees)0 30 60 900.000.250.500.751.00 H2 1x1015 W cm-2, 20 fs L, ND L, D NL3, DCounts (arb.units) Angle (degrees)

arXiv:physics/0004070v1 [physics.gen-ph] 26 Apr 2000A Brief Note on Discrete Space Effects B.G. Sidharth Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Hyderabad 500 063 Abstract We discuss briefly discrete space effects and show that this le ads to space reflection asymmetry and also a minor modification of Ei nstein’s energy-mass formula. Snyder[1] had shown how it is possible to consider discrete s pace time con- sistently with the Lorentz transformation. Discrete space time continues to receive attention over the years from several scholars (C f. for example [2, 3, 4, 5, 6, 7] for details). Indeed in the Dirac’s relativi stic theory of the electron, this discretization is evident - averages over th e Compton scale are required to eliminate Zitterbewegung effects and non Hermit ian position op- erators and to recover meaningful physics[8, 9] - and that in cludes special relativity. It is in this light that a recent formulation of a n electron in terms of a Kerr-Newman metric becomes meaningful[9, 10, 11] and fu rther this leads to a meaningful, if phenomenological mass spectrum [1 2, 13, 14]. All this pleasingly dovetails with the fact that if the minim um space time cut offs are taken at the Compton scale, then we have a non commu tative geometry viz., [x,y] = 0(l2) (1) and similar equations, and which further, leads directly to the Dirac equation (Cf.[15] for details). This can be easily seen from the fact t hat, given (1), the usual infinitessimal coordinate shift in Minkowski spac e, is, ψ′(xj) = [1 +ıǫ(ıǫljkxk∂ ∂xj) + 0(ǫ2)]ψ(xj) (2) 1The choice t=/parenleftBigg 1 0 0−1/parenrightBigg ,/vector x=/parenleftBigg 0/vector σ /vector σ0/parenrightBigg provides a representation for the coordinates, as can be eas ily verified and then from (2) we recover the Dirac equation. If on the other hand terms ∼l2are neglected, then (1) gives the usual com- mutation relations of Quantum Theory. Discrete space time i s therefore a higher order correction to usual Quantum Theory. In this context it has been pointed out that the discrete time provides an ex- planation for the puzzling Kaon decay which violates time re versal symmetry and also the decay of a pion into an electron and a positron[16 , 8, 9, 12, 13]. We now observe that from an intuitive point of view space or ti me reversal symmetries based on space time points theory cannot be taken for granted if space time is discrete. This can immediately be seen from (1) : If we retain terms∼l2, then there is no invariance under space reflections. Indeed in the same vein, as discussed earlier[9, 10], the fac t that the Compton wavelength of the nearly massless neutrino is very large pro vides an expla- nation of its handedness. We finally point out another 0( l2) effect, which can be demonstrated in a simple way by invoking the derivation of the wave equation of a particle by replacing the continuum by a set of ”lattice points”[9, 17]. In this case we have an equation like Ea(xn) =E0a(xn)−Aa(xn+b)−Aa(xn−b). wherebis the distance between successive ”lattice points”. This l eads to E=E0−2Acoskb. (3) We can choose the zero of energy in such a way that when b→0, we have E= 2Awhich is then identified with the rest energy mc2of the particle(Cf.[9] for details). However if we do not neglect terms ∼b2=l2, then we have from (3) /vextendsingle/vextendsingle/vextendsingle/vextendsingleE mc2−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼0(l2) It must be reiterated that when terms ∼l2are neglected, we recover the usual theory. Finally, the above conclusions are true with minor m odifications in case the minimum cut off is non-zero but not the Compton wavele ngth. 2References [1] Snyder, H.S., Phys. Rev. 71(1), 1947, 38ff. [2] Kadyshevskii, V.G., Soviet Physics Doklady 7(11), 1963, 1030. [3] Kadyshevskii, V.G., Soviet Physics Doklady 7(12), 1963, 1138ff. [4] Wolf C., Nuovo. Cim. B 109 (3), 1994, 213. [5] Bombelli, L., Lee, J., Meyer, D., Sorkin, R.D., Phys. Rev . Lett. 59, 1987, 521. [6] Caldirola, P., Lettere Al Nuovo Cimento, Vol.16, N.5, 19 76, 151ff. [7] Lee, T.D., Phys. Lett. 12 (2B), 1983, 217. [8] Dirac, P.A.M., ”Principles of Quantum Mechanics”, Clar endon Press, Oxford, 1958. [9] Sidharth, B.G., Ind. J. Pure and Applied Phys., 35 (7), 19 97, 456. [10] Sidharth, B.G., IJMPA, 13 (15), 1998, 2599. Also xxx.la nl.gov quant-ph 9808031. [11] Sidharth, B.G., Gravitation and Cosmology 4(2) (14), 158ff (1998) and references therein. [12] Sidharth, B.G., Mod.Phys. Lett. A., Vol. 12 No.32, 1997 , pp2469-2471. [13] Sidharth, B.G., Mod.Phys. Lett. A., Vol. 14 No. 5, 1999, pp387-389. [14] Sidharth, B.G., and Lobanov, Yu Yu, Proceedings of Fron tiers of Fun- damental Physics, (Eds.)B.G. Sidharth and A. Burinskii, Un iversities Press, Hyderabad, 1999. [15] Sidharth, B.G., Chaos, Solitons and Fractals, 11 (8), 2 000, 1269-1278. [16] Sidharth, B.G., Chaos, Solitons and Fractals, 11 (8), 2 000, 1171-1174. [17] Feynman, R.P., ”The Feynman Lectures on Physics”, (Vol .2), Addison- Wesley, Mass, 1965. 3

arXiv:physics/0004071v1 [physics.gen-ph] 26 Apr 2000Space-time as a lattice of harmonic oscillators: quantitatively deriving relativity, mass, and gravitatio n Richard Lieu1 1Department of Physics, University of Alabama, Huntsville, AL 35899, U.S.A. Received ; accepted– 2 – ABSTRACT The microscopic structure of space and time is investigated . It is proposed that space and time of an inertial observer Σ are most conveni ently described as a crystal lattice Λ, with nodes representing measurement ‘tickmarks’ and connected by independent quantized harmonic oscillators w hich vibrate more severely the faster Σ moves with respect to the object being m easured (as due to the Uncertainty Principle). The Lorentz transformation of Special Relativity is be derived. Further, mass is understood as a localized regio n ∆Λ having higher vibration temperature than that of the ambient lattice. The effect of relativistic mass increase may then be calculated without appealing to en ergy-momentum conservation. The origin of gravitation is shown to be simpl y a transport of energy from the boundary of ∆Λ outwards by lattice phonon con duction, as the system tends towards equilibrium. Application to a sing le point mass leads readily to the Schwarzschild metric, while a new solution is available for two point masses - a situation where General Relativity is too co mplicated to work with. The important consequence is that inertial observers who move at relative speeds too close to care no longer linked by the Lorentz transformation, because the lattice of the ‘moving’ observer has already disintegra ted into a liquid state. In this work I propose a model for the microscopic structure o f space and time to understand relativity and the origin of mass and gravitat ion in terms of statistical mechanics. The underlying ideas originated from an examination of Spec ial Relativity (Einstein– 3 – 1905), which does not treat distances and times as absolute. For a uniformly ‘moving’ observer, one can disregard spatial dimensions perpendicu lar to the velocity of motion vand elaborate the previous statement as meaning that if Σ′measures: (i) position differences between events at the same time, or (ii) time diffe rences at the same position, the resulting distances and intervals are smaller than thos e of a ‘stationary’ observer Σ who measures the same events from a frame at rest with respect to them, by a common factor γ= (1−v2/c2)−1 2. Since each of the two effects (i) and (ii) was presented witho ut interference from the other, they can in general be superpos ed to form the Lorentz transformation (i.e. rotation) of space-time (Lorentz 192 3). One could also describe the phenomenon as due to a measuring apparatus having different i ntrinsic properties when set in ‘motion’. Thus another way of expressing (i) is to visu alize the ‘tickmarks’ which calibrate the ruler of Σ′as more widely separated than those of Σ, as illustrated in Fi gure 1. If a unit of length for Σ is xmthen the same for Σ′will be xmγ, which is of the form xm√ N (N≥1), suggesting that microscopically space-time might be un dergoing a ‘random walk’ process of some kind. Similarly, (ii) may also be viewed in th is way, because the Lorentz transformation of time is the same as that of space, except ti me is being re-scaled by the speed c= 1/√ǫoµoto form a different dimension, see Figure 2. To avoid duplicat ion, the ensuing treatment will emphasize effect (i) only. The particular manner of considering Special Relativity as changes in the measuring devices of a travelling observer matches well with a statist ical approach: an obvious way of further separating the ‘tickmarks’ of a ‘moving’ ruler, i s if they are not part of a rigid body, but their relative positions can fluctuate while in mot ion, maintaining the mean translational velocities = v. I shall pursue the fluctuations specifically as oscillation s, the amplitude of which increase with the relative speed v. The spirit of approach is in accordance with the Heisenberg Uncertainty Principle (Hei senberg 1930): the higher the velocity (or momentum) being measured the larger the inhere nt uncertainties (deviations)– 4 – in the ability of an apparatus to determine pandx, with the product ∆ p∆xreaching the smallest value ∼¯honly for ground states where motion of the object is minimize d. I therefore let the natural separation of the ‘tickmarks’ be xo(note that xo< xm, for reasons to be explained). To this I add a sinusoidal variable x1, which has a mean < x1>= 0. The total separation x=xo+x1, then, has a mean < x > =xo, but the r.m.s. value is given by: xrms=xo/parenleftBigg 1 +< x2 1> x2 o/parenrightBigg1 2 (1) and is > xo. For the purpose of distance measurements it is the magnitud e ofx, not the sign, which calibrates the length between a pair of ‘tickmar ks’ at any phase of the cycle. Ifxoscillates rapidly1and substantially about xo, it is then xrmswhich defines a unit of length. An analogy is that of our mains AC power supply. Mos t electric appliances recognize only the magnitude of Vand have response times slow compared with typical AC cycle frequencies of ∼a few×10 Hz, so that the relevant output voltage is Vrms, even though < V > = 0. The fundamental postulates of the theory are stated here. Th e starting point concerns the first two of them. (a) Space andtime form a crystal lattice, the lattice points (hereafter s imply referred to as ‘nodes’, which represent the ‘tickmarks’ of an observer) are connected by independent harmonic oscillators with quantized energy levels E= (n+ 1/2)ǫ. All inertial observers Σ at rest relative to each other are assigned a common rest fra me lattice Λ with which 1Caution should be exercised while interpreting this word, b ecause not only do the spatial ‘tickmarks’ oscillate, but the same applies to time which be haves just like space (see earlier). Thus the period of a cycle cannot be defined with respect to ord inary time. The quantum mechanical approach of this paper alleviates somewhat the u rgency of settling the issue, however, since one considers < x2>as an expectation value, proportional to energy.– 5 – measurements are naturally performed. (b) The degree of oscillation of Λ is parametrized by a temper ature which may be defined not only with respect to Σ, but also anyinertial observer Σ′. If Σ′= Σ, this will become the rest temperature, which is taken to have an ambient value of 0 K. If a relative speed v∝negationslash= 0 exists between the two frames, this temperature will be >0 K, by an amount T which increases with vin a simple manner to be formulated below; as a result the quan tity xrmsis larger for Σ′than for Σ. (c) Matter owes its origin to a localized region ∆Λ of Λ having a rest temperature >0 K. The quantity we define as rest mass is proportional to this surplus of energy within ∆Λ. An oscillator in ∆Λ has larger than ambient amplitude, i.e. xrms→Xrms. With respect to Σ′this amplitude is further increased, by the same factor as xrmsdoes in Postulate (b), see Figure 3. The situation may be likened to Σ being already in a l ocal Lorentz frame (which widens xrmstoXrms), and Σ′moving relative to Σ at the stipulated speed v. (d) The effect of gravitation is due to conduction of energy by the oscillator waves (phonons) from the hotter region of ∆Λ to the cooler ambient region. The process causes a gradual downward trend in the temperature as one moves away from ∆Λ. T here is consequently a distribution of oscillator lengths, thereby affecting the separation of the measurement ‘tickmarks’, see Figure 3. This is the reason for the curvatu re of space-time responsible for the existence of universal gravitation in the lattice surro unding ∆Λ. None of the above is in conflict with the First Relativity Post ulate, viz. that no preferred frame of rest exists. There is also no violation of the Second Postulate. I now focus on Special Relativity, viz. Postulates (a) and (b ). When applying to inertial observers in relative motion, note that even for Σ t he ambient separation between nodes of Λ is larger than xo, due to the zero point energy which leads to a finite < x2 1>– 6 – given by < x2 1>=E(n= 0)/κ=ǫ/2κwhere κis the oscillator (spring) constant. The absolute minimum unit of length is then obtained from Equ (1) as xm=xo/parenleftBigg 1 +ǫ 2κx2o/parenrightBigg1 2 (2) Since energy is additive, for finite v(ambient T >0)xmincreases to: xrms=xm/parenleftBigg 1 +< x2 1> x2m/parenrightBigg1 2 (3) where < x2 1>=¯E(T)/κis the mean energy at temperature T, calculated with the lowest level now having energy E=ǫ(i.e.n= 1). The relevant Partition Function is Z= 1/[1−exp(−ǫ/kT)], leading to a mean energy of ¯E=ǫZe−ǫ kT. Substituting these into Equ (3), one obtains: xrms=xm/parenleftBigg 1 +ǫ 2κx2oe−ǫ kT 1−e−ǫ kT/parenrightBigg1 2 (4) At this point I complete postulate (b) above with the followi ng quantitative relationship between vandT: v2 c2=e−ǫ kT/parenleftBigg =¯E ǫ+¯E/parenrightBigg (5) Combining (4) and (5), one gets: xrms=xm 1 +ǫ 2κx2ov2 c2 1−v2 c2 1 2 (6) Finally, postulate (a) is made more specific by invoking: ǫ= 2κx2 o, (7) which simply states that the zero point fluctuations double t he mean-square separation of nodes from the natural value of x2 o. Equs (4), (5) and (6) altogether tell us that the unit length for observer Σ′is not xm, butxrms=xm/radicalBig 1/(1−v2/c2), implying smaller measured distances for Σ′than those for Σ, in accordance with Lorentz contraction.– 7 – There remains the possibility that the suceess achieved so f ar is an illusion, viz. statistical mechanics of harmonic oscillators is a wrong an d completely irrelevant approach to questions concerning the nature of space and time. Althou gh it so happens that Special Relativity can be explained in this way, the procedure could be merely formal: a wide variety of mathematical formulae describe the many diverse phenomena of the known physical world, the fact that some of the formulae resemble e ach other in appearance does not immediately imply a parallel in the physics involved. In this instance, however, such concerns are settled by the theory’s ability to go beyond Spe cial Relativity, to shed light on the issues of mass and gravitation. I first discuss the relativity of mass. According to Postulat e (c), mass is an energized lattice region ∆Λ wherein an oscillator has large r.m.s. amp litude, as illustrated in Figure 2. Upon transformation from Σ to Σ′(so that Λ is no longer the rest lattice) X, likexm, is required by (c) and (b) to increase by a factor γ. Thus the oscillator energy is increased by γ2. The number of oscillators is, of course, decreased by γ, see Figure 2. As a result, the total energy within ∆Λ is higher with respect to Σ′byγ. This then affords an exceedingly simple derivation of the relativistic increase of the energ y within ∆Λ. By (c) the same effect applies to mass. Next, I demonstrate how equally straightforward it is to exp lain gravitation as an energy transport effect. Let us center ∆Λ at the origin of the r est lattice Λ, and consider the 1-D conduction of energy in the +x-direction from some po sition x=xmin, Figure 3. Note that the notion of an inner boundary from which conducti on commences is an abstract one: the boundary does not have to represent the physical siz e of the mass depicted in Figure . The transport equation is: σthdT dx=n¯E¯v. (8) Hereσth=n¯vλd¯E/dT is the thermal conductivity of phonons: nis the linear phonon– 8 – density, ¯ vis the ‘speed of sound’ in the lattice2,¯E(T) is the mean energy of a phonon, and λis the phonon mean free path, which is the size of the availabl e lattice (since phonons do not interact), i.e. λ=x. Thus equ (8), together with the meaning of the various symbo ls involved, imply that −xd¯E dx=¯E,or¯E =1 αx(9) where αis a constant of integration. Combining equs (5) and (9), one obtains v2 c2=1 αǫx+ 1(10) as equation giving the speed of the local Lorentz (or ‘free fa ll’) frame at x. Since the x-axis can represent a radial direction, we now wri tex=r−rgandrg=xmin. In the limit of r≫rg(v≪c) there should be agreement with Newtonian gravity. This requires αǫ=c2/2GM, which removes the arbitrariness of the solution. Returnin g to equ (9), one clearly sees that ¯E∝M, consistent with the proposition in Postulate (c) that mass is proportional to incumbent energy. At high speeds the role of rgmust be taken into account ( v→casr→rg). In the case of spherical symmetry there is only one free parameter in the problem, i.e. rgmust depend on α. By setting rg= 1/αǫ= 2GM/c2, equ (10) reduces to v2/c2=rg/r, orγ= (1−rg r)−1 2. It is therefore apparent that the ‘tickmarks’ ofr(orx) are non-uniformly distributed throughout Λ, due to the ene rgy outflow, in such a way that if they are used to measure r(orx) the result is radius in Euclidean (flat) space-time. In fact, the expression for γcontains all the information one needs to construct the full Schwarzschild metric (Schwarzschild 1916) of Gene ral Relativity. For example, as a falling object approaches the gravitational radius (or ev ent horizon) r=rgtime dilation 2The conduction of energy takes place in Λ, which is a lattice o f space andtime. Thus, like the oscillations, there is the need to introduce a new ‘t ime’ axis when defining the propagation speed of these phonons - signature of a fifth dime nsion.– 9 – becomes infinite. The notion of a ‘boundary’ for ∆Λ is now clea r: phonon energy transport to other parts of the lattice takes place only beyond the even t horizon. Apart from its ability to offer insightful, even elegant, exp lanations of fundamental laws of physics, the model presented here can also make predictio ns. One immediate application is the problem of two point masses, where no known solution of the Einstein Field Equations exist. The phonon conduction approach reduces th e mathematical complexities by manifold. A member of the pair may be taken as the mass Mdescribed above (except nowM→M1) while the other (mass M2) is positioned at Euclidean distance Rfrom the first. The separation between their event horizons is then R−rg−r′ g, where rg= 2GM1/c2 andr′ g= 2GM2/c2. Since the presence of mass M2cannot alter the boundary conditions for the outward energy transport from mass M13and vice versa, one simply solves for the oscillator (phonon) energy E1due to mass M1, likewise E2due to M2, for any position along the line joining the two masses and distance rfrom the first. The total energy at this position is ¯E=¯E1+¯E2=ǫ[rg/(r−rg) +r′ g/(R−r−r′ g)]. By using equ (5), we have the speed vof the local Lorentz frame at r: v2 c2=rg r−rg+r′ g R−r−r′g 1 +rg r−rg+r′g R−r−r′g(11) This gives for the first time the space-time metric at any posi tion between two masses. For more complicated mass distributions, the metric at any poin t is given by the superposition of all the outwardly conducted energy distributions, in the same manner as above. In conclusion, I found that a simple microscopic model of spa ce-time can explain the Lorentz transformation and the origin of mass and gravitati on very naturally. In this model, space and time form a crystal lattice, with nodes conn ected by harmonic oscillators. 3Otherwise one will not be able to restore the single mass solu tion by letting the other mass tend to zero– 10 – The notion of distances and times has no meaning unless measu rements are performed using the nodes as ‘tickmarks’ to define unit intervals. The f aster the speed of an object, the ‘fuzzier’ the measurement, and this is reflected in equ (5 ) wherein the nodes vibrate with a temperature Twhich increases with the relative speed vbetween a rest lattice and a ‘moving’ object. When v=c,Treaches infinity, implying that the lattice has already disintegrated before that. Thus there must exist a threshol dv(< c) which corresponds to a critical T, observers who move with respect to each other at speeds high than this value are no longer connected by the Lorentz transformation. In other words, while Relativity links the macroscopic properties of crystal lattices at different vibration temperatures, here we are comparing two fundamentally different lattices, viz. th ose of a crystal and a liquid. I am indebted to Dr Massimilano Bonamente for suggesting a ha rmonic oscillator model for the space-time lattice of a moving observer.– 11 – Reference Einstein, A., 1905, Annalen der Physik ,18, 891. Heisenberg, W., 1930, Physical Principles of the Quantum Th eory, Dover, NY. Lorentz, H.A., Einstein, A., Minkowski, H., & Weyl, H., 1923 , The Principles Relativity - a Collection of Original Memoirs, trans. W. Per ret & G.B. Jeffrey, London: Methuen & Co. Ltd., paperback reprint, Dove r Publ. (1958). Schwarzschild, K., 1916, ¨Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzber Preuss. Acad. Wiss. Berlin 189 – 196.– 12 – Figure Captions Figure 1: The space-time lattice of stationary (Σ) and movin g (Σ′) observers are illustrated here for the case of distance measurements. The ‘tickmarks’ of the ruler of Σ are marked as the topmost set of black dots. The rod to be measured is the s hort bar immediately beneath, and is at rest with respect to Σ. Observer Σ′measures the length of this rod while in motion, by simultaneously acquiring data on the pos itions of the front and rear end of the rod. It is postulated that effectively Σ′is using a moving set of ‘tickmarks’, and if microscopically these are connected by oscillators w hich vibrate while in motion, the ‘tickmarks’ widen as depicted in the lower half of the diagra m. Consequently Σ′obtains a smaller value for the length of the rod. Figure 2: The space-time lattices of inertial observers Σ (t op) and Σ′(bottom), the latter ‘moving’ at velocity vwith respect to the former, who is regarded as ‘stationary’. If Σ′ measures ∆ xbetween two ‘stationary’ events at the same time, or ∆ tat the same position, in each case the result is less than that obtained by Σ. This me ans for ‘orthogonal’ measurements of space and time performed by Σ′, the ‘tickmarks’ of Σ are more widely separated, as indicated by the lower grid. Figure 3: Illustrating the origin of mass and gravitation. Top:a region ∆Λ of Λ (the rest frame lattice of observer Σ) has higher than ambient tempera ture. Mass is proportional to the incumbent extra energy. The physical boundary of the m ass (shown here in the space dimension only) is drawn as a rectangular box, inside o f which all the energy surplus resides, as a result the nodes are much more widely spaced tha n outside. Middle: The same region as it appears in the lattice of observer Σ′who ‘moves’ with respect to Σ at velocity v. Separation between any pair of nodes is now increased by the factor γ, meaning fewer oscillators within ∆Σ, but more energy per oscillator . The net increase in enclosed energy (hence mass) is γ(see text). Bottom: Energy is conducted outwards from ∆Λ to the– 13 – ambient lattice by phonons, which causes the separation bet ween nodes to gradually reach the natural minimum at asymptotically large distances. Not e that the inner boundary from which the energy transport commences is an abstract qua ntity which need not be the physical size of the mass; in fact, the former is usually with in the latter.

arXiv:physics/0004072v1 [physics.ed-ph] 27 Apr 2000Los Alamos Electronic ArXives http://xxx.lanl.gov/physics/0004072 ELEMENTARY QUANTUM MECHANICS HARET C. ROSU e-mail: rosu@ifug3.ugto.mx fax: 0052-47187611 phone: 0052-47183089 h/2π 1Copyright c∝ci∇cleco√†∇t2000 by the author. All commercial rights are reserved. April 2000 Abstract This is the first graduate course on elementary quantum mecha nics in Inter- net written for the benefit of undergraduate and graduate stu dents. It is a translation (with corrections) of the Romanian version of t he course, which I did at the suggestion of several students from different cou ntries. The top- ics included refer to the postulates of quantum mechanics, o ne-dimensional barriers and wells, angular momentum and spin, WKB method, h armonic oscillator, hydrogen atom, quantum scattering, and partia l waves. 2CONTENTS 0. Forward ... 4 1. Quantum postulates ... 5 2. One-dimensional rectangular barriers and wells ... 23 3. Angular momentum and spin ... 45 4. The WKB method ... 75 5. The harmonic oscillator ... 89 6. The hydrogen atom ... 111 7. Quantum scattering ... 133 8. Partial waves ... 147 There are about 25 illustrative problems. Spacetime nonrelativistic atomic units aH= ¯h2/mee2= 0.529·10−8cm tH= ¯h3/mee4= 0.242·10−16sec Planck relativistic units of space and time lP= ¯h/mPc= 1.616·10−33cm tP= ¯h/mPc2= 5.390·10−44sec 30. FORWARD The energy quanta occured in 1900 in the work of Max Planck (No bel prize, 1918) on the black body electromagnetic radiation. Planck’ s “quanta of light” have been used by Einstein (Nobel prize, 1921) to expl ain the pho- toelectric effect, but the first “quantization” of a quantity having units of action (the angular momentum) belongs to Niels Bohr (Nobel P rize, 1922). This opened the road to the universalization of quanta, sinc e the action is the basic functional to describe any type of motion. However , only in the 1920’s the formalism of quantum mechanics has been develope d in a system- atic manner. The remarkable works of that decade contribute d in a decisive way to the rising of quantum mechanics at the level of fundame ntal theory of the universe, with successful technological applicatio ns. Moreover, it is quite probable that many of the cosmological misteries may b e disentan- gled by means of various quantization procedures of the grav itational field, advancing our understanding of the origins of the universe. On the other hand, in recent years, there is a strong surge of activity in t he information aspect of quantum mechanics. This aspect, which was general ly ignored in the past, aims at a very attractive “quantum computer” techn ology. At the philosophical level, the famous paradoxes of quantum mechanics, which are perfect examples of the difficulties of ‘quantum’ th inking, are actively pursued ever since they have been first posed. Perha ps the most famous of them is the EPR paradox (Einstein, Podolsky, Rosen , 1935) on the existence of elements of physical reality , or in EPR words: “If, without in any way disturbing a system, we can predict with certainty (i.e. , with probability equal to unity) the value of a physical quantity, then there e xists an element of physical reality corresponding to this physical quantit y.” Another famous paradox is that of Schr¨ odinger’s cat which is related to the fundamental quantum property of entanglement and the way we understand a nd detect it. What one should emphasize is that all these delicate poin ts are the sourse of many interesting and innovative experiments (suc h as the so-called “teleportation” of quantum states) pushing up the technolo gy. Here, I present eight elementary topics in nonrelativistic quantum me- chanics from a course in Spanish (“castellano”) on quantum m echanics that I taught in the Instituto de F´ ısica, Universidad de Guanaju ato (IFUG), Le´ on, Mexico, during the semesters of 1998. Haret C. Rosu 41. THE QUANTUM POSTULATES The following six postulates can be considered as the basis f or theory and experiment in quantum mechanics in its most used form, which is known as the Copenhagen interpretation. P1.- To any physical quantity L, which is well defined at the class ical level, one can associate a hermitic operator ˆL. P2.- To any stationary physical state in which a quantum system c an be found one can associate a (normalized) wavefunction. ψ(∝ba∇dblψ∝ba∇dbl2 L2= 1). P3.- In (appropriate) experiments, the physical quantity L can take only the eigenvalues of ˆL. Therefore the eigenvalues should be real, a condition which is fulfilled only by hermitic operators. P4.- What one measures is always the mean value Lof the physical quantity (i.e., operator) ˆLin a stateψn, which, theoretically speaking, is the corresponding diagonal matrix element ∝an}b∇acketle{tψn|ˆL|ψn∝an}b∇acket∇i}ht=L. P5.- The matrix elements of the operators corresponding to the c artesian coordinate and momentum, /hatwidexiand/hatwiderpk, when calculated with the wave- functionsfandgsatisfy the Hamilton equations of motion of classical mechanics in the form: d dt∝an}b∇acketle{tf|/hatwidepi|g∝an}b∇acket∇i}ht=−∝an}b∇acketle{tf|∂/hatwideH ∂/hatwidexi|g∝an}b∇acket∇i}ht d dt∝an}b∇acketle{tf|/hatwidexi|g∝an}b∇acket∇i}ht=∝an}b∇acketle{tf|∂/hatwideH ∂/hatwidepi|g∝an}b∇acket∇i}ht, where/hatwideHis the hamiltonian operator, whereas the derivatives with respect to operators are defined as at point 3 of this chapter. P6.- The operators /hatwidepiand/hatwiderxkhave the following commutators: [/hatwidepi,/hatwiderxk] =−i¯hδik, [/hatwidepi,/hatwiderpk] = 0, [/hatwidexi,/hatwiderxk] = 0 5¯h=h/2π= 1.0546×10−27erg.sec. 1.- The correspondence between classical and quantum quant ities This can be done by substituting xi,pkwith/hatwidexi/hatwiderpk. The function L is supposed to be analytic (i.e., it can be developed in Taylor s eries). If the L function does not contain mixed products xkpk, the operator ˆL is directly hermitic. Exemple: T= (/summationtext3 ip2 i)/2m−→/hatwideT= (/summationtext3 i/hatwidep2)/2m. If L contains mixed products xipiand higher powers of them, ˆLis not hermitic, and in this case L is substituted by ˆΛ, the hermitic part of ˆL(ˆΛ is an autoadjunct operator). Exemple: w(xi,pi) =/summationtext ipixi−→/hatwidew= 1/2/summationtext3 i(/hatwidepi/hatwidexi+/hatwidexi/hatwidepi). In addition, one can see that we have no time operator. In quan - tum mechanics, time is only a parameter that can be introduce d in many ways. This is so because time does not depend on the canon ical variables, merely the latter depend on time. 2.- Probability in the discrete part of the spectrum Ifψnis an eigenfunction of the operator ˆL, then: L=<n|ˆL|n>=<n|λn|n>=λn<n|n>=δnnλn=λn. Moreover, one can prove that Lk= (λn)k. If the function φis not an eigenfunction of ˆL, one can make use of the expansion in the complete system of eigenfunctions of ˆLto get: ˆLψn=λnψn,φ=/summationtext nanψn and combining these two relationships one gets: ˆLφ=/summationtext nλnanψn. 6In this way, one is able to calculate the matrix elements of th e operator ˆL: ∝an}b∇acketle{tφ|ˆL|φ∝an}b∇acket∇i}ht=/summationtext n,ma∗ manλn∝an}b∇acketle{tm|n∝an}b∇acket∇i}ht=/summationtext m|am|2λm, telling us that the result of the experiment is λmwith a probability |am|2. If the spectrum is discrete, according to P4this means that|am|2, that is the coefficients of the expansion in a complete set of ei genfunc- tions, determine the probabilitities to observe the eigenv alueλn. If the spectrum is continuous, using the following definitio n φ(τ) =/integraltexta(λ)ψ(τ,λ)dλ, one can calculate the matrix elements in the continuous part of the spectrum ∝an}b∇acketle{tφ|ˆL|φ∝an}b∇acket∇i}ht =/integraltextdτ/integraltexta∗(λ)ψ∗(τ,λ)dλ/integraltextµa(µ)ψ(τ,µ)dµ =/integraltext/integraltexta∗a(µ)µ/integraltextψ∗(τ,λ)ψ(tau,µ )dλdµdτ =/integraltext/integraltexta∗(λ)a(µ)µδ(λ−µ)dλdµ =/integraltexta∗(λ)a(λ)λdλ =/integraltext|a(λ)|2λdλ. In the continuous case, |a(λ)|2should be understood as the probabil- ity density for observing the eigenvalue λbelonging to the continuous spectrum. Moreover, the following holds L=∝an}b∇acketle{tφ|ˆL|φ∝an}b∇acket∇i}ht. One usually says that ∝an}b∇acketle{tµ|Φ∝an}b∇acket∇i}htis the representation of |Φ∝an}b∇acket∇i}htin the reprezentation µ, where|µ∝an}b∇acket∇i}htis an eigenvector of ˆM. 73.- Definition of the derivate with respect to an operator ∂F(ˆL) ∂ˆL= limǫ→∞F(ˆL+ǫˆI)−F(ˆL) ǫ. 4.- The operators of cartesian momenta Which is the explicit form of /hatwiderp1,/hatwiderp2and/hatwiderp3, if the arguments of the wavefunctions are the cartesian coordinates xi? Let us consider the following commutator: [/hatwidepi,/hatwidexi2] =/hatwidepi/hatwidexi2−/hatwidexi2/hatwidepi =/hatwidepi/hatwidexi/hatwidexi−/hatwidexi/hatwidepi/hatwidexi+/hatwidexi/hatwidepi/hatwidexi−/hatwidexi/hatwidexi/hatwidepi = (/hatwidepi/hatwidexi−/hatwidexi/hatwidepi)/hatwidexi+/hatwidexi(/hatwidepi/hatwidexi−/hatwidexi/hatwidepi) = [/hatwidepi,/hatwidexi]/hatwidexi+/hatwidexi[/hatwidepi,/hatwidexi] =−i¯h/hatwidexi−i¯h/hatwidexi=−2i¯h/hatwidexi. In general, the following holds: /hatwidepi/hatwidexin−/hatwidexin/hatwidepi=−ni¯h/hatwidexin−1. Then, for all analytic functions we have: /hatwidepiψ(x)−ψ(x)/hatwidepi=−i¯h∂ψ ∂xi. Now, let/hatwidepiφ=f(x1,x2,x3) be the manner in which /hatwidepiacts onφ(x1,x2,x3) = 1. Then: /hatwidepiψ=−i¯h∂ψ ∂x1+f1ψand similar relationships hold for x2andx3. From the commutator [ /hatwidepi,/hatwiderpk] = 0 it is easy to get ∇×/vectorf= 0 and therefore fi=∇iF. The most general form of /hatwidepiis/hatwidepi=−i¯h∂ ∂xi+∂F ∂xi, whereFis an ar- bitrary function. The function Fcan be eliminated by the unitary transformaton/hatwideU†= exp(i ¯hF). 8/hatwidepi=/hatwideU†(−i¯h∂ ∂xi+∂F ∂xi)/hatwideU = expi ¯hF(−i¯h∂ ∂xi+∂F ∂xi)exp−i ¯hF =−i¯h∂ ∂xi leading to /hatwidepi=−i¯h∂ ∂xi−→/hatwidep=−i¯h∇. 5.- Calculation of the normalization constant Any wavefunction ψ(x)∈L2of variablexcan be written in the form: ψ(x) =/integraltextδ(x−ξ)ψ(ξ)dξ that can be considered as the expansion of ψin eigenfunction of the operator position (cartesian coordinate) ˆ xδ(x−ξ) =ξ(x−ξ). Thus, |ψ(x)|2is the probability density of the coordinate in the state ψ(x). From here one gets the interpretation of the norm ∝ba∇dblψ(x)∝ba∇dbl2=/integraltext|ψ(x)|2dx= 1. Intuitively, this relationship tells us that the system des cribed byψ(x) should be encountered at a certain point on the real axis, alt hough we can know only approximately the location. The eigenfunctions of the momentum operator are: −i¯h∂ψ ∂xi=piψ, and by integrating one gets ψ(xi) =Aexpi ¯hpixi.xandp have continuous spectra and therefore the normalization is performed by means of the “Dirac delta function. Which is the explicit way of getting the normalization const ant ? This is a matter of the following Fourier transforms: f(k) =/integraltextg(x) exp−ikxdx,g(x) =1 2π/integraltextf(k)expikxdk. It can also be obtained with the following procedure. Consid er the unnormalized wavefunction of the free particle φp(x) =Aexpipx ¯hand the formula δ(x−x′) =1 2π/integraltext∞ −∞expik(x−x′)dx . 9One can see that /integraltext∞ −∞φ∗ p′(x)φp(x)dx =/integraltext∞ −∞A∗exp−ip′ x ¯hAexpipx ¯hdx =/integraltext∞ −∞|A|2expix(p−p′ ) ¯hdx =|A|2¯h/integraltext∞ −∞expix(p−p′ ) ¯hdx ¯h = 2π¯h|A|2δ(p−p′) and therefore the normalization constant is: A=1√ 2π¯h. Moreover, the eigenfunctions of the momentum form a complet e sys- tem (in the sense of the continuous case) for all functions of theL2 class. ψ(x) =1√ 2π¯h/integraltexta(p) expipx ¯hdp a(p) =1√ 2π¯h/integraltextψ(x)exp−ipx ¯hdx. These formulae provide the connection between the x and p rep resen- tations 6.- The momentum (p) reprezentation The explicit form of the operators ˆ piand ˆxkcan be obtained either from the commutation relationships or through the usage of t he kernels 10x(p,β) =U†xU=1 2π¯h/integraltextexp−ipx ¯hxexpiβx ¯hdx =1 2π¯h/integraltextexp−ipx ¯h(−i¯h∂ ∂βexpiβx ¯h). The integral is of the form: M(λ,λ′) =/integraltextU†(λ,x)/hatwiderMU(λ′,x)dx, and using ˆxf=/integraltextx(x,ξ)f(ξ)dξ, the action of ˆ xon a(p)∈L2is: ˆxa(p) =/integraltextx(p,β)a(β)dβ =/integraltext(1 2π¯h/integraltextexp−ipx ¯h(−i¯h∂ ∂βexpiβx ¯h)dx)a(β)dβ =−i 2π/integraltext/integraltextexp−ipx ¯h∂ ∂βexpiβx ¯ha(β)dxdβ =−i¯h 2π/integraltext/integraltextexp−ipx ¯h∂ ∂βexpiβx ¯ha(β)dx ¯hdβ =−i¯h 2π/integraltext/integraltextexpix(β−p) ¯h∂ ∂βa(β)dx ¯hdβ =−i¯h/integraltext∂a(p) ∂βδ(β−p)dβ=−i¯h∂a(p) ∂p, where δ(β−p) =1 2π/integraltextexpix(β−p) ¯hdx ¯h. The momentum operator in the p reprezentation is defined by th e ker- nel: p(p,β) =/hatwideU†p/hatwideU =1 2π¯h/integraltextexp−ipx ¯h(−i¯h∂ ∂x)expiβx ¯hdx =1 2π¯h/integraltextexp−ipx ¯hβexpiβx ¯hdx=βλ(p−β) 11leading to ˆpa(p) =pa(p). It is worth noting that ˆ xand ˆp, although hermitic operators for all f(x)∈L2, are not hermitic for their own eigenfunctions. If ˆpa(p) =poa(p) and ˆx= ˆx†ˆp= ˆp†, then <a|ˆpˆx|a>−<a|ˆxˆp|a>=−i¯h<a|a> po[<a|ˆx|a>−<a|ˆx|a>] =−i¯h<a|a> po[<a|ˆx|a>−<a|ˆx|a>] = 0 The left hand side is zero, whereas the right hand side is inde finite, which is a contradiction. 7.- Schr¨ odinger and Heisenberg representations The equations of motion given by P5have different interpretations because in the expressiond dt∝an}b∇acketle{tf|ˆL|f∝an}b∇acket∇i}htone can consider the temporal dependence as belonging either to the wavefunctions or oper ators, or both to wavefunctions and operators. We shall consider here in only the first two cases. •For an operator depending on time/hatwideO=/hatwidestO(t) we have: ˆpi=−∂/hatwideH ∂ˆxi, ˆxi=∂/hatwideH ∂ˆpi [ˆp,f] = ˆpf−fˆp=−i¯h∂f ∂ˆxi [ˆx,f] = ˆxf−fˆx=−i¯h∂f ∂ˆpi and the Heisenberg equations of motion are easily obtained: ˆpi=−i ¯h[ˆp,/hatwideH], ˆxi=−i ¯h[ˆx,/hatwideH]. 12•If the wavefunctions are time dependent one can still use ˆ pi= −i ¯h[ˆpi,/hatwideH], because being a consequence of the commutation rela- tions it does not depend on representation d dt<f|ˆpi|g>=−i ¯h<f|[ˆp,/hatwideH]|g>. If now ˆpiand/hatwideHdoes not depend on time and taking into account the hermiticity one gets: (∂f ∂t,ˆpig) + (ˆpif,∂g ∂t) =−i ¯h(f,ˆpiˆHg) +i ¯h(f,ˆHˆpig) =−i ¯h(ˆpf,ˆHg) +i ¯h(ˆHf,ˆpig) (∂f ∂t+i ¯hˆHf,ˆpig) + (ˆpif,∂g ∂t−i ¯hˆHg) = 0 The latter relationship holds for any pair of functions f(x) and g(x) at the initial moment if each of them satisfy the equation i¯h∂ψ ∂t=Hψ. This is the Schr¨ odinger equation. It describes the system b y means of time-independent operators and makes up the so-cal led Schr¨ odinger reprezentation. In both reprezentations the temporal evolution of the syste m is char- acterized by the operator/hatwideH, which can be obtained from Hamilton’s function of classical mechanics. Exemple:/hatwideHfor a particle in a potential U(x1,x2,x3) we have: /hatwideH=ˆp2 2m+U(x1,x2,x3), and in the x reprezentation is: /hatwideH=−¯h2 2m∇2 x+U(x1,x2,x3). 8.- The connection between the S and H reprezentations P5is correct in both Schr¨ odinger’s reprezentation and Heise nberg’s. This is why, the mean value of any observable coincides in the two 13reprezentations. Thus, there is a unitary transformation t hat can be used for passing from one to the other. Such a transformation is of the form ˆs†= exp−iˆHt ¯h. In order to pass to the Schr¨ odinger reprezenta- tion one should use the Heisenberg transform ψ=ˆs†fwithfand ˆL, whereas to pass to Heisenberg’s reprezentation the Schr¨ o dinger transform ˆΛ =ˆs†ˆLˆswithψandˆΛ is of usage. One can obtain the Schr¨ odinger equation as follows: since in the transformat ionψ=ˆs†f the function fdoes not depend on time, we shall derivate the trans- formation with respect to time to get: ∂ψ ∂t=∂s† ∂tf=∂ ∂t(exp−i/hatwideHt ¯h)f=−i ¯h/hatwideHexp−i/hatwideHt ¯hf=−i ¯h/hatwideHˆs†f=−i ¯h/hatwideHψ. ¸ si deci, avem: i¯h∂ψ ∂t=/hatwideHψ. Next we get the Heisenberg equations: putting the Schr¨ odin ger trans- form in the form ˆ sˆΛˆs†=ˆLand performing the derivatives with respect to time one gets Heisenberg’s equation ∂ˆL ∂t=∂ˆs ∂tˆΛˆs†+ ˆsˆΛ∂ˆs† ∂t=i ¯h/hatwideHexpi/hatwideHt ¯hˆΛˆs†−i ¯hˆsˆλexp−iˆHt ¯h/hatwideH =i ¯h(/hatwideHˆsˆΛˆs†−ˆsˆΛˆs†/hatwideH) =i ¯h(/hatwideHˆL−ˆL/hatwideH) =i ¯h[/hatwideH,ˆL]. Thus, we have: ∂ˆL ∂t=i ¯h[/hatwideH,ˆL]. Moreover, Heisenberg’s equation can be written in the form: ∂ˆL ∂t=i ¯hˆs[/hatwideH,ˆΛ]ˆs†. 14ˆLis known as an integral of motion, which, ifd dt<ψ|ˆL|ψ >= 0, is characterized by the following commutators: [/hatwideH,ˆL] = 0, [/hatwideH,ˆΛ] = 0. 9.- Stationary states The states of a quantum system described by the eigenfunctio ns of/hatwideH are called stationary states and the corresponding set of ei genvalues is known as the energy spectrum of the system. In such cases, t he Schroedinger equation is: i¯h∂ψn ∂t=Enψn=/hatwideHψn. The solutions are of the form: ψn(x,t) = exp−iEnt ¯hφn(x). •The probability is the following: δ(x) =|ψn(x,t)|2=|exp−iEnt ¯hφn(x)|2 = expiEnt ¯hexp−iEnt ¯h|φn(x)|2=|φn(x)|2. Thus, the probability is constant in time. •In the stationary states, the mean value of any commutator of the form [/hatwideH,ˆA] is zero, where ˆAis an arbitrary operator: <n|/hatwideHˆA−ˆA/hatwideH|n>=<n|/hatwideHˆA|n>−<n|ˆA/hatwideH|n> =<n|EnˆA|n>−<n|ˆAEn|n> =En<n|ˆA|n>−En<n|ˆA|n>= 0. •The virial theorem in quantum mechanics - if/hatwideHis a hamiltonian operator of a particle in the field U(r), using ˆA= 1/2/summationtext3 i=1(ˆpiˆxi−ˆxiˆpi) one gets: 15<ψ|[ˆA,/hatwideH]|ψ>= 0 =<ψ|ˆA/hatwideH−/hatwideHˆA|ψ> =/summationtext3 i=1<ψ|ˆpiˆxi/hatwideH−/hatwideHˆpiˆxi|ψ> =/summationtext3 i=1<ψ|[/hatwideH,ˆxi]ˆpi+ ˆxi[/hatwideH,ˆpi]|ψ>. Using several times the commutators and ˆ pi=−i¯h∇i,ˆH= /hatwideT+U(r), one can get: <ψ|[ˆA,/hatwideH]|ψ>= 0 =−i¯h(2<ψ|/hatwideT|ψ>−<ψ|/vector r·∇U(r)|ψ>). This is the virial theorem. If the potential is U(r) =Uorn, then a form of the virial theorem similar to that in classical mech anics can be obtained with the only difference that it refers to mean values T=n 2U. •For a Hamiltonian/hatwideH=−¯h2 2m∇2+U(r) and [/vector r,H] =−i¯h m/vector p, calcu- lating the matrix elements one finds: (Ek−En)<n|/vector r|k>=i¯h m<n|ˆp|k>. 10.- The nonrelativistic probability current density The following integral: /integraltext|ψn(x)|2dx= 1, is the normalization of an eigenfunction of the discrete spe ctrum in the coordinate reprezentation. It appears as a condition on the micro- scopic motion in a finite region of space. For the wavefunctions of the continuous spectrum ψλ(x) one cannot give a direct probabilistic interpretation. Let us consider a given wavefunction φ∈L2, that we write as a linear combination of eigenfunctions of the continuum: 16φ=/integraltexta(λ)ψλ(x)dx. One says that φcorresponds to an infinite motion. In many cases, the function a(λ) is not zero only in a small neighbor- hood of a point λ=λo. In such a case, φis known as a wavepacket. We shall calculate now the rate of change of the probability o f finding the system in the volume Ω. P=/integraltext Ω|ψ(x,t)|2dx=/integraltext Ωψ∗(x,t)ψ(x,t)dx. Derivating of the integral with respect to time one finds: dP dt=/integraltext Ω(ψ∂ψ∗ ∂t+ψ∗∂ψ ∂t)dx. Using now the Schr¨ odinger equation in the integral of the ri ght hand side, one gets: dP dt=i ¯h/integraltext Ω(ψˆHψ∗−ψ∗ˆHψ)dx. Using the identity f∇2g−g∇2f=div[(f)grad(g)−(g)grad(f)] and also the Schr¨ odinger equation in the form: ˆHψ=¯h2 2m∇2ψ and subtituting in the integral, one gets: dP dt=i ¯h/integraltext Ω[ψ(−¯h2 2m∇ψ∗)−ψ∗(−¯h2 2m∇ψ)]dx =−/integraltext Ωi¯h 2m(ψ∇ψ∗−ψ∗∇ψ)dx =−/integraltext Ωdivi¯h 2m(ψ∇ψ∗−ψ∗∇ψ)dx. By means of the divergence theorem, the volume integral can b e trans- formed in a surface one leading to: 17dP dt=−/contintegraltexti¯h 2m(ψ∇ψ∗−ψ∗∇ψ)dx. The quantity /vectorJ(ψ) =i¯h 2m(ψ∇ψ∗−ψ∗∇ψ) is known as the probability density current, for which one can easily get the following c ontinuity equation dρ dt+div(/vectorJ) = 0. •Ifψ(x) =AR(x), where R(x) is a real function, then: /vectorJ(ψ) = 0. •For momentum eigenfunctions ψ(x) =1 (2π¯h)3/2expi/vector p/vector x ¯h, one gets: J(ψ) =i¯h 2m(1 (2π¯h)3/2expi/vector p/vector x ¯h(i/vector p ¯h(2π¯h)3/2exp−i/vector p/vector x ¯h) −(1 (2π¯h)3/2exp−i/vector p/vector x ¯hi/vector p ¯h(2π¯h)3/2expi¯h/vector p/vector x ¯h)) =i¯h 2m(−2i/vector p ¯h(2π¯h)3) =/vector p m(2π¯h)3, which shows that the probability density does not depend on t he coordinate. 11.- Operator of spatial transport If/hatwideHis invariant at translations of arbitrary vector /vector a, /hatwideH(/vector r+/vector a) =/hatwideH/vector(r) , then there is an operator/hatwideT(/vector a) which is unitary/hatwideT†(/vector a)/hatwideH(/vector r)/hatwideT(/vector a) = /hatwideH(/vector r+/vector a). Commutativity of translations /hatwideT(/vector a)/hatwideT(/vectorb) =/hatwideT(/vectorb)/hatwideT(/vector a) =/hatwideT(/vector a+/vectorb), implies that/hatwideTis of the form/hatwideT= expiˆka, where ˆk=ˆp ¯h. In the infinitezimal case: /hatwideT(δ/vector a)/hatwideH/hatwideT(δ/vector a)≈(ˆI+iˆkδ/vector a)/hatwideH(ˆI−iˆkδ/vector a), 18/hatwideH(/vector r) +i[ˆK,/hatwideH]δ/vector a=/hatwideH(/vector r) + (∇/hatwideH)δ/vector a. Moreover, [ˆ p,/hatwideH] = 0, where ˆ pis an integral of the motion. The sistem of wavefunctions of the form ψ(/vector p,/vector r) =1 (2π¯h)3/2expi/vector p/vector r ¯hand the unitary transformation leads to expi/vector p/vector a ¯hψ(/vector r) =ψ(/vector r+/vector a). The operator of spatial transport/hatwideT†= exp−i/vector p/vector a ¯his the analog of ˆ s†= exp−iˆHt ¯h, which is the operator of time ‘transport’ (shift). 12.- Exemple: The ‘crystal’ (lattice) Hamiltonian If/hatwideHis invariant for a discrete translation (for exemple, in a cr ystal lattice)/hatwideH(/vector r+/vector a) =/hatwideH(/vector r), where/vector a=/summationtext i/vector aini,ni∈Nandaiare baricentric vectors, then: /hatwideH(/vector r)ψ(/vector r) =Eψ(/vector r), /hatwideH(/vector r+/vector a)ψ(/vector r+/vector a) =Eψ(/vector r+/vector a) =ˆH(/vector r)ψ(/vector r+/vector a). Consequently, ψ(/vector r) andψ(/vector r+/vector a) are wavefunctions for the same eigen- value of/hatwideH. The relationship between ψ(/vector r) andψ(/vector r+/vector a) can be saught for in the form ψ(/vector r+/vector a) = ˆc(/vector a)ψ(/vector r), where ˆc(/vector a) is a gxg matrix (g is the order of degeneration of level E). Two column matrices, ˆ c(/vector a) and ˆc(/vectorb) commute and therefore they are diagonalizable simultaneo usly. Moreover, for the diagonal elements, cii(/vector a)cii(/vectorb) =cii(/vector a+/vectorb) holds for i=1,2,....,g, having solutions of the type cii(a) = expikia. Thus, ψk(/vector r) =Uk(/vector r)expi/vectork/vector a, where/vectorkis a real arbitrary vector and the func- tionUk(/vector r) is periodic of period /vector a,Uk(/vector r+/vector a) =Uk(/vector r). The assertion that the eigenfunctions of a periodic ˆHof the lattice type ˆH(/vector r+/vector a) =ˆH(/vector r) can be written ψk(/vector r) =Uk(/vector r)expi/vectork/vector a, where Uk(/vector r+/vector a) =Uk(/vector r) is known as Bloch’s theorem. In the continuous case,Ukshould be constant, because the constant is the only functio n periodic for any /vector a. The vector /vector p= ¯h/vectorkis called quasimomentum (by analogy with the continuous case). The vector /vectorkis not determined univoquely, because one can add any vector /vector gfor whichga= 2πn, wheren∈N. The vector /vector gcan be written /vector g=/summationtext3 i=1/vectorbimi, wheremiare integers and biare given by 19/vectorbi= 2πˆaj×/vector ak /vector ai(/vector aj×/vector ak), fori∝ne}ationslash=j∝ne}ationslash=k./vectorbiare the baricentric vectors of the lattice. Recommended references 1. E. Farhi, J. Goldstone, S. Gutmann, “How probability aris es in quantum mechanics”, Annals of Physics 192, 368-382 (1989) 2. N.K. Tyagi in Am. J. Phys. 31, 624 (1963) gives a very short proof of the Heisenberg uncertainty principle, which asserts that the s imultaneous mea- surement of two noncommuting hermitic operators results in an uncertainty given by the value of their commutator. 3. H.N. N´ u˜ nez-Y´ epez et al., “Simple quantum systems in th e momentum representation”, physics/0001030 (Europ. J. Phys., 2000) . 4. J.C. Garrison, “Quantum mechanics of periodic systems”, Am. J. Phys. 67, 196 (1999). 5. F. Gieres, “Dirac’s formalism and mathematical surprise s in quantum mechanics”, quant-ph/9907069 (in English); quant-ph/990 7070 (in French). 1N. Notes 1. For “the creation of quantum mechanics...”, Werner Heise nberg has been awarded the Nobel prize in 1932 (delivered in 1933). The pape r “Z¨ ur Quan- tenmechanik. II”, [“On quantum mechanics.II”, Zf. f. Physi k35, 557- 615 (1926) (received by the Editor on 16 November 1925) by M. B orn, W. Heisenberg and P. Jordan, is known as the “work of the three pe ople”, be- ing considered as the work that really opened the vast horizo ns of quantum mechanics. 2. For “the statistical interpretation of the wavefunction ” Max Born was awarded the Nobel prize in 1954. 1P. Problems Problema 1.1 : Let us consider two operators, A and B, which commutes by hypothesis. In this case, one can derive the following rel ationship: eAeB=e(A+B)e(1/2[A,B]). Solution 20Defining an operator F(t), as a function of real variable t, of the form: F(t) =e(At)e(Bt), then:dF dt=AeAteBt+eAtBeBt= (A+eAtBe−At)F(t). Applying now the formula [ A,F(B)] = [A,B]F′(B), we have [eAt,B] =t[A.B]eAt, and therefore: eAtB=BeAt+t[A,B]eAt. Multiplying both sides of the latter equation by exp−Atand substituting in the first equation, we get: dF dt= (A+B+t[A,B])F(t). The operators A , B and [A,B] commutes by hypothesis. Thus, we can integrate the differential equation as if A+Band [A,B] would be scalar numbers. We shall have: F(t) =F(0)e(A+B)t+1/2[A,B]t2. Puttingt= 0, one can see that F(0) = 1 and therefore : F(t) =e(A+B)t+1/2[A,B]t2. Putting now t= 1, we get the final result. Problem 1.2 : Calculate the commutator [ X,Dx]. Solution The calculation is performed by applying the commutator to a n arbitrary functionψ(/vector r): [X,Dx]ψ(/vector r) = (x∂ ∂x−∂ ∂xx)ψ(/vector r) =x∂ ∂xψ(/vector r)−∂ ∂x[xψ(/vector r)] =x∂ ∂xψ(/vector r)−ψ(/vector r)−x∂ ∂xψ(/vector r) =−ψ(/vector r). Since this relationship is satisfied for any ψ(/vector r), one can conclude that [ X,Dx] = −1. Problem 1.3 : Check that the trace of a matrix is invariant of changes of discrete orthonormalized bases. Solution The sum of the diagonal elements of a matrix representation o f an operator A in an arbitrary basis does not depend on the basis. 21This important property can be obtained by passing from an or thonormal- ized discrete basis |ui>to another orthonormalized discrete basis |tk>. We have:/summationtext i<ui|A|ui>=/summationtext i<ui|(/summationtext k|tk><tk|)A|ui> (where we have used the completeness relationship for the st atestk). The right hand side is: /summationtext i,j<ui|tk><tk|A|ui>=/summationtext i,j<tk|A|ui><ui|tk>, (the change of the order in the product of two scalar numbers i s allowed). Thus, we can replace/summationtext i|ui>< ui|with unity (i.e., the completeness relationship for the states |ui>), in order to get finally: /summationdisplay i<ui|A|ui>=/summationdisplay k<tk|A|tk> . Thus, we have proved the invariance property for matriceal t races. Problem 1.4 : If for the hermitic operator Nthere are the hermitic oper- atorsLandMsuch that : [ M,N] = 0, [L,N] = 0, [M,L]∝ne}ationslash= 0, then the eigenfunctions of Nare degenerate. Solution Letψ(x;µ,ν) be the common eigenfunctions of MandN(since they com- mute they are simultaneous observables). Let ψ(x;λ,ν) be the common eigenfunctions of LandN(again, since they commute they are simulta- neous observables). The Greek parameters denote the eigenv alues of the corresponding operators. Let us consider for simplicity sa ke thatNhas a discrete spectrum. Then: f(x) =/summationdisplay νaνψ(x;µ,ν) =/summationdisplay νbνψ(x;λ,ν). We calculate now the matrix element <f|ML|f >: <f|ML|f >=/integraldisplay/summationdisplay νµνaνψ∗(x;µ,ν)/summationdisplay ν′λν′bν′ψ(x;λ,ν′)dx . If all the eigenfunctions of Nare nondegenerate then <f|ML|f >=/summationtext νµνaνλνbν. But the same result can be obtained if one calculates <f|LM|f >and the commutator would be zero. Thus, at least some of the eigenfun ctions ofN should be degenerate. 222. ONE DIMENSIONAL RECTANGULAR BARRIERS AND WELLS Regions of constant potential In the case of a rectangular potential, V(x) is a constant function V(x) =Vin a certain region of the one-dimensional space. In such a region, the Schr¨ odinger eq. can be written: d2 dx2ψ(x) +2m ¯h2(E−V)ψ(x) = 0 (1) One can distinguish several cases: (i)E >V Let us introduce the positive constant k, defined by k=/radicalbig 2m(E−V) ¯h(2) Then, the solution of eq. (1) can be written: ψ(x) =Aeikx+A′e−ikx(3) whereAandA′are complex constants. (ii)E <V This condition corresponds to segments of the real axis whic h would be prohibited to any particle from the viewpoint of classical m echanics. In this case, one introduces the positive constant qdefined by: q=/radicalbig 2m(V−E) ¯h(4) and the solution of (1) can be written: ψ(x) =Beqx+B′e−qx, (5) whereBandB′are complex constants. (iii)E=V In this special case, ψ(x) is a linear function of x. 23The behaviour of ψ(x)at a discontinuity of the potential One might think that at the point x=x1, where the potential V(x) is discontinuous, the wavefunction ψ(x) behaves in a more strange way, maybe discontinuously for example. This is not so: ψ(x) anddψ dxare continuous, and only the second derivative is discontinuous at x=x1. General look to the calculations The procedure to determine the stationary states in rectang ular poten- tials is the following: in all regions in which V(x) is constant we write ψ(x) in any of the two forms (3) or (5) depending on application; ne xt, we join smoothly these functions according to the continuity condi tions forψ(x) and dψ dxat the points where V(x) is discontinuous. Examination of several simple cases Let us make explicite calculations for some simple stationa ry states according to the proposed method. The step potential xV(x) V0 0III Fig. 2.1 24a.E >V 0case;partial reflexion Let us put eq. (2) in the form: k1=√ 2mE ¯h(6) k2=/radicalbig 2m(E−V0) ¯h(7) The solution of eq. (1) has the form of eq. (3) in the regions I(x <0) andII(x>0): ψI=A1eik1x+A′ 1e−ik1x ψII=A2eik2x+A′ 2e−ik2x In region I eq. (1) takes the form ψ′′(x) +2mE ¯h2ψ(x) =ψ′′(x) +k2ψ(x) = 0 and in the region II: ψ′′(x)−2m ¯h2[V0−E]φ(x) =ψ′′(x)−q2ψ(x) = 0 If we limit ourselves to the case of an incident particle ‘com ing’ fromx= −∞, we have to choose A′ 2= 0 and we can determine the ratios A′ 1/A1and A2/A1. The joining conditions give then: •ψI=ψII, atx= 0 : A1+A′ 1=A2 (8) •ψ′ I=ψ′ II, atx= 0 : A1ik1−A′ 1ik1=A2ik2 (9) Substituting A1andA′ 1from (8) in (9): A′ 1=A2(k1−k2) 2k1(10) A1=A2(k1+k2) 2k1(11) 25From the two expressions of the constant A2in (10) and (11) one gets A′ 1 A1=k1−k2 k1+k2(12) and from (11) it follows: A2 A1=2k1 k1+k2. (13) ψ(x) is a superposition of two waves. The first (the A1part) corresponds to an incident wave of momentum p= ¯hk1, propagating from the left to the right. The second (the A′ 1part) corresponds to a reflected particle of momentum−¯hk1propagating in opposite direction. Since we have already chosenA′ 2= 0, it follows that ψII(x) contains a single wave, which is as- sociated to a transmitted particle. (We will show later how i t is possible by employing the concept of probability current to define the transmission coefficient T as well as the reflection coefficient R for the step p otential). These coefficients give the probability that a particle comin g fromx=−∞ can pass through or get back from the step at x= 0. Thus, we obtain: R=|A′ 1 A1|2, (14) whereas for T: T=k2 k1|A2 A1|2. (15) Taking into account (12) and (13) one is led to: R= 1−4k1k2 (k1+k2)2(16) T=4k1k2 (k1+k2)2. (17) It is easy to check that R+T= 1. It is thus sure that the particle will be either transmitted or reflected. Contrary to the predicti ons of classical mechanics, the incident particle has a nonzero probability of not going back. It is also easy to check using (6), (7) and (17), that if E≫V0then T≃1: when the energy of the particle is sufficently big in compari son with the height of the step, everything happens as if the step does not exist for the particle. 26Consider the following natural form of the solution in regio n I: ψI=A1eik1x+Ae−ik1x j=−i¯h 2m(φ∗▽φ−φ▽φ∗) (18) withA1eik1xand its conjugate A∗ 1e−ik1x: j=−i¯h 2m[(A∗ 1e−ik1x)(A1ik1eik1x)−(A1eik1x)(−A∗ 1ik1e−ik1x)] j=¯hk1 m|A1|2. Now withAe−ik1xand its conjugate A∗eik1xone is led to: j=−¯hk1 m|A|2. In the following we wish to check the proportion of reflected c urrent with respect to the incident current (or more exactly, we wan t to check the relative probability that the particle is returned back): R=|j(φ−)| |j(φ+)|=|−¯hk1 m|A|2| |¯hk1 m|A1|2|=|A A1|2. (19) Similarly, the proportion of transmission with respect to i ncidence (that is the probability that the particle is transmitted) is, tak ing now into account the solution in the region II: T=|¯hk2 m|A2|2| |¯hk1 m|A1|2|=k2 k1|A2 A1|2. (20) b.E <V 0case;total reflection In this case we have: k1=√ 2mE ¯h(21) q2=/radicalbig 2m(V0−E) ¯h(22) 27In the region I(x<0), the solution of eq. (1) [written as ψ(x)′′+k2 1ψ(x) = 0] has the form given in eq. (3): ψI=A1eik1x+A′ 1e−ik1x, (23) whereas in the region II(x>0), the same eq. (1) [now written as ψ(x)′′− q2 2ψ(x) = 0] has the form of eq. (5): ψII=B2eq2x+B′ 2e−q2x. (24) In order that the solution be kept finite when x→+∞, it necessary that: B2= 0. (25) The joining condition at x= 0 give now: •ψI=ψII, atx= 0 : A1+A′ 1=B′ 2 (26) •ψ′ I=ψ′ II, atx= 0 : A1ik1−A′ 1ik1=−B′ 2q2. (27) Substituting A1andA′ 1from (26) in (27) we get: A′ 1=B′ 2(ik1+q2) 2ik1(28) A1=B′ 2(ik1−q2) 2ik1. (29) Equating the expressions for the constant B′ 2from (28) and (29) leads to: A′ 1 A1=ik1+q2 ik1−q2=k1−iq2 k1+iq2, (30) so that from (29) we have: B′ 2 A1=2ik1 ik1−q2=2k1 k1−iq2. (31) 28Therefore, the reflection coefficient Ris: R=|A′ 1 A1|2=|k1−iq2 k1+iq2|2=k2 1+q2 2 k2 1+q2 2= 1. (32) As in classical mechanics, the microparticle is always refle cted (total re- flexion). However, there is an important difference, namely, because of the existence of the so-called evanescent wave e−q2x, the particle has a nonzero probability to find itself in a spatial region which is classi caly forbidden. This probability decays exponentially with xand turns to be negligible when x overcome 1/q2corresponding to the evanescent wave. Notice also that A′ 1/A1 is a complex quantity. A phase difference occurs as a conseque nce of the reflexion, which physically is due to the fact that the partic le is slowed down when entering the region x>0. There is no analog phenomenon for this in classical mechanics (but there is of course such an analog in optical physics). Rectangular barrier 0lxV(x) V0 IIIIII Fig. 2.2 a.E >V 0case;resonances Here we put eq. (2) in the form: k1=√ 2mE ¯h(33) k2=/radicalbig 2m(E−V0) ¯h. (34) 29The solution of eq. (1) is as in eq. (3) in the regions I(x <0),II(0< x<a) andIII(x>a) : ψI=A1eik1x+A′ 1e−ik1x ψII=A2eik2x+A′ 2e−ik2x ψIII=A3eik1x+A′ 3e−ik1x. If we limit ourselves to the case of an incident particle comi ng from x=−∞, we have to choose A′ 3= 0. •ψI=ψII, atx= 0 : A1+A′ 1=A2+A′ 2 (35) •ψ′ I=ψ′ II, atx= 0 : A1ik1−A′ 1ik1=A2ik2−A′ 2ik2 (36) •ψII=ψIII, atx=a: A2eik2a+A′ 2e−ik2a=A3eik1a(37) •ψ′ II=ψ′ III, atx=a: A2ik2eik2a−A′ 2ik2e−ik2a=A3ik1eik1a. (38) The joining conditions at x=agiveA2andA′ 2as functions of A3, whereas those atx= 0 giveA1andA′ 1as functions of A2andA′ 2(thus, as functions ofA3). This procedure is shown in detail in the following. Substituting A′ 2from eq. (37) in (38) leads to: A2=A3eik1a(k2+k1) 2k2eik2a. (39) Substituting A2from eq. (37) in (38) leads to: A′ 2=A3eik1a(k2−k1) 2k2e−ik2a. (40) Substituting A1from eq. (35) in (36) leads to: A′ 1=A2(k2−k1)−A′ 2(k2+k1) −2k1. (41) 30Substituting A′ 1from eq. (35) in (36) gives: A1=A2(k2+k1)−A′ 2(k2−k1) 2k1. (42) Now, substituting the eqs. (39) and (40) in (41), we have: A′ 1=i(k2 2−k2 1) 2k1k2(sink2a)eik1aA3. (43) Finally, substituting the eqs. (39) and (40) in (42) we get: A1= [cosk2a−ik2 1+k2 2 2k1k2sink2a]eik1aA3. (44) A′ 1/A1andA3/A1[these ratios can be obtained by equating (43) and (44), and by separating, respectively, in eq. (44)] allow the calc ulation of the reflexion coefficient Ras well as of the transmission one T. For this type of barrier, they are given by the following formulas: R=|A′ 1/A1|2=(k2 1−k2 2)2sin2k2a 4k2 1k2 2+ (k2 1−k2 2)2sin2k2a, (45) T=|A3/A1|2=4k2 1k2 2 4k2 1k2 2+ (k2 1−k2 2)2sin2k2a. (46) It is easy to see that they check R+T= 1. b.E <V 0case;the tunnel effect Now, let us take the eqs. (2) and (4): k1=√ 2mE ¯h(47) q2=/radicalbig 2m(V0−E) ¯h. (48) The solution of eq. (1) has the form given in eq. (3) in the regi onsI(x< 0) andIII(x>a), while in the region II(0<x<a ) has the form of eq. (5): ψI=A1eik1x+A′ 1e−ik1x ψII=B2eq2x+B′ 2e−q2x ψIII=A3eik1x+A′ 3e−ik1x. The joining conditions at x= 0 andx=aallow the calculation of the transmission coefficient of the barrier. As a matter of fact, i t is not necessary to repeat the calculation: merely, it is sufficient to replace k2by−iq2in the equation obtained in the first case of this section. 31Bound states in rectangular well a. Well of finite depth x aV(x) V0 Fig. 2.3 Finite r ectangular well We first study the case 0 <E <V 0(E >V 0is similar to the calculation in the previous section). For the exterior regions I, ( x<0) and III, ( x>a) we employ eq. (4): q=/radicalbig 2m(V0−E) ¯h. (49) For the central region II (0 <x<a ) we use eq. (2): k=/radicalbig 2m(E) ¯h. (50) The solution of eq. (1) has the form of eq. (5) in the exterior r egions and of eq. (3) in the central region: ψI=B1eqx+B′ 1e−qx 32ψII=A2eikx+A′ 2e−ikx ψIII=B3eqx+B′ 3e−qx In the region (0 <x<a ) eq. (1) has the form: ψ(x)′′+2mE ¯h2ψ(x) =ψ(x)′′+k2ψ(x) = 0 (51) while in the exterior regions: ψ(x)′′−2m ¯h2[V0−E]φ(x) =ψ(x)′′−q2ψ(x) = 0. (52) Becauseψshould be finite in the region I, we impose: B′ 1= 0. (53) The joining conditions give: ψI=ψII, atx= 0 : B1=A2+A′ 2 (54) ψ′ I=ψ′ II, atx= 0 : B1q=A2ik−A′ 2ik (55) ψII=ψIII, atx=a: A2eika+A′ 2e−ika=B3eqa+B′ 3e−qa(56) ψ′ II=ψ′ III, atx=a: A2ikeika−A′ 2ike−ika=B3qeqa−B′ 3qe−qa(57) Substituting the constants A2andA′ 2from eq. (54) in eq. (55) we get A′ 2=B1(q−ik) −2ik A2=B1(q+ik) 2ik, (58) respectively. 33Substituting the constant A2and the constant A′ 2from eq. (56) in eq. (57) we get B′ 3e−qa(ik+q) +B3eqa(ik−q) +A′ 2e−ika(−2ik) = 0 2ikA2eika+B′ 3e−qa(−ik+q) +B3Eqa(−ik−q) = 0, (59) respectively. EquatingB′ 3from eqs. (59) and taking into account the eqs (58) leads to B3 B1=e−qa 4ikq[eika(q+ik)2−e−ika(q−ik)2]. (60) Sinceψ(x) should be finite in region III as well, we require B3= 0. Thus [q−ik q+ik]2=eika e−ika=e2ika. (61) Becauseqandkdepend onE, eq. (1) can be satisfied for some particular values ofE. The condition that ψ(x) should be finite in all spatial regions imposes the quantization of the energy. Two cases are possib le: (i)if q−ik q+ik=−eika, (62) equating in both sides the real and the imaginary parts, resp ectively, we have tan(ka 2) =q k. (63) Putting k0=/radicaligg 2mV0 ¯h=/radicalig k2+q2 (64) one gets 1 cos2(ka 2)= 1 + tan2(ka 2) =k2+q2 k2= (k0 k)2(65) Eq. (63) is therefore equivalent to the system of eqs. |cos(ka 2)|=k k0 tan(ka 2)>0 (66) 34The energy levels are determined by the intersection of a str aight line of slope1 k0with the first set of cosinusoides ˆ ıntrerupte in fig. 2.4. Thu s, we get a certain number of energy levels whose wavefunctions are ev en. This fact becomes clearer if we substitute (62) in (58) and (60). It is e asy to check thatB′ 3=B1andA2=A′ 2leading toψ(−x) =ψ(x). (ii)if q−ik q+ik=eika, (67) a similar calculation gives |sin(ka 2)|=k k0 tan(ka 2)<0. (68) The energy levels are in this case determined by the intersec tion of the same straight line with the second set of dashed cosinusoide s in fig. 2.4. The obtained levels are interlaced with those found in the ca se (i). One can easily show that the corresponding wavefunctions are odd. ky 0π π π π /a23/a/a/aPIP 4I k0 Fig. 2.4 b. Well of infinite depth 35In this case it is convenient to put V(x) to zero for 0 <x<a and to infinity for the rest of the real axis. Putting k=/radicaligg 2mE ¯h2, (69) ψ(x) should be zero outside the interval [0 ,a] and continuous at x= 0 and x=a. For 0≤x≤a: ψ(x) =Aeikx+A′e−ikx. (70) Sinceψ(0) = 0, one can infer that A′=−A, leading to: ψ(x) = 2iAsin(kx). (71) Moreover,ψ(a) = 0 and therefore k=nπ a, (72) wherenis an arbitrary positive integer. If we normalize the functi on (71), taking into account (72), then we obtain the stationary wave functions ψn(x) =/radicalbigg2 asin(nπx a) (73) with the energies En=n2π2¯h2 2ma2. (74) The quantization of the energy levels is extremely simple in this case. The stationary energies are proportional with the natural numb ers squared. 2P. Problems Problem 2.1: The attractive δpotential Suppose we have a potential of the form: V(x) =−V0δ(x);V0>0;x∈ℜ. 36The corresponding wavefunction ψ(x) is assumed continuous. a) Obtain the bound states ( E <0), if they exist, localized in this type of potential. b) Calculate the dispersion of a plane wave falling on the δpotential and obtain the reflexion coefficient R=|ψrefl|2 |ψinc|2|x=0, whereψrefl,ψincare the reflected and incoming waves, respectively. Suggestion : To determine the behavior of ψ(x) in x=0, it is better to proceed by integrating the Schr¨ odinger equation in the interval ( −ε,+ε), and then to apply the limit ε→0. Solution. a) The Schr¨ odinger eq. is: d2 dx2ψ(x) +2m ¯h2(E+V0δ(x))ψ(x) = 0. (75) Far from the origin we have a differential eq. of the form d2 dx2ψ(x) =−2mE ¯h2ψ(x). (76) Consequently, the wavefunctions are of the form ψ(x) =Ae−qx+Beqxforx>0 and x<0,(77) whereq=/radicalig −2mE/¯h2∈ℜ.Since|ψ|2should beL2integrable , we cannot accept that a part growths exponentially. Moreover, the wav efunction should be continuous at the origin. With these conditions, we have ψ(x) =Aeqx; (x<0), ψ(x) =Ae−qx; (x>0). (78) Integrating the Schr¨ odinger eq. between −εand +ε, we get −¯h2 2m[ψ′(ε)−ψ′(−ε)]−V0ψ(0) =E/integraldisplay+ε −εψ(x)dx≈2εEψ(0) (79) Introducing now the result (78) and taking into account the l imitε→0, we have −¯h2 2m(−qA−qA)−V0A= 0, (80) 37orE=−m(V2 0/2¯h2) [−V2 0 4in units of¯h2 2m]. Clearly, there is a single discrete energy. The normalization constant is found to be A=/radicalig mV0/¯h2. The wavefunction of the bound state will be ψo=AeV0|x|/2, whereV0is in¯h2 2m units. b) Take now the wavefunction of a plane wave ψ(x) =Aeikx, k2=2mE ¯h2. (81) It moves from the left to the right and is reflected by the poten tial. IfBand Care the amplitudes of the reflected and transmitted waves, re spectively, then we have ψ(x) =Aeikx+Be−ikx; (x<0), ψ(x) =Ceikx; ( x>0). (82) The joining conditions and the relationship ψ′(ε)−ψ′(−ε) =−fψ(0) cu f= 2mV0/¯h2lead to A+B=C B =−f f+ 2ikA, ik(C−A+B) =−fC C =2ik f+ 2ikA. (83) The reflection coefficient will be R=|ψrefl|2 |ψinc|2|x=0=|B|2 |A|2=m2V2 0 m2V2 0+ ¯h4k2. (84) If the potential is very strong ( V0→∞), one can see that R→1, i.e., the wave is totally reflected. The transmission coefficient , on the other hand, will be T=|ψtrans|2 |ψinc|2|x=0=|C|2 |A|2=¯h4k2 m2V2 0+ ¯h4k2. (85) Again, if the potential is very strong ( V0→∞) thenT→0,i.e., the trans- mitted wave fades rapidly on the other side of the potential. In addition, R+T= 1 as expected, which is a check of the calculation. 38Problem 2.2: Particle in a 1D potential well of finite depth Solve the 1D Schr¨ odinger eq. for a finite depth potential wel l given by V(x) =/braceleftigg −V0dac˘ a|x|≤a 0 dac˘ a|x|>a. Consider only the bound spectrum ( E <0). EV −V0−a +ax Fig. 2.5 Solution. a) The wavefunction for |x|<aand|x|>a. The corresponding Schr¨ odinger eq. is −¯h2 2mψ′′(x) +V(x)ψ(x) =Eψ(x). (86) Defining q2=−2mE ¯h2, k2=2m(E+V0) ¯h2, (87) 39we get: 1) for x<−a :ψ′′ 1(x)−q2ψ1= 0, ψ1=A1eqx+B1e−qx; 2) for−a≤x≤a :ψ′′ 2(x) +k2ψ2= 0, ψ2=A2cos(kx) +B2sin(kx); 3) for x>a :ψ′′ 3(x)−q2ψ3= 0, ψ3=B3eqx+B3e−qx. b) Formulation of the boundary conditions. The normalization of the bound states requires solutions go ing to zero at infinity. This means that B1=A3= 0. Moreover, ψ(x) should be continu- ously differentiable. All the particular solutions are fixed in such a way that ψandψ′are continuous for that value of x corresponding to the bound ary between the interior and exterior regions. The second deriv ativeψ′′displays the discontinuity that the ‘box’ potential imposes. Thus we are led to: ψ1(−a) =ψ2(−a), ψ 2(a) =ψ3(a), ψ′ 1(−a) =ψ′ 2(−a), ψ′ 2(a) =ψ′ 3(a). (88) c) The eigenvalue equations. From (88) we get four linear and homogeneous eqs for the coeffic ients A1,A2,B2andB3: A1e−qa=A2cos(ka)−B2sin(ka), qA1e−qa=A2ksin(ka) +B2kcos(ka), B3e−qa=A2cos(ka) +B2sin(ka), −qB3e−qa=−A2ksin(ka) +B2kcos(ka). (89) Adding and subtracting one gets a system of eqs. which is easi er to solve: (A1+B3)e−qa= 2A2cos(ka) q(A1+B3)e−qa= 2A2ksin(ka) (A1−B3)e−qa=−2B2sin(ka) q(A1−B3)e−qa= 2B2kcos(ka). (90) AssumingA1+B3∝ne}ationslash= 0 andA2∝ne}ationslash= 0, the first two eqs give q=ktan(ka), (91) which inserted in the last two eqs gives A1=B3;B2= 0. (92) 40The result is the symmetric solution ψ(x) =ψ(−x), also called of positive parity. A similar calculation for A1−B3∝ne}ationslash= 0 andB2∝ne}ationslash= 0 leads to q=−kcot(ka)y A 1=−B3;A2= 0. (93) The obtained wavefunction is antisymmetric, correspondin g to a negative parity d) Quantitative solution of the eigenvalue problem. The equation connecting qandk, already obtained previously, gives the condition to get the eigenvalues. Using the notation ξ=ka, η =qa, (94) from the definition (87) we get ξ2+η2=2mV0a2 ¯h2=r2. (95) On the other hand, using (91) and (93) we get the equations η=ξtan(ξ), η =−ξcot(ξ). Thus, the sought energy eigenvalues can be obtained from the intersections of these two curves with the circle defined by (95) in the plane ξ-η(see fig. 2.6). 1 32 4η ξη ξη = −ξ cot ξξ2 +η2=r2 η = ξ tan ξξ2+ η =2r2 Fig. 2.6 There is at least one solution for arbitrary values of the par ameterV0, in the positive parity case, because the tangent function pa sses through the 41origin. For the negative parity, the radius of the circle sho uld be greater than a certain lower bound for the two curves to intersect. Thus, t he potential should have a certain depth related to a given spatial scale aand a given mass scalem, to allow for negative parity solutions. The number of energ y levels grows with V0,a, andm. For the case in which mVa2→∞, the intersections are obtained from tan(ka) =∞ −→ ka=2n−1 2π, −cot(ka) =∞ −→ ka=nπ, (96) wheren= 1,2,3, ...; by combining the previous relations k(2a) =nπ. (97) For the energy spectrum this fact means that En=¯h2 2m(nπ 2a)2−V0. (98) Widening the well and/or the mass of the particle m, the diference between two neighbourhood eigenvalues will decrease. The lowest le vel (n= 1) is not localized at−V0, but slightly upper. This ‘small’ difference is called zero point energy . e) The forms of the wavefunctions are shown in fig. 2.7. 1 32 4 x xψψ Fig. 2.7: Shapes of wave functions 42Problem 2.3: Particle in 1D rectangular well of infinite dept h Solve the 1D Schr¨ odinger eq. for a particle in a potential we ll of infinite depth as given by: V(x) =/braceleftigg 0 forx′<x<x′+ 2a ∞forx′≥xox≥x′+ 2a. The solution in its general form is ψ(x) =Asin(kx) +Bcos(kx), (99) where k=/radicaligg 2mE ¯h2. (100) Sinceψshould fulfill ψ(x′) =ψ(x′+ 2a) = 0, we get: Asin(kx′) +Bcos(kx′) = 0 (101) Asin[k(x′+ 2a)] +Bcos[k(x′+ 2a)] = 0. (102) Multiplying (101) by sin[ k(x′+2a)] and (102) by sin( kx′) and next subtract- ing the latter result from the first we get: B[ cos(kx′)sin[k(x′+ 2a)]−cos[k(x′+ 2a)]sin(kx′) ] = 0,(103) and by means of a trigonometric identity: Bsin(2ak) = 0 (104) Multiplying (101) by cos[ k(x′+ 2a)] and subtracting (102) multiplied by cos(kx′) leads to: A[ sin(kx′)cos[k(x′+ 2a)]−sin[k(x′+ 2a)]cos(kx′) ] = 0,(105) and by means of the same trigonometric identity: Asin[k(−2ak)] =Asin[k(2ak)] = 0. (106) Since we do not take into account the trivial solution ψ= 0, using (104) and (106) one has sin(2 ak) = 0 that takes place only if 2 ak=nπ, withnan 43integer. Accordingly, k=nπ/2aand sincek2= 2mE/¯h2then it comes out that the eigenvalues are given by the following expression: E=¯h2π2n2 8a2m. (107) The energy is quantized because only for each kn=nπ/2aone gets a well- defined energy En= [n2/2m][π¯h/2a]2. The general form of the solution is: ψn=Asin(nπx 2a) +Bcos(nπx 2a), (108) and it can be normalized 1 =/integraldisplayx′+2a x′ψψ∗dx=a(A2+B2), (109) wherefrom: A=±/radicalig 1/a−B2. (110) Substituting this value of Ain (101) one gets: B=∓1√asin(nπx′ 2a), (111) and plugging Bin (110) we get A=±1√acos(nπx′ 2a). (112) Using the upper signs for AandB, by substituting their values in (108) we obtain: ψn=1√asin(nπ 2a)(x−x′). (113) Using the lower signs for AandB, one gets ψn=−1√asin(nπ 2a)(x−x′). (114) 443. ANGULAR MOMENTUM AND SPIN Introduction It is known from Classical Mechanics that the angular momentum lfor macroscopic particles is given by l=r×p, (1) whererandpare the radius vector and the linear momentum, respectively . However, in Quantum Mechanics , one can find operators of angular mo- mentum type (OOAMT), which are not compulsory expressed onl y in terms of the coordinate ˆ xjand the momentum ˆ pkand acting only on the eigenfunc- tions in the x reprezentation. Consequently, it is very impo rtant to settle first of all general commutation relations for the OOAMT comp onents. InQuantum Mechanics lis expressed by the operator l=−i¯hr×∇, (2) whose components are operators satisfying the following co mmutation rules [lx,ly] =ilz,[ly,lz] =ilx,[lz,lx] =ily. (3) Moreover, each of the components commutes with the square of the angular momentum, i.e. l2=l2 x+l2 y+l2 z,[li,l2] = 0, i = 1,2,3. (4) These relations, besides being correct for the angular mome ntum, are ful- filled for the important OOAMT class of spin operators, which miss exact analogs in classical mechanics . These commutation relations are fundamental for getting th e spectra of the aforementioned operators as well as for their differential r eprezentations. The angular momentum For an arbitrary point of a fixed space (FS), one can introduce a function ψ(x,y,z), for which let’s consider two cartesian systems Σ and Σ′, where Σ′ is obtained by the rotation of the zaxis of Σ. 45In the general case, a FS refers to a coordinate system, which is different of Σ and Σ′. Now, let’s compare the values of ψat two points of the FS with the same coordinates (x,y,z) in Σ and Σ′, which is equivalent to the vectorial rotation ψ(x′,y′,z′) =Rψ(x,y,z) (5) whereRis a rotation matrix in R3  x′ y′ z′ = cosφ−sinφ0 sinφcosφ0 0 0 z  x y z . (6) Then Rψ(x,y,z) =ψ(xcosφ−ysinφ,xsinφ+ycosφ,z). (7) On the other hand, it is important to recall that the wavefunc tions do not depend on the coordinate system and that the transformation at rotations within the FS is achieved by means of unitary operators. Thus , to determine the form of the unitary operator U†(φ) that passes ψtoψ′, one usually considers an infinitesimal rotation dφ, keeping only the linear terms in dφ when one expands ψ′in Taylor series in the neighborhood of x ψ(x′,y′,z′)≈ψ(x+ydφ,xdφ +y,z), ≈ψ(x,y,z) +dφ/parenleftbigg y∂ψ ∂x−x∂ψ ∂y/parenrightbigg , ≈(1−idφlz)ψ(x,y,z), (8) where we have used the notation1 lz= ¯h−1(ˆxˆpy−ˆyˆpx). (9) As one will see later, this corresponds to the projection ope rator ontozof the angular momentum according to the definition (2) unless t he factor ¯h−1. In this way, the rotations of finite angle φcan be represented as exponentials of the form ψ(x′,y′,z) =eilzφψ(x,y,z), (10) where ˆU†(φ) =eilzφ. (11) 1The proof of (8) is displayed as problem 3.1 46In order to reassert the concept of rotation, we will conside r it in a more general approach with the help of the vectorial operatorˆ/vectorAacting onψ, assuming that ˆAx,ˆAy,ˆAzhave the same form in Σ and Σ′, that is, the mean values ofˆ/vectorAas calculated in Σ and Σ′should be equal when they are seen from the FS /integraldisplay ψ∗(/vector r′)(ˆAxˆı′+ˆAyˆ′+ˆAzˆk′)ψ∗(/vector r′)d/vector r =/integraldisplay ψ∗(/vector r)(ˆAxˆı+ˆAyˆ+ˆAzˆk)ψ∗(/vector r)d/vector r, (12) where ˆı′= ˆıcosφ+ ˆsinφ, ˆ′= ˆısinφ+ ˆcosφ, ˆk′=ˆk. (13) Thus, by combining (10), (12) and (13) we get eilzφˆAxe−ilzφ=ˆAxcosφ−ˆAysinφ, eilzφˆAye−ilzφ=ˆAxsinφ−ˆAycosφ, eilzφˆAze−ilzφ=ˆAz. (14) Again, considering infinitesimal rotations and expanding t he left hand sides in (14), one can determine the commutation relations o fˆAx,ˆAyand ˆAzwithˆlz [lz,Ax] =iAy,[lz,Ay] =−iAx,[lz,Az] = 0, (15) and similarly for lxandly. The basic conditions to obtain these commutation relations are ⋆The eigenfunctions transform as in (7) when Σ →Σ′. ⋆The components ˆAx,ˆAy,ˆAzhave the same form in Σ and Σ′. ⋆The kets corresponding to the mean values of ˆAin Σ and Σ′coincide (are the same) for a FS observer. One can also use another representation in which ψ(x,y,z) does not change when Σ→Σ′and the vectorial operators transform as ordinary vectors. In order to pass to such a representation when we rot ate byφ aroundzone makes use of the operator ˆU(φ), that is 47eilzφψ′(x,y,z) =ψ(x,y,z), (16) and therefore e−ilzφˆ/vectorAeilzφ=ˆ/vectorA. (17) Using the relationships (14) we obtain ˆA′ x=ˆAxcosφ+ˆAysinφ=e−ilzφˆAxeilzφ, ˆA′ y=−ˆAxsinφ+ˆAycosφ=e−ilzφˆAyeilzφ, ˆA′ z=e−ilzφˆAzeilzφ. (18) Since the transformations of the new reprezentation are per formed by means of unitary operators, the commutation relations do no t change. Remarks ⋆The operator ˆA2is invariant at rotations, that is e−ilzφˆA2eilzφ=ˆA′2=ˆA2. (19) ⋆It follows that [ˆli,ˆA2] = 0. (20) ⋆If the Hamiltonian operator is of the form ˆH=1 2mˆp2+U(|/vector r|), (21) then it remains invariant under rotations in any axis passin g through the coordinate origin [ˆli,ˆH] = 0, (22) where ˆliare integrals of the motion. Definition IfˆAiare the components of a vectorial operator acting on a wavefu nction depending only on the coordinates and if there are operators ˆlithat satisfy the following commutation relations [ˆli,ˆAj] =iεijkˆAk,[ˆli,ˆlj] =iεijkˆlk, (23) 48thenˆliare known as the components of the angular momentum operator and we can infer from (20) and (23) that [ˆli,ˆl2] = 0. (24) Consequently the three operatorial components associated to the com- ponents of a classical angular momentum satisfy commutatio n relations of the type (23), (24). Moreover, one can prove that these relat ions lead to specific geometric properties of the rotations in a 3D euclid ean space. This takes place if we adopt a more general point of view by defining an angular momentum operator J(we shall not use the hat symbol for simplicity of writing) as any set of three observables Jx,Jy¸ siJzwhich fulfill the com- mutation relations [Ji,Jj] =iεijkJk. (25) Moreover, let us introduce the operator J2=J2 x+J2 y+J2 z, (26) the scalar square of the angular momentum J. This operator is hermitic becauseJx,JyandJzare hermitic and it is easy to show that J2commutes with the three components of J [J2,Ji] = 0. (27) SinceJ2commutes with each of the components it follows that there is a complete system of eigenfunctions, i.e. J2ψγµ=f(γ2)ψγµ, Jiψγµ=g(µ)ψγµ, (28) where, as it will be shown in the following, the eigenfunctio ns depend on two subindices, which will be determined together with the form of the functions f(γ) andg(µ). The operators JiandJk(i∝ne}ationslash=k) do not commute, i.e. they do not have common eigenfunctions. For physical and mathemati cal reasons, we are interested to determine the common eigenfunctions of J2andJz, that is, we shall take i=zin (28). Instead of using the components JxandJyof the angular momentum J, it is more convenient to work with the following linear combi nations J+=Jx+iJy, J −=Jx−iJy. (29) 49Contrary to the operators aanda†of the harmonic oscillator (see chapter 5), these operators are not hermitic, they are only adjunct t o each other. The following properties are easy to prove [Jz,J±] =±J±,[J+,J−] = 2Jz, (30) [J2,J+] = [J2,J−] = [J2,Jz] = 0. (31) Jz(J±ψγµ) ={J±Jz+ [Jz,J±]}ψγµ= (µ±1)(J±ψγµ). (32) ThereforeJ±ψγµare eigenfunctions of Jzcorresponding to the eigenval- uesµ±1, that is these functions are identical up to the constant fa ctorsαµ andβµ(to be determined) J+ψγµ−1=αµψγµ, J−ψγµ=βµψγµ−1. (33) On the other hand α∗ µ= (J+ψγµ−1,ψγµ) = (ψγµ−1J−ψγµ) =βµ. (34) Therefore, taking a phase of the type eia(whereais real) for the function ψγµone can put αµreal and equal to βµ, which means J+ψγ,µ−1=αµψγµ,J−ψγµ=αµψγ,µ−1, (35) and therefore γ= (ψγµ,[J2 x+J2 y+J2 z]ψγµ) =µ2+a+b, a= (ψγµ,J2 xψγµ) = (Jxψγµ,Jxψγµ)≥0, b= (ψγµ,J2 yψγµ) = (Jyψγµ,Jyψγµ)≥0. (36) The normalization constant cannot be negative. This implie s γ≥µ2, (37) for a fixed γ; thus,µhas both superior and inferior limits (it takes values in a finite interval). Let Λ andλbe these limits, respectively, for a given γ J+ψγΛ= 0, J −ψγλ= 0. (38) 50Using the following operatorial identities J−J+=J2−j2 z+Jz=J2−Jz(Jz−1), J+J−=J2−j2 z+Jz=J2−Jz(Jz+ 1), (39) acting onψγΛas well as on ψγλone gets γ−Λ2−Λ = 0, γ−λ2+λ= 0, (λ−λ+ 1)(λ+λ) = 0. (40) In addition, Λ≥λ→Λ =−λ=J→γ=J(J+ 1). (41) For a given γthe projection µof the momentum takes 2 J+1 values that differ by unity, from Jto−J. Therefore, the difference Λ −λ= 2Jshould be an integer and consequently the eigenvalues of Jzthat are labelled by m are integer m=k, k = 0,±1,±2, ... , (42) or half-integer m=k+1 2, k = 0,±1,±2, ... . (43) A state having a given γ=J(J+ 1) presents a degeneration of order g= 2J+ 1 with regard to the eigenvalues m(this is so because Ji, Jk commute with J2but do not commute between themselves. By a “state of angular momentum J” one usually understands a state ofγ=J(J+ 1) having the maximum projection of its momentum, i.e. J. Quite used notations for angular momentum states are ψjmand the Dirac ket one|jm∝an}b∇acket∇i}ht. Let us now obtain the matrix elements of Jx, Jyin the representation in whichJ2andJzare diagonal. In this case, one obtains from (35) and (39) the following relations J−J+ψjm−1=αmJ−ψjm=αmψjm−1, J(J+ 1)−(m−1)2−(m−1) =α2 m, αm=/radicalig (J+m)(J−m+ 1). (44) 51Combining (44) and (35) leads to J+ψjm−1=/radicalig (J+m)(J−m+ 1)ψjm. (45) It follows that the matrix element of J+is ∝an}b∇acketle{tjm|J+|jm−1∝an}b∇acket∇i}ht=/radicalig (J+m)(J−m+ 1)δnm, (46) and analogously ∝an}b∇acketle{tjn|J−|jm∝an}b∇acket∇i}ht=−/radicalig (J+m)(J−m+ 1)δnm−1. (47) Finally, from the definitions (29) for J+, J−one easily gets ∝an}b∇acketle{tjm|Jx|jm−1∝an}b∇acket∇i}ht=1 2/radicalig (J+m)(J−m+ 1), ∝an}b∇acketle{tjm|Jy|jm−1∝an}b∇acket∇i}ht=−i 2/radicalig (J+m)(J−m+ 1). (48) Partial conclusions αProperties of the eigenvalues of JandJz Ifj(j+ 1)¯h2andm¯hare eigenvalues of JandJzassociated to the eigenvectors|kjm∝an}b∇acket∇i}ht, thenjandmsatisfy the inequality −j≤m≤j. βProperties of the vector J−|kjm∝an}b∇acket∇i}ht Let|kjm∝an}b∇acket∇i}htbe an eigenvector of J2andJzwith the eigenvalues j(j+1)¯h2 andm¯h –(i) Ifm=−j, thenJ−|kj−j∝an}b∇acket∇i}ht= 0. –(ii) Ifm>−j, thenJ−|kjm∝an}b∇acket∇i}htis a nonzero eigenvector of J2and Jzwith the eigenvalues j(j+ 1)¯h2and (m−1)¯h. γProperties of the vector J+|kjm∝an}b∇acket∇i}ht Let|kjm∝an}b∇acket∇i}htbe a (ket) eigenvector of J2andJzfor the eigenvalues j(j+ 1)¯h2andm¯h ⋆Ifm=j, thenJ+|kjm∝an}b∇acket∇i}ht= 0. ⋆Ifm < j , thenJ+|kjm∝an}b∇acket∇i}htis a nonzero eigenvector of J2andJz with the eigenvalues j(j+ 1)¯h2¸ si (m+ 1)¯h 52δConsequences of the previous properties Jz|kjm∝an}b∇acket∇i}ht=m¯h|kjm∝an}b∇acket∇i}ht, J+|kjm∝an}b∇acket∇i}ht=m¯h/radicalig j(j+ 1)−m(m+ 1)|kjm+ 1∝an}b∇acket∇i}ht, J−|kjm∝an}b∇acket∇i}ht=m¯h/radicalig j(j+ 1)−m(m−1)|kjm+ 1∝an}b∇acket∇i}ht. Applications of the orbital angular momentum Until now we have considered those properties of the angular momentum that could be derived only from the commutation relations. L et us go back to the orbital momentum Lof a particle without intrinsic rotation and let us examine how one can apply the theory of the previous sectio n in the important particular case [ˆli,ˆpj] =iεijkˆpk. (49) First, ˆlzand ˆpjhave a common system of eigenfunctions. On the other hand, the Hamiltonian of a free particle ˆH=/parenleftiggˆ/vector p√ 2m/parenrightigg2 , being the square of a vectorial operator has a complete syste m of eigenfunc- tions withˆL2andˆlz. In addition, this implies that the free particle can be found in a state of well-defined E,l, andm. Eigenvalues and eigenfunctions of L2and L z It is more convenient to work in spherical coordinates becau se, as we will see, various angular momentum operators act only on the angl e variables θ, φand not on r. Thus, instead of describing rby its cartesian compo- nentsx, y, z we determine the arbitrary point Mof vector radius rby the spherical 3D coordinates x=rcosφsinθ, y =rsinφsinθ, z =rcosθ, (50) where r≥0,0≤θ≤π, 0≤φ≤2π. 53Let Φ(r,θ,φ) and Φ′(r,θ,φ) be the wavefunctions of a particle in Σ and Σ′, respectively, in which the infinitesimal rotation is given byδαaround thezaxis Φ′(r,θ,φ) = Φ(r,θ,φ +δα), = Φ(r,θ,φ) +δα∂Φ ∂φ, (51) or Φ′(r,θ,φ) = (1 +iˆlzδα)Φ(r,θ,φ). (52) It follows that∂Φ ∂φ=iˆlzΦ,ˆlz=−i∂ ∂φ. (53) For an inifinitesimal rotation in x Φ′(r,θ,φ) = Φ + δα/parenleftbigg∂Φ ∂θ∂θ ∂α+∂Φ ∂θ∂φ ∂α/parenrightbigg , = (1 +iˆlxδα)Φ(r,θ,φ), (54) but in this rotation z′=z+yδα;z′=z+yδα;x′=x (55) and from (50) one gets rcos(θ+dθ) =rcosθ+rsinθsinφδα, rsinφsin(θ+dθ) =rsinθsinφ+rsinθsinφ−rcosθδα, (56) i.e. sinθdθ= sinθsinφδα→dθ dα=−sinφ, (57) and cosθsinφdθ+ sinθcosφdφ =−cosθδα, cosφsinθdφ dα=−cosθ−cosθsinφdθ dα.(58) Substituting (57) in (56) leads to dφ dα=−cotθcosφ . (59) 54With (56) and (58) substituted in (51) and comparing the righ t hand sides of (51) one gets ˆlx=i/parenleftbigg sinφ∂ ∂θ+ cotθcosφ∂ ∂φ/parenrightbigg . (60) For the rotation in y, the result is similar ˆly=i/parenleftbigg −cosφ∂ ∂θ+ cotθsinφ∂ ∂φ/parenrightbigg . (61) Using ˆlx,ˆlyone can also obtain ˆl±,ˆl2 ˆl±= exp/bracketleftbigg ±iφ/parenleftbigg ±∂ ∂θ+icotθ∂ ∂φ/parenrightbigg/bracketrightbigg , ˆl2=ˆl−ˆl++ˆl2+ˆlz, =−/bracketleftigg 1 sin2θ∂2 ∂φ2+1 sinθ∂ ∂θ/parenleftbigg sinθ∂ ∂θ/parenrightbigg/bracketrightigg . (62) From (62) one can see that ˆl2is identical up to a constant to the angular part of the Laplace operator at a fixed radius ∇2f=1 r2∂ ∂r/parenleftbigg r2∂f ∂r/parenrightbigg +1 r2/bracketleftigg 1 sinθ∂ ∂θ/parenleftbigg sinθ∂f ∂θ/parenrightbigg +1 sin2θ∂2 ∂φ2/bracketrightigg .(63) The eigenfunctions of lz ˆlzΦm=mΦ =−i∂Φm ∂φ, Φm=1√ 2πeimφ. (64) Hermiticity conditions of ˆlz /integraldisplay2π 0f∗ˆlzgdφ=/parenleftbigg/integraldisplay2π 0g∗ˆlzfdφ/parenrightbigg∗ +f∗g(2π)−f∗g(0). (65) It follows that ˆlzis hermitic in the class of functions for which f∗g(2π) =f∗g(0). (66) 55The eigenfunctions Φ mofˆlzbelong to the integrable class L2(0,2π) and they fulfill (66), as it happens for any function that can be ex panded in Φm(φ) F(φ) =k/summationdisplay akeikφ, k = 0,±1,±2, ... , G(φ) =k/summationdisplay bkeikφ, k =±1/2,±3/2,±5/2... , (67) withkonly integers or half-integers, but not for combinations of F(φ) and G(φ). The correct choice of mis based on the common eigenfunctions of ˆlz andˆl2. Spherical harmonics In the{/vectorr}representation, the eigenfunctions associated to the eige nvalues l(l+1)¯h2ofL2andm¯hoflzare solutions of the partial differential equations −/parenleftigg ∂2 ∂θ2+1 tanθ∂ ∂θ+1 sin2θ∂2 ∂φ2/parenrightigg ψ(r,θ,φ) =l(l+ 1)¯h2ψ(r,θ,φ), −i∂ ∂φψ(r,θ,φ) =m¯hψ(r,θ,φ). (68) Taking into account that the general results presented abov e can be applied to the orbital momentum, we infer that lcan be an integer or half- integer and that, for fixed l,mcan only take the values −l,−l+ 1, ... ,l−1,l. In (68),ris not present in the differential operator, so that it can be considered as a parameter. Thus, considering only the depen dence onθ, φ ofψ, one uses the notation Ylm(θ,φ) for these common eigenfunctions of L2 andlz, corresponding to the eigenvalues l(l+ 1)¯h2,m¯h. They are known as spherical harmonics. L2Ylm(θ,φ) =l(l+ 1)¯h2Ylm(θ,φ), lzYlm(θ,φ) =m¯hYlm(θ,φ). (69) For more rigorousness, one should introduce one more index i n order to distinguish among the various solutions of (69) correspond ing to the same l, m pairs. Indeed, as one will see next, these equations have a un ique 56solution (up to a constant factor) for each allowed pair of l, m; this is so because the subindices l, m are sufficient in this context. The solutions Ylm(θ, φ) have been found by the method of the separation of variables in sperical variables (see also the chapter The hydrogen atom ) ψlm(r,θ,φ) =f(r)ψlm(θ,φ), (70) wheref(r) is a function of r, which looks as an integration constant from the viewpoint of the partial differential equations in (68). The fact that f(r) is arbitrary proves that L2andlzdo not form a complete set of observables2 in the space εr3of functions of /vector r(or ofr,θ,φ). In order to normalize ψlm(r,θ,φ), it is convenient to normalize Ylm(θ,φ) andf(r) separately /integraldisplay2π 0dφ/integraldisplayπ 0sinθ|ψlm(θ,φ)|2dθ= 1, (71) /integraldisplay∞ 0r2|f(r)|2dr= 1. (72) The values of the pair l, m (α):l, mshould be integers Usinglz=¯h i∂ ∂φ, we can write (69) as follows ¯h i∂ ∂φYlm(θ,φ) =m¯hYlm(θ,φ). (73) Thus, Ylm(θ,φ) =Flm(θ,φ)eimφ. (74) If 0≤φ<2π, then we should tackle the condition of covering all space according to the requirement of dealing with a function cont inuous in any angular zone, i.e. c˘ a Ylm(θ,φ= 0) =Ylm(θ,φ= 2π), (75) implying eimπ= 1. (76) 2By definition, the hermitic operator A is an observable if the orthogonal system of eigenvectors form a base in the space of states. 3Each quantum state of a particle is characterized by a vector ial state belonging to an abstract vectorial space εr. 57As has been seen, mis either an integer or a half-integer; for the appli- cation to the orbital momentum, mshould be an integer. ( e2imπwould be −1 ifmis a half-integer). (β): For a given value of l, all the corresponding Ylmcan be obtained by algebraic means using l+Yll(θ,φ) = 0, (77) which combined with eq. (62) for l+leads to /parenleftbiggd dθ−lcotθ/parenrightbigg Fll(θ) = 0. (78) This equation can be immediately integrated if we notice the relationship cotθdθ=d(sinθ) sinθ. (79) Its general solution is Fll=cl(sinθ)l, (80) whereclis a normalization constant. It follows that for any positive or zero value of l, there is a function Yll(θ,φ), which up to a constant factor is Yll(θ,φ) =cl(sinθ)leilφ. (81) Using repeatedly the action of l−, one can build the whole set of functions Yll−1(θ,φ), ... ,Yl0(θ,φ), ... ,Yl−l(θ,φ). Next, we look at the way in which these functions can be put into correspondence with the eige nvalue pair l(l+ 1)¯h,m¯h(wherelis an arbitrary positive integer such that l≤m≤l ). Using (78), we can make the conclusion that any other eigen function Ylm(θ,φ) can unumbigously be obtained from Yll. Properties of spherical harmonics αIterative relationships From the general results of this chapter, we have l±Ylm(θ,φ) = ¯h/radicalig l(l+ 1)−m(m±1)Ylm±1(θ,φ). (82) 58Using (62) for l±and the fact that Ylm(θ,φ) is the product of a θ-dependent function and e±iφ, one gets e±iφ/parenleftbigg∂ ∂θ−mcotθ/parenrightbigg Ylm(θ,φ) =/radicalig l(l+ 1)−m(m±1)Ylm±1(θ,φ) (83) βOrthonormalization and completeness relationships Equation (68) determines the spherical harmonics only up to a constant factor. We shall now choose this factor such that to have the o rthonormal- ization ofYlm(θ,φ) (as functions of the angular variables θ, φ) /integraldisplay2π 0dφ/integraldisplayπ 0sinθdθY∗ lm(θ,φ)Ylm(θ,φ) =δl′lδm′m. (84) In addition, any continuous function of θ, φcan be expressed by means of the spherical harmonics as follows f(θ,φ) =∞/summationdisplay l=0l/summationdisplay m=−lclmYlm(θ,φ), (85) where clm=/integraldisplay2π 0dφ/integraldisplayπ 0sinθdθY∗ lm(θ,φ)f(θ,φ). (86) The results (85), (86) are consequences of defining the spher ical harmon- ics as an orthonormalized and complete base in the space εΩof functions of θ, φ. The completeness relationship is ∞/summationdisplay l=0l/summationdisplay m=lYlm(θ,φ)Y∗ lm(θ′,φ) =δ(cosθ−cosθ′)δ(φ,φ), =1 sinθδ(θ−θ′)δ(φ,φ). (87) The ‘function’ δ(cosθ−cosθ′) occurs because the integral over the variable θis performed by using the differential element sin θdθ=−d(cosθ). Parity operator Pfor spherical harmonics The behavior ofPin 3D is rather close to that in 1D. When it is applied to a function of cartesian coordinates ψ(x,y,z) changes the sign of each of the coordinates Pψ(x,y,z) =ψ(−x,−y,−z). (88) 59Phas the properties of a hermitic operator being also a unitar y operator as well as a projector since P2is an identity operator ∝an}b∇acketle{tr|P|r′∝an}b∇acket∇i}ht=∝an}b∇acketle{tr|−r′∝an}b∇acket∇i}ht=δ(r+−r′), P2|r∝an}b∇acket∇i}ht=P(P|r∝an}b∇acket∇i}ht) =P|−r∝an}b∇acket∇i}ht=|r∝an}b∇acket∇i}ht. (89) Therefore P2=ˆ1, (90) for which the eigenvalues are P=±1. The eigenfunctions are called even ifP= 1 and odd if P=−1. In nonrelativistic quantum mechanics, the operator ˆHfor a conservative system is invariant with regard to discre te unitary transformations, i.e. PˆHP=P−1ˆHP=ˆH. (91) Thus, ˆHcommutes withPand the parity of the state is a constant of the motion. In addition, Pcommutes with the operators ˆl [P,ˆli] = 0,[P,ˆl±] = 0. (92) IfˆHis even and we consider an eigenfunction |Φn∝an}b∇acket∇i}htthat has a well-defined parity (P|Φn∝an}b∇acket∇i}ht), ‘noncolinear’ with |ψn∝an}b∇acket∇i}ht, then becausePcommutes with ˆH we infer that (P|Φn∝an}b∇acket∇i}ht) is an eigenvector of ˆHwith the same eigenvalue as |Φn∝an}b∇acket∇i}ht). Ifψis a common eigenfunction of the triplet P,ˆl¸ siˆlz, it follows from (92) that the paritities of the states which difer only in ˆlzcoincide. In this way, one can identify the parity of a particle of orbital angu lar momentum ˆl. In spherical coordinates, we shall consider the following c hange of vari- ables r→r, θ→π−θ φ→π+φ. (93) Thus, using a standard base in the space of wavefunctions of a particle without ‘intrinsic rotation’, the radial part of the base fu nctionsψklm(/vector r) is not changed by the parity operator. Only the spherical har monics will change. The transformations (93) are from the trigonometri c standpoint as follows sin(π−θ)→sinθ, cos(π−θ)→−cosθ eim(π+φ→(−1)meimφ(94) leading to the following transformation of the function Yll(θ,φ) Yll(φ−θ,π+φ) = (−1)lYll(θ,φ). (95) 60From (95) it follows that the parity of Yllis (−1)l. On the other hand, l− (as well asl+is invariant to the transformations ∂ ∂(π−θ)→−∂ ∂θ,∂ ∂(π+φ)→∂ ∂φei(π+φ)→−eiφcot(π−θ)→−cotθ. (96) In other words, l±are even. Therefore, we infer that the parity of any spherical harmonics is ( −1)l, that is it is invariant under azimuthal changes Ylm(φ−θ,π+φ) = (−1)lYlm(θ,φ). (97) In conclusion, the spherical harmonics are functions of wel l-defined parity, which is independent of m, even iflis even and odd if lis odd. The spin operator Some particles have not only orbital angular momentum with r egard to external axes but also a proper momentum , which is known as spindenoted here by ˆS. This operator is not related to normal rotation with respec t to ‘real’ axes in space, although it fulfills commutation relat ions of the sme type as those of the orbital angular momentum, i.e. [ˆSi,ˆSj] =iεijkˆSk, (98) together with the following properties (1). For the spin operator all the formulas of the orbital ang ular momentum from (23) till (48) are satisfied. (2). The spectrum of the spin projections is a sequence of eit her integer or half-integer numbers differing by unity. (3). The eigenvalues of ˆS2are the following ˆS2ψs=S(S+ 1)ψs. (99) (4). For a given S, the components Szcan take only 2 S+ 1 values, from −Sto +S. (5). Besides the usual dependence on /vector rand/or/vector p, the eigenfunctions of the particles with spin depend also on a discrete variable, (cha racteristic for the spin) σdenoting the projection of the spin on the zaxis. 61(6). The wavefunctions ψ(/vector r,σ) of a particle with spin can be expanded in eigenfunctions of given spin projection Sz, i.e. ψ(/vector r,σ) =S/summationdisplay σ=−Sψσ(/vector r)χ(σ), (100) whereψσ(/vector r) is the orbital part and χ(σ) is the spinorial part. (7). The spin functions (the spinors) χ(σi) are orhtogonal for any pair σi∝ne}ationslash= σk. The functions ψσ(/vector r)χ(σ) in the sum of (100) are the components of a wavefunction of a particle with spin. (8). The function ψσ(/vector r) is called the orbital part of the spinor, or shortly orbital. (9) The normalization of the spinors is done as follows S/summationdisplay σ=−S||ψσ(/vector r)||= 1. (101) The commutation relations allow to determine the explicit f orm of the spin operators (spin matrices) acting in the space of the eig enfunctions of definite spin projections. Many ‘elementary’ particles, such as the electron, the neut ron, the pro- ton, etc. have a spin of 1 /2 (in units of ¯ h) and therefore the projection of their spin takes only two values, ( Sz=±1/2 (in ¯hunits), respectively. They belong to the fermion class because of their statistics when they form many-body systems. On the other hand, the matrices Sx, Sy, Szin the space of ˆS2,ˆSzare Sx=1 2/parenleftigg 0 1 1 0/parenrightigg , S y=1 2/parenleftigg 0−i i0/parenrightigg , Sz=1 2/parenleftigg 1 0 0−1/parenrightigg , S2=3 4/parenleftigg 1 0 0 1/parenrightigg . (102) Definition of the Pauli matrices The matrices σi= 2Si (103) 62are called the Pauli matrices. They are hermitic and have the same charac- teristic eq. λ2−1 = 0. (104) Therefore, the eigenvalues of σx, σyandσzare λ=±1. (105) The algebra of these matrices is the following σ2 i=ˆI, σkσj=−σjσk=iσz, σjσk=i/summationdisplay lεjklσl.+δjkI .(106) In the case in which the spin system has spherical symmetry ψ1(r,+1 2), ψ 1(r,−1 2), (107) are different solutions because of the different projections Sz. The value of the probability of one or another projection is determine d by the square moduli||ψ1||2or||ψ2||2, respectively, such that ||ψ1||2+||ψ2||2= 1. (108) Since the eigenfunctions of Szhave two components, then χ1=/parenleftigg 1 0/parenrightigg , χ 2=/parenleftigg 0 1/parenrightigg , (109) so that the eigenfunction of a particle of spin 1 /2 can be written as a column matrix ψ=ψ1χ1+ψ2χ2=/parenleftigg ψ1 ψ2/parenrightigg . (110) In the following, the orbitals will be replaced by numbers be cause we are interested only in the spin part. Transformations of spinors to rotations Letψbe the wavefunction of a spin system in Σ. We want to determine the probability of the spin projection in a arbitrary direct ion in 3D space, which one can always chose as the z′of Σ′. As we have already seen in the case of the angular momentum there are two viewpoints in tryi ng to solve this problem 63(α)ψdoes not change when Σ →Σ′and the operator ˆΛ transforms as a vector. We have to find the eigenfunctions of the projection sS′ z and to expand ψin these eigenfunctions. The square moduli of the coefficients give the result ˆS′ x=ˆSxcosφ+ˆSysinφ=e−ilφˆSxeilφ, ˆS′ y=−ˆSxsinφ+ˆSycosφ=e−ilφˆSyeilφ, ˆS′ z=−ˆSz=eilφˆSz, (111) for infinitesimal rotations and from the commutation relati ons one can find ˆL=ˆSz, (112) where ˆLis the infinitesimal generator. (β) The second reprezentation is: ˆSdoes not change when Σ →Σ′and the components of ψdoes change. The transformation to this reprezentation can be performed through a unitary transformation of the form ˆV†ˆS′ˆV=ˆΛ,/parenleftigg ψ′ 1 ψ′ 2/parenrightigg =ˆV†/parenleftigg ψ1 ψ2/parenrightigg . (113) Using (111) and (113) one gets ˆV†e−iˆSzφˆSeiˆSzφˆV=ˆS, ˆV†=eiˆSzφ, (114) and from (114) we are led to /parenleftigg ψ′ 1 ψ′ 2/parenrightigg =eiˆSzφ/parenleftigg ψ1 ψ2/parenrightigg . (115) Using the explicit form of ˆSzand the properties of the Pauli matrices one can find the explicit form of ˆV† z, such that ˆV† z(φ) =/parenleftigg ei 2φ0 0e−i 2φ/parenrightigg . (116) 64A result of Euler One can reach any reference frame Σ′of arbitrary orientation with regard to Σ through only three rotations; the first of angle φaroundz, the next of angleθaroundx′and the last of angle ψaaroundz′, i.e. This important result belongs to Euler. The parameters ( ϕ,θ,ψa) are called Euler’s angles. Thus ˆV†(ϕ,θ,ψa) =ˆV† z′(ψa)ˆV† x′(θ)ˆV† z(ϕ). (117) The matrices ˆV† zare of the form (116), whereas ˆV† xis of the form ˆV† x(ϕ) =/parenleftigg cosθ 2isinθ 2 isinθ 2cosθ 2/parenrightigg , (118) so that ˆV†(ϕ,θ,ψa) =/parenleftigg eiϕ+ψa 2cosθ 2ieiψa−ϕ 2sinθ 2 ieiϕ−ψa 2sinθ 2e−iϕ+ψa 2cosθ 2/parenrightigg . (119) It comes out in this way that by the rotation of Σ, the componen ts of the spinorial function transforms as follows ψ′ 1=ψ1eiϕ+ψa 2cosθ 2+iψ2eiψa−ϕ 2sinθ 2, ψ′ 2=iψ1eiϕ−ψa 2sinθ 2+ψ2e−iϕ+ψa 2cosθ 2. (120) From (120) one can infer that there is a one-to-one mapping be tween any rotation in E3and a linear transformation of E2, the two-dimensional Eu- clidean space. This mapping is related to the two components of the spinorial wavefunction. The rotation in E3does not imply a rotation in E2, which means that ∝an}b∇acketle{tΦ′|ψ′∝an}b∇acket∇i}ht=∝an}b∇acketle{tΦ|ψ∝an}b∇acket∇i}ht= Φ∗ 1ψ1+ Φ∗ 2ψ2. (121) From (119) one finds that (121) does not hold; nevertheless th ere is an invariance in the transformations (119) in the space E2of spinorial wave- functions {Φ|ψ}=ψ1Φ2−ψ2Φ1. (122) The linear transformations that preserve invariant biline ar forms invari- ant are called binary transformations. A physical quantity with two components for which a rotation of the coordinate system is a binary transformation is know as a spin of first order or shortly spin. 65The spinors of a system of two fermions The eigenfunctions of iˆs2iˆsz, withi= 1,2 have the following form i|+∝an}b∇acket∇i}ht=/parenleftigg 1 0/parenrightigg i, i|−∝an}b∇acket∇i}ht=/parenleftigg 0 1/parenrightigg i. (123) A very used operator in a two-fermion system is the total spin ˆS=1ˆS+2ˆS . (124) The spinors of ˆ s2ˆszare kets|ˆS,σ∝an}b∇acket∇i}ht, which are linear combinations of iˆs2iˆsz |+ +∝an}b∇acket∇i}ht=/parenleftigg 1 0/parenrightigg 1/parenleftigg 1 0/parenrightigg 1,|+−∝an}b∇acket∇i}ht=/parenleftigg 1 0/parenrightigg 1/parenleftigg 0 1/parenrightigg 2, |−+∝an}b∇acket∇i}ht=/parenleftigg 0 1/parenrightigg 2/parenleftigg 1 0/parenrightigg 1,|−−∝an}b∇acket∇i}ht =/parenleftigg 0 1/parenrightigg 2/parenleftigg 0 1/parenrightigg 2.(125) The spinorial functions in (125) are assumed orthonormaliz ed. InEn the ket|+ +∝an}b∇acket∇i}hthasSz= 1 and at the same time it is an eigenfunction of the operator ˆS=1ˆs2+ 2(1ˆs)(2ˆs) +2ˆs2, (126) as one can see from ˆS2=|+ +∝an}b∇acket∇i}ht=3 2|+ +∝an}b∇acket∇i}ht+ 2(1ˆsx·2ˆsx+1ˆsy·2ˆsy+1ˆsz·2ˆsz)|+ +∝an}b∇acket∇i}ht,(127) ˆS2=|+ +∝an}b∇acket∇i}ht= 2|+ +∝an}b∇acket∇i}ht= 1(1 + 1)|+ +∝an}b∇acket∇i}ht. (128) If we introduce the operator ˆS−=1ˆs−+2ˆs−, (129) one gets [ˆS−,ˆS2] = 0. (130) Then ( ˆS−)k|1,1∝an}b∇acket∇i}htcan be written in terms of the eigenfunctions of the operator ˆS2, i.e. ˆS−|1,1∝an}b∇acket∇i}ht=ˆS−|+ +∝an}b∇acket∇i}ht=√ 2|+−∝an}b∇acket∇i}ht+√ 2|−+∝an}b∇acket∇i}ht. (131) 66Thus,Sz= 0 in the state ˆS−|1,1∝an}b∇acket∇i}ht. On the other hand, from the normaliza- tion condition, we have |1,0∝an}b∇acket∇i}ht=1√ 2(|+−∝an}b∇acket∇i}ht+|−+∝an}b∇acket∇i}ht) (132) ˆS−|1,0∝an}b∇acket∇i}ht=|−−∝an}b∇acket∇i}ht +|−−∝an}b∇acket∇i}ht =α|1,−1∝an}b∇acket∇i}ht. (133) In addition, the normalization condition gives |1,−1∝an}b∇acket∇i}ht=|−,−∝an}b∇acket∇i}ht. (134) There is only one other linear-independent combination of f unctions of the type (125), which is different of |1,1∝an}b∇acket∇i}ht,|1,0∝an}b∇acket∇i}htand|1,−1∝an}b∇acket∇i}ht, which is ψ4=1√ 2(|+−∝an}b∇acket∇i}ht−|− +∝an}b∇acket∇i}ht), (135) ˆSzψ4= 0, ˆS2ψ4. (136) Therefore ψ4=|0,0∝an}b∇acket∇i}ht. (137) ψ4describes the state of a system of two fermions having the tot al spin equal to zero. The latter type of state is called singlet . On the other hand, the state of two fermions of total spin one can be called triplet having a degree of degeneration g= 3. Total angular momentum The total angular momentum is an operator defined as the sum of the an- gular and spin momenta, i.e. ˆJ=ˆl+ˆS, (138) where ˆlandˆS, as we have seen, act in different spaces, though the square ofˆlandˆScommute with ˆJ [ˆJi,ˆJj] =iεijkˆJk,[ˆJi,ˆl2] = 0,[ˆJi,ˆS2] = 0, (139) From (139) one finds that ˆl2andˆS2have a common eigenfunction system with ˆJ2andˆJz. 67Let us determine the spectrum of the projections ˆJzfor a fermion. The state of maximum projection ˆJzcan be written ¯ψ=ψll/parenleftigg 1 0/parenrightigg =|l,l,+∝an}b∇acket∇i}ht (140) ˆzψ= (l+1 2)¯ψ,→j=l+1 2. (141) We introduce the operator ˆJ−defined as ˆJ−=ˆl−+ˆS−=ˆl−+/parenleftigg 0 0 1 0/parenrightigg . (142) On account of the normalization α=/radicalbig (J+M)(J−M+ 1), one gets ˆJ−ψll/parenleftigg 1 0/parenrightigg =√ 2l|l,l−1,+∝an}b∇acket∇i}ht+|l,l−1,−∝an}b∇acket∇i}ht, (143) so that the value of the projection of ˆj−inˆj−¯ψwill be ˆz= (l−1) +1 2=l−1 2. (144) It follows that ˆ −lowers by one unit the action of ˆJz. In the general case we have ˆk −=ˆlk −+kˆlk−1 −ˆS−. (145) One can see that (145) is obtained from the binomial expansio n considering that ˆs2 −and all higher-order powers of ˆ sare zero. ˆk −|l,l,+∝an}b∇acket∇i}ht=ˆlk −|l,l,+∝an}b∇acket∇i}ht+kˆlk−1 −|l,l,−∝an}b∇acket∇i}ht. (146) Using (ˆl−)kψl,l=/radicalbigg k!(2l)! (2l−k)!ψl,l−k we get ˆk −|l,l,+∝an}b∇acket∇i}ht=/radicalbigg k!(2l)! (2l−k)!|l,l−k,+∝an}b∇acket∇i}ht+/radicalbigg (k+1)!(2l)! (2l−k+1)!k|l,l−k+ 1,−∝an}b∇acket∇i}ht.(147) Now noticing that m=l−k ˆl−m −|l,l,+∝an}b∇acket∇i}ht=/radicalbigg (l−m)!(2l)! (l+m)!|l,m,+∝an}b∇acket∇i}ht+/radicalbigg (l−m−1)!(2l)! (2l+m+1)!(l−m)|l,m+1,−∝an}b∇acket∇i}ht.(148) 68The eigenvalues of the projections of the total angular mome ntum are given by the sequence of numbers differing by one unit from j=l+1 2pˆ ın˘ a to j=l−1 2. All these states belong to the same eigenfunction of ˆJas|l,l,+∝an}b∇acket∇i}ht because [ ˆJ−,ˆJ2] = 0: ˆJ2|l,l,+∝an}b∇acket∇i}ht= (ˆl2+ 2ˆlˆS+ˆS2)|l,l,+∝an}b∇acket∇i}ht, = [l(l+ 1) + 2l1 2+3 4]|l,l,+∝an}b∇acket∇i}ht (149) wherej(j+ 1) = (l+1 2)(l+3 2). In the left hand side of (149) a contribution different of zero gives only j=ˆlzˆSz. Thus, the obtained eigenfunctions correspond to the pair j=l+1 2, mj=m+1 2; they are of the form |l+1 2,m+1 2∝an}b∇acket∇i}ht=/radicaligg l+m+ 1 2l+ 1|l,m,+∝an}b∇acket∇i}ht+/radicaligg l−m 2l+ 1|l,m+ 1,−∝an}b∇acket∇i}ht.(150) The total number of linearly independent states is N= (2l+ 1)(2s+ 1) = 4l+ 2, (151) of which in (150) only (2j+1)=2l+3 have been built. The rest o f 2l−1 eigenfunctions can be obtained from the orthonormalizatio n condition: |l−1 2,m−1 2∝an}b∇acket∇i}ht=/radicalig l−m 2l+1|l,m,+∝an}b∇acket∇i}ht−/radicalig l+m+1 2l+1|l,m+ 1,−∝an}b∇acket∇i}ht. (152) If two subsystems are in interaction in such a way that each of the angular momenta ˆjiis conserved, then the eigenfunctions of the total angular momentum ˆJ= ˆ1+ ˆ2, (153) can be obtained by a procedure similar to the previous one. Fo r fixed eigen- values of ˆ1and ˆ2there are (2 j1+1)(2j2+1) orthonormalized eigenfunctions of the projection of the total angular momentum ˆJz; the one corresponding to the maximum value of the projection ˆJz, i.e.MJ=j1+j2, can be built in a unique way and therefore J=j1+j2is the maximum value of the total angular momentum of the system. Applying the operator ˆJ= ˆ1+ ˆ2 repeatingly to the function |j1+j2,j1+j2,j1+j2∝an}b∇acket∇i}ht=|j1,j1∝an}b∇acket∇i}ht·|j2,j2∝an}b∇acket∇i}ht, (154) one can obtain all the 2( j1+j2) + 1 eigenfunctions of ˆJ=j1+j2with differentMs: −(j1+j2)≤M≤(j1+j2). 69For example, the eigenfunction of M=j1+j2−1 is |j1+j2,j1+j2−1,j1,j2∝an}b∇acket∇i}ht=/radicaligg j1 j1+j2|j1,j1−1,j2,j2∝an}b∇acket∇i}ht+/radicaligg j2 j1+j2|j1,j1,j2,j2−1∝an}b∇acket∇i}ht. (155) Applying iteratively the operator ˆJ−, all the 2(j1+j2−1)−1 eigenfunc- tions ofJ=j1+j2−1 can be obtained. One can prove that |j1−j2|≤J≤j1+j2, so that maxJ/summationdisplay minJ(2J+ 1) = (2J1+ 1)(2J2+ 1). (156) Thus |J,M,j 1,j2∝an}b∇acket∇i}ht=/summationdisplay m1+m2=M(j1m1j2m2|JM)|j1,m1,j2,m2∝an}b∇acket∇i}ht, (157) where the coefficients ( j1m1j2m2|JM) determine the contribution of the various kets|j1,m1,j2,m2∝an}b∇acket∇i}htto the eigenfunctions ofˆJ2,ˆJzhaving the eigen- valuesJ(J+ 1),M. They are called Clebsch-Gordan coefficients. 70References : 1. H.A. Buchdahl, “Remark concerning the eigenvalues of orb ital angular momentum”, Am. J. Phys. 30, 829-831 (1962) 3N. Note : 1. The operator corresponding to the Runge-Lenz vector of t he classical Kepler problem is written as ˆ/vectorA=ˆ r r+1 2/bracketleftigg (ˆl׈p)−(ˆp׈l)/bracketrightigg , where atomic units have been used and the case Z= 1 (hydrogen atom) was assumed. This operator commutes with the Hamiltonian of the atomic hydrogen ˆH=ˆp2 2−1 r, that is it is an integral of the atomic quantum motion. Its components have commutators of the type [ Ai,Aj] =−2iǫijklk·H; the commutators of the Runge-Lenz components with the componen ts of the angular momentum are of the type [ li,Aj] =iǫijkAk. Thus, they respect the conditions (23). Proving that can be a useful exercise. 713P. Problems Problem 3.1 Show that any translation operator, for which ψ(y+a) =Taψ(y), can be written as an exponential operator. Apply the result for y=/vector rand for a the finite rotation αaroundz. Solution The proof can be obtained expanding ψ(y+a)) in Taylor series in the in- finitezimal neighborhood around x, that is in powers of a ψ(y+a) =∞/summationdisplay n=0an n!dn dxnψ(x) We notice that∞/summationdisplay n=0andn dxn n!=ead dx and therefore one has Ta=ead dxin the 1D case. In 3D, y=/vector randa→/vector a. The result is T/vector a=e/vector a·/vector∇. For the finite rotation αaroundzwe hasy=φanda=α. It follows Tα=Rα=eαd dφ. Another exponential form of the rotation around zis that in terms of the angular momentum operator as was already commented in th is chapter. Letx′=x+dxand consider only the first order of the Taylor series ψ(x′,y′,z′) =ψ(x,y,z) + (x′−x)∂ ∂x′ψ(x′,y′,z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectorr′=/vector r +(y′−y)∂ ∂y′ψ(x′,y′,z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectorr′=/vector r +(z′−z)∂ ∂z′ψ(x′,y′,z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectorr′=/vector r. Taking into account ∂ ∂x′ iψ(/vector r′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle /vector r′=∂ ∂xiψ(/vector r), x′=x−ydφ, y′=y+xdφ, z′=z, 72one can reduce the series from three to two dimensions ψ(/vector r′) =ψ(/vector r) + (x−ydφ−x)∂ψ(/vector r) ∂x+ (y+xdφ−y)∂ψ(/vector r) ∂y′, =ψ(/vector r)−ydφ∂ψ(/vector r) ∂x+xdφx∂ψ(/vector r) ∂y, =/bracketleftbigg 1−dφ/parenleftbigg −x∂ ∂y+y∂ ∂x/parenrightbigg/bracketrightbigg ψ(/vector r). Sinceiˆlz=/parenleftig x∂ ∂y−y∂ ∂x/parenrightig it follows that R=/bracketleftig 1−dφ/parenleftig x∂ ∂y−y∂ ∂x/parenrightig/bracketrightig .In the second order one can get1 2!(iˆlzdφ)2, and so forth. Thus, Rcan be written as an exponential R=eiˆlzdφ. Problem 3.2 Based on the expressions given in (14) show that one can get (1 5). Solution Let us consider only linear terms in the Taylor expansion (in finitesimal ro- tations) eiˆlzdφ= 1 +iˆlzdφ+1 2!(iˆlzdφ)2+... , so that (1 +iˆlzdφ)ˆAx(1−iˆlzdφ) = ˆAx−ˆAxdφ, (ˆAx+iˆlzdφˆAx)(1−iˆlzdφ) = ˆAx−ˆAxdφ, ˆAx−ˆAxiˆlzdφ+iˆlzdφˆAx+ˆlzdφˆAxˆlzdφ=ˆAx−ˆAxdφ, i(ˆlzˆAx−ˆAxˆlz)dφ=−ˆAydφ. We easily arrive at the conclusion [ˆlz,ˆAx] =iˆAy. In addition, [ ˆlz,ˆAy] =iˆAxcan be obtained from (1 +iˆlzdφ)ˆAy(1−iˆlzdφ) = ˆAxdφ−ˆAy, (ˆAy+iˆlzdφˆAy)(1−iˆlzdφ) = ˆAxdφ−ˆAy, ˆAy−ˆAyiˆlzdφ+iˆlzdφˆAy+ˆlzdφˆAyˆlzdφ=ˆAxdφ−ˆAy, i(ˆlzˆAy−ˆAyˆlz)dφ=−ˆAxdφ. 73Problem 3.3 Determine the operatordˆσx dtbased on the Hamiltonian of an electron with spin in a magnetic field of induction /vectorB. Solution The Hamiltonian in this case is ˆH(ˆp,ˆr,ˆσ) =ˆH(ˆp,ˆr) + ˆσ·/vectorB, where the latter term is the Zeeman Hamiltonian of the electron. Since ˆσxcommutes with the momenta and the coordinates, applying the Heisenbe rg equation of motion leads to dˆσx dt=i ¯h[ˆH,ˆσx] =−i ¯he¯h 2me((ˆσyBy+ ˆσzBz)ˆσx−ˆσx(ˆσyBy+ ˆσzBz)). Using [σx,σy] =iσz, one gets : dˆσx dt=e me(ˆσyBz−ˆσzBy) =e me(/vector σ×/vectorB)x. 744. THE WKB METHOD In order to study more realistic potentials with regard to rectangular barriers and wells , it is necessary to employ approximate methods allowing to solve the Schr¨ odinger equation for more general classes of potentials and at the same time to give very good approximations of the exact solutions. The aim of the various approximative methods is to offer solut ions of acceptable precision and simplicity that can be used for und erstanding the behaviour of the system in quasianalytic terms. Within quantum mechanics, one of the oldest and efficient appr oximate method for getting rather good Schr¨ odinger solutions was d eveloped almost simulataneously by G. Wentzel, H. A. Kramers and L. Brillouin in 1926, hence the acronym WKB under which this method is known (or JWKB as is more correctly used by many authors, see note 4N). It is worth mentioning that the WKB method applies to 1D Schr¨ odinger equations and that there are serious difficulties when trying to generalize it to more dimensions. In order to solve the Schr¨ odinger equation −¯h2 2md2ψ dy2+u(y)ψ=Eψ (1) with a potential of the form u(y) =u0f/parenleftigy a/parenrightig , (2) we first perform the changes of notations and of variable ξ2=¯h2 2mu0a2(3) η=E u0(4) x=y a. (5) From eq. (5) we get d dx=dy dxd dy=ad dy(6) d2 dx2=d dx/parenleftig ad dy/parenrightig =/parenleftig ad dx/parenrightig/parenleftig ad dx/parenrightig =a2d2 dy2(7) 75and the Schr¨ odinger eq. reads −ξ2d2ψ dx2+f(x)ψ=ηψ . (8) Multiplying by−1/ξ2and defining r(x) =η−f(x), it is possible to write it as folows d2ψ dx2+1 ξ2r(x)ψ= 0. (9) To solve (9), the following form of the solution is proposed ψ(x) = exp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg . (10) Therefore d2ψ dx2=d dx/parenleftbiggdψ x/parenrightbigg =d dx/braceleftigg i ξq(x)exp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg/bracerightigg =⇒d2ψ dx2=i ξ/braceleftigg i ξq2(x)exp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg +∂q(x) ∂xexp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg/bracerightigg . Factorizing ψ, we have d2ψ dx2=/bracketleftigg −1 ξ2q2(x) +i ξdq(x) dx/bracketrightigg ψ . (11) Discarding for the time being the dependence of x, the Schr¨ odinger eq. can be written /bracketleftigg −1 ξ2q2+i ξ∂q ∂x+1 ξ2r/bracketrightigg ψ= 0 (12) and since in general ψ∝ne}ationslash= 0, we get: iξdq dx+r−q2= 0, (13) which is a nonlinear differential eq. of the Riccati type whos e solutions are sought in the form of expansions in powers of ξunder the assumption that ξis very small. More precisely, the series is taken of the form q(x) =∞/summationdisplay n=0(−iξ)nqn(x). (14) 76Plugging it into the Riccati eq., we get iξ∞/summationdisplay n=0(−iξ)ndqn dx+r(x)−∞/summationdisplay µ=0(−iξ)µqµ∞/summationdisplay ν=0(−iξ)νqν= 0. (15) By a rearrangement of the terms one is led to ∞/summationdisplay n=0(−1)n(iξ)n+1dqn dx+r(x)−∞/summationdisplay µ=0∞/summationdisplay ν=0(−iξ)µ+νqµqν= 0. (16) Double series have the following important property ∞/summationdisplay µ=0∞/summationdisplay ν=0aµν=∞/summationdisplay n=0n/summationdisplay m=0am,n−m, whereµ=n−m ,ν =m. Thus ∞/summationdisplay n=0(−1)n(iξ)n+1dqn dx+r(x)−∞/summationdisplay n=0n/summationdisplay m=0(−iξ)n−m+mqmqn−m= 0.(17) Let us see explicitly the first several terms in each of the ser ies in eq. (17): ∞/summationdisplay n=0(−1)n(iξ)n+1dqn dx=iξdq0 dx+ξ2dq1 dx−iξ3dq2 dx+... (18) ∞/summationdisplay n=0n/summationdisplay m=0(−iξ)nqmqn−m=q2 0−i2ξq0q1+... (19) Asking that the first terms in both series contain iξ, one should write them as ∞/summationdisplay n=1(−1)n−1(iξ)ndqn−1 dx+r(x)−q2 0−∞/summationdisplay n=1n/summationdisplay m=0(−iξ)nqmqn−m= 0, which leads to ∞/summationdisplay n=1/bracketleftigg −(−iξ)ndqn−1 dx−n/summationdisplay m=0(−iξ)nqmqn−m/bracketrightigg +/bracketleftigg r(x)−q2 0/bracketrightigg = 0.(20) 77In order that this equation be right the following condition s should be satisfied r(x)−q2 0= 0⇒q0=±/radicalig r(x) (21) −(−iξ)ndqn−1 dx−n/summationdisplay m=0(−iξ)nqmqn−m= 0 ⇒dqn−1 dx=−n/summationdisplay m=0qmqn−mn≥1. (22) The latter is a recurrence relatioship, which occurs natura lly in the WKB method. Recalling that we have defined r(x) =η−f(x), η=E u0&f(x) =u u0, by means of eq. (21) we get q0=±/radicalig η−f(x) =±/radicaligg E u0−u u0=±/radicaligg 2m(E−u) 2mu0. (23) This clearly indicates the classical nature of the WKB momen tum of the particle of energy Ein the potential uand units of√2mu0. Thus q0=p(x) =/radicalig η−f(x) is not an operator . If we approximate till the second order, we get q(x) =q0−iξq1−ξ2q2 and using the WKB recurrence relationship (22) we calculate q1andq2 dq0 dx=−2q0q1⇒q1=−1 2dq0 dx q0=−1 2d dx(ln|q0|) ⇒q1=−1 2d dx(ln|p(x)|) (24) dq1 dx=−2q0q2−q2 1⇒q2=−dq1 dx−q2 1 2q0. (25) A glance to eq. (24), affords us to consider q1as the slope, up to a change of sign, of ln|q0|; whenq0is very small, then q1≪0⇒ −ξq1≫0 and therefore the series diverges. To avoid this the following WKB condition is imposed |q0|≫|−ξq1|=ξ|q1|. 78It is worth noting that this WKB condition WKB is not fulfilled at those pointsxkwhere q0(xk) =p(xk) = 0. Sinceq0=p=/radicalig 2m(E−u) 2mu0the previous equation leads us to E=u(xk). (26) In classical mechanics the points xkthat satisfies (26) are called turning points because the change of the sense of the motion of a macroscopic particle takes place there. By means of these arguments, we can say that q0is a classical solution of the problem under examination; also that the quantities q1&q2are the first and the second quantum corrections, respectively, in the WK B problem. To obtain the WKB wavefunctions we shall consider only the cl assical solution and the first quantum correction that we plug in the W KB form of ψ ψ= exp/bracketleftigg i ξ/integraldisplayx aq(x)dx/bracketrightigg = exp/bracketleftigg i ξ/integraldisplayx a(q0−iξq1)dx/bracketrightigg ⇒ψ= exp/parenleftigg i ξ/integraldisplayx aq0dx/parenrightigg ·exp/parenleftigg/integraldisplayx aq1dx/parenrightigg . For the second factor, we get exp/parenleftigg/integraldisplayx aq1dx/parenrightigg = exp/bracketleftigg −1 2/integraldisplayx ad dx(ln|p(x)|)dx/bracketrightigg = = exp/bracketleftigg −1 2(ln|p(x)|)/vextendsingle/vextendsingle/vextendsinglex a/bracketrightigg =A/radicalbig p(x), whereAis a constant, whereas for the first factor we get exp/parenleftigg i ξ/integraldisplayx aq0dx/parenrightigg = exp/bracketleftigg ±i ξ/integraldisplayx ap(x)dx/bracketrightigg . Thus, we can write ψin the following form ψ±=1/radicalbig p(x)exp/bracketleftigg ±i ξ/integraldisplayx ap(x)dx/bracketrightigg . (27) 79The latter are known as the WKB solutions of the 1D Schr¨ odinger equation . The general WKB solution in the region in which the WKB condit ion is satisfied is written down as ψ=a+ψ++a−ψ−. (28) As already mentioned there is no WKB solution at the turning p oints. This raises the question of the manner in which one has to do the pas sing from ψ(x < xk) toψ(x > xk). The solution of this difficulty is achieved by introducing the WKB connection formulas. The connection formulas We have already seen that the WKB solutions are singular at th e clas- sical turning points; however, these solutions are correct both on the left and right side of these turning points xk. A natural question is how do we changeψ(x < xk) inψ(x > xk) when passing through the turning points. The explicit answer is given by the connection formulas. From the theory of differential equations of complex variabl e it can be proved that really there are such connection formulas and th at they are the following ψ1(x) =1 [−r(x)]1 4exp/parenleftbigg −/integraldisplayxk x/radicalig −r(x)dx/parenrightbigg → →2 [r(x)]1 4cos/parenleftbigg/integraldisplayx xk/radicalig r(x)dx−π 4/parenrightbigg , (29) whereψ1(x) has only an attenuated exponential behavior for x <xk. The first connection formula shows that the function ψ(x), which at the left of the turning point behaves exponentially decaying, turns at the right of xk into a cosinusoide of phase φ=π 4and double amplitude with regard to the amplitude of the exponential. In the case of a more general function ψ(x), such as a function with both rising and decaying exponential behavior, the connection f ormula is sin/parenleftbigg φ+π 4/parenrightbigg1 [−r(x)]1 4exp/parenleftbigg/integraldisplayxk x/radicalig −r(x)dx/parenrightbigg ← ←1 [r(x)]1 4cos/parenleftbigg/integraldisplayx xk/radicalig r(x)dx+φ/parenrightbigg , (30) 80under the condition that φs˘ a do not take a value that is too close to −π 4. The reason is that if φ=−π 4, then the sinus function is zero . The latter connection formula means that a function whose behavior is o f the cosinusoid type at the right of a turning point changes into a growing exp onential with sinusoid-modulated amplitude at the right of that point. In order to study the details of the procedure of getting the c onnection formulas we recommend the book Mathematical Methods of Physics by J. Mathews & R.L. Walker. Estimation of the WKB error We have found the solution of the Schr¨ odinger equation in th e regions where the WKB condition is satisfied. However, the WKB soluti ons are divergent at the turning points. We thus briefly analyze the e rror introduced by using the WKB approximation and tackling the connection formulas in a close neighbourhood of the turning points. Considering x=xkas a turning point, we have q0(xk) =p(xk) = 0⇒ E=u(xk). At the left of xk, that is on the ‘half-line’ x < xk, we shall assumeE <u (x) leading to the WKB solution ψ(x) =a /bracketleftigu(x)−E u0/bracketrightig1 4exp −1 ξ/integraldisplayxk x/radicaligg u(x)−E u0dx + +b /bracketleftigu(x)−E u0/bracketrightig1 4exp 1 ξ/integraldisplayxk x/radicaligg u(x)−E u0dx . (31) Similarly, at the right of xk(on the ‘half-line x>xk) we assume E >u (x); therefore the WKB solution in the latter region will be ψ(x) =c /bracketleftigE−u(x) u0/bracketrightig1 4exp i ξ/integraldisplayx xk/radicaligg E−u(x) u0dx + +d /bracketleftigE−u(x) u0/bracketrightig1 4exp −i ξ/integraldisplayx xk/radicaligg E−u(x) u0dx . (32) 81Ifψ(x) is a real function, it will have this property both at the rig ht and the left ofxk. It is usually called the “reality condition” . It means that if a,b∈ℜ, thenc=d∗. Our problem consists in connecting the approximations on th e two sides ofxksuch that they refer to the same solution. This means to find canddif one knowsaandb, as well as viceversa. To achieve this connection, we have to use an approximate solution, which should be correct alon g a contour connecting the regions on the two sides of xk, where the WKB solutions are also correct. A method proposed by Zwann andKemble is very useful in this case. It consists in going out from the real axis in the ne ighbourhood ofxkon a contour around xkin the complex plane. It is assumed that on this contour the WKB solutions are still correct. Here, we sh all use this method as a means of getting the estimation of the error produ ced by the WKB method. The estimation of the error is always an important matter for any ap- proximate solutions. In the case of the WKB procedure, it is m ore significant because it is an approximation on large intervals of the real axis that can lead to the accuulation of the errors as well as to some artefa cts due to the phase shifts that can be introduced in this way. Let us define the associated WKB functions as follows W±=1 /bracketleftigE−u(x) u0/bracketrightig1 4exp ±i ξ/integraldisplayx xk/radicaligg E−u(x) u0dx , (33) that we consider as functions of complex variable. We shall u se cuts in order to avoid the discontinuities in the zeros of r(x) =E−u(x) u0. These functions satisfy the differential equation that is obtaine d by differentiating with respect to x, leading to W′ ±=/parenleftbigg ±i ξ√r−1 4r′ r/parenrightbigg W± W′′ ±+/bracketleftigg r ξ2+1 4r′′ r−5 16/parenleftbiggr′ r/parenrightbigg2/bracketrightigg W±= 0. (34) Let us notice that s(x) =1 4r′′ r−5 16/parenleftbiggr′ r/parenrightbigg2 , (35) 82thenW±are exact solutions of the equation W′′ ±+/bracketleftbigg1 ξ2r(x) +s(x)/bracketrightbigg W±= 0, (36) although they satisfy only approximately the Schr¨ odinger equation, which is a regular equation in x=xk, whereas the same equation for the associate WKB functions is singular at that point. We shall now define the functions α±(x) satisfying the following two relationships ψ(x) =α+(x)W+(x) +α−(x)W−(x) (37) ψ′(x) =α+(x)W′ +(x) +α−(x)W′ −(x), (38) whereψ(x) is a solution of the Schr¨ odinger equation. Solving the pre vious equations for α±, we get α+=ψW′ −−ψ′W− W+W′−−W′+W−α−=−ψW′ +−ψ′W+ W+W′−−W′+W−, where the numerator is just the Wronskian ofW+andW−. It is not difficult to prove that this takes the value −2 ξi, so thatα±simplifies to the following form α+=ξ 2i/parenleftbigψW′ −−ψ′W−/parenrightbig(39) α−=−ξ 2i/parenleftbigψW′ +−ψ′W+/parenrightbig. (40) Doing the derivative in xin the eqs. (39) and (40), we have dα± dx=ξ 2i/parenleftbigψ′W′ ∓+ψW′′ ∓−ψ′′W∓−ψ′W′ ∓/parenrightbig. (41) In the brackets, the first and the fourth terms are zero; recal ling that ψ′′+1 ξ2r(x)ψ= 0 &W′′ ±+/bracketleftbigg1 ξ2r(x) +s(x)/bracketrightbigg W±= 0, we can write eq. (41) in the form dα± dx=ξ 2i/bracketleftbigg −ψ/parenleftbiggr ξ2+s/parenrightbigg W∓+r ξ2ψW∓/bracketrightbigg dα± dx=∓ξ 2is(x)ψ(x)W∓(x), (42) 83which based on eqs. (33) and (37) becomes dα± dx=∓ξ 2is(x) [r(x)]1 2/bracketleftbigg α±+α∓exp/parenleftbigg ∓2 ξi/integraldisplayx xk/radicalig r(x)dx/parenrightbigg/bracketrightbigg . (43) Eqs. (42) and (43) are useful for estimating the WKB error in t he 1D case. The reason for whichdα± dxcan be considered as a measure of the WKB er- rors is that in the eqs. (31) and (32) the constants a,bandc,d, respectively, give only approximate solutions ψ, while the functions α±when introduced in the eqs. (37) and (38) produce exact ψsolutions. From the geometrical viewpoint the derivative gives the slope of the tangent to th ese functions and indicates the measure in which α±deviates from the constants a,b,c andd. 4N. Note : The original (J)WKB papers are the following: G. Wentzel, “Eine Verallgemeinerung der Wellenmechanik”, [“A generaliza- tion of wave mechanics”], Zeitschrift f¨ ur Physik 38, 518-529 (1926) [received on 18 June 1926] L. Brillouin, “La m´ ecanique ondulatoire de Schr¨ odinger: une m´ ethode g´ en´ erale de resolution par approximations successives”, [“Schr¨ od inger’s wave mechan- ics: a general method of solving by succesive approximation s”], Comptes Rendus Acad. Sci. Paris 183, 24-26 (1926) [received on 5 July 1926] H.A. Kramers, “Wellenmechanik und halbzahlige Quantisier ung”, [“Wave mechanics and half-integer quantization”], Zf. Physik 39, 828-840 (1926) [received on 9 Sept. 1926] H. Jeffreys, “On certain approx. solutions of linear diff. eqs . of the second order”, Proc. Lond. Math. Soc. 23, 428-436 (1925) 4P. Problems Problem 4.1 Employ the WKB method for a particle of energy Emoving in a potential u(x) of the form shown in fig. 4.1. 84Eu(x) x x x 1 2 Fig. 4.1 Solution The Schr¨ odinger equation is d2ψ dx2+2m ¯h2[E−u(x)]ψ= 0. (44) As one can see, we have r(x) =2m ¯h2[E−u(x)]/braceleftigg is positive for a<x<b is negative for x<a,x>b . Ifψ(x) corresponds to the region x < a , when passing to the interval a<x<b , the connection formula is given by eq. (29) telling us that ψ(x)≈A [E−u]1 4cos /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4 , (45) whereAis an arbitrary constant. Whenψ(x) corresponds to the region x>b, when passing to the segment a<x<b , we have in a similar way ψ(x)≈−B [E−u]1 4cos /integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4 , (46) 85whereBis another arbitrary constant. The reason why the connectio n formula is again given by eq. (29) is easily understood exami nining what happens when the particle reaches the second classical turn ing point at x= b. This produces the inversion of the direction of motion. Thu s, the particle appears to come from the right toward the left. In other words , we are in the first case (from the left to the right), only that as seen in a mirror placed at the point x=a. These two expressions should be the same, independently of t he con- stantsAandB, so that cos /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4 =−cos /integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4  ⇒cos /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4 + cos /integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4 = 0. (47) Recalling that cosA+ cosB= 2cos/parenleftbiggA+B 2/parenrightbigg cos/parenleftbiggA−B 2/parenrightbigg , eq. (47) can be written 2cos 1 2 /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4+/integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4  · ·cos 1 2 /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4−/integraldisplayb x/radicaligg 2m ¯h2(E−u)dx+π 4  = 0,(48) which implies that the arguments of the cosinusoids are inte ger multiples of π 2. On the other hand, the argument of the second cosinusoid do n ot lead to a nontrivial result. Therefore, we pay attention only to the argument of the first cosinusoid, which prove to be essential for getting an i mportant result 1 2 /integraldisplayx a/radicaligg 2m ¯h2(E−u)dx−π 4+/integraldisplayb x/radicaligg 2m ¯h2(E−u)dx−π 4 =n 2πfor n odd ⇒/integraldisplayb a/radicaligg 2m ¯h2(E−u)dx−π 2=nπ 86⇒/integraldisplayb a/radicaligg 2m ¯h2(E−u)dx= (n+1 2)π ⇒/integraldisplayb a/radicalig 2m(E−u)dx= (n+1 2)π¯h . (49) This result is very similar to the Bohr - Sommerfeld quantization rules . We recall that Bohr’s postulate says that the orbital angula r momen- tum of an electron moving on an “allowed atomic orbit” is quan tized as L=n¯h,n= 1,2,3,.... We also recall that the Wilson - Sommerfeld quan- tization rules assert that any coordinate of a system that va ries periodically in time should satisfy the ‘quantum’ condition:/contintegraltextpqdq=nqh, whereqis a periodic coordinate, pqis the associated momentum, nqis an integer, and h is Planck’s constant. One can see that the WKB result is indee d very similar. Problem 4.2 Estimate the error of the WKB solution WKB at a point x1∝ne}ationslash=xk, where xkis a classical turning point for the differential equation y′′+xy= 0.The solution of this problem is of importance in the study of unif orm fields, such as the gravitational and electric fields generated by large p lanes. Solution: For this differential equation we have ξ= 1, r (x) =x &s(x) =−5 16x−2. r(x) =xhas a single zero at xk= 0, therefore for x≫0: W±=x−1 4exp/parenleftbigg ±i/integraldisplayx 0√xdx/parenrightbigg =x−1 4exp/parenleftbigg ±2 3ix3 2/parenrightbigg . (50) DerivatingW±up to the second derivative in x, we realize that the following differential equation is satisfied W′′ ±+ (x−5 16x−2)W±= 0. (51) The exact solution y(x) of the latter differential equation can be written as a linear combination of W±, as it has been shown in the corresponding 87section where the WKB error has been tackled; recall that the following form of the linear combination was proposed therein y(x) =α+(x)W+(x) +α−(x)W−(x) For largex, the general solution of the differential equation can be wri t- ten in the WKB approximation as follows y(x) =Ax−1 4cos/parenleftbigg2 3x3 2+δ/parenrightbigg forx→∞. (52) Thus,α+→A 2eiδandα−→A 2e−iδforx→∞. We want to calculate the error due to these WKB solutions. A simple measure of this err or is the deviation of α+and ofα−relative to the constants A. Using the equation dα± dx=∓ξ 2is(x)/radicalbig r(x)/bracketleftbigg α±+α∓exp/parenleftbigg ∓2i/integraldisplayx xk/radicalig r(x)dx/parenrightbigg/bracketrightbigg and doiing the corresponding substitutions, one gets dα± dx=∓i 2/parenleftbigg −5 16x−2/parenrightbigg x−1 2/bracketleftbiggA 2e±iδ+A 2e∓iδexp/parenleftbigg ∓2i2 3x3 2/parenrightbigg/bracketrightbigg .(53) Taking ∆α±as the changes displayed by α±whenxvaries between x1and ∞, we can do the required calculation by means of ∆α± A/2=2 A/integraldisplay∞ x1dα± dxdx= =±i5 32e±iδ/bracketleftbigg2 3x−3 2 1+e∓2iδ/integraldisplay∞ x1x−5 2exp/parenleftbigg ∓i4 3x3 2/parenrightbigg dx/bracketrightbigg . (54) The second term in the parentheses is less important than the first one because the complex exponential oscillates between 1 and −1 and therefore x−5 2<x−3 2. Consequently ∆α± A/2≈±5 48ie±iδx−3 2 1, (55) and as we can see the error introduced by the WKB approximatio n is in- deed small if we take into account that the complex exponenti al oscillates between−1 and 1, while x−3 2 1is also small. 885. THE HARMONIC OSCILLATOR (HO) The solution of the Schr¨ odinger eq. for HO The HO can be considered as a paradigm of Physics. Its utility is manifest in many areas from classical physics until quantum electrod ynamics and theories of gravitational collapse. From classical mechanics we know that many complicated pote ntials can be well approximated near their equilibrium positions by HO po tentials V(x)∼1 2V′′(a)(x−a)2. (1) This is a 1D case. For this case, the classical Hamiltonian fu nction of a particle of mass m, oscillating at the frequency ωhas the following form: H=p2 2m+1 2mω2x2(2) and the quantum Hamiltonian corresponding to the space of co nfigurations is given by ˆH=1 2m(−i¯hd dx)2+1 2mω2x2(3) ˆH=−¯h2 2md2 dx2+1 2mω2x2. (4) Since we consider a time-independent potential, the eigenf unctions Ψ n and the eigenvalues Enare obtained by means of the time-independent Schr¨ odinger equation ˆHΨn=EnΨn. (5) For the HO Hamiltonian, the Schr¨ odinger eq. is d2Ψ dx2+/bracketleftigg 2mE ¯h2−m2ω2 ¯h2x2/bracketrightigg Ψ = 0. (6) We cancealed the subindices of Eand Ψ because they are not of any importance here. Defining k2=2mE ¯h2(7) λ=mω ¯h, (8) 89the Schr¨ odinger eq. becomes d2Ψ dx2+ [k2−λ2x2]Ψ = 0, (9) which is known as Weber’s differential equation in mathemati cs. We shall make now the transformation y=λx2. (10) In general, by changing the variable from xtoy, the differential operators take the formd dx=dy dxd dy(11) d2 dx2=d dx(dy dxd dy) =d2y dx2d dy+ (dy dx)2d2 dy2. (12) Applying this obvious rule to the proposed transformation w e obtain the following differential eq. in the yvariable yd2Ψ dy2+1 2dΨ dy+ [k2 4λ−1 4y]Ψ = 0, (13) and, by definind : κ=k2 2λ=¯k2 2mω=E ¯hω, (14) we get yd2Ψ dy2+1 2dΨ dy+ [κ 2−1 4y]Ψ = 0. (15) Let us try to solve this equation by first doing its asymptotic analysis in the limity→∞. We first rewrite the previous equation in the form d2Ψ dy2+1 2ydΨ dy+ [κ 2y−1 4]Ψ = 0. (16) We notice that in the limit y→∞ the equation behaves as follows d2Ψ∞ dy2−1 4Ψ∞= 0. (17) This equation has as solution Ψ∞(y) =Aexpy 2+Bexp−y 2. (18) 90TakingA= 0, we eliminate expy 2since it diverges in the limit y→∞, keeping only the attenuated exponential. We can now suggest that Ψ has the following form Ψ(y) = exp−y 2ψ(y). (19) Plugging it in the differential equation for y( eq. 15) one gets: yd2ψ dy2+ (1 2−y)dψ dy+ (κ 2−1 4)ψ= 0. (20) The latter is a confluent hypergeometric equation4: zd2y dz2+ (c−z)dy dz−ay= 0. (21) The general solution of this equation is y(z) =A1F1(a;c,z) +B z1−c1F1(a−c+ 1;2−c,z), (22) where the confluent hypergeometric equation is defined by 1F1(a;c,z) =∞/summationdisplay n=0(a)nxn (c)nn!. (23) Comparing now our equation with the standard confluent hyper geomet- ric equation, one can see that the general solution of the firs t one is ψ(y) =A1F1(a;1 2,y) +B y1 21F1(a+1 2;3 2,y), (24) where a=−(κ 2−1 4). (25) If we keep these solutions in their present form, the normali zation con- dition is not satisfied for the wavefunction because from the asymptotic behaviour of the confluent hypergeometric function5it follows ( taking into account ony the dominant exponential behavior ) : Ψ(y) =e−y 2ψ(y)→const. ey 2ya−1 2. (26) 4It is also known as Kummer’s differential equation. 5The asymptotic behavior for |x|→ ∞ is 1F1(a;c, z)→Γ(c) Γ(c−a)e−iaπx−a+Γ(c) Γ(a)exxa−c. 91The latter approximation leads to a divergence in the normal ization in- tegral, which physically is not acceptable. What one does in this case is to impose the termination condition for the series6, that is , the series has only a finite number of terms and therefore it is a polynomial o fnorder. We thus notice that asking for a finite normalization constan t (as already known, a necessary condition for the physical interpretati on in terms of prob- abilities), leads us to the truncation of the series, which s imultaneously gen- erates the quantization of energy. In the following we consider the two possible cases 1)a=−nandB= 0 κ 2−1 4=n . (27) The eigenfunctions are given by Ψn(x) =Dnexp−λx2 21F1(−n;1 2,λx2) (28) and the energy is: En= ¯hω(2n+1 2). (29) 2)a+1 2=−nandA= 0 κ 2−1 4=n+1 2. (30) The eigenfunctions are now Ψn(x) =Dnexp−λx2 2x1F1(−n;3 2,λx2), (31) whereas the stationary energies are En= ¯hω[(2n+ 1) +1 2]. (32) The polynomials obtained by this truncation of the confluent hypergeo- metric series are called Hermite polynomials and in hyperge ometric notation they are H2n(η) = (−1)n(2n)! n!1F1(−n;1 2,η2) (33) 6The truncation condition of the confluent hypergeometric se ries1F1(a;c, z) isa=−n, where nis a nonnegative integer ( i.e., zero included). 92H2n−1(η) = (−1)n2(2n+ 1)! n!η1F1(−n;3 2,η2). (34) We can now combine the obtained results ( because some of them give us the even cases and the others the odd ones ) in a single expre ssion for the eigenvalues and eigenfunctions Ψn(x) =Dnexp−λx2 2Hn(√ λx) (35) En= (n+1 2)¯hω n = 0,1,2... (36) The HO energy spectrum is equidistant, i.e., there is the sam e energy difference ¯hωˆ ıbetween any consequitive neighbour levels. Another rema rk refers to the minimum value of the energy of the oscillator; s omewhat sur- prisingly it is not zero. This is considered by many people to be a pure quantum result because it is zero when ¯ h→0. It is known as the zero point energy and the fact that it is different of zero is the main characteri stic of all confining potentials. The normalization constant is easy to calculate Dn=/bracketleftigg/radicaligg λ π1 2nn!/bracketrightigg1 2 . (37) Thus, one gets the following normalized eigenfunctions of t he 1D opera- tor Ψn(x) =/bracketleftigg/radicaligg λ π1 2nn!/bracketrightigg1 2 exp(−λx2 2)Hn(√ λx). (38) Creation and anihilation operators: ˆa†andˆa There is another approach to deal with the HO besides the conv entional one of solving the Schr¨ odinger equation. It is the algebraic me thod, also known as the method of creation and annihilation (ladder) operato rs. This is a very efficient procedure, which can be successfully applied to man y quantum- mechanical problems, especially when dealing with discret e spectra. Let us define two nonhermitic operators aanda†: a=/radicalbiggmω 2¯h(x+ip mω) (39) 93a†=/radicalbiggmω 2¯h(x−ip mω). (40) These operators are known as annihilation operator and creation operator , respectively (the reason of this terminology will be seen i n the following, though one can claim that it comes from quantum fie ld theories). Let us calculate the commutator of these operators [a,a†] =mω 2¯h[x+ip mω,x−ip mω] =1 2¯h(−i[x,p] +i[p,x]) = 1, (41) where we have used the commutator [x,p] =i¯h . (42) Therefore the annihilation and creation operators do not co mmute, since we have [a,a†] = 1. (43) Let us also introduce the very important number operator ˆN: ˆN=a†a . (44) This operator is hermitic as one can readily prove using ( AB)†=B†A† : ˆN†= (a†a)†=a†(a†)†=a†a=ˆN . (45) Considering now that a†a=mω 2¯h(x2+p2 m2ω2) +i 2¯h[x,p] =ˆH ¯hω−1 2(46) we notice that the Hamiltonian can be written in a quite simpl e form as a function of the number operator ˆH= ¯hω(ˆN+1 2). (47) The number operator bear this name because its eigenvalues a re precisely the subindices of the eigenfunctions on which it acts ˆN|n∝an}b∇acket∇i}ht=n|n∝an}b∇acket∇i}ht, (48) where we have used the notation Psin=|n∝an}b∇acket∇i}ht. (49) 94Applying this fact to (47), we get ˆH|n>= ¯hω(n+1 2)|n> . (50) On the other hand, from the Schr¨ odinger equation we know tha tˆH| n >=E|n >. In this way, it comes out that the energy eigenvalues are given by En= ¯hω(n+1 2). (51) This result is identical (as it should be) to the result (36). We go ahead and show why the operators aanda†bear the names they have. For this, we calculate the commutators [ˆN,a] = [a†a,a] =a†[a,a] + [a†,a]a=−a , (52) which can be obtained from [ a,a] = 0 and (43). Similarly, let us calculate [ˆN,a†] = [a†a,a†] =a†[a,a†] + [a†,a†]a=a†. (53) Using these two commutators, we can write ˆN(a†|n>) = ([ ˆN,a†] +a†ˆN)|n> = (a†+a†ˆN)|n> (54) =a†(1 +n)|n>= (n+ 1)a†|n> . By a similar procedure, one can also obtain ˆN(a|n>) = ([ ˆN,a] +aˆN)|n>= (n−1)a|n> . (55) The expression (54) implies that one can consider the ket a†|n >as an eigenket of that number operator for which the eigenvalue is raised by one unit. In physical terms, this means that an energy quanta has been produced by the action of a†on the ket. This already expains the name of creation operator. Similar comments with corresponding conclusion can be infered for the operator a, originating the name of annihilation operator (an energy quanta is eliminated from the system when this operator is pu t in action). Moreover, eq. (54) implies the proportionality of the kets a†|n >and |n+ 1>: a†|n>=c|n+ 1> , (56) 95wherecis a constant that should be determined. Considering in addi tion (a†|n>)†=<n|a=c∗<n+ 1|, (57) one can perform the following calculation <n|a(a†|n>) =c∗<n+ 1|(c|n+ 1>) (58) <n|aa†|n>=c∗c<n + 1|n+ 1> (59) <n|aa†|n>=|c|2. (60) But from the commutation relation for the operators aanda† [a,a†] =aa†−a†a=aa†−ˆN= 1, (61) we have aa†=ˆN+ 1. (62) Substituting in (60), we get <n|ˆN+ 1|n>=<n|n>+<n|ˆN|n>=n+ 1 =|c|2.(63) Asking conventionally for a positive and real c, the following value is obtained c=√ n+ 1. (64) Consequently, we have a†|n>=√ n+ 1|n+ 1> . (65) For the annihilation operator, following the same procedur e one can get the following relation a|n>=√n|n−1> . (66) Let us show now that the values of nshould be nonnegative integers. For this, we employ the positivity requirement for the norm, applying it to the state vector a|n >. The latter condition tells us that the interior product of the vector with its adjunct (( a|n>)†=<n|a†) should always be nonnegative (<n|a†)·(a|n>)≥0. (67) This relationship is nothing else but <n|a†a|n>=<n|ˆN|n>=n≥0. (68) 96Thus,ncannot be negativ. It should be an integer since were it not by applying iteratively the annihilation operator we would be lead to negative values ofn, which would be a contradiction to the previous statement. It is possible to express the state n(|n∝an}b∇acket∇i}ht) directly as a function of the ground state (|0∝an}b∇acket∇i}ht) using the creation operator. Let us see how proceeds this important iteration |1∝an}b∇acket∇i}ht=a†|0∝an}b∇acket∇i}ht (69) |2∝an}b∇acket∇i}ht= [a† √ 2]|1∝an}b∇acket∇i}ht= [(a†)2 √ 2!]|0∝an}b∇acket∇i}ht (70) |3∝an}b∇acket∇i}ht= [a† √ 3]|2∝an}b∇acket∇i}ht= [(a†)3 √ 3!]|0∝an}b∇acket∇i}ht (71) ... |n∝an}b∇acket∇i}ht= [(a†)n √ n!]|0∝an}b∇acket∇i}ht. (72) One can also apply this method to get the eigenfunctions in th e config- uration space. To achieve this, we start with the ground stat e a|0∝an}b∇acket∇i}ht= 0. (73) In thexrepresentation, we have ˆaΨ0(x) =/radicalbiggmω 2¯h(x+ip mω)Ψ0(x) = 0. (74) Recalling the form of the momentum operator in the xrepresentation, we can obtain a differential equation for the wavefunction of th e ground state. Moreover, introducing the definition x0=/radicalig ¯h mω, we have (x+x2 0d dx)Ψ0= 0. (75) The latter equation can be readily solved, and normalizing ( its integral from −∞to∞should be equal to unity), we obtain the wavefunction of the ground state Ψ0(x) = (1/radicalig√πx0)e−1 2(x x0)2 . (76) 97The rest of the eigenfunctions, which describe the HO excite d states, can be obtained employing the creation operator. The procedure is the following Ψ1=a†Ψ0= (1√ 2x0)(x−x2 0d dx)Ψ0 (77) Ψ2=1√ 2(a†)2Ψ0=1√ 2!(1√ 2x0)2(x−x2 0d dx)2Ψ0. (78) By mathematical induction, one can show that Ψn=1/radicalig√π2nn!1 xn+1 2 0(x−x2 0d dx)ne−1 2(x x0)2 . (79) Time evolution of the oscillator In this section we shall illustrate on the HO example the way o f working with the Heisenberg representation in which the states are fixed i n time and only the operators evolve. Thus, we shall consider the operators as functions of time and obtain explicitly the time evolution of the HO pos ition and momentum operators, aanda†, respectively. The Heisenberg equations of the motion for pandxare dˆp dt=−∂ ∂ˆxV(ˆ x) (80) dˆx dt=ˆp m. (81) Hence the equations of the motion for xandpin the HO case are the following dˆp dt=−mω2ˆx (82) dˆx dt=ˆp m. (83) These are a pair of coupled equations, which are equivalent t o a pair of uncoupled equations for the creation and annihilation operators. Exp licitly, we have da dt=/radicalbiggmω 2¯hd dt(ˆx+iˆp mω) (84) da dt=/radicalbiggmω 2¯h(dˆx dt+i mωdˆp dt). (85) 98Substituting (82) and (83) in (85), we get da dt=/radicalbiggmω 2¯h(ˆp m−iωˆx) =−iωa . (86) Similarly, one can obtain a differential equation for the cre ation operator da† dt=iωa†. (87) The differential evolution equations for the creation and an nihilation opera- tors can be immediately integrated leading to the explicit e volution of these operators as follows a(t) =a(0)e−iωt(88) a†(t) =a†(0)eiωt. (89) It is worth noting based on these results and eqs. (44) and (47 ) that both the Hamiltonian and the number operator are not time dep endent. Using the latter two results, we can obtain the position and m omentum operators as functions of time as far as they are expressed in terms of the creation and annihilation operators ˆx=/radicaligg ¯h 2mω(a+a†) (90) ˆp=i/radicaligg m¯hω 2(a†−a). (91) Substituting them, one gets ˆx(t) = ˆx(0)cosωt+ˆp(0) mωsinωt (92) ˆp(t) =−mωˆx(0)sinωt+ ˆp(0)cosωt . (93) The time evolution of these operators is the same as for the cl assical equations of the motion. Thus, we have shown here the explicit evolution form of the fo ur HO ba- sic operators, and also we illustrated the effective way of wo rking in the Heisenberg representation. 99The 3D HO We commented on the importance in physics of the HO at the very beginning of our analysis of the quantum HO. If we will consider a 3D anal og, we would be led to study a Taylor expansion in three variables7retaining the terms up to the second order, we get a quadratic form in the most gene ral case. The problem at hand in this approximation is not as simple as i t might look from the examination of the corresponding potential V(x,y,z) =ax2+by2+cz2+dxy+exz+fyz . (94) There are however many systems with spherical symmetry or fo r which this symmetry is sufficiently exact. ˆIn acest caz: V(x,y,z) =K(x2+y2+z2), (95) which is equivalent to saying that the second unmixed partia l derivatives have all the same value, denoted by Kin our case). We can add that this is a good approximation in the case in which the values of the mix ed second partial derivatves are small in comparison to the unmixed on es. When these conditions are satisfied and the potential is give n by (95), we say that the system is a 3D spherically symmetric HO . The Hamiltonian in this case is of the form ˆH=−¯h2 2m▽2+mω2 2r2, (96) where the Laplace operator is given in spherical coordinate s andris the spherical radial coordinate. Since the potential is time independent the energy is conser ved. In addition, because of the spherical symmetry the orbital momentum is al so conserved. having two conserved quantities, we may say that to each of it one can associate a quantum number. Thus, we can assume that the eige nfunctions depend on two quantum numbers (even though for this case we sh all see that another one will occur). Taking care of these comments, the equation of interest is 7It is possible to express the Taylor series in the neighbourh ood of r0as an exponential operator e[(x−xo)+(y−yo)+(z−zo)](∂ ∂x+∂ ∂y+∂ ∂z)f(ro). 100ˆHΨnl=EnlΨnl. (97) The Laplace operator in spherical coordinates reads ▽2=∂2 ∂r2+2 r∂ ∂r−ˆL2 ¯h2r2(98) and can be also inferred from the known fact ˆL2=−¯h2[1 sinθ∂ ∂θ(sinθ∂ ∂θ) +1 sinθ2∂2 ∂ϕ2]. (99) The eigenfunctions of ˆL2are the spherical harmonics , i.e. ˆL2Ylml(θ,ϕ) =−¯h2l(l+ 1)Ylml(θ,ϕ) (100) The fact that the spherical harmonics ‘wear’ the quantum num berml introduces it in the total wavefunction Ψ nlml. In order to achieve the separation of the variables and funct ions, the follow- ing substitution is proposed Ψnlml(r,θ,ϕ) =Rnl(r) rYlml(θ,ϕ). (101) Once this is plugged in the Schr¨ odinger equation, the spati al part is separated from the angular one; the latter is identified with an operator that is proportional to the square of the orbital momentum, f or which the eigenfunctions are the spherical harmonics, whereas for th e spatial part the following equation is obtained R′′ nl+ (2mEnl ¯h2−m2ω2 ¯h2r2−l(l+ 1) r2)Rnl(r) = 0. (102) Using the definitions (7) and (8), the previous equation is pr ecisely of the form (9), unless the angular momentum term, which is commonl y known as the unghiular, careˆ ın mod comun se cunoa¸ ste ca angular momentum barrier R′′ nl+ (k2−λ2r2−l(l+ 1) r2)Rnl= 0. (103) To solve this equation, we shall start with its asymptotic an alysis. If we shall consider first r→∞, we notice that the orbital momentum term is negligible, so that in this limit the asymptotic behavior is similar to that of (9), leading to 101Rnl(r)∼exp−λr2 2for limr→∞. (104) If now we pass to the behavior close to zero, we can see that the dominant term is that of the orbital momentum, i.e., the differential e quation (102) in this limit turns into R′′ nl−l(l+ 1) r2Rnl= 0. (105) This is a differential equation of the Euler type8, whose two independent solutions are Rnl(r)∼rl+1orr−lfor limr→0. (106) The previous arguments lead to proposing the substitution Rnl(r) =rl+1exp−λr2 2φ(r). (107) One can also use another substitution Rnl(r) =r−lexp−λr2 2v(r), (108) which, however, produces the same solutions as (107) (showi ng this is a help- ful exercise). Substituing (107) in (103), the following di fferential equation forφis obtained φ′′+ 2(l+ 1 r−λr)φ′−[λ(2l+ 3)−k2]φ= 0. (109) Using now the change of variable w=λr2, one gets wφ′′+ (l+3 2−w)φ′−[1 2(l+3 2)−κ 2]φ= 0, (110) 8An equation of the Euler type has the form xny(n)(x) +xn−1y(n−1)(x) +···+xy′(x) +y(x) = 0 . Its solutions are of the type xαthat are plugged in the equation obtaining a polynomial inα. 102whereκ=k2 2λ=E ¯hωhas been introduced. We see that we found again a dif- ferential equation of the confluent hypergeometric type hav ing the solutions (see (21) and (22)) φ(r) =A1F1[1 2(l+3 2−κ);l+3 2,λr2]+B r−(2l+1)1F1[1 2(−l+1 2−κ);−l+1 2,λr2]. (111) The second particular solution cannot be normalized becaus e diverges strongly in zero. This forces one to take B= 0, therefore φ(r) =A1F1[1 2(l+3 2−κ);l+3 2,λr2]. (112) Using the same arguments as in the 1D HO case, that is, imposin g a regular solution at infinity, leads to the truncation of the series, w hich implies the quantization of the energy. The truncation is explicitly 1 2(l+3 2−κ) =−n , (113) where introducing κwe get the energy spectrum Enl= ¯hω(2n+l+3 2). (114) One can notice that for the 3D spherically symmetric HO there is a zero point energy3 2¯hω. The unnormalized eigenfunctions are Ψnlm(r,θ,ϕ) =rle−λr2 21F1(−n;l+3 2,λr2)Ylm(θ,ϕ). (115) 5P. Problems Problem 5.1 Determine the eigenvalues and eigenfunctions of the HO in th e momentum space . The quantum HO Hamiltonian reads ˆH=ˆp2 2m+1 2mω2ˆx2. In the momentum space, the operators ˆ xand ˆphave the following form ˆp→p 103ˆx→i¯h∂ ∂p. Thus, the HO quantum Hamiltonian in the momentum representa tion is ˆH=p2 2m−1 2mω2¯h2d2 dp2. We have to solve the eigenvalue problem (i.e., to get the eige nfunctions and the eigenvalues) given by (5), which, with the previous Hami ltonian, turns into the following differential equation d2Ψ(p) dp2+ (2E m¯h2ω2−p2 m2¯h2ω2)Ψ(p) = 0. (116) One can see that this equation is identical, up to some consta nts, with the differential equation in the space of configurations (eq. (6) ). Just to show another way of solving it, we define two parameters, which are analogous to those in (7) and (8) k2=2E m¯h2ω2λ=1 m¯hω. (117) With these definitions, we get the differential eq. (9) and the refore the solution sought for (after performing the asymptotic analy sis) is of the form Ψ(y) =e−1 2yφ(y), (118) wherey=λp2andλis defined in (117). Substitute (118) in (116) taking care to put (118) in the variable p. One gets a differential equation in the variableφ d2φ(p) dp2−2λpdφ(p) dp+ (k2−λ)φ(p) = 0. (119) We shall now make the change of variable u=√ λpthat finally leads us to the Hermite equation d2φ(u) du2−2udφ(u) du+ 2nφ(u) = 0, (120) wherenis a nonnegative integer and where we have put k2 λ−1 = 2n . 104From here and the definitions given in (117) one can easily con clude that the eigenvalues are given by En= ¯hω(n+1 2). The solutions for (120) are the Hermite polynomials φ(u) =Hn(u) and the unnormalized eigenfunctions are Ψ(p) =Ae−λ 2p2Hn(√ λp). Problem 5.2 Prove that the Hermite polynomials can be expressed in the fo l- lowing integral representation Hn(x) =2n √π/integraldisplay∞ −∞(x+iy)ne−y2dy . (121) This representation of Hermite polynomials is not really us ual, though it can prove useful in many cases. In order to accomplish the p roof, we shall expand expand the integral and next prove that what we’ve got is identical to the series expansion of the Hermite polynomials that read s [n 2]/summationdisplay k=0(−1)kn! (n−2k)!k!(2x)n−2k, (122) where the symbol [ c], indicating where the series terminates, denotes the greatest integer less or equal to c. The first thing we shall do is to expand the binomial in the inte gral by using the well-known binomial theorem (x+y)n=n/summationdisplay m=0n! (n−m)!m!xn−mym. Thus (x+iy)n=n/summationdisplay m=0n! (n−m)!m!imxn−mym, (123) 105which plugged in the integral leads to 2n √πn/summationdisplay m=0n! (n−m)!m!imxn−m/integraldisplay∞ −∞yme−y2dy . (124) Inspecting of the integrand we realize that the integral is n ot zero when mis even, whereas it is zero when mis odd. Using the even notation m= 2k, we get 2n √π[n 2]/summationdisplay k=0n! (n−2k)!(2k)!i2kxn−2k2/integraldisplay∞ 0y2ke−y2dy . (125) Under the change of variable u=y2, the integral turns into a gamma function 2n √π[n 2]/summationdisplay k=0n! (n−2k)!(2k)!i2kxn−2k/integraldisplay∞ 0uk−1 2e−udu , (126) more precisely Γ( k+1 2), which can be expressed in terms of factorials ( of course forkan integer) Γ(k+1 2) =(2k)! 22kk!√π . Plugging this expression in the sum and using i2k= (−1)k, one gets [n 2]/summationdisplay k=0(−1)kn! (n−2k)!k!(2x)n−2k, (127) which is identical to (122), hence completing the proof. Problem 5.3 Show that Heisenberg’s uncertainty relation is satisfied by doing the calculation using the HO eigenfunctions . We have to show that for any Ψ n, we have <(∆p)2(∆x)2>≥¯h2 4, (128) where the notation <>means the mean value. We shall separately calculate <(∆p)2>and<(∆x)2>, where each of these expressions is 106<(∆p)2>=<(p−<p> )2>=<p2−2p<p> +<p>2>=<p2>−<p>2, <(∆x)2>=<(x−<x> )2>=<x2−2x<x> +<x>2>=<x2>−<x>2. First of all, we shall prove that both the mean of xas well as of pare zero. For the mean of x, we have <x> =/integraldisplay∞ −∞x[Ψn(x)]2dx . This integral is zero because the integrand is odd. Thus <x> = 0. (129) The same argument holds for the mean of p, if we do the calculation in the momentum space, employing the functions obtained in pro blem 1. It is sufficient to notice that the functional form is the same (only the symbol does change). Thus <p> = 0. (130) Let us now calculate the mean of x2. We shall use the virial theorem9. We first notice that <V > =1 2mω2<x2> . Therefore, it is possible to relate the mean of x2directly to the mean of the potential for this case (implying the usage of the virial the orem). <x2>=2 mω2<V > . (131) We also need the total energy <H > =<T > +<V > , 9We recall that the virial theorem in quantum mechanics asser ts that 2< T > =<r· ▽V(r)> . For a potential of the form V=λxn, the virial theorem gives 2< T > =n < V > , where Tis the kinetic energy and Vis the potential energy. 107for which again one can make use of the virial theorem (for n= 2) <H > = 2<V > . (132) Thus, we obtain <x2>=<H > mω2=¯hω(n+1 2) mω2(133) <x2>=¯h mω(n+1 2). (134) Similarly, the mean of p2can be readily calculated <p2>= 2m<p2 2m>= 2m<T > =m<H > =m¯hω(n+1 2).(135) Employing (133) and (135), we have <(∆p)2(∆x)2>= (n+1 2)2¯h2. (136) Based on this result, we come to the conclusion that in the HO s tationary states that actually have not been directly used, Heisenber g’s uncertainty relation is satisfied and it is at the minimum for the ground st ate,n= 0. Problem 5.4 Obtain the matrix elements of the operators a,a†,ˆx, and ˆp. Let us first find the matrix elements for the creation and annih ilation operators, which are very helpful for all the other operator s. We shall use the relatinships (65) and (66), leading to <m|a|n>=√n<m|n−1>=√nδm,n−1. (137) Similarly for the creation operator we have the result <m|a†|n>=√ n+ 1<m|n+ 1>=√ n+ 1δm,n+1. (138) Let us proceed now with the calculation of the matrix element s of the position operator. For this, let us express this operator in terms of creation 108and annihilation operators. Using the definitions (39) and ( 40), one can immediately prove that the position operator is given by ˆx=/radicaligg ¯h 2mω(a+a†). (139) Employing this result, the matrix elements of the operator ˆ xcan be readily calculated <m|ˆx|n> =<m|/radicaligg ¯h 2mω(a+a†)|n> =/radicaligg ¯h 2mω[√nδm,n−1+√ n+ 1δm,n+1].(140) Following the same procedure we can calculate the matrix ele ments of the momentum operator, just by taking into account that ˆ pis given in terms of the creation and annihilation operators as follows ˆp=i/radicaligg m¯hω 2(a†−a). (141) This leads us to <m|ˆp|n>=i/radicaligg m¯hω 2[√ n+ 1δm,n+1−√nδm,n−1]. (142) One can realize the ease of the calculations when the matrix e lements of the creation and annihilation operators are used. Finally, we remark on the nondiagonality of the obtained matrix elements. This is not so much of a surprise because the employed representation is that of the number operator and none of the four operators do not commute with it. Problem 5.5 Find the mean values of ˆx2andˆp2for the1D HO and use them to calculate the mean (expectation) values of the kinetic and p oten- tial energies. Compare the result with the virial theorem. First of all, let us obtain the mean value of ˆ x2. For this, we use eq. (139) that leads us to ˆx2=¯h 2mω(a2+ (a†)2+a†a+aa†). (143) 109Recall that the creation and annihilation operators do not c ommute. Based on (143), we can calculate the mean value of ˆ x2 <ˆx2>=<n|ˆx2|n> =¯h 2mω[/radicalig n(n−1)δn,n−2+/radicalig (n+ 1)(n+ 2)δn,n+2 +nδn,n+ (n+ 1)δn,n], (144) which shows that <ˆx2>=<n|ˆx2|n>=¯h 2mω(2n+ 1). (145) In order to calculate the mean value of ˆ p2we use (141) that helps us to express this operator in terms of the creation and annihilat ion operators ˆp2=−m¯hω 2(a2+ (a†)2−aa†−a†a). (146) This leads us to <ˆp2>=<n|ˆp2|n>=m¯hω 2(2n+ 1). (147) The latter result practically gives us the mean kinetic ener gy <ˆT >=<ˆp2 2m>=1 2m<ˆp2>=¯hω 4(2n+ 1). (148) On the other hand, the mean value of the potential energy <ˆV >=<1 2mω2ˆx2>=1 2mω2<ˆx2>=¯hω 4(2n+ 1), (149) where (145) has been used. We can see that these mean values are equal for any n, which confirms the quantum virial theorem, telling us that for a quadratic ( HO) potential, the mean values of the kinetic and potential energies should be equal and therefore be half of the mean value of the total energy. 1106. THE HYDROGEN ATOM Introduction In this chapter we shall study the hydrogen atom by soving the time- independent Schr¨ odinger equation for the potential due to two charged par- ticles such as the electron and the proton, and the Laplace op erator in spherical coordinates. From the mathematical viewpoint, t he method of separation of variables will be employed, and a physical int erpretation of the wavefunction as solution of the Schr¨ odinger equation i n this important case will be provided, together with the interpretation of t he quantum num- bers and of the probability densities. The very small spatial scale of the hydrogen atom is a clue tha t the related physical phenomena enter the domain of applicability of the quantum me- chanics, for which the atomic processes have been a successf ul area since the early days of the quantum approaches. Quantum mechanics , as any other theoretical framework, gives relationships between observable quan- tities. Since the uncertainty principle leads to a substant ial change in the understanding of observables at the conceptual level, it is important to have a clear idea on the notion of atomic observable. As a matter of fact, the real quantities on which quantum mechanics offers explicit a nswers and con- nections are always probabilites. Instead of saying, for ex ample, that the radius of the electron orbit in the fundamental state of the h ydrogen atom is always 5.3×10−11m, quantum mechanics asserts that this is a truly mean radius (not in the measurable sense). Thus, if one performs a n appropriate experiment, one gets, precisely as in the case of the common a rrangement of macroscopic detectors probing macroscopic properties o f the matter, ran- dom values around the mean value 5 .3×10−11m. In other words, from the viewpoint of the experimental errors there is no essential d ifference with re- gard to the classical physics. The fundamental difference is in the procedure of calculating the mean values within the theoretical frame work. As is known, for performing quantum-mechanical calculatio ns, one needs a corresponding wave function Ψ. Although Ψ has no direct phy sical inter- pretation, the square modulus |Ψ|2calculated at an arbitrary position and given moment is proportional to the probability to find the pa rticle in the infinitesimal neighbourhood of that point at the given time. The purpose of quantum mechanics is to determine Ψ for a specified particle i n the prepared experimental conditions. 111Before proceeding with the rigorous approaches of getting Ψ for the hydrogen electron, we will argue on several general require ments regarding the wave function. First, the integral of |Ψ|2over all space should be finite if we really want to deal with a localizable electron. In addi tion, if /integraldisplay∞ −∞|Ψ|2dV= 0, (1) then the particle does not exist. |Ψ|2cannot be negative or complex because of simple mathematical reasons. In general, it is co nvenient to identify|Ψ|2with the probability P not just the proportionality. In orde r that|Ψ|2be equal to P one imposes /integraldisplay∞ −∞|Ψ|2dV= 1, (2) because /integraldisplay∞ −∞PdV= 1 (3) is the mathematical way of saying that the particle exists at a point in space at any given moment. A wave function respecting eq. 2 is said to be normalized. Besides this, Ψ should be single valued, becaus e P has a unique value at a given point and given time. Another condition is th at Ψ and its partial first derivatives∂Ψ ∂x,∂Ψ ∂y,∂Ψ ∂zshould be continuous at any arbitrary point. The Schr¨ odinger equation is considered as the fundamental equation of nonrelativistic quantum mechanics in the same sense in whic h Newton’s force law is the fundamental equation of motion of newtonian mechanics. Notice however that we have now a wave equation for a function Ψ which is not directly measurable. Once the potential energy is given, one can solve the Schr¨ od inger equa- tion for Ψ, implying the knowledge of the probability densit y|Ψ|2as a function of x,y,z,t . In many cases of interest, the potential energy does not depend on time. Then, the Schr¨ odinger equation simplifies c onsiderably. Notice, for example, that for a 1D free particle the wave func tion can be written Ψ(x,t) =Ae(−i/¯h)(Et−px) =Ae−(iE/¯h)te(ip/¯h)x =ψ(x)e−(iE/¯h)t, (4) 112i.e., Ψ(x,t) is the product of a time-dependent phase e−(iE/¯h)tand a sta- tionary wave function ψ(x). In the general case, the stationary Schr¨ odinger equation c an be solved, under the aforementioned requirements, only for certain va lues of the energy E. This is not a mathematical difficulty, but merely a fundamen tal physi- cal feature. To solve the Schr¨ odinger equation for a given s ystem means to get the wave function ψ, as a solution for which certain physical bound- ary condition hold and, in addition, as already mentioned, i t is continuous together with its first derivative everywhere in space, is fin ite, and single valued. Thus, the quantization of energy occurs as a natural theoretical element in wave mechanics, whereas in practice as a universa l phenomenon, characteristic for all stable microscopic systems. Schr¨ odinger equation for the hydrogen atom In this section, we shall apply the Schr¨ odinger equation to the hydrogen atom, about which one knows that it is formed of a positive nuc leus/proton of charge + eand an electron of charge - e. The latter, being 1836 times smaller in mass than the proton, is by far more dynamic. If the interaction between two particles is of the type u(r) =u(|/vector r1−/vector r2|), the problem of the motion is reduced both classically and qua ntum to the motion of a single particle in a field of spherical symmetry. I ndeed, the Lagrangian L=1 2m1˙/vector r2 1+1 2m2˙/vector r2 2−u(|/vector r1−/vector r2|) (5) is transformed, using /vector r=/vector r1−/vector r2 (6) and /vectorR=m1/vector r1+m2/vector r2 m1+m2, (7) in the Lagrangian L=1 2M˙/vectorR2+1 2µ˙/vector r2−u(r), (8) where M=m1+m2 (9) and µ=m1m2 m1+m2. (10) 113On the other hand, the momentum is introduced through the Lag range formula /vectorP=∂L ∂˙/vectorR=M˙/vectorR (11) and /vector p=∂L ∂˙/vector r=m˙/vector r , (12) that allows to write the classical Hamilton function in the f orm H=P2 2M+p2 2m+u(r). (13) Thus, one can obtain the hamiltonian operator for the corres ponding quantum problem with commutators of the type [Pi,Pk] =−i¯hδik (14) and [pi,pk] =−i¯hδik. (15) These commutators implies a Hamiltonian operator of the for m ˆH=−¯h2 2M∇2 R−¯h2 2m∇2 r+u(r), (16) which is fundamental for the study of the hydrogen atom by mea ns of the stationary Schr¨ odinger equation ˆHψ=Eψ . (17) This form does not include relativistic effects, i.e., elect ron velocities close to the velocity of light in vacuum). The potential energy u(r) is the electrostatic one u=−e2 4πǫ0r(18) There are two possibilities. The first is to express uas a function of the cartesian coordinates x,y,z, substituing rby/radicalbig x2+y2+z2. The second is to write the Schr¨ odinger equation in spherical polar coo rdinatesr,θ,φ. Because of the obvious spherical symmetry of this case, we sh all deal with the latter approach, which leads to considerable mathematical simplifications. 114In spherical coordinates, the Schr¨ odinger equation reads 1 r2∂ ∂r/parenleftbigg r2∂ψ ∂r/parenrightbigg +1 r2sinθ∂ ∂θ/parenleftbigg sinθ∂ψ ∂θ/parenrightbigg +1 r2sin2θ∂2ψ ∂φ2+2m ¯h2(E−u)ψ= 0 (19) Substituing (18), and multiplying the whole equation by r2sin2θ, one gets sin2θ∂ ∂r/parenleftbigg r2∂ψ ∂r/parenrightbigg +sinθ∂ ∂θ/parenleftbigg sinθ∂ψ ∂θ/parenrightbigg +∂2ψ ∂φ2+2mr2sin2θ ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg ψ= 0. (20) This equation is a partial differential equation for the elec tron wavefunction ψ(r,θ,φ) ‘within’ the atomic hydrogen. Together with the various co nditions that the wavefunction ψ(r,θ,φ) should fulfill [for example, ψ(r,θ,φ) should have a unique value at any spatial point ( r,θ,φ)], this equation specifies in a complete manner the behavior of the hydrogen electron. T o see the explicit behavior, we shall solve eq. 20 for ψ(r,θ,φ) and we shall interpret appropriately the obtained results. Separation of variables in spherical coordinates The real usefulness of writing the hydrogen Schr¨ odinger eq uation in spherical coordinates consists in the easy way of achieving the separa tion procedure in three independent equations, each of them being one-dime nsional. The separation procedure is to seek the solutions for which the w avefunction ψ(r,θ,φ) has the form of a product of three functions, each in one of th e three spherical variables, namely R(r), depending only on r; Θ(θ) depending only onθ, and Φ(φ) that depends only on φ. This is quite similar to the separation of the Laplace equation. Thus ψ(r,θ,φ) =R(r)Θ(θ)Φ(φ). (21) TheR(r) function describes the differential variation of the elect ron wave- functionψalong the vector radius coming out from the nucleus, with θand φassumed to be constant. The differential variation of ψwith the polar angleθalong a meridian of an arbitrary sphere centered in the nucle us is described only by the function Θ( θ) for constant randφ. Finally, the func- tion Φ(φ) describes how varies ψwith the azimuthal angle φalong a parallel of an arbitrary sphere centered at the nucleus, under the con ditions that r andθare kept constant. 115Usingψ=RΘΦ, one can see that ∂ψ ∂r= ΘΦdR dr, (22) ∂ψ ∂θ=RΦdΘ dθ, (23) ∂ψ ∂φ=RΘdΦ dφ. (24) Obviously, the same type of formulas are maintained for the u nmixed higher- order derivatives. Subtituting them in eq. 20, and after dev iding byRΘΦ, we get sin2θ Rd dr/parenleftbigg r2dR dr/parenrightbigg +sinθ Θd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg +1 Φd2Φ dφ2+2mr2sin2θ ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg = 0. (25) The third term of this equation is a function of the angle φonly, while the other two terms are functions of randθ. We rewrite now the previous equation in the form sin2θ R∂ ∂r/parenleftbigg r2∂R ∂r/parenrightbigg +sinθ Θ∂ ∂θ/parenleftbigg sinθ∂Θ ∂θ/parenrightbigg +2mr2sin2θ ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg =−1 Φ∂2Φ ∂φ2. (26) This equation can be correct only if the two sides are equal to the same constant, because they are functions of different variables . It is convenient to denote this (separation) constant by m2 l. The differential equation for the Φ function is −1 Φ∂2Φ ∂φ2=m2 l. (27) If one substitutes m2 lin the right hand side of eq. 26 and devides the resulting equation by sin2θ, after regrouping the terms, the fllowing result is obtained 1 Rd dr/parenleftbigg r2dR dr/parenrightbigg +2mr2 ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg =m2 l sin2θ−1 Θ sinθd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg . (28) Once again, we end up with an equation in which different varia bles occur in the two sides, thus forcing at equating of both sides to the same constant. 116For reasons that will become clear later on, we shall denote t his constant by l(l+ 1). The equations for the functions Θ( θ) andR(r) reads m2 l sin2θ−1 Θ sinθd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg =l(l+ 1) (29) and 1 Rd dr/parenleftbigg r2dR dr/parenrightbigg +2mr2 ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg =l(l+ 1). (30) The equations 27, 29 and 30 are usually written in the form d2Φ dφ2+m2 lΦ = 0, (31) 1 sinθd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg +/bracketleftigg l(l+ 1)−m2 l sin2θ/bracketrightigg Θ = 0, (32) 1 r2d dr/parenleftbigg r2dR dr/parenrightbigg +/bracketleftigg 2m ¯h2/parenleftigg e2 4πǫ0r+E/parenrightigg −l(l+ 1) r2/bracketrightigg R= 0. (33) Each of these equations is an ordinary differential equation for a function of a single variable. In this way, the Schr¨ odinger equation for the hydrogen electron that initially was a partial differential equation for a function ψof three variables got a simple form of three 1D ordinary differe ntial equations for unknown functions of one variable. Interpreting the separation constants: the quantum numbers The solution for the azimuthal part Eq. 31 is readily solved leading to the following solution Φ(φ) =Aφeimlφ, (34) whereAφis the integration constant. One of the conditions that a wav e- functions should fulfill and therefore also Φ, being a compon ent of the full wavefunction ψ) is to have a unique value for any point in space. For example, one should notice that φandφ+ 2πare identical in the same meridional plane. Therefore, one should have Φ( φ) = Φ(φ+ 2π), i.e., Aeimlφ=Aeiml(φ+2π). This can be fulfilled only if mlis zero or a positiv 117or negative integer ( ±1,±2,±3,...).mlis known as the magnetic quantum number of the atomic electron and is related to the direction of the projec- tion of the orbital momentum Lz. It comes into play whenever the effects of axial magnetic fields on the electron may show up. There is a lso a deep connection between mland the orbital quantum number l, which in turn determines the modulus of the orbital momentum of the electr on. The interpretation of the orbital number ldoes not miss some problems. Let us examine eq. 33 that corresponds to the radial wavefunc tionR(r). This equation rules only the radial motion of the electron, i .e., with the relative distance with respect to the nucleus along some gui ding ellipses. However, the total energy of the electron Eis also present. This energy includes the kineticelctron energy in its orbital motion th at is not related to the radial motion. This contradiction can be eliminated b y the following argument. The kinetic energy Thas two parts: Tradial due to the radial oscillatory motion and Torbital, which is due to the closed orbital motion. The potential energy Vof the electron is the electrostatic energy. Therefore, its total energy is E=Tradial+Torbital−e2 4πǫ0r. (35) Substituting this expression of Ein eq. 33 we get with some regrouping of the terms 1 r2d dr/parenleftbigg r2dR dr/parenrightbigg +2m ¯h2/bracketleftigg Tradial+Torbital−¯h2l(l+ 1) 2mr2/bracketrightigg R= 0. (36) If the last two terms in parentheses compansates between the mselves, we get a differential equation for the pure radial motion. Thus, we impose the condition Torbital =¯h2l(l+ 1) 2mr2. (37) However, the orbital kinetic energy of the electron is Torbital =1 2mv2 orbital (38) and since the orbital momentum of the electron Lis L=mvorbitalr , (39) we can express the orbital kinetic energy in the form Torbital =L2 2mr2. (40) 118Therefore, we have L2 2mr2=¯h2l(l+ 1) 2mr2(41) and consequently L=/radicalig l(l+ 1)¯h . (42) The interpretation of this result is that since the orbital q uantum number l is constrained to take the values l= 0,1,2,...,(n−1), the electron can only have orbital momenta Lspecified by means of eq. 42. As in the case of the total energy E, the angular momentum is conserved and gets quantized. Its natural unit in quantum mechanics is ¯ h=h/2π= 1.054×10−34J.s. In the macroscopic planetary motion (putting aside the many -body fea- tures), the orbital quantum number is so large that any direc t experimental detection is impossible. For example, an electron with l= 2 has an angular momentum L= 2.6×10−34J.s., whereas the terrestrial angular momentum is 2.7×1040J.s.! A common notation for the angular momentum states is by means of the lettersforl= 0,pforl= 1,dforl= 2, and so on. This alphabetic code comes from the empirical spectroscopic classification in terms of the so-called series, which was in use before the advent of quant um mechanics. The combination of the principal quantum number with the lat ter cor- responding to the angular momentum is another frequently us ed notation in atomic and molecular physics.. For example, a state for wh ichn= 2 and l= 0 is a state 2 s, while a state n= 4 andl= 2 is a state 4 d. On the other hand, for the interpretation of the magnetic qua ntum num- ber, we shall take into account, as we did for the linear momen tum, that the orbital momentum is a vector operator and therefore one h as to specify its direction, sense, and modulus. L, being a vector product, is perpendic- ular on the plane of rotation. The geometric rules of the vect orial products still hold, in particular the rule of the right hand: its dire ction and sense are given by the right thumb whenever the other four fingers po int at the direction of rotation. But what significance can be associated to a direction and sen se in the limited space of the atomic hydrogen ? The answer may be quick if we think that the rotating electron is nothing but a one-electr on loop current that considered as a magnetic dipole has a corresponding mag netic field. Consequently, an atomic electron will always interact with an applied mag- neticB. The magnetic quantum number mlspecifies the spatial direction ofL, which is determined by the component of Lalong the direction of the 119external magnetic field. This effect is commonly known as the q uantization of the space in a magnetic field. If we choose the direction of the magnetic field as the zaxis, the com- ponent ofLalong this direction is Lz=ml¯h . (43) The possible values of mlfor a given value of l, go from + lto−l, passing through zero, so that there are 2 l+ 1 possible orientations of the angular momentum Lin a magnetic field. When l= 0,Lzcan be only zero; when l= 1,Lzcan be ¯h, 0, or−¯h; whenl= 2,Lztakes only one of the values 2¯ h, ¯h, 0,−¯h, or−2¯h, and so forth. It is worth mentioning that Lcannot be put exactly parallel or anti-parallel to B, becauseLzis always smaller than the modulus/radicalbig l(l+ 1)¯hof the total orbital momentum. The spatial quantization of the orbital momentum for the hyd rogen atom is shown in fig. 6.1 in a particular case. Fig. 6.1: The spatial quantization of the electron angular m omentum for states l= 2,L=√ 6¯h. One should consider the atom/electron characterized by a gi venmlas having the orientation of its angular momentum Ldetermined relative to the external applied magnetic field. In the absence of the external magnetic field, the direction o f thezaxis is fully arbitrary. Therefore, the component of Lin any arbitrary chosen 120direction is ml¯h; the external magnetic field offers a preferred reference direction from the experimental viewpoint. Why is quantized only the component Lz? The answer is related to the fact that Lcannot be put along a direction in an arbitrary way. Its ‘vectorial arrow’ moves always along a cone centered on the q uantization axis such that its projection Lzisml¯h. The reason why such a phenomenon occurs is due to the uncertainty principle. If Lwould be fixed in space, in such a way that Lx,LyandLzwould have well-defined values, the electron would have to be confined to a well-defined plane. For example, ifLwould be fixed along the zdirection, the electron tends to maintain itself in the planexy(fig. 6.2a). Fig. 6.2: The uncertainty principle forbids a fixed directio n in space of the angular momentum. This can occur only in the case in which the component pzof the elec- tron momentum is ‘infinitely’ uncertain. This is however imp ossible if the electron is part of the hydrogen atom. But since in reality ju st the compo- nentLzofLtogether with L2have well-defined values and |L|>|Lz|, the electron is not constrained to a single plane (fig. 6.2b). If t his would be the case, an uncertainty would exist in the coordinate zof the electron. The direction of Lchanges continuously (see fig. 6.3), so that the mean values ofLxandLyare zero, although Lzkeeps all the time its value ml¯h. 121Fig. 6.3: The angular momentum displays a constant precessi on around the zaxis. The solution for Φ should also fulfill the normalization cond ition given by eq. 2. Thus, we have /integraldisplay2π 0|Φ|2dφ= 1 (44) and substituting Φ, one gets /integraldisplay2π 0A2 φdφ= 1. (45) It follows that Aφ= 1/√ 2π, and thefore the normalized Φ is Φ(φ) =1√ 2πeimlφ. (46) Solution for the polar part The solution of the Θ( θ) equation is more complicated. It is expressed in terms of the associated Legendre polynomials Pml l(x) = (−1)ml(1−x2)ml/2dml dxmlPl(x) = (−1)ml(1−x2)ml/2 2ll!dml+l dxml+l(x2−1)l. (47) 122Their orthogonality relationship is /integraldisplay1 −1[Pml l(cosθ)]2dcosθ =2 2l+ 1(l+ml)! (l−ml)!. (48) For the case of quantum mechanics, Θ( θ) is given by the normalized associ- ated Legendre polynomials. Thus, if Θ(θ) =AθPml l(cosθ), (49) then the normalization condition is /integraldisplay1 −1A2 θ[Pml l(cosθ)]2dcosθ = 1. (50) Therefore, the normalization constant for the polar part is given by Aθ=/radicaligg 2l+ 1 2(l−ml)! (l+ml)!(51) and consequently, the function Θ( θ) already normalized reads Θ(θ) =/radicaligg 2l+ 1 2(l−ml)! (l+ml)!Pml l(cosθ). (52) For our purposes here, the most important property of these f unctions is that they exist only when the constant lis an integer number greater or at least equal to |ml|, which is the absolute value of ml. This condition can be written in the form of the set of values available for ml ml= 0,±1,±2,...,±l . (53) Unification of the azimuthal and polar parts: spherical har- monics The solutions of the azimuthal and polar parts can be unified w ithin spher- ical harmonics functions that depend on both φandθ. This simplifies the algebraic manipulations of the full wave functions ψ(r,θ,φ). Spherical har- monics are introduced as follows Yml l(θ,φ) = (−1)ml/radicaligg 2l+ 1 4π(l−ml)! (l+ml)!Pml l(cosθ)eimlφ. (54) 123The supplementary factor ( −1)mldoes not produce any problem because the Schr¨ odinger equation is linear and homogeneous. This f actor is added for the sake of convenience in angular momentum studies. It i s known as the Condon-Shortley phase factor and its effect is to introduce a n alternance of the signs±for the spherical harmonics. Solution for the radial part The solution for the radial part R(r) of the wave function ψof the hydrogen atom is somewhat more complicated. It is here where significa nt differences with respect to the electrostatic Laplace equation do occur . The final result is expressed analytically in terms of the associated Laguer re polynomials (Schr¨ odinger 1926). The radial equation can be solved in ex act way only when E is positive or for one of the following negative values En(in which cases, the electron is in a bound stationary state within ato mic hydrogen) En=−me4 32π2ǫ2 0¯h2/parenleftbigg1 n2/parenrightbigg , (55) wherenis an integer number called the principal quantum number. It gives the quantization of the electron energy in the hydroge n atom. This discrete atomic spectrum has been first obtained in 1913 by Bo hr using semi- empirical quantization methods and next by Pauli and Schr¨ o dinger almost simultaneously in 1926. Another condition that should be satisfied to solve the radia l equation is thatnhave to be strictly bigger than l. Its lowest value is l+ 1 for a givem l. Vice versa, the condition on lis l= 0,1,2,...,(n−1) (56) for givenn. The radial equation can be written in the form r2d2R dr2+ 2rdR dr+/bracketleftigg 2mE ¯h2r2+2me2 4πǫ0¯h2r−l(l+ 1)/bracketrightigg R= 0, (57) Dividing by r2and using the substitution χ(r) =rRto eliminate the first derivativedR dr, one gets the standard form of the radial Schr¨ odinger equat ion displaying the effective potential U(r) =−const/r+l(l+ 1)/r2(actually, electrostatic potential plus quantized centrifugal barri er). These are nec- essary mathematical steps in order to discuss a new boundary condition, 124since the spectrum is obtained by means of the Requation. The difference between a radial Schr¨ odinger equation and a full-line one i s that a suppli- mentary boundary condition should be imposed at the origin ( r= 0). The coulombian potential belongs to a class of potentials that a re called weak singular for which lim r→0=U(r)r2= 0. In these cases, one tries solutions of the type χ∝rν, implyingν(ν−1) =l(l+ 1), so that the solutions are ν1=l+ 1 andν2=−l, just as in electrostatics. The negative solution is eliminated for l∝ne}ationslash= 0 because it leads to a divergent normalization constant, nor did it respect the normalization at the delta function fo r the continuous part of the spectrum. On the other hand, the particular case ν2= 0 is elmin- inated because the mean kinetic energy is not finite. The final conclusion is thatχ(0) = 0 for any l. Going back to the analysis of the radial equation for R, first thing to do is to write it in nondimensional variables. This is performe d by noticing that the only space and time scales that one can form on combining t he three fundamental constants entering this problem, namely e2,meand ¯hare the Bohr radius a0= ¯h2/me2= 0.529·10−8cm. andt0= ¯h3/me4= 0.24210−16 sec., usually known as atomic units. Employing these units, one gets d2R dr2+2 rdR dr+/bracketleftbigg 2E+2 r−l(l+ 1) r2/bracketrightbigg R= 0, (58) where we are especially interested in the discrete part of th e spectrum ( E < 0). The notations n= 1/√ −Eandρ= 2r/nleads us to d2R dρ2+2 ρdR dρ+/bracketleftbiggn ρ−1 4−l(l+ 1) ρ2/bracketrightbigg R= 0. (59) Forρ→∞, this equation reduces tod2R dρ2=R 4, having solutions R∝e±ρ/2. Because of the normalization condition only the decaying ex ponential is acceptable. On the other hand, the asymptotics at zero, as we already com- mented on, should be R∝ρl. Therefore, we can write Ras a product of three radial functions R=ρle−ρ/2F(ρ), of which the first two give the asymptotic behaviors, whereas the third is the radial funct ion in the inter- mediate region. The latter function is of most interest beca use its features determine the energy spectrum. The equation for Fis ρd2F dρ2+ (2l+ 2−ρ)dF dρ+ (n−l−1)F= 0. (60) 125This is a particular case of confluent hypergeometric equati on for which the two ‘hyper’geometric parameters depend on the pair of qu antum num- bersn,l. It can be identified as the equation for the associated Lague rre polynomials L2l+1 n+l(ρ). Thus, the normalized form of Ris Rnl(r) =−2 n2/radicaligg (n−l−1)! 2n[(n+l)!]3e−ρ/2ρlL2l+1 n+l(ρ), (61) where the following Laguerre normalization condition has b een used /integraldisplay∞ 0e−ρρ2l[L2l+1 n+l(ρ)]2ρ2dρ=2n[(n+l)!]3 (n−l−1)!. (62) We have now the solutions of all the equations depending on a s ingle variable and therefore we can build the wave function for any electronic state of the hydrogen atom. The full wave function reads ψ(r,θ,φ) =NH(αr)le−αr/2L2l+1 n+l(αr)Pml l(cosθ)eimlφ, (63) whereNH=−2 n2/radicalbigg 2l+1 4π(l−ml)! (l+ml)!(n−l−1)! [(n+l)!]3andα= 2/na0. Using the spherical harmonics, the solution is written as fo llows ψ(r,θ,φ) =−2 n2/radicaligg (n−l−1)! [(n+l)!]3(αr)le−αr/2L2l+1 n+l(αr)Yml l(θ,φ).(64) The latter formula may be considered as the final result for th e Schr¨ odinger solution of the hydrogen atom for any stationary electron st ate. Indeed, one can see explicitly both the asmptotic dependence and the two orthogonal and complete sets of functions, i.e., the associated Laguer re polynomials and the spherical harmonics that correspond to this particu lar case of lin- ear partial second-order differential equation. The parabo lic coordinates [ξ=r(1−cosθ),η=r(1 + cosθ),φ=φ], are another coordinate system in which the Schr¨ odinger hydrogen equation is separable (E. S chr¨ odinger, Ann. Physik 80, 437, 1926; P.S. Epstein, Phys. Rev. 28, 695, 1926; I. Waller, Zf. Physik 38, 635, 1926). The final solution in this case is expressed as th e product of factors of asymptotic nature, azimuthal harmoni cs, and two sets of associate Laguerre polynomials in the variables ξandη, respectively. The energy spectrum ( −1/n2) and the degeneracy ( n2) of course do not depend on the coordinate system. 126Electronic probability density In the Bohr model of the hydrogen atom, the electron rotates a round the nu- cleus on circular or elliptic trajectories. It is possible t o think of appropriate experiments allowing to “see” that the electron moves withi n experimental errors at the predicted radii r=n2a0(wherenis the principal quantum number labeling the orbit and a0= 0.53˚Ais the Bohr radius) in the equa- torial plane θ= 90o, whereas the azimuthal angle may vary according to the specific experimental conditions. The more rigorous quantum theory changes the conclusions of the Bohr model in at least two important aspects. First, one cannot sp eak about exact values ofr,θ,φ, but only of relative probabilities to find the electron with in an infinitesimal given region of space. This feature is a cons equence of the wave nature of the electron. Secondly, the electron does not move around the nucleus in the classical conventional way because the pr obability density |ψ|2does not depend on time but can vary substantially as a functi on of the relative position of the infinitesimal region. The hydrogenic electron wave function ψisψ=RΘΦ, whereR=Rnl(r) describes the way ψchanges with rwhen the principal and orbital quantum numbers have the values nandl, respectively. Θ = Θ lml(θ) describes in turn howψvaries with θwhen the orbital and magnetic quantum numbers have the values landml, respectively. Finally, Φ = Φ ml(φ) gives the change ofψwithφwhen the magnetic quantum number has the value ml. The probability density |ψ|2can be written |ψ|2=|R|2|Θ|2|Φ|2. (65) Notice that the probability density |Φ|2, which measures the possibility to find the electron at a given azimuthal angle φ, is a constant (does not depend on φ). Therefore, the electronic probability density is symmet ric with respect to the zaxis and independent on the magnetic substates (at least until an external magnetic field is applied). Conseque ntly, the electron has an equal probability to be found in any azimuthal directi on. The radial partRof the wave function, contrary to Φ, not only varies with r, but it does it differently for any different combination of quantum n umbersnand l. Fig. 6.4 shows plots of Ras a function of rfor the states 1 s, 2s, and 2p.Ris maximum at the center of the nucleus ( r= 0) for all the sstates, whereas it is zero at r= 0 for all the states of nonzero angular momentum. 1272p1s 2s51015r(a )0Rnl(r) Fig. 6.4: Approximate plots of the radial functions R1s,R2s,R2p; (a0= 0.53˚A). 1s 2s 2p 10155 r(a )0dP/dr=4 πr R2 2 Fig. 6.5: Probability density of finding the hydrogenic elec tron between rand r+drwith respect to the nucleus for the states 1 s, 2s, 2p. The electronic probability density at the point r,θ,φ is proportional to |ψ|2, but the real probability in the infinitesimal volume elemen tdVis |ψ|2dV. In spherical coordinates dV=r2sinθdrdθdφ , (66) 128and since Θ and Φ are normalized functions, the real numerica l probability P(r)drto find the electron at a relative distance with respect to the nucleus betweenrandr+dris P(r)dr=r2|R|2dr/integraldisplayπ 0|Θ|2sinθdθ/integraldisplay2π 0|Φ|2dφ =r2|R|2dr (67) P(r) is displayed in fig. 6.5 for the same states for which the radi al func- tionsRappear in fig. 6.4. In principle, the curves are quite differen t. We immediately see that P(r) is not maximal in the nucleus for the states s, as happens for R. Instead, their maxima are encountered at a finite distance from the nucleus. The most probable value of rfor a 1selectron is exactly a0, the Bohr radius. However, the mean value of rfor a 1selectron is 1 .5a0. At first sight this might look strange, because the energy lev els are the same both in quantum mechanics and in Bohr’s model. This apparent unmatch- ing is eliminitated if one takes into account that the electr on energy depends on 1/rand not on r, and the mean value of 1 /rfor a 1selectron is exactly 1/a0. The function Θ varies with the polar angle θfor all the quantum num- berslandml, unlessl=ml= 0, which are the sstates. The probability density|Θ|2for asstate is a constant (1/2). This means that since |Φ|2 is also a constant, the electronic probability density |ψ|2has the same value for a given rvalue, not depending on the direction. In other states, the electrons present an angular behavior that in many cases may be quite complicated. This can be seen in fig.6.5, where the electroni c probability densities for different atomic states are displayed as a func tion ofrandθ. (The plotted term is |ψ|2and not|ψ|2dV). Because|ψ|2is independent ofφ, a three-dimensional representation of |ψ|2can be obtained by rotat- ing a particular representation around a vertical axis. Thi s can prove that the probability densities for the sstates have spherical symmetry, while all the other states do not possess it. In this way, one can get mor e or less pronounced lobes of characteristic forms depending on stat e. These lobes are quite important in chemistry for specifying the atomic i nteraction in the molecular bulk. 6N. Note : 1. In 1933, E. Schr¨ odinger has been awarded the Nobel Prize i n Physics (to- gether with Dirac) for the “discovery of new productive form s of atomic the- ory”. Schr¨ odinger wrote a remarkable series of four papers “Quantisierung 129als Eigenwertproblem” [“Quantization as an eigenvalue pro blem”] (I-IV, re- ceived by Annalen der Physik on 27 January, 23 February, 10 Ma y and 21 June 1926, respectively). 6P. Problems Problem 6.1 - Obtain the formulas for the stable orbits and the energy levels of the electron in the atomic hydrogen using only argu ments based on the de Broglie wavelength associated to the electron and the empirical value 5.3·10−11m for the Bohr radius. Solution : The electron wavelength is given by λ=h mv, whereas if we equate the electric force and the centripetal forcemv2 r=1 4πǫ0e2 r2we obtain the electron ‘velocity’ v=e√4πǫ0mr.Thus, the wavelength of the electron isλ=h e/radicalig 4πǫ0r m. If we now use the value 5 .3×10−11m for the radius rof the electron orbit, we can see that the wavelength of the el ectron is λ= 33×10−11m. But this is exactly the same value as of the circumference of the orbit, 2 πr= 33×10−11m. One may say that the electron orbit in the atomic hydrogen corresponds to a wave “closing into it self” (i.e., stationary). This fact can be compared to the vibrations of a metallic ring. If the wavelengths are multiples of the circumference, the rin g goes on with its vibrations for a long time with very small dissipation If, on the other hand, the number of wavelengths making a circumference is not an in teger, the interference of the waves is negative and they dissapear in a short period of time. One may say that the electron will rotate around the nuc leus without radiating its energy for an infinite time as far as its orbit co ntains an integer number of de Broglie wavelengths. Thus, the stability/stat ionary condition is nλ= 2πrn, wherernis the radius of the electron orbit containing nwavelengths. Sub- stitutingλ, we have nh e/radicalbigg4πǫ0rn m= 2πrn, and therefore the stationary electron orbits are rn=n2¯h2ǫ0 πme2. 130To get the energy levels, we use E=T+Vand substituting the kinetic and potential energies leads to E=1 2mv2−e2 4πǫ0r, or equivalently En=−e2 8πǫ0rn. Plugging the value of rninto the latter equation, we get En=−me4 8ǫ2 0¯h2/parenleftbigg1 n2/parenrightbigg . Problem 6.2 - Uns¨ old’s theorem tells that for any value of the orbital numberl, the probability densities, summed over all possible subst ates, from ml=−ltoml= +lgive a constant that is independent of the angles θand φ, i.e. +l/summationdisplay ml=−l|Θlml|2|Φml|2=ct. This theorem shows that any atom or ion with closed (occupied ) sublevels has a spherically-symmetric charge distribution. Check Un s¨ old’s theorem for l= 0,l= 1, andl= 2. Solution : Forl= 0, Θ 00= 1/√ 2 and Φ 0= 1/√ 2π, so that |Θ0,0|2|Φ0|2=1 4π. Forl= 1, we have +1/summationdisplay ml=−1|Θlml|2|Φml|2=|Θ1,−1|2|Φ−1|2+|Θ1,0|2|Φ0|2+|Θ1,1|2|Φ1|2. On the other hand, the wave functions are given by Θ 1,−1= (√ 3/2)sinθ, Φ−1= (1/√ 2π)e−iφ, Θ1,0= (√ 6/2)cosθ, Φ0= 1/√ 2π, Θ1,1= (√ 3/2)sinθ, Φ1= (1/√ 2π)eiφ, which plugged into the previous equation give +1/summationdisplay ml=−1|Θlml|2|Φml|2=3 8πsin2θ+3 4πcos2θ+3 8πsin2θ=3 4π 131and again we’ve got a constant. Forl= 2, we have +2/summationdisplay ml=−2|Θlml|2|Φml|2=|Θ2,−2|2|Φ−2|2|Θ2,−1|2|Φ−1|2 +|Θ2,0|2|Φ0|2+|Θ2,1|2|Φ1|2+|Θ2,2|2|Φ2|2, and the wave functions are Θ 2,−2= (√ 15/4)sin2θ, Φ−2= (1/√ 2π)e−2iφ, Θ2,−1= (√ 15/2)sinθcosθ , Φ−1= (1/√ 2π)e−iφ, Θ2,0= (√ 10/4)(3cos2θ− 1), Φ 0= 1/√ 2π, Θ2,1= (√ 15/2)sinθcosθ , Φ1= (1/√ 2π)eiφ, Θ2,2= (√ 15/4)sin2θ, Φ2= (1/√ 2π)e2iφ, Plugging them into the previous equa- tion give +2/summationdisplay ml=−2|Θlml|2|Φml|2=5 4π, which again fulfills Uns¨ old’s theorem. Problem 6.3 - The probability to find an atomic electron whose radial wave functions is that of the ground state R10(r) outside a sphere of Bohr radius a0centered on the nucleus is /integraldisplay∞ a0|R10(r)|2r2dr . Obtain the probability to find the electron in the ground stat e at a distance from the nucleus bigger than a0. Solution : The radial wave function corresponding to the ground state is R10(r) =2 a3/2 0e−r/a0. Substituting it in the integral, we get/integraltext∞ a0|R(r)|2r2dr=4 a3 0/integraltext∞ a0r2e−2r/a0dr,or /integraldisplay∞ a0|R(r)|2r2dr=4 a3 0/bracketleftigg −a0 2r2e−2r/a0−a2 0 2re−2r/a0−a3 0 4e−2r/a0/bracketrightigg∞ a0. This leads us to /integraldisplay∞ a0|R(r)|2r2dr=5 e2≈68% !!, which is the result asked for in this problem. 1327. QUANTUM SCATTERING Introduction One usually begins the quantum theory of scattering by refer ring to results already known from the classical scattering in central field s with some sim- plifying assumptions helping to avoid unnecessary calcula tions in getting basic results. It is generally known that studying scatteri ngs in the labo- ratory provides information on the distribution of matter i n the target and other details of the interaction between the incident beam a nd the target. The hypotheses that we shall assume correct in the following are i) The particles are spinless. This, of course, does not mean that spin effects are not important in quantum scatterings. ii) We shall study only elastic scattering for which the inte rnal structure of the particles is not taken into account. iii) The target is sufficiently thin to neglect multiple scatt erings. iv) The interactions are described by a potential that depen ds only on the relative distance between the particles (central poten tial). These hypotheses eliminate some quantum effects that are mer ely details. They also represent conditions for getting the quantum anal ogs of basic classical results. We now define dσ dΩ∝I(θ,ϕ) I0, (1) wheredΩ is the solid angle infinitesimal element, I0is the number of inci- dent particles per unit transverse area, and IdΩ is the number of scattered particles in the solid angle element. Employing these well-known concepts, together with the asy mptotic no- tion of impact parameter bassociated to each classical incident particle, one gets in classical mechanics the following important formul a dσ dΩ=b sinθ|db dθ|. (2) If one wants to study the scattering phenomenology in quantu m ter- minology, one should investigate the time evolution of a ‘sc attering’ wave packet. Let Fibe the flux of incident particles, i.e., the number of parti- cles per unit of time passing through the unit of transverse s urface onto the propagation axis. An appropriate detector configuration is usually placed 133far away from the effective interaction region, ‘seeing’ a so lid angledΩ of that region. In general, the number of particles dn/dt scattered per unit of time indΩ in the direction ( θ,ϕ) is detected. 0 V(r)zDetector D (dn/dt ~ F d Ω)i Inc. flux FidΩ Fig. 7.1θ dn/dt is proportional to dΩ andFi. Let us call σ(θ,ϕ) the coefficient of proportionality between dnandFidΩ: dn=σ(θ,ϕ)FidΩ, (3) which is by definition the differential cross section. The number of particles per unit of time reaching the detecto r is equal to the number of particles crossing the surface σ(θ,ϕ)dΩ, which is perpen- dicular to the beam axis. The total section is by definition σ=/integraldisplay σ(θ,ϕ)dΩ. (4) To further simplify the calculation, we choose the z axis alo ng the inci- dent beam direction. On the negative side of the axis, for large negative t, the particle is prac- tically free: it is not affected by V(r) and its state can be represented by plane waves. Therefore, the wave function contains terms of the formeikz, wherekis the constant ocurring in the Helmholtz equation. By analo gy with optics, the form of the scattered wave is f(r) =eikr r. (5) 134Indeed (∇2+k2)eikr∝ne}ationslash= 0 (6) and (∇2+k2)eikr r= 0 (7) forr>r 0, wherer0is any positive number. We assume that the motion of the particle is described by the H amilto- nian H=p2 2µ+V=H0+V . (8) Vis different of zero only in a small neighbourhood close to the origin. A wave packet at t= 0 can be written ψ(r,0) =1 (2π)3 2/integraldisplay ϕ(k)exp[ik·(r−r0)]d3k, (9) whereψis a function that is nonzero in a ‘width’ ∆ kcentered on k0. We also assume that k0is antiparallel to r0. In order to see quantitatively what happens to the wave packet when scatters the target, one can use the expansion of ψ(r,0) in the eigenfunctions ψn(r) ofH, i.e.,ψ(r,0) =/summationtext ncnψn(r). Thus, the wave packet at time tis ψ(r,t) =/summationdisplay ncnϕn(r)exp(−i ¯hEnt). (10) This is an eigenfunction of the operator H0, not ofH, but we can sub- stitute these eigenfunctions by eigenfunctions of H, which we denote by ψ(+) k(r). The asymptotic form of the latter is ψ(+) k(r)≃eik·r+f(r)eikr |r|, (11) where, as usually p= ¯hkandE=¯h2k2 2m. This corresponds to a plane wave of the incident beam type and a di- vergent spherical wave resulting from the interaction betw een the incident beam and the target. One can expand ψ(r,0) in plane waves and ψk(r) ψ(r,0) =/integraldisplay ϕ(k)exp(−ik·r0)ψk(r)d3k , (12) where ¯hω=¯h2k2 2m. The divergent spherical wave does not contribute to the initial wave packet because it is an additive part. 135Scattering of a wave packet Any wave is dispersed during its propagation. This is why one cannot ignore the effect of the divergent wave from this viewpoint. One can m ake use of the following trick ω=¯h 2mk2=¯h 2m[k0+ (k−k0)]2=¯h 2m[2k0·k−k2 0+ (k−k0)2],(13) pentru a neglija ultimul termen ˆ ın paranteze. Substitutin gωinψ, we ask that¯h 2m(k−k0)2T≪1, whereT≃2mr0 ¯hk0. Therefore (∆k)2r0 k0≪1. (14) This condition tells us that the wave packet does not dispers e significantly even when it moves over amacroscopic distance r0. Choosing the direction of the vector kof the incident wave along one of the three cartesian directions (we use the zone), we can write in spherical coordinates the following important formula ψk(r,θ,ϕ)≃eikz+f(k,θ,ϕ)eikr r. Since the Hamiltonian H, up to now not considered as an operator (the class of the results presented are the same both at the classi cal and quantum level), is invariant under zrotations, we can choose boundary conditions of spherical symmetry too. Thus ψk(r,θ,ϕ)≃eikz+f(θ)eikr r. This type of functions are known as scattering wave function s. The coeffi- cientf(θ) of the spherical wave is known as the scattering amplitude. It is a basic concept in the formal theory of quantum scatterings. Probability amplitude in scattering We write the Schr¨ odinger equation as follows i¯h∂ψ ∂t=−¯h2 2m∇2ψ+V(r,t)ψ . (15) Recall that the expression P(r,t) =ψ∗(r,t)ψ(r,t) =|ψ(r,t)|2(16) 136can be interpreted, cf. Max Born, as a probability density un der normaliza- tion conditions of the type /integraldisplay |ψ(r,t)|2d3r= 1. (17) This normalization integral should be time independent. Th is can be noted by writing I=∂ ∂t/integraldisplay ΩP(r,t)d3r=/integraldisplay Ω(ψ∗∂ψ ∂t+∂ψ∗ ∂tψ)d3r , (18) and from Schr¨ odinger’s equation ∂ψ ∂t=i¯h 2m∇2ψ−i ¯hV(r,t)ψ (19) one gets I=i¯h 2m/integraldisplay Ω[ψ∗∇2−(∇2ψ∗)ψ]d3r=i¯h 2m/integraldisplay Ω∇·[ψ∗∇ψ−(∇ψ∗)ψ]d3r= =i¯h 2m/integraldisplay A[ψ∗∇ψ−(∇ψ∗)ψ]ndA , (20) where the Green theorem has been used to evaluate the volume i ntegral. dAis the infinitesimal surface element on the boundary of the in tegration region and [ ] ndenotes the component along the normal direction to the surface element dA. Defining S(r,t) =¯h 2im[ψ∗∇ψ−(∇ψ∗)ψ], (21) we get I=∂ ∂t/integraldisplay ΩP(r,t)d3r=−/integraldisplay Ω∇·Sd3r=−/integraldisplay ASndA , (22) for well-bahaved wave packets (not funny asymptotically) s o that the nor- malization integral converges. The surface integral is zer o when Ω covers the whole space. One can prove (see P. Dennery & A. Krzywicki, Mathematical methods for physicists ) that the surface integral is zero. Therefore, the nor- malization integral is constant in time and the initial cond ition holds. From the same equation for S, we get ∂P(r,t) ∂t+∇·S(r,t) = 0, (23) 137which is the continuity equation for the density flux Pand the current density Sin the absence of any type of sources or sinks. If we interpret¯h im∇ as a sort of velocity ‘operator’ (as for time, it is difficult to speak rigorously about a velocity operator in quantum mechanics!), then S(r,t) =Re(ψ∗¯h im∇ψ). (24) To calculate the quantum current density for a scattering wa ve function is a tricky and inspiring (not illustrative) exercise! The fi nal result is jr= ¯hk mr2|f(θ)|2, where the direction θ= 0 should not be included. Green’s function in scattering theory Another way of writing the Schr¨ odinger equation at hand is ( −¯h2 2m∇2+V)ψ= Eψ, or (∇2+k2)ψ=Uψ, wherek2=2mE ¯h2, ¸ siU=2mV ¯h2. It follows that it is more convenient to put this equation in a n integral form. This can be done if we consider Uψin the right hand side of the equation as a inhomogeneity. This allows to build the soluti on by means of Green’s function (integral kernel), which, by definition, i s the solution of (∇2+k2)G(r,r′) =δ(r−r′). (25) One can write now the Schr¨ odinger solution as the sum of the h omogeneous equation and the inhomogeneous one of Green’s type ψ(r) =λ(r)−/integraldisplay G(r,r′)U(r′)ψ(r′)d3r′. (26) We seek now a Gfunction in the form of a product of linear independent functions, for example, plane waves G(r,r′=/integraldisplay A(q)eiq·(r−r′)dq . (27) Using eq. 25, we have /integraldisplay A(q)(k2−q2)eiq·(r−r′)dq=δ(r−r′), (28) which turns in an identity if A(q) = (2π)−3(k2−q2)−1. (29) 138Thus G(r,r′) =1 (2π)3/integraldisplayeiqR k2−q2d3q , (30) whereR=|r−r′|. After performing a calculation of complex variable10, we get G(r) =−1 4πeikr r. (31) This function is not determined univoquely since the Green f unction can be any solution of the eq. 25. The right particular solution i s chosen by imposing boundary conditions on the eigenfunctions ψk(r). The Green function obtained in this way is G(r,r′) =−/parenleftigg eik|r−r′| 4π|r−r′|/parenrightigg . (32) Thus, we finally get the integral equation for the scattering wave function ψ(k,r) =ϕ(k,r)−m 2π¯h2/integraldisplayeik|r−r′| r−r′U(r′)ψ(k,r)dr, (33) whereϕis a solution of the Helmholtz equation. Noticing that |r−r′|=R, then (∇2+k2)ψ= (∇2+k2)[ϕ+/integraldisplay G(r,r′)U(r′)ψ(r′)d3r′] (34) and assuming that we can change the order of operations and pu t the∇ operator inside the integral, we get (∇2+k2)ψ=/integraldisplay (∇2+k2)G(r,r′)U(r′)ψ(r′)d3r′=U(r)ψ(r),(35) which shows us that G(R) =1 4πeikR Ris indeed a solution. Optical theorem The total cross section is given by σtot(k) =/integraldisplaydσ dΩdΩ. (36) 10See problem 7.1. 139Let us express now f(θ) as a function of the phase shift Sl(k) =e2iδl(k) in the form f(θ) =1 k∞/summationdisplay l=0(2l+ 1)eiδi(k)sinδl(k)Pl(cosθ). (37) Then σtot=/integraldisplay [1 k∞/summationdisplay l=0(2l+ 1)eiδl(k)sinδl(k)Pl(cosθ)] [/integraldisplay [1 k∞/summationdisplay l′=0(2l′+ 1)eiδl′(k)sinδl′(k)Pl′(cosθ)]. (38) Using now/integraltextPl(cosθ)Pl′(cosθ) =4π 2l+1δll′, we get σtot=4π k2∞/summationdisplay l=0(2l+ 1)sinδl(k)2. (39) Of interest is the relationship Imf(0) =1 k∞/summationdisplay l=0(2l+ 1)Im[eiδl(k)sinδl(k)]Pl(1) =1 k∞/summationdisplay l=0(2l+ 1)sinδl(k)2= k 4πσtot, (40) which is known as the optical theorem . Its physical significance is related to the fact that the interference of the incident wave with the d ispersed wave at zero/forward angle produces the “getting out” of the part icle from the incident wave, allowing in this way the conservation of the p robability. Born approximation Let us consider the situation of Fig. 7.2: 140LO Pur |r − r |’M Fig. 7.2 The observation point M is far away from P, which is in the rang e of the potentialU. The geometrical conditions are r≫L,r′≪l. The segment MP that corresponds to |r−r′|is in the aforementioned geometrical conditions approxiamtely equal to the projection of MP onto MO |r−r′|≃r−u·r′, (41) where uis a unit vector (versor) in the rdirection. Then, for large r G=−1 4πeik|r−r′| |r−r′|≃r→∞−1 4πeikr re−iku·r. (42) We now substitute Gin the integral expression for the scattering wave func- tion ψ(r) =eikz−1 4πeikr r/integraldisplay e−iku·rU(r′)ψ(r′)d3r′. (43) The latter is already not a function of the distance r=OM, but only of θ andψ. Thus f(θ,ψ) =−1 4π/integraldisplay e−iku·rU(r′)ψ(r′)d3r′. (44) We define now the incident wave vector kias a vector of modulus kdirected along the polar axis of the beam. Then eikz=eiki·r. Similarly, kd, of moduluskand of direction fixed by θandϕ, is called the shifted wave vector in the direction ( θ,ϕ):kd=ku. Themomentum transfer in the direction ( θ,ϕ) is introduced as the vec- torial difference K=kd−ki. 141kd kiθ Fig. 7.3K Hence we can write the integral equation in the form ψ(r) =eiki·r+/integraldisplay G(r,r′)U(r′)ψ(r′)d3r′(45) One can try to solve this equation iteratively. Putting r→r′;r′→r′′, we can write ψ(r′) =eiki·r′+/integraldisplay G(r′,r′′)U(r′′)ψ(r′′)d3r′′. (46) Substituting in 45, we get ψ(r) =eiki·r+/integraldisplay G(r,r′)U(r′)eiki·r′d3r′+ /integraldisplay /integraldisplay G(r,r′)U(r′)G(r′,r′′)U(r′′)ψ(r′′)d3r′′d3r′. (47) The first two terms in the right hand side are known and it is onl y the third one that includes the unknown function ψ(r). We can repeat the procedure: substituting rbyr′′, andr′byr′′′, we getψ(r′′) , that we can reintroduce in the eq. 47 ψ(r) =eiki·r+/integraldisplay G(r,r′)U(r′)eiki·r′+ /integraldisplay /integraldisplay G(r,r′)U(r′)G(r′,r′′)U(r′′)eiki·r′′d3r′d3r′′+ /integraldisplay /integraldisplay /integraldisplay G(r,r′)U(r′)G(r′,r′′)U(r′′)eiki·r′′G(r′′,r′′′)U(r′′′)ψ(r′′′).(48) The first three terms are now known and the unknown function ψ(r) has been sent to the fourth term. In this way, by succesive iterat ions we can build the stationary dispersed wave function. Notice that e ach term of the series expansion has one more power in the potential with res pect to the previous one. We can go on until we get a neglijible expressio n in the right hand side, obtaining ψ(r) as a function of only known quantities. 142Substituting the expression of ψ(r) inf(θ,ϕ), we get the expansion in Born series of the scattering amplitude. In first order in U, one should replaceψ(r′) byeiki·r′in the right hand side to get f(B)(θ,ϕ) =−1 4π/integraldisplay eiki·r′U(r′)e−iku·r′d3r′=−1 4π/integraldisplay e−i(kd−ki)·r′U(r′)d3r′= −1 4π/integraldisplay e−iK·r′U(r′)d3r′(49) Kis the momentum transfer vector. Thus, the differential cros s section is simply related to the potential, V(r) =¯h2 2mU(r). Sinceσ(θ,ϕ) =|f(θ,ϕ)|2, the result is σ(B)(θ,ϕ) =m2 4π2¯h4|/integraldisplay e−iK·rV(r)d3r|2(50) The direction and modulus of Kdepends on the modulus kofkiandkdas well as on the scattering direction ( θ,ϕ). For given θandϕ, it is a function ofk, the energy of the incident beam. Analogously, for a given en ergy,σ(B) is a function of θandϕ. Born’s approximation allows one to get information on the potential V(r) from the dependence of the differential cross section on the scattering direction and the incident energy. 7N. Note - The following paper of Born was practically the first dealin g with quantum scattering: M. Born, “Quantenmechanik der Stossvorg¨ ange” [“Quantum m echanics of scattering processes ”], Zf. f. Physik 37, 863-867 (1926) 7P. Problems Problem 7.1 Calculus of complex variable for the scattering Green funct ion We recall that we already obtained the result G(r,r′) =1 (2π)3/integraltexteiqR k2−q2d3q ,cuR=|r−r′|. Sinced3q=q2sinθdqdθdφ , we get after integrating in angular variables G(r,r′) =i 4π2R/integraltext∞ −∞(e−iqR−eiqR) k2−q2qdq . PuttingC=i 4π2R, we separate the integral in two parts C(/integraltext∞ −∞e−iqR k2−q2qdq−/integraltext∞ −∞eiqR k2−q2qdq). Let us make now q→−qin the first integral 143/integraltext∞ −∞e−i(−q)R k2−(−q)2(−q)d(−q) =/integraltext−∞ ∞eiqR k2−q2qdq=−/integraltext∞ −∞eiqR k2−q2qdq , so that G(r,r′) =−2C(/integraltext∞ −∞qeiqR k2−q2dq). Substituting C, leads to G(r,r′) =−i 2π2R/integraltext∞ −∞qeiqR k2−q2dq In this form, the integral can be calculated by means of the th eorem of residues of its poles. Notice the presence of simple poles at q=+ −k. Fig. 7.4: Contour rules around the poles for G+andG− We use the contour of fig. 7.4 encircling the poles as shown, be cause in this way we get the physically correct effect from the theorem of residues G(r) =−1 4πeikr r(Imk>0) , G(r) =−1 4πe−ikr r(Imk<0). The solution of interest is the first one, because it provides divergent waves, whereas the latter solution holds for convergent wav es (propagating towards the target). Moreover, the linear combination 1 2limǫ→0[Gk+iǫ+Gk−iǫ] =−1 4πcoskr r corresponds to stationary waves. The formal calculation of the integral can be performed by ta kingk2−q2→ k2+iǫ−q2, so that:/integraltext∞ −∞qeiqR k2−q2dq→/integraltext∞ −∞qeiqR (k2+iǫ)−q2dq .This is possible forR > 0. This is why the contour for the calculation will be placed in the upper half plane. Thus, the poles of the integrand are l ocated at q=±√ k2+iǫ≃±(k+iǫ 2k). The procedure of taking the limit ǫ→0 should 144be applied aftercalculating the integral. Problem 7.2 Asymptotic form of the radial function As we have already seen in the chapter Hydrogen atom the radial part of the Schr¨ odinger equation can be written (d2 dr2+2 rd dr)Rnlm(r)−2m ¯h2[V(r) +l(l+1)¯h2 2mr2]Rnlm(r) +2mE ¯h2Rnlm(r) = 0. n,l,m are the spherical quantum numbers. For the sake of convenien ce of writing we shall discard them hereafter. Ris the radial wave function (i.e., depends only on r). We assume that the potential goes to zero stronger than 1/r, and that lim r→0r2V(r) = 0. Usingu(r) =rR, since (d2 dr2+2 rd dr)u r=1 rd2 dr2u, we have d2 dr2u+2m ¯h2[E−V(r)−l(l+1)¯h2 2mr2]u= 0. Notice that the potential displays a supplementary term V(r)→V(r) +l(l+1)¯h2 2mr2, which corresponds to a repulsive centrifugal barrier. For a free particle V(r) = 0, and the equation becomes [d2 dr2+2 rd dr)−l(l+1) r2]R+k2R= 0. Introducing the variable ρ=kr, we get d2R dρ2+2 ρdR dρ−l(l+1) ρ2R+R= 0. The solutions are the so-called spherical Bessel functions . The regular solu- tion is jl(ρ) = (−ρ)l(1 ρd dρ)l(sinρ ρ), while the irregular one nl(ρ) =−(−ρ)l(1 ρd dρ)l(cosρ ρ). For largeρ, the functions of interest are the spherical Hankel functio ns h(1) l(ρ) =jl(ρ) +inl(ρ) ¸ sih(2) l(ρ) = [h(1) l(ρ)]∗. The behaviour for ρ≫lis of special interest jl(ρ)≃1 ρsin(ρ−lπ 2) (51) nl(ρ)≃−1 ρcos(ρ−lπ 2). (52) Then h(1) l≃−i ρei(ρ−lπ/2). The solution regular at the origin is Rl(r) =jl(kr). The asymptotic form is (using eq. 51) 145Rl(r)≃1 2ikr[e−ikr−lπ/2−eikr−lπ/2]. Problem 7.3 Born approximation for Yukawa potentials Let us consider the potential of the form V(r) =V0e−αr r, (53) whereV0andαare real constants and αis positive. The potential is ei- ther attractive or repulsive depending on the sign of V0; the larger|V0|, the stronger the potential. We assume that |V0|is sufficiently small that Born’s approximation holds. According to a previous formula, the s cattering am- plitude is given by f(B)(θ,ϕ) =−1 4π2mV0 ¯h2/integraltexte−iK·re−αr rd3r . Since this potential depends only on r, the angular integrals are trivial leading to the form f(B)(θ,ϕ) =1 4π2mV0 ¯h24π |K|/integraltext∞ 0sin|K|re−αr rrdr . Thus, we obtain f(B)(θ,ϕ) =−2mV0 ¯h21 α2+|K|2. From the figure we can notice that |K|= 2ksinθ 2. Therefore σ(B)(θ) =4m2V2 0 ¯h41 [α2+4k2sinθ 22]2. The total cross section is obtained by integrating σ(B)=/integraltextσ(B)(θ)dΩ =4m2V2 0 ¯h44π α2(α2+4k2). 1468. PARTIAL WAVES Introduction The partial waves method is quite general and applies to part icles interact- ing in very small spatial regions with another one, which is u sually known as scattering center because of its physical characteristi cs. (for example, because it can be considered as fixed). Beyond the interactio n region, the interaction between the two particles is usually negligibl e. Under this cir- cumstances, it is possible to describe the scattered partic le by means of the Hamiltonian H=H0+V , (1) whereH0corresponds to the free particle Hamiltonian. Our problem i s to solve the equation (H0+V)|ψ∝an}b∇acket∇i}ht=E|ψ∝an}b∇acket∇i}ht. (2) Obviously, the spectrum will be continuous since we study th e case of elastic scattering. The solution will be |ψ∝an}b∇acket∇i}ht=1 E−H0V|ψ∝an}b∇acket∇i}ht+|φ∝an}b∇acket∇i}ht. (3) It is easy to see that for V= 0 one can obtain the solution |φ∝an}b∇acket∇i}ht, i.e., the solution corresponding to the free particle. It is worth noting that in a certain sense the operator1 E−H0is anomalous, because it has a continuum of poles on the real axis at positions coinciding with the eig envalues of H0. To get out of this trouble, it is common to produce a small shif t in the imaginary direction ( ±iǫ) of the cut on the real axis |ψ±∝an}b∇acket∇i}ht=1 E−H0±iεV|ψ±∝an}b∇acket∇i}ht+|φ∝an}b∇acket∇i}ht (4) This equation is known as the Lippmann-Schwinger equation. Finally, the shift of the poles is performed in the positive sense of th e imaginary axis because in this case the causality principle holds (cf. Feynman). Let us consider the x representation ∝an}b∇acketle{tx|ψ±∝an}b∇acket∇i}ht=∝an}b∇acketle{tx|φ∝an}b∇acket∇i}ht+/integraldisplay d3x′/angbracketleftbigg x|1 E−H0±iε|x′/angbracketrightbigg ∝an}b∇acketle{tx′|V|ψ±∝an}b∇acket∇i}ht.(5) The first term on the right hand side corresponds to a free part icle, while the second one is interpreted as a spherical wave getti ng out from the 147scattering center. The kernel of the previous integral can b e considered as a Green function (also called propagator in quantum mechani cs). It is a simple matter to calculate it G±(x,x′) =¯h2 2m/angbracketleftbigg x|1 E−H0±iε|x′/angbracketrightbigg =−1 4πe±ik|x−x′| |x−x′|, (6) whereE= ¯h2k2/2m. Writing the wave function as a plane wave plus a divergent spherical one (up to a constant factor), ∝an}b∇acketle{tx|ψ+∝an}b∇acket∇i}ht=ek·x+eikr rf(k,k′). (7) the quantity f(k,k′) is known as the scattering amplitude and is explicitly f(k,k′) =−1 4π(2π)32m ¯h2∝an}b∇acketle{tk′|V|ψ+∝an}b∇acket∇i}ht. (8) Let us now define an operator Tsuch that T|φ∝an}b∇acket∇i}ht=V|ψ+∝an}b∇acket∇i}ht (9) If we multiply the Lippmann-Schwinger equation by Vand make use of the previous definition, we get T|φ∝an}b∇acket∇i}ht=V|φ∝an}b∇acket∇i}ht+V1 E−H0+iεT|φ∝an}b∇acket∇i}ht. (10) Iterating this equation (as in perturbation theory) we can g et the Born approximation and its higher-order corrections. Partial waves method Let us now consider the case of a central potential. In this ca se, using the definition (9), it is found that the operator Tcommutes with /vectorL2and/vectorL; it is said thatTis a scalar operator. To simplify the calculations it is conv enient to use spherical coordinates, because of the symmetry of the problem that turns theToperator diagonal. Let us see now a more explicit form of the scattering amplitude f(k,k′) = const./summationdisplay lml′m′/integraldisplay dE/integraldisplay dE′∝an}b∇acketle{tk′|E′l′m′∝an}b∇acket∇i}ht∝an}b∇acketle{tE′l′m′|T|Elm∝an}b∇acket∇i}ht∝an}b∇acketle{tElm|k∝an}b∇acket∇i}ht, (11) 148where const .=−1 4π2m ¯h2(2π)3. After some calculation, one gets f(k,k′) =−4π2 k/summationdisplay l/summationdisplay mTl(E)Ym l(k′)Ym∗ l(k). (12) Choosing the coordinate system such that the vector khave the same direction with the z axis, one infers that only the spherical harmonics of m= 0 will contribute to the scattering amplitude. If we define b yθthe angle between kandk′, we will get Y0 l(k′) =/radicaligg 2l+ 1 4πPl(cosθ). (13) Employing the following definition fl(k)≡−πTl(E) k, (14) eq. (12) can be written as follows f(k,k′) =f(θ) =∞/summationdisplay l=0(2l+ 1)fl(k)Pl(cosθ). (15) Forfl(k) a simple interpretation can be provided, which is based on the expansion of a plane wave in spherical waves. Thus, we can write the function∝an}b∇acketle{tx|ψ+∝an}b∇acket∇i}htfor large values of rin the following form ∝an}b∇acketle{tx|ψ+∝an}b∇acket∇i}ht=1 (2π)3/2/bracketleftigg eikz+f(θ)eikr r/bracketrightigg = 1 (2π)3/2/bracketleftigg/summationdisplay l(2l+ 1)Pl(cosθ)/parenleftigg eikr−ei(kr−lπ) 2ikr/parenrightigg +/summationdisplay l(2l+ 1)fl(k)Pl(cosθ)eikr r/bracketrightigg =1 (2π)3/2/summationdisplay l(2l+ 1)Pl(cosθ) 2ik/bracketleftigg [1 + 2ikfl(k)]eikr r−ei(kr−lπ) r/bracketrightigg .(16) This expression can be interpreted as follows. The two expon ential terms correspond to spherical waves: the first to a divergent wave, and the latter to a convergent one. Moreover, the scattering effect is conveni ently displayed in the coefficient of the divergent wave, which is unity when th ere are no scattering centers. 149Phase shifts We consider now a surface enclosing the scattering center. A ssuming that there is no creation and annihilation of particles, one has /integraldisplay j·dS= 0, (17) where the integration region is the aforementioned surface , and jis the probability current density. Moreover, because of the cons ervation of the orbital momentum, the latter equation should hold for each p artial wave. The theoretical formulation of the problem does not change i f one assumes the wave packet as a flux of noninteracting particles propaga ting through a region of central potential for which the angular momentum o f each particle is conserved, so that the ‘particle’ content of the wave pack et really does not change. Thus, one may think even intuitevely that only phase factor effects can be introduced under these circumstances. Thus, if one de fines Sl(k)≡1 + 2ikfl(k) (18) we should have |Sl(k)|= 1. (19) These results can be interpreted using the conservation of p robabilities. They are natural and expected because we assumed that there i s no particle creation and annihilation. Therefore, the effects of the sca ttering center is reduced to adding a phase factor in the components of the dive rgent wave. Taking into account the unitarity of the phase factor, we can write it in the form Sl=e2iδl, (20) whereδlis real and a function of k. Taking into account the definition (18), we can write fl=e2iδl−1 2ik=eiδlsin(δl) k=1 kcot(δl)−ik. (21) The total cross section has the following form σtotal=/integraldisplay |f(θ)|2dΩ = 1 k2/integraldisplay2π 0dφ/integraldisplay1 −1d(cos(θ))/summationdisplay l/summationdisplay l′(2l+ 1)(2l′+ 1)eiδlsin(δl)eiδl′sin(δl′)PlPl′ 150=4π k2/summationdisplay l(2l+ 1)sin2(δl′). (22) Getting the phase shifts Let us consider now a potential V that is zero for r>R , where the parameter Ris known as the range of the potential. Thus, the region r>R corresponds to a spherical unperturbed/free wave. On the other hand, the general form of the expansion of a plane wave in spherical ones is ∝an}b∇acketle{tx|ψ+∝an}b∇acket∇i}ht=1 (2π)3/2/summationdisplay lil(2l+ 1)Al(r)Pl(cosθ) (r>R ), (23) where the coefficient Alis by definition Al=c(1) lh(1) l(kr) +c(2) lh(2) l(kr). (24) h(1) landh(2) lare the spherical Hankel functions whose asymptotic forms are the following h(1) l∼ei(kr−lπ/2) ikr h(2) l∼−e−i(kr−lπ/2) ikr. Inspecting the following asymptotic form of the expression (23) 1 (2π)3/2/summationdisplay l(2l+ 1)Pl/bracketleftigg eikr 2ikr−e−i(kr−lπ) 2ikr/bracketrightigg , (25) one can see that c(1) l=1 2e2iδlc(2) l=1 2. (26) This allows to write the radial wave function for r>R in the form Al=e2iδl[cosδljl(kr)−sinδlnl(kr)]. (27) Using the latter equation, we can get the logarithmic deriva tive inr=R, i.e., at the boundary of the potential range βl≡/parenleftbiggr AldAl dr/parenrightbigg r=R=kR/bracketleftigg j′ lcosδl−n′ l(kR)sinδl jlcosδl−nl(kR)sinδl/bracketrightigg . (28) 151j′ lis the derivative of jlwith respect to revaluated at r=R. Another important result that can be obtained from the knowledge of t he previous one is the phase shift tanδl=kRj′ l(kR)−βljl(kR) kRn′ l(kR)−βlnl(kR). (29) To get the complete solution of the problem in this case, it is necessary to make the calculations for r < R , i.e., within the range of the potential. For a central potential, the 3D Schr¨ odinger equation reads d2ul dr2+/parenleftbigg k2−2m ¯h2V−l(l+ 1) r2/parenrightbigg ul= 0, (30) whereul=rAl(r) is constrained by the boundary condition ul|r=0= 0. Thus, one can calculate the logarithmic derivative, whic h, taking into account the continuity of the log-derivative (equivalent t o the continuity condition of the derivative at a discontinuity point) leads to βl|in=βl|out. (31) An example: scattering on a hard sphere Let us now consider an important illustrative case, that of t he hard sphere potential V=/braceleftigg ∞r<R 0r>R .(32) It is known that a particle cannot penetrate into a region whe re the potential is infinite. Therefore, the wave function should b e zero atr=R. Since we deal with an impenetrable sphere we also have Al(r)|r=R= 0. (33) Thus, from eq. (27), we get tanδl=jl(kR) nl(kR). (34) One can see that the phase shift calculation is an easy one for anyl. In thel= 0 case (s wave scattering), we have δl=−kR 152and from eq. (27) Al=0(r)∼sinkr krcosδ0+coskr krsinδ0=1 krsin(kr+δ0). (35) We immediately see that there is an additional phase contrib ution with regard to the motion of the free particle. It is also clear tha t in more general cases the various waves will have different phase shifts lead ing to a transient distortion of the scattered wave packet. At small energies, i.e.,kR << 1, the spherical Bessel functions (entering the formulas for t he spherical Hankel functions) are the following jl(kr)∼(kr)l (2l+ 1)!!(36) nl(kr)∼−(2l−1)!! (kr)l+1, (37) leading to tanδl=−(kR)2l+1 (2l+ 1)[(2l−1)!!]2. (38) From this formula, one can see that a substantial contributi on to the phase shift is given by the l= 0 waves. Moreover, since δ0=−kRthe cross section is obtained as follows σtotal=/integraldisplaydσ dΩdΩ = 4πR2. (39) One can see that the total scattering cross section is four ti mes bigger than the classical one and coincides with the total area of th e impenetrable sphere. For large values of the incident energy, one can work in the hypoth- esis that all values of lup to a maximum value lmax∼kRcontribute to the total cross section σtotal=4π k2l∼kR/summationdisplay l=0(2l+ 1)sin2δl. (40) In this way, from eq. (34), we have sin2δl=tan2δl 1 + tan2δl=[jl(kR)]2 [jl(kR)]2+ [nl(kR)]2∼sin2/parenleftbigg kR−lπ 2/parenrightbigg ,(41) where the expressions jl(kr)∼1 krsin/parenleftbigg kr−lπ 2/parenrightbigg 153nl(kr)∼−1 krcos/parenleftbigg kr−lπ 2/parenrightbigg . have been used. Inspection of δlshows a negative jump ofπ 2wheneverlis augmented by a unity. Thus, it is clear that sin2δl+ sin2δl+1= 1 holds. Approximating sin2δlby its mean value1 2over a period and using the sum of odd numbers, one gets σtotal=4π k2(kR)21 2= 2πR2. (42) Once again the quantum-mechanical result, although quite s imilar to the corresponding classical result is nevertheless different. What might be the origin of the factor of two that makes the difference ? To get an explanation, we first separate eq. (15) in two parts f(θ) =1 2ikl=kR/summationdisplay l=0(2l+1)e2iδlPlcos(θ)+i 2kl=kR/summationdisplay l=0(2l+1)Plcos(θ) =frefl+fshadow. (43) Calculation of/integraltext|frefl|2dΩ gives /integraldisplay |frefl|2dΩ =2π 4k2lmax/summationdisplay l=0/integraldisplay −11 (2l+ 1)2[Plcos(θ)]2d(cosθ) =πl2 max k2=πR2. (44) Analysing now fshadowat small angles, we get fshadow∼i 2k/summationdisplay (2l+ 1)J0(lθ)∼ik/integraldisplayR 0bJ0(kbθ)db=iRJ1(kRθ) θ.(45) This formula is rather well known in optics. It corresponds t o the Fraun- hofer diffraction. Employing the change of variable z=kRθone can calcu- late the integral/integraltext|fshadow|2dΩ /integraldisplay |fshadow|2dΩ∼2πR2/integraldisplay∞ 0[J1(z)]2 zdz∼πR2. (46) Finally, neglecting the interference between freflandfshadow(since the phase oscillates between 2 δl+1= 2δl−π), one gets the result (42). The label ‘shadow’ for one of the terms is easily explained if one thinks of the wavy behaviour of the scattered particle (from the physical viewpoint there is no difference between a wave packet and a particle in this ca se). Its origin can be traced back to the backward-scattered components of t he wave packet leading to a phase shift with respect to the incident waves an d destructive interference. 154Coulomb scattering In this section we briefly consider the Coulomb scattering in the quantum- mechanical approach. For this case, the Schr¨ odinger equat ion is /parenleftigg −¯h2 2m∇2−Z1Z2e2 r/parenrightigg ψ(r) =Eψ(r), E > 0, (47) wheremis the reduced mass of the system, E >0 since we deal with the simple scattering case where no kind of bound states are allo wed to form. The previous equation is equivalent to the following expres sion (for adequate values of the constants kandγ) /parenleftbigg ∇2+k2+2γk r/parenrightbigg ψ(r) = 0. (48) If we do not consider the centrifugal barrier, i.e., we look o nly to thes waves, we really deal with a pure coulombian interaction, fo r which one can propose a solution of the following form ψ(r) =eik·rχ(u), (49) where u=ikr(1−cosθ) =ik(r−z) =ikw , k·r=kz . ψ(r) is the complete solution of the Schr¨ odinger equation with an asymptotic ‘physical’ behaviour to which a plane wave eik·rand a spherical wave are expected to contribute r−1eikrare expected to contribute. Defining new variables z=z w =r−z λ =φ , and by employing of previous relationships, eq. (48) takes t he form /bracketleftigg ud2 du2+ (1−u)d du−iγ/bracketrightigg χ(u) = 0. (50) To solve this equation, one should first study its asymptotic behaviour. Since we have already tackled this issue, we merely present t he asymptotic normalized wave function that is the final result of all previ ous calculations ψk(r) =1 (2π)3/2/parenleftigg ei[k·r−γln(kr−k·r)]+fc(k,θ)ei[kr+γln2kr] r/parenrightigg . (51) 155As one can see, this wave function displays terms that turns i t quite dif- ferent from the form in eq. (7). This is due to the fact that the Coulomb po- tential is of infinite range. Performing the exact calculati on for the Coulomb scattering amplitude is not an easy matter. Here we give only the final result for the ‘normalized’ wave function ψk(r) =1 (2π)3/2/parenleftigg ei[k·r−γln(kr−k·r)]+g∗ 1(γ) g1(γ)γ 2ksin(θ/2)2ei[kr+γln2kr] r/parenrightigg , (52) whereg1(γ) =1 Γ(1−iγ). In addition, we reduce the partial wave analysis to a clear cu t presenta- tion of the results, of which some have already been mentione d. First of all, we write the wave function ψ(r) in (49) as follows ψ(r) =eik·rχ(u) =Aeik·r/integraldisplay Ceuttiγ−1(1−t)−iγdt , (53) whereAis a ‘normalization’ constant, while all the integral part i s the inverse Laplace transform of the direct transform of eq. (50 ). A convenient form of the latter equation is ψ(r) =A/integraldisplay Ceik·r(1−t)eikrt(1−t)d(t,γ)dt (54) where d(t,γ) =tiγ−1(1−t)−iγ−1. (55) Within the partial wave analysis we proceed by writing ψ(r) =∞/summationdisplay l=0(2l+ 1)ilPl(cosθ)Al(kr), (56) where Al(kr) =A/integraldisplay Ceikrtjl[kr(1−t)](1−t)d(t,γ). (57) Applying the relationships between the spherical Bessel fu nctions and the Hankel functions, we get Al(kr) =A(1) l(kr) +A(2) l(kr). (58) We shall not sketch here how these coefficients are obtained (t his is quite messy). They are A(1) l(kr) = 0 (59) 156A(2) l(kr)∼−Aeπγ/2 2ikr[2πig1(γ)]/parenleftig e−i[kr−(lπ/2)+γln 2kr]−e2iηl(k)ei[kr−(lπ/2)+γln2kr]/parenrightig (60) where e2iηl(k)=Γ(1 +l−iγ) Γ(1 +l+iγ). (61) Calculation of the Coulomb scattering amplitude If we perform the Laplace transform of eq. (50), we get χ(u) =A/integraldisplay Ceuttiγ−1(1−t)−iγdt . (62) The contour Cgoes from−∞to∞on the real axis and closes through the upper half-plane. There are two poles in this case at t= 0 andt= 1. By the change of variable s=ut, we get χ(u) =A/integraldisplay C1essiγ−1(u−s)−iγ. (63) χ(u) should be regular in zero. Indeed, we get χ(0) = (−1)−iγA/integraldisplay C1es sds .= (−1)−iγA2πi (64) Performing now the limit u→∞, let’s do an infinitesimal shift to avoid the location of the poles on the contour. Moreover, by the cha nge of variable s u=−(s0±iε) iκ, we see that this expression goes to zero when u→∞. Thus, we can expand ( u−s) in power series ofs ufor the pole with s= 0. This expansion is not the right one in s= 1, because in this case s=−s0+i(κ±ε). It comes out thats u= 1−(s0±iε) κtends to 1 when κ→∞. If instead we do the change of variable s′=s−u, we get rid of this difficulty χ(u) =A/integraldisplay C2/parenleftbigg [essiγ−1(u−s)−iγ]ds+ [es′+u(−s′)iγ(u+s′)iγ−1]ds′/parenrightbigg . (65) Expanding the power series, it is easy to calculate the previ ous integrals, but one should take the limits u→0 in the result in order to get the correct asymptotic forms for the Coulomb scattering χ(u)∼2πiA/bracketleftig u−iγg1(γ)−(−u)iγ−1eug2(γ)/bracketrightig 1572πg1(γ) =i/integraldisplay C2essiγ−1ds 2πg2(γ) =i/integraldisplay C2ess−iγds . (66) After all this chain of variable changes, we get back to the or iginalsone to obtain (u∗)iγ= (−i)iγ[k(r−z)]iγ=eγπ/2eiγlnk(r−z) (u)−iγ= (i)−iγ[k(r−z)]−iγ=eγπ/2e−iγlnk(r−z). (67) The calculation of χ, once effected, is equivalent with having ψk(r) start- ing from (49). Eikonal approximation We shall briefly expound on the eikonal approximation whose p hilosophy is the same to that used when one wants to pass from the wave opt ics to the geometrical optics. Therefore, it is the right approxim ation when the potential varies slowly over distances comparable to to the wavelength of the scattered wave packet, i.e., for the case E >>|V|. Thus, this approx- imation may be considered as a quasiclassical one. First, we propose that the quasiclassical wave function has the known form ψ∼eiS(r)/¯h, (68) whereSsatisfies the Hamilton-Jacobi equation, having the solutio n S ¯h=/integraldisplayz −∞/bracketleftbigg k2−2m ¯h2V/parenleftig/radicalbig b2+z′2/parenrightig/bracketrightbigg1/2 dz′+ constant . (69) The additive constant is chosen in such a way to fulfill S ¯h→kz forV→0. (70) The term multiplying the potential can be interpreted as a ch ange of phase of of the wave packet, having the following explicit fo rm ∆(b)≡−m 2k¯h2/integraldisplay∞ −∞V/parenleftig/radicalbig b2+z2/parenrightig dz . (71) Within the method of partial waves, the eikonal approximati on has the following application. We know it is correct at high energie s, where many 158partial waves do contribute to the scattering. Thus, we can c onsiderlas a continuous variable and by analogy to classical mechanics we letl=bk. Moreover, as we already mentioned lmax=kR, which plugged into eq. (15) leads to f(θ) =−ik/integraldisplay bJ0(kbθ)[e2i∆(b)−1]db . (72) 8P. Problems Problem 8.1 Obtain the phase shift and the differential cross section at s mall angles for a scattering centre of potential U(r) =α r2. It should be taken into account that for low-angle scattering the main contribution is give n by the partial waves of large l. Solution : Solving the equation R′′ l+/bracketleftigg k2−l(l+ 1) r2−2mα ¯h2r2/bracketrightigg = 0 with the boundary conditions Rl(0) = 0,Rl(∞) =N, whereNis a finite number, we get Rl(r) =A√rIλ(kr), whereλ=/bracketleftigg (l+1 2)2+2mα ¯h2/bracketrightigg1/2 andIis the first modified Bessel function. To determine δl, one should use the asymptotic expression of Iλ: Iλ(kr)∝/parenleftbigg2 πkr/parenrightbigg1/2 sin(kr−λπ 2+π 4). Therefore δl=−π 2/parenleftbigg λ−l−1 2/parenrightbigg =−π 2 /bracketleftigg (l+1 2)2+2mα ¯h2/bracketrightigg1/2 −/parenleftbigg l+1 2/parenrightbigg . 159The condition of large lleads us to δl=−πmα (2l+ 1)¯h2, whence one can see that |δl|≪1 for largel. From the general expression of the scattering amplitude f(θ) =1 2ik∞/summationdisplay l=0(2l+ 1)Pl(cosθ)(e2iδl−1), at small angles one gets e2iδl≈1 + 2iδl, so that ∞/summationdisplay l=0Pl(cosθ) =1 2sinθ 2. Thus f(θ) =−παm k¯h21 2sinθ 2. The final result is dσ dΩ=π3α2m 2¯h2Ectgθ 2. 160