Landau's Major Contribution to Physics A Brief Overview

C. SlYARAMI AND KENATH ARUN2 1. Indian Institute of Astrophysics, Bangalore c. mail: [email protected] 2. Christ Junior College, Banga/ore

ABSTRACT Lev Davidovich Landau was arguably one of the greatest and most versatile physicists. His work spans a very wide range and has had a considerable impact on ail areas of physics including condensed matter physics, plasma, nigh energy and particle physics, fluid ~namics, astrophysics, gravitation, elasticity, etc. A long with his former fellow student, E Eifshitz, he wrote the set of a dozen volumes, the magnificent world-renowned Course of Theoretical Physics which covered topics ranging from mechanics, classical theory offields, quantum mechanics, electrodynamics of conti1lUous media, flUid dynamics, kinetic theory, theory ofelasticity, statistical physics and more.

Lev Davidovich Landau studied physics at Cambridge University to work with Paul Dirac. Baku University and in Leningrad and On his return in 1932, he became head of graduated in 1927. From 1929-31, he visited the theoretical division at the Ukrainian various centres in Europe, including Gottingen, Institute at Kharkov, establishing the first the Bohr Institute at Copenhagen and then school of theoretical physics in the former Physics Education • April - June 2009 85 Soviet Union, which soon received 8. High energy physics: Landau- international reputation. In 1937, the eminent Pomeranchuk theorem; particle physics: Russian physicist, Peter Kapitza who was then two component theory of neutrinos, CP the director of the Institute for Physical violation, Landau gauge, Landau poles. Problems in Moscow, persuaded Landau to 9. Gravitation: Landau-Lifshitz pseudo- head the theory division at the Institute. tensor for calculating energy of the Landau's work ,spans a very wide range and gravitational field. has had a considerable impact on all areas of 10. Hydrodynamics: Theory of explosion, physics including condensed matter physics, Landau-Darrieus instability (now applied plasma, high energy and particle physics, fluid to type Ia supernova). dynamics, astrophysics, gravitation, elasticity, etc. Along with his former fellow student, E 1_ Theory of Liquid Helium Lifshitz, he wrote the set of a dozen volumes, I Landau thought of superfluid helium 4 as a the magnificent world-renowned 'Course of quantum liquid so that its atoms behave as if Theoretical Physics' which covered topics they were in the same quantum state. Their ranging from mechanics, classical theory of motions are highly correlated enhancing the fields, quantum mechanics, electrodynamics of physical properties of the superfluid state. He continuous media, fluid dynamics, kinetic introduced the notion of quasipartic\es, which theory, theory of elasticity, statistical physics act as carriers of motion in the liquid. This is and more. an important concept applied in many other "There is hardly anything in physics phenomena in condensed matter physics. (For which the authors have not mastered!" instance, polarons in a crystal is electron 'dressed' by the surrounding lattice - Nature deformation). Some well known results of Landau's work The energy-momentum (dispersion) include: I ,2 relation for elementary excitations in superfluid helium-4 has the following form: 1. Theory of liquid helium. Introduction of notion of quasiparticies. Superfiuid helium as a quantum liquid. Phonons and rotons. Prediction of second, third, fourth and fifth sounds, zero sound. E 2. Ginzburg-Landau theory of super­ conductivity. Type II superconductors, maximum magnetic field, led to concept of magnetic vortices carrying quantized flux. 3. Landau damping in plasma. 4. Landau-Lifshitz equations for evolution of q spin fields in ferromagnets. General theory of phase transitions. 5. Landau levels in diamagnetism. The long wavelength excitations are 6. Elasticity: Attenuation of shear waves in phonons with dispersion nCtJ, =c.q , where c". is crystals, Landau-Rumer theory. the sound speed and q is the magnitude of the 7. Neutron star mass limit. wave vector. These are the phonons. 86 Physics Education • April - June 2009 In case of shorter wavelengths, the 2 P. Ts 2 excitation spectrum has a local minimum. The c2.=--- excitations near this minimum are known as Pn C rotons and their dispersion relation is roughly where r:is the temperature, S is the specific described by: entropy and C is the specific heat.

Ii coq ~ A + a(q - qoi, a is a constant. Above the so called lambda (J.) point, when there is no superfluid. p. = 0 , so C2~ = O. Pn Density of states peE) ocq'l ~~I Below about lK, there is a rapid increase to value C2S = CI •.f J3 , where CIS is the usual first For wavevector near (minimum) sound velocity having a value c~. = B , B is the qo, E ~ A + ak! , P where, bulk modulus and p the total density. The third sound occurs in very thin films k=q-qo and is given by: Defining c2 _P. 3fJ Ji2 3.,- P d 3 m=-, 2a where d is the film thickness, ~ is the Van der by analogy with the non-relativistic relation Waal coefficient. Typically it is 140m/s. The fourth sound is a pressure wave Ji2k2 £::.-- , propagating in superfluid helium confined to a 2a porous material (like packed powders) and is given by: we see that this describes free quasiparticles of mass m in a constant potential A. One can write a single roton wave function (using the NR Schrodinger equation). The binding of rotons is related to As P. --+ 1 , C4s --+ C1.I phenomenon of Cooper pairs in p superconductors, Landau first predicted existence of a second sound in the This measurement (of C4s) can be used to superfluid. Unlike the usual first sound give precise values of the superfluid fraction (which propagates pressure oscillations) second sound involves oscillations in Ps as a function of T. We also have the fifth entropy and is a temperature wave in the P sound which is a temperature wave which can bulk superfluid. propagate in helium confined to a narrow . This was observed experimentally in 1944, orifice. in Moscow, confirming many aspects of the It is analogous to the second sound except two fluid model of liquid helium. The second that only a superfluid component can flow and sound velocity Cz., involves the ratio of the is a low velocity mode with a maximum of superfJuid to normal component Pn densities, 12m/s at about 2K and vanishes at the lambda that is P"lPn. and is given by: Physics Education • April - June 2009 87 point where Pn == 0 is the coherence length, A(x) is the vector potential. At the onset of superconductivity,

2 Pn 2 number density of Cooper pairs, n.f ex: \\,*\\" is Cs• =-c2s P small, so the last term in the Ginzburg-Landau equation (so called cubic term in \jI(x), which is Again when the sound frequency is a novel feature of the model, the relativistic increased to a value greater than the collision version leading later to the so called Higgs rate between helium atoms the wave changes to model with its famous still sought after zero sound from first sound as first predicted particle!) can be neglected. by Landau (for He-3) and has quite recently So we have domination by the external been discovered. (For an account of·the above field with the substitution phenomena, one can refer, for example, to Superfluid Hydrodynamics by S. Putterman If! (x) = rp( x )e1k)'Y eik,Z , (1974 ).) the equation can be shown to reduce to: 2. Ginzburg-Landau Theory of Super­ conductivity Another famous contribution to condensed matter physics is the Ginzburg-Landau theory the equation for a SHO, with of superconductivity which is a mathematical 2eB way of describing properties of Q)=-- superconductors with the help of general cm thermodynamic arguments. It is especially Given A, the maximum allowed value of useful for understanding Type II super­ the magnetic field Bmax (corresponding to n =0, conductors which can be thought of as a Ie. = 0) is second order transition from normal to superconducting phase in a uniform magnetic field. (IiC) 1 lic ( TJ The phase boundary line for the penetrating He' = Bma.x = 2e :i2=2e~; 1-7;. .• magnetic field Hc(1) as a function of temperature can be calculated using the When the field exceeds Bmax, sample is no Ginzburg-Landau equation for the longer superconducting. The field is confined superconducting order parameter \jI(X), valid at to vortices (the theory was later refined by all temperatures T < 1({so called field Abrikosov), the radius of the vortex being A. transition temperature). The equation is the quantum of flux being lic (in fact the 2e factor of 2 was already noted before Cooper pairs were postulated). So

where B =..L-~ I max n:ii. 2 - 2e1l"A. 2 • 2(T)=Au (1_~)-2 7;.. This has also been applied to neutron star

88 Physics Education • April - June 2009 magnetic fields (with pion condensation tran~ition for example which gives a maximal interior field of 10 16G). The Ginzburg-Landau theory is still of great relevance in at Vo = OJI k, the particles give their energy to understanding the new high temperature the wave and the wave amplitude increases. superconductors. This showed that energy exchange processes Following the tradition of Landau being are possible even in colIisionless plasma. This awarded the Nobel Prize in 1962, for his theory phenomenon has found wide applications of liquid helium, Ginzburg and Abrikosov ranging from quark-gluon plasma fo shared the 2003 Nobel Prize with Anthony synchronous light flashing of fireflies! Legett, for their detailed theoretical explanation of He-3 superfluidity. Also we note above that the electron 4. Landau-Lifshitz Equations for motion in the z-direction given rise to Evolution of Spin Fields in Ferromagnets quantized energy levels (n+.!.) eB, the so Again we have the famous Landau-Lifshitz 2 me equations to describe the evolution of spin called ubiquitous Landau levels. fields in continuum ferromagnets. They playa A recent application has been to estimate basic role in understanding non-equilibrium neutron star magnetic fields from cyclotron x- magnetism analogous to that if Navier-Stokes equation in fluid dynamics. · . h h ray Imes seen In t e compact stars, -eB t e The equation has the form: me gyro-frequency being in kilo-electron volts for iJu =~uxH""t -A-zUX(UXHe

Physics Education • April - June 2009 89 6. Landau-Rumer Theory component neutrino theory~ where t~e Weyl equation (rather than the Dirac equation) was In elasticity we have the Landau-Rumer theory the appropriate one for neutrinos. which deals with the direct interaction of Landau also elegantly showed that an acoustic waves with thermal phonons. They electric dipole moment for an elementru:y showed that attenuation for propagating slow particle can exist only in the case when ~here IS shear waves increases as the fourth power of a violation of not only parity conservatIOn (P) the temperature. but also time reversal (T). This can be seen by The attenuation coefficient was defining electric dipole moment of a particle as . kT (T)3 1 follows: proportional Mv2 B ~' where A is the Let CIJ be the charge density inside a wavelength, e the Debye temperature and v the particle with angular momentum J, with sound velocity. quantum number 1, whose orientation is given Experiments were in good agreement with by m = J relative to the z-axis through the the theory. centre of mass. Then ed= faJdVz, 7. Neutron Star Mass Limit dV is the volume element. 6 Landau also obtained a maSs limit for neutron If particle is charged then this implies stars which was the same as the Chandrasekhar charge centroid does not coincides with mass limit. It was given as centroid when d '* O. For a particle like a neutron (uncharged), then d*,O corresponds to excess positive or negative charge in one hemisphere. To connect with CP, we see that if d = 0 there is symmetry under P and T (d does Indeed ma~y pulsars (including the Hulse not change under T, but J does, so d must Taylor binary) have just this mass! vanish if there is T symmetry). So if CPT is a It is said that Landau got this result in 1932, good symmetry, T violation implies CP very soon after he heard of Chadwick's violation. Thus d,* 0 only if CP is violat~. discovery of the neutron. He realized that this Landau pole occurs in the evolution of the would be the 'ultimate' stellar core! (Matter (electromagnetic) coupling constant at a only made of neutrons). Many Russian articles particular value of the momentum scale when call this the Landau-Chandrasekhar limit. perturbation theory breaks down and the coupling diverges. Thus: 8. High Energy Physics 2 In high energy physics, we have the Landau­ -e (p) 1-2f3e2 p =-1--- -;z-2/3op Pomeranchuk theorem, which states that the e cross-sections for particle and antiparticle interactions should (with increasing energy) As p increases, i grows and diverges when asymptotically approach the same value. After 1 P~--2 . the discovery of parity violation in weak 2j3()e interactions, Landau (and independently Lee and Yang and Salam) proposed the two

90 Physics Education • April-June 2009 For QeD, the behaviour is opposite, the In the TT gauge (transverse-traceless coupling vanishes as the momentum tends to gauge, where there are only two componeQ.ts) infinity. the second term in the angle brackets vanishes. This is the analogue of the Poynting vector 2 describing the energy carried by 2(t)- g . 7 g -1-21/3,1() g 2 P e I ectromagnettc waves. The energy flux density of the wave is given as: (p ~ co, g ~ 0) T __1_1';:; hap) Landau gauge: For the gauge fixing term 01 - 32.1Z'G \ alJ,O ,I

_0/ 21 oi 2 - --x2h x-=--h 32nG 2 32nG (~a is the auxiliary field, A. is the gauge fixing parameter) Factor Yz comes from taking time average of square of an oscillating quantity, h is the A. = 0, corresponds to the Landau gauge. Again for the propagator wave amplitude, ro the frequency. c~ / D;~(p)=- i~:b (TIp v +(iL-l) P;;v ) If 1';,1 = ~1Z'r2 (maximal flux), then at ro = 1kHz, r =lkpc, we have h - 10- 1\ r = IMpc. A. = 0 is the Landau gauge, when propagator is we have. h _10-17• transverse A. = 1 is the Feynman gauge, which makes the form simple, that is: Reference 1. Landau and Lifshitz, Course of Theoretical PhYSics, Vol. 1-10, Pergamon Press, Oxford. 2. Collected Papers of L D Landau. Ed. D ter Haar, NY, 1965 (Reprint of Landau's papers). 9. Landau-Lifshitz Pseudo-Tensor 3. Feynman R.P., Statistical PhYSics. Wiley Interscience, New York. 1972. A common prescription for the energy 4. Landau L.D., 1. Phys. USSR, 10,45.1946 momentum tensor describing the energy 5. J.M. Malemberg and C.B. Wharton. Nucl. carried by gravitational waves is the Landau­ Fusion Research (Conf. Proc. IAEA Vienna. Lifshitz pseudo-tensor 1965),1,485,1966. 6. Landau, L.D., Physikalische Zeitschrift dur Sowjetunion, 285-288, 1932. 7. C. Misner, K. Thorne. J. Wheeler. Gravitatioll. Freeman. New York, 1973.

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