VOLUME PHYSICAL REVIEW LETTERS 35, NUMBER 23 8 DECEMBER 1975 have (d„s)„=cosmos cosy(d„a)„i+ sings(d~s), i —costs siny(d s), ~, (d„s), = —sings cosy(d„&)„i+ cosmos(d„&), + sings siny(d &), , (d 6), =siny(d„s)„. +cosy(d„s), . For transitions with (d „s)„.=(d „a),.=0 and (d s), . c0, none of the three components of cT s is zero. In view that q. B = gB sin6, cos(y~- y, ), where 6, and y, are the polar and azimuthal angles of the scattered beam wave vector ~, none of N „, N„, and M, can in general be zero through the integration of ps in Eg. (11). If we integrate over the scattered beam direction dQ= sine, ds, dy„ the y dependence of the spontaneous radiations drops out. The double differential cross section for spontaneous dipole ~ radiations takes the (nonuniform) d'a, & /der d~~a+b sin'8, form with a=2fdQ~MJ', b = fdQ([M, (' —[M„)'). *Research supported in part by the National Science Foundation (Grant No. MPS-74-22259), and in part by the U. S. Air Force Office of Scientific Research (Contract No. 742716). 'F. W. Saris, W. F. van der Weg, H. Tawara, and R. Laubert, Phys. Rev. Lett. 28, 717 (1972). 2B. Muller and W. Greiner, Phys. Rev. Lett. 33, 469 (1974). 3See, for example, R. S. Thoe, I. A. Sellin, M. D. Brown, J. P. Forester, P. M. Griffin, D. J. Pegg, and R. S. Peterson, Phys. Rev. Lett. 34, 64 (1975), and references contained therein. See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962). ~J. C. Y. Chen, V. H. Ponce, and K. M. Watson, J. Phys. B: At. Mol. Phys. 6, 965 (1973); J. C. Y. Chen and K. M. Watson, Phys. Rev. 174, 152 (1968). Hydrodynamics Physik-Department of 3He near the A Transition Mario Liu der Technischen Universitat, MN'nt. "hen, Germany (Received 10 July 1975) The hydrodynamics of He-A fn the vicinity of the phase transition is generalized to include an extra variable, the magnitude of the order parameter. Its equation of motion is derived and added to the set of hydrodynamic equations. The effect on the dissipation and dispersion of first, second, and fourth sound is calculated. First sound is found to be anisotropic and a pair of new propagating modes is predicted. Approaching the X point of 4He, p' is determined by the local temperature and pressure only in the low-frequency limit. The situation is quite analogous in 'He, the only difference being the complexity of the order parameter D~;, which has eighteen real components. Five of them" correspond to different spontaneously broken symmetries; i.e. , their spatial derivatives are to the superfluid velocity in 4He. The rest become independent variables in the vicinity. of the phase .transition for all but the lowest frequencies. They do not obey conservation laws and relax with different characteristic times, all diverging with the same exponent as T approaches T, . But, as will be shown below, only the magnitude of the order parameter, which is the exact counterpart of p', needs to be considered in the context of mass-transporting hydrodynamequivalent ics. I shall denote the deviation of the magnitude from its equilibrium value, as defined in Eqs. (1), by b. With the inclusion of & as an additional thermodynamic variable and assuming the order-parameter symmetry of the axial state, ~'3 one can derive the equation of motion for 4 using standard procedures4 of hydrodynamic theory for anisotropic systems. With addition of this equation of motion to the set of hydrodynamic equations of 'He-A, ' first, second, and fourth sound have been recalculated. An additional pair of propagating modes due to the coupling of the fluctuation of & and the density arises in the highfrequency regime; their existence may be verif ied experimentally. Out of nine complex components of D„; only the three in the preferred spin direction may couple to hydrodynamic variables of ordinary space. PHYSICAL RKVIKW LKTTKRS VQI. UME 35' NUMBER 23 orbital rotation, respectively. 4 is the deviation from the overall magnitude of D~;, which is a measure of the ordering. 6 and H describe the deformation of the internal structure of D„~. A fluctuating 6 is equivalent to the oscillation of the angle between the real and imaginary parts of the order parameter around the equilibrium value of m/2. II allows each component to have a different magnitude. Note that 6 is odd under time reversal and neither 6 nor H is gauge invariant. y and l; have been treated within the framework of hydrodynamics'; I introduce p. &, p, g, and p, ~ as the respective conjugate variables of H, 6, and 4. Thus, the thermodynamic identity now reads These six independent variables form an axial vector /; and four scalars: H = (D„;d„;+D„g*d«*)/2N, N =D„;*D„;, 8-;*d„;*)/2¹, G = (D;d„; „;*)/2N, y=(D„)+d„; -D )d«*)/2Ni. b, = (D«*d«+D„gd The equilibrium value of the order parameter its deviation are denoted by D~q and d«, and re- spectively. y and l; correspond to the spontaneously broken symmetries under gauge transformation and dE =T dS+ p dp+v" dg+A. "dv'+4;~dVql;+ pgd&+ where p is the mass density, S is the entropy density, g is the total momentum density, v' is the superfluid velocity, and 1 is the axial vector pointing in the preferred direction in equilibrium. @;~, &', p. , v", and T are the respective conjugate variables. With the proper symmetry under time, space, we have and gauge transformation, Pa =X6 p~ & p) d p ~ Considering for the moment only the homogeneous terms in the equations of motion f or the three scalars, we have, with the irreversible expressions on the right-hand side, re/ 6 —cp~ = —Hag, 4= —8P, z, . Equations (4) have three solutions: - w| 2 = (+ o. —i8) X and co, = i 8g . Wolf le' has treated the vibration of the internal structure of D~; on a Fermi-liquid level in a theory of zero-sound absorption. Since 6 and H are connected to his variables, it is reasonable to identify u» with his "clapping mode, " ar„=1.234, whose damping resulting from the relaxation has been estimated to be negligible. However, because of a substantial amount of pair-breaking processes, it seems quite pointless to extend the hydrodynamics to this frequency range, even if ~~7, «1, where v, denotes the average quasiparticle collision time. As long as the frequency is restricted to ~ «~„, H and G do not become 1578 l p, „dII+ p, g dG, (2) soft and need not be included in the thermodynamic identity. The equations of motion of H and 6 may be combined to form one for D«d„&. To the next order in wave number k, we have D„qd«+ (8 —ia)XD«d„;+PD«DS;V„vs" =0. Re(P)D„;DB;V~vs" is reversible and i Im(P)D« is not. Consistent with the result of &&D~;&~as" = X+ + (& u b/8T ) dT + (& u ~/8 H+cxpg = —HAH, 8 DECEMBER 1975 Wolf le, the coupling to collective modes like the sound occurs only when k is not parallel to 1. This still holds for terms of higher order in k (e g. , D.;DB*.V Vap) With the inclusion of only & as a new variable and to first order in wave number k the conservation laws for momentum and density yield g;+V;P+rj;yV&pz —0, p+V;g& —0, (6) with gf =Vj (Pity —Ply q;q = qual;i(+ pi(5;q )+Vg Pig —l;lq), ) ~'4) . . — p s' = p|t'«4+ pi'(~* ; The equations of motion for the remaining variables have the following form: S+pV;v "=8p 2/T, 0 v 4+qg~V;vy" + pp;g V((vg +V (p+ vp )=0, (7) . (8) —vg") = —8p, h The form of the fluxes has been determined by the correct symmetry under time and space transformation and the absence of entropy production for the reversible part. The only irreversible part is the relaxation term Op, ~. This VOLUME I HYSrCwr. REVIEW I. ZTTERS 35, NUMBER 23 suggests the definition of the relaxation time for the order parameter as v= (8g) '. The positive definiteness of entropy production requires 0 ~ 0. At the present level of accuracy the orbital variables l& are decoupled. In an isotropic superfluid such as 'He, Eq. (8) should reduce to the corresponding equation of motion'. p'+V, (p'v, ') = —(2Amp'/8) BE/Bp', A simple homogeneous Galilean transformation would transport the superfluid part in a different direction from the normal part and thereby build up an entropy gradient. Expansion in powers of k is restricted to the where $ is the critical correlacondition k$ «1, tion length, becoming infinite at T,. Also, assuming the local equilibrium value, except for the order parameter and the conserved quantities, i.e. , v«10' Hz. b, is an indemeans mv, pendent variable only for ~» ~„. with 7 taken to be of the same form as in an uncharged superconductor, ' i. e. , 7 =k[32k~(T T,)]-', it is equivalent to e=(T, T)/T—«10 '. These three con= 1.5 ditions with the one mentioned above, co x 10'e' ', specify the temperature, wave number, and frequency range, for which the following predictions are made. second and fourth sound are found With w7. to be " «1, ' with p'= gb, near the phase transition. It does so under the additional requirement that the superfluid part of the liquid is not transported by the normal velocity. This has also been used by Khalatnikov. The additional requirement means Howthat v=(2t;b, ) ', q=-6/2, and 8=Am(2&5) ever, this requirement is not compatible with the known property of 'He having an anisotropic p': '. ' =k'(S'T/p 8 DECEMBER 1975 , «1, C„)(p ']'11 —(i(0T/8C„)[- (Bk/BT )& & +C„/ST (6 —vp + 1gllp ll ' —(ice/8)[(BA/ap) v- (ST/PC„)(aa/BT) „]'). (u, '=k'(p„')((p'~) to „- ll We denote the compressibility and specific heat by ~ and C„, respectively, and abbreviate Allcos'8 +A~ sin'e by (A, l] with 8 as the angle between k and l. Approaching T, , we arrive eventually at a temperature where, for a given frequency, w7. =1 (e.g. , ~ = 10' and e = 10 ') and, with second sound being connected to the critical mode, k$ = l. Here, second sound ceases to propagate. First sound has the same dissipative mechanism and is therefore also characterized by 7. But +v»1 is and since" U, »U, (U, equivalent to kU, $/U, and U, are the velocity of first and second sound, still holds. Hence first respectively), k$ sound even propagates in a high-frequency regime as a compressional mode with a small disturbance, i.e. , ' »1 «1 w' —k'/zp «k'/~p, arising from the presence of a nonconserved but slowly relaxing Landau-type quantity. With Eq. (11) we have (12) co,'=k'[Co'+ i&f7(Co' —C„')(1 —icos) '], where C,'=(p~) ', C„=CO +[bjll] -P(ab/Bp)q p For mT» 1, there is virtually ]8P~, no relaxation of «e ' ]'/(p '))]'}, ll (10) 6 and in Eq. (8) only the reactive part remains. Therefore the diffusive b, mode, w = —i/v, becomes a propagating mode. Because k$ «1, the velocities in the solutions of the following equations must be much greater than that of second sound: (u+bll]p~) =0 P+ + (&Ill]P + &Ill ]O'4) & (13) Equations (13) have two pairs of propagating modes as solutions: +,'=C, 'k'. Assuming C+ »C [which is equivalent to Eq. (11)], we can expand them, yielding C+ =C„and C '=[(C,/C )(q, —q~) cos8 sin8]'/p8v. l (14) Since in an isotropic superfluid g„=@~, 6 sound cannot propagate in ~He. Using the Ginzburg criterion" e, = ~3' (k~/wb, C),')', taking" $, = 200 A and b, C = 23x 10 'nR, I estimate the mean-field region to extend to about e ~ 10 '. Therefore I expect the critical exponents to assume the mean-field values: and ah/Bp = (Bh/BT, ) BT, /Bp- e"'. Besides, being Onsager coefficients, 8, g~, , and g~ are expected to be smooth functions of temperature. Consequently, Co and C „do not exhibit critical behavior and one may hope to detect b, sound where first sound has a measurable anisotropic '" 1579 VOLUME 35, NUMBER 23 PHYSICAL REVIEW LETTERS term proportional to sin48. I am indebted to Professor Hartwig Schmidt for suggesting this work and gratefully acknowledge both his and Professor Peter %olfle's numerous helpful and illuminating discussions. I. M. Khalatnikov, Zh. Eksp. Teor. Fiz. 57, 489 (1969) ISov. Phys. JETP 30, 268 (1970)]. A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975). P. W. Anderson and W. F. Brinkman, "Theory of Anisotropic Superfluid in He" (to be published). L. D. Landau and E. M. Lifshitz, Eluid Mechanics (Addison-Wesley, Reading, Mass. , 1959). F. Jahnig and H. Schmidt, Ann. Phys. (N. Y.) 71, 129 1580 8 DECEMBER 1975 (1972); D. Forster, T. C. Lubensky, P. C. Martin, and P. C. Pershan, Phys. Bev. Lett. 26, 1016 J. Swift, (1971). R. Graham, Phys. Rev. Lett. 33, 1431 (1974). P. Wolfle, in Quantum Statistics and th, e Many-Body Problem, edited by S. B. Trickey et al. (Plenum, New York, 1975), and Phys. Rev. Lett. 31, 1437 (1973). A. Schmid, Phys. Kondens. Mater. 5, 302 (1966). R. A. Ferrell, N. Menyhard, H. Schmidt, F. Schwabl, and P. Szepfalusy, Ann. Phys. ) 47, 565 (1968). L. P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Palciauskas, N. Bayl, J. Swift, D. Asphes, and J. Kane, Rev. Mod. Phys. 39, 395 (1967). B. A. Webb, T. C. Greytak, R. T. Johnson, and J. C. Wheatley, Phys. Rev. Lett. 30, 210 (1973). (¹Y.