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ATTENUATION OF SECOND SOUND IN SOLIDS
K. Weiss
To cite this version:
JOURNAL DE PHYSIOUE
CoZZoque C6, supplSment au n o 12, Tome 42, de'cembre 1981 page C6-531
A T T E N U A T I O N OF SECOND SOUND I N S O L I D S K. ~ e i s s *
Philips Research Laboratories, 5600 MD Eindhoven, The Netherlands.
Abstract. - The small-amplitude plane-wave solutions of the Boltzmann-Peierls equation for phonons in solids are classified and the propagation properties of second sound are discussed in a simple way. Specifically the transition (e.g. with risingtemperature) from t h e case where momentum non-conserving phonon collison processes are much slower than t h e N-processes (second sound regime) t o the case where t h e U-processes dominate (heat diffusion regime) is examined in detail.
0. Introduction. - Second sound in solids [ l ] is a density wavc of t h e phonons which can be detected as a temperature wavc propagating through the sample. It is n o t easy t o create t h e experimental conditions [ 2 ]
which allow these observations because t h e rate 7: of phonon-phonon collisions which conserve momen- tum and energy must be large compared t o the frequency C2 of the wave (local thermal equilibrium) while
the rate T& of thc collisions which d o not conserve these four quantities must be low compared t o
a.
This "window condition" 7; ,> C2 5 r& puts rather severe restrictions on t h e experimental conditions: thesample must only contain very few impurities, dislocations, etc., it must be fairly large t o avoid excessive boundary scattering, and t h e temperature mast be low (but not t o o low because otherwise ballistic propa- gation results in a pure sample) in order t o ensure that Umklapp (U) processes are much less frequent than Normal (N) processcs.
The consequences of this last condition arc t h e topic of t h e present paper. We will therefore assume that t h e U-processes arc faster than all t h e other dissipative processes which we will ignore. Our discussion will be concerned with t h e transition (with rising temperature) from the regime where 7; S 7 2 (second sound)
t o the regime 7 2 T $ (diffusion). It is based on t h e Boltzmann-Peierls equation [ 3 ] in t h e relaxation time approximation t o which we will apply a general method [4] which allows the small-amplitude plane-wave solutions of such an equation t o b c classified.
1. T h e Boltzmann-Peierls Equation in Relaxation Time Approximation. - In t h e absence of external fo,.ces t h e kinetic equation for phonons with group velocity v($) = ae/a$ ( E = phonon energy) in a pure and large crystal in the relaxation timc approximation is [ S ]
fl.,N($) and fid,U(i?) arc the local equilibrium distribution functions towards which f($) develops as a function of timc in t h e two limiting cases t h a t all collisions conserve momentum and energy (N) or energy only (U). They arc
*
Present address: HILT1 A.G. FL-9494 Schaan, Fiirstentum Liechtenstein.JOURNAL DE PHYSIQUE
and
Here T l ( 2 , t ) and ~ ~ ( 2 , t ) are t h e local temperatures in t h e t w o cases and Z(?,t) is t h e local drift velocity established in the case t h a t only N-processes are driving t h e system towards equilibrium. The relaxation times TN and TU are assumed n o t t o depend on the phonon energy and we will restrict the discussion t o phonons of one polarization. A t t h e end we will comment o n both of these assumptions.
The five unknown local parameters
T1,
T Z , and2
(the hydrodynamic variables) can be determined for a given f($) from the five conservation laws which must be imposed o n t h e collision terms. If we define for any function g($) the average<
g>
= ( 2 ~ h ) - ~ j d 3 these conservation laws can be written asand
In what follows we will be interested in small-amplitude plane-wavc solutions about total equilibrium fo(T)
-+
with (complex) frequency Q and real wave vector K and we will restrict the discussion t o t h e hydrodynamic regime, i.e.
1 S L I ~ < l a n d ~ ~ ~ < l w i t h ? - ' = T : +r; (5) whcrc
v
=<
vfo> / <f,> is the mean phonon velocity.After expanding the hydrodynamic variables about their total equilibrium values we are now ready t o apply the classification scheme of Ref. 4 t o discuss the physical properties of t h e solutions of Eq. (1) as the ratio T ~ / T " varies from values much smaller than one t o very large values.
2. Transition from second sound t o the diffusive regime.- As shown in Ref. 4 t h e relevant information nccdcd t o interpret t h e modes resulting from an analysis of t h e Bolrzmann equation as sketched in the pre- vious section is the number o f collisional invariants, i.e. conserved quantities belonging t o t h e fastest relax- ation mechanism. If T: $7;: this number is four and therefore the analysis in Ref. 4 implies that four of the five nontrivial solutions belonging t o the five conscrvation laws Eqs (4) arc compatible with t h e con- ditions of Eqs (5). The fifth solution belongs t o a collisional mode which implies that ~ ~ ( ? , t ) = ~ ~ ( ? , t ) on a timc scalc T TN which is must faster than the time scalc of t h e hydrodynamic modes. Solving t h e fourth order polynomial for t h e four solutions t o second order in t h e small quantitics o f Eqs (5) yields t w o purely damped transverse modes and, more interestingly, the t w o propagating damped second sound modes L . .
2i ' 1
SL=SLo(* 1 +--QOTN + L P )
5 2 a o 7 u ( 6 )
As the temperature of the sample is increased the rate of U-processes increases much faster than the rate of N-processes. Therefore as 7" becomes of the same order as T N , the damping
$
(S~,,T)-') becomes compa- rable t o the velocity (which is lower in this case) of the strongly damped wave. At still higher temperatures we have T; :7 and only one (i.e. the number of collisional invariants of the U-processes) of the fivesolutions remains compatible with the conditions (5). The other four solutions imply (on a time scale of the order of TU <CL-1) ~ ~ ( T t , t ) = ~ ~ ( T f , t ) as before and in addition z(?,t)= 0, which is plausible from the physics involved. The only remaining solution describing a collective mode is
i.e. a purely diffusive mode in the phonon gas: heat diffusion in the crystal.
Concluding Remarks. -We have made two assumptions which are not in fact necessary for the argument as we have developed it: we have considered phonons of only one polarization and relaxation times which were independent of the phonon energies (and of the polarization, of course). Dropping these assumptions still allows t o carry the argument through, and as far as phonons of more than one polarization are con- cerned earlier results (obtained by different methods) are reproduced [1,6,7].
Admitting energy dependent relaxation times introduces the interesting possibility of time-dependent local equilibrium distribution functions [4] but does not change the classififcation of the modes as given in this paper.
References
1. H. Beck, P.F. Meier, and A Thellung, Phys. Stat. Sol. A29, 1 1 (1974).
2. in NaF: T.F. McNelly, S.J. Rogers, D.J. Channin, R.J. Rollefson, W.M. Gobau, G.E. Schmidt,
J.A. Krumhansl, and R.O. Pohl, Phys. Rev. Lett., 24, 100 (1970); H.E. Jackson, C.T. Walker, and T.F. McNelly, Phys. Rev. Lett., 25, 26 (1970); S.J. Rogers, Phys. Rev., 3, 1440 (1971).
in solid helium: C.C. Ackermann, B. Brrtram, H.A. Fairbank, and R.A. Guyer, Phys. Rev. Lett., 16, 789 (1966); C.C. Ackermann and R.A. Guyer, Ann. Phys. (New York) 50, 128 (1968); C.C. Acker- mann and W.C. Overton, Phys. Rev. Lett., 22, 764 (1969). in bismuth: V. Narayanarnurti and R.C. Dynes, Phys. Rev. Lett., 28, 1461 (1972), V. Narayanamurti, R.C. Dynes, and K. Andres, Phys. Rev., B11,2500 (1975).
3 . R.E. Peier!~, Quantum Theory of Solids, Oxford 1955. 4. D. Polder and K. Weiss, Phys. Rev., 17, 1478 (1978).
