PHONON - PHONON INTERACTIONSECOND SOUND IN SOLIDS : THE EFFECTS OF COLLINEAR AND NON-COLLINEAR THREE PHONON PROCESSES

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PHONON - PHONON INTERACTIONSECOND SOUND IN SOLIDS : THE EFFECTS OF COLLINEAR

AND NON-COLLINEAR THREE PHONON PROCESSES

S. Rogers

To cite this version:

S. Rogers. PHONON - PHONON INTERACTIONSECOND SOUND IN SOLIDS : THE EFFECTS

OF COLLINEAR AND NON-COLLINEAR THREE PHONON PROCESSES. Journal de Physique

Colloques, 1972, 33 (C4), pp.C4-111-C4-117. �10.1051/jphyscol:1972424�. �jpa-00215101�

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SECOND SOUND IN SOLIDS : THE EFFECTS OF COLLINEAR AND NON-COLLINEAR THREE PHONON PROCESSES

S. J. ROGERS

The Physics Laboratory, The University of Kent at Canterbury, Kent, England RbumB. - Nous passons ici en revue les etudes deja faites sur l'intensite de processus normaux a trois phonons dans NaF et helium solide. Les resultats des experiences faites sur I'onde ther- mique et 1'Ccoulement de chaleur de Poiseuille ne concordent pas avec les analyses faites sur les experiences de conductibilite calorifique. Les faits rassembles font penser que ces differences pro- viennent des r6les differents joues par les processus de diffusion a grand angle et les processus de diffusion co-lineaire. Le taux de diffusion a grand angle semble representer environ un dixikme du taux de diffusion totale. Dans NaF on estime que la deviation angulaire moyenne par interaction est de 6 , 5 O .

L'analyse des donnees sur la conductibilite calorifique de melanges isotopiques de neon etaye cette these.

Abstract. ^-A review is presented of the available estimates of the strength of three phonon normal processes in NaF and solid helium. The results of experiments on second sound and Poiseuille flow are seen to be at variance with the analyses of thermal conductivity experiments.

Evidence is adduced to suggest that the differences arise because of the differing roles of wide angle and collinear scattering processes. The rate for wide angle scattering appears to be approximately one tenth of the total scattering rate. In NaF it is estimated that the mean angular deviation per interaction is 6 . 5 O .

An analysis of the thermal conductivity data for solid isotopic mixtures of neon lends support to this thesis.

I. Introduction. - The intrinsic three phonon normal processes (N processes) play an important if elusive role in the transport of heat in dielectric crystals. For low frequency phonons these processes are most directly studied in ultrasonic attenuation measurements. The scattering cross-section at thermal frequencies has been inferred in analyses of steady state thermal conductivity data [I].

The prediction [2] that a second sound or temperature wave would propagate in a sufficiently pure dielectric solid offered a third possible approach to the study of the N processes. It was, however, realised early in the search for second sound that wide angle scattering processes were needed to sustain the disturbance [3]. It would not be observed if, as had been suggested [4]-[6], the N processes were mainly collinear in character.

In the euphoria which followed the initial observation of second sound in solid 4He by Ackerman et al. [7], the distinction between the effects of collinear and non-collinear interactions was not always clearly main- tained and, as we shall suggest, this has given rise to apparent contradictions in the interpretation of second sound and thermal conductivity data. In the development of the paper we shall consider in turn this data for NaF, solid helium and solid neon.

those phenomenological models which have proved useful in interpreting the experimental data. It is implicitly assumed that the thermal excitations in a dielectric crystal can be represented adequately as an isotropic gas of localized weakly interacting phonons, and that the processes by which the phonons are scattered can be described in terms of relaxation rates.

Three such relaxation rates are of particular impor- tance : 7, the rate for N processes ; 7; = 7; the

j

combined rate for all otherprocesses ; 7;' = z i l

+

^{7 i 1}

the combined rate for all processes.

In the analysis of second sound experiments it has been generally assumed [8], [9] that the motion of the interacting phonon gas can be described by hydrodynamic equations modified by the addition of a further dissipative term to represent the resistive scattering of phonons. The equation for the conser- vation of quasi-momentum can be written in terms of F, the heat flux vector, and E, the phonon energy density as

11. Theoretical summary. - The limited compass where T is the temperature, C, the specific heat per of this paper allows time for only a brief summary unit volume and c , is the average phonon velocity.

of the theoretical concepts. We confine ourselves to q and [ the first and second viscosities are both func-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972424

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C4-112 S. J. ROGERS tions of ^2,. The corresponding equation for the conser-

vation of energy is

C V x aT = - div F . (2) The solutions of these equations for a periodic thermal disturbance of angular frequency o have been considered in the two limits wz,

<

1 and o z ,

r

1.

In the low frequency limit [ / 2 x q = E2,/3 [ l o ] . The response of the system at x = L and time t when a temperature pulse of amplitude 6T0 and width At is input in the x ⁼0 plane at time zero is given by

Dl.

where c, is the second sound velocity. The derivation of this solution presupposes that o r , % l 1111. In the high frequency limit w z ,

r

1 the first viscosity of the phonon gas tends to zero. The dispersion relation is then determined by the second viscosity which may be written as

where i =

-

1 . Numerical solutions have been gene- rated in this limit [9] ; representative solutions for the case z, + co are shown in figure 1 [12].

L . . . . I . . . a I . . . t 1

1 2 3

D E L A Y TIMES (in units of Llc,)

FIG. 1. - Detected heat pulse shapes in computer model at x = L = 1 : ZR = CO. ZN, in units of Llcl, is for various curves : (a) 0.4 ; (b) 0.2 ; (c) 0.14 ; (d) 0.1 ; (e) 0.07 ; ( f ) 0.05 ; ^(g)0.02 ;

(h) 0.001 7.

Similar hydrodynamic considerations apply to the steady state transport of heat in a dielectric solid in which phonons are only scattered by N processes and by crystal boundaries. The thermal conductivity in this Poiseuille flow region in a cylindrical sample of radius R is given by

In this region the N processes give rise to a direct enhancement of the heat flow. We shall be more concerned with the indirect effect of the N processes in a situation in which phonons are also subject to resistive scattering. If the resistive scattering cross- section increases with phonon frequency, in the absence of N processes the low frequency phonons will be long lived and carry a disproportionately large fraction of the heat flux. By shortening the life times of these phonons the N processes enhance the scattering efficiency of the resistive processes. In this situation the thermal conductivity can be represented by the Callaway [13] equation

where C = ( k / 2 n2 e l ) (k/R)3. The equation is written in terms of the dimensionless parameter x = ( f i o / k T ) ; F(x) = x4 ex/(ex - 1)'.

Analyses using this formalism yield x-dependent rates for ^7,' ; the rates ^{2 ,} ax2 T~ and z i l axT5 will be of particular interest to us. To make comparison with the hydrodynamic results these rates must be averaged over x. Estimates of 5 and 2can be obtain- ed by replacing the x dependence in x i 1 in eq. (6) by a constant factor which generates similar values of rc. The estimates of mean values vary somewhat with z,', z i l and T, but reasonable averages for our _- purpose are : i? = 6.5 ; x2 = 23.

111. Sodium fluoride. - In our discussion of the experimental data we consider first NaF [9], [14]-[17].

Two approaches to the analysis of the data yield rather different estimates of 7,'. Using the Callaway formalism to analyse the thermal conductivity data for their second sound crystals Jackson and Walker [I61 obtained the result

7,' = 10-lo W T ~ M 60 T 5 S - I

.

(7) In an analysis of heat pulse profiles the present author found

2,' = 260 T3." ^M 120 T~ ^S-'

.

(8) In attempting to reconcile these two results we ask first whether it is possible to fit the thermal conductivity data using eq. (8). A good fit can be obtained for the

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curve for the crystal of highest purity. The solid curve in figure 2 represents the fit obtained using eq. (7) ; the solid circles are the best fit for the relaxation rate

TEMPERATURE ( K )

FIG. 2. - Comparison of computed thermal conductivity curves for NaF. The solid line is the best fit of [16] using eq. (7) : the solid circles are the fit obtained using eq. (8). The effect of resistive scattering in these two cases is shown respectively by

the dotted line and open circles.

of eq. (8). However, when point defect scattering is introduced into the model the two scattering rates for the N processes have a markedly different influence.

The dashed curve in figure 2 which is obtained using eq. (7), furnishes an excellent representation of the data for a crystal containing chemical impurities [16].

The open circles are an attempt to realise the same curve using eq. (8). This weaker N process term does not afford an adequate fit of the conductivity curves for the less pure crystals. The magnitude of 7,' is all important ; a change in the assumed temperature dependence from T~ to T~ makes only a marginal difference.

We must now consider whether the faster relaxation rate of eq. (7) is compatible with the heat pulse observations. Figure 3 shows the variation with temperature of the profile of a heat pulse propagated in the [loo] direction in NaF [9]. (The peak thermal conductivity for this sample was 150 watts cm-' deg- ^l.)

Consider the situation at 13.5 OK. At this tempera- ture the ratio of the crystal length, L, to the N process mean free path, l,, given by Jackson and Walker [16], is 50. This ratio is similar to those deduced from the solid helium data in the second sound regime. For example, for solid 4He with 0, = 43 OK the analysis of Ackerman and Guyer [8] yields this ratio for their sample at 0.78 OK. However, at this temperature the second sound pulse is fully developed and propagates with a velocity with is nearly independent of temperature ; the width of the pulse decreases with increa- sing temperature. In NaF at 13.5 OK the pulse velocity

1

8 12 16 20

TEMPERATURE (K)

FIG. 3.

-

Heat pulse propagation in [I001 direction in N a F : L = 6.19 m m ; input pulse width = 0.3 ps. Curves mark : leading edge and peak of longitudinal pulse; leading edge, forward half height, peak and backward half height of main pulse whih divides from the transverse pulse at high temperatures; peak of first echo. The dotted line corresponds t o the

second sound velocity c l 1171.

is decreasing whilst the width of the pulse is increasing with increasing temperature. These effects are characteristic of the transitional region between ballistic and second sound propagation. If these data are inter- preted in terms of the hydrodynamic model the 15

%

delay in the arrival of the peak of the pulse is consistent with a value of LIZ,

-

^8.

For such short delays, the hydrodynamic model does not provide a particularly good representation of the observed pulse shapes ; the experimental pulses are narrower than those simulated on the computer.

It is clear, nevertheless, that the faster relaxation rate of eq. (7) is incompatible with calculations based on phonon hydrodynamics.

In this impasse we are led to question the assumption that the two types of experiment determine the same parameter. We recall the distinction that has been made between collinear and non-collinear N processes and ask what will be their respective roles in the experiments we are considering. We have seen that the N processes affect the conduction of heat by shortening the lifetimes of long-lived low frequency phonons. This purpose is equally well served by collinear and non-collinear interactions. For second sound, and Poiseuille flow, however, the relevant N process relaxation rate is that for large angle scattering. If small angle scattering predominates it will be similar in its effect in the hydrodynamic region to large angle scattering with a slower relaxation rate. Its effect on a ballistically propagated pulse will be that of an angular random walk.

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C4-114 S. J. ROGERS

We have calculated the pulse velocity for such a three dimensional angular walk for the sample of figure 3 using the N process relaxation rate of Jackson and Walker. The one adjustable parameter in the analysis is

a,

the mean angular deviation per scattering process. The experimental delay times for the main pulse in figure 3 are replotted in figure 4 ; the solid

8 12 16 20

TEMPERATURE (K)

FIG. 4. - Comparison between angular random walk calculation and data for main' pulse of figure 3. Computed curves are for the case Z = 6.5O : the solid line represents the calculated peak of the pulse and the dotted lines its forward and backward

half heights.

line represents the calculated delay of the peak for the case E = 6.5O. A good fit is obtained of the temperature variation of the delay in the peak of the experimental pulse. We have estimated the width of the pulse by calculating the root mean square deviation of the velocity in the random walk. Assuming that the pulse profile will be Gaussian, the predicted forward and backward half heights are as represented by the dotted lines. The curves are fairly consistent with the observed widths at temperatures below 13 ^OK.

At this temperature L/lN = 40 in the calculation.

After 40 interactions the phonons that carry the heat flux have a mean angular deviation from the direction of propagation of 300. As the mean deviation becomes large this random walk treatment will break down, and it will become more appropriate to discuss the pulse propagation in hydrodynamic terms.

The unrealistic increase in width of the calculated pulses for T

>

13 OK may indicate this point of tran- sition. Neither of these complementary descriptions is, however, able to account satisfactorily for the narrow puke widths observed in NaF, and it is tempting to make comparison with the narrowing of ballistic pulses observed at lower temperatures [18].

If small angle scattering predominates the crystal symmetry may play an important role [19].

IV. Solid helium.

-

The most complete data on second sound and Poiseuille flow in solids has been obtained in solid 4He [7], [S], [20] and solid 3He 1211-1231. The ordinary thermal conductivity of

these solids and of solid isotopic mixtures has also been studied at Oxford [24]-[26] and at Duke [27], [28].

These various experiments have all yielded estimates of 7,l.

We consider first solid 4He. When the second sound and Poiseuille flow data are written in terms of the reduced temperature parameter (TI@) it is found that for pressures in the range 50-200 atm the relaxation rate for solids of various densities can be represented by the single expression

The relaxation rates obtained in the two independent studies of the thermal conductivity in the same pressure range differ, and neither is entirely compatible with eq. (9). In these experiments ^2,' and the relaxation rate for the scattering of phonons by isotopic impurities, z;', are determined simultaneously. The analysis is complicated by the possibility that, in addition to the mass difference scattering, there may also be a point scattering contribution from the strain field surrounding an isotopic impurity. The two analyses differ essentially in their estimates of the magnitude of this strain field component.

Berman et al. 1241 inferred that at low pressures the strain field enhanced the phonon scattering by a factor

-

3. The analysis of the low pressure data has been confirmed in experiments up to 1 700 atm 1251 ; as predicted [29], 1301 the strain field effect becomes negligible at high pressures. In the pressure range of the second sound and Poiseuille flow data the analysis is consistent with the single expression

Fairbank et al. [27], however, found it necessary to assume up to a 35-fold strain field enhancement of 7; l .

There is some variation of

TN'

within the analysis, but average values of this parameter can be represented by

Although eq. (1 1) is in better accord with eq. (9) in the second sound region, the faster relaxation rate of eq. (10) is able to provide a more consistent account of the thermal conductivity data and the, effects of the strain field. If this choice of 7,' for the conductivity data is allowed the situation in solid 4He is essentially similar to that in NaF and invites a similar explanation. It must be noted, however, that at the lowest temperatures the 4He and NaF heat pulse data are different [31], [32]. In 4He only a small fraction of the input energy propagates at the phonon mode velocities even at T < 0.1 ^OK.The main pulse is broad and travels at roughly the second sound

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velocity. If this pulse is identified as second sound a larger value of z i l is indicated than the extrapolated value of eq. (9).

In the Iow pressure body centred cubic phase of solid 3He the thermal conductivity is anomalous 1331 and estimates of 7,' are not easily made [34]. Esti- mates based on the second sound data alone suggest that in the temperature range 0.4-0.6 OK the scattering rate is in the range 4 x lo6-lo7 s-I for a 96 atm solid [35].

V. Solid neon.

-

We turn finally to consider Npro- cess scattering and the prospects for the observation of second sound in solid neon. We have made thermal conductivity measurements on solids of five different isotopic compositions and have also studied the propagation of heat pulses. Although a wide range of growth conditions was employed it has not proved possible to grow large defect free crystals. Details of this work will be presented elsewhere 1361, and we consider here only the results of the thermal conductivity measure- ments. A selection of the data is presented in figure 5.

The solid lines represent the best fit of the data in the Callaway formalism.

1 2 4 6 10

T E M P E R A T U R E (K)

FIG. 5. -Thermal conductivity of solid isotopic mixtures of

?oNe and 22Ne. 22Ne concentrations in atomic % : A, .05 ; B, 1 ; C , 3.2 ; D, 8.9 ; E, 40. The solid lines are computed using

the Callaway equation, eq. (6).

For our present discussion we are concerned only with ^2,' and 2':'. We have chosen to represent the N process scattering in the calculation by the functional form

7,' = BN a2 T~ = bN x 2 T ~ . As in the case of solid helium the analysis of the phonon scattering by isotopes is complicated by uncertainty in the strain field scattering. If the mass difference scattering is represented by the expression

- 1

zi = AK.f (c) w4 [37], where f (c) represents the dependence of the scattering upon the isotopic impurity concentration c, the contribution of the strain field can be allowed for by substituting a variable A for the multiplicative constant A,. A/AK is a measure of the enhancement of the scattering by the strain field.

The computed curves in figure 5 represent the choice of constants : A/A, = 1.4 ; b, = 4.5 x lo5 s-I deg-4.

The overall quality of the fit is appreciably impaired for choices of b, outside of the range 4 x l o 5 ' to 5 x lo5 s-' deg-4. The average relaxation rate can be represented by

This relaxation rate lends support to the choice of eq. (10) for solid 4He ; it is unlikely that the N process scattering rate in solid neon is greater than in solid helium since anharmonic effects are thought to be less in solid neon. The estimate of strain field scattering in solid neon is also in reasonable accord with the strain field estimates for solid helium of Berman et al.

The low temperature thermal conductivities of the samples of figure 5 are severely limited by dislocations and crystallite boundaries ; the limiting phonon mean free paths are typically M 1/20 mm. Although second sound cannot be expected to propagate in these samples we can examine the possibility of its future observation by << switching off ^)> the boundary and dislocation scattering in the calculation. The resulting mean free paths for resistive and N process scattering are

- 1

shown in figure 6. At 20K : z, ^M200 z i l . If the

TEMPERATURE ( K )

FIG. 6. - Estimates of resistive and N process mean free paths for a hypothetical 2 mm dislocation free crystal. The estimates reveal a possible ((window >> for the propagation of

second sound.

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C4-116 S. J. ROGERS

wide angle scattering rate for N processes is 10

%

of eq. (12) the ratio of wide angle scattering to resistive scattering would be 20. Second sound propagation might well be observed in a 2 mm crystal.

VI. Conclusions.

-

We have examined the available estimates of N process scattering in NaF, solid helium and solid neon. The relaxation rates obtained from the analyses of ordinary thermal conductivity data 'are seen t o be a t variance with the results of second sound and Poiseuille flow experiments. The different relaxation rates are consistent with the assumption that, whilst the observation of second sound and Poiseuille flow requires the wide angle scattering of phonons the steady state transport of heat is also sensitive to collinear and near collinear scattering.

The rate for wide angle scattering appears to be appro-

ximately one tenth of the total scattering rate, a factor which may account for the failure of some experiments (e. g. in Al,O,). I t will be of interest to see if the recent experiments on bismuth [38] are consistent with these conclusions.

Note added in proof. - The now published data lend suppose to the present thesis [see NARAYANAMURTI (V.) and DYNES (R. C.), Phys. Rev. Lett., 1972, 28, 14611. The values of ⁷^,^' derived from the second sound data are some six times less than those obtained from the analysis of thermal conductivity data. These observations in bismuth bear a marked similarity to those in N a F and are consistent with a similar angular deviation per N process interaction As in N a F the characteristics of the second sound propagation carry somethat with crystallographic direction.

References [I] See for example BERMAN (R.) and BROCK (J. C. F.),

Proc. R. Soc., London, 1965, A289, 46.

[2] WARD (J. C.) and WILKS (J.), Phil. Mag., 1951, 42, 314.

[3] VON GUTFELD (R. J.) and NETHERCOT, Jr (A. H.), Phys. Rev. Letters, 1966, 17, 868.

[4] SIMONS (S.), Proc. Phys. SOC., London, 1963, 82, 401.

[5] MARIS (H. J.), Phil. Mag., 1964, 9 , 901.

[6] CICCARELLO (I. S.) and DRANSFELD (K.), Phys. Rev., 1964, 134, A 1517.

[7] ACKERMAN (C. C.), BERTMAN (B.), FAIRBANK (H. A.) and GUYER (R. A.), Phys. Rev. Lett., 1966, 22, 764.

[8] ACKERMAN (C. C.) and GUYER (R.A.), Ann. Phys., N. Y., 1968, 50, 128.

[9] ROGERS (S. J.), Phys. Rev., 1971, B 3 , 1440.

[lo] See eq. (4) in the limit ^{z c l}z 7;' ;'7;'.

[ l l ] Exact numerical solutions which include the effects of resistive scattering are reported in reference [22].

No account is taken, however, of the complex form of the second viscosity in eq. (4).

[12] Since for all frequencies [ > q the treatment which considers only the second' viscosity should be essentially correct, apart from a small numerical factor, even for ^{W Z N}4 1.

[13] CALLAWAY (J.), Phys. Rev., 1959, 113, 1046.

1141 MCNELLY (T. F.), ROGERS (S. J.), CHANNIN (D. J.), ROLLEFSON (R. J.), GOUBAU (W. M.), SCHMIDT (G. E.), KRUMHANSL (J. A.) and POHL (R. O.), Phys. Rev. Lett., 1970, 24, 100.

[15] JACKSON (H. E.), WALKER (C. T.) and MCNELLY (T. F.), Phys. Rev. Lett., 1970, 25, 26.

[16] JACKSON (H. E.) and WALKER (C. T.), Phys. Rev., 1971, B 3, 1428.

[17] HARDY (R. J.) and JASWAL (S. S.), Phys. Rev., 1971, B 3 , 4385.

[18] NARAYANAMURTI (V.) and VARMA (C. M.), Phys.

Rev. Lett., 1970, 25, 1105.

[19] PURDOM (R. C.) and PROHOFSKY (E. W.), Phys. Rev., 1972, B 5 , 617.

[20] MEZOV-DEGLIN (L. P.), SOV. Phys. JETP, 1967, 25, 568 (see also Ref. [28]).

[21] ACKERMAN (C. C.) and OVERTON (W. C.), Phys. Rev.

Lett., 1969, 22, 764.

[22] OVERTON (W. C.) and ACKERMAN (C. C.), in (<Pro- ceedings of the Twelfth International Conference on Low-Temperature Physics ^)), Kyoto, Japan, 1970 (Academic, Tokyo, 1971,) p. 133.

[23] TOMLINSON (W. C.), Phys. Rev. Lett., 1969,23, 1330.

[24] BERMAN (R.), BOUNDS (C. L.) and ROGERS (S. J.), Proc. R. Soc., London, 1965, A289, 66.

1251 BERMAN (R.), BOUNDS (C. L.), DAY (C. R.) and SAMPLE (H. H.), Phys. Lett., 1968, 26A, 185.

[26] BERMAN (R.) and DAY (C. R.), Phys. Lett., 1970, 33A, 329.

[27] BERTMAN (B.), FAIRBANK (H. A.), GUYER (R. A.) and WHITE (C. W.), Phys. Rev., 1966, 142, 79.

[28] HOGAN (E. M.), GUYER (R. A.) and FAIRBANK (H. A.), Phys. Rev., 1969, 185, 356.

[29] KLEMENS (P. G.), ^DE BRUYN OUBOTER (R.) and LE PAIR (C.), Physica, 1964, 30, 1863.

[30] GUYER (R. A.), Phys. Lett., 1968, 27A, 452.

[31] MUELLER (K. H.) and FAIRBANK (H. A.), in (<Pro- ceedings of the Twelfth International Conference on Low-Temperature Physics D, Kyoto, Japan, 1970 (Academic, Tokyo, 1971), p. 135.

[32] Fox (J. N.), TREFNY (J. U.), BUCHANAN (J.), SHEN (L.) and BERTMAN (B.), Phys. Rev. Lett., 1972, 28, 16.

1331 WALKER (E. J.) and FAIRBANK (H. A.), Phys. Rev.

Lett., 1960, 5 , 139.

[34] For data on isotopic mixtures in this phase see ROGERS (S. J.), 1965, doctoral dissertation, Oxford University, Oxford, England (Unpu- blished).

[35] In reference [22] Overton and Ackerman argue for smaller values of ^7;'. It is not clear that their arguement concerning resistive processes can be self consistent.

[36] KIMBER (R. M.) and ROGERS (S. J.), to be presented at the International Conference on Phonon Scat- tering in Solids, Paris, 1972.

[37] KLEMENS (P. G.), PYOC. Phys. SOC., 1955, A 6 8 , 11 13.

[38] DYNES (R. C.) and NARAYANAMURTI (V.), Bull. Am.

Phys. SOC., 1972, 17, 72.

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DISCUSSION

A. ZYLBERSZTEJN. - What are the simplest criteria S. J. ROGERS. - Such experimental evidence as to tell whether one is dealing with second sound or there is from data such as I have presented suggests a with the onset of diffusion, since the received heat T 4 rather than T5 dependence for z i l but the diffe- pulses look very much alike in a number of cases ? rence is not very significant over the range of tempe-

S. J. ROGERS.

-

The scattering cross-section for almost all resistive scattering processes varies as some positive power of the phonon wave vector. Since on average the transverse phonons have larger wave vectors than the longitudinal phonons, one would expect the ratio of the longitudinal to transverse pulse amplitudes to increase with diffusive scattering.

Precisely the opposite effect is observed in the approach to second sound in both NaF and Bi.

ratures where the analyses are clear cut.

As far as the angular averages are concerned the present thesis is advanced somewhat tentatively and the point is well taken. It may be worth pointing out, however, that the term cr colinear ⁾⁾as I have used it is not very precisely used. The case of angles envisaged is not small but is very different from the essentially isotropic scattering assumption of the hydrodynamic model. By analogy with the kinecti theory of real gases perhaps the term cc persistence of velocity N

H. J. MARIS. - Comment : 1) Your first equation would be a happier one.

is valid only for oz,

<

1. It does not go over correctly to the ballistic results when wz,

+

^1.

2) There is a difficulty with the model for collisions in which one assumes diffusion (random walk) in angle space. This model does not conserve momentum.

3) Solid helium may have positive (anomalous) dispersion. In this case three phonon processes in which all modes have the same polarization are allowed.

S. J. ROGERS. - 1) What you say is true but for the purpose of this analysis the inequality z, 2 l suffices.

2) There is no real difficulty. The angular random walk relaxes the phonon distribution to a displaced Planck distribution with a net drift velocity.

3) Such an anomalous dispersion clearly changes the selection rules.

R. ORBACH. ^-For what reasons do you propose 117, # on

T5-"

? This is a general result when o,, , z,

+

1 which is certainly the case for kT phonons.

There is no problem with conventional 3 phonon processes for kT phonons - the colinear processes are of the same order or smallev than the usual three phonon relation processes ( t

+

t o 1, t ^f1

-

^1).

rr Proper ^)>computations of second sound and thermal conductivity presumably would account for the apparent discrepancy because of (as you noted) differing angular averages.

R. J. VON GUTFELD. - The collinear process refers mainly to longitudinal phonon scattering. How can this be reconciled with the predominantly transverse collisions required for second sound ?

S. J. ROGERS. - The longitudinal processes you envisage are only allowed by virtue of the uncertainty principle. Longitudinal phonons of the frequencies involved here must decay by processes involving transverse phonons : t

+

^t^t^,¹^;t

+

¹^o1.

D. OSBORNE. - I n a second sound pulse (eg. in NaF) is the drift velocity amplitude of the phonon gas comparable with the velocity of second sound ? If so you might see shock wave effects.

S. J. ROGERS. - We were aware of this danger and looked quite carefully for an amplitude dependence which would have been characteristic of such a shock wave. We observed none within the accuracy of the experiments.

C. ELBAUM. ^-YOU mentioned the possible need to take account of phonon focusing in some situations.

In the case of small angle scattering this effect may be very important, but in the case of high angle scattering, when considerable redistribution of wave vector direc- tions occurs, focusing is unlikely to be significant.

S. J. ROGERS. ^-ForE

-

⁶⁰the effect would certainly not be negligible.