Temperature pulses in dielectric solids

ANNALS

OF

PHYSICS:

50,

128-185 (1968)

Temperature

Pulses in Dielectric

Solids*

C. C. ACKERMAN' Department of Physics, Duke University, Durham, North Carolina 27701 and Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544 AND

R. A. GUYER~ Department of Physics, Duke University, Durham, North Carolina 27701

We discuss the criteria for second sound propagation in dielectric solids and apply these criteria to solid helium and LiF crystals. Second sound should be observed in solid helium and LiF crystals free from dislocations and chemical and isotopic impurities. The best LiF crystal of Thacher will not exhibit second sound. We critically discuss several second sound experiments which have achieved null results. Only the experiments of Rogers and Rollefson have been done with sufficient care to warrant attention. We report the result of an extensive investigation of temperature pulse propagation in helium crystals and demonstrate that a variety of information may be found from such an experiment. (1) Temperature pulse data taken at temperatures above that of the thermal conductivity peak at T/O N 0.025 yield values of thermal conductivity in agreement with those obtained by steady state measurements. (2) At temperatures below T/O eO.025 the pulse transits the chamber with a shape and velocity consistent with the requirements for second sound. In this temperature region the observed pulse velocity is approximately that calculated for second sound using the experimentally determined first sound velocities. Multiple reflections of the heat pulse are seen, a phenomenon which indicates the temperature pulse obeys a wave equation, a further requirement for second sound propagation. (3) In the second sound region, the finite N-process collision rate causes a broadening of the temperature pulse. The magnitude of broadening is temperature dependent and this effect is used to calculate 7N. We find the N-process relaxation time is given by 7N = 2 x lo-la (T/@)-s set for Oo between 21.5 ’ and 43.0 “K. * This work was supported in part by the National Science Foundation, the Army Research Office (Durham), the Office of Naval Research and the U. S. Atomic Energy Commission. + Postdoctoral Appointee at the Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544. * A. P. Sloan Foundation Fellow.

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The values of 7N found are in good agreement with the results of Poiseuille flow experiments. They suggest that the lifetime of phonons in solid helium is about an order of magnitude shorter than those in LiF. The phonons are still a very well-defined excitation.

I. INTRODUCTION

The earliest recorded speculation on the existence of a propagating temperature wave, second sound, was advanced by Nernst (I) in 1917. He suggestedthat in good thermal conductors, at low-temperature, heat may have sufficient “inertia” to give rise to an “oscillatory discharge”. In 1944 Peshkov (2) discovered second seond sound in superfluid 4He. This phenomenon, which provided dramatic confirmation of Tisza’s and Landau’s basicideasabout the superfluid phaseof liquid helium, owes its existence to the special nature of superfluid helium, in particular the presence of the extra degree of freedom manifested in the superfluid phase. In 1947 Peshkov (3) suggestedthat second sound might also exist in crystals in which the scattering of phonons by impurities and inhomogeneities is a minimum. Following this speculation by Peshkov considerable effort has been expended to establish a theoretical foundation for “second sound in solids” and to make an observation of this phenomena. In the early fifties it was shown by Dingle (4) Ward and Wilks (.5), (6), and London (7) that a density fluctuation in a phonon gas would propagate as a second sound wave provided that “losses” from the wave were negligible. In one of their papers, Ward and Wilks (6) indicated they would attempt to look for second sound in sapphire crystals. No results of their experiments were published. Then, for nearly a decade, the subject of “second sound in solids” lay dormant. Interest was revived in the early sixties primarily through the efforts of J. A. Krumhansl. In addition to stimulating interest in this subject among many people, he and his associateshave made a number of basic contributions to the understanding of this phenomena (8)-(11). In 1966 Guyer and Krumhansl (II) described in detail the physical requirements which must be met by a solid helium sample in order to permit the observation of second sound. Shortly thereafter, second sound was observed in single crystals of helium by Ackerman et al. (12). This experiment, which employed a pulse technique similar to those used in the study of liquid helium (Z.?), established secondsound in solids as a subject worthy of serious consideration and has led to considerable further theoretical work at a more fundamental level (14)-(17). Further, the successof a pulse experiment has revealed how powerful pulse techniques can be in the study of transport phenomena in solids. At about the sametime second sound was observed in solid helium, a number of other investigators reported null results for second sound experiments in other 595/50/I-9

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solids (18)-(22). These experiments do not demonstrate the failure in principle of second sound to exist in LiF, sapphire, etc., but rather, that the very stringent conditions for observing second sound which Krumhansl and co-workers (8)-(II) have repeatedly emphasized have not been met. The purpose of this paper is threefold: (1) to make a quantitative critique of the several unsuccessful attempts to observe second sound in solids, (2) to report the details of the extensive investigation of temperature pulse (TP) propagation in hcp 4He which followed the initial observation of second sound, (3) to demonstrate the usefulness of TP data for obtaining a wide variety of quantitative information about the properties of a phonon gas. In Section II we write down the macroscopic equations of Guyer and Krumhansl (IO), (II) on which we base the discussion of TP propagation. Our purpose is to carefully delineate the criteria for observation of second sound and to exhibit the results of their strict application. We discuss in detail the experiments of Rogers and Rollefson (22) on LiF and the experiments of von Gutfeld and Nethercot on sapphire (18), (19). In Section III the details of the low-temperature apparatus and sample chamber design are discussed. As we point out in Section 11, the sample chamber geometry is very important to the success of a TP experiment. The results of temperature pulse measurements on single crystals of hcp 4He grown at constant pressures of 33, 54, 100, and 130 atm are presented in Section IV (23). These measurements are discussed thoroughly in Section V. The second sound velocity, thermal conductivity, N-process collision rate (23), and crystal orientation are determined from the data. Where possible our measurements of the properties of the phonon gas are compared with more conventionally obtained experimental values. The major results of this investigation are summarized in Section VI along with a number of speculations about future experimental work. In the Appendices many lesser details of the investigation not included in the main body of the paper are presented. In Appendix A we discuss the results obtained for temperature pulse experiments on polycrystalline samples. Appendix B contains a discussion of the effects of large temperature transients on temperature pulse propagation. This discussion suggests an explanation of at least one of the “exotic” effects reported by von Gutfeld and Nethercot (19). In Appendix C the crystal growth procedure, in particular the growth rate calculation, is outlined. The solution to the linearized phonon Boltzmann equation for a multipolarization system is set down in Appendix D. Dispersion relations for drifting and driftless second sound are derived. The many equations of motion for temperature pulses that are employed in the analysis of TP data are discussed in Appendix E.

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BACKGROUND

Quantitative discussions of temperature pulse experiments in a dielectric solid are most easily accomplished in terms of the system of macroscopic equations which describe the thermodynamic, as opposed to thermostatic, properties of the phonon gas in the solid, [see Ref. (II), hereafter referred to as I]. These equations, the statement of energy conservation and the statement of momentum conservation, have been derived by a number of authors. Here we employ the equations and notation of I. We apply these equations to solid helium for which the usual harmonic approximation makes no sense (24) and for which there is no simple justification for writing a Peierls’ equation (25). The recent progress that has been made by Koehler, Nosanow, Werthamer and collaborators (26) in treating the problem of quantum solids suggests that a harmonic-like approximation is valid for treating the phonons in solid helium. We make no further comment in this paper about the special treatment required in dealing with phonons in quantum solids. However we assume that we can talk about the transport properties in terms of phonons which weakly interact with one another in a 3-phonon interaction. The quantitative estimates of phonon lifetime we obtained in Section V provide an experimental proof that phonons in hcp 4He are not in a very anharmonic environment. For an isotropic, one polarization, Debye-approximated (c = cq/l q 1) phonon gas the statement of energy conservation is

where Cv is the specific heat per unit volume and T temperature. The heat current Q is proportional to the momentum current under the above conditions, Q = c2P. The statement of momentum conservation takes the form 4

0

Q + $ Q + ; Cvc2VT - ; c~T~[V~ + 2V@ .)]Q = o,

(2)

0

where c is the velocity of sound and TN and rRz are the relaxation times which characterize the scattering processesin the phonon gas (27). This momentum conservation law is valid in the low-temperature limit when TN, the Normal process relaxation time, is much shorter than the resistive relaxation time, TV=, which must be computed in the Ziman limit (28). These relaxation times (29) are best regarded as parameters which are to be extracted from experimental data. Calculation of TNand 7 Rz from first principles requires detailed information about the harmonic phonon spectrum and the physical condition of the solid at the microscopic level. This kind of information is not easily obtained. In this

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paper we rely on auxillary experimental data to provide a measure of TN and rRz or we extract these quantities from the TP data employing the solutions to Eqs. (1) and (2). The times rRz and ?-Nare the transport and equilibrium times respectively (8, IO). Equations (1) and (2) give a valid description of the properties of the phonon gas if &, 0, 7Rz --+ + co, Eqs. (1) and (2) lead to $+;ceV2T=

0

(3)

for which, if T = T,, + 6Texp i(k * x - L?t), we have Q2 = k2c2/3; second sound propagates in the phonon gas at all frequencies. For finite values of the relaxation times, Eqs. (1) and (2) lead to a dispersion relation of the form (8), (9) (4) Second sound will propagate if &N < 1 and Qn,,= > 1, or TN < G-l < rRz. We would expect this separation of the relaxation times to occur at low temperature, e.g., near the low-temperature thermal conductivity maximum. The transport relaxation time ~~~ depends up on the scattering rates of the known phonon momentum-destroying scattering mechanisms in the solid according to the prescription

where 7;’ is the scattering rate for the ith momentum-destroying scattering mechanism. 7Rz is optimized by eliminating all scattering mechanisms (point defects, dislocations, etc.) except the intrinsic momentum nonconserving 3-phonon process, the Umklapp process (25), and the effects of the finite crystal size, boundary scattering. The scattering rate for U-processes goes to zero exponentially with T, T;;’ cc exp -a/T, whereas the scattering rate for the N-processes goes to zero as a polynomial in T, T;’ Oc T”. Thus, as temperature is lowered, TRz = Tuz becomes much greater than TN until huZ = CTU= M d (d is a typical transverse dimension of the sample), whence 7Rz = d/c and remains constant at this value as the temperature is further reduced. TN however, continues to increase with

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decreasing temperature. The thermal conductivity maximum occurs at a temperature, Tmax , imposed on the system by the geometry of the sample since it occurs at hUZ m Ct. Tmax is rather insensitive to d since Xuz is a strong function of T at low temperatures. A typical steady-state thermal conductivity (K) measurement is made on a sample with d w 1 cm for which ~~~~ occurs at T = Tmax M 0.025 0, . A temperature pulse measurement performed across a space & = 1 cm would lead to diffusive propagation if the ambient temperature were above Tmax ; since Xuz < r”, momentum would be lost from the TP in crossing 8. For temperatures below Tmax , TP’s would traverse G as second sound if the Normal process relaxation time is short enough to allow local thermal equilibrium in (a)

pulse

cspcrinifnt

(b)

CW experiment

(c)

light

scattering

FIG. 1. Criteria for second sound propagation. Temperature pulse experiments and continuous wave experiments can be performed if a criteria dependent upon sample geometry is satisfied, (a) and (b). For an “in situ” experiment, the criteria is unreferenced to sample geometry, e.g., light scattering.

134

ACKERMAN

lo-;‘o

’

’

AND

I.0

GUYER

2.0 T (OK)

FIG. 2. Information from temperature pulse experiments. The N-process and U-process mean free paths for a 4He crystal (P = 85 atm) are plotted against temperature. The N-process mean free path (dashed line) is computed from Eq. (26) of the text. The two extreme U-process mean free paths (solid lines) are taken from the thermal conductivity data of Guyer and Hogan (42). Thermal conductivity in hcp ‘He is anisotropic; K, is the thermal conductivity I to the c-axis and KiI is the thermal conductivity 11to the c-axis. The anisotropy in K results in a spread in the value of U shown as the shaded area in the figure. When a temperature pulse is propagated I to the c-axis across a l-cm crystal, it propagates diffusively from 1 to 2, as second sound from 3 to 4 and ballistically below the temperature corresponding to 4. For a temperature pulse propagated 1)to the c-axis across a 0.05 cm sample chamber, diffusive propagation occurs from 5 to 6, second sound propagation occurs from 7 to 8. Ballistic propagation occurs at temperatures below 8.

the TP; Qn7N< 1 must hold. In a pulseexperiment Q is determined from Q w 25-/d t where At is the duration of the TP; At must satisfy the condition At Q tIl m 2/3//c for second sound propagation. SeeFig. l(a). A TP experiment across a space & = 1 cm explores the properties of a phonon gas in the temperature range illustrated in Fig. 2. Along 1 + 2 the TP propagates

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diffusively and measures K or TV=, along 3 + 4 it propagates as second sound and measures TN and uII . Below 4 the phonons in the TP cross the sample ballistically. A TP experiment across a space 8 = 0.05 cm explores rRz along 5 + 6, 7N and vu along 7 --f 8 and propagates ballistically below 8. Thus the properties of a phonon gas are explored in different temperature ranges with different geometry samples (31). This is true for pulse experiments as well as for continuous wave experiments. See Fig. l(b) and Fig. 2. The usually stated criteria for second sound propagation involving the wavelength of the second sound wave unreferenced to a geometrical feature of the sample is relevant if second sound is detected “in situ”, for example in the bulk crystal by a light scattering experiment (14, 32). See Fig. l(c). Regardless of the second sound experiment being considered, its evaluation depends upon a knowledge of the relaxation times rRz and TN. The transport time 7Rz is directly accessible from steady-state thermal conductivity data, since, from Eq. (2) with TN + 0, a/at + 0, Q

=

-KVT

=

-

fCvc2rRZVT.

As remarked above rRz is optimized for a second sound experiment by removing isotopic and chemical impurities, dislocations, etc. Whether such a program of removal or crystalline imperfections can be carried out is a major consideration in choosing a subject crystal for a second sound investigation. Information about the equilibrium time TN is much more difficult to obtain than that about TV” . There are five methods by which information about TN can be obtained: 1. direct computation; 2. a Callaway analysis of conventional thermal conductivity 3. analysis of Poiseuille flow data; 4. interpolation of acoustic attenuation data; 5. analysis of second sound data. Direct computation of TN This kind of information Recently however, Jones argon crystals using ideas Tuz

from

first

data (TC data);

requires a detailed knowledge of the phonon spectrum. is far harder to obtain than TN itself by methods 2-5. (33) has carried out a computation for TN and TU= in similar to those developed by Julian (34) for computing

principkS.

The equilibrium time TN does not enter directly in determining the transport properties of a phonon gas. Nonetheless, if thermal conductivity data is taken over a temperature range such that TV < TN in one temperature interval and 7R > TN in another, then the Callaway equation (35) or a similar equation (10) (II) can be used to fit the data and find TN . To date this kind of analysis of steady

136

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AND GUYER

state thermal conductivity data has been carried out carefully for only two materials, solid helium (36), (37) and LiF (38), (39). We discuss the results of these analysesbelow. The Poiseuille flow (16), (40) of a phonon gas and its appearance in TC data has been discussedin great detail in I. When pure Poiseuille flow occurs in a phonon gas the thermal conductivity is given by

for cylindrical samplesof radius R. If Poiseuille flow is observed in a steady state thermal conductivity experiment, then second sound can be observed; the criteria for Poiseuille flow, h,z > R2/crN and CTN< R, are more stringent than those for second sound (11). TN can be computed from thermal conductivity data in the Poiseuille region using Eq. (7). Once TNis known, it may be usedin conjunction with ~~~ from (6) and the conditions for second sound stated above to define the temperature frequencies which will propagate as secondsound. To date, Poiseuille flow has been observed only in solid helium crystals (41), (42). The impurities and defects which make second sound experiments difficult exert a negative effect on Poiseuille flow experiments as well. Acoustic attenuation measurements at acoustic frequencies 52, such that G,TN > 1 measureTNat the frequence Qn,. From thesemeasurementsit is possible to estimate TNat thermal phonon frequencies by extrapolation, i.e., rN(T, w M k,T/h) w 7N(T,n,)(+$)-

for

?-N Cc a-%.

In the one case where this procedure has been verifiable (43), LiF, the acoustic attenuation value of TN interpolated to thermal phonon frequencies agreed reasonably well with the value of TNdetermined by method (2) above (38). The N-process relaxation time can also be measured in a second sound experiment. This is discussedin detail in Section V. The remainder of this section is devoted to a quantitative discussion of the application of the criterion for second sound propagation to solid helium and LiF. A. SOLID HELIUM When the second sound experiment in solid helium was originally conceived information about TN for solid helium was available only by method (2). Callaway (44) analyzed the thermal conductivity data of Walker and Fairbank (4.5) on solid 3He-4He isotopic mixtures and extracted T;’ = 2 x 106T5set-l from the analysis. With this value of TN and 7Rz from the data of Bertman et al. (36)

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IC

I.(

IO- I_

-z u

lo- 2

2 5 Id 3

IO-

4-

IO- 5;

I.” T(“K)

FIG. 3. Mean free path vs. temperatures, 4He. The mean free path for N-process and Uprocess scattering are plotted against temperature. The N-process mean free paths are from: (1) the analysis of thermal conductivity data by Bertman et al., (2) Eq. (26) of this paper, and (3) the analysis of thermal conductivity data by Rogers et al. The U-process mean free paths are from the results of Hogan and Guyer (42).

it was possibleto construct a mean free path diagram and determine the feasibility of a second sound experiment in hcp 4He. See Fig. 3. Indications were that the experimental situation was favorable and the sample chamber was designed to have a l-cm path length as suggestedby the Callaway value of TN. At the sametime it was decided an independent check of the Callaway result would be useful. Thus Bertman and one of us (RAG) undertook an extensive analysis of the data from Bertman’s thesis of 3He-4He mixtures. These data covered a wider and more relevant temperature range (46) than the data analyzed by Callaway. The results of this analysis were not completely satisfactory. However it led to -rr;’ = (BNw2T3)average = 3 x 105T5set-l,

(8)

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ACKERMAN

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GUYER

where (7N)average = 1; T~(T, co) A(x) dx/j”;

A(x) dx, A(x) = x4er/(ex -

1)”

and x = fi(olkgT. Simultaneously Rogers et al. (37) undertook the same kind of analysis of their 3He-4He mixture data (47). Their results were quantitatively and qualitatively different from those reported by Bertman et al. Rogers et al. found ~;l = (BNw2T2)average = 3 x 107T4 set-l,

(9)

a result much more favorable to a second sound experiment than TN given by Eq. (8) or Callaway. In November of 1965 the extensive measurements of Poiseuille flow in helium crystals by Mezov-Deglin (41) became known to us. Poiseuille flow measurements are not entirely unambiguous, nonetheless the order of magnitude of TN determined from the raw data of Mezov-Deglin using Eq. (7) supported the basic design criteria for the second sound experiment (48) and was in better agreement with the value of TN from Bertman et al. than that of Rogers et al. Finally, the analysis of second sound data also permits a direct determination of TN. (See Section V.) When values of TN from the second sound experiment and the Poiseuille flow experiment are combined an unambiguous value for TN results (23) which is rather different from that of Bertman et al. and Rogers et al. We have recited this history to illustrate two points. (1) A wealth of data and analysis are available in support of a TP experiment in solid helium. This is the only solid for which this is true. (2) Determinations of 7N by a Callaway-like analysis of thermal conductivity data can support an independent determination of TN but cannot easily stand alone. B. LIF Excluding solid helium, the best understood dielectric solid is LiF. Two excellent sets of thermal conductivity data on sLi-7Li solid mixtures exist, those of Berman and Brock (38) and Thacher (49). These data have been subjected to a Callawaylike analysis to determine TN. The data of Berman and Brock, analyzed by them using the Callaway equation, yielded 7~~ = (BNwT3)avwerage = 12T4(sec)-l.

(10)

As pointed out above this value of TN is in reasonable agreement with the interpolation of the acoustic attenuation value. Guyer and Sarkissian (39) have analyzed the data of Thacher employing the thermal conductivity equation

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SOLIDS

proposed in I. This equation contains the possibility of Poiseuille flow and thus offers a serious constraint to TN since no Poiseuille flow is observed in Thacher’s data (50). The value of TN from this analysis was (51) or;’ = (B,wT~),,,,~,~

= 0.5T5(sec)-l,

(11)

in substantial agreement with the value found by Berman and Brock. Note that TN given by Eq. (10) is equal to TN given by Eq. (11) at T ru 25°K. This is consistent with what we believe to be a valid generalization; a Callaway-like analysis will give the correct magnitude for TN near Tmax where thermal conductivity data is most sensitive to TN . A mean free path diagram for LiF is shown in Fig. 4.

l.O-

IO2

5

1 50

IO

100

T (“Kl

FIG. 4. Mean free path vs. temperature, LiF. The mean free path for N-process and U-process scattering are plotted against temperature. The N-process mean free path is from the analysis of Thacher’s (49) data by Guyer and Sarkissian (39). The plot of R2/hNagainst T leads to the conclusion that Poiseuille flow cannot be observed in Thacher’s best LiF sample which was 0.4 cm in diameter (the dashed horizontal line). Second sound propagation might be possible across a l-mm space at temperatures near 20°K. The Umklapp mean free path is calculated in the Ziman limit. If isotopic and chemical impurities and dislocations could be removed from the sample, this would be the limiting resistive mean free path. In such a perfect single crystal Poiseuille flow and second sound propagation would occur.

140

ACKERMAN

LL-L’

1.0

AND

’ “‘lb

GUYER

’ ’ L “if40

T (OK)

’

FIG. 5. Thermal conductivity of LiF. The data on Thacher’s best LiF crystal is shown as open circles. The curves are computed using the thermal conductivity expression proposed by Guyer and Krumhansl, Eq. (15) of Ref. (II). The dashed curves are computed using Eq. (I-15) with the value of 7~ from Berman and Brock with all other parameters adjusted to give the best fit to Thacher’s data. The lower dashed curve is the best fit obtained in this way. The upper dashed curve is the prediction of Eq. (I-15) for K in Thacher’s best crystal if all defects and dislocations are removed (using 7~ from Berman and Brock). The solid curves are computed using Eq. (I-l 5) with the value of ‘N determined by Guyer and Sarkissian, Eq. (8). The lower solid curve is the fit to the data. The upper solid curve is a prediction for K in a perfect single crystal using the Guyer and Sarkissian value of TN . The dip in the computed curves below the conductivity maximum is due to a defect in the Guyer and Krumhansl equation as the boundary region is approached.

It is clear from this diagram that neither second sound or Poiseuille flow can be detected in the best LiF crystal of Thacher due to the presence of dislocations and point defects. In Fig. 5 we have plotted the “theoretical” thermal conductivity for a LiF crystal with no dislocations and no point defects. Such a crystal has a thermal conductivity maximum about a factor of five higher than Thacher’s best crystal, exhibits Poiseuille flow and will propagate secondsound.

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Rogers and Rollefson (22) have made a careful attempt to observe second sound in LiF, NaI, and NaF crystals. They looked for second sound in the best LiF crystal of Thacher without success as we would expect from Fig. 4. (This crystal has exhibited the largest thermal conductivity of any dielectric solid ever measured, 170 W/cm”K.) Following the work on LiF, Rogers and Rollefson performed a series of experiments on NaF and NaI in which they employed the following procedures: (1) grow a single crystal starting with Merck ultrapure powder; (2) measure its thermal conductivity over the full temperature range; (3) do a temperature pulse experiment over the relevant temperature range; (4) regrow the crystal and repeat steps (l)-(3). By regrowing an NaF crystal twice (3 crystals in all), the thermal conductivity at the low-temperature maximum was improved from 60 W/cm”K to 100 W/cm”K to 150 W/cm”K. In this best NaF crystal somewhat unusual behavior was observed for temperature pulses propagated at temperatures just below the thermal conductivity maximum (52). However, the departures from expected behavior were not sufficiently great to permit a favorable conclusion as regards second sound propagation. More recently Rogers and Rollefson have shortened the length of the best NaF crystal and continued the temperature pulse measurements. No new results have been reported. C.

OTHERS

Three other experimental groups have looked for second sound in solids, (a) von Gutfeld and Nethercot (I??), (19) in sapphire and many other substances (.53), (b) Brown et al. (20) in sapphire, and (c) M. Chester (30). These experiments were performed neither on a solid having the desirable physical properties of solid helium nor with the care of the experiments of Rogers and Rollefson. For example, the sapphire sample investigated by von Gutfeld and Nethercot (29) was very dirty as evidenced by (a) the serious disruption of ballistic flight of the phonons at temperatures as low as 18°K and (b) the mean free paths computed in their Table II. In general quantitative commentary on this experiment and the others above is difficult. However since we wish to maintain that the phonon gas in solid helium is not grossly different from that in other dielectric solids, we dismiss these null results as being due to the physical condition of the crystals investigated and not to the absence of a second sound mode from them.

III.

APPARATUS

FOR

THE

4HE EXPERIMENT

The apparatus used in our experiments is not fundamentally different from that used by others for experiments at 3He temperatures. There are certain features, however, which may be of interest.

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In Subsection A the pertinent details of the cryostat are described. The sample system and sample chambers are described in Subsections B and C, respectively. In Subsection D, we give some details of the electronics and carbon films used in the experiments. A.

THE

CRYOSTAT

An illustration of the cryostat showing the pumped 4He and 3He systems is given in Fig. 6. These systems are enclosed in an evacuated “exchange gas” can which isolates them from the 4.2OK outer bath. The 4He container is 250 cm3 in volume and may be filled from the 4.2”K bath through a valve in the manometer line. With this arrangement it is possible to fill the 4He volume while it remains below the X point. Thus, a completed crystal may be kept well below its melting temperature for extended periods. The 3He container is 6 cm3 in volume. The

4 He%4Tp’h VALVE FILL

MANOMETER

LINE

STEM

VALVE

q

e

HEATER

MAN%EETE

TEMP RESIST0

VACUUM 4.2 ‘K BATH

FIG.

crystals.

6. A The

schematic diagram of the cryogenic apparatus used for heat pulse studies in ‘He sample chamber shown here is illustrated in greater detail in Fig. 9.

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SOLIDS

3He system has a provision for continuous recondensation which enabled it to be operated for prolonged periods at temperatures down to 05°K. With the recondensation line operating continuously, heat loads up to 7 mW at 0.75”K were possible. Both helium volumes had heaters for temperature control. B. THE SAMPLE SYSTEM The sample system is illustrated in Fig. 7. Helium gas for the crystal was obtained from a liquid 4He storage dewar with a transfer line similar to that used for nitrogen dewars with the addition of a porous stainless steel plug at the inlet. The gaswas taken from the storage dewar at about 6 psi. and condensed into a 50-cm3volume at 4.2”K. When sufficient gas had been condensed into this TO SAMPLE

CHAMBER -‘V

77’K COLD

TRAP

8 4.2’K COLD TRAP

LEVEL DETECTOR /x)Occ STORAGE n JjERMOCOUPLE ‘4 He GAS INLET VACUUM

FIG. 7. A schematic diagram of the sample system. In this figure: TV = toggle valve (Hoke #450), NV = needle valve (3000 psi).

volume (after one hour), the pressurization system was isolated and the 50-cm3 volume was warmed to room temperature. This procedure gave a 30-atm starting pressure when the charge condensed into the 5-cm3 volume of the second sample chamber. Further pressurization was achieved by an oil over mercury hydraulic system. The fill line from the 300 cc storage volume to a point just inside the exchange gas can was .050-in.-i.d. stainless steel tubing. From that point, to the sample chamber, the fill line was O.OlO-in.-i.d. Cu-Ni capillary. The sample system was evacuated through this line for at least 48 hours prior to cool down. No exchange gas was used during the transfer of helium into the dewar. Since the sample chamber was thermally isolated, any residual gas in the chamber was cryopumped into the fU line during the transfer of helium into the dewar

144 C.

ACKERMAN SAMPLE

AND

GUYER

CHAMBERS

Two different chamber designs were used in the course of the experiments. The basic design of the first chamber was dictated by the conditions for second sound given in Section II and the considerations of a growth rate calculation. (See Appendix C.) For this reason it is not the optimum chamber for observation of a clean second sound pulse. TO PULSE GENERATOR.

FIG.

8.

The

first

sample

.OlO”l.D.

FILL

LINE

chamber. The active areasof the carbonfilm are cross-hatched.

Fig. 8 illustrates the first chamber. The chamber body is g-in.-o.d., O.OlO-in. wall stainlesstubing. End flanges are brass and a pressure seal is made at both ends using a 0.030-in. indium-wire O-ring. Sealsare also necessaryon the coaxial leads from the carbon transducers. They are 50-Q impedence hermetic seals manufactured by Sealectro Corporation (55). A g-in. diameter pointed copper rod projects into the chamber and serves as a nucleation point 1or the crystal. Thermal isolation of the rod from the sample chamber walls is provided by an insert of #2850 Ft. Stycast epoxy (56). Thermal isolation of the point tends to minimize the possibility of stray crystal nucleation. The sample volume is about 0.5 cm3.

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The temperature detector is mounted on the epoxy insert with GE 7031 varnish. The temperature pulse generator is affixed to the brass plug in the top of the chamber in the same manner. The active areas in these films are shown as slashed lines in the inset of Fig. 8. Both tranducers were 19 52/n carbon resistor board from I.R.C. (57). The generator and detector were a nominal 50 and 100 Q, respectively, at 3He temperatures. Attention was given to the alignment of the transducers and they were parallel to about O.OOl-in. The transducer separation was 0.80 cm. Resistance wire wound around the top of the sample chamber and along the fill line to a point above the 4He volume served as a heater during the crystal growth process. Plastic standoffs served to thermally isolate the fill line from the pumped 4He volume. This was necessary as this volume was below the melting temperature during the growth process. Because the fill line runs from the 4.2”K bath to the sample chamber directly, the fill line imposed the largest thermal load on the 3He system and limited the lowest attainable temperature to 0.41”K. Crystals were grown in this chamber which displayed a reflection of the first received second sound pulse without being annealed. The pulse shapes in this chamber, however, were distorted by resistive collisions of phonons in the heat pulse with the walls. We compare the pulse shapes in the two chambers in Section V. To eliminate undersirable wall effects, a second chamber with suitable alterations in geometry was constructed.

FIG. 9. The second sample chamber. The generator and detector geometry are shown in the lower right-hand corner. 595/50/I-10

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The second chamber has a volume of about 5 cm3 and is illustrated in Fig. 9. This chamber, including the transducer mounts, was constructed entirely of stainless steel. The generator and detector resistances in both chambers were nominally 50 and 200 Q, respectively, at operating temperatures. As may be seen from the figures, the geometry of this chamber was an improvement over that of the first chamber in that the crystal boundaries are far removed from the transducers. The copper hemisphere on the chamber bottom was used to start the crystal and it was grown in the samemanner as those in the small chamber. The massive side walls required by this design made the growth of single crystals less likely than was the casefor the first chamber, as there was a greater possibility of stray nucleation at the side walls. With this chamber, good crystals were obtained only after they were carefully annealed. The geometry of this chamber enabled us to achieve a better transducer alignment than did that of the first chamber. In this chamber the transducers were parallel to within 0.0005-in. Excessive transducer tilt will limit the upper thermal frequency responseof the system. The tilt jn this chamber placed a ~-MHZ upper limit on thermal frequency, well above observed frequencies. D. ELECTRONICS Detection and generation of temperature pulses requires electronjcs capable of broadband frequency response. The system was designed for a 50-nsec rise time, but in practice the broadening of the pulse at the helium-transducer interface limited the thermal responsetime to a few microseconds. Stainless steel was used to construct the 50 I2 impedance coax lines into the cryostat. Two sizes of coax were used. That from the dewar head to a point above the pumped 4He volume utilized a vacuum insulated &-in.-o.d. O.OlO-in. wall stainlesstube outer conductor and a 0.072-in., O.OlO-in. wall stainlesstube inner conductor. Small nylon spacers were used to maintain alignment. A smaller coax with less heat leak was used from the larger coax to the sample chamber. This coax had a .083-in.-o.d., .OlO-in. wall outer conductor, and a O.OlO-in-diameter stainlesssteel wire inner conductor. Stycast was used as a dielectric in this coaxial line. Both types of coax had transmission characteristics similar to commercial 50-Q impedance cable. The Stycast filled coax could be bent to a &-in. radius without obviously affecting its electrical characteristics, a feature which greatly facilitated the assembly of the sample chamber. The coax from the chamber was sealed at the end with the hermetic sealsmentioned previously. These seals underwent repeated thermal cycling and pressurization to 150atm without failure. Connections from the inner coax to the resistor board were made with #40 copper wire, one end of which was soldered to the coax line while the other

PULSES

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147

was cemented to the carbon resistor board using conductive paint (58). This method was also used for the first chamber. The voltage pulse to the thermal generator was supplied by a 200-W peakpower pulse generator (Hewlett Packard #214A, rise time ~10 nsec) and the received pulse was amplified (Tektronix 1121 Amplifier, 18-nsec rise time) before being observed on an oscilloscope (Tektronix 545). There was one serious defect in this system. The pulse amplifier overloaded when the sample was pulsed, and took about 17psec to clear. This overloading was principally due to a ground loop in the dewar which caused the voltage across the generator to be transmitted to the detector circuit. The carbon detectors used in these experiments had a relatively low sensitivity: AR/RAT = l/4. The biasing currents commonly used were never more than 25 PA. Since the temperature sensitivity of these devices is directly proportional to the biasing current, the films should be operated with the highest biasing currents possible. It was found, however, that increasing the biasing current beyond a certain value (-25 PA) lead to a nonlinear detector response, i.e., doubling detector current did not double the detector response to a thermal pulse. The maximum “linear response” biasing current, then, is achieved by using the maximum detector area. It is best from a physical standpoint, however, to sample only a small portion of the crystal face. No systematic effort was made to determine the optimum transducer size. The active areas of our films were about .5 cm2. When data was taken in the second sound region the temperature excursions at the generator were usually kept within limits which prevented driving the crystal temperature into the diffusion region. The temperature rise in the crystal was calculated by assuming the pulse coupled perfectly into the crystal and propagated at velocities appropriate to the temperature in consideration. For most runs ATpulse/Tam,,ient was about .I or less. The effects of large temperature excursions on pulse shapes in the second sound region is discussed in Appendix B.

IV.

EXPERIMENTAL

RESULTS

Here we present the data taken in the best crystals grown and annealed in the second sample chamber, Fig. 9. Additional results, which are not relevant to the analysis of the next section, are in the appendices. In the first sample chamber a sizable fraction of the temperature pulse was diffusively scattered by the side walls which were close to the transducers. This greatly influenced the detailed structure of the arriving temperature pulse in the second sound region. This was not the case with the second sample chamber where side walls were far from the transducers so the pulse was transmitted from generator to detector without interaction with the walls.

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FIG. 10. Comparison of pulse shapes in the two chambers. For both traces P = 54atm, sweep speed = 20 psec/cm. The top trace is representative of thermal pulse shapes in the small chamber (Fig. 8). Ambient crystal temperature SW0.6” K. The pulse input occurs at t, . Second sound is first seen as I~ (the amplifier overload is seen just preceding the second sound trace). The lower trace in this picture is obtained by removing the bias on the detector. An oscilloscope trace representative of pulse shapes in the second chamber (Fig. 9) is shown in the lower trace. Here the reflection of second sound is clearly evident. In this chamber the walls cause no distortion of the received pulse. Hence, temperature pulses in crystals grown in the second chamber are affected only by phonon-phonon collision mechanisms acting in the bulk crystal.

A comparison of the received pulse in the second sound region in the two samplechambersin crystals grown at 54 atm is given in Fig. 10. In both chambers the leading edge arrival times are about 50 psec, although the pulsesare radically different from that point on. The boundary scattering which occurred in the first sample chamber made quantitative analysis of pulse shapesdifficult, hence only the data taken in the secondchamber is used in our analysis.

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149

SOLIDS

Temperature pulse data was taken in the second chamber in crystals grown at constant pressures of 33, 54, 100, and 130 atm. Oscilloscope traces of typical temperature pulses in crystals at ambient temperatures between 0.5” and 1.2”K are shown in Fig. 11.

(a) T=l

1°K

(e) T= 0.91 “K

mu 0

:t,

(b) T=0.63”K

(f) T=O

Cc) T=0.50”K

(cl) T=0.86

89OK

i5m !t,

‘tat,

tt,

tt 2

ft 3

“K

(i)

FIG. 11. Representativescopetraces taken in crystals at ambient temperatures ranging between 0.5” and 1.2” K. The experimental conditions for these photographs are given in tabular form in the text directly below. The sketch at the bottom of this figure identifies the points on the second sound pulse used in analysis. Data pertinent to these photographs are as follows. Trace

Pressure (atm) 130 130 130 54.2 54.2 54.2 54.2 54.2

Hor. Sweep Speed Vert. Sens. (“K x 10a/cm) (rsecicm) 10 10 10 50 20 20 20 20

Ambient temperatures are given on the figure.

Input Energy (ergs) 240 240 240 100 200 200 200 200

Input pulse width (psec)

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The relaxation times and collision frequencies are: Trace

AR(m)

0.2 12 40 12 0.32 0.40 0.82 1.10

AN(cII1)

5.5 2.6 5 1.05 2.97 3.18 3.61 4.2

x x x x x x x x

/\RihN

10-s 1O-2 10-8 lo-* lo-* 1OF 1O-8 1O-8

36 460 670 1140 108 126 239 262

AR/O.11

0.26 15.6 52 15.6 0.415 0.52 1.06 1.42

SN(MHZ) 8.33 1.6 0.78 3.32 11.7 10.9 9.8 8.33

J&(MHz) 0.5 0.5 0.5 0.1 0.1 0.1 0.1 0.1

QR(MHz 0.23 0.0038 0.0011 0.0029 0.11 0.088 0.0427 0.032

The times I, , t, , t, , t, denote the time of injection of the heat pulse, and the times for first arrival and first and second second-sound reflections, respectively. We have also included the calculated phonon-phonon mean free paths and collision frequencies, as well as the input temperature frequencies needed for the discussion which follows. The horizontal line seen on some pictures was obtained by turning off the bias on the detector; any deviation from this line seen when bias is applied to the detector is due to temperature excursions in the crystal. First consider Fig. 11(e)-(h) which show the transition from pure diffusion, Fig. 1 l(e), to mixed mode heat propagation, second sound and diffusion, Fig. 1l(h), in a crystal grown at 54.2 atm. This transition occurs in the region of the thermal conductivity maximum. We will show that over the temperature interval covered by these traces, the experimentally determined relaxation times and the conditions for diffusion and second sound predict this is the region of transition from diffusion to second sound. The mean free paths h, and hN are determined from pulse data in the diffusion region, T 3 0.91”K, and second sound region. (See Section. V B and C.) The collision frequency Q, is obtained from the mean free path data and Debye velocity. G’, is found from Eq. (18); X, is determined from Eq. (18) and the Debye velocity. The input electronic pulse in these traces was 2 psec. We were unable to detect any change in the received pulse shape in the second sound region for input pulses up to 5-psec wide. We believe this is due to pulse broadening at the crystalgenerator interface. Hence, for this discussion we assume an input thermal pulse width of 5 psec, which corresponds to an input temperature frequency, fir, of 0.1 MHz. The spatial condition for diffusion is AR/L < 1, where L is the transducer separation, 0.77 cm in this instance. The frequency condition for diffusion is I& < QR. For Fig. 1I(e), AR/L = 0.415 and Qr < 1;2, is also satisfied (0.10 < 0.11). Also, the pulse has a shape characteristic of diffusion. At this temperature the first

151

PULSES IN DIELECTRIC SOLIDS

arrival time is greater than that in the second sound region (see Fig. 12). We believe heat flow is diffusive at this temperature. The conditions for second sound are X, > L; hN < L, and AT pulse Q tII or, alternatively, !& > fir > Q, . The condition At pulse < tII is met over the entire temperature range of the experiment. The first evidence for second sound propagation is seen in Fig. 11(f). While here the pulse structure is primarily diffusive, the first arrival time corresponds to that seen at lower temperatures and the pulse arrival is more abrupt than in Fig. 1l(e). The slope of the pulse is steeper, indicating an increase in the rate of arrival of the pulse energy. At this temperature X,/L = 0.52 (conditions for diffusion are satisfied), but the frequency conditions for second sound are nearly met (see the caption of Fig. 11). Figure II(g) shows an even greater initial pulse slope and there is also a flat spot on the trace which is not characteristic of diffusion. The first arrival time

400

.

t

.

';i i?i *ZOOw z i=

a a

.

a

. 0

2 1004 00a

a

4

.

-I

a 0

60

i

. 31.6 aA 30.7 27.6 0 43.0

IO 04

06

06

AMBIENT

I ::

1.0

1.2

TEMPERATURE

I .4

54 33 I00 I30

1.6

1

I.6

(OKI

FIG. 12. First arrival time of TP’s in the second chamber. No diffusion data was taken in the lOO-atm crystal. The dotted horizontal line for each pressure marks the arrival time of the pulse peak, This time is used for the calculation of observed second sound velocity.

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ACKERMAN

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is characteristic of second sound. At this temperature, X,/L = 1.06 and the relation !& > Qr > J& yields 9.8 > 0.10 > 0.0427. There is not a sufficiently large difference in J& and 9, for second sound propagation without attending diffusion at this temperature. Figure 11(h) shows a second sound pulse super-posed on a diffusion pulse. A reflection of the second sound pulse is seen. It is at this temperature that there is clear evidence of second sound propagation. Note that in this trace the maximum temperature excursion of the crystal (vertical trace deflection) occurs shortly after the initial pulse arrival is observed. This is not seen in any traces at higher temperatures where diffusion dominates heat propagation. At this temperature the conditions for second sound are nearly satisfied (see text directly below Fig. II). The resistive collision rates are still high enough to cause some diffuse scattering with an attending asymmetry in the received pulse. Figure 10, lower trace, and Fig. 1 l(d) show the second sound pulse at temperatures well into the region for second sound. Both traces were taken at an ambient crystal temperature of 0.6”K, the sweep speeds for Fig. 10 and lld are 20 and 50 psec/cm, respectively. Figure 10 shows the first received pulse

70 60

v~xP=206m/sec QD= 43OK

0.40

I 0.60

I 0.90

AMBIENT

I I .o

TEMPERATURE

/ 1.4

I 1.2

1 1.5

(*K)

FIG. 13. The arrival time of various features of the received temperature pulse vs ambient temperature. The symbols in this figure are defined in Fig. 11. Data was taken on a single crystal grown at a constant pressure of 130 atm. The arrival time of the leading edge, maximum, forward and back half-heights are plotted. tg is the expected arrival time for second sound computed from the Debye temperature and t, D is the expected arrival tune for first sound computed from the Debye temperature.

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153

and first reflection, while Fig. 11(d) shows two reflections. At this temperature X, = 12 cm X, = 1.05 x 1O-2 cm, X,/L = 15.6 and h,/X, = 1140. Hence the conditions for diffusion are definitely not satisfied. The ratio X,/h, indicates that there are numerous normal process collisons occurring within the temperature wave, a requirement for local thermal equilibrium. The frequency conditions for second sound are well satisfied at this temperature, 3.32 > 0.10 > 0.0029. Immediately after the passage of the first pulse, the trace returns to a point slightly above the initial base line, indicating the crystal temperature near the detector relaxes to a value very slightly above ambient temperature. At later times a slow rise of the trace is seen, indicating an overall crystal temperature rise. We feel that in these two traces the input temperature pulse indeed transits the chamber as second sound. A detailed analysis of these pulses is given in the following sections and in the appendices. Figure 1 l(a)-(c) shows second sound pulses taken in a crystal grown at 130 atm. These traces are all taken at temperatures where the frequency conditions for second sound are satisfied (see text directly below Fig. 11). This crystal was evidently of poorer quality that the 54.2-atm crystal illustrated, as no reflections were present. These three traces show the N-process broadening of the second sound pulse (see also Fig. 13). Several important features of the temperature pulses can be seen by displaying the data as illustrated in Figs. 12 and 13. In Fig. 12 the leading-edge arrival time of the received pulse is plotted versus the temperature for the four pressures employed in the experiment. Figure 13 shows the temperature dependence of the arrival times of the leading edge, forward half-height, maximum and back halfheight of temperature pulses propagated in a 130 atm. crystal. V. DISCUSSION

The general features of the data presented in the preceding section are: (1) At temperatures above T/O = 0.025 the temperature pulse shape is characteristic of diffusive propagation, Fig. lle. The leading edge arrival time, ta, , increases with increasing T and is linear on a log ta, vs T-l plot. The diffusion process in strain free single crystals is characterized by the Umklapp scattering rate. No reflections are observed. In Appendix A, we present diffusion region data where scattering mechanisms other than Umklapp processes determine arrival times. (2) At temperatures below T/O m 0.025 the detected temperature pulse has a reasonably symmetric shape, Fig. II(b). The arrival time of the TP varies mildly with T, becoming shorter as T decreases, Fig. 1 I(c). In a number of cases reflections of the pulse are detected, Fig. 1l(d).

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ACKERMAN

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(3) At the lowest temperatures, T w 0.5”K, the arrival time of all features of the TP moves toward shorter times, the leading edge moving faster than the maximum; Fig. 13. The TP broadens by almost 100 “/, w 0.4”K. This broadening is due to the decreased rate of N-process collissions as T--f OOK. (4) At low temperatures the velocity with which the leading edge of the TP crosses the sample chamber is comparable to the first sound velocity computed from On . In this section we discuss the quantitative information which can be ascertained from this data about the properties of the phonon gas. In particular, we deal with (A) the second sound velocity, (B) the thermal conductivity or resistive process mean free path, (C) the N-process collision rate and (D) the first sound velocity. A.

SECOND

SOUND

VELOCITY

For a multiple polarization phonon gas in the Debye c,, = c,,q/l q 1the second sound velocity is given by (40), (64)

approximation,

The equation is specialized to the case of isotropic velocities, although it is well known that the velocity of sound in solid helium is highly anisotropic. Sound velocity measurements in solid helium have been made by Vignos and Fairbank (65) (longitudinal) and Lipshultz and Lee (66) (transverse). The experiment of Vignos and Fairbank was done over the pressure range 25-140 atm, whereas that of Lipshultz and Lee was done at a single pressure, 26 atm. In Table I we have tabulated in column (b) the range of longitudinal velocities from the experiment TABLE

(4 P

(b)

(cl

(atm)

(m/set)

Ct (misec)

33 54 100 130

510-540 590620 700-740 760-810

235-330 270-390 330470 370-530

cc

(4 @I,

“K 27.5 31.8 38.7 43.0

I

(e)

(f)

(4

CSlC VP

VII

(m/se4

~m/sec)

310 350 420 460

140-200 160-240 200-290 220-330

expt. V::

b/se4 200 230 280 300

VII

(m/s4 130 160 180 210

PULSES IN DIELECTRIC SOLIDS

155

of Vignos and Fairbank for each of the pressures at which the second sound measurementswere done [column (a)]. Column (c) contains the range of transverse velocities, found by interpolating the measurements of Lipshultz and Lee using the Debye temperature (67) given in column (d). The second sound velocity can be computed using Eq. (13) and various pairs of velocities from columns (b) and (c); the result for the extreme casesis in column (f). It is also possible to compute a second sound velocity from 0, and the relation between c, and ct suggestedby the experimental data, ct = $cc. We have computed the first and second sound velocities, L$’ and rl: respectively, from 0, for ct = gc, , columns (e> and Cd. These computed velocities of second sound are to be compared to the experimental second sound velocity, ~7~;~.We compute 2’zp from the known path length between transducers, 0.77 cm., and the arrival time of the maximum of the TP in the second sound temperature range, Fig. 12. The maximum is used because we believe the TP is considerably broadened by N-process collisions. The velocities determined in this way are recorded in column (h) of Table I. Agreement between the calculated and experimental second sound velocities is qualitatively quite good. It is as good as we can expect since (a) the velocity

FIG. 14. Second sound velocity vs. pressure. The experimental second sound velocity at four pressures is shown as dark circles (arrival of peak of first pulse) and as an open circle (velocity from peak to peak for reflected pulse). The shaded area is the range of second sound velocities computed from first sound velocity measurements using Eq. (18). The dashed line is the second sound velocity computed from the Debye temperature and the assumption ct = 1j2 cc

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of sound in solid helium is highly anisotropic and Eq. (13) does not take this into account, (b) the velocity measurements of Vignos and Fairbank and Lipshultz and Lee were probably on polycrystalline samples, and (c) if the first sound velocity is anisotropic so is the second sound velocity (69), but less so. The comparisons made in Table I are shown in Fig. 14. In Fig. 14 we also show the results of a second sound velocity measurement made using the time between successive reflections of the temperature pulse. This second sound velocity is slightly faster (7 %) than that found using the maximum of the first arriving TP. This is in the direction we would expect if the normal processes were causing pulse broadening, but the velocity discrepancy may be due in part to non perturbative temperature excursions in the crystal. (See Appendix B.)

B. THE DIFFUSION REGION For temperatures above T/Ou w 0.025 the TP propagates diffusively. As we stated earlier (12) TP propagation in the diffusion region provides an alternative to steady state thermal conductivity measurements for finding K and rRz. In this temperature range, TV --f 0, Eqs. (1) and (2) have the solution (20), (70) T(8, t) =

f

[8T,,e-LljzT 2qt

GT,,At L+‘~~R=’ - t,) e(t, + At - t) + d(t - tn) 27Rz

?l=O

x Io--(t2- t,2)1/2 + ) I i TIC=

t (t2

_

&2)1,2

4

p2

;Rk2)1’2)]p

(13)

where t, = (2n $- 1) Q/u,, , (see Appendix E for further details). At t = 0 the temperature pulse has heated a region of the solid of volume hIIAt to temperature ST,. (A is the area of the generator, At is the duration of the input heat pulse.) The first term in Eq. (13) is the direct pulse of phonons which cross the sample chamber in time to = k/on, reflects and returns at t, = 3&n , etc. This second sound pulse is rapidly damped by the factor exp( -t/2TRz) in the diffusion region. The second term represents the diffusion with finite velocity of the heat energy which decays out of the second sound pulse. The sum of many terms is due to image sources which preserve the correct boundary condition at x = 0 and x = 8. For t - tlI > ~a” the solution for 7(/, t) is

PULSES

which is a superposition

IN

of solutions 8T _-_ at

DIELECTRIC

SOLIDS

157

of the Fourier heat law

c" V2T== 0. v

All traces of vu disappear in this equation. Solutions of Eq. (15) can be used in the real space form Eq. (14) or in a Fourier analyzed form. In either case it can be shown that the temperature profile at G evolves in time as shown in Fig. 15. The time t, determined by the extrapolation of the slope (shown on Fig. 15) is related to the thermal conductivity by (71)

(164 Since

K = 1/3Cvch,,

(16b)

the resistive mean free path X, is given by A, -

1.44c2 . 772t,c

(164

FIG. 15. An idealized dimensionless plot of the detector temperature history. The one dimensional solution of the Fourier heat law shown assumes the crystal is thermally isolated. T/T, is the ratio of the detector temperature to the maximum temperature. In practice the crystal is not thermally isolated and the crystal temperature slowly decays (a few thousand psec) to the temperature of the ‘He system. The extrapolated slope arrival time, ts , should be little affected by the nonideality of our system. ts occurs at w = (T?/@)& = 0.48. Here a is the thermal diffusivity K/C(cma/sec).

158

ACKERMAN

AND GUYER

.

FIG. 16. Comparison and pulse experiments.

of the resistive process mean free path obtained from steady-state

We have applied Eq. (16~) to the TP data at temperatures well above the onset of second sound and determined the resistive process mean free path in several crystals. See Fig. 16. Since the thermal conductivity and hence the mean free path for R-processesin hcp 4He is highly anisotropic, we have also indicated on the figure the extremes found for AR from steady state measurements(42). The steady-state measurements were performed at P = 85 atm. so we have plotted A, against reduced temperature, T/O. The values AR from our experiment fall within the range of those obtained from the steady state experiment; the qualitative behavior of our A, as a function of T is also correct. The thermal conductivity K can be obtained using Eq. (16a). If K scalesas T/@ and Krp is reliable, it is possible to determine the angular orientation of the crystals investigated in the TP experiment, since the correspondence of ~~~~ and Kmin with angle is known (72). We obtain the correspondence shown in Table II. It is possible to use the full solution to Eq. (1) and (2), i.e., Eq. (19) and extend the analysis of the TP data to lower temperatures. When this is done 7N becomes finite and the solution will have to be modified to include the effect of the viscosity term. So far we have found no reasonable form for the solution when this term

159

PULSES IN DIELECTRIC SOLIDS TABLE P (atm)

Angle __-

33 54 100 130

0 30 15”

11 zpt.

8(130)/O(P) (m/W 200 216 200 210

is included. It is possible to estimate the effect of the viscosity term and roughly determine the temperature range over which Eq. (13) should be valid without modification. A pulse in flight for time t is broadened by the viscosity term by At, - (tTN)1/2and by the diffusion mechanism by At, - (tTRZ)1/2. If we require At, < At, we must have (7Ji2 < (TVz ) l12. These considerations are relevant if the solution of Eqs. (1) and (2) is extended toward times on the order of tI1 and t, to detect incipient second sound by looking for departures from the predictions of Eq. (14). Equation (19) has been used successfully by Ziman (72) in discussing TP propagation in superfluid “He and by Brown et al. (20) in discussingTP propagation in sapphire at temperatures where ballistic propagation and diffusion are competitive. C. N-PROCESS SCATTERING The most important characteristic of the received TP in the second sound region, T/O < 0.025, asidefrom its velocity, is the observed temperature dependent broadening. The pulse broadening can be understood in terms of the solution of Eqs. (1) and (2) in the limit QrRZ >. 1, for then we find

(17) This equation represents a temperature pulse whose center of mass travels with velocity vn , which is broadened by N-process scattering. As T is lowered and 7N becomes large, there are not enough collisions among the phonons to keep them in the TP; as 7N -+ 0 the viscosity term drops out of Eq. (2) and the TP travels with unchanging shape as second sound. Dramatic evidence for broadening appears in Fig. 13. The data displayed in Fig. 13 can be analyzed using Eq. (17) to find 7N . When the TP first appears as second sound, at T = l.O”K in the 130-atm crystal, the

160

ACKERMAN

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received pulse is 12 psec wide; the input pulse is typically 1 psec wide. We could assign all of this broadening to the N-process mechanism or we might assume that another (unknown) mechanism is responsible for some or all of it. We used values of rN from the analysis of Mezov-Deglin’s Poiseuille flow data to estimate the magnitude of the broadening of the TP we should expect at T w l.O”K, (i.e., near the onset of second sound propagation). The estimated broadening agreed well with what we observed. Thus we have assumed that all of the broadening of the TP at any temperature below l.O”K is due to the N-process mechanism. We have analyzed the second sound data using Eq. (17) and the forward half-height (Fig. 11) of the received TP which arrives at the detector when

or t,/,

=

t11 -

1.57(~~t,,,)~/~

(18)

where t,,, is the arrival time of the forward half-height and tII is the peak arrival time of the TP. The results of this analysis are plotted in Fig. 17 as a function of T/O along with rN computed from the Poiseuille flow data of Mezov-Deglin. In the analysis of the second sound data and the data of Mezov-Deglin the Debye velocity c,, from the equation CD=-- k&ID n

113 - 1 i 67r2N 1 ’

(19)

and the values of On from Edwards and Pandorf (67) were used. A reduced plot was used because there is not enough second sound data or Poiseuille flow data at any single pressure to have it stand alone. From Fig. 17 we see that TV determined from the second sound data and the Poiseuille flow data are in excellent agreement. The agreement of these two different kinds of data enhances our confidence that we have the correct view of the second sound data. The aggregate of the data presented in Fig. 17 can be represented by 7N = 2 x 10-12(T/O)-3 (set).

(18)

Before we discuss the consequences of this result two remarks are in order: (1) The quantities TN entering the Poiseuille flow formula and the second sound formula are the same quantity, i.e., (a) q-space matrix element of X-l which measures the N-process viscosity of the phonon gas (10). As such the temperature dependence of 7N (the matrix element) should be the same as that of 7N determined for example

PULSES

IN

DIELECTRIC

161

SOLIDS

l

54atm

. .

100 130

POlSEUlLLE FLOW OATA c 60 v 65

, .6

FIG.

.B I.0

2.0

3.0 4.0

, , 6.0

, , 8.0

3

17. Normal process relaxation time vs. T/B.

by Herring (74); (2) Poiseuille flow data and second sound data are complementary in that pure Poiseuille flow occurs at temperatures just below Max (at lower temperatures boundary scattering interferes with Poiseuille flow), whereassecond sound is most broadened at lower temperatures (75). The 3-phonon processlifetime rN given by Eq. (18) can be compared with the value of TV from the analysis of steady state TC data by Bertman et al. (36) and Rogers et al. (37). SeeFig. 17. We first note that we find a T3 law whereasBertman et al. assumeda T5 law and Rogers et al. found T4 to give a better fit than T5. The result of this analysis agrees in magnitude with that of Bertman et al. for T GZTmsx , i.e., near the thermal conductivity maximum. This is consistent with the previous remark about a Callaway analysis. The results disagree with those of Rogers et al. over the full temperature range of experiment. The data of Rogers et al. goes only down to T = 1°K; the limited data available to them could well be the reasonfor their results. The temperature law which we find, T~I a T3, is different from the predictions of 7;1 oc T5 for the Herring mechanism (74). This prediction of Herring is a low

162

ACKERMAN

AND GUYER

temperature limit, it is not clear whether this limit applies or not. At higher temperatures the power law should be less than T5. It is of some interest to estimate the lifetime of a phonon in solid helium due to N-process scattering. Because of the way in which this lifetime enters the transport properties, i.e., averaged over the phonon spectrum, it is not possible to unravel the w and T dependence. However for a phonon at frequency w w kBT/fi we can write fi AW - w = Tiil W

1 T3 zi 2 x IO-12 t 0 i- kBT

(19)

or Am/w m 3T2/Q3, Au/ w M 3 x 1O-5 at T = 1°K and P = 130 atm. A similar calculation for LiF yields AW n - 6J = 1’35T5 kJ or Am/w M lo-‘rT4, Aw/,a~ m 1O-s at T = 18”K, the equivalent reduced temperature. These phonons are at T/OH = 0.025, i.e., about l/40 of the way out to the Brillouin zone boundary. At this point in the Brillouin zone the phonons are well defined excitations in solid helium and LiF. Those in LiF are more long lived by an order of magnitude. A word of caution is necessary at this point. The two relaxation times which we have employed in our discussion of TP data are a matrix element of R*, the transport time, and a single eigenvalue of M*, the equilibrium time T$. It is not correct without justification to assume that the eigenvalue T$ and the phonon lifetime are simply related. If the eigenvalue spectrum of .k’“* is dominated by a single eigenvalue, then the identification is tenable. There is no evidence in the transport properties for unusual effects in these crystals attributable to its quantum crystal properties. This result is consistent with measurements of the thermostatic properties of solid helium and the speculations which recent theoretical progress on quantum solids have permitted. D. FIRST SOUND

Von Gutfeld and Nethercot (28) and Rogers and Rollefson (22) have observed ballistic phonon propagation in crystals at ambient temperatures well below that of the thermal conductivity maximum. Their experiments provide a good measure of the first sound velocity of all polarization components. In the solid helium experiment it was not possible to reach sufficiently low temperatures to see ballistic phonons. However at the lowest temperatures reached in the experiments, a small fraction of the phonons crossed the sample chamber with essentially first

PULSES

IN

DIELECTRIC

SOLIDS

163

sound velocity, see Fig. 13. We estimate from the value of 7N given by Eq. (18) that at T = 0.20X, P = 130 atm, AN would be about 1 cm, the distance across the sample chamber and ballistic phonons should be observed. In a sample chamber 0.1 cm across the ballistic region is reached at T = 0.41 “K.

VI.

CONCLUSION

We believe these experiments demonstrate that second sound exists in dielectric crystals. In the temperature region where the experimentally determined relaxation rates satisfy the requirements for second sound propagation we observe the following: (1) Pulse propagation is no longer diffusive as it is at higher temperatures where the conditions for second sound are not satisfied. (2) The pulse velocity is nearly constant and close to the calculated value of second sound velocity. Multiple reflections are observed. The observed pressure dependence of the velocity agrees with calculated values. (3) The region of second sound propagation covers the temperature range over which Poiseuille flow is observed in similar crystals. The presence of Poiseuille flow ensures the existence of second sound. (4) At the lowest temperatures, the finite N-process rates causes the second sound pulse to broaden. This N-process broadening is used to determine values of 7N which are in agreement with those obtained by steady-state thermal conductivity experiments. In the temperature region T/O > 0.025 temperature pulses propagate diffusively and pulse arrival times measure the diffusivity of the crystal. The diffusivity data are used to obtain thermal conductivities which are in agreement with steady-state measurements. We briefly recall the experimental situation with respect to solid helium to indicate the contribution which can be made by temperature pulse techniques. The thermostatic properties of solid helium have been rather carefully explored (67) (76). Thermal conductivity has been measured numerous times in unoriented and probably polycrystalline samples (36), (37), (45), (77). First sound measurements have been made by Vignos and Fairbank (65) and by Lipshultz and Lee (66) on samples of unknown quality. Determination of the orientation of single crystals has been accomplished by means of optical birefringence as was recently reported by Vos et al. (78) and Heybey and Lee (79). The early thermal conductivity measurements will presumably be superseded by those of Mezov-Deglin (II) and Hogan (42), since experiments on carefully grown crystals yield a great deal

164

ACKERMAN

AND GWER

of information about phonon-phonon scattering processes, i.e., rN and rRz . It would be useful to have simulataneous studies of thermal conductivity and first sound velocity, first sound velocity and orientation, or similar coupled experiments. It is with respect to the latter that we believe TP experiments to be unrivaled in neatness. A single set of temperature transducers can measure K or TV”, In a carefully designed sample chamber these TN, 'II 2 c,, and orientation. parameters can be measured in several directions simultaneously. ACKNOWLEDGMENTS

We would like to give particular thanks to Dr. H. A. Fairbank for his help and support during the course of these investigations and to Dr. B. Bertman who suggested this experiment to one of us (C.C.A.) as a thesis topic. Further we have benefited from discussions with many people about second sound in solids; Drs. R. 0. Pohl, John Rogers, P. A. Griffin, G. V. Chester, A. Thellung and W. Overton. One of us (R.A.G.) owes a particular debt of gratitude to Dr. J. A. Krumhansl for having introduced him to this subject. We would like to thank Dr. L. Goldstein for critically reading the manuscript.

APPENDIX

A. TEMPERATURE

PULSES IN POLYCRYSTALLINE

STRAINED SAMPLES

SINGLE

CRYSTALS

AND

Many of the initial attempts to grow single crystals resulted in polycrystalline samples. In Fig. 18 we show the arrival time curve for a typical polycrystalline sample grown at 54 atm. In polycrystalline samples the received pulse is diffusive in character down to the lowest temperatures. The low temperature arrival time can be used to estimate polycrystal size using Eq. (22~). For the sample illustrated, the limiting polycrystal dimension is about 1 mm. Polycrystalline samples in

,,I , , , / , , I 08

IO

1.2

I.4

, ?!Fr,

1.6

1.8

20

A

22

I/T(“K-‘)

FIG.

18. Arrival time curves for polycrystallme

and strained single crystal samples.

PULSES IN DIELECTRIC SOLIDS

165

which pulse arrival times were within a few microseconds of the second sound arrival time were very likely to display second sound if carefully annealed. Some single crystals were obtained in which the temperature onset of second sound was depressedto temperatures below TJO H NN0.025. The arrival time curves obtained using these samples did not show a sharp knee at the transition of second sound. The R-process mean free path for one of these crystals (33 atm) is shown in Fig. 16. For T/O < 0.028, the effective resistive process mean free path is lessthan that determined from the pure Umklapp mean free path obtained from steady state thermal conductivity measurements,but at high temperatures the resistive mean free path closely coincides with that obtained from steady state data; i.e., in this region the Umklapp processesdominate the total R-processrate.

TAMB~E~T

(“K)

FIG. 19. Input temperature excursions in a 54-atm crystal. The input pulses are O.Spsec, 100 ergs, and 3-4 psecs, 16 ergs in run I and II respectively. The solid horizontal line marks the transition temperature from diffusion to second sound.

166

ACKERMAN AND GUYER

This behavior is consistent with the argument that there is an additional scattering mechanism present due to some defect in crystal structure which affects h, at the lowest temperatures where the Umklapp scattering rate is very low. This sample fractured at about 0S”K and the second sound pulse disappeared and was replaced by a diffusion pulse. The effect of a serious defect in a 54-atm crystal in which second sound propagated at low temperatures is shown in Fig. 18. In the best crystals grown at this pressure, the knee in the arrival time curve occurred at T-l M 1.2”K-l and second sound reflections were observed at that temperature. In this crystal however, reflections of second sound were not seen until T-l m 1.8’K-l. Thus, in this sample an additional resistive scattering mechanism depressed the onset temperature for second sound from .8” to about .55”K. If one assumes additivity of the R-process mean free paths, X&l = hi’ + Xi’, then the overall temperature dependence of the additional scattering mechanism or mechanisms may be found. ho is determined from an extrapolation of the high-temperature arrival times and ha is found from the measured arrival times. Xo is then determined by the additivity relation. For this particular crystal it was found the relaxation time of the unknown resistive mechanism varied as Q CC TY5. If the defect had a stronger effect on the total R-process rate, the onset of second sound would have been further depressed in temperature. Had we been able to reach temperatures where ballistic phonon propagation occurs, the ballistic phonons would first be seen accompanying either a second sound or diffusion pulse, depending on crystal quality. The failure of Von Gutfeld and Nethercot to observe second sound is undoubtedly due to crystal imperfections. In crystals with high non-Umklapp resistive scattering rates, the second sound frequency window is closed and diffusion of the heat pulse would be observed until the R-process mean free path became comparable to the transducer separation, where upon ballistic phonons would be observed. The magnitude of the diffusive pulse accompanying the ballistic pulses would depend upon the strength of resistive processes. The apparent resistive process relaxation rate can be increased if the input pulse powers are large enough to heat a portion of the crystal far above ambient temperature. These effects are discussed in Appendix B.

APPENDIX

B. THE EFFECT OF LARGE INPUT TEMPERATURE EXCURSIONS ON SECOND SOUND PULSE SHAPE AND VELOCITY

As mentioned previously it is desirable to keep input temperature excursions in the crystal to a minimum. In the first chamber input power was adjusted to give 20-millidegree temperature excursions at the receiver. Wall effects dominated

PULSES

IN

DIELECTRIC

SOLIDS

167

the structure of the pulse in this chamber and we were not able to detect power dependent pulse distortion. Much more attention was given to input power levels in the second chamber where the effect of high input powers were observed. Two passes, designated runs I and II, were made on a 54-atm crystal at different input powers. The subject crystal was the best obtained in the large chamber (multiple second sound reflections were present). Run I was made with 100 erg, 0.5-psec input pulses. Assuming perfect coupling of the heat pulse into the crystal, the temperature excursions of the crystal in the region of the generator were large enough to heat this portion of the crystal well into the diffusion region. With these high input power levels, the temperature pulse traversed a portion of the crystal with a velocity and broadening characteristic of diffusion until it cooled sufficiently to propagate through the remainder of the crystal as second sound. This is evident when the pulse velocity and halfwidth obtained in this run is compared with the data of run II where the input pulses (334 psec, 16 ergs) were a minimum for a usable received pulse amplitude. The temperature excursions in the crystal for runs I and II are shown in Fig. 19. The pulse velocity in these runs was obtained from the arrival time of the peak of the first received pulse of runs I and II, with a correction for the finite pulse width of run II, and from the arrival time of the peak of the first received pulse and first reflection seen in run I. The latter velocity measurement is obtained with pulses which are, at most, 30 mdeg above ambient temperature. Hence these data are obtained with the smallest pulse temperature transients. A comparison of the measured pulse velocities is as follows: Run I:

148-160 m/set

Run II:

157- 164 m/set

Run II, reflections:

170 m/set

The latter figure, 170 m/set, compares most favorably with estimates of the second sound velocity (see Fig. 14). Further evidence of pulse distortion is seen in a plot of the temperature dependence of the second sound pulse width for run I and II given in Fig. 20. Note that the 0.5-psec input pulse undergoes a minimum broadening of about 14 psec, whereas the low-power pulses of run II have a minimum breath of about 10 psec. The minimum in halfwidth occurs at about 0.60” and 0.65”K, respectively, for runs II and I. The minimum for run I is much broader than the minimum for run II. The pulse broadening caused by the high-power inputs is most evident at higher temperatures, where the Umklapp scattering rate determines the structure of the received pulse. In this region the crystal heating greatly increases the resistive process rate. Both the high- and low-power pulses are broadened by N-processes at the lowest temperatures.

ACKERMAN

168

AND GUYER

.

. 0’8 l

.15-

. :

.

Hz .

.

K_I-

. . I

I

.03-

.4

.5

I

I

I

.6

.?

.8

T (OKI

FIG. 20. Second sound attenuation. Here we have plotted the ratio of the height of the first second sound reflection to the height of the first received pulse versus ambient temperature. The attenuation is due to R-processes at high temperatures and N-processes at low temperatures. The minimum in attenuation occurs at the point where X&N is a maximum.

Hz

HI

.

.4

I

I

.5

.6

I

I

.8

.9

T (‘Kj7

FIG. 21. Width at half height of the temperature pulses of runs I and II.

PULSES IN DIELECTRIC SOLIDS

169

Effects related to these have been observed by Nethercot and Von Gutfeld. In their pulse experiments using sapphire crystals, they obtained first sound pulsessuperposedon a diffusive ramp in one instance (19) but not in another (18) They attributed the presence of the diffusive ramp in one case to high-input powers. This is undoubtedly the case,but calculations of their crystal temperature excursions have not been made. The persistence of ballistic phonons at higher temperatures also is probably a consequenceof high-input pulse power. Experiments are now underway at Los Alamos in which the transition from second sound to first sound in 4He crystals will be investigated. The results of these experiments may shed somelight on power dependenceeffects in the ballistic region. In addition to the pulse width and velocity measurements,the multiple pulse reflections seenin run I, enabled us to measure the attenuation of second sound, (Fig. 21). The data obtained are plotted as the ratio of the height of the first reflection Hz to the height of the first received pulse, H1 . These data show a minimum in attenuation at T = 0.7”K, which roughly corresponds to the minimum in pulse width at 0.65”K shown in Fig. 20. The increase in attenuation at temperatures above 0.7”K is strongly temperature-dependent aswould be expected if the Umklapp processeswere responsible. Below 0.7”K the attenuation has a weaker temperature dependence as would be expected if the normal processes dominated the pulse behavior at these temperatures.

APPENDIX

C. CRYSTALGROWTH

Here we outline the growth rate calculation and techniques used in growing and annealing single crystals of 4He. Due to the difficulty of visual observation of crystal growth and quality as was first done by Shal’nikov (59), (60) and later by Mezov-Deglin (41) and others (79), the crystals were grown by carefully controlling the temperature at various points on the sample chamber in a manner consistent with the results of a growth rate calculation given below. We considered the sample to be a single crystal when second sound was observed. Of twelve crystals grown in the first chamber, three were believed to be single crystals. Mezov-Deglin experienced approximately the same percentage of single crystals with his technique. He used Poiseuille flow as an indicator of crystal quality, a criterion which is more stringent than ours (61). In the second chamber we were unable to obtain good crystals unless they were carefully annealed. The large cross section of both chambers used in these experiments ruled out the use of a steady state thermal conductivity measurementsto determine crystal quality (62).

170

ACKERMAN

AND

GUYER

A. THE GROWTH RATE CALCULATION The growth rate calculation enabled us to program crystal growth at constant rates using a modified Bridgeman technique. The calculation assumed the conductance of the walls to be negligible, a condition which is most nearly met in the first chamber. A further requirement is that the temperature gradient in the melt be zero. This condition was met experimentally by maintaining the top of the sample chamber slightly above (-. 0.01 OK) the crystal melting temperature while growing the crystal. Under these conditions the temperature gradient in the solid is maintained by the latent heat QL required to propagate the interface. The growth rate of the crystal is characterized by the amount of heat absorbed at the interface in the solidification process, QL = LpAi-2 (C-1) and the amount of heat carried through the completed portion of the crystal eL =&AFT

(C-2)

In the above equations VT is the temperature gradient in the solid, L the latent heat of the phase change, p the density appropriate to the pressure used, A the interface area (which is assumed to be planar), and ff the interface velocity. R, is the averaged thermal conductivity of the solid. Some sort of average is required, as the temperature dependence of the thermal conductivity of the solid cannot be ignored in the calculation. We averaged the conductivity in the solid by assuming R, = I& exp@/T) (C-3) K,, and /I are determined by fitting Eq. (19) to thermal conductivity at or near the desired molar volume. Tis given by I/T=

1/2(1/T, + l/T,)

= T;T,Ts

data (51) (C-4)

where Ts is the temperature of the cold end of the crystal (the copper rod) and TM the melting temperature. Defining 7 = Ts/TM , l/T=

1/2T, (+).

Since the crystal interface propagates at velocity ff, at time I the length of the crystal is jil; also dT = TM - T, = T,(l - 7). Hence Eqs. (C-2) and (C-3), when combined, become .t+t = 9

(1 -

7) exp [/3/2TM (+)I.

(C-6)

171

PULSES IN DIELECTRIC SOLIDS

This equation describes crystal growth. The rates considered are 0.1 cm/hr to 10 cm/hr. Once sink temperatures have been calculated as a function of time for a specific growth rate and molar volume using Eq. (C-6) they are converted to carbon resistancethermometer values and a graph of thermometer resistancevs. time is constructed. When growing the crystal experimental rates are plotted against calculated rates. B. EXPERIMENTAL

TECHNIQUES FOR GROWING

AND ANNEALING

CRYSTALS

The simplest and most reliable technique for growing and annealing crystals utilized the convective cooling of the gas in the 3He system which occurs when the pumped 4He volume is dropped in temperature. Other methods were tried with varying success,but all were much more difficult to execute and did not produce good crystals as consistently as this method which is outlined below. It is desirable to have the 3He and 4He pot well coupled thermally during the growth process. For this reason the 4He volume was first pumped down to l.l”K and all the 3He is condensedinto the 3He lines. Then the 4He system was brought to a temperature slightly above the crystal melting temperature and controlled with a pressure referenced regulator. Heaters on the sample chamber and the 3He volume were used to bring these systems into approximate equilibrium with the 4He container. After 6-10 hours equilibrium was achieved, with the sample chamber at some temperature slightly higher than the 4He volume temperature becauseof the heat leak down the fill line. Then the pressurereference volume on the temperature regulator on the 4He system was slowly evacuated through a 20-turn needle valve. The resultant pressure drop in the regulator controls the temperature in the 4He system and the crystal growth proceeds at a constant rate with a single valve setting. When the top of the crystal had cooled to about O.OlO”K above the melting temperature, the heater on the top of the sample chamber was usedto regulate the temperature slightly above the solidification temperature. As the crystal grew it was necessary to adjust the sample pressure to compensate for density changes as the sample solidified. The pressure was kept constant to within fO.l atm while growing crystals. In the first sample chamber completion of the crystal was marked by a rapid increase in the amount of heat required to maintain the top of the chamber above the melting temperature. This effect was causedby the higher thermal conductivity of the solid helium in comparison to that of the liquid. In the second chamber the walls have a much higher conductance and this effect was not observed. Hence, in the second chamber, the crystal was grown for about one hour after there was no further need to increase the sample pressure to compensate for the density changescaused by crystal growth.

172

ACKERMAN

AND GUYER

The crystal growth was terminated in both chambers by slowly turning down the heater power. Crystals were grown at rates from 0.1 to 3 cm/hr. Crystals were annealed by setting the 4He temperature regulator to an empirically determined temperature which brought the sample chamber to within O.l”K or less of melting temperature. Annealing periods were from 6-36 hours long. Some crystals were improved greatly after careful annealing. After annealing, the crystals were cooled at about O.O2”K/min until the temperatures were reached where data was taken.

APPENDIX For a multipolarization is (IO), (79): ($+

D. MULTIPOLARIZATION phonon

SYSTEMS

gas the system of Boltzmann

equations

($k’ . v n(k) = Lgl C’kq+e’ 1

G = 1, 2,..., s. There are s such equations; C(“sd) may be decomposed momentum conserving and a momentum nonconserving part;

(D-1) into a

The N-process part of C generates the eigenvalue problem

The eigenvectors n&“’ and 7:“’ correspond to a local energy density and a local “drift” of the phonon gas in the kth mode. If the local heating and “drift” of all modes are the same, then P-3)

The eigenvector problem has as solution the set of vectors {7Lk’> which obey the equations

il
(D-4)

PULSES

IN

The two null space eigenvectors

DIELECTRIC

173

SOLIDS

are -1

TOR)

= pxk

2 sinh 9)

_

(2 sinh $-)

,

(D-5)

-1 (k)

%a

- &

B

,

(D-6)

which have the normalization (D-7) (D-8) and obey the equation

The solution to Eq. (D-l)

is sought in the form

n(k)= c a,(~, t) qkk). Substitution

of the expression

The energy and momentum (#’ 1and (~1~’ 1respectively;

for rick) into Eq. (D-l)

equations these are

are found

(D-10) leads to the equation

upon

multiplying

Eq. by

(D-13) where we have used Eq. (D-9) and have approximated the matrix elements of the non-momentum conserving collision operator by the leading matrix element.

ACKERMAN

174

AND GUYER

Equations (D-12) and (D-13) may be employed in discussing second sound propagation, anisotropy of the thermal conductivity, etc. A. SECOND SOUND VELOCITY In the limit 9Ck*p)---f 0 Eqs. (D-12, 13) can be combined to give

(D-14)

The energy in the kth phonon mode is given by

&4= c nwckwk) =

(D-15)

*

so that Eq. (D-14) becomes an equation for the energy upon multiplying by (k,T/p)(#’ / qr’) and summing on all k. For a, varying as exp i(k * v - at), Eq. (D-14) leads to the dispersion relation (D-15) This is the correct velocity for drifting second sound. We can derive the driftless second sound dispersion relation by modifying the above discussion. So far we have regarded the s-polarization branches as strongly coupled so that the drift of the phonon gas is communicated to all polarization branches and we write dk) = a, I ~6~)) + ala I ~:t)>

(D-16)

where a, and a,, which measure the local energy density and local drift are the same for all polarization branches. For times long compared to M-l (or rN the N-process relaxation time) this is necessaryin order to preserve the drift of the phonon gas as a whole in the presence of N-process collisions. See Eq. (D-3). For times short compared to rN , there is no communication between the polarization branches. The fact that they are not all “drifting” together does not matter becauseJV does not couple them. In this limit Eqs. (D-12) and (D-13) may be written for each polarization branch separately. For R --f 0 we have (D-17a)

175

PULSES IN DIELECTRIC SOLIDS

and “y’ +

$

f

fic’k’2

=

(D-17b)

0.

These equations relate the amplitude of the local drift in a particular polarization branch to the local energy density in that branch. Equations (D-12) and (D-13) relate the local drift of the phonon gas as a whole to the local energy density as a whole. If the amplitude of the local energy density of all polarization branches is the same, then the amplitude of the drift in each polarization branch depends on the phonon velocity in that branch; thus

for ~2:“’ cc up’ cc exp i(kx - Qt). The energy density amplitude

obeys the equation (D-19)

So the energy density in the whole phonon gas obeys a similar equation (D-20) For E(~) cc exp i(kx - J2t) this leads to the dispersion relation 79 II’

=-=Q2 k2

CL c(k)2@ c”,=, Cik’

’

(D-21)

in agreement with the result of Griffin and Enz for driftless second sound (14), (17). B. ANISTROPY OF THE THERMAL

CONDUCTIVITY

Equations (D-12) and (D-13) above with slight modification may be employed to discuss the anisotropy of the steady-state thermal conductivity. Our purpose here is to explicitly show how the scattering anisotropy of the thermal conductivity arises. Thus we ignore the effect of velocity anisotropy and employ Eq. (D-13) in the form

176

ACKERMAN

AND

GUYER

or (D-23) Recalling that (D-24) and using Eq. (15), we can write (D-25) where (D-26) is the thermal resistivity. The thermal conductivity i v=l

Jp)W’f’ LYY ?a = aas .

is the tensor my’ for which (D-27)

We note the following: (a) The thermal conductivity and thermal resistivity can exhibit scattering anisotropy because they are related to q-space matrix elements of R weighted in the direction of the heat current flow. (b) The actual structure of the tensors & and W,, depends on the crystal symmetry since they are tensors of rank 2. For cubic crystals, e.g., bee 3He, K and Ware isotropic. For hcp crystals, the form

where the x-axis is parahel to the c-axis of the crystal. (c) The thermal resistivity of the various polarization branches and various phonon groups in a given polarization branch is additive. This is true because we are considering the Ziman limit. In the Debye limit the thermal conductivities are additive. Inclusion of velocity anisotropy effects are not difficult in principle but lead to formulas much more cumbersome than those above. The observations (a)-(c) are not changed by the presence of velocity anisotropy.

177

PULSES IN DIELECTRIC SOLIDS C. ANISOTROPY

OF THE SECOND SOUND VELOCITY

The presenceof anisotropy in the first sound velocity gives rise to an anisotropy of the second sound velocity. However since the second sound velocity involves a momentum spaceaverage and polarization average it is clear that the anisotropy of the second sound velocity will be considerably lessthan that of the first sound velocity. This problem has been treated by Kwok in somedetail (67).

APPENDIX

E. PULSE ANALYSIS

EQUATIONS

Consider a cylindrical sample of dielectric solid which is driven at one end by a temperature transducer; the temperature at the other end of the sample is detected as a function of time. The purpose of this Appendix is to discuss the system of equations which will relate the signal at the detector, T(L’, t), to the input T(0, t). The most general equation of motion for T(x, t) which is easily solved is

(E-1) where K = $CVc2r, is the steady state thermal conductivity, CV is the specific heat, c is the velocity of sound and T is the transport relaxation time. Eq. (E-l) follows from Eqs. (1) and (2) of the text if we neglect the viscosity term in Eq. (2). The solution to Eq. (E-l) for a cylinder of infinite length is given by T(x, t) = &

j dx’ T(x’, 0) [e-t/2T * 8(t - to) + e-t/27 &

x

1,

1

(f2

(

-

bY2

27

+

i

(t2 -ft,2)',2

z1 ft2 -2:2)1'2)1

O(t -

I,)]

(E-2)

where u,,~= ~~7,437,t, = (x - x)/v, and I,-, and Z1 are modified Besselfunctions. We are concerned with the solution to Eq. (E-l) when a nearly delta function temperature pulse drives the transducer. Suppose the transducer is driven at the energy rate e for duration dt and that the transfer of energy from the transducer to the solid is 100% efficient. Then at t rv 0, the temperature distribution in the solid is AT, =

595/50/I-=

cAt CvAv,At ’

0 < x < v,At

178

ACKERMAN

With this t = 0 temperature

AND

GUYER

profile, Eq. (E-2) yields

T(l, t) = &A Toe-t/2Te(t - to) e(t, + dt - t)

+ 2 dT0e-t/2T /Z(z)+ & i

r,(z)/

e(t

-

to),

(E-3)

where z = (t2 - t,2)1/2/2~ and t, = //iv,, . The first term in this solution corresponds to ballistic propagation of the temperature pulse damped by exp(--t/27) across the sample. The second term corresponds to the propagation of the energy scattered out of the ballistic pulse by diffusion. In the limit 7 ---f + co the ballistic pulse alone arrives at the detector. In the limit T ---f 0, the ballistic pulse is completely damped and the second term takes an asymptotic form which is the solution to the conventional diffusion equation. In particular in this limit we have z --+ co,

Z,(z)-&m

4(z)- &

so that the second term in Eq. (E-3) becomes T(/,+2f+o

e-t/2T exp[-(t2 - t,2)lj2/27] [(Tr/T)(P - t,yl1/2 -

Now for t > f,, we can write (t” - t,2)1/2 - t - t(to2/t) and thus obtain

$I

7+00

T(/, t) = AT,

dt

(4mt)1/2

exp (-

-$-),

(E-4)

which is the solution to the diffusion equation (see below). It is a correct asymptotic form for times long compared to the direct flight time across the sample chamber. The solution to Eq. (E-l) when there are reflecting boundaries on the sample chamber is the superposition of the temperature at Gfrom the original temperature transducer and from image heat sources at f2nL’. This solution is T(/, t) = jJ A Toe-t/2TO(t - ti)

i=O

+ A To 2 e-t’2r

lZo(~d

e(t, + dt - t) +

&

$

E

I,/

e(t - ti),

(E-5)

ti = I, ) 3t, ) 5t, )...) t, = Quo . For short times only the i = 0 term in the sum is important. Each source begins to contribute at x = L’ at a time (a) proportional to its distance (in the ballistic limit) or (b) proportional to the square of its

PULSES

IN

DIELECTRIC

179

SOLIDS

distance (in the diffusion limit). After sufficiently long times the diffusion limit always obtains. At time t, > t, all sources at distances greater than t1 given by

do not contribute to the temperature

& 2T_ w 7 ( 0071 temperature at / at t, . Each

to the at e contributes

source which

contributes

the same amplitude

Ai = AT&(mt,)-l/2 so that the total heat at /is N AToAt WC t> = i; (7rTtp

(E-6)

where the number of contributing sources N is given by (2N + 1) 8 = t1 or N = (uo/l)[&(Tt1)]1~2, so that Eq. (E-6) becomes T(lf, t) cz N

AT&

(E-7)

(57Tt,)1/2

which is proportional to the ratio of the volume of the sample initially heated by the temperature source to the total volume. The above equations can be used to analyze temperature pulse data in the diffusion region, the diffusion to second sound region and diffusion to ballistic region. It may be possible to detect incipient second sound by looking for departures from the predictions of these equations. In the diffusion limit, T + 0, t > to, the asymptotic form of Eq. (E-5) is

where the expansion of zi in Eq. (E-5) in terms of tilt is only valid for t > ti . Since the ith term contributes to T(/, t) only for t - ti2/T, it is valid to use the asymptotic expansion of each term. Eq. (E-8) is the solution of the diffusion equation, Eq. (E-l) with 7 = 0, i.e., i3T --at

c” V2T = 0. Y

(E-9)

This equation may also be solved in the form T(x, t) = I,,: T(x, 0) dx + sngl exp (-

y]

nnx 6 cos 7 so 7(X’, 0) cos ‘$

dx’, (E-10)

180

ACKERMAN

AND

GUYER

where we have used the set of functions which naturally incorporate the boundary condition at the sample ends. For an initial temperature distribution like that discussed above we find T(l, t) = AT, f-$$ [l + 2 ntl (-1)”

exp (-

F)].

(E-l 1)

Note the prefactor is just the final ambient temperature given by Eq. (E-7). In this form it is particularly easy to see how to use measurements of r(t, t) to learn the thermal conductivity or 7. At long times, Eq. (E-10) is asymptotically

TV, t> - Tt [l - 2 exp (- $,)I so that Tf(l) - T(/, t) may be used to measure T. Von Gutfeld and Nethercot (29) have used Eq. (E-S) (the first item only) to find K for sapphire crystals. In the text we have used Eq. (E-IO), the method of Parker et al. (70), to determine K for solid helium. Time dependent methods of measuring the thermal conductivity are not uncommon. In the remainder of this appendix we write out the steps by which Eqs. (E-2,4, 10) and Eq. (23) of the text are obtained. EQ. (E (a)

1) Fourier-Laplace-transform T(x, t) = &

T(x, t), s dk eikx &

/

dz eztri’(k, z),

(E-13)

B.C.

T(k, t) = 8-l I dx e-iKrT(x, t), F(k, z) = Jrn dt’ e-zt’T(k, cl

t’),

and B.C. denotes a Bromewich contour c - iR --f c + iR, R + + co, such that all the poles of F(k, z) are to the left of c. (b) Fourier-Laplace-transform Eq. (E-l); obtain F(k, z) = 8-l 1 dx e-iksT(x, X

=

KjCy

.

0) 7z2 l+T yk2X

, (E-14)

PULSES

IN

DIELECTRIC

181

SOLIDS

(c) Insert this result into Eq. (E-13); change the order of the k and z integrations; integrate on k and obtain dx’ T(x’, 0) &

dz ezt 1 + T&,

j B.C.

X

exp[ -~(z2+$)1’z] (z2

+

(E-15)

y2

-

(d) The inverse Laplace transform called for here is given by formula 29.3.91 in Abramowitz and Stegun (80). It is

ds,stexpbWs+ dl”“>= e-at,2I ,, [& (t2 - k2)1/2] e(t - k). MS

+

(E-16)

w2

(e) Using this result in Eq. (E-15) and carrying out the differentiation called for, leads to Eq. (E-2). The ballistic pulse part of Eq. (E-2) comes from differentiation of the 8 function. EQ. (E - 9) (a)

Fourier-analyze

T(x, t), T(x, t) = &

(b)

(E-17)

j dk T(k, t) eikx.

Substitute the solution into Eq. (E-9) and obtain (E-18)

T(k, t) = T(k, 0) e-kBXt.

(c) Substitute terms of T(x, 0),

T(k, t) into Eq. (E-17) and invert (E-17) to find T(k, 0) in T(k, 0) = f-l j T(x’, 0) cikx

(d)

(E-19)

Combine Eqs. (E-17,-19) to obtain T(x, t) = &

(e)

dx’.

Do the k-integration T(x, t) =

j dk j dx’ T(x’, 0) eik(x-s’)e-k2xt,

by completing

(E-20)

the square; obtain

’ j dx’ T(x’, 0) exp ((4?rxt)l/2

(x yX:‘)I).

(E-21)

182

ACKERMAN

AND GUYER

(1 AND 2)

EQS.

(a) obtain

Fourier-analyze

T(x, t) in space as above, carry out steps (a) -+ (d);

T(x, t) = j dx’ / $

T(x’, 0) exp{i[k(x

- x’) - Q(k)t]},

(E-22)

(b) Insert this dispersion for second sound into the exponential the k-integration as in (e) above; obtain T(x, t) = e-t/2T j dx’ W’, 0) ( 12:TNt )l” exp [ RECEIVED: March

(x - x’ - zl# QC%Nt

I’

and do

(E-24)

19, 1968

REFERENCES

AND FOOTNOTES

I. W. NERNST, “Die Theoretischen . .. Grundlagen des n W&mestatzes” (Halle: Knapp, 1917); as referenced in (5). 2. V. PESHKOV, J. Whys. (USSR) 8, 381 (1944). 3. V. PESHKOV, in “Report on an International Conference on Fundamental Particles and Low Temperature Physics.” Vol. II, p. 19. The Physical Society of London, 1947. 4. R. B. DINGLE, Proc. Phys. Sot. (London) A65, 374 (1952). 5. J. C. WARD ANLI J. WILKS, Phil. Mug. 42, 314 (1951). 6. J. C. WORD AND J. WILKS, Phil. Mug 43, 48 (1952). 7. F. LONDON, “Superfluids,” Vol. II, p. 101. Wiley, New York, 1954. 8. E. W. PROHOF~KYAND J. A. KRUMHANSL, Phys. Rev. 133, A1403 (1964). 9. R. A. GUYER AND J. A. KRUMHANSL, Phys. Rev. 133, A141 1 (1964). IO. R. A. GUYER AND J. A. KRUMHANSL, Phys. Rev. 148, 766 (1966). Il. R. A. GUYER AND J. A. KRUMHANSL, Phys. Rev. 148, 778 (1966) (hereafter Referred to as I). 12. C. C. ACKERMAN, B. BERTMAN, H. A. FAIRBANK, AND R. A. GUYIXR,Phys. Rev. Letters 16, 789 (1966).

T. LANE, “Superfluid Physics,” McGraw-Hilt, New York, 1962. 14. A. GRIFFIN, Phys. Letters 17, 208 (1965); and an article to be published in Rev. Mod. Physics. 15. P. C. KWOK, P. C. MARTIN, AND P. B. MILLER, SoIid State Commun. 3, 181 (1965); P. C. 13. C.

KWOK

16.

AND P. C. MARTIN, Phys. Rev. 142,495 (1966). Tver. Tela 7, 3515 (1965) [English transl.: Soviet Phys. Solid State 7, 2838 (1966)]; Yu. A. TSERKOWIKOV, Dokl. Akud. Nuuk SSR 169, 1064 (1966) [English transl.: Soviet Phys. Dokl. 7, 723 (1967)]; L. J. SHIM, Phys. Rev. 156, 494 (1967).

R. N. GURZHI, Fiz.

17. An excellent review of the rather extensive theoretical literature is contained in the article by C. P. ENZ (to be published).

PULSES

18.

IN

DIELECTRIC

183

SOLIDS

R. J. VON GUTFELD AND A. H. NETHERCOT, JR., Phys. Rev. Letters 12, 641 (1964); also in “Proceedings of the Ninth International Conference on Low Temperature Physics,” pp. 1189-92. Plenum Press, New York, 1965.

19. R. J. VON GUTFELD AND A. H. NETHERCOT, JR., Phys. Rev. Letters 17, 868 (1966) 20. J. B. BROWN, D. Y. CHUNG, AND P. W. MATTHEWS, Phys. Lefters 21,241(1966); D.

V. CHUNG, Unpublished doctoral dissertation, University of British Columbia (1966). 21. J. M. ANDREWS, JR. AND M. W. D. STRANDBERG, Proc. IEEE 54, 523 (1966). 22. S. J. ROGERS AND R. J. ROLLEFSON, Bull. APTI. Phys. Sot. 12, 339 (1967). 23. A preliminary report of this work has been published: C. C. ACKERMAN AND R. A. GIJYER, Solid State Commun. 5,671 (1967). 24. F. W. DEWETTE AND B. R. A. NIEJBOER, Phys. Letters l&19(1965). 25. R. E. PEIERLS, “Quantum Theory of Solids,” Clarendon Press, Oxford, 1955. 26. F. W. DEWETTE, L. H. NOSANOW, AND N. R. WERTHAMER (to be published); T. R. KOEHLER, Phys. Rev. Letters 17,89 (1966); Phys. Rev. Letters 18,654 (1967). In the harmonic approxi-

27. 28.

29.

30. 31. 32. 33. 34.

35.

mation, the second derivative of the potential is evaluated at the equilibrium lattice sites to give the spring constant. The spring constant calculated in this way for solid helium leads to imaginary frequencies [Ref. (24)]. The work of these authors leads to a harmoniclike approximation in which the spring constant is related to the second derivative of an effective potential averaged over space with a weighting factor which is the ground-state wavefunction. The phonon spectrum which results from the harmonic-like approximation agrees well with experiment. J. M. ZIMAN, “Electrons and Phonons,” Clarendon Press, Oxford, 1960; see also Ref. (25). J. M. ZIMAN, Can. J. Phys. 34, 1256 (1956); J. CALLAWAY AND H. C. VON BAEYER, Phys. Rev. 120, 1149 (1960). For some resistive relaxation mechanisms exact formulas are available. For example, for mass fluctuation scattering (isotope scattering) an exact expression for the scattering cross-section exists. See also P. G. KLEMENS, Proc. Phys. Sot. (London) A68, 1113 (1955). An excellent review of phonon scattering mechanisms and low-temperature thermal conductivity is contained in the article by P. CARR~THERS, Rev. Mod. Phys. 33, 92 (1961). M. CHESTER,Phys. Rev. 131, 2013 (1963). An extensive discussion of the propagation of temperature pulses in the various regions is given in I. R. A. GUYER, Phys. Letters 19,261 (1965); Phys. Rev. 148,789 (1966). H. JONES,unpublished doctoral dissertation, Cornell University (1967). C. L. JULIAN, Phys. Rev. 137, Al28 (1965). J. CALLAWAY, Phys. Rev. 113, 1046 (1959).

36. B.BERTMAN, H. A. FAIRBANK, B. BERTMAN, unpublished

37. 38. 39.

40.

41.

R. A. GUYER,

AND C. W. WHITE,

Phys.

Rev.

142, 79 (1966);

doctoral dissertation, Duke University (1965). R. BERMAN, C. L. BOUNDS, AND S. J. ROGERS, Proc. Roy. Sot. (London) A289, 66 (1965), hereafter referred to as Rogers et aI. to distinguish it from Bertman et al, Ref. (36); S. J. ROGERS, unpublished doctoral dissertation, Oxford University (1965). R. BERMAN AND J. C. F. BROCK, Proc. Roy. Sot. (London) A289,46 (1965). R. A. GUYER AND B. SARK~SSIAN (unpublished calculations). In this work both the T* dependence of Ref. (38) and a TS dependence were tried for 7~~. No significant difference was found between the two fits. J. A. SUSSMANAND A. THELLUNG, Proc. Phys. Sot (London) 81,1122 (1963). L. P. MEZOV-DEGLIN, Zh. Eksperim. i. Teor. Fiz. 46, 1926 (1964) [English transl.: Soviet Phys.-JETP 19,1297 (1964)]; ibid. 49,66 (1965) [English transl.: Soviet Phys.-JETP 22, 47 (1966); ibid. 52, 866 (1967).

184

ACKERMAN

AND

GUYER

42. R. A. GUYER AND E. M. HOGAN, Solid State Commun. 5, 909 (1967). 43. J. DEKLERK AND P. G. KLEMENS, Phys. Rev. 147, 585 (1966). 44. J. CALLAWAY, Phys. Rev. 122, 787 (1961). 45. E. J. WALKER AND H. A. FAIRBANK, Phys. Rev. 118,913 (1960). 46. The data of Refs. (37) and (45) are taken down to only 1°K. The thermal

conductivity maximum and the region of greatest sensitivity to thechoiceof 7~ occurs below this temperature. 47. Recently, Agrawal has analyzed the 20.2-cml/mole data from Ref. (36). He finds a result intermediate between that of Rogers er al. and Bertman et al.: B. AGRAWAL, Phys. Rev. 162, 731 (1967). 48.

The preliminary analysis of Mezov-Deglin data which was reported in I was carried out using Eq. (lo), which is wrong by a factor of 2. The analysis also employed the Debye temperatures of E. C. HELTEMES AND C. A. SWENSON, Phys. Rev. 128, 1512 (1962). These are now superseded by the results of D. 0. EDWARDS AND R. C. PANDORF, Phys. Rev.

140, A816 49. P. THACHER,

50. 51.

52.

53. 54. 55.

56. 57. 58. 59.

(1965). Phys.

Rev. 156, 975 (1967); also unpublished doctoral dissertation, Cornell University (1965). The Callaway equation does not permit Poiseuille flow to occur. See Ref. (39). By incipient second sound we mean second sound propagation which is not clean. For example, Rogers and Rollefson, Ref. (22), reported “unusual” behavior for their temperature pulses as they approached the thermal conductivity maximum from below. It is possible to interpret their results as some combination of second sound and diffusive propagation. R. J. VON GUTFELD AND A. H. NETHERCOT, JR., IBM Research paper, RC-1267 (1964). B. BERTMAN, H. A. FAIRBANK, C. W. WHITE, AND M. J. CROOKS, Phys. Rev. 142, 74 (1966). Conhex Division, Sealectro Corporation, Mamaroneck, N. Y. Emmerson-Cummings, Inc., Canton, Mass. International Resistance Corp., Boone, North Carolina. Conductive paint SC-12, Microcircuits Co., New Buffalo, Mich. A. I. SHAL’NIKOV, Zh. Eksperim. i Teor. Fir. 41, 1056 (1961) [English transl.: Soviet Phys.-

JETP 14, 753 (1962)]. 60. A. I. SHAL’NIKOV, Zh. Eksperim. JETP 14, 735 (1962)]. 61. As discussed in I, the criteria

i Tcor. Fiz. 41, 1059 (1961) [English transl.: Soviet Phys.-

for Poiseuille flow and second sound are similar. In fact, the observation of Poiseuille flow guarantees that second sound can be observed but not vice versa. 62. In order to establish measurable temperature gradients across a sample of large cross section, very large heat currents must be conducted. These currents are hard to carry away. 63. The aggregate of data obtained in solid helium is similar to that obtained in liquid helium in careful experiments, like that of H. C. KRAMERS, T. VAN PESKI-TINBERGEN, J. WEB=, F. A. W. VAN DEN BURG, AND C. J. GORTER,

Physica

20,743

(1954).

64. There are, in principle, two kinds of second sound, “drifting” and driftless. The second sound observed in solid helium at low temperature is drifting second sound. See Ref. (17). 65. 66. 67. 68.

J. F. D. P.

H. P. 0. C.

VIGNOS AND H. A. FAIRBANK, Phys. Rev. 147, 185 (1966). LIPSHULTZ AND D. M. LEE, Phys. Rev. Letters 14, 1017 (1965). EDWARDS AND R. C. PANDORF, Phys. Rev. 140, A816 (1965). KWOK, IBM intralaboratory research report. This report

anisotropy on the velocity of second sound. 69. R. A. GUYER (unpublished).

deals with the effect of

PULSES

IN

DIELECTRIC

SOLIDS

185

70. D. V. OSBORNE, “NBS Semicentennial Symposium on Low Temperature Physics,” p. 139 (National Bureau of Standards Circular 519 (1952). U. S. Government Printing Office, Washington, D. C. 71. W. J. PARKER, R. J. JENKINS, C. P. BUTLER, AND G. L. ABBOTT, J. Appl. Phys. 32, 1679 (1961). 72. The analysis of steady state thermal conductivity data to determine crystal orientation is carried out in detail in Ref. (42). 73. J. M. ZIMAN, Phil. Msg. 45, 100 (1954). 74. C. HERRING, Phys. Rev. 95, 954 (1954). 75. As the temperature is lowered, 7N becomes longer, thus the SZrN << 1 condition begins to break down. We estimate the corrections to the dispersion relation due to the need to keep higher-order terms [see Ref. (9)] are about 5%. 76. J. JARVI~ AND H. MEYER (to be published in Phys. Rev.); also Ref. (66), for example. 77. F. J. WEBB AND J. WILKS, Phil. Mug. 44, 664 (1953). 78. J. E. Vos, R. EENENGA KINGMA, F. J. VAN DER GAAG, AND B. S. BLAISSE, Phys. Letters 24A, 738 (1967). 79. 0. W. HEYBEY AND D. M. LEE, Phys. Rev. Letters 19, 106 (1967). 80. I. A. STEGUN AND M. ABRAMOWITZ, “Handbook of Mathematical Functions,” p. 355. Dover, New York, 1965.