Effect of hydrostatic pressure on the elastic properties of some rare earth-iron laves phase compounds

.I Phys Chem So/Ids Vol PnntcdmtheUSA

46, No

2, pp

157-163,

1985

OOZZ-3697/85 $3 00 + 00 0 1985 Pergamon Press Ltd

EFFECT OF HYDROSTATIC PRESSURE ON THE ELASTIC PROPERTIES OF SOME RARE EARTH-IRON LAVES PHASE COMPOUNDS H. KLIMKER and Y. GEFEN Physics Department, Nuclear Research Center, Negev, POB9001, Beer Sheva, Israel

and M. ROSEN Matenals Scrence and Engmeenng Department, The Johns Hopkms Umverslty, Baltimore, MD 21218, U S A (Recerved 30 January 1984, aceepled 3 May 1984)

Abstract-The pressure dependence of the Young’s and shear moduh of RFe2 (R = Sm, Gd, Tb, Dy, Ho and Er) has been determmed at room temperature m the pressure range between 0 and 1 GPa The elastic moduh of GdFe*, DyFez, HoFe* and ErFe* show a moderate Increase wtth mcreasmg hydrostatic pressure However, the elastic moduh of SmFez and TbFe2 exhlbrt an mmally drastic increase followed by a high, and linear, pressure dependence From the pressure and temperature denvatlves of the elastic moduh of these RFe2 Laves phase compounds the equations of state and the Grunelsen parameters have been denved The vanatlon of the elastic properties with hydrostatic pressure 1s compared with the effect of magnetic fields The anomalous behavior of SmFe2 and TbFez 1s dIscussed

1. INTRODUmION

The magnetic behavior of the cubic rare earth-iron Laves-phase compounds (type MgCu,) has been extensrvely investigated [l-4] Then Cune temperatures are relattvely high [2] and the amsotropy properties are quote remarkable [5, 61. A vanety of different magnetic structures IS displayed [I, 71, and magnetocrystallme amsotroptes were found m several compounds (e g DyFeJ to be as high as that of some umaxtal magnetic structures [8]. The magnetostnctton at room temperature m TbFez and SmFe2 IS of the same order of magnitude as that of rare earth metals at low temperatures [3, 8-101. Sound veloctty measurements for the determinatron of the elastic moduh, adiabatic compresslbthty and Debye temperature of these compounds exhibited existence of substanttal magnetoelastic effects [4, 10, 1 l] TbFez and SmFel show remarkable lattice softerring. Furthermore, applied magnettc fields strongly affect the temperature dependence and the absolute values of the elastic moduh Particularly noteworthy IS the AE effect m TbFe* and SmFez AE 1s the difference between the magnetized and unmagnetized Young’s moduh This magnetomechamcal loss 1s associated with the vibration of the magnetic domain walls m the ferromagnettc phase under the influence of a propagating high-frequency stress wave. The applied stress alters the local magnetization through the magnetostncttve coupling The observed values of 42 and 55%, respecttvely, for the AE effect m SmFez [lo] and TbFeZ [4] at 300 K, a magnetic field of 25 kGe and 10 MHz ultrasonic frequency, are considered to be extremely high. The AE effect can be quahtatrvely deduced from the expression of the

total energy that mcludes the tsotroptc exchange magnetic energy, the magnetocrystalhne anisotropy energy, the elastic energy of the nonmagnettc lattice and the magnetoelastic energy that depends on the devtatton from sphenctty of the 4f electronic shell, and thus vanes wtth spm dtrecnon. The grant AE effect IS a result of the mteractton between the magnetoelastic and the magnetocrystalhne anisotropy observed in these compounds. The tmportance of the detennmatron of the elastic moduh as a function of temperature and magnetic field was demonstrated in the study of the stability of magnetic structures m RlR2Fe2 and RlR2Cq systems [5, 6, 121, where RI and Rz denote different rare earth elements. The ObJective of the present work was to study the pressure dependence of the elasuc and magnetoelasnc properties of RFe2 compounds (R = Sm, Gd, Tb, Dy, Ho and Er). From the pressure and temperature denvanves of the longitudinal and shear-wave sound velocities, the equations of state of these compounds and the Gnmetsen parameters were determined.

2. EXPERIMENTAL

The binary Laves compounds RFe2 (R = Sm, Gd, Tb, Dy, Ho and Er) were prepared by arc-meltmg 99 9% pure rare earth and 99.99% pure iron under an argon atmosphere. The heavy rare earth-non arccast samples were annealed m evacuated quartz capsules at 900°C for 200 h. Sm-Fe was annealed at 700°C for 500 h Metallographic, X-ray and electron microprobe analysis m~cated that after annealing the samples were homogeneous and essentially smglephase, foreign phases never amounting to more than

157

158

H KLIMKER,Y GEFFN and M ROSEN

5% Lattice parameter determmatlons, using DebyeScherrer diffraction patterns, yielded results m good agreement with previously published data Flat and parallel disks. 5 mm m diameter and about 4 mm thick, were prepared for ultrasonic measurements. The parallehsm of the sample faces was better than two parts m IO4 The thickness of the samples was determmed to withm ?5 X 10e4 mm The longttudmal and transverse sound-wave velocIties were measured by means of an ultrasonic pulse technique at a frequency of 10 MHz Pulses of 0 5ps duration were generated by an Arenberg PG-650C pulse generator Reflected pulses were amplified by an Arenberg WA-600 wide-band amplifier An automatic ultrasonic pulse analyzer (own design and construction) determined the transit times between any two arbltranly selected rf cycles m the reflected successive echoes A locking pulse system followed the zero cross of these cycles, and generated a time gate which monitored an analog time-measunng system The transit time resolution was 2 ns The absolute values of the sound-wave velocltles of the samples were determined employing a frequencyvanatlon method The estimated expenmental error m the absolute values of the elastic moduh is less than 0 4%, the relative point-to-pomt values are better by a factor of 10 Specimen densities were determined by means of a fluid-displacement technique, using monobromobenzene The devlatlon from the calculated X-ray densities [ 131 was less than 0 5% The hydrostatic pressure system [I41 consisted of a pnmary Ammco pump (200 MPa), a pressure intensifier with a ratio of 1 10, and a pressure cell (20-mm 1d and 100-mm o d ) of Grade 300 Maragmg steel heat treated for maximum ultimate tensile strength The liquid pressure-transmlttmg medium m the cell was a mixture of kerosene and 011 A Harwood 12-H high-pressure hne was used to separate the pressure intensifier from the cell m order to permit lmmerslon of the cell m a thermostatic bath The ultrasomc coaxial feedthroughs, as well as those for the thermocouples and for the mangamn pressuremeasunng cod were stainless steel alumina-insulated, copper coaxial cables The hydrostatic pressure determination was camed out by means of the mangamn coil and a Cary-Foster pressure measunng bndge calibrated by a Harwood dead-weight tester agamst the freezing point of mercury at 0°C The hydrostatic pressure was increased, and decreased, m steps of 50 MPa, up to I GPa The accuracy of the pressure measurement was within +I MPa The thermostatic bath, mto which the high-pressure cell was immersed. was controlled to within +O 1°C of the nominal 22°C In order to reach temperature equlhbratlon, It was necessary to wait for IO mm between the apphcation of pressure and the ultrasonic measurement 3. ANALYSIS OF ULTRASONIC

DATA

The pressure dependence of the iongtudmal transverse wave velocltles V, and V,, respectively

been determined up to 1-GPa pressure The velocities were calculated consldenng the changes occumng m the acoustic path length due to apphcatlon of hydrostatic pressure VP = Vg(/p/&), where i0 and lp are the path lengths at atmosphenc and at pressure p, respectively, Vg 1s the approximate velocity value at pressure P computed with the ongmal acoustic path length I,, and VP 1s the corrected sound velocity The pressure dependence of the ratlo I,/lp was determined from Cook’s relation [ 151 l+A

l,llp = 1 + -

PO

p dp s 0 3(V;)2 - 4(VJZ

where p. 1s the density at atmosphenc pressure A 1s the adiabatic-to-isothermal correction

and

(A = P2T&IC,~d CD ISthe heat capacity at constant pressure, P 1s the volume thermal expansion coeffiaent, T ISthe absolute temperature, BH ISthe bulk modulus at atmosphenc pressure A slmdar procedure was used to correct for the pressure dependence of the specimen density The heat capacity C, for HoFez and ErFe2 [ 161 and the thermal expansion coefficient p for ErFe2 [ 171 were taken from the literature Slmllar corrections were applied to the other rare earth-iron Laves compounds studied m this research The estimated error m the calculation of the elastic moduh of the compounds using the published data of C, and p for HoFez and ErFe2 IS less than 0 0 1%

4. RESULTS

AND DISCUSSION

The pressure dependence of the Young’s (E), shear (G) and bulk (B)moduh of SmFe, at room temperature m the pressure range between 0 and 1 GPa 1s shown m Fig 1 The elastic moduh of the compounds

I

I

I

I

I

02

04

08

08

10

PRESSUREP (GPa) and has

Rg 1 Vanation of the elastx moduh (Young’s, shear and bulk) of SmFe, as a function of hydrostatic pressure

159

Effect of hydrostatic pressure on elastic propertles

75 -

PRESSURE

P (GPa)

FIN 2 Vanatlon of the Young’s elastic modulus of RFez (R = Gd, Tb, Dy, Ho and Er) as a function of hydrostatic pressure

I

I

I

I

02

04

03

08

PRESSURE

10

P(GPa)

Fig 4 Vanatlon of the adiabatic bulk modulus of RFez (R = Gd, Tb, Dy, Ho and Er) as a function of hydrostatic

pressure GdFel, TbFeZ, DyFez, HoFeZ and ErFez are shown in Figs. 2-4. In the absence of magnettc or crystallographic phase changes the elastic moduh are expected to mcrease wtth pressure wtth a moderate slope. Such a normal behavior 1s displayed by GdFe2, DyFeZ, ErFez and HoFeZ (Ftgs. 2-4) A monoatomically linear increase m the moduh wtth mcreasmg hydrostatic pressure, up to 1 GPa, was found. The pressure denvattves of the elastic moduh of GdFer, DyFeZ and HoFez are nearly identical, whereas the slope of ErFez m E and G deviates from hneanty m the pressure range between 0 and 1 GPa thus exhtbttmg a constant pressure dependence of the bulk modulus (Fig. 4) up to 0 3 GPa, followed by an mcrease in B wtth increasing pressure

i/:-; 02

0.

PRESSURE

06

08

P (GPd

Rg 3 Vanafion of the shear elastic modulus of RFez (R = Gd, Tb, Dy, Ho and Er) as a function of hydrostatic presSllre

The pressure dependence of the elastic moduh of SmFer (Fig. 1) and TbFe2 (Figs. 2-4) is completely dtfferent. An mmally drastic increase m slope IS abserved, followed by a relattvely high, and linear, pressure dependence of the elasttc moduh In neither of the mvesttgated Laves phase compounds was saturatton of the values of the elastic moduh achieved at pressures up to 1 GPa. It appears that the linear dependence of the elastrc moduh wrth pressure extends well beyond 1 GPa. It 1s noteworthy to compare the “magnetic” AE of these Laves-phase comeffect (EH - E~/Eo&,, pounds [4-lo] with the hydrostatic AE effect (E, - Eo/Eo)hydm obtained m the present study Results are summanzed m Table 1. The comparison between and (AE)hydm can be made m a quahtattve (AE),,, manner only The magnetic AE effect was determined by comparmg the elastrc mod& at zero and saturatton (25 kOe) fields, respectively [4, lo] The hydrostatic AE, presented m Table 1, compares the change m the elastic moduh at pressures m the range between 0 and I GPa. As was mentioned previously, the elastic mod& did not achieve saturation at the maximal pressure However, tt 1s instructive to make this comparison Both (AE),,, and (AEhydro are prommently higher m SmFez and TbFer, for all elastic moduh, 1.e E, G and B. As may be noted, these two compounds possess very high magnetostnctton constants, hs. For GdFer , DyFe2, HoFez and to some extent for ErFez, (AE),, is of the same order of magnitude as (AE),,,,, . A am&r relation 1s found between the elastic effects of the other moduh, 1 e. AG, AB. However, the magnetic field effects, e.g. (AE),, m SmFez and TbFez are much greater than the correspondmg hydrostatic pressure effects, namely (A E)hydro One obvtous reason for this dtscrepancy 1s the fact that the maximal applied magnetic field (25 kOe) achieved saturatton

160 Table

I

Magnetostrlctlon

H

KLIMKER,

constants.

magnetic

Y GEFFN and M field

RFez

effects.

(at

SlllFC2 GdFe2

MAGNETOSTRICTION CONSTANT X sx 10-6 -1,560 0

pressure

effects

on

the

elastic

moduh

of

compounds

MAGNETIC

COUPOUND

and

ROSEN

AE;$”

FIELD

25

EFFECT

HYDROSTATIC

KOe)

AG;$O

EFFECT A”;‘;”

AE;hl@

PRESSURE

(at

1 GPs)

AG”/GO (%a)

66 7 1 1

600

17 0

27 0

($1 12 1

01

28

24

45

660

48

22 4

23 2

22 4

18

28

=0

27

25

36

345

08

10

20

24

22

38

-239

60

68

17

31

37

2?

TbFe2

2,630

57 0

DyFe2

433

HoFe2 ErFe2

m the elastic moduh, whereas the highest hydrostatic pressure apphed m this research (1 0 GPa) did not raise the values of the elastic moduh to saturation Table 1 indicates that the magnetic field effect on the bulk modulus (AB,/B,) of the SIX Laves phase compounds mvestlgated IS about an order of magmtude smaller than (AE),,, and (AG),, In contrast, the hydrostatic pressure effect on the bulk modulus (ABdB,) m these compounds IS almost identical with the effect on the correspondmg moduh, e g (A-%ydro The reason for this difference may be due to the mtnnslc nature of the two externally applied fields Both tend to harden the lattice of the rare earth-iron Laves phase compounds, [4, lo] and Figs l-4 The elastic hardenmg of the lattice, due to increasing magnetic fields, IS related to the operation of a magnetoelastlc interaction as a result of a coupling between lattice strams and onentatlon of the magnetic moment. Therefore, the easy direction of magnetlzatlon becomes strain dependent The rare-earth localized 4f magnetic shell strains the lattice m the course of a spm reonentatlon of any kmd The lattice hardening, or softening, m the RFez compounds, bemg dependent on the strength of this magnetoelastic interaction finds its expression m the anomalously large shear magnetostnction constants, XIII, as 1s observed m SmFez and TbFe2, Table 1 This lattice softening occurs when the magnetoelastlc interaction and the magnetocrystalhne amsotropy energy are of the same order of magnitude Then substantial vanatlons m the values of the elastic moduh are expected Indeed, large (AE),,,, , (AG),,, and (As),,, have been observed (Table 1) On the other hand, m GdFez, DyFez and HoFeZ where the magnetocrystalline amsotropy energy IS substantially greater than the magnetoelastlc interaction, there 1s no lattice softening Apphcatlon of an external magnetic field can also weaken this magnetoelastlc interaction between magnetic moments and lattice strains by rotating the magnetic moments towards the direction of the apphed magnetic field Consequently, the elastic softening will be dlmmlshed thus substantially varymg the elastic moduh

1 1

73

A@/60

In the case of an externally apphed hydrostatic pressure, the relevant strams have no stram component but are volumetnc in nature The strains are homogeneous and lsotroplc, where c,, = L, , = t,, are the components of a compressive hydrostatic stram Therefore, m contrast to the effect of an external magnetic field where the mteractlon IS through the shear magnetoelastlc constant, corresponding to the magnetostnctlon constant, X, , , , the effect of pressure IS determmed by the existence of a volume strain magnetoelastlc constant that affects all the elastic moduh, mcludmg the bulk modulus B The threedimensional effect of hydrostatic pressure appears to reconcile for the dlscrepancles between the observed with (AB).,,,, , as values of (A Bhydrost compared shown m Table 1

Equation of state The equation of state of solids at very high pressures, m the range of 100 GPa IS usually computed from shock wave expenments [ 181 The ultrasomc equation of state, based on the value of the isothermal bulk modulus and Its temperature and pressure denvatlves, IS determined by measurmg the sound wave velocltles as functions of pressure m a more accessible pressure range (0 1 GPa) [ 191 than shock compression measurements The ultrasonic equation of state, extrapolated to very high pressures, m the lOO-GPa range, agrees to within a few percent with the shock compresslon equation of state Thus, the volume change of a solid at very high pressures can be estimated from ultrasomc data m the low-pressure range Two semi-empmcal equations are commonly employed, Murnaghan’s [20] and Birch’s [2 1) equations of state Murnaghan’s equation 1s based on the assumption that the adiabatic bulk modulus 1s linear with pressure up to the value of the extrapolating pressure From this follows Murnaghan’s equation of state V/V,, = (1 + B'm/Bm)-"BTo, where V and V. 1s the volume of the sohd at pressures

I61

Effect of hydrostatic pressure on elastic properties P and PO (atmospheric), respectively, and Blo and B;D are the isothermal bulk modulus and its pressure

(a&&P),

derivative at atmospheric pressure. Birch proposed that the stram energy be expanded m terms of linear strams By retaining terms up to the thud power of strain, Birch’s equation of state becomes P = ;B,[(

V,/V)“’

dVn, dP = 1 + A + 2p, V, ---

4 dV,c, . 3 Vm dP >

In order to obtain the pressure denvative of the tsothermal bulk modulus at atmosphenc pressure (aB&W), , which appears m both Murnaghan’s and Birch’s equations of state, the followmg relationship due to Overton [22] is used:

- ( V,/V)5”] {I - (3 - ~Bro)W’4’)2”

-

II}

-a&

(ap1 7

Both Murnaghan’s and Birch’s equations of state assume that no phase transformation occurs within the extrapolation region It follows, from Murnaghan’s equation of state V/ V0 = [1 + (aB70/aP)/B~]-“(JBdBmIaP), that (V/V,) can be evaluated provided BTo and @B&W) can be expenmentally determined Determmation of the longitudmal and shear wave velocities V, and V, by ultrasonic measurements is performed under adiabatic, rather than isothermal, conditions. Therefore, the followmg relation should be applied B70

=

Bal(l

+

A),

where B,. 1sthe adiabatic bulk modulus at atmospheric pressure, and A is

In the room-temperature range the coefficient of thermal expansion @can be assumed to be constant, and its temperature denvative (&3/dT), vamshes. The temperature denvauves of the adtabattc bulk modulus (aBJaT), IS obtained from the slope of the temperature dependence of the bulk modulus, at constant pressure [23]. The temperature denvatrve of the isothermal bulk modulus, applying the correctton terms to the denvattve of the adiabatic bulk modulus 1s

(%),.=ib(X),.

A = 8=TB,olC,,~,po, where /3 IS the volume coefficient of thermal expansion, T absolute temperature, C,, is the heat capacity

---

at constant pressure and p0 is the mass density BJo IS obtained from velocity measurements in the following manner.

A

(I+A)2

Ro T

B,o = PO(V: - ; V:) The pressure derivative of the adiabatic bulk moclulus, (dB,o/aP)7, may be obtained from the following relationship, using the appropnate denvatives, with respect to pressure, of the longitudinal and transverse sound-wave velocities, dV&dP and dV,/dP, respectively

Since the correction IS less than 0 2’70, it can be assumed that @B&IT), z (aB,/aT)p. Table 2 shows the values of the ultrasonic velocities at atmosphenc pressure Vm and V,, their pressure denvattves, the computed adiabatic and isothermal bulk moduh Ba and Bm and the pressure denvattve

Table 2 Ultrasomc velocltles, adiabatic and lsothennal bulk moduh at atmosphenc pressure, and their pressure denvatlves, of RFe2 compounds

I

COMPOUND

SmFe2

L (103m s-1)

bYe/OP (103,

s-

I03

A s*GPa

B;

(GPa)

3 378

1.415

2.25

67.3

bBWbP (GPa/GPal 111.0

GdFe2

4.045

2 159

0 048

(19 0

TbFe2

3.509

1.748

1 000

76.0

OyFe2

4.060

2.165

0.049

93.6

3.86

H0Fe2

4.074

2.197

0.048

94.1

3.92

ErFe2

3.983

2 121

0.049

93.4

4 01

3.04 43.1

H KIIMKER, Y GEFEN and M ROSEV

162

0

PRESSURE P (GPa) Fig 5 Equations of state of RFe: (R = Cd, Dy. Ho) culnc Lam

of the adiabatic

bulk modulus

at room temperature

(a&/a% The very large values of (aB,/aP), of SmFez and TbFe2 are related with the anomalously low values of the elastic propertles of these compounds along with their anomalous pressure dependence The pressure denvatlves of the adlabatlc bulk moduh of GdFez, DyFez HoFez are very small That of ErFez was found to be zero Figure 5 displays the equation of state ( V/Vo) vs (P) of GdFe,, DyFez and HoFe, compounds accordmg to Murnaghan’s relationship The extrapolation IS shown up to pressures of 1000 GPa, m the pressure range of shock compression expenments The ratio ( V/Vo) increases as the atomic number increases In these rare earth-Iron compounds the lattice parameter follows the lanthamde contractlon, I e the decrease of the lattice parameter with Increasing atomic number, m accordance with the decrease of the lomc radius The result of this behavior IS that as the lattice parameter decreases the mass density of the compound Increases and the matenal becomes less compressible, I e higher bulk modulus The anomalous, non-linear, behavior of the bulk moduh of SmFez and TbFe2 (Figs 1 and 4) does not allow computations of the equation of state of these compounds smce the vanatlon of the adlabatlc bulk modulus IS not linear with pressure, as required by Murnaghan‘s relatlonshlp However, an approxlmatlon can be made The bulk modulus of these two compounds IS very low at atmosphenc pressure (Table 2) As pressure increases (Fig 4), the bulk modulus of TbFe* increases rapldly towards the value of the other compounds At a pressure of 1 0 GPa the bulk modulus of TbFeZ IS close to that of GdFep The ratlo V/V0 at 1 0 GPa IS 0 998, slmllar to the magmtude of this ratio for GdFe2, DyFez and HoFe, The computed extrapolation for very h:gh pressures of TbFez falls between the values for GdFe, and DyFe2 For SmFez, V/V, = 0 987 at 1 0 GPa A

phase: compounds

possible explanation for the lower volume-change ratlo IS that Sm IS a hght rare-earth metal, thus the lattice parameter of SmFe* , followmg the lanthamde contractlon, IS greater than that of GdFez

The Grunelsen parameter The Grunelsen parameter, Y, IS of crucial Importance m computation of equations of state The Grunelsen parameter IS generally determined using the relation denved from Mle-Grunelsen equation of state Ythcrm

=

PB,

POC,>

Another way to compute y IS from the sound wave velocltles and then pressure denvatlves Slater [24] assumed that Ponson’s ratlo IS independent of pressure, and that the pressure denvatlves of all vlbratlonal modes are equal From these assumptions Slater obtamed the followmg expression

ysr=_l+i 6

dB_m 2 (

ap,1

Followmg Anderson [20] the Grunelsen parameter m a polycrystalhne matenal can be evaluated from the sound velocity measurements by means of the two modes of vlbratlon, namely the longtudmal ye, and the transveme one Y,, usmg the followmg expresslons

and

y,z;+Bm !!A

(>

vt ap

The acoustic Grunelsen as y. = tcr, + 2YJ

parameter

T

yU IS defined

[25]

Effect of hydrostatic pressure on elastic propertres

163

Table 3 Grunelsen parameters of RFe2 compounds

Y, 45.20 1.400 21.72 1.463 1.44 1.482

I r,

I

Y SL

13 7

24.15

55.35

1014

1.14

165

22.54

22 26

21.36

1012

1.162

1 76

0 92

1 09

1 79

1.037

1.185

1.835

Table 3 gtves the computed values of the Grunetsen parameters for the compounds SmFe2, GdFez , TbFe2, DyFez, HoFez and ErFer The thermal Grimetsen parameters ythennalare gtven for HoFez and ErFez. Only for HoFez a good agreement IS found between the acousttc and thermal Grunetsen parameters, 7a and ytiermal,respecuvely. Slater’s -yr is generally greater than the acousttc parameter y.. Conststent wtth the behavior of the elasttc moduh as a function of hydrostatrc pressure, the Grunersen parameters of SmFez and TbFez are anomalously large.

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8 Clark A E , Belson H S and Tamagawa N, Phys Lett _. A 42,_160 (1972) 9 Koon N , Schmdler A and Carter F , Phys Lett A 37, 413 (1971) 1o Rosen M , Khmker H , Atzmony U and Dane1 M P , Phys Rev B 9,254 (1974) 11 Clark A E, Belson H S and Strakna R E, J Appl Phys 44,2913 (1973) Khmker H, Dane1 M P and Rosen M , J Phys l2 Chem Soltds 41, 215 (1980) 13 Buschow K H J and Van Stapele R P, J de Phys Colloq 32, Cl-672 (1971) 14 Gefen Y and Rosen M , J Phys Chem Soltds 37,669 (1976) I5 Cook R K ,J Acoust Sot Am 29,445 (1957) 16 German D J , Butera R A, Sankar S G and Gschnelder K A, Jr, J Appl Phys 50,7495 (1979) 17 Buschow K H J and Mledema A R, Soled St Commun 13, 367 (1973) 18 Altshuler L V , Sov Phys Usp 8, 52 (1965) 19 Swanson C A, J Phys Chem Solrds, 29, 1337 (1968) 20 Mumaghan F D , Proc Nat1 Acad SCI 30,244 (1944) 21 Birch F , J Geophys Res 57, 227 (1952) 22 Overton W L , J Chem Phys 37, 116 (1962) 23 Anderson 0 L , J Phys Chem Sohds, 27, 547 (1966) 24 Slater J C , Introductronto Chemrcal Phystcs, p 239 McGraw-Hill, New York (1939) 25 Schuele D E and Smith C S , J Phys Chem Soltds, 25,801 (1964)