Elastic constants, acoustic mode vibrational anharmonicity and Grüneisen parameters of hexahalometallate crystals

0022-3697/85

J, Phys. Chem. Solids Vol.46, No. I I. pp. 1235-1242. 1985 Printed in Great Britain.

53.00 + .OO

e 1985Per&rdmon PrexLtd.

ELASTIC CONSTANTS, ACOUSTIC MODE VIBRATIONAL ANHARMONICITY AND GRiiNEISEN PARAMETERS OF HEXAHALOMETALLATE CRYSTALS and J. F'ELZL

N. WRUK

lnstitut fir Experimentalphysik IV, Ruhr Universitiit, 4630 Bochum, West Germany G. A. SAUNDERS and Tu HAILING School of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom (Received 17 September 1984; accepted in revised form 11 April 1985)

Abstract-The room-temperature elastic constants of a number of hexahalometallate A&fXe single crystals [K2SnC16,K2ReC16,(NH&SnC&, (NH&SnBr6, (NH&SiF6, Rb*SnBq , K2SeBrs, (NH&TeBr6, K$tBr6 and (NHJ2PtBr6] have been measured either by Brillouin scattering or by the ultrasonic pulse echo overlap technique. Refractive indices have also been determined. These antifluorite structure compounds contain large MXa- ions and the interionic spacings are much greaterthan those of the alkaline-earthfluorite structure halides: their elastic stiffnesses are correspondingly smaller. Hydrostatic pressure derivatives of the elastic stiffness constants have been measured for K$SnC&, (NH&SnBr6 and (NH.,)rSnC&and are found to be positive; there is no marked softening of the long-wavelength acoustic-phonon modes at room temperature. The vibrational anharmonicities of these long-wavelength modes are discussed in terms of the acousticmode Griineisen parameters, which are compared with the thermal Grtineisen parameters. For K2SnC16a mean of optic- and acoustic-mode Griineisen parametersis shown to correlatewell with the thermal Griineisen parameter.

1. INTRODUCTION

The elastic constants of a number of hexahalometallates have been measured, several for the first time. In addition the hydrostatic pressure dependences of the velocities of ultrasonic waves propagated in K&C&, (NH&SnBr6 and (NH4)zSnC16 have been measured at room temperature. These results are used to obtain the hydrostatic pressure derivatives (aCI,/tW)p=O of the second-order elastic stiffness constants: they complement previous measurements made on several other isomorphous crystals: K2SnC16 [ 1, 21, (NH4)$SiF6 [I, 21, Ni(N03)z-6NH3 [2], Mg(Br03)z-6Hz0 [2]. The pressure derivatives (K,J/dP)p=O define the effect of hydrostatic pressure on slopes of the acoustic-mode phonon-dispersion curves at the Brillouin-zone centre and the vibrational anharmonicity of these modes, which is discussed here in terms of mode Grtineisen parameters. Many hexahalometallates of the general type A2A4X6 (A = K+,NH:, . . . , M = Sn, Re, Si, . . . , X = halogens), including the compounds in this study, crystallize in the face-centred-cubic lattice of the antifluorite structure (space-group Fm3m). A characteristic feature of a number of such compounds is that they undergo structural phase transitions, at least some of which are linked with softening of rotary phonons of the MX, octahedra [3-71. The temperatures at which structural phase transitions have been observed in several of these compounds are listed in Table 1. A number of transitions have been reported in other crystals of this type, transition temperatures for some of them

can be found in Ref. [8], which points out that phasetransition temperatures of A2MX6 compounds can vary extensively with composition, although the origin of this is not established. The phase transitions can be attended by elastic anomalies [ 1, 14, 151 which originate from coupling between a critical optical-Qhonon mode and the elastic strain. The A&fX6 compounds constitute ideal model systems for investigations of the elastic effects of such coupling. In the vicinity of the upper phase transitions in K2SnC16 [ 1, 141 and (NH&SnBr6 [ 151 the elastic stiffness in the cubic phase is reduced by anharmonic coupling between the acoustic and soft rotary modes. Therefore elastic constant measurements can provide useful physical insight into the lattice dynamics and its relation to transition mechanism in these compounds. Different kinds of displacive phase transitions take place in these antifluorite salts depending upon which particular MXarotary mode softens [3]; interpretations of the spectrascopic information and of the phonon-dispersion curves for a given crystal usually depend upon lattice dynamical models based upon parameterised force fields between rigid ions [3, 16-181. Thus extensive lattice-dynamical information is needed as a foundation for understanding the behaviour of the hexahalometallates: the present work provides details of the elastic behaviour and the vibrational anharmonicity of the long wavelength acoustic modes for several of these compounds. The elastic anomalies induced by coupling with softening optic phonons in those crystals which undergo phase transitions should influence the

1235

1236

N. WRUK et al.

Table 1. Phase-transition

temperatures metallates

of some hexahaloReferences

TcW

I'31

262,255 I1 I, 103, 76, I1 145, 124 169, 143, 137, 105,78 59 214, 194 249,22 I, 209 no transition no transition no transition

mode Griineisen parameters. For KzSnCls it has been shown [ 191 that the thermal Griineisen parameter yth (=pV&/C,) shows an inverse X-point behaviour which is directly associated with the soft phonon contributions to the specific heat (C,). One objective of the present study has been to find out whether the long wavelength acoustic-mode Griineisen gammas for several other A&IX6 compounds in the high-temperature cubic phase contain any indications of incipient lattice instabilities.

2. EXPERIMENTAL

Single crystals were grown from the crystals were IO MHz. When

RESULTS

of a number of hexahalometallates aqueous solutions. In some instances large enough for ultrasonic studies at small crystals only were grown, the

elastic constants were measured by Brillouin scattering (scattering angle 0 = 180”). The results, collected in Table 2, include the first measurements of elastic properties for a number of hexahalometallates. The results obtained in a number of previous studies of the elastic constants of hexahalometallate crystals are also given for comparison in this table; the present data agree reasonably well with the earlier results when available. Elastic constants obtained here by both Brillouin scattering and ultrasonic wave velocity measurements were found to be in reasonable agreement within experimental error. The refractive indices, also determined here (Table 2), fall into the range normally expected for ionic crystals. The temperature dependenccs ofthe elastic constants have also been measured, the temperature coefficients at 300 K being given in Table 3. For those compounds for which large single crystals were grown, the effects of hydrostatic pressure upon the velocities of enough ultrasonic modes were measured to enable determination of the initial slopes of the hydrostatic pressure derivatives of the elastic constants. Results are compared in Table 4 with those of CaFz [25] and Ni(N0&.6NHJ [2]. The pressure derivatives B,, = (B,,f) of the thermodynamic secondorder elastic constants (261 are also given in Tabie 4. Three combinations of third-order elastic constants (TOEC) can be obtained from the hydrostatic pressure derivatives of the elastic constants; these are also included in Table 4. For K2SnC16 complete sets of the TOEC have been published elsewhere [ 191as a function of temperature down to the upper-phase-transition temperature of - lO.S”C.

Table 2. Elastic constants of hexahalometallates at room temperature. B = (C,, + 2Cj2)/3. Units: density

Abbreviations: C’ = (C,, - C&2, bulk modulus in kg/m3, lattice constant in nm, elastic constants in GPa, compressibility K in I/TPa. source of elastic

Lattice

Refractive index

Density

constant

C,,

Cl2

Cu

C

B

x

K&C16

I.657

2710

0.9986

KzReCls

1.788

3340

0.984

WLkSnCb

1.690

2410

I .OO60

1.861

3500

I .059

1.369

12001

0.8395

19.7 19.7 19.23 15.6 19.0 19.12 22.8 23.5 18.25 22.0 21.9 21.7 18.4 18.8 18.7 18.43 21.7 21.5 21.80 19.0 23.2 17.6 21.6 22.1 9.265 90.4 164

12.0 11.4 11.17 10.2 11.5 11.17 12.8 12.9 8.84 10.6 10.6 9.5 10.0 11.1 9.7 9.84 7.8 7.2 7.86 10.4 13.5 9.9 12.0 11.6 8.6 40.6 44

8.1 8.4 8.37 8.7 8.4 8.30 8.5 8.4 6.95 9.7 9.7 10.6 8.4 8.2 7.3 6.96 7.5 6.9 6.96 8.4 9.3 7.0 8.5 9.0 0.884 25.3 34

3.85 4.2 4.03 2.7 3.8 3.98 5.0 5.3 4.7 1 5.7 5.7 6.1 4.2 3.9 4.5 4.30 7.0 7.2 6.97 4.3 4.9 3.8 4.8 5.3 0.333 24.9 60

14.6 14.2 13.85 12.0 14.0 13.82 16.1 16.4 11.98 14.4 14.4 13.6 12.8 13.7 12.7 12.7 12.4 12.0 12.57 13.3 16.7 12.5 15.2 15.1 8.82 57.2 84

68.6 70.4 72.16 83.3 71.4 72.36 62.1 60.9 83.4 69.6 69.6 73.5 73.0 73.0 78.7 78.7 80.4 83.3 77.10 75.2 59.9 80. I 65.8 66.2 113 17.5 11.9

Crystal

RbSnBr6 K&eBre (NH.kTeBrs KIPtBrs (NHdkPtBrs Ni(N01)2. 6NH1 fJ& C&l

I.825 2.15 2.17 2.11 2.1

4250 3790 3360 4660 4265 1475

3180

I .064

1.0363 1.0728

1.0293 1.037

1.098 0.6196 0.5462

&Kg

2.10 2.0 2.08 3.22 2.24 2.09 1.70 1.58 I .48 I .70 I .70 I .74 2.1 2.1 1.62 I .62 I .07 0.96 1.0 1.92 1.89 I .84 1.77 1.70 2.65 I .02 0.57

constant data

Present work 1141 El PO1 t11 t211 Present work 1141

1221 Present work [I41 1231 WI Present work 1201

1221 Present work

111 PI Present work Present work Present work Present work Present work t21 (241 1241

Elastic constants, anhannonicity

and Griineisen parameters

1237

3. DISCUSSION

mmoo-rwor-099c?o?o?9?N.c!

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The elastic stiffness constants of the antifluorite structure hexahalometallates are several times smaller than those of the fluorite structure alkaline-earth fluorides (Table 2). This can be readily understood qualitatively. In both types of materials the binding forces are probably dominantly ionic. However, due to the large size of the octahedral MXE- anion, the interionic separations are much larger in the hexahaIometaIlates; for example, the lattice parameter (Q, = 9.986 X lO-‘o m) of K2SnC16 is much greater than that (Q, = 5.462 X lo-” m) of CaF2. Hence the Coulombic interactions in the hexahalometalles are much smaller than in the alkaline-earth fluorites, so that the elastic constants are correspondingly smaller. A rough estimate of the differences to be expected in the elastic stiffness due to the difference in lattice spacing can be obtained from the Madelung contribution to C,, which for the fluorite structure is given by 3.27e2/2ri (where r. = $6); this implies that this dominant term in CII should bk an order of magnitude smaller for K2SnC16 than for CaF2-consistent with the relative experimental magnitudes of C, , in these two materials. The elastic constants of CaF2 have been calculated [27, 281 using a rigid-ion central-force model with Coulomb forces and short-range repulsive forces of the form xi” between Ca-F and F-F. Although the calculated values of CII and Cl2 were found to be in good agreement with the experimental values, the Cauchy divergence C12-Ca was found to be only about a third of the experimental quantity. This discrepancy may be due to neglect in the model of covalent and many-electron forces [28]. We find that this rigid-ion model does not give a satisfactory fit to the measured elastic constants of the hexahalometallates, a failure which can be ascribed to neglect of the large polarisability of the MXa- group. Inspection of the elastic stiffness constants of the hexahalometallates in Table 2 reveals the striking feature that they have closely similar values for crystals comprised of a wide range of elements. This can be readily understood on the basis of an ionic model of binding which in the simplest case can be considered to comprise a Madelung attractive energy term and a repulsive term. In general in crystals the second-order elastic constants are determined largely by the Madelung term which provides an order of magnitude greater contribution, to the bulk modulus for instance, than does the repulsive term in the lattice potential. The Madelung contribution to a second-order elastic constant depends upon the ionic charge (2e in the case of these hexahalometallates), and Madelung constant and the interionic spacing. Now the lattice parameters in all these compounds, determined by the large MXZ- ions, are surprisingly similar (Table 2), and hence the Madelung terms in the lattice potential are also much the same-and as this term dominates the second-order elastic constants, these also do not vary greatly from one compound to the next. Since there are a number ofatoms per formula unit, the lattice vibrational spectrum of the hexahalometal-

N. WRUKet

1238

al.

Table 4. Elastic constants and their pressure derivatives of hexahalometallates at room temperature in comparison with those of CaF*. Data sources: K&C&, and (NH&SF6 [I], CaFz [25], Ni(NO& * 6NHj [2]

Pressure derivative

K2ReC16 K&Cl6 8.24 5.80 2.67 1.22 6.61 9.75 5.05 3.86

dC,,IdP dCddP dC,ldP dC’fdP dB/dP BII 812 844 TOEC

X

8.4 7.6 3.6 0.4 7.9 9.9 6.9 4.8

(NH.&SnBre 9.66 7.22 3.13 1.22 8.03 11.14 6.48 4.31

(NH&SnC&

(NH4)$iF6

9.26 6.16 3.92 1.25 7.59 10.77 6.00 5.14

CaF2

8.5 4.7 1.9 1.9 6.0 10.1 3.9 3.1

Ni(N03)r - 6NHs 5.73 6.71 -0.82 -0.42 6.38 7.08 6.03 0.21

6.05 4.35 1.31 0.85 4.92 7.7 3.52 2.44

IO” Pa

Cl,, + 2C112 Cl23 + 2CI 12 C144f x166

-3.50 -1.81 -1.39

-4.1 -2.9 -2.0

-4.25 -2.47 -1.64

-4.64 -2.59 -2.22

-3.6 -1.4 -1.1

Temp @I-! Extremal angle

2.1 102 K 43.9O

2.5 112K 43”

2.6 91 K 43.7”

2.5 133 K 43.3”

1.7 167 K 45.1°

gbye

lates is complex. To interrelate properties such as the specific heat with the sound velocities, it is always useful to turn first to the Debye model. If the Debye distribution were to be taken to represent the entire phonon density of states, then each atom would need to be considered as a vibrating unit. However, this would not give a reasonable estimation of the specific heat at ordinary temperatures where it is difficult to assess the contributions from modes on the different optic branches of the phonon dispersion curves. However, the elastic constant data allow an estimation of the acoustic-mode contributions. It is useful to model the acoustic-mode phonon density of states by a Debye distribution, leaving the optic modes to be treated separately. Therefore the Debye temperatures of several hexahalometallates have been calculated from the ultrasonic wave velocities by treating them as comprising three vibrating entities per formula unit, namely two cations A+ and a rigid anion MXZ- i.e. N = 3. Such a

Table 5. Lkbye temperatures Sample &%lc16 &Rec16

W-LkSnC&

K2PtBr6 (NH&hBG K&Bq (NH4)JeBr6

-18.9 -8.66 -5.98

-1.88 -1.60 -0.06

1.2 724 K 45.1”

-2.5 65 K 44.6”

vibrational model represents the low-temperature condition when the internal modes of the octahedral complex anion do not contribute to the specific heat. The elastic Debye temperature can then be found from the integration:

efj=

(~~3(~)/[f(~+~+f)$l’19(1)

The vi have been obtained as the eigenvalues of the Christoffel equations; the integration has been performed over the whole of velocity space as a summation over a solid angle of 1.2I8 X 10m3stemdians. The results obtained are given in Table 5. Also included is the case for N = 9 which corresponds to a Debye temperature applicable at very high temperatures. To avoid the necessity of carrying out the integration over velocity space, a number of approximate methods for calculating the elastic Debye temperature of a compound

t3$ (in K) of hexahalometallates temperature elastic constant data

Elastic constant data

Method

Ultrasonic Brillouin scattering Ultrasonic Ultrasonic Brillouin scattering Ultrasonic Brillouin scattering Ultrasonic Brillouin scattering Ultrasonic Ultrasonic Brillouin scattering Ultrasonic Brillouin scattering Brillouin scattering Brillouin scattering Brillouin scattering

de Launav de fauna; Integration de Launay de Launay Integration de Launay Integration de Launay de Launay Integration de Launay Integration de Launay de Iaunay de launay de Launay

calculated from roomN=9

N=3

161.6 165.5 161 147.6 158.3 147 194.6 191 134.3 130.9 131 243.3 241 126.9 136.0 144.2 129.0

112.2 114.9 111.8 102.5 109.9 102.1 135.2 132.6 93.3 90.9 90.6 169.0 167.3 88.1 94.4 100.1 89.6

Elastic constants, anharmonicity and Griineisen parameters

1239

tures near that T,., at which the upper phase transition onsets in K2SnC16 [32]. It is found that this acoustic branch shows no temperature dependence in the low wave-vector region but stiffens slightly near the X point of the fee Brillouin zone as the temperature is reduced towards T,., .These observations substantiate the conclusion drawn from earlier work [6, 2 l] that the phase has also been used: here f is a function of CU. This transition is driven by softening of the X point optical approximate method gives closely similar values of rotational mode rather than being due to condensation @Ato those given by the integration (Table 5). This is of transverse-acoustic phonons. because the elastic anisotropy of these hexahalomeApplication ofhydrostatic pressure shifts T,, by ( I .35 +- 0.10) K kbar-’ ( 1 kbar = 10’ Pa) [33] and this induces tallates is not pronounced. One source of error in these K2SnC16 at room temperature towards its upper phase results for @j stems from use of room temperature instead of low-temperature elastic data. For comtransition. In these circumstances the derivative aC’/ pounds such as K2SnC16 and Ni(NO& - 6NHs near to aP of the elastic constant C’ which softens markedly in the vicinity of the transition, would be expected to a displacive phase transition, the reduction of the elastic constants arising from the coupling of the acoustic be small, as found. The third-order elastic constant (TOEC) combinamode to the softening rotary optic mode could result in an unrealistically small @A. In general nevertheless tions (C, I I + X112), (Cl23 + 2C, 12)and (CM + 2C166), which can be obtained from the KIJ/aP, are substanrelatively low Debye temperatures would be expected for salts with comparatively large anions--8$ is much tially smaller than those of CaF2 (Table 2) (and those lower for the hexahalometallates than for the alkalineof BaF2 [34]). An elastic constant C,,,,. . . of order u earth fluorides, a direct consequence of their larger latis the ath derivative [&#@)l&,a~,,&j,,,~ - -) of the tice spacings, smaller Coulombic interionic binding interatomic potential b(R) with respect to Lagrangian forces and correspondingly smaller elastic stiffnesses. strain. Since the repulsive term is of shorter range than Specific-heat data are available for several hexahalthe attractive contribution to the interatomic potential, the TOEC tend to be determined more by the repulsive ometallates [30]. Extraction of an accurate Debye forces than the SOEC. Thus the small values of the temperature from these specific-heat measurements TOEC combinations of the hexahalometallates as does not prove possible (it is noteworthy that the aucompared with those of CaF2 or BaF, imply that the thors themselves did not attempt to do so). However, a rough estimate of @d from the low-temperature spe- interionic repulsive forces are smaller in the comcific heat for (NH&SnC& gives 120 + 20 K. The corpounds with the larger ions. respondence of this value with that obtained from the An important feature of the hydrostatic pressure deelastic constant data of 132.6 K (Table 5) using the rivatives aC,,/aP of the SOEC is that they provide invibrating model with N = 3 provides some confirmaformation of the dependence of the mode frequency tion that at low temperatures the specific-heat contriwi with volume, expressable as a mode Griineisen butions arise from the three ion model and that the gamma yi [=(-alnw,/alnV),]. These mode Grtineisen contributions from the internal modes of the SnClagammas have been computed for K2ReC16, ion are negligible. (NH4)2SnBr6. (NH4)2SiF6, (NH4)2SnC16 and Haussiihl[2] has noted that the hydrostatic pressure Ni(N03)2 - 6NH3 and their dependences upon mode derivatives K,.,JaP of antifluorite structure crystals re- propagation direction in the high-symmetry planes are flect the trend typical of fluorite structure materials: plotted in Fig. I. For K2SnCle it has been shown preac,,/ap> ac,2/ap> acMIap>actlap. The results viously that all the acoustic-mode Griineisen gammas obtained here on other antifluorite crystals conform are positive at room temperature [ 11. Several characwith this pattern (Table 4). KzSnCls [ 1,2] and especially teristic features become apparent. The yi for the quasiNi(NOsh - 6NH3 [2], for which i(C,, - C',z) is particlongitudinal modes are not very dependent upon mode ularly small near T, (-34.2”C) [31], exhibit propropagation direction in any of the salts and range benounced thermoelastic anomalies over a wide range of tween about 2 and 3.5. For one shear mode there is a temperatures above their transition temperatures. Both pronounced minimum in the [I IO] direction, which aC'lc?P and t3C44/aP are anomalously negative for also occurs in CaF2 [25] and BaF2 1341. In general for Ni(NO&* 6NHs: there is marked acoustic shear mode cubic crystals, in addition to those in the twofold and softening under pressure in this material [2, 311. As fourfold directions, there is another extremum in the the temperature of KrSnC&, is reduced towards Tc, shear mode yi in direction given by [35] acoustic-mode softening also occurs [I] as a result of interaction with the softening A& rotary phonon (1 - N:)/2 = N: = N: = (2 - R)l(8 - 5R), (3) mode; due to the connectivity with the TA mode prop agated along [ 1 IO] and polarised [ 1TO], it is $(C,, where R is (C,, - Cl2 - 2C44)/(C,, - CM). Usually - C12) that is particularly affected 114, I]. Recently this extremum, whose direction is related to the shear the dispersion curve for [I IO] propagation [ 1iO] polar&d transverse-acoustic phonons has been determode anisotropy (C.JC’) and the Cauchy relation mined using inelastic neutron scattering at tempera(C4C,2), tends to be in a similar direction for crystals from its elastic constants are available. One of these, namely that due to deLaunay [29] who gives

1240

N. WRUK et al.

1

INt+,l15n~r6 -11

1 II

’

I

Ill11 CRYSTRL

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1

4

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timl ORIENTATION

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Fig. I. The orientation dependence of the acoustic mode Griineisen parameters in the long-wavelength limit for (a) K&5& (b) (NHASnBr6, (C) (NH4kSiF6, (d) (NH&SnC16 and (e) Ni(NO&-6NH3. The labels L and S refer to pure longitudinal and shear modes along crystallographic symmetry axes and to quasilongitudinal and quasishear modes for other directions.

Elastic constants, anharmonicity and Gtineisen

having the same structure; this proves to be so for the hexahalometallate and the fluorite structure crystals [Table 21. For those hexahalometallates, which at room temperature are not near a displacive phase transition, the yi are not anomalous or particularly anisotropic; thus for (NH&SnCl,j, (NH&SnBre and KzReC& the longitudinal and the degenerate shear yi along the [00 l] axis are similar. However, for Ni(NO& - 6NH3 the y, for the acoustic modes at the zone centre are negative for all the shear modes; this is a consequence of the negative-pressure derivatives aC, JaP and aC’/aP of both shear moduli: the softness of the shear modes is enhanced by application of pressure. The anharmonicity of lattice vibrations is responsible for thermal expansion fl as well as the nonlinear elastic behaviour of a crystal under a finite strain. The thermal Grime&n parameter yth is defined by Y th =

pms/c*.

Values of y” obtained from experimental data are given in Table 6. yth stems from summation (C C’iri/ZZ Ci) over all the individual modes, including those on both optic and acoustic branches. To assess the role played by the long wavelength acoustic modes, and thus interrelate the influence of vibrational anharmonicity on thermal expansion with that on elastic behaviour, it is useful to obtain values of the mean acoustic-mode Griineisen gamma. When T > @A, the heat capacity per mode Ci becomes equal to Boltzmann’s constant, and a high-temperature limit 7 $ of this mean acousticmode Grtineisen parameter can be found at the longwavelength limit by using

i4 = -LC 3N

Yi.

The mean acoustic-mode Griineisen gammas r$ for the hexahalometallates, computed by using the same grid as for @A, are compared with yth in Table 6. For Ni(NO& - 6NH3 T# and Y’~differ in sign: contributions to the specific heat and thermal expansion from either acoustic phonons in states away from the Brillouin zone centre or optic modes (or phonons from both) having positive Gruneisen parameters must outweigh the negative ones from the zone-centre acoustic phonons. This suggests for Ni(NO& - 6NHS that the acoustic-mode softening under pressure may be largely confined to the long-wavelength modes. The thermal Table 6. Thermal Griineisen parameters ylh for K&Q,,

1241

parameters

Griineisen parameters y” of KzSnC&, KzReCla and (NH4)2SnC16 (Table 6) are substantially smaller than the acoustic mean 5;. This result implies that at roomtemperature optic phonons with a somewhat smaller yi than the acoustic modes contribute substantially to the thermal expansion and specific heat of these materials. The Griineisen parameters of the Raman-active optic modes in KzSnC& have been estimated with the following results (381: T4:

V =

170.1 cm-‘,

yi = 0.61,

Ci = 0.94kB,

AI,:

V = 322.0 cm-‘,

yi = 0.45,

C’i = 0.81ke,

V = 242.0 cm-‘,

yi = 0.38,

Ci = 0.89ke,

E*:

Ci being the weighting factor due to the specific-heat contribution. These would lead to a mean optic-mode Grtineisen parameter 70 of 0.62 at room temperature assuming them to be representative values for the optic modes. Taking a mean weighted value of this optic mode gamma with that 48 for the acoustic modes gives a value of 1.24 for Yn for K2SnCla which lies satisfyingly close to the room-temperature thermal value y” of 1.48. Therefore, measurements of the effect of hydrostatic pressure on the Raman-active mode frequencies and the elastic stiffness constants correlate well with the thermal Griineisen parameter y” of the cubic hexahalometallate K2SnCb. 4. SUMMARY 1. The first measurements of the elastic stiffnesses of several hexahalometallates have been reported. The elastic constants obtained are similar for a wide range of crystals; these materials have similar lattice spacings so that the contributions from the Madelung term (and also the repulsive interactions) to the elastic constants are much the same for each compound. 2. The elastic stiffnesses and Debye temperatures of the compounds are small compared with the alkalineearth fluorides due to their larger lattice spacings and hence weaker Coulombic binding forces. 3. Compounds near to a displacive phase transition can exhibit elastic softening due to coupling with the soft rotary optic mode. 4. The hydrostatic pressure derivatives aC&3P of these antifluorite structure crystals follow the same trend as that of fluorite crystals, namely aC,,/aP > ac,2fap> acujap>acrap.

(NH4)2SnC16,K2ReC&and Ni(NO&-6NHs (determined at room

temperature from elastic constant, specific heat and thermal expansion data) in comparison with q$. Coefficient (8) of volume thermal expansion (deg-‘) (15.6) X lo-’ [2] (15.7 f 2.1) x 10-S (15.0 + 1.5) x 10-s [36] (21.0) x lo-’ [2]

Molar specific heat at constant pressure (CP) J/mole deg 220.1 263.6 214.0 400.8

[30] [30] [9] [37]

Thermal Griineisen parameter yfb 1.48 k 1.31 + 1.24 + 0.89 +

0.14 0.17 0.12 0.04

?!! 2.5 2.5 2.1 -2.5

1242

N. WRUK et al

5. The mode Griineisen gammas in the long-wavelength limit are all positive at room temperature for each of crystals except Ni(NO& * 6NH3. For &SnQ, the Griineisen parameter yu ,e).(,io),which corresponds to the shear acoustic mode that couples most strongly to the AA”, softening optic phonon which drives the transition at -10.8”C, is small at room temperature and as shown recently becomes large and negative (- 11 at - 10.5”C) as the temperature is reduced towards the transition point; softening of this shear acoustic mode contributes to the X-point anomaly in the specific heat [ 191. Thus in antifluorite structure crystals which are near to a structural phase transition the acoustic-mode Griineisen gammas respond to lattice instabilities. The normal positive values (in the range of about 2 to 3.5) of the acoustic-mode gammas near k = 0 found for K2ReC16, (NH&SnBr6, (NH&SiF, and (NH&SnC& at room temperature [Fig. 1(a)- 1(d)] show that in these particular crystals zone-centre acoustic modes do not evidence a tendency towards softening. 6. For KzSnC&, (NH&SnC& and K2ReC16 the mean high-temperature long-wavelength acousticmode parameter +$ is substantially larger than the thermal Griineisen parameter yfh, implying that at room temperature optic phonons with rather smaller mode gammas yi than those ofthe zone-centre acoustic modes contribute substantially to the thermal expansion and specific heat. This is confirmed for K2SnC16 by reference to the Griineisen parameters of the Raman-active optic modes. Acknowledgments-Support by the British Council and the Deutsche Forschungsgemeinschah is gratefully acknowledged.

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