Analysis of anharmonic elastic properties of anti-fluorite structure A2MX6 single crystals

~-~~

INFRARED PHYSICS &TECHNOLOGY

FI.$EVIER

Infrared Phys. Technol. 36 (1995) 743-748

Analysis of anharmonic elastic properties of anti-fluorite structure A 2 M X 6 single crystals PradeeptiChaudhry,S.P. Singh, B.R.K. Gupta Department of Physics, G.B. Pant Unirersity, Panmagar 263 145 (U.P.), India Received 9 June 1994

Abstract

In the present paper an analysis of vibrational anharmonic properties of a number of antifluorite structure crystals [K2SnCI6, K2ReCI6, K:PtBr6, (NH4):SnC16, (NH4)2PtBr6 and (NH4):TeCI6] has been performed in terms of higher order elastic constants and their linear combinations, pressure and temperature derivatives of their second order elastic constants (SOEC) and Gruneisen parameters. These physical quantities have been calculated by making use of a phenomenological potential model. It has been found that the three-body interactions considered in the present potential model plays an important and significant role in explaining the characteristic behaviour of the crystals under study. Good agreement between the experimental and theoretical values has been obtained.

1. Introduction In recent years the studies of the lattice dynamics and the pressure derivatives of the elastic constants of hexahaiometallate (antifluorite structure) crystals have been quite useful in view of their phase transition properties [1-5]. Many hexahalometallates of the general type A2MX6 (A = K ÷, NH~" . . . . . M = Sn, Re, Pt . . . . . X = halogens) 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 transition, at least some of which are linked with softening of rotary phonons of the MX6 octahedra [6-8]. The transition can be attended by elastic Elsevier Science B.V. SSDI 1350-4495(95)011-9

anomalies which originate from coupling between a critical optical-phonon mode and the elastic strain [9,10]. The A2MX 6 compounds constitute ideal model systems for investigations of the elastic effects of such coupling. In the vicinity of the upper phase transitions in these compounds the elastic stiffness in the cubic phase is reduced by anharmonic coupling between the acoustic and soft rotary modes. Therefore the theoretical study on higher order elastic constants and the temperature derivatives of elastic constants can provide useful physical insight in the lattice dynamics (anharmonicity) and its relation to the transition mechanism in these compounds. The temperature dependence of elastic constants of four hexahalometallate compounds (NH4)2TeCIr, (ND4)zTeCIr,

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P. Chaudhrv et al./ Infrared Physics and Technology 36 (1995) 743-748

(NH4)! SnCl6, and (ND4)., SnCI 6 have recently been analysed experimentally [6] as well as theoretically [6] by making use of Brillouin scattering technique and three-body force shell model respectively. A good agreement between the experimental and theoretical values of the temperature derivatives of the second order elastic constants (SOEC) at room temperature has been obtained by these workers. In the present paper we consider the extended three-body-force shell model [11] (ETSM) to calculate the third order elastic constants (TOEC) pressure and temperature derivatives of SOEC and Gruneisen parameters of seven hexhalometallate crystals. The inclusion of three-body interactions [12] and second-neighbour interactions in the potential model of Woods, Cochran, and Brockhouse [13] has led to the development of an extended three-body force shell model (ESTM). A brief theory of the model and method of computation is given in Section 2. Section 3 is a discussion of the results achieved in the present investigation.

Within the framework of the extended threebody-force shell model, the potential energy expression for the hexahalometallates crystallizing in the cubic antifluorite structure can be written as follows:

~ 'e'-f(r~.~.,)+¢R(r), kk'k"

(1)

where

(2)

~ba(r) = ~b~(r) + ~b~(r).

__Nbexp (r t + r~ -- r)

~btR(r)

rkk"

¢~(r) = g ' b rkk"

(3)

P +f2rt - 1"1 Nb

exp/--/+ \ P++ ]

['2r 2 - r'~

-- exp/~/, rk~' \ P-- ]

=

~o + I

+ ~.

1

c ~

+ T., c ~ , ~ , ~

G~,,.u~u,,u,.

+

K, SnCI 6

K,. Re4Sl 6 K: PtBr6 (NH4)2SnC16 (NH4).,SnBr6 (NH4).,PtBr6 (NH4).,TeCI6

(5)

The strain parameter npq is defined as

~pq=~(Rpq-.bRqp-}-~r RrqRqr ).

(6)

In view of Eqs (1) and (5), (6) the expression of various order elastic constant obtained in the present study are reported elsewhere [15]. The expression for pressure derivatives of SOEC can be obtained in a straightforward manner using

Table I The value of input data: linear parameter (a), second order elastic constants, coefficientsof volume thermal expansion (/~) Crystals

(4)

where r I and r 2 are the radii o f A + and (MX6)-ions. N represents the total number of ion pairs, b and pu are, respectively, the repulsive and hardness parameters determined from the equilibrium condition and overlap integrals [14, 15]. Following the Wallace theory [14] the potential energy expression in terms of various elastic constants is given by

~(~)

2. Theory and method of computation

i~'Z~Zke"+-q~(r) = 2 kk" rkz

Here, the first and second terms in (I) are, respectively, the long-range Coulomb and threebody interaction energies, f (r) is a charge-transfer parameter calculated from the Cauchy relation. In (2) erR(r) and ~ ( r ) represent the Born-Mayer type short-range overlap repulsive interactions between nearest and next-nearest neighbour iens, i.e.

a (nm)

Ctt (GPa)

G: (GPa)

G, (GPa)

( x 8 sdeg-t

0.43240 0.42608 0.44569 0.43561 0.45856 0.44903 0.44253

19.7 22.8 21.6 22.0 18.8 22.1 21.2

12.0 12.8 12.0 10.6 I 1.1 I 1.6 10.6

8.1 8.5 8.5 9.7 8.2 9.0 7.8

14.2 16.5 18.5 18.5 14.8 18.0 17.8

fl

Table 2 Liner combinations of third order elastic constants (x IO"Pa) Crystal -3.56 (-5.10) -2.83 (- 3.50) -2.56 (--) -3.73 (-5.64) -3.90 (-5.25) -2.56 (-) -2.15 (-) -9.26 (- 18.9)

K,SnCI, K, ReCl, K,PtBr, (NHMnCl, (NH,)?SnBr, (NH,)2PtBr, (NH,),TeCI, CaF,

-3.05 (-2.90) -2.10 (-1.81) -1.75 (-) - 1.71 (-2.59) -2.37 (-2.57) -1.59 (-) -1.21 (-) -3.05 (-8.66)

- I .79 (-2.0) - 1.57 (-1.30) 1.57 (-) -1.66 (-2.22) - 1.52 (- 1.65) - 1.55 (-) - 1.33 (-) -1.89 (-5.98)

ac,,

( > -K,, (dT,) ac, 27, ( ) dT,

ac12

Eqs. (l-6) in view of Birch’s theory [16]. These expression are -2c,,

de,, ap=ac,, -=

ap

ac,

-c,,

ap

(7)

G 6p

K,SnCI,

9.61 (8.40) 7.33 (8.25) 7.10 (-) 7.83 (9.26) 8.72 (9.66) 6.91 (-) 6.56 (-) (6.05)

7.93 (7.60) 5.07 (5.80) 5.58 (-) 5.71 (6.76) 6.51 (7.22) 4.03 (-) 3.61 (-) (5.35)

K, ReCI, K,PtBr, (NH,), SnCI, (NH,)$nBr, (NHMtBr, (NHATeCl, CaF,

+ zc,,j + c,, + 3c,JI,

= -!L

+ 2C,66- c,, + c,z + 3&]/I,

(11)

(12) where /I is the volume thermal expression coefficient. The values of Gruneisen parameters have been determined using the expression derived by Wong and Schele [I81 as

y=_!+&aci

(13)

2C,.-@

where + 2Cu).

(14)

(9)

The model parameters are calculated using the input data which involve the values of lattice parameter, second order elastic constants (C,, , Cu, C,) and volume thermal expansion

Table 3 First order pressure derivatives of second order elastic costants. and Gruneiwt 6p

= -jL

B, = -f(C,,

cc,, + 2c,2

G

+ 2C,I2+ C,,lB,

63)

- 2c,z + c, + 2C,M

Crystals

=

6

- 2C,* + C,,, + 2C112 C,, +2Cu ’

- -C,, - Cl2+ Cl23+ 2c,,* G, +2c,* ’

-=-

The expressions for temperature derivatives of SOEC have been obtained by including the phonon-lattice interactions arising from the thermal vibrations [17]. These expressions achieved in the present study are given as follows:

G 6P 5.26 (3.60) 3.87 (2.67) 5.04 (-) 5.63 (3.92) 5.50 (3.13) 4.21 (-) 3.97 (-) (1.31)

parameter

6BT 6P

dC, bp

8.44 (7.90) 5.83 (6.61) 5.52 (-) 5.75 (7.59) 1.25 (8.03) 5.99 (-)

0.83 (0.50) I.13 (1.22) 1.26 (-) 1.55 (1.25) 1.10 (1.22) 1.45 (-) 1.57 (-) (0.85)

(‘14” (5.92)

T I.51 (1.58 kO.15) 1.65 (1.25~0.12) 1.82 (-) 1.80 (I.31 kO.17) 1.79 (-) (!!Y 1.80 (-) (1.2)

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P. Chaudhry et al./ Infrared Physics and Technology 36 (1995) 743-748

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P. Chaudhry et al./ Infeartd P/iyslcs~md Technology36 (1995) 743-748

coefficient (#). The input data have been taken from the experimental investigations made by Wruk et al [3]. The values of input data are reported in Table 1. The calculated values of the linear combinations of third order elastic constants are given in Table 2 along with the available experimental values for the sake of comparison. The values of pressure derivative and temperature derivatives of SOEC predicted using Eqs. (7)-(12) have been shown in Table 3. and Table 4 respectively. The values of Gruneisen parameters determined for all the crystals are also shown in Table 3. It is found that the results achieved in the present investigation agree well with the experimental values.

3. Discussions It is noted from Tables 2 and 3 that the linear combination of TOEC and the first pressure derivatives of TOEC of antifluorite structure crystals considered in the present study reflect the trend typical of fluorite structure materials like CaF2. The observation of this kind of trend in antifluorite structure crystals has also been noted by Haussuhl [19]: dCIt/dP > dC,z/Op > 0C44/~p > OC/dp. The results obtained on these crystals by us in the present investigation thus confirm with this pattern. However, the third order elastic constants (TOEC) combinations (Clll + C,2), (Cm + C.2) and (Ct44+ C ~ ) are substantially smaller than those of CaF2 and BaF2. Since the repulsive term in the expression of lattice energy is of shorter range than the attractive contribution to the interatomic potential, the TOEC tend to be determined more by the repulsive forces than the SOEC. Thus the smaller values of the TOEC combinations of the hexahalometallates as compared to those of CAF2 or BaF2 imply that the interionic repulsive forces are smaller in the compounds with the large ions (MXr)--. An important feature of the hydrostatic pressure derivative dCu/dp of the SOEC is that they provide information of the dependences of the mode frequency with volume which has been determined in terms of the Gruneisen parameters. To assess the role played by the long wave length acoustic

747

modes, and thus interrelate the influence of vibrational anharmonicity on thermal expansion with that on elastic behaviour, it is useful to determine the volume of Gruneisen parameters. We have calculated the values of Gruneisen parameters from Eq. (13) and obtained a fairly good agreement with the experimental values. A look at Table 4 indicates that the calculated values of the temperature derivatives of the SOEC are in good agreement with experimental values. The important feature of the temperature derivatives of the elastic constants, i.e. (dCJOT) is that they provide the details of the elastic behaviour and the vibrational anharmonicity of the long wave length acoustic modes in hexahalometaUate crystals. On the basis of the overall descriptions, it may be emphasized here that the extended three-body force shell model considered in the present investigation is capable of predicting fairly well the anharmonic properties of all the seven haxahalometallate crystals.

Acknowledgement The authors would like to thank Dr. P. Singh, Associate Professor of Physics, for valuable discussion and encouragement.

References [I] N. Wruk, J. Pelzl, K.H. Hock and G.A. Saunders, Phil. Mag. B61 (1990) 67. [2] R.L. Armstrong, D. Mintz, B.M. Powell and W.L. Buyers. Phys. Rev 319 (1979) 448. [3] N. Wruk, J. Pelzl, G.A. Saunders and T. Hailing, J. Phys. Chem. Evlids. 46 (1985) 1235. [4] U. Kawald, S. Muller, J. Pelzl and C. Dimitropoulous, Solid State Commun. 67 (1988) 239. [5] B.R.K. Gupta, R.S. Adhikari, O.P. Gupta and R.P. Goyal, J. Phys. C: Solid State Phys. 21 (1988) L353. [6] B.R.K Gupta, U. Kawald, H. Johannsmann, J. Pelzl and Y.C, Xu, J. Phys. Condens. Matter 4 (1992) 6879. [7] G.P. O'Leary and R.G. Wheeler, Phys. Rev BI, 11 (1970) 4409. [81 J. Pelzl, P. Engels and R. Florian, Phys. Stat. Sol. (b), 82 (1977) 145. [9] M. Sutton, R.L. Armstrong, B.M. Powell and W.L. Buyers, Phys. Rev. 327 (1983) 380.

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P. Chaudhry et al./ Infrared Physics and Technology 36 (1995) 743-748

[10] J. Pelzl, K.H. Hock, A.J. Miller, P.J. Ford and G.A. Saunders,.Z. Phys. B40 (1981) 321. [11] B.R.K. Gupta, Philos. Mag. Lett. 55 (1987) 231. [12] M.P. Verma and R.K. Singh, Phys. Stat. Sol. (b) 33 (1969) 769. [13] A.D.B. Woods, W. Cochran and B.N. Brockhouse, Phys. Rev. 119, 3 (1960) 980. [14] D.C. Wallace, Solid State Physics. Vol. 25 ~Academic. New

[15] [16] [17] [18] [19]

York, 1972) p. 301. Thermodynamics of Crystals (Wiley, New York). Pradeepti Chaudhary, M.Sc. thesis (awarded, G.B. Pant University, Pantnagar, 1994). F. Birch, Phys. Rev. 71 809 (1942) U.C. Sdvastava, Phys. Stat. Sol. (b), 150 (1980) 641. C. Wong and D.E. Schuele, J. Phys. Chem. Solids 28 (1967) 1225. S. Haussuhl, Solid State Commun. 38 (1981) 329.