phys. stat. sol. (b) 244, No. 9, 3099–3104 (2007) / DOI 10.1002/pssb.200642463

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

Influence of temperature on elastic properties

of caesium cyanide

Preeti Singh*, 1, N. K. Gaur1, and R. K. Singh1, 2

1 Department of Physics, Barkatullah University, Bhopal 462026, India

2 Institute of Professional & Scientific Studies & Research, Chaudhary Devi Lal University,

Sirsa 125055, India

Received 4 September 2006, revised 13 February 2007, accepted 4 April 2007

Published online 5 June 2007

PACS 62.20.Dc, 65.40.–f, 78.35.+c

An extended three body force shell model (ETSM), which incorporates the effects of translational–

rotational (TR) coupling, three body interactions (TBI) and anharmonicity, has been applied to investigate

the temperature dependence of the second order elastic constants (cij, i, j = 1, 2) of CsCN. The elastic con-

stant c44

obtained by us shows an anomalous behaviour with the variation of temperature. The variations

of elastic constants (c11

, c12

, c44

) with temperature are almost in excellent agreement with Brillouin scatter-

ing measured data. We have also evaluated the temperature variations of the third order elastic constants

(cijk) and the pressure derivatives of the cij in the CsCN material. However, their values could not be com-

pared due to lack of experimental data.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The cesium cyanide (CsCN) is a molecular ionic crystal with cubic (CsCl) structure [1, 2] in which the spherical (CN–) ions are orientationally disordered. Also, the neutron scattering and the ultrasonic meas-urement results have proved the existence of translational–rotational (TR) coupling due to the interac-tion of low lying excitations of the CN-dumbbells with transverse acoustic phonons [3, 4]. Furthermore, the CsCN exhibits the strong Cauchy discrepancy (c12–c44) [5, 6], whose magnitude is many times larger than that of the caesium chloride [7]. These Cauchy violations have been ascribed to the existence of three body interactions [8–11], whose effects have been included in the of rigid shell model [12] frame-work by some of us [10, 11] to develop the three body force shell model (TSM). This TSM has been extended to incorporate the effects of these TR couplings [3, 4] and anharmonicity. Consequently an extended TSM (ETSM) has been developed by Singh and Gaur [13, 14]. This ETSM has earlier been successfully applied to describe the lattice dynamics of CsCN. This ETSM has also been applied to pre-dict the lattice dynamics and the temperature dependence of the second order elastic constants (SOECs) for MCN (M=Na, K, Rb) [14–20] having the rocksalt structure. Currently, the temperature dependence of the elastic constants of copper halides has been studied in detail by Mouffok et al. [21] in their zincblende, rocksalt and intermediate (tetragonal, SC-16) crystal structures by using the two-body poten-tial of Buckingham type [22]. Motivated from the versatility of the ETSM, we thought it pertinent to apply this model to predict the influence of temperature on the second order elastic constants (SOECs) measured [5] from the Brillouin Scattering technique for CsCN. This ETSM has also been used to compute the temperature dependence of the third order elastic constants (TOECs) and the pressure derivatives of the SOECs. The essential

* Corresponding author: e-mail: preetisingh2006@gmail.com, Phone: +91-0755-2489028, Fax: +91-0755-2672223

3100 P. Singh et al.: Influence of temperature on elastic properties of caesium cyanide

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formulation for the ETSM is presented in the next section and its application to evaluate the temperature dependence of the elastic properties of CsCN in the subsequent sections.

2 Essential formulation

The interaction potential used to derive the framework of the present ETSM can be expressed as [12, 13]

2

TR1 6 8 an1 ( ) exp2

k k kk

k k kk k kk kk kk kk kk kk

kk kk kk kk kk

r r reZ Z r f r c r d r bφ β φφ

ρ

- - - ¢ ¢

¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢

¢ ¢ ¢ ¢ ¢

+ -Ï ¸È ˘= - + - - - + +Ì ˝Í ˙Î ˚ Ó ˛

     . (1)

Here, k (k′) denote the positive (negative) ions and sum is taken over all the ions (kk′). In the above ex-pression, the first two terms represent the long-range Coulomb and three-body interactions (TBI) [8–11]. The third and fourth terms are the van der Waals (vdW) attraction due to the dipole–dipole and dipole–quadrupole (d–q) interactions. The fifth term is Hafemeister and Flygare (HF) type short-range (SR) overlap repulsion [23] extended upto the second neighbour ions. βkk′ are the Pauling coefficients given by

1 ( / ) ( / )kk k k k k

z n z nβ¢ ¢ ¢

= + + (2)

with zk (zk′) and nk (nk′) as the valency and the number of electrons in the outermost orbit of the cations (anions). b and ρ are the hardness and range parameters as defined in our earlier papers [15, 16], φTR and φ an are the contributions due to the TR coupling and anharmonicity [24, 25]. The hardness and range parameters are determined from the equilibrium condition:

0

[d ( )/d ] 0r r

r rφ=

= (3)

and the bulk modulus:

0

1 2 2

0(9 ) [d ( )/d ]

r rB Kr r rφ-

=

= , (4)

where r is the nearest neighbour (nn) seperation and r0 is the equilibrium nn separation. K as the crystal structure constant (= 1.333 for CsCl structure). The values of TBI parameters f(r0) and its derivatives (df/dr)

r=r0, are obtained from the Cauchy discrepancy [5] and the analytical expression as given by Coch-

ran [26],

0

( ) exp ( / )f r f r ρ= - . (5)

Using the Eq. (1) and the straight forward method [19], the second order elastic constants (SOECs) are obtained as

2

TR an1 2

11 T 1 2 11 1144.133 1.7333 2.0666 3.133 ( ) δ δ

4 6 2

e A Ac T B B B f r c c

rβ

È ˘= + - + - + + +¢Í ˙Î ˚ , (6)

2

TR an1 2

12 T 1 2 12 1243.0666 1.3666 1.5333 3.1336 ( ) δ δ

4 6 2

e A Ac T B B B f r c c

rβ

È ˘= - + - + - + + +¢Í ˙Î ˚ , (7)

2

TR an1 2

44 T 1 2 44 4442.0666 1.3666 1.5333 δ δ

4 6 2

e A Ac T B B B c c

rβ

È ˘= - + - + - + +Í ˙Î ˚ , (8)

where

2

T 1 24[0.3392 { 16 ( )}]

4

eT B f r B B

rβ ε ε= + + + . (9)

The terms an

δ ijc and TRδ ijc are the anharmonicity and TR coupling contribution [27–29] respectively. A1,

A2, B1 and B2 are the short-range force constants [12, 13], β is the volume thermal expansion coefficient

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and BT is the isothermal bulk modulus as defined in Ref. [19]. The Elastic constant depend upon the short-range repulsion and quadrupole contribution to the translational-rotational coupling and the rota-tional susceptibility (χij) obtained in the presence of direct interaction. For the calculation of TR

δ ijc , we have used the expressions given by

TR

11 eff 11

2δ ( ) ,c A T

aχ= - (10)

TR

12 eff 11

1δ ( ) ,c A T

aχ= (11)

TR

44 eff 44

1δ ( )

2c B T

aχ= - . (12)

Here, 2a is the lattice constant of the CsCN lattice. The effective coupling constant Aeff and Beff are given by

Aeff = AR + AQ and Beff = BR + BQ (13)

with AR and BR as the constant associated with short-range repulsive interactions and AQ and BQ being associated with the quadrupole electric field gradient interaction given by [25, 27, 29].

2 2 1/2

R 2 0 1 35π ( ) [ (3 ) ( 3 )]

bA d a a f f d f f

ρ

-

= + - + - (14)

and

2 2 1/2

R 1 330π ( ) [ ( )]

bB d a d f f

ρ

-

= - + - , (15)

where

1

( 1) 2

0

1

(1 ) e d

g

n n yh

g

f g y y

+

+ -

-

= - -Ú (16)

with

g = 2 2

2

( )

da

d a+ and h = 2 21

( )d aρ

+ . (17)

The various symbols appearing in these expressions are explained in Refs. [27, 29]. The quadrupole electric field gradient interaction AQ and BQ are also given in Refs. [25, 27, 29]. We have taken these values directly from Ref. [29]. These two contributions oppose each other and the relevant expressions are same as those given by Michael and Naudts [25], and Sahu and Mahanti [27, 29]. The susceptibility χ11 is related to c11 and c12 through Aeff, while χ11 is to c44 through Beff. The last term in Eqs. (6)–(8) is the anharmonicity contribution and it has been approximated to γijT, for T > 100 for all the alkali halides, which is an important factor for the study of temperature dependent elastic constants i.e. c11 and c44. It is well known that the anharmonic effect gives rise to phonon–phonon interaction and renormalize the elastic constants. We have used the values of γij for CsBr due to lack of their data on CsCN as it is closest to CsBr in its lattice size. For proper understanding of the temperature dependence of c11(T) it is neces-sary to include anharmonicity. This shed light on the softening of elastic constants in Orientationally Disordered Phase of molecular crystals. The inclusion of the anharmonic effect provides good resem-blance between the theoretical and experimental results.

3102 P. Singh et al.: Influence of temperature on elastic properties of caesium cyanide

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

180 200 220 240 260 280 300

0.30

0.33

0.36

0.39

1.32

1.38

1.44

1.50

1.56

1.96

2.00

2.04

2.08

2.12

C44

T(K)

C12

Cij(1

011dy

ne/c

m2 )

C11

a)

180 200 220 240 260 280 300

0.300.330.360.390.420.450.481.301.401.501.601.701.801.902.002.202.402.602.803.003.203.403.60

C44

T(K)

C12

Cij(

1011

dyne

/cm

2 )

C11

b) Fig. 1 Temperature variation of the second order elastic constants a) with TR and anharmonicity, b) calculated

without TR and anharmonicity of CsCN. The experimental [6] values of c11

, c12

and c44 are represented, respectively,

by circles (�), squares (�) and triangles (�). The solid lines represent our ETSM result.

4 Results and discussion

Using the input data listed in Table 1 and the values of the vdW coefficients taken directly from an ear-lier paper [14], we have calculated the model parameters b, ρ, and f(r) using the Eqs. (3) to (5) at 300 K and listed them in Table 1. The values of model parameters at different temperatures are calculated on the lines of our earlier papers [16–19]. The TR coupling and anharmonicity effects are included on the lines of Sahu and Mahanti [29]. The model parameters thus obtained are given in Table 2. We have cal-culated the second order elastic constants (c11, c12 and c44) as a function of temperature. While computing these elastic constants, the contributions from the Translational-Rotational coupling of the CN– and an-haronicity are taken into account. We have used the Eqs. (6) to (8) to calculate the SOECs. The ETSM results on the temperature dependence of c11, c12 and c44 are depicted in Fig. 1(a). It is seen from this figure that the values of c11 and c44 increases with the increase in temperature, while the same for c12 de-creases with the increase in temperature. The increasing trend of c44 indicates the existence of softening over the lower range of temperature. This is an anomalous behaviour exhibited by c44 as a result of the influence of temperature on them. However, the temperature variations of the elastic constants (c11, c12,

and c44) are almost in excellent agreement with their Brillouin scattering measured data [6] except for the temperature range 200 K ≤ T ≤ 240 K particularly in the case of c11. These results are interpreted on the basis of Landau theory of the phase transition with a linear coupling between the strain and the order parameter. Here, both c44 and (c11–c12)/2 become soft in a wide range of temperature, whereas the bulk modulus B = (c11 + 2c12)/2 is becoming stiffer on reaching the phase transition from the above. This fea-ture is in keeping with the conclusions drawn by Mouffok et al. [21].

Table 1 Input data, TR coupling and anharmonicity coefficient for CsCN.

properties values TR coupling coefficient

–values anharmonicity coefficient

values

r0 (Å) 3.68a AR (10–4 dyn) –0.098 γ11 (109 dyn cm2 K) 0.120

c11 (1011 dyn cm–2) 2.11b BR (10–4 dyn) –0.403 γ12 (109 dyn cm2 K) 0.048

c12 (1011 dyn cm–2) 1.18b AQ (10–4 dyn) –0.578 γ44 (109 dyn cm2 K) 0.106

c44 (1011 dyn cm–2) 0.40b BQ (10–4 dyn) –0.472 α+(Å)3 3.34c χ44 (1012 dyn cm–1) –0.122e α– (Å)3 1.80d χ11 (1012 dyn cm–1) –3.68e a [30], b [6], c [31], d [32], e [29]

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Table 2 Model parameters for CsCN.

model parameters ρ (Å) b (10–12 erg) – f(r)

values 0.3248 0.6579 –0.0465

Table 3 Values of TOEC (1012 dyn cm–2)

T (K) –c111 –c112 –c166 –c123 –c144 –c456

193 –7.0725 –0.8029 –0.5970 –0.8985 –0.6661 –0.5531

253 –7.0357 –0.8054 –0.5996 –0.9008 –0.6683 –0.5551

300 –7.0081 –0.8075 –0.6016 –0.9027 –0.6702 –0.5568

In order to demonstrate a visual comparison of the contribution from TR coupling and anharmonicity effect, we have calculated the SOEC’s with TR coupling and anharmonicity effect, and without the in-clusion of TR coupling and anharmonicity effects and plotted them in Fig. 1(a) as depicted in Fig. 1(b). It has been found that the agreement improves quite significantly in the case of ETSM (see Fig. 1(a)) with the inclusion of TR coupling and an harmonicity effects. It shows an obvious necessity of the inclusion of TR coupling and anharmonic effects. Besides, we have calculated the temperature dependence of the Third Order Elastic Constants (TOECs) of CsCN since the values obtained at different temperature show only marginal difference; therefore, we reported their values only for three temperatures in Table 3. We have also calculated the pressure derivatives of SOECs (dcij/dP) using their expressions from Ref. [3] but they are found to be almost independent of temperature. Therefore, only the room temperature results on dc11/dP = 11.6, dc12/dP = 7.59 and dc44/dP = 5.87 are being reported. The present results on the TOECs and the pressure derivative of SOECs are, probably, the first report on them and hence their comparison is not possible in the absence of experimental data. However, it is interesting to indicate that their values show a close resemblance in magnitude and sign with those obtained for the alkali cyanides NaCN, KCN and RbCN available at 300 K [13] from a similar model potential. The overall agreements achieved show that the present ETSM may be regarded as an appropriate model for the description of the temperature dependence of the elastic properties of the Orientationally Disordered CsCN. This is not surprising as ETSM frame-work is capable of taking the realistic account of the Cauchy violation (in terms of three-body interaction effects), the polarizabilities (dielectric proper-ties), the TR coupling and the anharmonicity for the present material. The ETSM is expected to reveal the influence of temperature on the elastic properties of other materials of the cyanide family.

Acknowledgements The authors are thankful to the University Grant Commission (UGC), New Delhi, for provid-

ing the financial support to this project.

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