phys. stat. sol. (b) 223, 767 (2001)

Subject classification: 65.40.De; S9.11

Study of Temperature Dependence of Bulk Modulusand Interatomic Separation for Ionic Solids

Qing He and Zu-Tong Yan

Department of Physics, Anhui Normal University, Wuhu, 241000, P.R. China

(Received June 19, 2000; in revised form November 24, 2000)

The temperature dependence of the bulk modulus BT(T ) and interatomic separation r(T ) are in-vestigated from room temperature up to the melting temperature for sixteen ionic solids. In thepresent paper, the relations for BT(T ) and r(T ) are developed with the approximation that theAnderson-Gruneisen parameter dT is independent of temperature (T ) when T is higher than theDebye temperature QD. A phenomenological equation of state is also obtained. These relationscan be used to predict the values of BT(T ) and r(T ) at different temperatures. The results arecompared with available experimental data and are discussed in view of recent research in thefield of high temperature physics. A very good agreement between theory and experiment suggeststhe validity of the present work.

1. Introduction

Recently, many workers [1–6] have made various efforts to exploit thermodynamicproperties of solids under the effect of high temperatures because they are of funda-mental interest to geophysical and geochemical theories of the interior of the earth [7].In these studies, workers have examined the thermodynamic properties such as thetemperature dependence of the volume thermal expansivity, interatomic separation andbulk modulus of solids from room temperature to the melting temperature with differ-ent approximations and best fit relations [4–6]. Kwon et al. [8] investigated the thermalproperties of KCl using a modified Einstein model. Sherry and Kumar [1] performedan analysis of the temperature dependence of the interatomic separations in a staticlattice up to the melting point based on the equation of state that depends on theHildebrand approximation.It is known that the adequate knowledge of the temperature dependence of the bulk

modulus BT(T ) is necessary for understanding the thermodynamic and anharmonicproperties of the crystals. A method to estimate the temperature dependence of thebulk modulus BT(T ) and interatomic separation r(T ) has been developed with the ap-proximation that the coefficient of volume thermal expansion a is linearly dependenton temperature (T ) [5]. This can be justified from the work of Spetzler et al. [9]. In thispaper, we present the relations for the temperature dependence of bulk modulus BT(T )and interatomic separation r(T ) by using the well known thermodynamic relations si-multaneously assuming that the Anderson-Gruneisen parameter dT is independent oftemperature (T ) above the Debye temperature QD [7, 10] and that the volume thermalexpansion coefficient a is a quadratic function of temperature [11, 12].The method of analysis is described in Section 2. The results and the discussion are

given in Section 3.

Qing He and Zu-Tong Yan: Study of Temperature Dependence of Bulk Modulus 767

2. Method of Analysis

The Anderson-Gruneisen parameter dT, which plays an important role in understandingthe anharmonic properties of ionic solids, is defined as follows [13]:

dT ¼ ––(1/aBT) (@BT/@T )P, (1)

where a and BT are the volume thermal expansion and bulk modulus, respectively.They are defined as

a ¼ (1/V) (@V/@T )P (2)

and

BT ¼ ––V(@P/@V)T . (3)

Assuming that a depends quadratically on temperature, we can write

a(T ) ¼ a0 þ a00 (T � T0) þ (1/2) a0

00 (T � T0)2 , (4)

where a0 is the value of a at T0. a00 and a0

00 are, respectively, the first- and second-order temperature derivatives of a at T0. Using the most useful thermodynamic approx-imation aBT ¼ const, we have [14]

dT ¼ (1/a2) (@a/@T )P. (5)

From this relationship we can get the following equation which is a well known result[15]:

a00 ¼ (@a/@T )P ¼ dTa0

2 , (6)

and then the following useful equation:

a000 ¼ (@2a/@T2)P ¼ 2dT

2a03 , (7)

in which we additionally use the assumption that the Anderson-Gruneisen parameterdT is independent of temperature (T ) above the Debye temperature QD. So, Eq. (4)can be rewritten as follows:

a(T ) ¼ a0 þ dTa02(T � T0) þ dT

2a03(T � T0)2 . (8)

2.1 Expression for temperature-dependent bulk modulus BT(T)

Substituting the value of a from Eq. (8) in Eq. (1), we get

dBT/BT ¼ �dT[a0 þ dTa02(T � T0) þ dT

2a03(T––T0)2] dT . (9)

Integrating Eq. (9), we finally get the following expression for BT(T ):

BT(T ) ¼ B0 exp [––dTa0(T––T0) f1 þ (1/2) dTa0(T � T0) þ (1/3) dT2a0

2(T � T0)2g] . (10)

2.2 Expression for temperature-dependent interatomic separation r(T)

Putting Eq. (8) into (2), we have

dV/V ¼ [a0 þ dTa02(T � T0) þ dT

2a03(T � T0)2] dT. (11)

768 Qing He and Zu-Tong Yan

Integrating Eq. (11), we get an isobaric volume-temperature relation,

V(T )/V0 ¼ exp [a0(T � T0) f1 þ (1/2) dTa0(T � T0) þ (1/3) dT2a0

2(T � T0)2g] .

(12)

This is a phenomenological equation of state which can be used to predict the compres-sion (V/V0) in the whole temperature range from room temperature up to the meltingtemperature along an isobaric. Moreover, considering the relation V/V0 ¼ (r/r0)3, Eq.(12) becomes the following expression for r(T ):

r(T )¼ r0 exp [(1/3)a0(T�T0) f1þ (1/2) dTa0(T�T0)þ (1/3) dT2a0

2(T�T0)2g] .

(13)

The Debye temperatures QD of solids considered in this paper have values close toroom temperature except that for LiF (QD ¼ 700 K). Equations (10) and (13) bothwork well above the Debye temperature; but for LiF, they may not be satisfied belowQD.

3. Results and Discussion

We calculated the values of bulk modulus BT(T ) and interatomic separation r(T ) as afunction of temperature (T ) from room temperature up to melting temperature makinguse of Eqs. (10) and (13), respectively, for 16 crystals with NaCl-type structure. Thevalues of a, dT, r and BT at room temperature and ambient pressure used in the calcula-tions as input data are given in Table 1 [1, 16]. The results along with the availableexperimental data [11, 17] are given in Table 2. A comparison of BT(T ) between thetheory and experiment is presented in Table 2 in the case of NaCl for which experimen-tal data are available. A similar comparison of r(T ) can be performed for those experi-

Study of Temperature Dependence of Bulk Modulus and Interatomic Separation 769

Tab l e 1Values of input parameters at room temperature along ambient pressure [1, 16]

crystal r (�A) a (10––4 K) dT BT (108 Pa) Tm (K)

LiF 2.013 0.999 6.15 665 1143LiCl 2.570 1.32 6.88 297 887LiBr 2.751 1.50 7.08 235 820LiI 3.006 1.80 7.47 172 719

NaF 2.317 0.96 5.84 465 1205NaCl 2.820 1.19 5.95 240 1050NaBr 2.989 1.26 6.30 199 1028NaI 3.236 1.37 6.46 151 928

KF 2.674 1.02 6.16 305 1124KCl 3.146 1.11 6.29 175 1043KBr 3.289 1.16 5.88 14 1006KI 3.525 1.23 5.83 17 957

RbF 2.826 0.94 6.80 67 1033RbCl 3.291 1.03 6.73 156 990RbBr 3.445 1.08 6.64 132 950RbI 3.668 1.23 6.53 105 913

770 Qing He and Zu-Tong Yan

Tab l e 2Calculated values of interatomic separation r (in�A) and bulk modulusBT (in units of 108 Pa)at different temperature T (in K). The experimental data are given in parentheses [11, 17]

a)

LiF LiCl LiBr LiI

T r BT r BT r BT r BT

300 2.013 (2.012) 665.0 2.570 297.0 2.751 235.0 3.006 172.0400 2.020 (2.019) 624.2 2.582 270.0 2.766 210.1 3.025 148.9500 2.027 (2.028) 583.3 2.595 243.1 2.782 185.2 3.048 126.0600 2.035 (2.037) 542.6 2.610 216.5 2.801 160.7 3.075 103.6700 2.044 (2.047) 502.1 2.626 190.3 2.822 136.9 3.106 82.5800 2.053 (2.058) 462.1 2.644 164.9 2.846 114.2 � �Tm 2.091 (2.101) 330.8 2.662 143.8 2.851 109.8 3.113 78.7

b)

NaF NaCl NaBr NaI

T r BT r BT r BT r BT

300 2.317 (2.318) 465.0 2.820 (2.820) 240.0 (240.0) 2.989 199.0 3.236 151.0400 2.325 (2.326) 438.9 2.832 (2.831) 223.0 (224.1) 3.002 183.2 3.252 137.6500 2.333 (2.335) 412.9 2.844 (2.845) 206.0 (205.0) 3.016 167.4 3.269 124.3600 2.341 (2.345) 386.9 2.858 (2.860) 189.1 (188.0) 3.032 151.8 3.288 111.1700 2.351 (2.356) 361.0 2.873 (2.877) 172.4 (174.0) 3.049 136.3 3.309 98.1800 2.361 (2.368) 335.3 2.889 (2.894) 155.9 (156.0) 3.069 121.1 3.332 85.5900 2.371 (2.382) 309.9 2.907 (2.914) 139.8 (140.9) 3.090 106.4 3.359 73.4Tm 2.409 (2.428) 235.7 2.936 (2.945) 116.6 (119.0) 3.120 88.6 3.367 70.1

c)

KF KCl KBr KI

T r BT r BT r BT r BT

300 2.674 305.0 3.146 (3.146) 175.0 3.289 (3.289) 148.0 3.525 (3.525) 117.0400 2.683 285.8 3.158 (3.158) 162.8 3.302 (3.302) 137.9 3.540 (3.540) 108.6500 2.693 266.7 3.171 (3.170) 150.6 3.318 (3.316) 127.8 3.556 (3.556) 100.2600 2.704 247.6 3.185 (3.185) 138.4 3.332 (3.331) 117.8 3.574 (3.578) 91.9700 2.716 228.6 3.201 (3.200) 126.4 3.349 (3.346) 107.8 3.593 (3.592) 83.6800 2.729 209.9 3.218 (3.220) 114.5 3.367 (3.364) 98.0 3.614 (3.612) 75.5900 2.742 191.4 3.236 (3.234) 102.9 3.387 (3.382) 88.4 3.637 (3.633) 67.5Tm 2.777 151.8 3.265 (3.259) 86.9 3.409 (3.401) 78.5 3.652 (3.646) 63.1

d)

RbF RbCl RbBr RbI

T r BT r BT r BT r BT

300 2.826 267.0 3.291 (3.291) 156.0 3.445 (3.442) 132.0 3.668 (3.668) 105400 2.835 249.9 3.303 (3.302) 145.2 3.458 (3.457) 122.5 3.684 (3.683) 96.6500 2.845 232.9 3.315 (3.316) 134.4 3.472 (3.472) 113.1 3.701 (3.699) 88.1600 2.856 215.9 3.329 (3.331) 123.6 3.487 (3.487) 103.7 3.720 (3.716) 79.8700 2.867 199.0 3.344 (3.346) 113.0 3.504 (3.502) 94.3 3.741 (3.734) 71.5800 2.879 182.3 3.360 (3.363) 102.4 3.522 (3.518) 85.2 3.764 (3.753) 63.4900 2.893 165.9 3.378 (3.379) 92.2 3.541 (3.535) 76.2 3.789 (3.774) 55.6Tm 3.092 144.7 3.395 (3.399) 83.2 3.552 (3.544) 71.8 3.792 (3.776) 54.6

mental data available. It is clear that the present work is in very good agreement withexperiment. Even for LiF, we have also found that Eqs. (10) and (13) both work wellfor either T above or below the Debye temperature (QD). The beauty of the newlydeveloped phenomenological equations for BT(T ) and r(T ) is that they have been de-veloped under assuming Eq. (4), i.e. the volume thermal expansion a(T ) is a quadraticfunction of temperature (T ). Thus, it can be concluded that the present formalism ismore logical and physically realistic. It can be useful to predict anharmonic propertiesof solids (up to melting temperature Tm).Summarily, the present work based on the assumptions that the Anderson-Gruneisen

parameter dT is independent of temperature (T ) above the Debye temperature QD andthe coefficient of volume expansion a is a quadratic function of T, is simple and applic-able. The very good agreement between theory and experiment within the whole tem-perature range, especially at high temperature, confirms the internal consistency of thethermodynamic relations from which the most useful relations have been derived.

Acknowledgements The authors are grateful to the referee for his valuable commentswhich have been found very useful in revising the manuscript.

References

[1] A.M. Sherry and M. Kumar, J. Phys. Chem. Sol. 52, 1145 (1991).[2] J.L. Tallon, J. Phys. Chem. Sol. 41, 837 (1980).[3] M.P. Verma and B. Dayal, phys. stat. sol. (b) 3, 901 (1963).[4] M. Kumar and S.P. Upadhyaya, phys. stat. sol. (b) 181, 55 (1994).[5] M. Kumar, Physica B 205, 175 (1995).[6] R.K. Pandey, J. Phys. Chem. Sol. 59, 1157 (1998).[7] O.L. Anderson, D. Isaak, and H. Oda, Rev. Geophys 3, 57 (1992).[8] T.H. Kwon, S.D. Kwon, Z.H. Yoon, Y.K. Sohn, and S.C. Kim, Physica B 183, 75 (1993).[9] H. Spetzler, C.G. Samis, and R.J. Connell, J. Phys. Chem. Sol. 33, 1727 (1972).[10] O.L. Anderson, A. Chopelas, and R. Boehler, Geophys. Res. Lett. 17, 685 (1990).[11] K.K. Srivastava and H.D. Merchant, J. Phys. Chem Sol. 34, 2069 (1973).[12] R.K. Pandey, J. Phys. Chem. Sol. 59, 989 (1998).[13] O.L. Anderson, J. Geophys. Res. 72, 3661 (1967).[14] J. Shanker and M. Kumar, phys. stat. sol. (b) 179, 351 (1993).[15] O.L. Anderson, Equations of State of Solids for Geophysics and Ceramic Science, Oxford

University Press, Oxford 1995.[16] A. Dhoble and M.P. Verma, phys. stat. sol. (b) 136, 497 (1986).[17] S. Yamamoto, I. Ohno, and O.L. Anderson, J. Phys. Chem. Sol. 48, 143 (1987).

Study of Temperature Dependence of Bulk Modulus and Interatomic Separation 771