Temperature dependence of interatomic separation

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Abstract

A simple unification of various relations for the temperature dependence of interatomic separation is presented. It is found that the relations reported by the earlier workers as new are contained in a simple thermodynamic relation. Some other relations based on different physical origins are also included in the study and compared with the thermodynamic relation. The results are reported for ten crystals, viz. KF, KCl, KBr, KI, RbF, RbCl, RbBr, RbI, MgO and CaO. A good agreement between unified relation and experiment demonstrates the superiority of the simple theory based on the thermodynamic analysis.

Introduction

The properties of solids at high pressures and high temperatures are of fundamental interest for the understanding of the Earth's deep interior. Considerable efforts have been made to determine the properties of solids under high pressures. Moreover, at room pressure, relatively fewer efforts have been made. The investigators have tried to study the temperature dependence of interatomic distances using different approaches. A detailed review of the experimental work has been provided by Fiquet et al. [1]. Kumar and Upadhyay [2] as well as Anderson [3] proposed that the coefficient of volume thermal expansion, α, depends on temperature, T, as follows:αα0=[1-α0δT(T-T0)]-1orα=α0+α02δT(T-T0)+α03δT2(T-T0)2+where δT is the Anderson–Gruneisen parameter and 0 refers to the initial condition, α is the coefficient of volume thermal expansion, which is defined asα=1V(dVdT)P

Using the definition of α, the integration of Eq. (1) gives the following relation, as reported by Kumar and Upadhyay [2]:rr0=[11-α0δT(T-T0)]1/3δT

Here, the relation V/V0=(r/r0)3 has been used, which is a mere scaling, coinciding with an actual interatomic distance just in some particular cases, those, for instance, in which atoms’ positions are primitive cubic lattice points. r is the interatomic distance.

A thermodynamic analysis of the material under high temperature–high pressure has been performed by Kumar [4], [5] by developing the theory of equation of state (EOS). It has been discussed that the theory may be used to study the properties of solids for a wide range of pressure and temperature, viz. from room temperature upto the melting temperature of solids and from atmospheric pressure upto the structural transition pressure of solids. The theory may be used to study the temperature dependence of interatomic separation.

Gruneisen theory of thermal expansion as formulated by Born and Huang [6] has been used by Shanker et al. [7] and claimed to report what became known as ‘Shanker formulation’, which has been widely used in the literature [7], [8], [9], [10], [11], [12], [13], [14], [15] for the determination of the temperature dependence of interatomic separation. Moreover, it has been found that the formulation does not work under high pressure and needs modification [16].

He and Yan [17] used the quadratic expansion of α with T, to study the temperature dependence of r, and claimed to have got a new relation. Singh and Chauhan [18] used the linear expansion of α with T and also claimed to have got a new relation for the temperature dependence of r. Kushwah and Shanker [19] studied the thermal expansion of MgO in the temperature range (300–1800 K) using Guillermet and Gustafson [20] relation, which is based on the linear expansion of α with T.

Thus, there are many formulations with their advocates. It is therefore legitimate and may be useful to discuss that these models are not different and can be unified. The unification may help the researchers to reach on a single platform to study the temperature–volume EOS for solids, which is the purpose of the present paper.

Section snippets

Method of analysis

Kushwah et al. [19], [21] as well as Chauhan and Singh [22] reported the following relation:VV0=[1-α0δT(T-T0)]-1/δT

A comparison of Eqs. (4), (5) shows that Eq. (5) is exactly same as Eq. (4).

He and Yan [17] assumed that α depends quadratically on T as given below:α=α0+α02δT(T-T0)+α03δT2(T-T0)2

Using the definition of α, the integration of Eq. (6) gives the following relation as presented by He and Yan [17]:rr0=exp[13{α0(T-T0)+12α02δT(T-T0)2+13α02δT3(T-T0)3}]

It should be mentioned here that Eq.

Results and discussion

We have thus presented a critical analysis of various relations reported in the literature to study the temperature dependence of r. It is found that the relations reported by He and Yan [17] as well as Singh and Chauhan [18] are the approximate forms of Eq. (4). The relation of Singh and Chauhan [18] is exactly same [23] as that of Guillermet and Gustafson [20] relation as used by Kushwah and Shanker [19]. Eq. (17) reported as ‘Shanker formulation’ in the literature [7], [8], [9], [10], [11],

Conclusion

We conclude that various relations that appeared in the literature are the approximate forms of Eq. (4) and therefore these relations may be unified as Eq. (4). The unified relation (Eq. (4)) has been found to work well for simple solids, viz. potassium and rubidium halides, as well as some important minerals of geophysical importance (MgO and CaO). The unification presented in the present paper makes the theoretical formulation as simple as possible. Such a unified analysis has not been

Acknowledgment

We are thankful to the referee for his valuable comments, which have been used in the revised manuscript.

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