On the Mie-Grüneisen and Hildebrand approximations to the equation of state of cubic solids

Abstract

An analysis is made of the conditions under which the equation of state of a cubic solid under hydrostatic pressure takes the form given by either the Mie-Grüneisen or the Hildebrand approximation. The condition of validity of the vibrational or thermal formulation of the Mie-Grüneisen approximation is reduced to the existence of a purely volume-dependent characteristic temperature for the vibrational or thermal free energy of the solid. The analysis consists then in a search for temperature ranges where these restrictions on the functional form of the free energy, and the restrictions on the form of the internal energy imposed by the Hildebrand approximation, are satisfied for a non-metal in the quasi-harmonic approximation. The main results are as follows: (1) At temperatures somewhat above the Debye characteristic temperature for the (quasi-harmonic) high-temperature heat capacity at constant volume, it is appropriate to take as equation of state the vibrational Hildebrand equation: (2) at somewhat lower temperatures, this Hildebrand equation is generally more inaccurate than the corresponding Mie-Grüneisen equation; and (3) in the low-temperature T 3 region of the heat capacity, the equation of state reduces to the thermal Mie-Grüneisen equation. The explicit forms of the vibrational and thermal Mie-Gruneisen equations of state, and of their volume derivatives at constant temperature, are reported together with the corresponding Hildebrand equations. Some corollary results are obtained, within the quasi-harmonic approximation, on the temperature variation at constant volume of the Grüneisen parameters relating the explicit volume and temperature dependence of the vibrational and thermal free energy and of the entropy of a cubic solid, and (in an Appendix) on the temperature variation of the Debye temperatures appropriate to the various thermodynamic functions of any non-metal. The available experimental and theoretical evidence on the anharmonic contributions to the thermodynamic functions of solids is briefly discussed, and points to the conclusion that their weight is quite small in the region of temperature of interest for our analysis.