Derivation of equations of state for ideal crystals of simple structure

Abstract

We consider an approach to the derivation of thermodynamic equations of state by averaging the dynamic equations of particles of the crystal lattice. Microscopic analogs of macroscopic variables such as pressure, volume, and thermal energy are introduced. An analysis of the introduced variables together with the equations of motion permits obtaining the equation of state. Earlier, this approach was used to obtain the equation of state in the Mie-Gruneisen form for a one-dimensional lattice. The aim of this paper is to develop and generalize this approach to the three-dimensional case. As a result, we obtain the dependence of the Gruneisen function on the volume, which is compared with the computations performed according to well-known models with experimental data taken into account. It is proved that the Gruneisen coefficient substantially depends on the form of the strain state. Moreover, we refine the equation of state; namely, we show that the Gruneisen coefficient depends on the thermal energy, but this dependence in the three-dimensional case is much weaker than in the one-dimensional case. A refined equation of state containing a nonlinear dependence on the thermal energy is obtained