A thermodynamically self-consistent quantum statistical theory of the crystalline state

The quantum hierarchy of equations for equilibrium density matrices obtained earlier by the author is extended to studies of the crystalline state. The thermodynamics of a crystal is constructed consistently with periodic solutions of the hierarchy. The structure of the pair correlation function is assumed to be the same as in a previous paper by the author on the classical case. The equations obtained are analysed in the quasiclassical approximation and in the low-temperature limit. The quasiclassical approximation gives quantum corrections to the classical formulae of the previous paper. The low-temperature limit leads to Debyeʹs and Grüneisenʹs laws without using the concept of phonons. The results obtained in this limit are compared with experimental data for rare gas solids. A method for solving the quantum equations based upon bifurcation theory is considered too. Results given by this method agree with the Landau theory of phase transitions