Self-consistent description of a system of interacting phonons

Abstract

A proposal for a method of self-consistent description of phonon systems. This method generalizes the Debye model to account for phonon-phonon interaction. The idea of “self-consistent” phonons is introduced; their speed depends on the temperature and is determined by solving a non-linear equation. The Debye energy is also a function of the temperature within the framework of the proposed approach. The thermodynamics of “self-consistent” phonon gas are built. It is shown that at low temperatures the cubic law temperature dependence of specific heat acquires an additional term that is proportional to the seventh power of the temperature. This seems to explain the reason why the cubic law for specific heat is observed only at relatively low temperatures. At high temperatures, the theory predicts a linear deviation with respect to temperature from the Dulong-Petit law, which is observed experimentally. A modification to the melting criteria is considered, to account for the phonon-phonon interaction.