Nonlinear Effects in the Phonon System of Diamond Crystal

Abstract

Thermodynamic properties of diamond are theoretically investigated on the ground of self-consistent description of a phonon gas in lattice, which generalizes the Debye model with taking into account the phonon-phonon interaction. In many cases properties of crystals of certain symmetry can be well approximated by a model of an isotropic continuous medium, if its elastic moduli are chosen optimally. They should be found for a crystal of each symmetry from the condition of their proximity to the exact elastic moduli, which are measured experimentally and are given in the corresponding tables. At high temperatures, the nonlinear phonon interaction takes into account both three- and four-phonon interactions. In this reason we take into account not only the second-order elastic moduli tensor in the reduced isotropic crystal model, but also the third- and fourth-order elastic moduli tensors, which are all together characterized by nine independent components. Account of the phonon-phonon interaction leads to the redefinition of the phonon’s speed and of the Debye energy. Their dependence on the temperature occurs. In the absence of interaction and neglecting the nonlinear effects, the phonons are the same as that of the Debye model. They are called "bare" or "Debye". Phonons whose speed is renormalized due to the interaction are called the “self-consistent” ones. It is shown that, at high temperatures, the theory predicts the linear in the temperature deviation of the isochoric heat capacity from the Dulong-Petit law. Unlike for the most crystals, where the decrease in the isochoric heat capacity is observed, our calculations for diamond and crystals with diamond structure predict the linear increase of the isochoric heat capacity with the temperature, viewed experimentally. The isobaric heat capacity of diamond, similar to other substances, linearly increases with the temperature.