We propose a method of statistical description of simple crystals and calculation of their thermodynamic functions and equation of state. The method is based on the derivation of an exact expression for the binary distribution function of atomic displacements and a variational procedure for the determination of an effective constant of the quasielastic bond of atoms of the crystal. For rare gas crystals with Morse and Lennard-Jones potentials, we obtained the equation of state and thermodynamic parameters of the solid-state, which are in agreement with experimental data. We also found that a solid-state instability occurs near the observed melting temperature of the crystal, corresponding to a point above which there is no more an equilibrium effective parameter of the quasielastic bond.

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We demonstrate that the discrete character of the vibrational spectrum of a small crystal accounts for size dependence of its thermodynamic properties and melting temperature. Using a self-consistent statistical method [Phys. Rev. B 66, 054302 (2002)] we derive the Gibbs free energy of free nanocrystalline plates and calculate the thermodynamic parameters as functions of plate thickness for Cu.

The self-consistent quasiharmonic modol is used to describe the equilibrium atomic structure of a crystal and the thermodynamical properties of vacancies at high temperature. It is shown that an increase in the amplitude of the thermal atomic vibrations with temperature leads to a considerable reduction of the vacancy formation energy in the high-temperature region, which results in the generation of a large number of defects near the melting point and, as a consequence, in the significant softening of the acoustic vibration modes of the crystal. The effective attraction between the vacancies through the phonon subsystem of the crystal is found to lead to the first-order phase transition with a discontinuous change in the defect conceritration at the transition point. The melting lines are calculated for the rare gas crystals. The effects of the solid-vapour interface on t,he kinetics of the solid-liquid transition are studied.