Self-consistent statistical theory of crystal structures

In this presentation I give a short review of the state of the art in the development of the statistical theory of crystal structures. It is a generalization of the unsymmetrized self-consistent field approximation proposed and applied earlier to crystals with Bravais lattices. This method is an alternative to the known self-consistent phonon theory of strongly anharmonic solids. It is free of any perturbative schemes and can be applied with in principle any type of empirical or semi-empirical interatomic potential functions. The basic equations are derived and solved for a few graphene-type systems. The solutions of these equations are used as a background for the investigation of some structural properties of these materials. We calculate the equilibrium nearest neighbor distances and the lattice parameters, the coefficient of thermal expansion, the effective amplitudes and other higher moments of the configurational atomic phase probability densities. We also investigate the thermodynamics of these systems, and calculate the Gruneisen parameter, the thermal coefficient of pressure, the thermal capacities at constant volume and pressure, the isothermal and adiabatic compressibilities. We show here also results for the energy of formation of defects like vacancies, and also for the surface free energy. For imperfect crystals we calculate the distortion of the lattice and the softening of the atomic vibrations at the vicinity of the surfaces. We also give comparisions to other available theoretical, experimental, and simulational results.