Statistical-thermodynamic description within the ring approximation. I. Lattice-gas model

The general method is elaborated for the statistical-thermodynamic description within the ring approximation of the lattice gas with a complex crystal lattice and with nonpair atomic interactions of any order and effective radii of action. The ring approximation corresponds to the first order of a modified thermodynamic perturbation theory under the choice of the inverse effective number of atoms interacting with one fixed atom as a small parameter of expansion. By the elaborated method one can calculate the complete phase diagram of the lattice gas as well as the correlation effects in both disordered and long-range ordered states of it. The elaborated method is general and analytically simple. The corresponding analytical expressions do not change their form at an increase of the effective radius of atomic interactions and are valid in case of any superstructure. The number of the "variational" parameters for minimization of the free energy is considerably fewer than that within the cluster-variation method and are determined by the type of the superstructure rather than by the value of the effective radius of atomic interactions. Due to the analytical nature of the ring approximation, the time for calculations within it is much less than that of the Monte Carlo simulation. By a comparison with the results of the Monte Carlo simulation the high numerical accuracy of the ring approximation is demonstrated in wide temperature-concentration intervals. The tendency of increase of the numerical accuracy of the ring approximation with increase of the effective radius of atomic interactions is shown. The applicability of the ring approximation is discussed. The obtained results may be useful for a description of solid solutions, alloys, magnetics, fluids, and amorphous materials.