Application of continuous displacement treatment of CVM to atomic structure of solid surface

Abstract

A new formulation of the cluster variation method (CVM) which allows atoms to displace continuously from reference lattice points is applied to finite temperature studies of atomic structure in surface layers of a two-dimensional square lattice. The Lennard-Jones potential is assumed for nearest-neighboring pairs, and further neighbor interactions are neglected. The basic variables in the formulation are: (i) the probability f( r ) of finding an atom displaced at r from the reference lattice point; and (ii) the probability g( r, r′ ) of finding a pair of atoms displaced at r and r′ from the respective reference lattice points. The basic point of view is to regard an atom displaced at r as ‘an r species’, and treat the problem as a system containing an infinite number of species. The natural iteration method (NIM) formulation of the CVM can accommodate any number of species with equal ease, and does not cause any numerical difficulty. The main difference in the formulation from the conventional CVM of atoms on fixed lattice points is in the constraints to require f( r ) and g( r, r′ ) to satisfy the lattice symmetry. The free energy is minimized with respect to g( r, r′ ) to obtain the basic equations, which are also used for determining the surface relaxation. In the numerical calculations, we limit the atomic displacements to 121 local points around a reference lattice point. The NIM is used in the numerical solution of the basic equations. We use the angular coordinates (ϱ, θ) for r . The surface relaxation can also be seen directly by plotting the free energy values against the change of the lattice constant a 12 between the first and second layers. We have found that the surface relaxation is very sensitive to the temperature; outward surface relaxation occurs for lower temperatures, 0 < kT ε 0 < 0.3 , while an inward surface relaxation of the order of Δa 12 a T = 0.025–0.04 occurs at the temperature range of kT ε 0 = 0.4–0.6 , where a T is the bulk lattice constant at temperature T. Since the surface does not relax at T = 0 in the nearest-neighbor interaction, the inward relaxation of the present result is purely the effect of entropy occuring at finite temperatures.