I. General theory

Abstract For alloys in which the numbers ot two kinds of atoms A and B are equal the statistical theory of superlattices was first developed by Bethe (1935), assuming interaction only between atoms which are nearest neighbours to one another. The theory was then extended by Peierls (1936) to the case of unequal concentrations of components still retaining the assumption of interaction between nearest neighbours, and by Chang (1937) to include the interaction between next nearest pairs of atoms in the case of equal concentartions of components. It is desirable to have a more general theory which includes long-range interactions of a general type and is applicable both to equal and unequal concentrations of components, but the interaction energy must fall off sufficiently rapidly with increasing distance to make the contribution of very distant atoms negligible so that the shape and the dimensions of the crystal block do not influence the state of the superlattice. For example, if the interaction energy is of the form V(r) = Ar-n, then n should be greater than 3 in order that the integral ∫∞0V(r) r2dr may be finite. A general theory of this kind has already been developed for a similar problem in adsorption (Wang 1938). The problem of adsorption is simpler than that of superlattices for two reasons, First, the former is a twodimensional problem while the latter is a three-dimensional one. Second, there is only one kind of atom in the adsorption case and therefore every lattice site is equivalent to every other, whereas in the superlattice case there are two kinds of atoms and hence two different kinds of lattice sites, usually called α sites and β sites respectively. This second difference between the problem of adsorption and that of superlattices is the one which makes the direct application of the adsorption theory impossible, because the device of replacing the contribution of distant adsorbed atoms by that of a continuous uniform distribution no longer works for the superlattice, since it would lead to a sort of amalgamation of the two kinds of lattice sites. This difficulty is overcome in the present theory as is explained in 2.