Distant-neighbour interactions in ferromagnetic systems (BPW approximation)

Abstract

A simple approximation is described which may be applied to Ising and classical Heisenberg ferromagnetic systems containing forces of arbitrary range. (In this paper this approximation is loosely referred to as an `extension' of the usual BPW (Bethe-Peierls-Weiss) approximation although this is not strictly true.) In the case of the S = ½ Ising model in the absence of an external field, it is shown that the present results reduce correctly to the first two terms (i.e. the `point' and `bond') of the linked cluster expansion developed by Hiley and Joyce. Specific calculations for the second-neighbour problem (S = ½ Ising model) are compared with results obtained using exact series expansions, and a formal similarity (within this approximation) between the S = ½ Ising and the classical Heisenberg models is pointed out.