On the Ising problem and Mayer's cluster sums

Abstract

Abstract When Mayer’s imperfect-gas formalism is applied to the Ising problem emphasis is focused on certain irreducible cluster sums, βk. It is shown that there is no contribution to βk unless the k +1 systems concerned occupy a set of lattice sites which are themselves either (a) a single point, (b) a pair of neighbouring points or (c) multiply-connected on the lattice. Each βk is the sum of such contributions. The familiar quasi-chemical approximation results from neglecting all contributions to the βk from multipty-connected sets of occupied lattice sites. A second approximation, involving the smallest of such multiply-connected sets of occupied sites, can always be carried through explicitly.