On the density and temperature dependence of the critical behavior in non-random frozen spin clusters

The statistical mechanics of clusters of a magnetic material (where the distribution function for the clusters is fixed and dependent on the history of the sample) in a non-magnetic matrix is studied for the plane-square lattice. The spin-configurational partition functions for small clusters are evaluated explicitly giving density and low-temperature series for the magnetic susceptibility, χ, for both random and non-random clusters. For random clusters with net density ϱ at the critical point of the pure magnetic material we find that χ goes to infinity as (1-ϱ)-32 while for non-random clusters (treated as a frozen Ising mixture) we find that χ is infinite at the double critical point (magnetic critical point and critical point for clustering).