Phase diagram of an Ising model with random sublattice vacancies

We use a modified Kadanoff's variational method to calculate the phase diagram of an Ising model with random vacancies on one of two interpenetrating sublattices of the isotropic square and body-centered-cubic lattices. We find second-order phase transitions only for $T>\mathrm{}0\mathrm{}$. The transition temperature to very good approximation decreases linearly with impurity (i.e., vacancy) concentration at small concentration. This agrees with the linear decrease observed in other systems. A plausible explanation of the absence of first-order transitions for $T>\mathrm{}0\mathrm{}$ is given. The relation of our models to certain percolation problems is similar to that of some spin models studied by Syozi. Thus our results allow an estimate of critical probabilities for percolation.