Phase diagrams of random Ising magnets

We examine the various kinds of phase diagrams that can occur in magnetic alloys when all the interactions have the same sign, i.e., all ferromagnetic. It is pointed out that these phase diagrams can be usefully cataloged in terms of the initial slope [$\frac{\partial \left(\ln T_c\right)}{\partial p}$] of the transition temperature $T_c$ with concentration $p$ at the two points $p=0\mathrm{}$ and $p=1\mathrm{}$. For the two-dimensional Ising model, these initial slopes can be obtained exactly, using perturbation theory, for alloys involving either bond or site disorder. For site disorder we obtain results for a spin $S$ in a host with spin ½. We show that mean-field theory is fairly reliable but fails to predict some kinds of phase diagrams that may occur.