The statistical mechanics of a general spin-$S$ magnetic system, which is described by a Heisenberg-Dirac isotropic exchange Hamiltonian with single-ion anisotropy included, has been studied with the aid of the Green-function technique. This problem has been set up in a new formalism in which it is not necessary to decouple the anisotropy Green functions. A scheme has been found for decoupling each of the exchange Green functions. For zero anisotropy, our results reduce to the usual random-phase approximation. For finite values of the anisotropy parameter $D$ the ensemble averages $〈{\left(S_g^z\right)}^n〉$ for $n$ integer, show a greater dependence on $D$ than they do in the molecular-field-theory (MFT) calculation. Unlike the results of some of the previous decoupling schemes used on this problem, our prediction for the transition temperature $T_c\mathrm{}\left(D\right)\mathrm{}$, as a function of $D$, remains finite as the anisotropy becomes infinite. The asymptotic value of $T_c\mathrm{}\left(D\right)\mathrm{}$ as $D\to \infty$ for our Green-function calculation is the same as the asymptotic value of the MFT prediction, $T_c^{\mathrm{MFT}}(D\to \infty )$. We present here the appropriate formalism for antiferromagnetic as well as for ferromagnetic systems.

• Received 10 December 1970

DOI:https://doi.org/10.1103/PhysRevB.4.136