Sensitivity of Curie temperature to single-ion anisotropy

Existing numerical estimates of the sensitivity of Heisenberg Curie temperature $T_C$ to crystal-field anisotropy have been obtained (primarily by Green's-function techniques) only for simple easy-axis anisotropy and low values of spin quantum number. In this paper we demonstrate the usefulness of the correlated-effective-field (CEF) theory in this context, showing that with modest computational effort numerical results for both easy-axis and easy-plane anisotropies can be obtained for any spin quantum number of physical relevance. Moreover, for cases where a comparison can be made with the existing literature, we show that the CEF results are more accurate than those obtainable from first-order equation-of-motion Green's-function techniques. Numerical calculations in the correlated-effective-field approximation are given for all three cubic lattices subject to the Hamiltonian $\mathcal{H}={\Sigma }_{i}D{S}_{\mathrm{iz}}^2-{\Sigma }_i{\Sigma }_{i}J{\overrightarrow{\mathrm{S}}}_i·{\overrightarrow{\mathrm{S}}}_j$, where $J$ is a nearest-neighbor exchange parameter. Results are given for the complete range $-\infty \le \frac{D}{J}\le \infty$ (where negative values refer to an easy-axis situation and positive values to a preferred easy plane) and for spin quantum numbers $S=1\mathrm{} \mathrm{to} \frac{7}{2}$ inclusive. For integer spin values and an easy-plane anisotropy we encounter the problem of singlet-ground-state ferromagnetism for which $T_C\to 0$ at a finite value of $\frac{D}{J}$.