Sensitivity of Curie Temperature to Crystal-Field Anisotropy. I. Theory

Abstract

This paper discusses the statistical mechanics of ferromagnetic and antiferromagnetic systems in the presence of uniaxial anisotropy, which is included both as anisotropic exchange $D_{\mathrm{ij}}S_i^{}{}_{}{}^{z}S_j^{}{}_{}{}^z$ and in the form of single-ion crystal-field terms $D_0{\left(S_i^{}{}_{}{}^z\right)}^2$. Emphasis is given to the calculation of magnetic transition temperatures $T_N$ and particularly to a discussion of the sensitivity of $T_N$ to crystal-field anisotropy. Earlier efforts in this direction have produced widely varying results, some Green's-function calculations predicting a sensitivity fully ten times the equivalent molecular-field result. A Green's-function theory is developed for which the decoupling of anisotropy terms is carried out in a manner which is essentially consistent with the random-phase decoupling of exchange terms, at least in the limit of small anisotropy. As such, the theory is an improvement on earlier decoupling schemes and allows, in particular, for a value of $〈{\left(S^z\right)}^2〉$ at $T_N$, which differs from the isotropic result $\frac{1}{3}S(S+1\mathrm{})\mathrm{}$. It indicates a sensitivity of $T_N$ to $D_0$ which is smaller than suggested by earlier Green's-function theories but still considerably larger than given by molecular-field theory. Quantitative calculations are carried out for simple cubic and body-centered cubic lattices and the detailed results for the different theories are compared. In Paper II the theory is applied to the salt Fe${\mathrm{F}}_2$ for which the spin Hamiltonian contains a sizeable crystal-field anisotropy of the form $D_0{\left(S_i^{}{}_{}{}^z\right)}^2$.