Effect of Crystal-Field Anisotropy on Magnetically Ordered Systems

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 〈(Sgᶻ)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 Tc(D), as a function of D, remains finite as the anisotropy becomes infinite. The asymptotic value of Tc(D) as D→∞ for our Green-function calculation is the same as the asymptotic value of the MFT prediction, TcMFT(D→∞). We present here the appropriate formalism for antiferromagnetic as well as for ferromagnetic systems.