Temperature dependence of the single-ion lattice anisotropy coefficients of a ferromagnetic Heisenberg monolayer

In investigations of the orientation of the magnetization of thin ferromagn etic films, the single-ion anisotropy coefficients play a crucial role. By applying a thermodynamic perturbation theory we calculate the temperature d ependence of the second- and fourth-order single-ion anisotropies for a Hei senberg monolayer with Tyablikov decoupling (the random-phase approximation , RPA) and compare with results obtained from mean-field theory. In order t o assess the accuracy of the Tyablikov (RPA) and also the Callen decoupling approximations in the Green's function many-body theory, we calculate the magnetization of a Heisenberg spin pair and of a ferromagnetic monolayer, a nd compare the results with exact solutions available for the spin pair wit h arbitrary spins, and a recent quantum Monte Carlo calculation (for spin 1 /2) for the monolayer. The RPA decoupling provides a fairly good approximat ion to the exact results for the magnetization over the whole temperature r ange of interest. Because of this, we expect the calculated anisotropy coef ficients to be approximated well by this method.