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The behaviour of ferromagnetic systems with single-ion anisotropies in morethan one direction is investigated with many-body Green's function theorygeneralizing earlier work with uniaxial anisotropies only. It turns out to beof advantage to construct Green's functions in terms of the spin operators S^x,S^y and S^z, instead of the commonly used S^+,S^- and S^z operators. Theexchange energy terms are decoupled by RPA and the single-ion anisotropy termsby a generalization of the Anderson-Callen decoupling. We stress that in thederivation of the formalism none of the three spatial axes is special, so thatone is always able to select a reference direction along which a magnetizationcomponent is not zero. Analytical expressions are obtained for all threecomponents of the magnetization and the expectation values <(S^x)^2>, <(S^y)^2>and <(S^z)^2> for any spin quantum number S. The formalism considers bothin-plane and out-of-plane anisotropies. Numerical calculations illustrate thebehaviour of the magnetization for 3-dimensional and 2-dimensional systems forvarious parameters. In the 2-dimensional case, the magnetic dipole-dipolecoupling is included, and a comparison is made between in-plane andout-of-plane anisotropies.Comment: 16 pages, 8 figures, missing figures adde

Many-body Green’s function theory of ferromagnetic Heisenberg systems with single-ion anisotropies in more than one direction