Series expansions for the magnetisation of a solid superparamagnetic system of non-interacting particleswith anisotropy

The calculation of the magnetisation curve of an assembly of non-interacting fine superparamagnetic particles, with uniaxial anisotropy and easy axes fixed in a solid non-magnetic matrix is considered. The presence of anisotropy complicates the calculation which otherwise would result in the Langevin function. The calculation for particles with anisotropy and easy axes fixed at arbitrary angles to the external field, requires the calculation of the partition function, which has previously been expressed exactly as a double integral or as a sum of single integrals. We have recently shown how the partition function can be reduced to a single integral and here we show how this can be expressed as a double infinite series containing known functions. Special cases are considered, some existing analytic formulae are reobtained, and some new analytic formulae are presented. For identical particles the deviation from the Langevin function is known to be considerable. The formulae presented should facilitate the incorporation of the effects of anisotropy.