Superparamagnetic magnetization equation in two dimensions
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An equation for the dependence of magnetization on magnetic field in the case of two-dimensional (base plane) anisotropy has been derived. The resulting equation is expressed as an infinite series of modified Bessel functions, unlike the elementary function expressions that are applicable to the one-dimensional (axially anisotropic) and three-dimensional (isotropic) cases. Nevertheless, in the low-field limit, the series can be effectively truncated to give an approximate solution, while, in the high-field limit, an alternative expression has been derived which represents the limiting function as the field strength tends to infinity. The resulting expressions can be used to describe the superparamagnetic magnetization and susceptibility as a function of magnetic field in situations where the magnetic moments are constrained to lie in a plane, with no preferred direction within the plane. This can therefore be applied to two-dimensional structures, such as magnetic thin films, where magnetostatic energy confines the moments to the plane of the film, or to three-dimensional structures with planar magnetocrystalline anisotropy.
The following article appeared in Applied Physics Letters 77 (2000): 1029 and may be found at http://dx.doi.org/10.1063/1.1288677.