Computational models

Equation 2.15 requires the evaluation of a number of sums. The computational effort for $n$ magnetic moments scales as $\mathcal{O}(n^2)$ as a result of the dipolar term (see section 2.3.4).

Brown's continuum approximation postulates that the magnetisation $\ensuremath{\mathbf{M}}$ (i.e. the magnetic moment per unit volume) can be regarded as a continuous function of space. This allows an approximation of equation 2.15 to be expressed as a partial differential equation (equation 2.32) for which the standard mathematical techniques for solving PDEs can be used.

The following sections describe different approaches attacking this challenge. In section 2.6.1 the Stoner-Wohlfarth model is described which reduced the number of degrees of freedom to tackle the reversal of small magnetic particles.

In section 2.6.2 we show how the Landau-Lifshitz-Gilbert (LLG) equations can be used to determine the time development of the magnetisation once the effective field is determined through Brown's static equations.

Section 2.7 introduces the simulation packages used n this work which solve the equations of Brown and Landau-Lifshitz-Gilbert numerically -- this is commonly referred to as computational micromagnetism.