Micromagnetic description

Since numerical computations based on the equations in section 2.3 are at an atomic level, they are historically limited to simple cases containing not too many degrees of freedom (Aharoni, 2000, p173). For larger problems other techniques must be used.

Brown (1963) suggested a theory which is referred to as micromagnetic theory. Instead of considering individual magnetic moments, a continuous magnetisation function $\ensuremath{\mathbf{M}}$ is used to approximate the atomic interaction described above. The magnetisation represents the locally averaged density of magnetic moments:

$\displaystyle \ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}})$

$\displaystyle = \displaystyle {1 \over V(\ensuremath{\mathbf{r}},\Delta r)} \displaystyle \sum_{i \in \mathbb{J}(\ensuremath{\mathbf{r}}, \Delta r)}$ $\displaystyle \mbox{\boldmath {$\mu$}}$ $\displaystyle _i$

(2.16)

where $V(\ensuremath{\mathbf{r}}, \Delta r)$ is a sphere of radius $\Delta r$ placed at $\ensuremath{\mathbf{r}}$ and $\mathbb{J}(\ensuremath{\mathbf{r}}, \Delta r)$ is the set of indices:

$\displaystyle \mathbb{J}$

$\displaystyle =$

$\displaystyle \left\{ i : \ensuremath{\mathbf{r}}_i \in V(\ensuremath{\mathbf{r}}, \Delta r) \right\}$

(2.17)

for magnetic moments $\mu$ that are located inside the volume $V(\ensuremath{\mathbf{r}}, \Delta r)$ .

This averaging can be performed over the scale of the exchange length (see equation 2.40) and will always contain many magnetic moments.

$\ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}})$ is assumed to be a continuous and differentiable function which allows the expression of the interactions described above using differential operators. The resulting equations can be solved analytically (if possible) or numerically.

**Figure:** The unit vectors of two moments **$\ensuremath{\mathbf{S}}_i$** and **$\ensuremath{\mathbf{S}}_j$**
$\includegraphics[width=1.0\textwidth,clip]{images/exchangecontinuum2}$