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Dipolar energy

Dipolar energy (often called magnetostatic or demagnetising energy) is the resultant energy from the interaction of magnetic moments with each other. Two magnetic moments at positions $ \ensuremath{\mathbf{r}}_i$ and $ \ensuremath{\mathbf{r}}_j$ have the dipolar energy:


$\displaystyle \mathcal{E}_{\mathrm{di}}^{i,j}$ $\displaystyle =$ $\displaystyle \mu_0 \left( {\mbox{\boldmath {$\mu$}}_i\cdot\mbox{\boldmath {$\m...
...math{\mathbf{r}}_{ij}) \over \vert\ensuremath{\mathbf{r}}_{ij}\vert^5 } \right)$ (2.12)

$ \mathcal{E}_{\mathrm{di}}^{i,j}$The dipolar energy between two magnetic moments $ \mu$$ _i$ and $ \mu$$ _j$

where


$\displaystyle \ensuremath{\mathbf{r}}_{ij}$ $\displaystyle =$ $\displaystyle \ensuremath{\mathbf{r}}_j - \ensuremath{\mathbf{r}}_i$ (2.13)

$ \ensuremath{\mathbf{r}}_{ij}$The distance between two magnetic moments $ \mu$$ _i$ and $ \mu$$ _j$ at positions $ \ensuremath{\mathbf{r}}_i$ and $ \ensuremath{\mathbf{r}}_j$

For $ N$ magnetic moments this becomes:


$\displaystyle \mathcal{E}_{\mathrm{di}}$ $\displaystyle =$ $\displaystyle {1 \over 2} \sum_{i=1}^{N}\sum_{j\neq i} \mathcal{E}_{\mathrm{di}}^{i,j}$ (2.14)

Computing the dipolar energy is the most expensive part of any micromagnetic simulation as the dipolar energy is a long-range interaction and therefore must consider the interaction of each magnetic moment $ \mu$$ _i$ with every other magnetic moment $ \mu$$ _j$.


next up previous contents
Next: Total energy Up: Interactions between atomic magnetic Previous: Zeeman energy   Contents
Richard Boardman 2006-11-28