3.3.3 Zeeman Energy

The Zeeman energy of a magnetic body $\boldsymbol{J}(\boldsymbol{x})$ in an external field $\boldsymbol{H}_\mathrm{ext}(\boldsymbol{x})$ is simply given by

$$E_\mathrm{ext}= \int_\Omega w_{\mathrm{ext}}(\boldsymbol{J}) d{v} = \int_\Omega (-\boldsymbol{J} \cdot \boldsymbol{H}_\mathrm{ext}) d{v} =$$ (3.29)

$$= \int_\Omega \sum_j J_{\mathrm{s},j} \sum_{k}^{\{x,y,z\}} u_{j,k} \eta_j H_{\mathrm{ext},k} d{v} \quad.$$ (3.30)

For the gradient we find

$$\frac{\partial E_\mathrm{ext}}{\partial u_{i,l} } = -\int_\Omega \sum_j J_{\mathrm{s},j} \frac{\partial}{\partial u_{i,l}} \sum_{k}^{\{x,y,z\}} u_{j,k} \eta_j H_{\mathrm{ext},k} d{v} =$$ (3.31)

$$= \int_\Omega \sum_j -J_{\mathrm{s},j} \sum_{k}^{\{x,y,z\}} \delta_{ij} \delta_{lk} \eta_j H_{\mathrm{ext},k} d{v} =$$ (3.32)

$$= -J_{\mathrm{s},i} \int_\Omega \eta_i H_{\mathrm{ext},l} d{v} \quad.$$ (3.33)

Since we know the external field explicitly, we can just simply add it to the other contributions to the effective field.