3.3.2 Magnetocrystalline Anisotropy Energy

The magnetocrystalline anisotropy energy for uniaxial anisotropy is given by

$$E_\mathrm{ani}= \int_\Omega \sum_j K_1(1-(\boldsymbol{a} \cdot \boldsymbol{u}_j \eta_j)^2) d{v} \quad.$$ (3.23)

The gradient is given by

$$\frac{\partial E_\mathrm{ani}}{\partial u_{i,l} }= \int_\O... ...dot u_{j,k}\eta_j) \right)^{\!\!\!\!2} \right) d{v}$$ (3.24)

$$\frac{\partial}{\partial u_{i,l}} \left( \sum_{k}^{\{x,y,z\}} (a_k \cdot u_{j,k}\eta_j) \right)^{\!\!\!\!2} = 2 \sum_{k}^{\{x,y,z\}} (a_k \cdot u_{j,k}\eta_j) \cdot \sum_{m}^{\{x,y,z\}} (a_m \delta_{ij} \delta_{lm}\eta_j) =$$

$$= 2 \sum_{k}^{\{x,y,z\}} (a_k \cdot u_{j,k}\eta_j) \cdot a_l \eta_i$$ (3.25)

and we get the result

$$\frac{\partial E_\mathrm{ani}}{\partial u_{i,l} }= -2 K_1 ... ...k}^{\{x,y,z\}} a_k u_{j,k}\eta_j \cdot \eta_i d{v} \quad.$$ (3.26)

This can be rewritten in matrix notation as

$$\boldsymbol{g}_\mathrm{ani} = G_\mathrm{ani} \cdot \boldsymbol{u}$$ (3.27)

with

$$G_{\mathrm{ani},i,l}= -2 K_1 a_l \int_\Omega \sum_{k}^{\{x,y,z\}} a_k \eta_j \cdot \eta_i d{v} \quad.$$ (3.28)