Uniaxial Anisotropy

The simplest form of crystal anistropy is uniaxial anisotropy. For cubic crystals the anisotropy energy can be expressed in terms of the direction cosines ($\alpha_1$, $\alpha_2$, $\alpha_3$) of the internal magnetisation with respect to the three cube edges. Due to the high symmetry of the cubic crystal this can be expressed in a simple manner as a polynomial series in the direction cosines. This can be simplified further[10] to

$$E_{a} = K_{1}\left( \alpha_{1}^{2}\alpha_{2}^{2} + \alpha_{2}^{2}... ...ight) + K_{2}\left( \alpha_{1}^{2}\alpha_{2}^{2}\alpha_{3}^{2} \right) + \cdots$$ (5.1)

When $K_{1} > 0$ the first term of Equation 5.1 becomes a minimum for $\langle100\rangle$ directions, whilst for $K_{1} < 0$ it is a minimum for $\langle111\rangle$ directions.