1.1.4 Anisotropy

In ferromagnetic bodies it is very frequent to deal with anisotropic effects, due to the structure of the lattice and to the particular symmetries that can arise in certain crystals. In fact, in most experiments one can generally observe that certain energy-favored directions exist for a given material, i.e. certain ferromagnetic materials, in absence of external field, tend to be magnetized along precise directions, which in literature are referred to as easy directions. The fact that there is a ``force'' which tends to align magnetization along easy directions can be taken into account, in micromagnetic framework, by means of an additional phenomenological term in the free energy functional. To this end, let us refer to an elementary volume $\Delta V$, uniformly magnetized and characterized by magnetization unit-vector $\textbf{{m}}=\textbf{M}/M_s$. The magnetization unit-vector $\textbf{{m}}=m_x\mathbf{e}_x+m_y\mathbf{e}_y+m_z\mathbf{e}_z$ can be expressed in spherical coordinates by means of the angles $\theta$ and $\phi$ such that:

$$m_x =\sin\theta \cos\phi$$

$$m_y =\sin\theta \sin\phi$$ (1.39)

$$m_z =\cos\theta \quad.$$

The anisotropy energy density $f_$an$(\textbf{{m}})$ can be seen as a function of the spherical angles $\theta$ and $\phi$, and the anisotropy energy as

$$F_ (\textbf{{m}})=\int_\Omega f_ (\textbf{{m}}) dV \quad.$$ (1.40)

In this phenomenological analysis, it turns out that the easy directions correspond to the minima of the anisotropy energy density, whereas saddle-points and maxima of $f_$an$(\textbf{{m}})$ determine the medium-hard axes and the hard axes respectively.