Magnetic anisotropy

Single-domain magnet

Suppose that a ferromagnet is single-domain in the strictest sense: the magnetization is uniform and rotates in unison. If the magnetic moment is ${\displaystyle \mathit{\mu }}$ and the volume of the particle is ${\displaystyle V}$, the magnetization is ${\displaystyle \mathbf{M}=\mathit{\mu }/V=M_s\left(\alpha ,\beta ,\gamma \right)}$, where ${\displaystyle M_s}$ is the saturation magnetization and ${\displaystyle \alpha ,\beta ,\gamma }$ are direction cosines (components of a unit vector) so ${\displaystyle {\alpha }^2+{\beta }^2+{\gamma }^2=1}$. The energy associated with magnetic anisotropy can depend on the direction cosines in various ways, the most common of which are discussed below.

Uniaxial

A magnetic particle with uniaxial anisotropy has one easy axis. If the easy axis is in the ${\displaystyle z}$ direction, the anisotropy energy can be expressed as one of the forms:

${\displaystyle E=KV\left(1-{\gamma }^2\right)=KV{\sin }^2\theta ,}$

where ${\displaystyle V}$ is the volume, ${\displaystyle K}$ the anisotropy constant, and ${\displaystyle \theta }$ the angle between the easy axis and the particle's magnetization. When shape anisotropy is explicitly considered, the symbol ${\displaystyle \mathcal{N}}$ is often used to indicate the anisotropy constant, instead of ${\displaystyle K}$. In the widely used Stoner–Wohlfarth model, the anisotropy is uniaxial.

Triaxial

A magnetic particle with triaxial anisotropy still has a single easy axis, but it also has a hard axis (direction of maximum energy) and an intermediate axis (direction associated with a saddle point in the energy). The coordinates can be chosen so the energy has the form

${\displaystyle E=K_{a}V{\alpha }^2+K_{b}V{\beta }^2.}$

If ${\displaystyle K_a>K_b>0,}$ the easy axis is the ${\displaystyle z}$ direction, the intermediate axis is the ${\displaystyle y}$ direction and the hard axis is the ${\displaystyle x}$ direction.

Cubic

A magnetic particle with cubic anisotropy has three or four easy axes, depending on the anisotropy parameters. The energy has the form

${\displaystyle E=KV\left({\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\right).}$

If ${\displaystyle K>0,}$ the easy axes are the ${\displaystyle x,y,}$ and ${\displaystyle z}$ axes. If ${\displaystyle K<0,}$ there are four easy axes characterized by ${\displaystyle x=\pm y=\pm z}$.