Suppose that a ferromagnet is
single-domain in the strictest sense: the magnetization is uniform and rotates in unison. If the
magnetic moment is \boldsymbol{\mu} and the volume of the particle is V, the magnetization is \mathbf{M} = \boldsymbol{\mu}/V = M_s \left(\alpha,\beta,\gamma\right), where M_s is the
saturation magnetization and \alpha, \beta, \gamma are
direction cosines (components of a
unit vector) so \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 z direction, the
anisotropy energy can be expressed as one of the forms: :E = KV \left(1 - \gamma^2 \right) = KV \sin^2\theta, where V is the volume, K the anisotropy constant, and \theta the angle between the easy axis and the particle's magnetization. When shape anisotropy is explicitly considered, the symbol \mathcal{N} is often used to indicate the anisotropy constant, instead of 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 :E = K_aV\alpha^2 + K_bV\beta^2. If K_a > K_b > 0, the easy axis is the z direction, the intermediate axis is the y direction and the hard axis is the 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 :E = KV \left(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\right). If K > 0, the easy axes are the x, y, and z axes. If K there are four easy axes characterized by x = \pm y = \pm z. == See also ==