Origin of magnetocrystalline anisotropy

There are two distinct sources of magnetocrystalline anisotropy:
• single-ion contributions;
• two-ion contributions.

Single-ion anisotropy The single-ion contribution is essentially due to the electrostatic interaction of the orbitals containing the magnetic electrons with the potential created at the atomic site by the rest of the crystal. This crystal-field interaction tends to stabilize a particular orbital, and by spin-orbit interaction the magnetic moment is aligned in a particular crystallographic direction. The single-ion contribution, where its effect on the paramagnetic susceptibility was described. In a ferromagnetic crystal, the contributions of all the ions are summed to produce a set of macroscopic energy terms with appropriate symmetry. The sum is straightforward when the local anisotropy axes of all the sites in the unit cell coincide. For example, a uniaxial crystal having n = 2 × 10²⁸ ions m⁻³, described by a spin Hamiltonian DS²_z with D/k_B = 1 K and S = 2 will have anisotropy constant K₁ = nDS² = 1.1 × 10⁶ J m⁻³. Broadside and head-to-tail configurations for a pair of ferromagnetically coupled magnetic moments. The latter is lower in energy.
Two-ion anisotropy The two-ion contribution often reflects the anisotropy of the dipole–dipole interaction. Comparing the broadside and head-to-tail configurations of two dipoles, each with moment m, that the head-to-tail configuration is lower in energy by 3μ₀m²/4πr³. Magnets tend to align head-to-tail. This anisotropy is of order 1 K per atom, or 100 kJ m⁻³. However, the dipole sum has to be extended over the entire lattice, and it vanishes for certain lattices (including all the cubic lattices). In noncubic lattices, the dipole interaction is an appreciable source of ferromagnetic anisotropy.
Another source of two-ion anisotropy is anisotropic exchange. The Heisenberg Hamiltonian is perfectly isotropic, but there are higher-order corrections involving spin-orbit coupling that lead to preferred orientations of the exchangecoupled pairs.