Temperature dependence

Any anisotropy originating from the magnetic dipole interaction will vary with temperature as M² because the dipole field H_d ∝ M_s and the energy density E_a = −1/2μ₀M_s · H_d ∝ M²_s .
Single-ion anisotropy due to the crystal field exhibits a j (j + 1)/2 power law at low temperatures for the second-, fourth- and sixth-order terms, which vary respectively as the third, tenth and twenty-first powers of the magnetization; K_j (T )/K_j (0) = M^{j (j+1)/2}. Close to T_C, these turn into M^j power laws. Here is a rare example of an effect whose temperature dependence is determined by symmetry alone.