General Formalism for Magnetic Anisotropy Constants

Direct expressions for the magnetic anisotropy constants are given at a finite temperature from microscopic viewpoints. In the present derivation, it is assumed that the Hamiltonian is a linear function with respect to the magnetization direction. We discuss in detail the first-order constant $K_1$ and show that the results reproduce previous results. We also apply our method to Nd$_2$Fe$_{14}$B compounds and demonstrate that the temperature dependencies of the magnetocrystalline anisotropy constants $K_1$, $K_2$, and $K_3$ are successfully computed.