Magnetic Anisotropy at 0°K

Abstract

The results of the spin-wave method of Holstein and Primakoff are extended to obtain approximate values for the dependence of the energy of magnetically saturated cubic crystals upon the direction of magnetization. The anisotropy at 0\ifmmode^\circ\else\textdegree\fi{}K is calculated for the cases of dipolar ferromagnetism and exchange ferromagnetism. In a nearest neighbor approximation, the results agree with those obtained by Van Vleck from a treatment of ferromagnetic anisotropy. However, in the case of dipolar interaction, the nearest neighbor approximation is not sufficient and more distant neighbors are included. For dipolar ferromagnetism, the method gives the following results: In a face-centered cubic (fcc) lattice of spins $S$, the anisotropy constant ${K}_{1}$ is a maximum when there is no external field and is then given by ${K}_{1}=\ensuremath{-}0.11$ $\frac{{{M}_{0}}^{2}}{S}$, where ${M}_{0}$ is the saturation magnetization. In a body-centered cubic (bcc) lattice, with no external field, ${K}_{1}=\ensuremath{-}0.15\frac{{{M}_{0}}^{2}}{S}$. An external field must be applied to obtain saturation in a simple cubic (sc) lattice; ${K}_{1}=+0.43\frac{{{M}_{0}}^{2}}{S}$ when the external field $H=0.6{M}_{0}$. More generally, in the presence of an external field $H$, then ${K}_{1}=\ensuremath{\kappa}(\frac{{{M}_{0}}^{2}}{S}){((\frac{4\ensuremath{\pi}}{3})+\frac{H}{{M}_{0}})}^{\ensuremath{-}1}$, where $\ensuremath{\kappa}=\ensuremath{-}0.46$ for an fcc lattice, -0.63 for a bcc lattice, and +2.06 for an sc lattice. All values given are for specimens having zero demagnetizing field.