Derivation of the resonance frequency from the free energy of ferromagnets

Abstract

The general form for the magnetic resonance frequency \ensuremath{\omega} of anisotropic ferromagnets as derived from the free energy F by Smit and Beljers (\ensuremath{\omega}/\ensuremath{\gamma}${)}^{2}$=(M sin\ensuremath{\theta}${)}^{\mathrm{\ensuremath{-}}2}$(${\mathit{F}}_{\mathrm{\ensuremath{\theta}}\mathrm{\ensuremath{\theta}}}$${\mathit{F}}_{\mathrm{\ensuremath{\varphi}}\mathrm{\ensuremath{\varphi}}}$-${\mathit{F}}_{\mathrm{\ensuremath{\theta}}\mathrm{\ensuremath{\varphi}}}^{2}$), although numerically correct, is physically not convenient, because the origin of the different terms in F is obscured by an angular-dependent mixing. This mixing is avoided by using the relation (\ensuremath{\omega}/\ensuremath{\gamma}${)}^{2}$=1/${\mathit{M}}^{2}$ [${\mathit{F}}_{\mathrm{\ensuremath{\theta}}\mathrm{\ensuremath{\theta}}}$(${\mathit{F}}_{\mathrm{\ensuremath{\varphi}}\mathrm{\ensuremath{\varphi}}}$ /${\mathrm{sin}}^{2}$\ensuremath{\theta}+cos\ensuremath{\theta}/sin\ensuremath{\theta}${\mathit{F}}_{\mathrm{\ensuremath{\theta}}}$)-(${\mathit{F}}_{\mathrm{\ensuremath{\theta}}\mathrm{\ensuremath{\varphi}}}$/sin\ensuremath{\theta}-cos\ensuremath{\theta}/sin\ensuremath{\theta} ${\mathit{F}}_{\mathrm{\ensuremath{\varphi}}}$/sin\ensuremath{\theta}${)}^{2}$]. .sp Explicit expressions will show the symmetry of each of the terms in F for all magnitudes and directions of H. In addition, an alternate method which uses only rectangular coordinates and which can easily be generalized for multisublattice systems is described.