General Spin-Wave Dispersion Relations

Abstract

The general spin-wave dispersion relation obtained by a simultaneous solution of the equation of motion of the magnetization and Maxwell's equations is given. In contrast to previous calculations, the effects of conductivity, relaxation, exchange, and propagation are all properly taken into account. The resulting algebraic equation, being biquadratic in the square of the wave number, $k^2$, has four possible pairs of solutions. Some of these solutions correspond to growing plane waves while others represent attenuated ones in the direction of propagation. Whereas an analytical solution for $k$ could be easily obtained for the special case where the wave vector k is in the direction of the static magnetic field (${\theta }_k=0\mathrm{}$), the solution for the cases where ${\theta }_k\ne 0$ could be conveniently obtained only by numerical solution. The solutions for the latter cases have been obtained by using an IBM 709 computer and some of the representative results are given in this paper in graphical form. When relaxation and eddy current damping are neglected, our result reduces to that of Herring and Kittel in the static limit ($\omega \to 0$). Furthermore, it was found that the uniform precessional mode ($k=0\mathrm{}$) can truly exist only under very special conditions, namely, under the condition of zero permeability for one of the two normal modes in a gyromagnetic medium.