Spin-Wave Interactions in an Anisotropic Ferromagnet

In the presence of anisotropic interactions the spin-wave dispersion parameter $\mathcal{D}\mathrm{}\left(T\right)\mathrm{}$ acquires an $a{T}^{\frac{3}{2}}$ dependence in addition to the $b{T}^{\frac{5}{2}}$ dependence due to isotropic exchange. We have used Van Vleck's anisotropic exchange, and the largest effect consistent with the magnetocrystalline anisotropy of a cubic ferromagnet is found to come from the pseudodipolar coupling. While the anisotropy appears only in the second order of perturbation theory, there is a first-order contribution to $\mathcal{D}\mathrm{}\left(T\right)\mathrm{}$ which varies as the third power of the magnetization, itself a function of temperature. Thus $a\approx C{\left(\frac{g\beta {H_A}^D}{k_B{T}_c}\right)}^{\frac{1}{2}}$, where $C$ is the coefficient in the Bloch law, ${H_A}^D$ is the pseudodipolar contribution to the anisotropy field, and $T_c$ is the Curie temperature. The coefficient $a$ depends upon the direction of spin-wave propagation and averages to zero over a sphere. In a first approximation, then, there is no $T^3$ term in the magnetization. In an experiment dealing with selected propagation directions, such as spin-wave resonance or inelastic neutron scattering, since $b\approx \frac{C}{T_c}$, an effect important when $\frac{T}{T_c}\lesssim {\left(\frac{g\beta {H_A}^D}{k_B{T}_c}\right)}^{\frac{1}{2}}$ is predicted.