Relaxation of Ferromagnetic Precession by Excitation of Spin Waves in Polycrystalline Nickel-Cobalt Ferrites

A detailed test has been made of the spin-wave theory for anisotropy broadening in polycrystalline ferrites. For this purpose, the effective linewidth $W$ and the effective line shift $S$ have been measured as a function of the applied magnetic field, at 9 GHz, for samples of the composition ${\mathrm{Ni}}_{1-x}{\mathrm{Co}}_x{\mathrm{Mn}}_{0.05}{\mathrm{Fe}}_{1.95}{\mathrm{O}}_4$, with $x=0, 0.015, 0.027, \mathrm{and} 0.05$. For these ferrites, the first-order cubic anisotropy field $\frac{2K_1}{M_s}$ varies from—540 Oe for $x=0\mathrm{} \mathrm{to} +460\mathrm{}$ Oe for $x=0.05\mathrm{}$, whereas $4\pi M_s=3400\mathrm{}$ G. The spin-wave theory may thus be expected to be valid. To reduce the influence of porosity on $W$ and $S$, very dense materials ($p<\mathrm{}0.8\%$) have been used. Well within the limits of the spin-wave manifold, the experimental data are in very good agreement with a theory by Schlömann, except for $x=0\mathrm{}$. Near the edges of the manifold, however, the predicted singularities and discontinuities in $W$ and $S$ are not found. It is suggested that this must be attributed to a broadening of the spin-wave frequencies themselves by the variations in the anisotropy field. The amount of this broadening is calculated for a simple model, where the anisotropy field varies sinusoidally. The result for the simple model is then generalized to the case where many Fourier components are present in the anisotropy field. The modifications in the theory lead to an improved agreement with the experiments, especially near the edges of the spin-wave manifold. A further modification of the theory is proposed for the case where the anisotropy field is comparable to or larger than the saturation magnetization. The distribution of anisotropy fields typical for cubic anisotropy must then be introduced into the theory. The theoretical predictions are in fair agreement with recent data of Patton. For pure nickel ferrite, none of the modified theories leads to a good fit with our experiments. This discrepancy remains unexplained. Outside the spin-wave manifold, $W$ has a constant value. At high fields, i.e., below the manifold, $W$ increases linearly with $x$ (16 Oe/% Co). At low fields, the values are higher and not linear in $x$.