Theory of Ferromagnetic Resonance in Rare Earth Garnets. I.gValues

Ferromagnetic resonance phenomena in rare earth iron garnets (except Gd iron garnet) are dominated at room temperature and above by the rapid spin relaxation of the rare earth ions. The rare earth ion relaxation controls the $g$ values and the line widths, as well as the temperature dependence of these quantities. It is shown that in the appropriate limit the $g$ value of the microwave spin resonance line satisfies the relation $g=\frac{g_A(M_A+M_B)}{M_A}$, where $g_A$ and $M_A$ refer to the ferric lattice and $M_A+M_B$ is the net saturation magnetization of the crystal. This relation obtains essentially when the $B$ lattice relaxation frequency is high in comparison with the $\mathrm{AB}$ exchange frequency and relaxation frequency of the $A$ lattice. The theory accounts quite well, with no disposable parameters, for the sequence and temperature variation of the $g$ values reported for Dy, Ho, Er, Yb, and Sm iron garnets. When at low temperatures the $B$ relaxation frequency becomes sufficiently low, the $g$ value should approach the usual result for two coupled undamped lattices, in the absence of anisotropy effects on the $B$ lattice. The theory predicts further that at ordinary temperatures the exchange frequency resonance will occur at the usual position and its width will be proportional to the damping constant of the $B$ lattice.