Primitive Theory of Ferrimagnetic Resonance Frequencies in Rare-Earth Iron Garnets

The effective $g$ factor for ferrimagnetic resonance frequencies in rare-earth iron garnets is calculated by direct inspection of eigenvalues rather than study of the equations of motion. The rare-earth ions are treated as captive in the exchange field from the iron, but subject to decomposition of their energy levels by crystalline fields and/or spin-orbit interaction. With a crystalline field, the problem is tractable in a simple way only if these decompositions are large or small compared to those which could be produced by the exchange field acting alone. Anisotropy, actually very important at low temperatures, is neglected except insofar as it can be represented by an anisotropy field. The concept of "fictitious spin" is useful, and the spectroscopic splitting factors turn out to be more relevant than the true gyromagnetic ratios. For europium garnet, our theory becomes essentially that of Wolf. It is shown that Kittel's formula $g_{\mathrm{eff}}=\frac{2(M_{\mathrm{Fe}}+M_{\mathrm{RE}})}{M_{\mathrm{Fe}}}$ has approximate validity if most of the magnetic moment of the rare earth arises from nondiagonal matrix elements joining ionic energy levels with separations large compared with the Zeeman energy in the exchange field. The fact that in certain cases the experimental results are represented fairly well by Kittel's formula is hence not necessarily to be construed as evidence that the rare-earth ion is highly damped by spin-lattice interaction as in his original model.