Fine-structure splitting of a localized moment in a metal: A diagrammatic analysis

Beginning with a generalized $s-d$ Hamiltonian and including a term which induces electron-lattice relaxation, the Abrikosov fermion representation of spins and the Feynman-diagram technique are used to investigate the dynamics of localized moments. The analysis is performed for arbitrary spin, and in the presence of a fine-structure splitting of the localized-moment resonance by the crystalline field. A set of coupled linear equations is obtained which determines the macroscopic dynamic transverse susceptibility and which, under certain circumstances, is equivalent to a set of $2S+1$ coupled Bloch-type equations. These equations are analyzed in some detail for the high-temperature regime $\mathrm{kT}\gg S{g}_s{\mu }_B{H}_0$. In contrast to the case of the hyperfine splitting of a localized moment, it is found that the Korringa process alone leads to a narrowing of the resonance linewidth, even in the absence of a bottleneck. However, when the bottleneck is present, these narrowing processes are even larger. For a reasonable value of the exchange constant, it is at least qualitatively possible to explain the anisotropic behavior observed in recent electron-spin-resonance experiments of Mg:Gd alloys in terms of unresolved fine structure.