Dynamic Susceptibility of Magnetic Ions in Metals

We consider a magnetic dilute alloy system. The interaction between the localized magnetic moment and the conduction electrons is described by an $s-d$ exchange Hamiltonian. The transverse susceptibility of the impurity spin is calculated in a Green's-function formalism. By employing the Wick and linked cluster theorems for spin systems developed by Wang and Callen, we have been able to analyze the Green's function diagrammatically. In the high-field low-temperature region ($\mu H\gg \mathrm{kT}$) a partial summation of the main diagrams and lock diagrams gives, in addition to diagrams with self-energy structure, a set of diagrams with terminal parts. We have calculated the self-energy and the terminal function to the third-order in $N\left(O\right)J$, where $N\left(O\right)\mathrm{}$ is the density of states at the Fermi surface and $J$ is the exchange parameter. It is shown that there is a $\ln H g$ shift in the resonance frequency, and an $H\ln H$ correction to the analogous Korringa linewidth (which is linear in $H$). The magnetization is also calculated and is consistent with the earlier perturbation calculation.