Dynamics of Localized Moments in Metals. I. Kubo Formalism and Relaxation

The dynamic transverse susceptibility ${\chi }^{-}\left(\omega \right)$ is calculated microscopically for a random array of localized spins in a metal. Relaxation of both local and conduction-electron spins towards their respective instantaneous local fields is included using a new formalism for such instantaneous relaxation. The resulting equations for the relevant propagators are solved to first order in the exchange coupling constant $J$, and it is demonstrated that to this order, the results are identical to those obtained using a molecular-field model. Careful attention is paid to the correct analytic form of the spin propagators and to the sum rules which they must satisfy. It is shown that in this formalism an additional inhomogenous contribution to ${\chi }^{-}\left(\omega \right)$ arises from the requirement of relaxation to the local field. When this term is added to the usual Lorentzian form for the spin propagators, a form for ${\chi }^{-}\left(\omega \right)$ results which is distinctly non-Lorentzian, but which satisfies the physical requirement that the static limit ($\omega =0\mathrm{}$) of the transverse susceptibility equal the longitudinal value.