Dynamics of Localized Moments in Metals. III. Exchange Vertex Corrections and the Bloch Equation Parameters

Abstract

The exchange vertex correction to the dynamic uniform transverse conduction-electron susceptibility $\ensuremath{\chi}_{\mathrm{ee}}^{}{}_{}{}^{\ensuremath{-}+}(\ensuremath{\omega})$ is calculated in a dilute magnetic alloy. The relation between the magnetic resonance linewidth $\frac{1}{{T}_{2}}$ and the longitudinal relaxation rate $\frac{1}{{T}_{1}}$, as calculated by Overhauser, is carefully examined. It is shown that $\frac{1}{{T}_{2}}$ equals the sum of the imaginary parts of the "up" and "down" conduction-electron self-energies plus the vertex correction. The latter equals in magnitude the frequency-modulation contribution to $\frac{1}{{T}_{2}}$ and its inclusion results in an equality of ${T}_{1}$ and ${T}_{2}$, guaranteeing rotational invariance. An additional feature of our results is that the form of $\ensuremath{\chi}_{\mathrm{ee}}^{}{}_{}{}^{\ensuremath{-}+}(\ensuremath{\omega})$ is that appropriate to exchange relaxation to the instantaneous local field.