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

The exchange vertex correction to the dynamic uniform transverse conduction-electron susceptibility $\chi _{\mathrm{ee}}^{}{}_{}{}^{-+}\left(\omega \right)$ 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 $\chi _{\mathrm{ee}}^{}{}_{}{}^{-+}\left(\omega \right)$ is that appropriate to exchange relaxation to the instantaneous local field.