Paramagnetic-Impurity Spin Resonance and Relaxation in Metals

The response of a paramagnetic spin in a metal to applied time-dependent magnetic fields can be described in terms of frequency-dependent susceptibilities of the Kubo type. The diagrammatic technique used here to evaluate these susceptibilities has the advantage that approximations are made by expanding in powers of a small parameter, so that corrections to the results are known to be small if certain conditions are satisfied. For example, corrections to the formula given below for the transverse susceptibility are of order $\beta (\omega -{\omega }_R)$ and $\beta {\Gamma }_2$, where $\omega$ is the applied frequency, ${\omega }_R$ is the resonance frequency, and ${\Gamma }_2={T_2}^{-1\mathrm{}}$ is the transverse relaxation rate ($h=1\mathrm{}$); corrections to the formula for the longitudinal susceptibility are of order $\beta \omega$ and $\beta {\Gamma }_1$, where ${\Gamma }_1={T_1}^{-1\mathrm{}}$. An effective spin $S=\frac{1}{2}$ is assumed. The results of the diagrammatic analysis are interpreted in terms of the Bloch equations, modified to include relaxation to the instantaneous value of the magnetic field. The resonance frequency is calculated to second order in the interaction between spins and conduction electrons, while the relaxation times $T_1$ and $T_2$ are calculated to third order, and to this order all quantities are found to have a Kondo-like dependence on the logarithm of the temperature. The Korringa relation between the shift of the resonance frequency and $T_1$ is found to break down as the Kondo temperature is approached from above, but to third order in the interaction, the relation $T_1=T_2$ is found to be true for an isotropic interaction. Departures from thermal equilibrium of the conduction-electron system have been neglected in this work.