Dynamics of Localized Moments in Metals. II. Second-Order Exchange Effects

Abstract

We have extended our previous microscopic treatment of the dynamic transverse susceptibility for a random array of localized spins in a metal to include terms of second order in the exchange coupling constant $J$. Lattice relaxation of the localized and conduction electrons is included, as before, in such a way as to ensure relaxation to the instantaneous local field. The results, in the limit of equal conduction-electron and localized-spin $g$ values and no lattice damping, reduce to the correct ("bottlenecked") limit. The linewidth for frequencies close to the localized-spin resonance frequency agrees with previous calculations. Bottlenecking of both the longitudinal (frequency-modulation) ($T_2^{}{}_{}{}^{\prime }$) and transverse (spin-flip) ($T_1^{}{}_{}{}^{\prime }$) parts of the localized-spin resonance linewidth is demonstrated for equal $g$ values and no lattice relaxation. Similarly, the line-width for frequencies close to the conduction-electron resonance frequency exhibits both $T_2^{}{}_{}{}^{\prime }$- and $T_1^{}{}_{}{}^{\prime }$-type terms, and again bottleneck effects are present. The results are compared with previous macroscopic treatments. It is demonstrated that it is unnecessary to introduce detailed balance conditions per se in the microscopic theory. The relation between the conduction-electron-hole relaxation width and the one-electron width calculated by Overhauser is examined in an appendix.