Self-Consistent Approximations for the Resonant Response of Magnetic Ions in Metals

The transverse response functions, which determine the resonant frequencies of the ions and electrons, are studied for spin-½ magnetic ions coupled to the conduction electrons of a host metal by the $s-d$ interaction. By means of Green's-function techniques, it is shown that the coupling of the ionic and electronic spin-wave modes in the equations for the response functions can be just as important as the shifts in the single-particle energies due to the interaction. Certain conserving properties, such as the existence of an undamped ionic mode whose frequency is unshifted by the interaction if the $g$ factors of the electrons and ions are equal, hold true if both the single-particle Green's functions and the equations for the response functions are calculated to the same order in perturbation theory. The exact equations to first order in the coupling strength are solved. Various approximations are made to the second-order equations.