Dynamical Properties of Magnetic Impurities in Transition Metals

The dynamical properties of a magnetic impurity in a metallic host are studied. We employ the Wolff model, which describes an impurity of the $3d$ series placed in a nonmagnetic $4d$ or $5d$ host. We calculate the dynamical susceptibility of a material in which a single impurity is imbedded. We imagine the system is placed in an external magnetic field at the absolute zero of temperature. It is assumed that the ground-state wave function is well approximated by the Hartree-Fock ground state in the presence of the external field. A generalized random-phase approximation (RPA) is employed in the equation of motion for the two-particle Green's function; this allows us to find and approximate expression for the two-particle correlation function. The same result may be obtained by a diagrammatic analysis, in which the contribution from a certain subset of diagrams is summed. The dynamic susceptibility obtained in this manner exhibits a resonance for frequencies in the vicinity of the free-election spin resonance frequency. The total transverse magnetic moment that arises in the system from the application of a field of fixed frequency and arbitrary spatial variation exhibits a resonance response at the free-electron spin resonance frequency, with vanishing width. The short-wavelength components of the induced spin density exhibit a resonance of finite width, shifted from the free-electron resonance frequency. The $g$ shift and width of the resonance in the short-wave-length response is independent of the wave vector of the component examined, so long as $v_{F}q$, $v_F{q}^{\prime }\gg \Omega$, where $v_F$ is the Fermi velocity of an electron at the Fermi surface, $\Omega$ and q' are the frequency and wave vector of the driving wave, and q is the wave vector of the component of the spin density in question.