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• W1975451839 abstract "Using the conventional exchange Hamiltonian ${mathcal{H}}_{mathrm{es}}$, together with an additional model Hamiltonian which provides a source of lattice relaxation for the conduction-electron magnetization, the coupled local-moment-conduction-electron transverse dynamic susceptibility is calculated. Feynman temperature-ordered Green's functions are used and are dressed with self-energies which are correct to second order in interaction parameters. A pair of coupled vertex equations are constructed which includes all diagrams correct to second order in the interaction parameters. In order to obtain the desired two-particle characteristics of the response function, some new manipulative and mathematical methods are introduced. These enable the vertex equations to be reduced to a coupled pair of linear equations which determine the coupled susceptibility. At sufficiently high temperatures the Kondo $g$ shifts and the weak frequency dependence of the self-energies can be ignored. These linear equations are then equivalent to the linearized version of the following Bloch equations: $frac{d}{mathrm{dt}}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s}={g}_{s}[{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s}ifmmodetimeselsetexttimesfi{}({stackrel{ensuremath{rightarrow}}{mathrm{H}}}_{mathrm{ext}}+ensuremath{lambda}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e})]ensuremath{-}(frac{1}{{T}_{mathrm{se}}})[{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s}ensuremath{-}{ensuremath{chi}}_{s}^{0}({stackrel{ensuremath{rightarrow}}{mathrm{H}}}_{mathrm{ext}}+ensuremath{lambda}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e})]+(frac{{g}_{s}}{{g}_{e}{T}_{mathrm{es}}})[{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e}ensuremath{-}{ensuremath{chi}}_{e}^{0}({stackrel{ensuremath{rightarrow}}{mathrm{H}}}_{mathrm{ext}}+ensuremath{lambda}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s})]$, $frac{d}{mathrm{dt}}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e}={g}_{e}[{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e}ifmmodetimeselsetexttimesfi{}({stackrel{ensuremath{rightarrow}}{mathrm{H}}}_{mathrm{ext}}+ensuremath{lambda}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s})]ensuremath{-}(frac{1}{{T}_{mathrm{es}}}+frac{1}{{T}_{mathrm{e}1}})[{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e}ensuremath{-}{ensuremath{chi}}_{e}^{0}({stackrel{ensuremath{rightarrow}}{mathrm{H}}}_{mathrm{ext}}+ensuremath{lambda}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s})]+(frac{{g}_{e}}{{g}_{s}{T}_{mathrm{se}}})[{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s}ensuremath{-}{ensuremath{chi}}_{s}^{0}({stackrel{ensuremath{rightarrow}}{mathrm{H}}}_{mathrm{ext}}+ensuremath{lambda}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e})]+D{ensuremath{nabla}}^{2}[{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e}ensuremath{-}{ensuremath{chi}}_{e}^{0}({stackrel{ensuremath{rightarrow}}{mathrm{H}}}_{mathrm{ext}}+ensuremath{lambda}{stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s})]$, where ${T}_{mathrm{se}}$, ${T}_{mathrm{es}}$, and ${T}_{mathrm{e}1}$ are the relaxation times for the local-moment-conduction-electron-spin, the conduction-electron-spin-local-moment, and the conduction-electron-spin-lattice systems. The diffusion constant is $D=frac{1}{3}{v}_{F}^{2}{T}_{i}$, where ${T}_{i}$ is the nonmagnetic-impurity scattering time, and ${stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{e}$ and ${stackrel{ensuremath{rightarrow}}{mathrm{M}}}_{s}$ are, respectively, the conduction-electron and local-moment magnetizations. The spin-orbit scattering in the presence of large-potential-impurity scattering results in relaxation effects identical to those obtained above with the model electron-lattice Hamiltonian. ${stackrel{ensuremath{rightarrow}}{mathrm{H}}}_{mathrm{ext}}$ is the external static and rf field, ${ensuremath{chi}}_{e}^{0}$ and ${ensuremath{chi}}_{s}^{0}$ are the conduction-electron and local-moment-unenhanced (by the $sensuremath{-}d$ exchange) susceptibilities, and ${g}_{e}$ and ${g}_{s}$ are the respective $g$ factors ($ensuremath{hbar}$ and ${ensuremath{mu}}_{B}$ have been set equal to unity). These Bloch equations show that the relaxation destination vectors are those appropriate to the total internal field. The presence of the $g$-factor ratios are consistent with the fact that ${mathcal{H}}_{mathrm{es}}$ transfers spin, rather than magnetization, from the conduction electrons to local moments and vice versa. For temperatures $mathrm{kT}<{ensuremath{omega}}_{s}$, where ${ensuremath{omega}}_{s}$ is the local-moment Zeeman energy, these equations fail, the self-energies become frequency dependent, and there are modifications to ${ensuremath{chi}}_{s}^{0}$. Most important, the electronic self-energy becomes dependent on the magnitude of the electronic momentum, so that a single equation for the total electron magnetization cannot be written. A microscopic derivation of the detailed-balance condition is given." @default.
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• W1975451839 date "1973-03-01" @default.
• W1975451839 modified "2023-10-14" @default.
• W1975451839 title "Diagrammatic Analysis of the Dynamics of Localized Moments in Metals" @default.
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• W1975451839 doi "https://doi.org/10.1103/physrevb.7.2163" @default.
• W1975451839 hasPublicationYear "1973" @default.
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