### Abstract

Using the conventional exchange Hamiltonian Hes, 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: ddtMs=gs[Ms×(HextMe)]-(1Tse)[Ms-s0(HextMe) ]+(gsgeTes)[Me-e0(HextMs)], ddtMe=ge[Me×(HextMs)]-(1Tes+1Te1)[Me- e0(HextMs)]+(gegsTse)[Ms-s0(HextMe)]+D2[Me-e0(HextMs)], where Tse, Tes, and Te1 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=13vF2Ti, where Ti is the nonmagnetic-impurity scattering time, and Me and Ms 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. Hext is the external static and rf field, e0 and s0 are the conduction-electron and local-moment-unenhanced (by the s-d exchange) susceptibilities, and ge and gs are the respective g factors (and 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 Hes transfers spin, rather than magnetization, from the conduction electrons to local moments and vice versa. For temperatures kT

Original language | English (US) |
---|---|

Pages (from-to) | 2163-2192 |

Number of pages | 30 |

Journal | Physical Review B |

Volume | 7 |

Issue number | 5 |

DOIs | |

State | Published - 1973 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

*Physical Review B*,

*7*(5), 2163-2192. https://doi.org/10.1103/PhysRevB.7.2163

**Diagrammatic analysis of the dynamics of localized moments in metals.** / Barnes, Stewart; Zitkova-Wilcox, J.

Research output: Contribution to journal › Article

*Physical Review B*, vol. 7, no. 5, pp. 2163-2192. https://doi.org/10.1103/PhysRevB.7.2163

}

TY - JOUR

T1 - Diagrammatic analysis of the dynamics of localized moments in metals

AU - Barnes, Stewart

AU - Zitkova-Wilcox, J.

PY - 1973

Y1 - 1973

N2 - Using the conventional exchange Hamiltonian Hes, 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: ddtMs=gs[Ms×(HextMe)]-(1Tse)[Ms-s0(HextMe) ]+(gsgeTes)[Me-e0(HextMs)], ddtMe=ge[Me×(HextMs)]-(1Tes+1Te1)[Me- e0(HextMs)]+(gegsTse)[Ms-s0(HextMe)]+D2[Me-e0(HextMs)], where Tse, Tes, and Te1 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=13vF2Ti, where Ti is the nonmagnetic-impurity scattering time, and Me and Ms 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. Hext is the external static and rf field, e0 and s0 are the conduction-electron and local-moment-unenhanced (by the s-d exchange) susceptibilities, and ge and gs are the respective g factors (and 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 Hes transfers spin, rather than magnetization, from the conduction electrons to local moments and vice versa. For temperatures kT

AB - Using the conventional exchange Hamiltonian Hes, 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: ddtMs=gs[Ms×(HextMe)]-(1Tse)[Ms-s0(HextMe) ]+(gsgeTes)[Me-e0(HextMs)], ddtMe=ge[Me×(HextMs)]-(1Tes+1Te1)[Me- e0(HextMs)]+(gegsTse)[Ms-s0(HextMe)]+D2[Me-e0(HextMs)], where Tse, Tes, and Te1 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=13vF2Ti, where Ti is the nonmagnetic-impurity scattering time, and Me and Ms 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. Hext is the external static and rf field, e0 and s0 are the conduction-electron and local-moment-unenhanced (by the s-d exchange) susceptibilities, and ge and gs are the respective g factors (and 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 Hes transfers spin, rather than magnetization, from the conduction electrons to local moments and vice versa. For temperatures kT

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U2 - 10.1103/PhysRevB.7.2163

DO - 10.1103/PhysRevB.7.2163

M3 - Article

AN - SCOPUS:3743150662

VL - 7

SP - 2163

EP - 2192

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 1098-0121

IS - 5

ER -