Electron-phonon interaction simple metals

It was noted in Section 19-D that in A15 compounds, the metallic bands may be thought of as being based entirely upon orbitals of the same symmetry. In such a one-band system or in a system we so approximate, with nearest-neighbor interactions, the electron-phonon interaction can be written in a very simple and informative way, following an analysis given by Barisic, Labbe, and Fricdel (1970). We... 

We shall make use of the effective Hamiltonian formalism [14] that enables us to isolate effects of interest from irrelevant complications. We divide the electronic Hamiltonian into a strong part H° and a weak part H, and we shall suppose that H° is simple enough to be solved exactly. The Hamiltonian including the cubic field and interelectronic repulsion only is the usual choice for H in the case of the 3d group ions. Then H should include all other interactions (spin-orbit coupling, lower symmetry fields, electron-phonon interaction, external fields, strain etc). The most important assumption is that the perturbations, described by the H Hamiltonian (in particular the JT interaction) must be smaller relative to the initial splitting due to H°. In the case of the 3d metal ions the assumption is usually well justified. 

In one-dimensional metals, the electron-phonon interaction produces much stronger Kohn anomalies than in three-dimensional metals. In fact, a giant Kohn anomaly has been observed in the one-dimensional conductor KCP [1.35]. For a more detailed discussion of the theory of phonons in metals, the reader is referred to [4.66] and to the work by HARRISON [4.71] in which the pseudopotential approximation is used. This latter method yields good agreement between calculated and observed phonon frequencies of simple metals such as aluminium with only two model parameters. 

Any temperature dependence of the CEF halfwidth can be attributed to either s-f interaction or the interaction of f states with phonons, s-f line broadening for simple metals has been attributed to the effects of the carrier spin dynamics on the CEF-state lifetimes and has been described by the Fermi liquid theory, introduced by Becker et al. (1977). The s-f interaction Hamiltonian describing the coupling between the 4f electrons and the conduction electrons can be written as... 

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