Adiabatic perturbation theory for metals and the problem of lattice stability

Electron-phonon interaction in metals is considered on the basis of quantum-mechanical perturbation theory, which is fully equivalent to the adiabatic expansion. An appropriate diagram technique is used. The dependence of the electron-phonon matrix elements on the phonon momentum is analyzed in various models. Results of calculations are presented for corrections to the vertices, for the energy spectra of the electrons and phonons, and for the phonon damping. It is shown that even though the adiabatic phonon frequency is renormalized very little as a result of nonadiabatic and anharmonic terms, its value depends significantly on the electron-phonon interaction. This dependence, however, does not lead to a possible lattice instability at a sufficiently large value of the electron-phonon interaction parameter, as in the Frohlich model, since it corresponds only to a transition from an optical dispersion law, in the absence of interaction of the electron and phonons, to an acoustic dispersion law when this interaction is taken into account. The Frohlich model in its literal form cannot be obtained from the exact Hamiltonian of the system, but it is possible to choose a zero-order Hamiltonian such that the form of the electron-phonon interaction Hamiltonian coincides, accurate to small terms, with the form of this operator in the Frohlich model. It turns out here that the nonrenormalized phonon frequency is described not by an acoustic dispersion law, as postulated in the Frohlich model, but by an optical law, and is equal to the ion plasma frequency, as in the Bohm–Staver model of “bare” ions. Therefore even in this model allowance for the electron-phonon interaction leads only to a transformation of the optical dispersion law into an acoustic one, and cannot lead to lattice instability, i.e., to a decrease of the acoustic frequency all the way to zero.