Electron-Phonon Interaction in Metals I: The Harmonic Approximation

The problem of electron-phonon coupling is investigated using the Born-Oppenheimer expansion. The expansion is carried out in this paper to second order in the expansion parameter (m/M)^¼. It is shown that to this order the expansion can be conveniently generated by a canonical transformation. The advantage of the canonical transformation is that a direct comparison can be made with the usual theories of the electron-phonon interaction. The frequency spectrum, and matrix elements of the electron-phonon interaction, of a monovalent metal are calculated and shown to be essentially identical with those derived by Nakajima and Bardeen and Pines. To second order in (m/M)^¼ the effects on the specific heat and spin susceptibility are calculated and found to be negligible at all temperatures.