Electron-phonon interaction in an impure metal

On the basis of the existing theory of ultrasonic attenuation, a model Hamiltonian is derived which has to replace the Fröhlich Hamiltonian, at least in an impure metal. Whilst the imaginary part of the phonon self-energy is easily connected with the ultrasonic attenuation constant, the same quantity of the electrons allows no conclusion on inelastic collisions in this case. Therefore, from the linear response function, a kinetic equation for the electrons is derived. It is found that the time between inelastic collisions with phonons increases if impurities are added to the metal.