Electron-phonon interaction, ultrasonic attenuation, and Eliashberg functionα2F(ω) in impure metals

A relation between the ultrasonic attenuation coefficient and the Eliashburg function ${\alpha }^2$F is derived, which holds up to leading order in the impurity density. This shows that all electron-phonon interaction models which yield Pippard’s result for the sound attenuation are equivalent with respect to calculation of electronic properties as well. The low-frequency behavior of ${\alpha }^2$F(ω) is calculated for a jellium model, and several mutually contradicting results in the literature are discussed.