Some effects of phonon dynamics on electron lifetime, mass renormalization, and superconducting transition temperature

Effects of the dynamical properties of phonons on the electronic properties of metals are treated through the use of the phonon spectral weight function. It is shown that the leading behavior of the electron quasiparticle lifetime is lowered to quadratic in the frequency or temperature from the cubic behavior which is well known to occur for infinitely sharp phonons. Explicit evaluations of $\frac{1}{\tau }$ are made using an extension of the Debye model which allows for a finite phonon lifetime. The mass-enhancement parameter $\lambda$ is discussed and is shown formally to be related to the static phonon propagator. Thus if there is a difference between the phonon relaxation at zero frequency and at the phonon frequency, it is possible for the phonon frequencies which determine $\lambda$ to differ significantly from the peak frequencies observed in inelastic neutron scattering. The implications of these results for superconducting transition temperatures are discussed with particular reference to recent neutron scattering experiments in ${\mathrm{Nb}}_3$Sn.