Theory of Raman scattering in superconductors

The electronic Raman scattering by pairs of quasiparticles is calculated at zero temperature, generalizing previous calculations that were based on the Bardeen-Cooper-Schrieffer model of a super-conductor. Analytical and numerical results are presented for the spectrum as a function of wave vector $\overrightarrow{q}$, and an integration is performed over $q_z$ to include the effect of a finite optical penetration depth. Allowing for gap anisotropy, we correct the results for vertex and Coulomb polarization effects. The theoretical results for finite $q$ are used to calculate spectra for ${\mathrm{Nb}}_3$Sn, ${\mathrm{V}}_3$Si, and Nb, neglecting gap anisotropy. Experimental data are presented for ${\mathrm{V}}_3$Si and Nb. The data for ${\mathrm{V}}_3$Si are fit to a zero-$q$ theory that includes gap anisotropy, with results similar to those presented earlier for ${\mathrm{Nb}}_3$Sn. The role of possible excitons on the Raman spectra is examined. These theoretical results are then used to discuss the self-energy of a Raman-active optical phonon in a superconductor.