On the Question of Coincidence Between Energy Gaps and Kohn Anomalies

Recently, neutron scattering spin echo measurements have provided high resolution data on the temperature dependence of the linewidth $\Gamma({\bf q},T)$ of acoustic phonons in conventional superconductors Pb and Nb. [P. Aynajian, et al, Science 319, 1509 (2008)]. At low temperatures the merging of the $2\Delta(T)$ structure in the linewidth with a peak associated with a low lying $\hbar\omega_{\bf q_{KA}}$ Kohn anomaly suggested a coincidence between $2\Delta(0)$ and $\hbar\omega_{\bf q_{KA}}$ in Pb and Nb. Here we carry out a standard BCS calculation of the phonon linewidth to examine its temperature evolution and explore how close $2\Delta(0)/\hbar\omega_{\bf q_{KA}}$ must be to unity in order to be consistent with the neutron data.