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 $2\mathrm{\Delta }(T)$ structure in the linewidth with a peak associated with a low lying $\mathrm{\hslash }{\omega }_{{\mathbf{q}}_{\mathbf{K}\mathbf{A}}}$ Kohn anomaly suggested a coincidence between $2\mathrm{\Delta }(0)$ and $\mathrm{\hslash }{\omega }_{{\mathbf{q}}_{\mathbf{K}\mathbf{A}}}$ 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 $2\mathrm{\Delta }(0)\mathrm{/}\mathrm{\hslash }{\omega }_{{\mathbf{q}}_{\mathbf{K}\mathbf{A}}}$ must be to unity in order to be consistent with the neutron data.