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 $Γ({\bf q},T)$ of acoustic phonons in conventional superconductors Pb and Nb. [P. Aynajian, et al, Science 319, 1509 (2008)]. At low temperatures the merging of the $2Δ(T)$ structure in the linewidth with a peak associated with a low lying $\hbarω_{\bf q_{KA}}$ Kohn anomaly suggested a coincidence between $2Δ(0)$ and $\hbarω_{\bf q_{KA}}$ 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 $2Δ(0)/\hbarω_{\bf q_{KA}}$ must be to unity in order to be consistent with the neutron data.

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