Coincidence between energy gaps and Kohn anomalies in conventional superconductors

Abstract

Recently, neutron scattering spin-echo measurements have provided high-resolution data on the temperature dependence of the linewidth $\ensuremath{\Gamma}(\mathbf{q},T)$ of acoustic phonons in conventional superconductors Pb and Nb [see P. Aynajian et al., Science 319, 1509 (2008)]. At low temperatures, the merging of the $2\ensuremath{\Delta}(T)$ structure in the linewidth with a peak associated with a low-lying $\ensuremath{\hbar}{\ensuremath{\omega}}_{{\mathbf{q}}_{KA}}$ Kohn anomaly suggested a coincidence between $2\ensuremath{\Delta}(0)$ and $\ensuremath{\hbar}{\ensuremath{\omega}}_{{\mathbf{q}}_{KA}}$ 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 $2\ensuremath{\Delta}(0)/\ensuremath{\hbar}{\ensuremath{\omega}}_{{\mathbf{q}}_{KA}}$ must be to unity in order to be consistent with the neutron data.