Neutron-scattering profile of $Q\ne 0$ phonons in BCS superconductors

Phonons in a metal interact with conduction electrons. In the normal state, this gives rise to a linewidth ${\gamma }_Q$ which is small compared with the frequency ${\omega }_Q.$ In the superconducting state, the line shape can be altered if $ħ{\omega }_Q\lesssim \mathrm{}(1+2r)2\Delta$ where $\Delta$ is the superconducting gap, and $r$ is the ratio ${\gamma }_Q/{\omega }_Q,$ which scales with the strength of the electron-phonon coupling $\lambda$. As long as ${\omega }_Q\ll {\mathrm{Qv}}_F$ where $v_F$ is the Fermi velocity, BCS theory predicts a line shape which is a universal function of the dimensionless parameters $r$, ${\omega }_Q/2\mathrm{}\Delta$, $\omega /2\mathrm{}\Delta$, and ${T/T}_c$ where $T_c$ is the superconducting transition temperature. Formulas and curves are given for the full range of these parameters. The BCS predictions correspond well to key features seen in recent experiments on YNi${}_2$B${}_2$C and LuNi${}_2$B${}_2$C.