Phonon lifetimes in BCS superconductors provide information on the details of the electron-phonon coupling mechanism. In 2008 high-resolution neutron inelastic scattering experiments revealed unexpected anomalies in the lifetimes of transverse acoustic modes in the elemental superconductors Pb and Nb. These experiments were performed using a resonant spin-echo technique. To date no quantitative explanation for these anomalies has been forthcoming. Here we propose to exploit the unprecedented high-resolution of LET to investigate phonon lifetimes effects arising from electron-phonon coupling in Nb. In addition to providing an important check of the surprising behaviour of the acoustic modes, our experiments will provide a high-resolution survey of the longitudinal acoustic modes where new features may be expected.