Neutron Spectroscopy of Superconductors

A complete understanding of the mechanism for superconductivity requires knowledge of the details of electrons, phonons, and their interactions, and can be summarized by the function ${\ensuremath{\alpha}}^{2}F(\ensuremath{\omega})$. This function is often very similar to the phonon density of states $F(\ensuremath{\omega})=\ensuremath{\Sigma}\ensuremath{\delta}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{Q})$, which can be derived from an analysis of neutron-scattering data. In this paper it is pointed out that the complete function ${\ensuremath{\alpha}}^{2}F(\ensuremath{\omega})$ is also (in principle) contained in neutron-scattering data if the intrinsic linewidth ${\ensuremath{\gamma}}_{Q}$ is measured as well as the dispersion relation ${\ensuremath{\omega}}_{Q}$. It is shown that ${\ensuremath{\alpha}}^{2}F$ differs from $F$ by having a weighting factor $\frac{2{\ensuremath{\gamma}}_{Q}}{\ensuremath{\pi}N{(0)}_{\ensuremath{\omega}}}$ inside the summation, where $N(0)$ is the electronic density of states at the Fermi surface for both spin orientations. The dimensionless coupling constant $\ensuremath{\lambda}$ can also be expressed in terms of $N(0)$, ${\ensuremath{\omega}}_{Q}$, and ${\ensuremath{\gamma}}_{Q}$. In practice, for most superconductors, the average widths ${\ensuremath{\gamma}}_{Q}$ are smaller than presently available resolution. However, for materials with a high density of states like $\ensuremath{\beta}\ensuremath{-}W$ superconductors, the widths ${\ensuremath{\gamma}}_{Q}$ may be measurable. Also, the question of whether superconductivity arises predominantly from coupling to certain groups of phonons can be answered experimentally by searching for anomalously large widths. Estimates of average phonon widths are given for a variety of metals.