Phonon Anomalies in Strongly Coupled Superconductors

We discuss the connection between phonon anomalies and high superconducting temperature $T_c$ in transition metals and their carbides, based on strong-coupling theory. Using the previously developed model theory for the lattice dynamics of these materials, we investigate the dependence on the phonon frequencies $\omega$ of the electron-phonon coupling constant $\lambda$, which essentially determines $T_c$. We find that the variation of the moment $〈 {\omega }^2 〉$ caused by the anomalies is not enough to explain the variation of $\lambda$. This result is interpreted by assuming that the electron-phonon matrix elements which enter for $\lambda$ are determined not only by a $p-d$ contact term but also rather strongly by $d-d$ overlap interaction.