The effect of strong electron-rattling phonon coupling on some superconducting properties

Using the Eliashberg theory of superconductivity we have examined several properties of a model in which electrons are coupled only to rattling phonon modes represented by a sharp peak in the electron-phonon coupling function. Our choice of parameters was guided by experiments on $\beta$-pyrochlore oxide superconductor KOs$_{2}$Os$_{6}$. We have calculated the temperature dependence of the superconducting gap edge, the quasiparticle decay rate, the NMR relaxation rate assuming that the coupling between the nuclear spins and the conduction electrons is via a contact hyperfine interaction which would be appropriate for the O-site in KOs$_{2}$Os$_{6}$, and the microwave conductivity. We examined the limit of very strong coupling by considering three values of the electron-phonon coupling parameter $\lambda=$ 2.38, 3, and 5 and {\em did not} assume that the rattler frequency $\Omega_{0}$ is temperature dependent in the superconducting state. We obtained a very unusual temperature dependence of the superconducting gap edge $\Delta(T)$, very much like the one extracted from photoemission experiments on KOs$_2$O$_6$.