Eliashberg theory of superconductivity and inelastic rare-earth impurity scattering in filled skutterudite La${}_{1-x}$Pr${}_x$Os${}_4$Sb${}_{12}$

We study the influence of inelastic rare-earth impurity scattering on electron-phonon mediated superconductivity and mass renormalization in (La${}_{1-x}$Pr${}_x$)Os${}_4$Sb${}_{12}$ compounds. Solving the strong coupling Eliashberg equations we find that the dominant quadrupolar component of the inelastic scattering on Pr impurities yields an enhancement of the superconducting transition temperature T${}_c$ in LaOs${}_4$Sb${}_{12}$ and increases monotonically as a function of Pr concentration. The calculated results are in good agreement with the experimentally observed T${}_c(x)$ dependence. Our analysis suggests that phonons and quadrupolar excitations cause the attractive electron interaction which results in the formation of Cooper pairs and singlet superconductivity in PrOs${}_4$Sb${}_{12}$.