We study the thermodynamic properties of impure superconductors by explicitlytaking into consideration the finiteness of electronic bandwidths within thephonon-mediated Eliashberg formalism. For a finite electronic bandwidth, thesuperconducting transition temperature, $T_c$, is suppressed by nonmagneticimpurity scatterings. This is a consequence of a reduction in the effectiveelectron-phonon coupling, $\lambda_{eff}$. The reduced $\lambda_{eff}$ isreflected in the observation that the coherence peak in $1/(T_1 T)$, where$T_1$ is the nuclear spin-lattice relaxation time and $T$ is the temperature,is enhanced by impurity scatterings for a finite bandwidth. Calculations arepresented for $T_c$ and $1/(T_1 T)$ as bandwidths and impurity scattering ratesare varied. Implications for doped C$_{60}$ superconductors are discussed inconnection with $T_c$ and $1/T_1$ measurements.
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