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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.Comment: 10 pages. REVTeX. 5 postscript figures. Scheduled to be published in  Physical Review B, March 1. The previous submission is revised and two  figures are adde

Finite bandwidth effects on the transition temperature and NMR relaxation rate of impure superconductors