In small Fermi energy metals, disorder can deeply modify superconducting state properties leading to a strong suppression of the critical temperature $T_c$. In this paper, we show that also normal state properties can be seriously influenced by disorder when the Fermi energy $E_{ \rm F}$ is sufficiently small. We calculate the normal state spin susceptibility $\chi$ for a narrow band electron-phonon coupled metal as a function of the non-magnetic impurity scattering rate $\gamma_{ \rm imp}$. We find that as soon as $\gamma_{ \rm imp}$ is comparable to $E_{ \rm F}$, $\chi$ is strongly reduced with respect to its value in the clean limit. The effects of the electron-phonon interaction including the nonadiabatic corrections are discussed. Our results strongly suggest that the recent finding on irradiated MgB$_2$ samples can be naturally explained in terms of small $E_{ \rm F}$ values associated with the $\sigma$-bands of the boron plane, sustaining therefore the hypothesis that MgB$_2$ is a nonadiabatic metal.
Categories: cond-mat.supr-con