T$_{c}$ vs Isotopic Mass and vs Residual Resistivity Investigation in MgB$_{2}$

Almost five years after the discovery of superconductivity in MgB$_{2}$ the isotope effect on T$_{c}$ is not yet understood (M. Calandra et al, Physica C456, 38 (2007) and references therein). The isotope effect is mainly due to the B atoms reflecting the important role of B vibrations in determining T$_{c}$. Detailed two bands calculation leads to $\alpha $(B) of the order of 0.4--0.45, in disagreement with experiments which evaluated $\alpha $(B) = 0.30. Anharmonicity was proposed as a possible explanation for the reduced B isotope coefficient, but recently it was emphasized that such an explanation needs to be reconsidered. On the other hand, recent investigations on the effect of disorder on T$_{c}$ pointed out that samples with residual resistivity ($\rho _{0})$ of few $\mu \Omega $cm present T$_{c}$ variations comparable with the intrinsic variations due to isotopic effect. This calls for new investigations of isotopic effect in samples with controlled amount of disorder. Ultra clean Mg$^{10}$B$_{2}$ and Mg$^{11}$B$_{2}$ samples ($\rho _{0}\sim $0.5 $\mu \Omega $cm) were damaged respectively with annealing and neutron irradiation and T$_{c}$ and resistivity were measured. T$_{c}$ vs $\rho _{0}$ plot shows in both cases a linear relationship allowing us to extrapolate T$_{c}$ ($\rho $=0)${\rm g}$ for both the sample series. $\alpha $(B) evaluated by these intrinsic T$_{c}$ values confirms results of previous report and the crucial role of disorder in determining T$_{c}$ has been proved.