Anomalous isotope effects of fulleride superconductors

The numerical calculations of the standard Eliashberg-Nambu strong coupling theory and a formula of isotope effect derived in this paper provide direct evidences that an-harmonic vibrations of lattice enhance isotope effect with anomalous coefficient $\alpha >1\mathrm{/}2$. The results in this paper explain very well the wide distributed $\alpha$ values for the samples with different ratios of substitutions of ${}^{12}$C by ${}^{13}$C of fulleride superconductor Rb${}_3$C${}_{60}$. The calculations of isotope effects indicate that the intra-molecule radial modes have more important contributions to superconductivity than the intra-molecule tangential modes with higher phonon frequencies