Anomalous Isotope Effect in Rattling-Induced Superconductor

In order to clarify that the Cooper pair in $\beta$-pyrochlore oxides is mediated by anharmonic oscillation of guest atom, i.e., rattling, we propose an experiment to detect anomalous isotope effect. In the formula of $T_{\rm c} \propto M^{-\eta}$, where $T_{\rm c}$ is superconducting transition temperature and $M$ denotes mass of the oscillator, it is found that the exponent $\eta$ is increased with the increase of anharmonicity of a potential for the guest atom. We predict that $\eta$ becomes larger than 1/2 in rattling-induced superconductor, in sharp contrast to $\eta=1/2$ for weak-coupling superconductivity due to harmonic phonons and $\eta<1/2$ for strong-coupling superconductivity with the inclusion of the effect of Coulomb interaction.