The Isotope Effect in Superconductivity

Abstract

The isotope effect in superconductors is usually summarized by giving the observed values of $p$ in the equation $M^p{T}_c=\mathrm{constant}$, where $M$ is the isotopic mass and $T_c$ the superconducting transition temperature. Fröhlich predicted the value $p=\frac{1}{2}$, but the measurements in some instances show deviations from this prediction. An explanation of the deviation of $p$ from ½ is offered based on an analog of Wien's displacement law applicable to the vibration spectrum of real crystal lattices. The departure of $p$ from the value ½ is attributed to the departure of the frequency spectrum from a simple power law. For many superconducting elements, $p$ may be estimated from specific heat data, when such data are available to the desired degree of accuracy. A value of $p$ is calculated for Sn which is in good agreement with some of the experiments. The large value 0.73 observed for Pb is shown to be reasonable. The values of $p$ for the other superconducting elements are discussed. It is concluded that the observed deviations of $p$ from ½ are not necessarily in conflict with the theories of Fröhlich and Bardeen.