The Isotope Effect in Superconductivity

The isotope effect in superconductors is usually summarized by giving the observed values of p in the equation MᵖTc=constant, where M is the isotopic mass and Tc the superconducting transition temperature. Fr\"ohlich predicted the value p=1/2, but the measurements in some instances show deviations from this prediction. An explanation of the deviation of p from {} 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 {} 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 {} are not necessarily in conflict with the theories of Fr\"ohlich and Bardeen.