Solutions of the BCS Integral Equation and Deviations from the Law of Corresponding States

Abstract

The BCS integral equation has been studied for nonseparable interactions of the Bardeen-Pines form, V(|εε’|). Numerical solutions were obtained using an IBM 7090 for a simple interaction of this form which included the effect of the Coulomb repulsion. The results for the ratio of the energy gap to the critical temperature and for the temperature dependence of the energy gap, the electronic specific heat, and the critical field in terms of the proper reduced quantities were rather insensitive to the form or strength of the interaction. This indicates that the BCS theory gives the law of corresponding states. The calculated ratio of energy gap to critical temperature varies with the ratio of critical temperature to the Debye temperature, and this variation is of the correct order of magnitude if the Coulomb interaction is included. The same model is used to study the isotope effect. With the plausible assumption that the Coulomb cutoff is independent of the ionic mass, there are deviations from the M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−½</sup> law that are larger for small T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> /θ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</inf> superconductors.