Gas anisotropy and T/sub c/ enhancement: general theory, and calculations for Nb, using Fermi surface harmonics

A general theory is given of the anisotropy of the energy gap and the resulting transition temperature (T/sub c/) enhancement of pure superconductors. The frequency dependence of the gap Δ(k,ω) is approximated by the two square well form of McMillan, but otherwise an exact algebraic solution of the strong-coupling Eliashberg equations is given, valid for arbitrarily large anisotropy. In the limit of weak anisotropy a simple perturbative formula is also derived. The method of solution relies on the use of expansion functions called Fermi surface harmonics (FSH's) which are velocity polynomials orthonormalized on the Fermi surface. Methods for explicit construction of these functions are described. As an application of these techniques the mass enhancement and gap anisotropy are calculated for Nb, in an approximation which includes all electronic anisotropy but neglects the contribution to the anisotropy which arises from phonons. The rms gap anisotropy in this model is 6 percent which is not inconsistent with most of the current experimental data. The resulting T/sub c/ enhancement is predicted to be 0.7 percent or .060K