Bulk (Hc2) and Surface (Hc3) Nucleation Fields of Strong-Coupling Superconducting Alloys

The linearized self-consistency equation, in the electron-phonon model, for the nucleation of superconductivity in the presence of a magnetic field is studied. It is found that near the transition temperature this integro-differential equation is susceptible to analysis. The main results are that the ratio of surface to bulk nucleation fields is predicted to have the weak-coupling value of 1.695, thus not confirming a recent experimental suggestion of a value near 1.9 for strong-coupling superconductors. The slope ${\left(\frac{\partial H_{c3\mathrm{}}}{\partial T}\right)}_{T_c}$ is worked out both in terms of integrals over the parameters of the bulk-strong-coupling theory and in terms of other experimental quantities. Comparison of this last result with existing experiments on pure lead is attempted, with good success. In the Appendix an implicit equation for $H_{c2\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$ for all $T$ is derived, but the numerical work necessary to solve this equation has not been undertaken.