Possible large relative enhancement of the superconductingTcby anisotropy

The enhancement of the superconducting transition temperature $T_c$ by anisotropy is studied as a function of the Coulomb repulsion ${\mu }^{*}$. Numerical solutions of the anisotropic Eliashberg equations are presented with isotropic ${\mu }^{*}$ ranging up to and beyond the average electron-phonon parameter $\lambda$, assuming separable anisotropy. An approximate anisotropic $T_c$ equation is developed, tested against the numerical results, and used to interpret them. It is found that the dependence of ${\mu }^{*}$ on the upper phonon cutoff plays an important role. Further, within BCS-like approximations, a general formalism is developed applicable to any anisotropy function. It is applied, in connection with calculated microscopic parameters in real metals, to develop additional insight into the role played by ${\mu }^{*}$ in criteria for superconductivity.