Band-Structure Effects in Superconductivity. I. Formalism

Dyson equations are derived for the interacting electrons and phonons of a perfect crystal. A separation of the electron and phonon fields which is exact within either the Hartree or the Migdal approximation is effected. All umklapp processes and local-field corrections are included, but the hybridization of electronic bands and of phonon modes is given only implicitly. Our equations reduce to previously obtained forms in the jellium approximation and to lowest order in $r_s$. The Nambu-Gor'kov formalism is used to derive equations for the superconducting energy gap. Since the Migdal approximation is not applicable to the treatment of the Coulomb interaction, we develop other approximations which yield either three-dimensional integral equations for an anisotropic energy gap $\mathbf{\Delta }(\mathbf{k},k_0)$ or one-dimensional integral equations for an isotropic energy gap $\Delta \left(k_0\right)$. An easy method for the calculation of the anisotropy of the energy gap $\mathbf{\Delta }(\mathbf{k},k_0)$ in the simple metals is presented. The validity of these approximations is discussed and our equations are compared with those of previous authors. The strongly nonlinear homogeneous integral equation for $\Delta \left(k_0\right)$ is transformed into a quasilinear, inhomogeneous integral equation. This transformation immediately displays several interesting properties of the solution $\Delta \left(k_0\right)$ and yields a method for the rapid numerical calculation of $\Delta \left(k_0\right)$.