Theory of the Anisotropic Energy Gap in Superconducting Lead

The energy gap of a model anisotropic superconductor is considered. This model calculation forms the basis of a more realistic theoretical consideration of the energy gap of superconducting Pb, one in which the phonon density of states is the principal source of gap anisotropy. The effect of energy-band structure, important only near Brillouin-zone boundaries, is included as a perturbation. The phonon density of states is calculated from the experimental dispersion curves and singularities—present in the special density of states entering the superconductivity problem—are discussed. The phonon density of states and the isotropic gap solution obtained by previous workers are used to calculate the anisotropic part of the energy gap. The double gap, $2\Delta$, is found to have an absolute maximum of 2.86 meV in the [100] direction, and an absolute minimum of 2.55 meV in the [110] direction. Ten other maxima, minima, and saddle points are listed. The effect of the energy-gap anisotropy on electron-tunneling, electromagnetic-absorption, and acoustic-attenuation experiments is predicted.