Wave Functions for Superconducting Electrons

The observed variation of the transition temperature of mercury with isotopic mass is evidence that the superconducting state arises from interaction of electrons with lattice vibrations. The interaction term which gives scattering of electrons at high temperatures contributes at low temperatures a term to the energy of the system of electrons plus normal modes. Fröhlich has calculated the interaction energy at $T=0\mathrm{}$ by second-order perturbation theory. The energy is calculated here by taking wave functions of superconducting electrons, which have energies near the Fermi surface, as linear combinations of Bloch functions whose coefficients are functions of coordinates of the normal modes. In an equivalent approximation, Fröhlich's expression for the interaction energy is obtained. When the energy is calculated directly rather than by perturbation theory, modified expressions are obtained for the energy and distribution of electrons in the superconducting state. The criterion for superconductivity is $\frac{\hslash }{\tau }>\sim 2\pi \kappa T$, where $\tau$ is the relaxation time for electrons at some high temperature $T$ where $\tau T$ is constant. It is shown that superconducting electrons have small effective mass.