Energy Gap in Nuclear Matter. I. Extended Theory

Abstract

The Bogoliubov condition of compensation of dangerous diagrams is invoked to derive a generalized energy-gap equation of the BCS form, but with a renormalized pairing interaction and renormalized single-particle energies. The second-order contributions to the renormalized pairing interaction are found to be significant, contrary to popular belief. It is stressed that the self-consistent solution of the energy-gap equation yields qualitatively and quantitatively different results than a perturbative evaluation of the energy spectrum following a Bogoliubov-Valatin transformation characterized by the lowest order BCS gap parameters. The dependence of the energy gap on high-order corrections is studied in one and three dimensions for simple potentials; for some values of the potential parameters, no solution to the gap equation exists. Finally, the energy gap is studied in nuclear matter. We include the scattering in both singlet and triplet states of particle-hole pairs which can be neutron-neutron, neutron-proton, or proton-proton. The interaction is taken to be a sum of separable potentials which reproduce the $s$-wave phase shift. Because of the short-range repulsion that is included, we sum an infinite set of particle-particle diagrams which replaces the second-order potential vertices by $T$ matrices. The higher order effects studied increase the energy gap by a large factor, especially when the lowest order BCS gap is calculated to be small. Nevertheless, the qualitative conclusion remains that the energy gap in infinite nuclear matter appears to be considerably smaller than that in the heaviest nuclei.