Effects of electron band structure on neutrino pair bremsstrahlung in neutron star crusts

We calculate the rate of energy emission by bremsstrahlung of neutrino pairs by electrons moving in the crystalline lattice of ions in dense matter in the crust of a neutron star. Since the periodic potential in the solid gives rise to electronic band gaps which can be as large as about 1 MeV, it is necessary in estimating the bremsstrahlung rate at low temperatures to take into account band structure in detail. We find that, in the densest parts of the inner crust of a neutron star, neutrino emission at temperatures of about $2\times {10}^9\mathrm{K}$ or less is much suppressed compared with earlier estimates that treated the electron-lattice interaction perturbatively, and conclude that neutrino pair bremsstrahlung by electrons in the crusts of neutron stars is much less important for neutron star thermal evolution than was previously thought.