Shell energies of non-spherical nuclei in the inner crust of a neutron star

Shell effects in the non-spherical nuclei, which may exist in the inner crust of a neutron star, are studied. Spherical as well as non-spherical single-particle potentials are constructed based on the results of Thomas-Fermi calculations. In order to take into account the spin-orbit forces and the finite-range effects of nuclear interactions, three parameters are introduced in these potentials and their values are determined so as to be consistent with the single-particle energies in normal nuclei. Shell energies of the non-spherical nuclei are extracted from the single-particle energies in the non-spherical single-particle potentials. Neutron-shell energies are found to be negligibly small compared with proton-shell energies, and the band-structure effects of protons are also very small at the densities of interest, (1.0-1.5)x1014 g/cm3. The proton-shell energies are found to be somewhat smaller than the energy difference between the successive nuclear shapes. Numerical calculations are performed for spherical, cylindrical and slab-shape nuclei while for cylindrical-hole and spherical-hole nuclei an argument is given that the shell effects are expected to be small. Therefore, we conclude that the shell effects do not change the qualitative features of the Thomas-Fermi results; namely, as the average matter density increases from subnuclear to normal nuclear density, the stable nuclear shape changes successively from sphere to cylinder, slab, cylindrical hole and spherical hole before going into uniform matter. Quantitatively, however, the shell effects cause some change; the transitions from sphere to cylinder and from cylinder to slab occur at somewhat higher densities than in the Thomas-Fermi calculation. ((orig.))