Self-Consistent Study of Nuclear Shapes

Self-consistent Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) calculations have been performed for the even Ti, Cr, and Fe isotopes. The residual two-body interaction is assumed to be a central potential with a Yukawa radial dependence and an exchange mixture which has been adjusted to fit the low-lying shell-model states of ${\mathrm{O}}_{10}^{18}$ and ${\mathrm{F}}_9^{18}$. It is found that the HF minima occur for prolate shapes, and hence the HFB calculations are done only for prolate deformations. The HFB calculation yields normal solutions for $N=Z$ nuclei and spherical shapes for $Z<N=28\mathrm{}$ nuclei. The effect of pairing on the ground-state properties is also studied. The value of the mass quadrupole moment for the HFB solution is smaller than that for the HF solutions for all the nuclei under consideration. A spherical BCS calculation has been performed to estimate the effect of deformation on pairing energy in the HFB calculation. We construct the excited states for the $Z<N=28\mathrm{}$ nuclei from the spherical BCS ground-state solutions in the random-phase approximation. This gives the values for the $0^{+}$ ${\to 2}^{+}$ energy separation and the $B\left(E2\mathrm{}\right)\mathrm{}$ rates which are in good agreement with experiment.