Projected Hartree-Fock Spectrum ofSi28Including Effects of Pairing and Shape Mixing

The energy spectrum of ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$ has been calculated using the angular-momentum states projected from the oblate and the prolate Hartree-Fock solutions. The Hartree-Fock solutions chosen by us take into account all 28 nucleons, and therefore the effect of the polarization of the ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ core has been incorporated into our calculations to some extent. The complexity of the projection calculation has been greatly simplified by suitably using the known symmetries of the Hartree—Fock solutions. The theoretically predicted energy spectrum is compressed by about a factor of 2 when compared with experiment. The small energy gap between the oblate and the prolate Hartree—Fock solutions suggested the possibility of admixing the two solutions by the two-body interaction. The two solutions differ in the four-particle—four-hole states in the intrinsic frame, and therefore the admixture was calculated in the projected basis. The mixing, however, turns out to be too small to affect the spectrum. Another attempt to improve our results was made by including the corrections due to the $T=1\mathrm{}$ pairing in the Hartree—Fock solutions, and then calculating the energy spectrum using the states projected from the corrected intrinsic states. The corrections due to the $T=1\mathrm{}$ pairing turn out to be of the order of 2% for both the oblate and the prolate Hartree-Fock solutions. This result is consistent with the earlier Hartree-Fock-Bogolyubov calculations on ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$, which predict no pairing affects. Because of the small size of the pairing corrections, the projected energy spectrum from the corrected intrinsic states does not show any significant improvement.