Deformed shell gaps

The ground state of the ³²S nucleus, obtained within the HF approach with the SLy4 force, corresponds to a spherical-shape configuration that contains, on top of the closed ¹⁶O core, the 1d_5/2 and 2s_1/2 orbitals filled, both for the neutrons and protons. With an increase in the prolate deformation, the negative-parity Nilsson orbitals originating from the spherical ¹⁶O core stay occupied (see Fig. 2.21a in Ref. [39] for a qualitative illustration). The same is true for the positive-parity valence orbitals, except for the orbital [202]5/2 (the up-sloping extruder orbital), which originates from the spherical 1d_5/2 shell, and rapidly grows up in energy with increasing deformation. After this orbital is crossed by the [330]1/2 orbital (the down-sloping intruder orbital), which originates from the spherical 1f_7/2 shell, one obtains a large, about 2.5MeV gap that corresponds to a SD configuration in the ³²S nucleus. Therefore, the SD states in such a light system as ³²S, formally correspond to the 4p-4h excitation with respect to the spherical ground state. However, our calculations presented in detail below indicate that the SD configurations have extremely large quadrupole deformations, $\beta$$\simeq$0.7, and the implied structures of the SD, ND (normal deformed), or spherical wave functions have so little similarities that the notion of particle-hole excitations with respect to the spherical ground-state is not very useful.

Figures 1 and 2 show the neutron and proton single-particle routhians, as functions of the cranking frequency $\hbar\omega$. One can see that over a very large range of the rotational frequencies, there exists an important gap in the single-particle HF spectrum at the neutron and proton numbers N=Z=16. By definition, in the underlying ³²S SD configuration all the neutron and proton levels lying below the gaps at N=16 and Z=16 are occupied, and all those above the gaps are empty.

As a result of the presence of those large gaps in the single-particle ³²S proton and neutron spectra, we refer to the corresponding lowest-energy SD state as to the magic SD configuration.

A characteristic result visible from Figs. 1 and 2 is that the over-all single-particle structure of the HF orbitals near the Fermi level is remarkably simple. First of all, the dependence of the single-particle routhians on the rotational frequency is very regular, and there is only one clear-cut crossing caused by the down-sloping routhians [440]1/2(r=-i), originating from the N₀=4 shell. Second, the density of levels appearing in the figures is very low as compared, e.g., to those in the mass A$\simeq$150 region of SD nuclei. The negative parity states are represented only by two N₀=3 intruder orbitals [330]1/2(r=$\pm {i}$) below, and two intruder orbitals [321]3/2(r=$\pm {i}$) above the Fermi level. Similarly, in the positive parity there are only two states [211]1/2(r=$\pm {i}$) below, and two extruder states [202]5/2(r=$\pm {i}$) above the Fermi level. Signature splitting of the extruder states [202]5/2(r=$\pm {i}$) is very weak, because they carry high K=5/2 angular momentum projection, whereas splitting between the intruder levels [321]3/2(r=$\pm {i}$) is more pronounced. It becomes well visible at rotational frequencies of about 0.8MeV. Below the Fermi level, orbitals [330]1/2(r=$\pm {i}$) and [211]1/2(r=$\pm {i}$) have K=1/2, hence both are strongly split.