Evolution of the $N=20$ and 28 shell gaps and two-particle-two-hole states in the FSU interaction

The connection between fundamental nucleon-nucleon forces and the observed many-body structure of nuclei is a main question of modern nuclear physics. Evolution of the mean field, inversion of traditional shell structures, and structure of high spin states in nuclei with extreme proton-to-neutron ratios are at the center of numerous recent experimental investigations targeting the matrix elements of the effective nuclear Hamiltonian that is responsible for these phenomena. The FSU $spsdfp$ cross-shell interaction for the shell model was successfully fitted to a wide range of mostly intruder negative parity states of the $sd$ shell nuclei. In this paper, we explore the evolution of nuclear structure in and around the island of inversion (IoI), where low-lying states involve cross-shell particle-hole excitations. We apply the FSU interaction to systematically trace out the relative positions of the effective single-particle energies of the $0f_{7/2}$ and $1p_{3/2}$ orbitals forming the $N=20$ and 28 shell gaps. We find that above a proton number of about 13, the $0f_{7/2}$ neutron orbital lies below that of $1p_{3/2}$, which is considered normal ordering but, systematically, for more exotic nuclei with lower $Z=12$ and 10 the order of orbitals reversed. The crossing of the neutron orbitals happens right near the neutron separation threshold. Our Hamiltonian reproduces remarkably well the absolute binding energies for a broad range of nuclei and the inversion in the configurations of nuclei inside the IoI. The effective interaction accounts well for the energies and variations with mass number $A$ of aligned high-spin states that involve nucleon pairs prompted across the shell gap. This paper puts forward an empirically determined effective Hamiltonian where data from many recent experiments allowed us to significantly improve our knowledge about cross-shell nuclear interaction matrix elements. The quality with which this Hamiltonian describes the two-particle, two-hole cross-shell excitations, binding energies, and the physics of aligned states that were not a part of the fit, is remarkable, making the FSU interaction an important tool for the future exploration of exotic nuclei.

Figure 1

Histogram of the differences in excitation energy between experiment and the FSU interaction fit. The root-mean-square deviation is 190 keV.

Figure 2

Neutron effective single particle energies (ESPEs) of $0f_{7/2}$ and $1p_{3/2}$ orbitals calculated with the FSU interaction. They represent the theoretical centroids of the energies of the $0f_{7/2}$ and $1p_{3/2}$ orbitals. In the “normal” ordering, the red diamonds ($1p_{3/2}$) lie above the black circles ($0f_{7/2}$).

Figure 3

(a) Average energy differences between the lowest $7/2^{-}$ and $3/2^{-}$ experimental levels in Table 1. The error bars give an indication of the range of values for different neutron numbers. Positive (negative) values of the ordinate correspond to the $3/2^{-}$ state above (below) the $7/2^{-}$ one. (b) Occupancies of the neutron $0f_{7/2}$ and $1p_{3/2}$ orbitals in neutron number $N=20$ nuclei as a function of proton number $Z$ for the lowest 2p2h states. The values are shown as filled circles for the cases where the lowest 2p2h state is the ground state (IoI) and as open circles where the lowest 2p2h state is excited above the ground state.

Figure 4

The experimentally known levels of ${}^{31}\mathrm{Na}$ compared to the lowest ones predicted using the FSU interaction for 0p0h, 1p1h, and 2p2h configurations. The experimental levels agree well with the 2p2h results while the 0p0h states start almost 2.5 MeV higher in excitation energy. Only the two lowest calculated 1p1h states are labeled because of the high-level density above this.

Figure 5

The lowest experimental energy levels of $N=20sd$-shell nuclei compared to those calculated using the FSU shell-model interaction for 0p0h and 2p2h configurations. The levels of the known IoI nuclei ${}^{30}\mathrm{Ne}$ and ${}^{32}\mathrm{Mg}$ agree well with the 2p2h results while the lowest states in the higher Z nuclei agree much better with the 0p0h results.

Figure 6

Experimental $\mathrm{E}(2_1^{+}$) and B(E2: $0_1^{+}\to 2_1^{+}$) values for the $N=20$ isotones are compared to those calculated by using the FSU interaction. The B(E2: $0_1^{+}\to 2_1^{+}$) value of ${}^{34}\mathrm{Si}$ has not been calculated because of the different configurations associated with the $0_1^{+}$ and $2_1^{+}$ states.

Figure 7

The number displayed inside a box corresponding to an isotope is the difference in binding energy between experiments and shell-model predictions using the FSU interaction with 0 or 2 particle-hole configurations. We call the states overbound where the calculated states are more tightly bound than that of the experimental ones and underbound when it is otherwise.

Figure 8

2p2h occupancies of the $\nu 0f_{7/2}$ and $\nu 1p_{3/2}$ orbitals for the first $0^{+}$, $2^{+}$, and $4^{+}$ calculated states using the FSU interaction for nuclei with $N=20$ and $Z$ between 10 and 18.

Figure 9

Comparisons of the energies of fully aligned states in $sd$-shell nuclei with those predicted employing the FSU interaction. Many of the experimental points are confirmed by both selective population in $(\alpha ,d)$ reactions and in high-spin $\gamma$ decay sequences and are displayed with solid black circles, while dotted black circles are used to represent states observed by only one of the two signatures. The structure of many of these aligned states involve the promotion of two (extra) nucleons to the $0f_{7/2}$ orbital and are shown with solid red diamonds. Those with at least one nucleon in the $0f_{7/2}$ orbital may require only one more promotion (1p1h excitation) and are shown with dotted red diamond symbols.