Evolution of the N=20 and 28 Shell Gaps and 2-particle-2-hole states in the FSU Interaction

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. This paper reports the application of 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, the evolution from normally ordered low-lying states to the"Island of Inversion"(IoI), and the behavior of a wide range of excited states with a $0f_{7/2}$ proton and neutron coupled to maximum spin of $7 \hbar$. Above a proton number of about 13 the $0f_{7/2}$ orbital lies below that of $1p_{3/2}$, which is considered normal ordering, but systematically at $Z = 10$ to $12$ the orbitals cross. The calculations reproduce well the 2p2h - 0p0h inversion in the configurations of nuclei inside the IoI, they reproduce the absolute binding energies and the transition to normal ordering as the proton number approaches that of the neutrons. The important role of $1p_{3/2}$ neutron pairs in the IoI is also demonstrated. The calculations account well for the energies of the fully aligned states with 0, 1, or 2 individual $sd$ nucleon aligned in spin with the aligned $\pi 0f_{7/2}$ - $\nu 0f_{7/2}$ pair and reproduce well their systematic variation with $A$ and number of aligned $sd$ nucleons. The results presented in this paper give hope for the predictive power of the FSU interaction for more exotic nuclei to be explored in near future.


I. INTRODUCTION
Recent experimental works in the 1s0d shell with large γ detector arrays and heavyion fusion reactions have substantially extended knowledge of relatively high spin states.
However, these do not form well-behaved rotational bands amenable to study by collective models because rotational energies are comparable to single-particle energies. On the other hand microscopic configuration-interaction model calculations are feasible in these lighter nuclei. The USD family of effective interactions [1,2] have been very successful in describing most lower-lying positive-parity states of nuclei with 8 ≤ (N, Z) ≤ 20. However, higher spin states involve excitations into the f p shell where orbitals contributing larger values of angular momentum are occupied, which is beyond the scope of the USD interaction. Also, neutron-rich isotopes quickly move beyond the sd shell boundaries [3][4][5][6][7][8]. Our experience with available cross-shell interactions to compare with our experimental results was not completely satisfactory because different interactions work better for different nuclei, as briefly discussed in Ref. [9].
In search for a single cross-shell interaction which works well over a wide range of nuclei, we have developed a new interaction [9] by fitting the energies of 270 states in nuclei from 13 C through 51 Ti and 49 V originated from the wbp interaction [10]. As already described in Ref. [9] the USDB Hamiltonians were used for the sd shell and were kept fixed whereas 40 linear combinations of 70 single-particle and two-body matrix elements were varied. The resulting root-mean-square (RMS) difference was 190 keV for all 270 states. For comparison, an RMS deviation of 130 keV was achieved with the USDB interaction [2] over 608 pure sd states in 77 nuclei where 56 linear combinations of the total 68 elements of the Hamiltonian were varied.
Fewer states, mostly cross-shell excited, were included in the FSU fit as few are known with firm spin assignments. This also limits the number of interaction parameters which can be determined reliably. All the shell model calculations in this paper were performed with the shell model code CoSMo [11]. A graph of the differences between experiment and theory is shown in Figure 1. No 0p0h sd states are included in this figure or the fit because the USDB interaction which was previously fitted to them was used unchanged. Predictions from the FSU interaction have compared well with experiment for 38 Cl [9] and 39 Ar [12]. In the present report we are applying the model to understand few interesting features of the sd-shell nuclei.

II. EFFECTIVE SINGLE PARTICLE ENERGY (ESPE)
A key question in any shell model is the single particle energies of the orbitals and their position change with the changing number of proton and neutron. This is a particularly interesting and non-trivial question in the strongly-interacting two-component many-body system of atomic nuclei because the ideal single-particle strength is distributed over many states. An experimental approach of determining the ESPEs has been to measure and sum up the energies of appropriate states (such as 7/2 − ) weighted by the reaction spectroscopic factors. This process is limited by decreasing cross sections for higher lying states, difficulties of making spin assignments and of determining what fraction of the cross sections come from direct reaction components, not to mention the availability of targets. Theoretical approaches do not suffer from most of these experimental limitations, but have their own uncertainties. Perhaps chief among them being the reliability of the determination of the interaction. The FSU interaction was fitted to a wide range of mostly negative-parity states in sd nuclei involving one particle in the f p shell. As such, it samples a broad spectrum of configurations not limited by those experimentally reachable with single-nucleon transfer reactions. The bare single-particle energies in this or any other such interaction tell only part of the story of the effective shell positions. The two-body matrix elements (TBME) between sd and f p nucleons have a major influence on the positions of the orbitals. In fact, the TBMEs shift the orbitals based on the number of particles in shells and are the major reason that one interaction could fit such a wide range of nuclei.
In order to determine the effective single-particle energies (ESPE) of the 0f 7/2 and 1p 3/2 orbitals, we have followed a procedure similar to the experimental approach, but using the theoretical state energies and calculated neutron spectroscopic factors using the following formula: Where, s.f. i is the spectroscopic factor and E * i is the excitation energy of the i-th state for a given spin calculated with the FSU interaction. It has been observed from the calculations that the s.f.s reach to a saturation within first 30 states.
The ESPEs obtained from the above formula across the sd shell are plotted in Figure 2 as a function of proton number Z. The points represent the ESPEs of the 0f 7/2 and 1p 3/2 orbitals above the ground state for one additional neutron added to the even-even nuclei indicated in the figure. The systematic crossing of the ESPEs of the 0f 7/2 and 1p 3/2 orbitals with increasing neutron excess is evident in the Figure. The cross occurs between Z = 10 and 12, suggesting that the N = 28 shell gap shifts to N = 24 with lower Z, which points to the inversion of 0f 7/2 and 1p 3/2 neutron orbitals. The ground state of 31 Ne is tentatively assigned 3/2 − as is the first excited state in 27 Ne. In 27 Mg the lowest 3/2 − and 7/2 − states are essentially degenerate [13].
This inversion of the 1p 3/2 and 0f 7/2 ESPE is related to the 2-body interactions between nucleons in the sd and f p shells whose strengths vary with shell filling. In the FSU interaction it appears that this is a consequence of fitting the sd -f p two-body matrix elements (TBME) to describe known cross-shell excitations in a wide range of nuclei. Over half a century ago Talmi and Unna [14] attributed the inversion of the 1s 1/2 and 0p 1/2 orbitals to the same principle. Alternate explanations, especially for the 1s 1/2 and 0p 1/2 case, have been given in terms of the effects of weak binding on the mean field of low ℓ orbitals. Hoffman et al . [15] have explored the weak binding effect for pure single-particle shells in a Woods-Saxon potential and have shown that it is large near the threshold for neutron s states. While much smaller for p states, there is still a crossing between the 0p 1/2 and 0d 5/2 orbitals at the threshold. A similar effect for the 1p 3/2 and 0f 7/2 orbitals could be a contributing factor to the inversion shown in Figure 2. If so, then it was incorporated through the fitting of the effective interaction. This inversion of the 1p 3/2 and 0f 7/2 ESPE at high neutron excess also has implications for the IoI phenomenon discussed below.

III. EVOLUTION OF THE N=20 SHELL GAP
One of the first indications that the pure sd shell model could not represent low-lying states in all sd nuclei came from the experimentally measured mass of 31 Na [16]. The experimental mass was ≈ 1.6 MeV lower than that predicted from the USD interaction [1] which did account for the other ground state masses. This was further clarified when many more sd states, but not those for the highest N -Z nuclei were fitted. A consistent overprediction of 1 to 2 MeV of the ground state energies of these nuclei can be seen in Figure   9 of Ref. [2]. This region of nuclei is now known as the "Island of Inversion" (IoI) and its origin has been discussed a lot. Most explanation center around the filled or almost filled neutron sd shell and f p intruder configurations leading, counter-intuitively, to lowering the energy of the 2p2h state below that of the "normal" 0p0h one through increased correlation energy or higher deformation, lowering Nilsson orbitals. However the effect fades away with filling of the proton sd shell, and this should also be accounted for in complete theoretical calculations.
We discuss now the IoI nuclei in the framework of the FSU interaction. None of the states discussed in this section were included in the fitting of the interaction. Also it is worth mentioning that all the 2p2h calculations performed with the FSU interaction involve the promotion of two nucleons from the sd to the f p shell, no mixing with the opoh configuration has been considered. We first discuss the case of 31 Na (N=20) [16]. As shown in Figure 3, the total binding energies for the first four 2p2h states were found to be below that of the lowest 0p0h state. The first three 2p2h states agree well with what is so far known experimentally, whereas the spin sequence of the first two 0p0h states is opposite to experiment. Within the limited experimental information available, the FSU interaction has depicted the correct picture of 31 Na as one with the inverted configuration. As mentioned above, only the low Z and N = 20 nuclei exhibit the IoI or inverted 2p2h -0p0h behavior. To explore the transition from IoI to "normal" behavior, Figure 4 compares experimentally measured energies and calculations with the FSU cross-shell interaction for the lowest levels in a sequence of N=20 even A sd nuclei. For Z = 10 and 12, not only do the lowest states have 2p2h character, but the whole 0 + , 2 + , 4 + 2p2h sequence agrees well with experiment. In addition to starting much higher in energy, the spacing between 0p0h states differs significantly from experiment.
The story changes for Z = 14 where the 0p0h 0 + state is substantially lower than the 2p2h one. Above this the second experimental 0 + and first 2 + states are much closer to the 2p2h ones while the second experimental 2 + level corresponds well with the 0p0h one in a clear illustration of shape coexistence. For Z = 16 and 18 both the first experimental 0 + and 2 + states correspond with the 0p0h calculations while the 4 + lies much closer to the calculated 2p2h ones. Note that the FSU cross-shell interaction describes the transition from inverted 0p0h-2p2h order to normal as a function of Z despite not having been fitted to any of these states.
This interpretation of the IoI does not involve any f p orbitals dropping below the sd shell, at least not for spherical shape. This lowering in energy of the 2p2h configurations does not extend so much to 1p1h ones, as shown for 31 Na in Figure 3. The lowest 1p1h state (3579 keV, 3/2 − ) lies over an MeV above the lowest 0p0h state. So it is the promotion of a neutron pair to the f p shell which lowers the 2p2h configuration so much. And the promotion of a neutron pair to the f p orbital appears to lower its energy because of correlation energy in the shell model. In a geometrical picture this corresponds to increased prolate deformation due to promotion of the pair into a downsloping Nilsson orbital whose excitation energy decreases rapidly with increasing deformation. An indication of this difference in deformation is shown in the lower panel of Figure 5.  Table   I. The agreement is quite good with an RMS deviation of 257 keV comparing the 2p2h results below A = 33 and with 0p0h for higher Z.
Since the IoI involves excitations into the f p shell, the question arises how the inversion of the 0f 7/2 and 1p 3/2 single particle energies at low Z discussed before affects our understanding of the IoI. The answer, within the context of the FSU interaction which predicts the IoI well without having been fitted to these nuclei, is shown in Table II. This table shows  occupancy is about twice that of ν0f 7/2 . With increasing Z, the ratio of ν1p 3/2 to ν0f 7/2 decreases steadily from about 200% to about 20%, throughout this region and the ν1p 3/2 orbital plays a role. Of course, the energies of the 2p2h configurations rise above that of the 0p0h ones around Z = 14. Note that for 34 Si the 2p2h 0 + state lies 2432 keV above the 0p0h ground state but the 2p2h 2 + level lies close in energy with the lowest experimental 2 + state. Together these calculations imply that the ν1p 3/2 orbital plays a larger role in the IoI phenomenon than does the ν0f 7/2 one.

IV. FULLY ALIGNED STATES
In describing the states used in the fit of the FSU interaction, we included only 0p0h(1p1h) configurations for natural(unnatural) parity sectors. In particular, no 2p2h configurations were used to adjust the interaction parameters. After the fitting, two early tests were performed to explore the predictive properties of the FSU interaction for 2p2h configurations.
One was a calculation of the lowest 2p2h 7 + states in 34 Cl and 36 Cl [9]. These agreed within 200 keV with the experimental states. The other test was performed on 38 Ar, since experimental states up to 8 + and (10 + ) are known [12]. Calculations using the USD family of interactions agree within 200 keV with the excitation energy of the lowest 2 + state of 38 Ar, but over-predict the lowest experimental 4 + level by over 3 MeV. With only two holes in the sd shell, the maximum spin from coupling two 0d 3/2 protons is 2 . The very high 4 + energy represents the cost of promoting a 0d 5/2 proton to 0d 3/2 , but nature finds another less energetic way of achieving 4 + . This must be by promoting an sd nucleon pair to the f p shell. A 2p2h calculation with the FSU interaction predicts the lowest 4 + level only 300 keV above the experimental one, and it predicts the 6+ state 200 keV below experiment, while the predicted 8 + state is 100 keV above experiment.
With this success we have searched for other states with confirmed 2p2h structure to compare with theory. One such group of excited states across the sd shell are often called the "fully aligned" states. One subgroup of fully-aligned states is the lowest J π = 7 + states.
These states have been suggested to have both odd nucleons in the highest spin orbital around -f 7/2 -and with their spins fully aligned, which, from the Pauli principle, is only possible for non-identical nucleons. These fully-aligned πf 7/2 ⊗ νf 7/2 are yrast and strongly populated in high-spin γ-decay sequences. Stronger evidence of their unique nature comes from (α, d) reactions [18][19][20][21][22][23][24] where they are the most strongly populated states with an orbital angular momentum transfer of ℓ = 6. In most cases such states involve two nucleons beyond those in the dominant ground state configuration outside the sd shell. The energies of these 7 + states (including those in 34 Cl and 36 Cl mentioned above) are graphed in Fig.   6 along with calculated results using the FSU interaction. The agreement is quite good both in value and in the trend which extends from 10 MeV for the lightest nuclei down to 2 MeV for the heaviest and from 2p2h to 1p1h excitations relative to the ground state The calculations also indirectly confirm the spin alignment with approximately equal proton and neutron occupancies in the 0f 7/2 orbitals, even though most 2p2h states in these neutron-rich nuclei as discussed in the IoI section involve predominantly two neutron configurations.
Fully aligned states are also known for some odd-A nuclei where an sd nucleon is also aligned in spin with the aligned 0f 7/2 nucleons. Five such cases in Figure 6 are known experimentally as the strongest states populated in (α, d) reactions. They have an unpaired nucleon in the 0d 3/2 orbital which contributes an extra spin of 3/2 . Again the 2p2h and 1p1h calculations with the FSU interaction agree well. In lighter odd-A nuclei the aligned sd nucleon could be in the 1s 1/2 or 0d 5/2 orbitals, leading to total spins of 15/2 or 19/2 and higher excitation energies. Their calculated energies are also shown in Figure 6 Figure 6. In the case of 42 Ca the analogous state would involve breaking a πd 3/2 pair, promoting one proton to 0f 7/2 , breaking the νf 7/2 pair and coupling them to maximum spin for a total of 11 − . This state has been seen in γ decay following fusion-evaporation and its energy agrees well with the FSU calculation. We hope that future experiments in the FRIB age will be able to test these predictions .

V. SUMMARY
This report has focused on a comprehensive study of 2p2h excitations outside the sd shell using the cross shell FSU interaction. In the FSU interaction, mostly the sd -f p two-body matrix elements had been adjusted to best describe experimental energy states as listed in Ref. [9]. This paper demonstrates that the effective cross shell matrix elements determined in 1p1h excitations are consistent in describing evolution of the ESPEs as well as the 2p2h states and the IoI phenomena. As such it is well positioned to predict how the positions of the lowest f p orbitals shift with the filling of the sd shells. The resulting ESPEs of the 0f 7/2 and 1p 3/2 (the ones best determined by the fitting process) show the expected normal ordering of 0f 7/2 below 1p 3/2 for Z > 12 and a consistent trend of a decreasing separation with decreasing Z until the energy order reverses around Z = 10 to 12. While there have been many indications of inverted shell ordering in the past, these results present a more systematic picture from a model very firmly rooted in data. Perhaps somewhat surprisingly, over the range explored here, the inversion appears to depend more on the proton number than the neutron excess.
Ideally if an interaction is well determined from states involving just one nucleon promoted from the ground state configuration leaving an sd hole behind (1p1h), it should also describe 2p2h excitations. The FSU interaction was applied to the Island of Inversion region, where nuclei are more tightly bound than predicted by the USD family of interactions in the pure emerges as transitional with a 0p0h ground state and a 2p2h lowest 2 + state. It would be interesting to locate experimentally the 4 + 1 state which is predicted as 2p2h at 5523 keV. Another implication of the FSU shell model calculations is that 1p 3/2 pairs dominate over 0f 7/2 ones in the IoI, but 0f 7/2 pairs dominate the lowest 2p2h states beyond the IoI.
The calculations with the 2p2h excitations have been performed to reproduce the fully aligned states which are best described experimentally populated via the (α,d) reactions and also often stand out in heavy-ion fusion-evaporation reactions. The agreement between the experimental energies and the ones calculated with the FSU interaction has been found very satisfactory.
Additionally, this work brings forward an interesting comparison between traditional shell model interactions, with those arising from first principles methods. While the former are obtained from simply fitting SPEs and TBMEs to experimental data, the latter require renormalizations, many-body forces and explicit inclusion of the reaction continuum to achieve agreement with experiment. This dichotomy, presents a modern challenge to nuclear theory and deserves a full investigation.
The capability of the FSU interaction to explain the exotic phenomena of the nuclei carries the prospect that the interaction will be successful for more exotic nuclei or states.
It is hoped that the interaction will prove valuable in the coming FRIB age.   with the FSU interaction. They represent the theoretical centroids of the energies of the 0f 7/2 and 1p 3/2 orbitals above the ground state in the the nuclei with one more neutron than the indicated even-even ones. In the "normal" ordering the red diamonds (1p 3/2 ) lie above the black circles (0f 7/2 ).