**2. The Region of 28O**

The oxygen isotopes provided the first complete testing ground for theory and experiment from the proton drip line to the neutron drip line [45]. The prediction by the "universal" *sd*-shell (USD) Hamiltonian [1,46], in the 1980s that 24O was a doubly-magic nucleus was later confirmed experimentally in 2009 [47–49].

For the one-neutron decay of 25O, the USD charge-dependent (USDC) Hamiltonian in the *sd* shell [19] gives *Q* = 1.15(15) MeV, to be compared to the experimental value of *Q* = 0.749(10) MeV [50]. The explicit addition of the continuum will lower the calculated energy [31]. The calculated value of the spectroscopic factor is (25/24)<sup>2</sup> C2S(0*d*3/2) = 1.01(1) (the error, shown in the parentheses for the value last decimal, comes from the comparison of the four *sd*-shell Hamiltonians developed in [19]). For the calculated decay width, one obtains Γ = (25/24)<sup>2</sup> C2S Γ*sp*(*Q*) = 75(1) keV. Γ*sp* = 74(1) keV is obtained using the experimental *Q* value and a Woods–Saxon potential. The experimental neutron decay width is Γ = 88(6) keV [50]. The theoretical error in the width is probably dominated by the uncertainty in the parameters of the Woods–Saxon potential.

The measured masses of the Na isotopes [51] found more binding near *N* = 20 than could be accounted for by the pure Δ = 0 configurations; here, the notation Δ = *n* is used where *n* is the number of neutrons excited from *sd* to *p f* . Hartree–Fock calculations [52] showed that these mass anomalies were associated with a large prolate deformation, where the 2Ω*<sup>π</sup>* [N,n*z*,Λ]=1<sup>−</sup> [3,3,0] and 3<sup>−</sup> [3,2,1] Nilsson orbitals from the *f p* shell cross the 1<sup>+</sup> [2,0,0] and 3<sup>+</sup> [2,0,2] orbitals from the *sd* shell near a value for the deformation paramater of *β* = +0.3. The anomaly was confirmed by Δ = 0, CI calculations in [53,54], where in [53] it was called the "collapse of the conventional shell-model". CI calculations that included Δ = 2 components [15,55] showed that nuclei in this region have ground-state wavefunctions dominated by the Δ = 2 component. This is due to a weakened shell gap at *N* = 20 below *Z* = 14, pairing correlations in the Δ = 2 configurations, and proton–neutron quadrupole correlations that give rise to the Nilsson orbital inversion. In [15], the region of nuclei below 34Si involved in this inversion was called the "island-of-inversion".

The Hamiltonian, used in [15], was appropriate for pure Δ = *n* configurations. This Hamiltonian was modified to account for more recent data related to the energies of Δ = 1 and Δ = 2 configurations resulting in the new Florida State University (FSU) Hamiltonian [56]. As examples of the type of predictions, results, obtained with the FSU Hamiltonian, are shown for 34Si in Figure 5, 32Mg in Figure 6, and 29F in Figure 7. All of these calculaitons were carried out with NuShellX [57] code and allowed only for neutron excitations from 1*s*–0*d* to 1*p*–0 *f* . Calculations in a full *nh*¯ *ω* basis (*h*¯ being the reduced Planck constant) with *n* > 0 also require the addition of proton excitations from 0*p* to 1*s*–0*d* and proton exicitations from 1*s*–0*d* to 1*p*–0 *f* . In full *nh*¯ *ω* basis, the 1*h*¯ *ω* spurious states can be removed with the Gloeckner-Lawson method [58]. Comparison to calculations in the full *nh*¯ *ω* basis with the Oxbash code [59] show that the energies are lowered relative to the Δ basis by up to approximately 200 keV. This shows the Δ = 1, 2 proton and proton–neutron components are small compared to the Δ = 1, 2 neutron components for the low-lying states in these neutron-rich nuclei. For nuclei with *N* ≈ *Z*, removal of the spurious states in the *nh*¯ *ω* basis is important.

**Figure 5.** Spectrum of 34Si obtained with the Florida State University (FSU) Hamiltonian [56] compared to experiment. The length of the horizontal lines are proportional to the the angular momentum, *J*. The experimental parity is indicated by blue for negative parity and red for positive parity. Experimental spin-parity, *Jπ*, values that are tentative are shown by "()", and those with multiple of no *J<sup>π</sup>* assignments are shown by the black points. The calculated results are obtained with the FSU Hamiltonian with pure Δ configurations. The parities are positive for Δ = 0 (green) and Δ = 2 (red) and negative for Δ = 1 (blue).

**Figure 6.** Spectrum of 32Mg obtained with the FSU Hamiltonian [56]. The results are obtained with pure Δ configurations. The spins are proportional to the length of the horizontal lines. The parities are positive for Δ = 0 (green) and Δ = 2 (red) and negative for Δ = 1 (blue).

**Figure 7.** Spectrum of 29F obtained with the FSU Hamiltonian [56]. The results are obtained with pure Δ configurations. The spins are proportional to the length of the horizontal lines. The parities are positive for Δ = 0 (green) and Δ = 2 (red) and negative for Δ = 1 (blue).

The barrier between the Δ = 0 (spherical) and Δ = 2 (deformed) configurations reduces the mixing between the lowest energy states of each configuration. When one combines the Δ = 0 and Δ = 2 configurations in CI calculations, the state that is dominated by Δ = 0 is pushed down in energy by the mixing with many Δ = 2 configurations mainly due to the increase in the pairing energy. If one were to start with the FSU Hamiltonian and add off-diagonal TBME of the type < *sd* | *V* | *f p* >, the components dominated by Δ = 0 would be pushed down in energy due to this increase in pairing. However, this results in a double-counting since the *sd* part FSU interaction is already implicitly renormalized for the *f p* admixtures. In addition, to achieve convergence in the mixed wavefunctions, one has to add Δ = 4 and higher. This results in large matrix dimensions.

When one mixes the Δ components, one has to modify parts of the Hamiltonian that are diagonal in Δ. This is sometimes performed by changing the pairing strength in the *J* = 0, *T* = 1 two-body matrix elements, so that the ground-state binding energies agree with experimental values. Hamiltonians that have been designed for mixed configurations are called SDPF-U-MIX [60] and SDPF-M [61,62]. Details about the modifications to SDPF- U to obtain SDPF-U-MIX are given in the Appendix of [60]. In the remainder of this section, I discuss some examples, obtained with the FSU Hamiltonian with pure Δ configurations. This provides a starting point for more complete calculations with mixed Δ and those explicitly involving the continuum.

The Δ = 0 (*sd*-shell) part of the FSU spectrum for 34Si (the green lines in Figure 5) has a simple interpretation. The ground state is dominated by the (0*d*5/2)<sup>6</sup> proton configuration. The 5.24 MeV 2<sup>+</sup> and the 6.47 MeV 3<sup>+</sup> states are dominated by the (0*d*5/2)5(1*s*1/2)<sup>1</sup> proton configuration. In the two-proton transfer experiment from 36S [63], a 2<sup>+</sup> state at 5.33 is observed that can be interpreted as two protons removed from (0*d*5/2)6(1*s*1/2)<sup>2</sup> to make (0*d*5/2)5(1*s*1/2)1. The (0*d*5/2)4(1*s*1/2)<sup>2</sup> 0+ state is predicted at 8.76 MeV. For the FSU Hamiltonian, all of these predictions are based on the USDB effective Hamiltonian [64]. The ESPE for the 0*d*5/2 and 1*s*1/2 proton states near 34Si are determined from the binding energies of 33Al, 34Si and 35P. Above 2.5 MeV the level density is dominated by the neutron Δ = 1 and Δ = 2 configurations. The Δ = 1 states can be interpreted in terms of the low-lying 3/2<sup>+</sup> and 1/2<sup>+</sup> 1*h* states of 33Si coupled to the low-lying 7/2<sup>−</sup> and 3/2<sup>−</sup> 1*p* states of 35Si. The state with maximum spin-parity *J<sup>π</sup>* of 5<sup>−</sup> predicted at 5.12 MeV can be compared to the proposed experimental 5− state at 4.97 MeV [65]. The theoretical spectra from the mixed SDPF-U-Mix shown in [65] is similar to the FSU unmixed spectrum in Figure 5.

The FSU results for 32Mg are shown in Figure 6. Compared to 34Si, there is an inversion of the low-lying Δ = 0 and Δ = 2 configurations. For pure Δ configurations, the reduced electric-quadrupole transition strength *B*(*E*2) for 2<sup>+</sup> <sup>1</sup> (<sup>Δ</sup> = 2) to 0<sup>+</sup> <sup>2</sup> (Δ = 0) is zero. Experimentally, *B*(*E*2, 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>2</sup> ) = 48+<sup>75</sup> <sup>−</sup><sup>20</sup> e2 fm4 compared to *<sup>B</sup>*(*E*2, 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) = 96(16) e<sup>2</sup> fm4; see Table 1 in [66]. An improved half-life for the 0<sup>+</sup> <sup>2</sup> is important since it helps to determine the Δ mixing.

One of the key experiments for 32Mg is the two-neutron transfer from 30Mg (t,p), where the first two 0<sup>+</sup> states were observed with approximately equal strength [67]. This observatiom has proven difficult to understand; see the references in [68]. Starting from a Δ = 0 configuration for the 30Mg ground state, one can populate the Δ = 0, 0<sup>+</sup> configuration in 32Mg by (*sd*)<sup>2</sup> transfer and the Δ = 2, 0<sup>+</sup> configuration by (*f p*)<sup>2</sup> transfer. Macchiavelli et al. [68] analyzed the (t,p) cross sections by used centroid energies for the Δ = 0,2,4 configurations of 1.4, 0.2 and 0.0 MeV, respectively, obtained with the SDPF-U-MIX Hamiltonian [60]. This three-level model could account for the experimental observation with a ground state that is 4% Δ = 0, 46% Δ = 2 and 40% Δ = 4 together with a ground-state wavefunction for 30Mg that has 97% Δ = 0 and 3% Δ = 2. In this three-level model for 32Mg, the main part of the Δ = 0 configuration is in the 0<sup>+</sup> <sup>3</sup> state predicted to be near 2.2 MeV; see Table 1 in [68].

Two-proton knockout from 34Si provides more information. Starting with a pure Δ = 0 configuration for the 34Si ground state, only Δ = 0, 0<sup>+</sup> configurations in 32Mg can be made. In the two-proton knockout experiment of [69,70], strong 0<sup>+</sup> strength is observed in the sum of the first two 0<sup>+</sup> states; see Figure 9 in [70]. The strength to the 0<sup>+</sup> <sup>1</sup> and 0<sup>+</sup> <sup>2</sup> states cannot be separated due to the long lifetime of the 0<sup>+</sup> <sup>2</sup> state. Significant strength to 0<sup>+</sup> states above 1.5 MeV was not observed, in contradiction to that predicted in the three-level model above [68] or the SDPF-M model. More needs to be done to understand the structure of 32Mg and how it connects to the experimental data discussed above.

Results from the FSU Hamiltonian provide an extrapolation down to 28O. 29F has been called a "lighthouse on the island-of-inversion" [71]. The FSU results for 29F are shown in Figure 7. The lowest state is 5/2<sup>+</sup> with a Δ = 2 configuration. The lowest 1/2+, 3/2+, 7/2<sup>+</sup> and 9/2<sup>+</sup> Δ = 2 states are dominated by the configuration with 0*d*5/2 coupled to the Δ = 2, 2<sup>+</sup> state in 28O at 1.26 MeV. The 0*d*5/2 coupled to 2+, 5/2<sup>+</sup> configuration is spread over many higher 5/2<sup>+</sup> states in 29F. The Δ = 3 states for 29F start at 3.9 MeV. An excited state in 29F at 1.080(18) MeV [72] made from proton knockout from 30Ne was suggested to be 1/2<sup>+</sup> on the basis comparisons to the SDPF-M calculations shown in [72].

With the FSU Hamiltonian, for 27F, the lowest Δ = 0, 5/2<sup>+</sup> state is 1.9 MeV below the Δ = 2, 5/2<sup>+</sup> state. The large FSU occupancy of 1.38 in 29F for the loosely bound 0*p*3/2 orbital may explain the observed neutron halo [73]. In particular, the two-neutron transfer amplitudes TNA[(0p3/2)]<sup>=</sup>0.62 for the 29F, Δ = 2, 5/2<sup>+</sup> ground state going to the 27F, Δ = 0, 5/2<sup>+</sup> ground state. Improved mass mesurements are needed for the neutron-rich fluorine and neon isotopes.

Results for these calculations depend on the ESPE extrapolation down to 28O contained in the FSU interaction. The ESPE for the neutron orbitals as a function of *Z* obtained with the FSU Hamiltonian with (Δ = 0) are shown in Figure 8 (for 34Si I assume a (0*d*5/2)<sup>6</sup> configuration for the protons). These are compared with the results from the Skyrme-x energy density functional (Skx EDF) calculations [74].

For unbound states, the energies can be approximated by first increasing the EDF central potential to obtain a wavefunction bound by, for example, 0.2 MeV, and then taking the expectation value of the wavefunction value with original EDF Hamiltonian. This method provides a practical approximation to the centroid energy. Results for the unbound resonances could be calculated more exactly from neutron scattering on the EDF potential.

The results in Figure 8 show that the *N* = 20 shell gap decreases from approximately 7.0 MeV in 34Si to approximately 2.7 MeV in 28O. The major part of this decrease is due to the lowering energy for 1*p*3/2 relative to 0 *f*7/2 as the states become more unbound. The energies for these two states cross at approximately *Z* = 10. Recent experimental information on the ESPE near 28Mg and their interpretation similar to those of Figure 8 with a Woods–Saxon potential is given in [75]. For the FSU Hamiltonian, the loose binding effects are implicitly built into the monopole components of the TBME from the SVD fit to data on the BE and excitations energies.

There is also an increase in the gap in 34Si due to the proton-neutron tensor interaction contribution to the spin–orbit splitting [5] that is built into the FSU Hamiltonian. The spin–orbit tensor interacton is zero in the double-*LS* closed shell nuclei such as 28O and 40Ca. The tensor interaction is important for changing the effective single–particle energies as a function of proton and/or neutron number [5] or as a function of the state-dependent orbital occupancies [76].

The *f p* ESPE obtained from the Skx EDF [74] from 30Ne to 78Ni are shown in Figure 9. The energies of 1*p* and 0 *f* systematically shift due to the finite-well potential.

**Figure 8.** Effective single-particle energies (ESPE) for neutron orbitals as a function of proton number. The lines are from the Skyrme-x energy-density functional (Skx EDF) [74] calculations. The crosses are from the FSU [56] Hamiltonian calculations.

**Figure 9.** ESPE for the *f p* proton (p) and neutron (n) orbitals obtained from the Skx EDF interaction [74] for a range of nuclei.

For nuclei near the neutron drip line, there are few bound states that can be studied by their gamma decay. States above the neutron separation energy neutron decay. These neutron decays can be complex both experimentally and theoretically. The neutron decay spectrum depends upon how the unbound states are populated. They are often made by proton and neutron knockout reactions. For one- and two-nucleon knockout, one can calculate spectroscopic factors that can be combined with a reaction model to find which states are most strongly populated. A recent example of this type of calculation was for two-proton knockout from 33Mg going to 31Ne [77]. One neutron decay can often go to excited states in the daughter [77]. Additionally, multi-neutron decay can occur. It is important to measure the neutrons in coincidence with the final nucleus and its gamma decays. On the theoretical side, one must use the calculated wavefunctions to obtain neutron decay spectra.

An example of multi-neutron decay is in the one-proton knockout from 25F to make 24O [78,79]. The calculated one-proton knockout spectroscopic factors showed that 0*d*5/2 knockout mainly leads to the ground state of 24O, and that 0*p* knockout leads to many negative-parity states above the neutron separation energy of 24O. These excited states multi-neutron decay to <sup>21</sup>−23O [78]. However, in the (p,2p) reaction [79], it was suggested from the momentum-distribution of 23O that a low-lying positive-parity excited state in 24O above the neutron separation energy was strongly populated by 0*d* removal, in strong disagreement with the calculations of [78]. This experimental result should be confirmed.

The two-neutron decay of 26O has a remarkably small *Q* value of 0.018(5) MeV [50]. The theoretical *Q* value from USDC Hamiltonian [19] is 0.02(15) MeV. The decay width depends strongly on the for the -<sup>2</sup> two-nucleon decay amplitude. From Figure 2b of [80], pure -<sup>2</sup> two-nucleon decays widths with the experimental *Q* value are approximately 10−4, 10−<sup>8</sup> and 10−<sup>14</sup> MeV for - = 0, 1 and 2, respectively. The calculated TNA in the *sd* model space with the USDC Hamiltonian are 0.99 for (0*d*3/2)<sup>2</sup> and 0.16 for (1*s*1/2)2. Thus, Γ = [TNA(1*s*1/2)2] <sup>2</sup> <sup>Γ</sup>*sp*(*Q*) ≈ 0.003 keV. The (1*p*3/2)<sup>2</sup> TNA will be on the order of TMBE < (0*d*3/2)<sup>2</sup> | *<sup>V</sup>* | (1*p*3/2)<sup>2</sup> > /2Δ*E*, where <sup>Δ</sup>*<sup>E</sup>* is the energy difference between the the 1*p*3/2 and 0*dd*3/2 states in 25O. With typical values of TMBE < (0*d*3/2)<sup>2</sup> | *<sup>V</sup>* | (1*p*3/2)<sup>2</sup> >≈ 2 MeV and <sup>Δ</sup>*<sup>E</sup>* ≈ 2 MeV [81] giving TNA = 0.5, the (1*p*3/2)<sup>2</sup> contribution to the two-neutron decay width will be small.

The nucleus 28O is unbound to four neutron decay. The theoretical understanding of this complex decay involves the four-body continuum [80]. These continuum calculations strongly depend upon the single-particle states involved; see Figure 2d in [80]. With the FSU Hamiltonian, the Δ = 2 configuration for 28O lies 0.8 MeV below the Δ = 0 (closedshell) configuration due to the pairing correlations. The calculated four-neutron decay energy is 1.5 MeV. The energy should be lowered by an explicit treatment of the many-body continuum. Thus, the "island of inversion" may be a "peninsula of inversion" extending from 32Mg all the way to the neutron drip line; below, I discuss what may be the first true "island of inversion" between 60Ca and 78Ni. There are many paths for the four-neutron decay of 28O. For example, in the FSU Δ = 2 model, it may proceed by a relatively fast (1*p*3/2)<sup>2</sup> decay to the 26O ground state followed by its decay to 24O.
