**1. Introduction**

The starting point for the nuclear shell model is the establishment of model spaces that allow for tractable configuration-interaction (CI) calculations from which we are able to understand and predict the properties of low-lying states [1–5]. This choice is based on the observation that a few even–even nuclei can be interpreted in terms of having magic numbers for *Z* (atomic number) or *N* (nucleon number) and doubly-magic numbers for a given (*Z*, *N*). These magic numbers can be inferred from experimental excitation energies of 2<sup>+</sup> states shown for the low end of the nuclear chart in Figure 1. Magic numbers are those values of *Z* or *N* for nuclei that have a relatively high 2<sup>+</sup> energy within a series of isotopes or isotones.

Another measure of magic numbers is given by the double difference in the binding energy, BE, defined by

$$D(q) = (-1)^q [2\text{BE}(q) - \text{BE}(q+1) - \text{BE}(q-1)] \tag{1}$$

for isotopes (*q* = *N* with *Z* held fixed) or isotones (*q* = *Z* with *N* held fixed) can also be used to measure shell gaps [6]. An example for the neutron-rich calcium isotopes is shown in Figure 2 (the dashed line extrapolation to *N* = 40 is discussed below.) The value of *D*(*N*) at these magic numbers gives the effective shell gap. In between the magic numbers, *D*(*N*) gives the pairing energy [6]. The excitation energies of the 2<sup>+</sup> states at *N* = 28, 32 and 34, also shown in Figure 2, are close to the *D*(*N*) values at these magic numbers. The neutron gaps at *N* = 32 and 34 are weaker than the gap at *N* = 28, but they are strong enough to allow the configurations to be dominated by the orbitals, shown in Figure 2.

In the simplest model, the magic number is associated with a ground state that has a closed-shell configuration for the given value of *Z* or *N*. The following is from footnote 9 in [7]. It was Eugene Paul Wigner who coined the term "magic number". Steven A. Moszkowski, who was a student of Maria Goeppert-Mayer, in a talk presented at the American Physical Society meeting in Indianapolis, 4 May 1996 said: "Wigner believed in the liquid drop model, but he recognized, from the work of Maria Mayer, the very strong evidence for the closed shells. It seemed a little like magic to him, and that is how the words 'Magic Numbers' were coined". The discovery of "magic numbers" lead M. Goeppert-Mayer, and independently J. Hans D. Jensen in Germany, one year later, in 1949, to the construction of the shell model with strong spin–orbit coupling, and to the Nobel Prize they shared with Wigner in 1963.

**Citation:** Brown, B.A. The Nuclear Shell Model towards the Drip Lines. *Physics* **2022**, *4*, 525–547. https:// doi.org/10.3390/physics4020035

Received: 7 March 2022 Accepted: 14 April 2022 Published: 12 May 2022

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**Figure 1.** Lower mass region of the nuclear chart. The colors indicate the energy of the first 2<sup>+</sup> state. In addition to the data from [8], recent data for 40Mn [9], 62Ti [10], 66Cr [11] and 70,72Fe [11] are added. The filled black circles show the doubly-magic nuclei associated with the most robust pairs of magic numbers 8, 20, 28 and 50. The small open circles show the doubly-magic nuclei associated with less robust magic numbers 6, 14, 16, 32, 34, and 40. The large open circles indicate the nuclei near the neutron drip lines that are the focus of this paper. The triangles are those nuclei observed to decay by two protons in the ground state. The cross indicates no magic number for protons or neutrons, and the question mark indicates that the doubly-magic status is not known.

**Figure 2.** *D*(*N*) as given by Equation (1). The black dots with error bars are the experimental data. The blue crosses are the excitation energies of the 2<sup>+</sup> <sup>1</sup> states. The orbitals that are being filled are shown. The red line is the results from the universal *f p* calcium (UFP-CA) Hamiltonian [12]. The dashed line is the extrapolation based on the universal nuclear energy density functional (version zero) (UNEDF0) binding energies. for 60,61,62Ca [13].

The nuclei marked with closed circles in Figure 1 are commonly used to define the boundaries of CI model spaces. Those indicated by small open circles are usually contained within a larger CI model spaces. Historically, the size of the assumed model space has depended on the computational capabilities. At the very beginning in the 1960s, they were the 0*p* model space bounded by 4He and 16O, and the 0 *f*7/2 model space bounded by 40Ca and 56Ni.

For heavy nuclei, doubly-magic nuclei are associated with the shell gaps at 28, 50, 82 and 126. These gaps are created by the spin–orbit splitting of the high orbitals, which lowers the the *j* = - + 1/2 single-particle energies for - = 3 (28), - = 4 (50), - = 5 (82) and - = 6 (126). Since the two *j* values for a given high value are split, 28, 50, 82 and 126 will be referred to as *jj* magic numbers. The nuclei with *jj* magic numbers for both protons and neutrons will be called double-*jj* closed-shell nuclei. These are shown by the red circles in Figure 3: 208Pb, 132Sn, 100Sn, 78Ni and 56Ni. The open red circle for 100Sn indicates that it is expected to be double-*jj* magic [14], but it has not yet been confirmed experimentally. The continuation of the double-*jj* sequence with - = 2 (14) and - = 1 (6) is shown by the open blue circles for 42Si, 28Si, 18C and 12C on the lower left-hand side of Figure 3. As discussed below, the calculations for these nuclei show rotational bands with positive quadrupole moments indicative of an oblate intrinsic shape.

**Figure 3.** The nuclear chart showing the *jj* magic numbers (see text for *jj* definition). The black lines show where the two-proton (upper) and two-neutron (lower) separation energies obtained with the universal nuclear energy density functional (version one) (UNEDF1) [13] cross 1 MeV. The filled red circles show the locations of double-*jj* magic nuclei established from experiment. The open red circle for 100Sn indicates a probably double-*jj* magic nucleus that has not been confirmed by experiment. The blue circles in the bottom left-hand side are nuclei in the double-*jj* magic number sequence that are oblate deformed.

In light nuclei, magic numbers 2, 8, 20 and 40 are associated with the filling of a major harmonic-oscillator shell with *No* = (2*nr* + -) (*nr* is the radial quantum number), where both members of the spin–orbit pair *j* = - ± 1/2 are filled. Since one can recouple the two orbitals with the same value to total angular momentum *L* and total spin *S*, 2, 8, 20 and 40 will be referred to as *LS* magic numbers.

The *LS* magic numbers for isotopes and isotones are shown by the thin brown lines in Figure 4. There are only three known double-*LS* magic nuclei, 4He, 16O and 40Ca shown by the filled red circles in Figure 4. The next one in the sequence would be 80Zr, but in this case the *Z* = *N* = 40 gap is too small due to the lowering of the 0*g*9/2 single-particle energy from the spin–orbit splitting. As will be discussed below, 60Ca (the red open circle with a question mark) could be a "fourth" double-*LS* magic nucleus. There are regions where the *LS* magic numbers for isotopes or isotones dissappear as shown by the blue lines in Figure 4. These will be referred to as "islands of inversion" [15].

**Figure 4.** Lower mass region of the nuclear chart showing the *LS* magic numbers, 2, 8, 20 and 40 (see text for *LS* definitioon). The black lines show where the two-proton (upper) and two-neutron (lower) separation energies obtained with the UNEDF1 [13] functional cross 1 MeV. The filled red circles show the double-*LS* magic nuclei 4He, 16O and 40Ca. The open red circle for 60Ca indicates a possible doubly-magic nucleus that has not been confirmed by experiment. The green circles are doubly-magic nuclei associated with the *j*-orbital fillings. The blue lines indicate isotopes or isotones where the *LS* magic number is observed to be broken.

The nuclei with green circles in Figure 4 also have doubly-magic properties. The pattern is that when one type of nucleon (proton or neutron) has an *LS* magic number, then the other one has a magic number for the filling of each *j* orbital. These are 6 (0*p*3/2), 8 (0*p*1/2), 14 (0*d*5/2), 16 (1*s*1/2), 20 (0*d*3/2), 28 (0 *f*7/2), 32 (1*p*3/2), 34 (1*p*1/2), 40 (0 *f*5/2), 50 (0*g*9/2) and 56 (1*d*5/2).

The only addition to the *jj* and *LS* closed-shell systematics discussed above is for 88Sr shown in Figure 4, where there is an energy gap between the proton 1*p*1/2 and 1*p*3/2,0 *f*5/2 states. In early calculations, 88Sr was used as the closed shell for the 1*p*1/2, 0*g*9/2 model space [16], but more recently the four-orbit model space of 0 *f*5/2,1*p*3/2,1*p*1/2,0*g*9/2 has been used for the *N* = 50 isotones [17,18].

For a given shell gap, the *LS* magic numbers are more robust than those for *jj*. The reason is that deformation for *jj* magic numbers starts with a one-particle one-hole (1*p*–1*h*) excitation of a nucleon in the *j* = - + 1/2 orbital to the other members of the same oscillator shell, *No* = (2*nr* + -). Since 1*p*–1*h* excitations across *LS* closed shell gaps change parity, ground-state deformation for *LS* magic numbers must come from *np*–*nh* (*n* ≥ 2) excitations across the *LS* closed shells as in the region of 32Mg [15].

Let us discuss here results, obtained with Hamiltonians. based on data-driven improvements to the two-body matrix elements, provided by ab initio methods. The ab initio methods are based on two-nucleon (NN) and three-nucleon (NNN) interactions obtained by model-dependent fits to nucleon-nucleon phase shifts and properties of nuclei with *A* = 2 to 4. For a given model space, these are renormalized for short-range correlations and for the truncations into the chosen model space to provide a set of two-body matrix elements (TBME) for nuclei near a chosen doubly-closed shell. From this starting point, one attempts to make minimal changes to the Hamiltonian to improve the agreement with energy data for a selected set of nuclei and states within the model space. A convenient way to do this is by using singular value decomposition (SVD) [19]. In many cases, one adjusts specific TBME or combinations of TBME. The most important are the monopole, pairing and quadrupole components. An important part of the universal Hamiltonian is in

the evolution of the effective single-particle energies (ESPE) as one changes the number of protons and neutrons. Starting with a closed shell with a given set of single-particle energies, the ESPE as a function of *Z* and *N* are determined by the monopole average parts of the TBME [5].

These methods provide "universal" Hamiltonians in the sense that a single set of single-particle energies and two-body matrix elements are applied to all nuclei in the model space, perhaps allowing for some smooth mass dependence. This has turned out to be a practical and useful approximation. As the ab initio, starting points are improved, these "universal" Hamiltonians were replaced by Hamiltonians for a more restricted set of nuclei, or even for individual nuclei as has been done in the valence-space in-medium similarity renormalization group (VS-IMSRG) method [4,20].

The empirical modifications to the effective Hamiltonian account for deficiencies in the more ab initio methods. Most ab initio calculations are carried out in a harmonic-oscillator basis due to its convenient analytical properties. Near the neutron drip lines, the radial wavefunctions become more extended, the single-particle energy spectrum becomes more compressed, and the continuum becomes explicitly more important. To take this into account, the ab initio methods require a very large harmonic-oscillator basis.

Due to the continuum, nuclei near the neutron drip line present a substantial theoretical challenge [2,21]. Methods have been developed that take the continuum into account explicitly. The density matrix renormalization group (DMRG) method [22,23] makes use of a single-particle potential together with a simplified interaction based on halo effective field theory [24,25]. In the Gamow shell model (GSM) [26–28], the many-body basis is constructed from a single-particle Berggren ensemble [29,30]. The DMRG and GSM methods rely on use of simplified two-body interactions with adjustable parameters. There is also the shell model embedded in the continuum formalism that can make use of the universal interactions [31]. Recent progress in the GSM method is presented in [32].

Ground-state nuclear halos are a unique feature of nuclei near the neutron drip line [33]. This is due to the loose binding of low- orbitals with extended radial wavefunctions. The most famous case is that for 11Li which was observed to have a rapid rise in the nuclear matter radius compared to the trends up to 9Li [34]. The wavefunction of 11Li is dominated by a pair of neutrons in the 1*s*1/2 orbital. As discussed below, halos in the region of 30Ne and 42Si are dominated by the 1*p*3/2 orbital. Proton halos are not so extreme due to the Coulomb barrier. The excited 1/2<sup>+</sup> (1*s*1/2) state of 17F is a good example of an excited-state halo as determined indirectly from its large Thomas–Ehrman energy shift of 0.87 MeV 17O to 0.49 MeV in 17F.

States above the (proton/neutron) separation energy have (proton/neutron) decay widths. In the conventional CI approach, one calculates states whose energy is taken to be the centroid energy of the decaying state. The decay width is calculated using the approximation Γ = C2S Γ*sp*(*Q*), where C2S is the spectroscopic factor and Γ*sp* is the singleparticle neutron decay width calculated with a a decay energy, *Q*, value taken from the shell-model centroid or the experimental centroid if known. The explicit addition of the continuum shifts down the energy relative to its CI energy [31]. Further, the continuum (finite-well potential) is responsible for the Thomas–Ehrman shift for states in proton-rich nuclei compared to those in the neutron-rich mirror nuclei [19].

In this review, I concentrate on four regions of neutron-rich "outposts" whose understanding are most important for future developments. These are shown in Figure 1: 28O, 42Si, 60Ca and 78Ni. 42Si is labeled by "×" since it does not have a magic number for protons or neutrons. 78Ni is labeled by a filled circle since it is now known to be doubly magic [35]. 60Ca is known to be inside the neutron drip line [36], but its mass and excited states have not yet been measured.

Nuclei that are observed to decay by two protons are shown by the triangles in Figure 1. The two-proton ground-state decays for 45Fe, 48Ni, 54Zn and 67Kr have half-lives on the order of ms and compete with the *β* decay of those nuclei. An experimental and theoretical summary of the results for those nuclei together with that of 19Mg has been given in [37]. There is qualitative agreement between experiment and theory. In order to become more quantitative, the experimental errors in the partial half-lives need to be improved. Theoretical models need to be improved to incorporate three-body decay dynamics (presently based on single-orbit configurations) with the many-body CI calculations for the two-nucleon decay amplitudes. The correlations for two-nucleon transfer amplitudes via (t,p) or (3He,n) are largely determined by the (*S*, *T*)=(0, 1) structure of the triton or 3He, whereas two-nucleon decay is determined by the decay through the Coulomb and angularmomentum barriers that are dominated by the low- components. For the lightest nuclei, multi-proton emissions (shown in Figure 1 of [38]) are observed as broad resonances.

Knockout reactions are used to produce nuclei further from stability. The cross sections for these reactions can be compared to theoretical models in terms of the cross-section ratio *Rs* = *σ*exp/*σ*th; see [39] for a recent summary. It is observed for nuclei far from stability where Δ*S* = | *S*1*<sup>p</sup>* − *S*1*<sup>n</sup>* | is large (*S*<sup>1</sup> is the one nucleon separation energy) that *Rs* is near unity when the knocked out nucleon is loosely bound but drops to approximately 0.3 for deeply bound nucleons. This has been attributed to the short- and long-ranged correlations that depletes the occupation of deeply-bound states [40]. The short-ranged correlations are connected to the high-momemtum tail observed in observed in high-energy electron scattering experiments [41]. The long-ranged correlations come fron particle-core coupling and pairing correlations beyond that included within the valence space. Another reason may be the approximations made in the sudden approximation for the dynamics used for the reaction [39]. In the analysis of [40], the *Rs* factor for loosely-bound nucleons that comes mainly from the long-ranged correlations is expected to be 0.6–0.7 rather than unity. The analysis of (*p*, 2*p*) experiments [42] find *Rs* values that depend less on the proton separation energy going from 0.6 to 0.7.

The *σ*th depends on the CI calculations for the spectroscopic factors. An approximation that is made in CI calculations is that only the change in configurations for the knocked out nucleon contributes to the spectroscopic factor. The radial wavefunctions for all other nucleons in the parent and daughter nuclei are assumed to be the same. However, consider, as an example, the knockout of a deeply bound proton from 30Ne to 29F. The size of the neutrons orbtials in 30Ne and 29F are changing due to the proximity to the continuum, and the overlap of the spectator neutrons in the nuclei with the atomic mass numbers *A* and *A* − 1 will be reduced from unity. This effect should be contained in ab initio and continuum models [43,44], but an understanding within these models requires an explicit separation of the one-nucleon removal overlaps in terms of the removed nucleon within the basis states for (*A*, *Z*) and the radial overlaps between the nuclei with *A* and *A* − 1.
