# The Nuclear Shell Model towards the Drip Lines

## Abstract

**:**

## 1. Introduction

## 2. The Region of ${}^{\mathbf{28}}$O

## 3. The Region of ${}^{\mathbf{42}}$Si

## 4. The Region of ${}^{\mathbf{60}}$Ca

## 5. The Region of ${}^{\mathbf{78}}$Ni

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Lower mass region of the nuclear chart. The colors indicate the energy of the first 2${}^{+}$ state. In addition to the data from [8], recent data for ${}^{40}$Mn [9], ${}^{62}$Ti [10], ${}^{66}$Cr [11] and ${}^{70,72}$Fe [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.**$D\left(N\right)$ 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${}_{1}^{+}$ states. The orbitals that are being filled are shown. The red line is the results from the universal $fp$ 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 ${}^{60,61,62}$Ca [13].

**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 ${}^{100}$Sn 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.

**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 ${}^{4}$He, ${}^{16}$O and ${}^{40}$Ca. The open red circle for ${}^{60}$Ca 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.

**Figure 5.**Spectrum of ${}^{34}$Si 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, ${J}^{\pi}$, values that are tentative are shown by “()”, and those with multiple of no ${J}^{\pi}$ assignments are shown by the black points. The calculated results are obtained with the FSU Hamiltonian with pure $\Delta $ configurations. The parities are positive for $\Delta =0$ (green) and $\Delta =2$ (red) and negative for $\Delta =1$ (blue).

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

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

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

**Figure 10.**Electric quadrupole ($E2$) maps for ${}^{28}$Si. The results shown are based on the universal $sd$-shell version-B (USDB) Hamiltonian with (

**a**) the $0d$ spin–orbit energy gap increased by 1 MeV, (

**b**) in the $sd$ model space, and (

**c**) with the $0d$ spin–orbit energy gap decreased by 1 MeV. For each J value, ten states were calculated. The widths of the lines are proportional to the reduced electric quadrupole transition strength, $B\left(E2\right)$. Lines for $B\left(E2\right)$ less than 5% of the largest value are not shown. The radius of the circles are proportional to spectroscopic quadrupole moment, ${Q}_{s}$ (2). To set the scale, for panel (

**b**) the 2${}_{1}^{+}$ to 0${}_{1}^{+}$$B\left(E2\right)$ = 82 e${}^{2}$ fm${}^{4}$ and ${Q}_{s}$ (2${}_{1}^{+}$) = +19 e fm are used.

**Figure 12.**Nilsson diagram for ${}^{42}$Si. At the deformation parameter $\beta $ = 0, the orbitals are labeled by ${n}_{r},\ell ,2j$ (see text for definitions), and, at larger deformation, the orbitals are labeled by the Nilsson quantum numbers 2$\mathrm{\Omega}$ [N,n${}_{z}$,$\mathrm{\Lambda}$]. The negative and positive parities is shown by the blue and red lines, respectively. The black dots show the highest Nilsson states occupied. This figure is made using the code WSBETA [87] with the potential choice ICHOIC = 3. The spin–orbit potential are reduced here for protons to make the spherical energies for the $0{d}_{3/2}$ and $1{s}_{1/2}$ orbitals approximately the same.

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**MDPI and ACS Style**

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

**AMA Style**

Brown BA.
The Nuclear Shell Model towards the Drip Lines. *Physics*. 2022; 4(2):525-547.
https://doi.org/10.3390/physics4020035

**Chicago/Turabian Style**

Brown, B. Alex.
2022. "The Nuclear Shell Model towards the Drip Lines" *Physics* 4, no. 2: 525-547.
https://doi.org/10.3390/physics4020035