# The Proxy-SU(3) Symmetry in Atomic Nuclei

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. SU(3) Symmetry in Nuclear Structure

## 3. Nucleon Pairs Favoring Deformation

## 4. The Proxy-SU(3) Approximation

## 5. Corroboration of Proxy-SU(3) through Nilsson Model Calculations

## 6. Proxy-SU(3) Symmetry in the Spherical Shell Model Basis

## 7. The Dominance of the Highest Weight Irreducible Representations of SU(3)

## 8. Physical Consequences of the Dominance of the Highest Weight Irreps

#### 8.1. Prolate over Oblate Dominance

**Table 6.**Highest weight SU(3) irreps (labeled by hw) for U(n), n = 6, 10, 15, 21, 28, 36, and highest ${C}_{2}$ irreps (labeled by C) for n = 6, 10, 15, 21. The highest weight (hw) irreps differing from their highest ${C}_{2}$ counterparts are shown in boldface. The results were obtained by the code of Reference [138]; moreover, a new version exists [139] (see also Reference [140]), and were presented in [141]. An analytic formula for the calculation of the hw irreps was given in Reference [142]. Adapted from Reference [143]. See Section 7 for further discussion.

8–20 | 8–20 | 28–50 | 28–50 | 50–82 | 50–82 | 82–126 | 82–126 | 126–184 | 184–258 | ||
---|---|---|---|---|---|---|---|---|---|---|---|

sd | sd | pf | pf | sdg | sdg | PFH | PFH | sdgi | PFHj | ||

M | irrep | U(6) | U(6) | U(10) | U(10) | U(15) | U(15) | U(21) | U(21) | U(28) | U(36) |

hw | C | hw | C | hw | C | hw | C | hw | hw | ||

0 | (0,0) | (0,0) | (0,0) | (0,0) | (0,0) | (0,0) | (0,0) | (0,0) | (0,0) | (0,0) | |

1 | [1] | (2,0) | (2,0) | (3,0) | (3,0) | (4,0) | (4,0) | (5,0) | (5,0) | (6,0) | (7,0) |

2 | [2] | (4,0) | (4,0) | (6,0) | (6,0) | (8,0) | (8,0) | (10,0) | (10,0) | (12,0) | (14,0) |

3 | [21] | (4,1) | (4,1) | (7,1) | (7,1) | (10,1) | (10,1) | (13,1) | (13,1) | (16,1) | (19,1) |

4 | [${2}^{2}$] | (4,2) | (4,2) | (8,2) | (8,2) | (12,2) | (12,2) | (16,2) | (16,2) | (20,2) | (24,2) |

5 | [${2}^{2}$1] | (5,1) | (5,1) | (10,1) | (10,1) | (15,1) | (15,1) | (20,1) | (20,1) | (25,1) | (30,1) |

6 | [${2}^{3}$] | (6,0) | (0,6) | (12,0) | (12,0) | (18,0) | (18,0) | (24,0) | (24,0) | (30,0) | (36,0) |

7 | [${2}^{3}$1] | (4,2) | (1,5) | (11,2) | (11,2) | (18,2) | (18,2) | (25,2) | (25,2) | (32,2) | (39,2) |

8 | [${2}^{4}$] | (2,4) | (2,4) | (10,4) | (10,4) | (18,4) | (18,4) | (26,4) | (26,4) | (34,4) | (42,4) |

9 | [${2}^{4}$1] | (1,4) | (1,4) | (10,4) | (10,4) | (19,4) | (19,4) | (28,4) | (28,4) | (37,4) | (46,4) |

10 | [${2}^{5}$] | (0,4) | (0,4) | (10,4) | (4,10) | (20,4) | (20,4) | (30,4) | (30,4) | (40,4) | (50,4) |

11 | [${2}^{5}$1] | (0,2) | (0,2) | (11,2) | (4,10) | (22,2) | (22,2) | (33,2) | (33,2) | (44,2) | (55,2) |

12 | [${2}^{6}$] | (0,0) | (0,0) | (12,0) | (4,10) | (24,0) | (24,0) | (36,0) | (36,0) | (48,0) | (60,0) |

13 | [${2}^{6}$1] | (9,3) | (2,11) | (22,3) | (22,3) | (35,3) | (35,3) | (48,3) | (61,3) | ||

14 | [${2}^{7}$] | (6,6) | (0,12) | (20,6) | (20,6) | (34,6) | (34,6) | (48,6) | (62,6) | ||

15 | [${2}^{7}$1] | (4,7) | (1,10) | (19,7) | (7,19) | (34,7) | (34,7) | (49,7) | (64,7) | ||

16 | [${2}^{8}$] | (2,8) | (2,8) | (18,8) | (6,20) | (34,8) | (34,8) | (50,8) | (66,8) | ||

17 | [${2}^{8}$1] | (1,7) | (1,7) | (18,7) | (3,22) | (35,7) | (35,7) | (52,7) | (69,7) | ||

18 | [${2}^{9}$] | (0,6) | (0,6) | (18,6) | (0,24) | (36,6) | (36,6) | (54,6) | (72,6) | ||

19 | [${2}^{9}$1] | (0,3) | (0,3) | (19,3) | (2,22) | (38,3) | (38,3) | (57,3) | (76,3) | ||

20 | [${2}^{10}$] | (0,0) | (0,0) | (20,0) | (4,20) | (40,0) | (40,0) | (60,0) | (80,0) | ||

21 | [${2}^{10}$1] | (16,4) | (4,19) | (37,4) | (4,37) | (58,4) | (79,4) | ||||

22 | [${2}^{11}$] | (12,8) | (4,18) | (34,8) | (0,40) | (56,8) | (78,8) | ||||

23 | [${2}^{11}$1] | (9,10) | (2,18) | (32,10) | (3,38) | (55,10) | (78,10) | ||||

24 | [${2}^{12}$] | (6,12) | (0,18) | (30,12) | (6,36) | (54,12) | (78,12) | ||||

25 | [${2}^{12}$1] | (4,12) | (1,15) | (29,12) | (7,35) | (54,12) | (79,12) | ||||

26 | [${2}^{13}$] | (2,12) | (2,12) | (28,12) | (8,34) | (54,12) | (80,12) | ||||

27 | [${2}^{13}$1] | (1,10) | (1,10) | (28,10) | (7,34) | (55,10) | (82,10) | ||||

28 | [${2}^{14}$] | (0.8) | (0,8) | (28,8) | (6,34) | (56,8) | (84,8) | ||||

29 | [${2}^{14}$1] | (0,4) | (0,4) | (29,4) | (3,35) | (58,4) | (87,4) | ||||

30 | [${2}^{15}$] | (0,0) | (0,0) | (30,0) | (0,36) | (60,0) | (90,0) | ||||

31 | [${2}^{15}$1] | (25,5) | (2,33) | (56,5) | (87,5) | ||||||

32 | [${2}^{16}$] | (20,10) | (4,30) | (52,10) | (84,10) | ||||||

33 | [${2}^{16}$1] | (16,13) | (4,28) | (49,13) | (82,13) | ||||||

34 | [${2}^{17}$] | (12,16) | (4,26) | (46,16) | (80,16) | ||||||

35 | [${2}^{17}$1] | (9,17) | (2,25) | (44,17) | (79,17) | ||||||

36 | [${2}^{18}$] | (6,18) | (0,24) | (42,18) | (78,18) | ||||||

37 | [${2}^{18}$1] | (4,17) | (1,20) | (41,17) | (78,17) | ||||||

38 | [${2}^{19}$] | (2,16) | (2,16) | (40,16) | (78,16) | ||||||

39 | [${2}^{19}$1] | (1,13) | (1,13) | (40,13) | (79,13) | ||||||

40 | [${2}^{20}$] | (0,10) | (0,10) | (40,10) | (80,10) | ||||||

41 | [${2}^{20}$1] | (0,5) | (0,5) | (41,5) | (82,5) | ||||||

42 | [${2}^{21}$] | (0,0) | (0,0) | (42,0) | (84,0) | ||||||

43 | [${2}^{21}$1] | (36,6) | (79,6) | ||||||||

44 | [${2}^{22}$] | (30,12) | (74,12) | ||||||||

45 | [${2}^{22}$1] | (25,16) | (70,16) | ||||||||

46 | [${2}^{23}$] | (20,20) | (66,20) | ||||||||

47 | [${2}^{23}$1] | (16,22) | (63,22) | ||||||||

48 | [${2}^{24}$] | (12,24) | (60,24) | ||||||||

49 | [${2}^{24}$1] | (9,24) | (58,24) | ||||||||

50 | [${2}^{25}$] | (6,24) | (56,24) | ||||||||

51 | [${2}^{25}$1] | (4,22) | (55,22) | ||||||||

52 | [${2}^{26}$] | (2,20) | (54,20) | ||||||||

53 | [${2}^{26}$1] | (1,16) | (54,16) | ||||||||

54 | [${2}^{27}$] | (0,12) | (54,12) | ||||||||

55 | [${2}^{27}$1] | (0,6) | (55,6) | ||||||||

56 | [${2}^{28}$] | (0,0) | (56,0) |

#### 8.2. Parameter-Free Predictions for the Collective Variables $\beta $ and $\gamma $

#### 8.3. Prolate to Oblate Shape/Phase Transition

## 9. Islands of Shape Coexistence

#### 9.1. Harmonic Oscillator (HO) and Spin–Orbit (SO) Magic Numbers

#### 9.2. A Dual Shell Mechanism for Shape Coexistence

#### 9.3. From Stripes to Islands of Shape Coexistence

#### 9.4. Multiple-Shape Coexistence

## 10. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Diagonal matrix elements (in units of $\hslash {\omega}_{0}$) of the Nilsson Hamiltonian for the 82–126 (

**a**) and PFH (

**c**) neutron shells compared to the results of the full diagonalization for the 82–126 (

**b**) and PFH (

**d**) neutron shells, as functions of the deformation parameter $\u03f5$. The intruder orbitals are indicated by dashed lines and their labels appear in boldface. Orbitals are grouped in color only to facilitate visualizing the patterns of orbital evolution, adapted from Reference [88]. See Section 5 for further discussion.

**Figure 4.**Proxy SU(3) predictions for Gd-Pt isotopes for the collective variable $\beta $, obtained from Equation (10), as described in detail in Reference [89], compared with results by the D1S–Gogny interaction (D1S–Gogny) [146] and by relativistic mean field theory (RMF) [147], as well as with empirical values (exp.) [148], adapted from Reference [157]. See Section 8.2 for further discussion.

**Figure 5.**The same as Figure 4, but for the collective variable $\gamma $, derived from Equation (8), adapted from Reference [157]. See Section 8.2 for further discussion.

**Figure 6.**Values of the square root of the second-order Casimir operator of SU(3), obtained from Equation (5), vs. particle number M, for different shells, obtained through proxy-SU(3) (columns hw in Table 6) or through the particle–hole symmetry assumption (columns C in Table 6), adapted from Reference [89]. See Section 8.2 for further discussion.

**Figure 7.**Proxy-SU(3) predictions for the collective variable $\beta $, obtained from Equation (10), as described in detail in Reference [89], adapted from Reference [89]. See Section 8.3 for further discussion.

**Figure 8.**The eigenvalues of the second-order Casimir operator of SU(3) versus the proton (Z) or neutron number (N). Islands of shape coexistence are predicted by the dual shell mechanism within proton or neutron numbers 7–8, 17–20, 34–40, 59–70, 96–112, 145–168, in which ${C}_{2,SO}\ge {C}_{2,HO}$, adapted from Reference [225]. See Section 9.2 for further discussion.

**Figure 9.**This map indicates which nuclei have to be examined both theoretically and experimentally for manifesting shape coexistence according to the proposed dual shell mechanism. The colored regions possess proton or neutron numbers (7–8, 17–20, 34–40, 59–70, 96–112, and 145–168). The horizontal stripes correspond to the neutron-induced shape coexistence, while the vertical stripes correspond to the proton-induced shape coexistence, adapted from Reference [225]. See Section 9 for further discussion.

**Table 1.**Pairs of orbitals playing a leading role in the development of deformation in different mass regions of the nuclear chart according to Federman and Pittel [96,97,98]. The pairs on the left part of the table contribute to the beginning of the relevant shell, while the pairs on the right become important further within the shell, as adapted from Reference [87]. See Section 3 for further discussion.

Protons | Neutrons | Protons | Neutrons | |
---|---|---|---|---|

light | 1d${}_{5/2}$ | 1d${}_{3/2}$ | 1d${}_{5/2}$ | 1f${}_{7/2}$ |

intermediate | 1g${}_{9/2}$ | 1g${}_{7/2}$ | 1g${}_{9/2}$ | 1h${}_{11/2}$ |

rare earth | 1h${}_{11/2}$ | 1h${}_{9/2}$ | 1h${}_{11/2}$ | 1i${}_{13/2}$ |

actinides | 1i${}_{13/2}$ | 1i${}_{11/2}$ | 1i${}_{13/2}$ | 1j${}_{15/2}$ |

**Table 2.**Nilsson Hamiltonian matrix elements (in units of $\hslash {\omega}_{0}$) with $\u03f5=0.3$ for Nilsson orbitals in the 82–126 neutron shell (upper part) and in the full PFH neutron shell (lower part). Matrix elements in the lower part of the table, which differ from their counterparts in the upper part, are shown in boldface, adapted from Reference [88]. See Section 5 for further discussion.

$\frac{1}{2}$[501] | $\frac{1}{2}$[521] | $\frac{3}{2}$[512] | $\frac{1}{2}$[510] | $\frac{3}{2}$[501] | $\frac{5}{2}$[503] | $\frac{1}{2}$[541] | $\frac{3}{2}$[532] | $\frac{5}{2}$[523] | $\frac{7}{2}$[514] | $\frac{1}{2}$[530] | $\frac{3}{2}$[521] | $\frac{5}{2}$[512] | $\frac{7}{2}$[503] | $\frac{9}{2}$[505] | $\frac{1}{2}$[660] | $\frac{3}{2}$[651] | $\frac{5}{2}$[642] | $\frac{7}{2}$[633] | $\frac{9}{2}$[624] | $\frac{11}{2}$[615] | $\frac{13}{2}$[606] | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1/2[501] | 7.44 | 0.19 | 0 | 0.16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1/2[521] | 6.46 | 0 | $-0.18$ | 0 | 0 | 0.26 | 0 | 0 | 0 | 0.22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

3/2[512] | 6.88 | 0 | $-0.13$ | 0 | 0 | 0.23 | 0 | 0 | 0 | 0.22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

1/2[510] | 6.86 | 0 | 0 | 0 | 0 | 0 | 0 | 0.26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||

3/2[501] | 7.31 | 0 | 0 | 0 | 0 | 0 | 0 | 0.19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||

5/2[503] | 7.35 | 0 | 0 | 0.15 | 0 | 0 | 0 | 0.18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||

1/2[541] | 5.92 | 0 | 0 | 0 | $-0.18$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||

3/2[532] | 6.12 | 0 | 0 | 0 | $-0.16$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||

5/2[523] | 6.38 | 0 | 0 | 0 | $-0.13$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||

7/2[514] | 6.69 | 0 | 0 | 0 | $-0.09$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||

1/2[530] | 6.10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||

3/2[521] | 6.34 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||

5/2[512] | 6.63 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||||

7/2[503] | 6.97 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||

9/2[505] | 7.05 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||

1/2[660] | 6.70 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||

3/2[651] | 6.67 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||

5/2[642] | 6.69 | 0 | 0 | 0 | 0 | |||||||||||||||||

7/2[633] | 6.77 | 0 | 0 | 0 | ||||||||||||||||||

9/2[624] | 6.90 | 0 | 0 | |||||||||||||||||||

11/2[615] | 7.08 | 0 | ||||||||||||||||||||

13/2[606] | 7.32 | |||||||||||||||||||||

1/2[501] | 7.44 | 0.19 | 0 | 0.16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

1/2[521] | 6.46 | 0 | $-0.18$ | 0 | 0 | 0.26 | 0 | 0 | 0 | 0.22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

3/2[512] | 6.88 | 0 | $-0.13$ | 0 | 0 | 0.23 | 0 | 0 | 0 | 0.22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||

1/2[510] | 6.86 | 0 | 0 | 0 | 0 | 0 | 0 | 0.26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||

3/2[501] | 7.31 | 0 | 0 | 0 | 0 | 0 | 0 | 0.19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||

5/2[503] | 7.35 | 0 | 0 | 0.15 | 0 | 0 | 0 | 0.18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||

1/2[541] | 5.92 | 0 | 0 | 0 | $-0.18$ | 0 | 0 | 0 | 0 | 0.20 | 0 | 0 | 0 | 0 | 0 | |||||||

3/2[532] | 6.12 | 0 | 0 | 0 | $-0.16$ | 0 | 0 | 0 | 0 | 0.25 | 0 | 0 | 0 | 0 | ||||||||

5/2[523] | 6.38 | 0 | 0 | 0 | $-0.13$ | 0 | 0 | 0 | 0 | 0.27 | 0 | 0 | 0 | |||||||||

7/2[514] | 6.69 | 0 | 0 | 0 | $-0.09$ | 0 | 0 | 0 | 0 | 0.25 | 0 | 0 | ||||||||||

1/2[530] | 6.10 | 0 | 0 | 0 | 0 | 0.24 | 0 | 0 | 0 | 0 | 0 | |||||||||||

3/2[521] | 6.34 | 0 | 0 | 0 | 0 | 0.26 | 0 | 0 | 0 | 0 | ||||||||||||

5/2[512] | 6.63 | 0 | 0 | 0 | 0 | 0.23 | 0 | 0 | 0 | |||||||||||||

7/2[503] | 6.97 | 0 | 0 | 0 | 0 | 0.15 | 0 | 0 | ||||||||||||||

9/2[505] | 7.05 | 0 | 0 | 0 | 0 | 0.20 | 0 | |||||||||||||||

1/2[550] | 6.57 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||

3/2[541] | 6.59 | 0 | 0 | 0 | 0 | |||||||||||||||||

5/2[532] | 6.67 | 0 | 0 | 0 | ||||||||||||||||||

7/2[523] | 6.80 | 0 | 0 | |||||||||||||||||||

9/2[514] | 6.98 | 0 | ||||||||||||||||||||

11/2[505] | 7.21 |

**Table 3.**Shell model orbitals of the original spin–orbit-like shells and of the proxy-SU(3) shells. The magic number 14 was proposed as a sub-shell closure in Reference [122]. The symmetry of each proxy-SU(3) shell is U($\Omega $) with $\Omega =(\mathcal{N}+1)(\mathcal{N}+2)/2$. Orbitals being replaced are indicated in boldface, adapted from Reference [119]. See Section 6 for further discussion.

Spin–Orbit | Proxy-SU(3) | 3D-HO | |||
---|---|---|---|---|---|

Magic Numbers | Original Orbitals | Proxy Orbitals | Proxy $\mathit{U}\left(\mathbf{\Omega}\right)$ Symmetry | Magic Numbers | Magic Numbers |

6–14 | $1{p}_{\pm 1/2}^{1/2}$ | $1{p}_{\pm 1/2}^{1/2}$ | $U\left(3\right)$ | 6–12 | 2–8 |

$\mathbf{1}{\mathbf{d}}_{\pm \mathbf{1}/\mathbf{2},\pm \mathbf{3}/\mathbf{2}}^{\mathbf{5}/\mathbf{2}}$ | $\mathbf{1}{\mathbf{p}}_{\pm \mathbf{1}/\mathbf{2},\pm \mathbf{3}/\mathbf{2}}^{\mathbf{3}/\mathbf{2}}$ | ||||

$\mathbf{1}{\mathbf{d}}_{\pm \mathbf{5}/\mathbf{2}}^{\mathbf{5}/\mathbf{2}}$ | - | ||||

14–28 | $2{s}_{\pm 1/2}^{1/2}$ | $2{s}_{\pm 1/2}^{1/2}$ | $U\left(6\right)$ | 14–26 | 8–20 |

$1{d}_{\pm 1/2,\pm 3/2}^{3/2}$ | $1{d}_{\pm 1/2,\pm 3/2}^{3/2}$ | ||||

$\mathbf{1}{\mathbf{f}}_{\pm \mathbf{1}/\mathbf{2},\pm \mathbf{3}/\mathbf{2},\pm \mathbf{5}/\mathbf{2}}^{\mathbf{7}/\mathbf{2}}$ | $\mathbf{1}{\mathbf{d}}_{\pm \mathbf{1}/\mathbf{2},\pm \mathbf{3}/\mathbf{2},\pm \mathbf{5}/\mathbf{2}}^{\mathbf{5}/\mathbf{2}}$ | ||||

$\mathbf{1}{\mathbf{f}}_{\pm \mathbf{7}/\mathbf{2}}^{\mathbf{7}/\mathbf{2}}$ | - | ||||

28–50 | $2{p}_{\pm 1/2}^{1/2}$ | $2{p}_{\pm 1/2}^{1/2}$ | $U\left(10\right)$ | 28–48 | 20–40 |

$2{p}_{\pm 1/2,\pm 3/2}^{3/2}$ | $2{p}_{\pm 1/2,\pm 3/2}^{3/2}$ | ||||

$1{f}_{\pm 5/2,\pm 3/2,\pm 1/2}^{5/2}$ | $1{f}_{\pm 5/2,\pm 3/2,\pm 1/2}^{5/2}$ | ||||

$\mathbf{1}{\mathbf{g}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{7}/\mathbf{2}}^{\mathbf{9}/\mathbf{2}}$ | $\mathbf{1}{\mathbf{f}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{7}/\mathbf{2}}^{\mathbf{7}/\mathbf{2}}$ | ||||

$\mathbf{1}{\mathbf{g}}_{\pm \mathbf{9}/\mathbf{2}}^{\mathbf{9}/\mathbf{2}}$ | - | ||||

50–82 | $3{s}_{\pm 1/2}^{1/2}$ | $3{s}_{\pm 1/2}^{1/2}$ | $U\left(15\right)$ | 50–80 | 40–70 |

$2{d}_{\pm 1/2,\pm 3/2}^{3/2}$ | $2{d}_{\pm 1/2,\pm 3/2}^{3/2}$ | ||||

$2{d}_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$ | $2{d}_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$ | ||||

$1{g}_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$ | $1{g}_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$ | ||||

$\mathbf{1}{\mathbf{h}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{9}/\mathbf{2}}^{\mathbf{11}/\mathbf{2}}$ | $\mathbf{1}{\mathbf{g}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{9}/\mathbf{2}}^{\mathbf{9}/\mathbf{2}}$ | ||||

$\mathbf{1}{\mathbf{h}}_{\pm \mathbf{11}/\mathbf{2}}^{\mathbf{11}/\mathbf{2}}$ | - | ||||

82–126 | $3{p}_{\pm 1/2}^{1/2}$ | $3{p}_{\pm 1/2}^{1/2}$ | $U\left(21\right)$ | 82–124 | 70–112 |

$3{p}_{\pm 1/2,\pm 3/2}^{3/2}$ | $3{p}_{\pm 1/2,\pm 3/2}^{3/2}$ | ||||

$2{f}_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$ | $2{f}_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$ | ||||

$2{f}_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$ | $2{f}_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$ | ||||

$1{h}_{\pm 1/2,\dots ,\pm 9/2}^{9/2}$ | $1{h}_{\pm 1/2,\dots ,\pm 9/2}^{9/2}$ | ||||

$\mathbf{1}{\mathbf{i}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{11}/\mathbf{2}}^{\mathbf{13}/\mathbf{2}}$ | $\mathbf{1}{\mathbf{h}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{11}/\mathbf{2}}^{\mathbf{11}/\mathbf{2}}$ | ||||

$\mathbf{1}{\mathbf{i}}_{\pm \mathbf{13}/\mathbf{2}}^{\mathbf{13}/\mathbf{2}}$ | - | ||||

126–184 | $4{s}_{\pm 1/2}^{1/2}$ | $4{s}_{\pm 1/2}^{1/2}$ | $U\left(28\right)$ | 126–182 | 112–168 |

$3{d}_{\pm 1/2,\pm 3/2}^{3/2}$ | $3{d}_{\pm 1/2,\pm 3/2}^{3/2}$ | ||||

$3{d}_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$ | $3{d}_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$ | ||||

$2{g}_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$ | $2{g}_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$ | ||||

$2{g}_{\pm 1/2,\dots ,\pm 9/2}^{9/2}$ | $2{g}_{\pm 1/2,\dots ,\pm 9/2}^{9/2}$ | ||||

$1{i}_{\pm 1/2,\dots ,\pm 11/2}^{11/2}$ | $1{i}_{\pm 1/2,\dots ,\pm 11/2}^{11/2}$ | ||||

$\mathbf{1}{\mathbf{j}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{13}/\mathbf{2}}^{\mathbf{15}/\mathbf{2}}$ | $\mathbf{1}{\mathbf{i}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{13}/\mathbf{2}}^{\mathbf{13}/\mathbf{2}}$ | ||||

$\mathbf{1}{\mathbf{j}}_{\pm \mathbf{15}/\mathbf{2}}^{\mathbf{15}/\mathbf{2}}$ | - |

**Table 4.**Expansions of Nilsson orbitals $K[N{n}_{z}\Lambda ]$ in the shell model basis $|Nlj{m}_{j}\rangle $ for three different values of the deformation $\u03f5$. The Nilsson orbitals shown possess the highest total angular momenta j in their shells. The existence of a leading shell model eigenvector is evident at all deformations, adapted from Reference [123]. See Section 6 for further discussion.

$\frac{3}{2}\left[541\right]$ | $\mathit{\u03f5}$ | $|{\mathit{Nljm}}_{\mathit{j}}\rangle $ | $\left|51\frac{3}{2}\frac{3}{2}\right.\u232a$ | $\left|53\frac{5}{2}\frac{3}{2}\right.\u232a$ | $\left|53\frac{7}{2}\frac{3}{2}\right.\u232a$ | $\left|55\frac{9}{2}\frac{3}{2}\right.\u232a$ | $\left|55\frac{11}{2}\frac{3}{2}\right.\u232a$ | |
---|---|---|---|---|---|---|---|---|

0.05 | 0.0025 | $-0.0015$ | 0.0641 | $-0.0122$ | 0.9979 | |||

0.22 | 0.0371 | $-0.0286$ | 0.2565 | $-0.0640$ | 0.9633 | |||

0.30 | 0.0601 | $-0.0506$ | 0.3287 | $-0.0922$ | 0.9366 | |||

$\frac{\mathbf{3}}{\mathbf{2}}\left[\mathbf{651}\right]$ | $\mathbf{\u03f5}$ | $|{\mathit{Nljm}}_{\mathbf{j}}\rangle $ | $\left|\mathbf{6}\mathbf{2}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{6}\mathbf{2}\frac{\mathbf{5}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{6}\mathbf{4}\frac{\mathbf{7}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{6}\mathbf{4}\frac{\mathbf{9}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{6}\mathbf{6}\frac{\mathbf{11}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{6}\mathbf{6}\frac{\mathbf{13}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.\u232a$ |

0.05 | $-0.0002$ | 0.0046 | $-0.0013$ | 0.0821 | $-0.0086$ | 0.9966 | ||

0.22 | $-0.0100$ | 0.0711 | $-0.0278$ | 0.3240 | $-0.0469$ | 0.9418 | ||

0.30 | $-0.0207$ | 0.1149 | $-0.0509$ | 0.4091 | $-0.0687$ | 0.9010 |

**Table 5.**Expansions of Nilsson orbitals $K[N{n}_{z}\Lambda ]$ in the shell model basis $|Nlj{m}_{j}\rangle $ for three different values of the deformation $\u03f5$. The Nilsson orbitals shown do not possess the highest total angular momenta j in their shells. The existence of leading shell model eigenvectors is evident in small deformations, but this is not the case anymore at higher deformations, at which several shell model eigenvectors make considerable contributions, adapted from Reference [123]. See Section 6 for further discussion.

$\frac{1}{2}\left[431\right]$ | $\mathit{\u03f5}$ | $|{\mathit{Nljm}}_{\mathit{j}}\rangle $ | $\left|40\frac{1}{2}\frac{1}{2}\right.\u232a$ | $\left|42\frac{3}{2}\frac{1}{2}\right.\u232a$ | $\left|42\frac{5}{2}\frac{1}{2}\right.\u232a$ | $\left|44\frac{7}{2}\frac{1}{2}\right.\u232a$ | $\left|44\frac{9}{2}\frac{1}{2}\right.\u232a$ | |
---|---|---|---|---|---|---|---|---|

0.05 | $-0.0213$ | 0.1254 | $-0.0702$ | 0.9893 | 0.0127 | |||

0.22 | $-0.2248$ | 0.4393 | $-0.2791$ | 0.8057 | 0.1717 | |||

0.30 | $-0.2630$ | 0.5003 | $-0.2458$ | 0.7447 | 0.2559 | |||

$\frac{\mathbf{1}}{\mathbf{2}}\left[\mathbf{541}\right]$ | $\mathbf{\u03f5}$ | $|{\mathit{Nljm}}_{\mathbf{j}}\rangle $ | $\left|\mathbf{5}\mathbf{1}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{5}\mathbf{1}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{5}\mathbf{3}\frac{\mathbf{5}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{5}\mathbf{3}\frac{\mathbf{7}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{5}\mathbf{5}\frac{\mathbf{9}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.\u232a$ | $\left|\mathbf{5}\mathbf{5}\frac{\mathbf{11}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.\u232a$ |

0.05 | $-0.0200$ | 0.1770 | $-0.0295$ | 0.9780 | $-0.0446$ | $-0.0944$ | ||

0.22 | $-0.2492$ | 0.4619 | $-0.3768$ | 0.5550 | $-0.4161$ | $-0.3185$ | ||

0.30 | $-0.3121$ | 0.4331 | $-0.4829$ | 0.3430 | $-0.4789$ | $-0.3671$ |

**Table 7.**Highest weight SU(3) irreps for nuclei with protons in the 82–126 shell and neutrons in the 126–184 shell. Oblate irreps are underlined, adapted from Reference [143]. See Section 8.1 for further discussion.

Rn | Ra | Th | U | Pu | Cm | Cf | Fm | No | Rf | Sg | Hs | Ds | Cn | Fl | Lv | Og | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Z | 86 | 88 | 90 | 92 | 94 | 96 | 98 | 100 | 102 | 104 | 106 | 108 | 110 | 112 | 114 | 116 | 118 | 120 | 122 | ||

${\mathit{Z}}_{\mathbf{val}}$ | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | ||||||||

N | ${N}_{val}$ | irrep | (16,2) | (24,0) | (26,4) | (30,4) | (36,0) | (34,6) | (34,8) | (36,6) | (40,0) | (34,8) | (30,12) | (28,12) | (28,8) | (30,0) | (20,10) | (12,16) | (6,18) | (2,16) | (0,10) |

130 | 4 | (20,2) | (36,4) | (44,2) | (46,6) | (50,6) | (56,2) | (54,8) | (54,10) | (56,8) | (60,2) | (54,10) | (50,14) | (48,4) | (48,10) | (50,2) | (40,12) | (32,18) | (26,20) | (22,18) | (20,12) |

132 | 6 | (30,0) | (46,2) | (54,0) | (56,4) | (60,4) | (66,0) | (64,6) | (64,8) | (66,6) | (70,0) | (64,8) | (60,12) | (58,12) | (58,8) | (60,0) | (50,10) | (42,16) | (36,18) | (32,16) | (30,10) |

134 | 8 | (34,4) | (50,6) | (58,4) | (60,8) | (64,8) | (70,4) | (68,10) | (68,12) | (70,10) | (74,4) | (68,12) | (64,16) | (62,16) | (62,14) | (64,4) | (54,14) | (46,20) | (40,22) | (36,20) | (34,14) |

136 | 10 | (40,4) | (56,6) | (64,4) | (66,8) | (70,8) | (76,4) | (74,10) | (74,12) | (76,10) | (80,4) | (74,12) | (70,16) | (68,16) | (68,12) | (70,4) | (60,14) | (52,20) | (46,22) | (42,20) | (40,14) |

138 | 12 | (48,0) | (64,2) | (72,0) | (74,4) | (78,4) | (84,0) | (82,6) | (82,8) | (84,6) | (88,0) | (82,8) | (78,12) | (76,12) | (76,8) | (78,0) | (68,10) | (60,16) | (54,18) | (50,16) | (48,10) |

140 | 14 | (48,6) | (64,8) | (72,6) | (74,10) | (78,10) | (84,6) | (82,12) | (82,14) | (84,12) | (88,6) | (82,14) | (78,18) | (76,18) | (76,14) | (78,6) | (68,16) | (60,22) | (54,24) | (50,22) | (48,16) |

142 | 16 | (50,8) | (66,10) | (74,8) | (76,12) | (80,12) | (86,8) | (84,14) | (84,16) | (86,14) | (90,8) | (84,16) | (80,20) | (78,20) | (78,16) | (80,8) | (70,18) | (62,24) | (56,26) | (52,24) | (50,18) |

144 | 18 | (54,6) | (70,8) | (78,6) | (80,10) | (84,10) | (90,6) | (88,12) | (88,14) | (90,12) | (94,6) | (88,14) | (84,18) | (82,18) | (82,14) | (84,6) | (74,16) | (66,22) | (60,24) | (56,22) | (54,16) |

146 | 20 | (60,0) | (76,2) | (84,0) | (86,4) | (90,4) | (96,0) | (94,6) | (94,8) | (96,6) | (100,0) | (94,8) | (90,12) | (88,12) | (88,8) | (90,0) | (80,10) | (72,16) | (66,18) | (62,16) | (60,10) |

148 | 22 | (56,8) | (72,10) | (80,8) | (82,12) | (86,12) | (92,8) | (90,14) | (90,16) | (92,14) | (96,8) | (90,16) | (86,20) | (84,20) | (84,16) | (86,8) | (76,18) | (68,24) | (62,26) | (58,24) | (56,18) |

150 | 24 | (54,12) | (70,14) | (78,12) | (80,16) | (84,16) | (90,12) | (88,18) | (88,20) | (90,18) | (94,12) | (88,20) | (84,24) | (82,24) | (82,20) | (84,12) | (74,22) | (66,28) | (60,30) | (56,28) | (54,22) |

152 | 26 | (54,12) | (70,14) | (78,12) | (80,16) | (84.16) | (90,12) | (88,18) | (88,20) | (90,18) | (94,12) | (88,20) | (84,24) | (82,24) | (82,20) | (84,12) | (74,22) | (66,28) | (60,30) | (56,28) | (54,22) |

154 | 28 | (56,8) | (72,10) | (80,8) | (82,12) | (86,12) | (92,8) | (90,14) | (90,16) | (92,14) | (96,8) | (90,16) | (86,20) | (84,20) | (84,16) | (86,8) | (76,18) | (68,24) | (62,26) | (58,24) | (56,18) |

156 | 30 | (60,0) | (76,2) | (84,0) | (86,4) | (90,4) | (96,0) | (94,6) | (94,8) | (96,6) | (100,0) | (94,8) | (90,12) | (88,12) | (88,8) | (90,0) | (80,10) | (72,16) | (66,18) | (62,16) | (60,10) |

158 | 32 | (52,10) | (68,12) | (76,10) | (78,14) | (82,14) | (88,10) | (86,16) | (86,18) | (88,16) | (92,10) | (86,18) | (82,22) | (80,22) | (80,18) | (82,10) | (72,20) | (64,26) | (58,28) | (54,26) | (52,20) |

160 | 34 | (46,16) | (62,18) | (70,16) | (72,20) | (76,20) | (82,16) | (80,22) | (80,24) | (82,22) | (86,16) | (80,24) | (76,28) | (74,28) | (74,24) | (76,16) | (66,26) | (58,32) | (52,34) | (48,32) | (46,26) |

162 | 36 | (42,18) | (58,20) | (66,18) | (68,22) | (72,22) | (78,18) | (76,24) | (76,26) | (78,24) | (82,18) | (76,26) | (72,30) | (70,30) | (70,26) | (72,18) | (62,28) | (54,34) | (48,36) | (44,34) | (42,28) |

164 | 38 | (40,16) | (56,18) | (64,16) | (66,20) | (70,20) | (76,16) | (74,22) | (74,24) | (76,22) | (80,16) | (74,24) | (70,28) | (68,28) | (68,24) | (70,16) | (60,26) | (52,32) | (46,24) | (42,32) | (40,26) |

166 | 40 | (40,10) | (56,12) | (64,10) | (66,14) | (70,14) | (76,10) | (74,16) | (74,18) | (76,16) | (80,10) | (74,18) | (70,22) | (68,22) | (68,18) | (70,10) | (60,20) | (52,26) | (46,28) | (42,26) | (40,20) |

168 | 42 | (42,0) | (58,2) | (66,0) | (68,4) | (72,4) | (78,0) | (76,6) | (76,8) | (78,6) | (82,0) | (76,8) | (72,12) | (70,12) | (70,8) | (72,0) | (62,10) | (54,16) | (48,18) | (44,16) | (42,10) |

170 | 44 | (30,12) | (46,14) | (54,12) | (56,16) | (60,16) | (66,12) | (64,18) | (64,20) | (66,18) | (70,12) | (64,20) | (60,24) | (58,24) | (58,20) | (60,12) | (50,22) | (42,28) | (36,30) | (32,28) | (30,22) |

172 | 46 | (20,20) | (36,22) | (44,20) | (46,24) | (50,24) | (56,20) | (54,26) | (54,28) | (56,26) | (60,20) | (54,28) | (50,32) | (48,32) | (48,28) | (50,20) | (40,30) | $\underline{(32,36)}$ | $\underline{(26,38)}$ | $\underline{(22,36}$) | $\underline{(20,30)}$ |

174 | 48 | (12,24) | (28,26) | (36,24) | (38,28) | (42,28) | (48,24) | (46,30) | (46,32) | (48,30) | (52,24) | (46,32) | (42,36) | (40,36) | (40,32) | (42,24) | $\underline{(32,34}$) | $\underline{(24,40)}$ | $\underline{(18,42)}$ | $\underline{(14,40)}$ | $\underline{(12,34)}$ |

176 | 50 | (6,24) | (22,26) | (30,24) | (32,28) | (36,28) | (42,24) | (40,30) | (40,32) | (42,30) | (46,24) | (40,32) | (36,36) | (34,36) | (34,32) | (36,24) | $\underline{(26,34)}$ | $\underline{(18,40)}$ | $\underline{(12,42)}$ | $\underline{(8,40)}$ | $\underline{(6,34)}$ |

178 | 52 | (2,20) | (18,22) | (26,20) | (28,24) | (32,24) | (38,20) | (36,26) | (36,28) | (38,26) | (42,20) | (36,28) | (32,32) | (30,32) | (30,28) | (32,20) | $\underline{(22,30)}$ | $\underline{(14,36)}$ | $\underline{(8,38)}$ | $\underline{(4,36)}$ | $\underline{(2,30)}$ |

180 | 54 | (0,12 ) | (16,14) | (24,12) | (26,16) | (30,16) | (36,12) | (34,18 | (34,18) | (36,18) | (40,12) | (34,20) | (30,24) | (28,24) | (28,20) | (30,12) | $\underline{(20,22)}$ | $\underline{(12,28)}$ | $\underline{(6,30)}$ | $\underline{(2,28)}$ | $\underline{(0,22)}$ |

**Table 8.**The highest weight SU(3) irreps for the spin–orbit (SO)-like magic numbers 6, 14, 28, 50, 82, and 126, according to the proxy-SU(3) symmetry and for the harmonic oscillator (HO) magic numbers 2, 8, 20, 40, 70, 112, and 168 according to the Elliott SU(3) symmetry are given for each nucleon number M, as obtained from Table 6, adapted from Reference [225]. See Section 9 for further discussion.

M | ${(\mathit{\lambda},\mathit{\mu})}_{\mathit{SO}}$ | ${(\mathit{\lambda},\mathit{\mu})}_{\mathit{HO}}$ | M | ${(\mathit{\lambda},\mathit{\mu})}_{\mathit{SO}}$ | ${(\mathit{\lambda},\mathit{\mu})}_{\mathit{HO}}$ | M | ${(\mathit{\lambda},\mathit{\mu})}_{\mathit{SO}}$ | ${(\mathit{\lambda},\mathit{\mu})}_{\mathit{HO}}$ | M | ${(\mathit{\lambda},\mathit{\mu})}_{\mathit{SO}}$ | ${(\mathit{\lambda},\mathit{\mu})}_{\mathit{HO}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | (0, 0) | (0, 0) | 1 | (0, 0) | (0, 0) | 94 | (36, 0) | (30, 12) | 93 | (33, 2) | (32, 10) |

4 | (0, 0) | (2, 0) | 3 | (0, 0) | (1, 0) | 96 | (34, 6) | (28, 12) | 95 | (35, 3) | (29, 12) |

6 | (0, 0) | (0, 2) | 5 | (0, 0) | (1, 1) | 98 | (34, 8) | (28, 8) | 97 | (34, 7) | (28, 10) |

8 | (2, 0) | (0, 0) | 7 | (1, 0) | (0, 1) | 100 | (36, 6) | (30, 0) | 99 | (35, 7) | (29, 4) |

10 | (0, 2) | (4, 0) | 9 | (1, 1) | (2, 0) | 102 | (40, 0) | (20, 10) | 101 | (38, 3) | (25, 5) |

12 | (0, 0) | (4, 2) | 11 | (0, 1) | (4, 1) | 104 | (34, 8) | (12, 16) | 103 | (37, 4) | (16, 13) |

14 | (0, 0) | (6, 0) | 13 | (0, 0) | (5, 1) | 106 | (30, 12) | (6, 18) | 105 | (32, 10) | (9, 17) |

16 | (4, 0) | (2, 4) | 15 | (2, 0) | (4, 2) | 108 | (28, 12) | (2, 16) | 107 | (29, 12) | (4, 17) |

18 | (4, 2) | (0, 4) | 17 | (4, 1) | (1, 4) | 110 | (28, 8) | (0, 10) | 109 | (28, 10) | (1, 13) |

20 | (6, 0) | (0, 0) | 19 | (5, 1) | (0, 2) | 112 | (30, 0) | (0, 0) | 111 | (29, 4) | (0, 5) |

22 | (2, 4) | (6, 0) | 21 | (4, 2) | (3, 0) | 114 | (20, 10) | (12, 0) | 113 | (25, 5) | (6, 0) |

24 | (0, 4) | (8, 2) | 23 | (1, 4) | (7, 1) | 116 | (12, 16) | (20, 2) | 115 | (16, 13) | (16, 1) |

26 | (0, 0) | (12, 0) | 25 | (0, 2) | (10, 1) | 118 | (6, 18) | (30, 0) | 117 | (9, 17) | (25, 1) |

28 | (0, 0) | (10, 4) | 27 | (0, 0) | (11, 2) | 120 | (2, 16) | (34, 4) | 119 | (4, 17) | (32, 2) |

30 | (6, 0) | (10, 4) | 29 | (3, 0) | (10, 4) | 122 | (0, 10) | (40, 4) | 121 | (1, 13) | (37, 4) |

32 | (8, 2) | (12, 0) | 31 | (7, 1) | (11, 2) | 124 | (0, 0) | (48, 0) | 123 | (0, 5) | (44, 2) |

34 | (12, 0) | (6, 6) | 33 | (10, 1) | (9, 3) | 126 | (0, 0) | (48, 6) | 125 | (0, 0) | (48, 3) |

36 | (10, 4) | (2, 8) | 35 | (11, 2) | (4, 7) | 128 | (12, 0) | (50, 8) | 127 | (6, 0) | (49, 7) |

38 | (10, 4) | (0, 6) | 37 | (10, 4) | (1, 7) | 130 | (20, 2) | (54, 6) | 129 | (16, 1) | (52, 7) |

40 | (12, 0) | (0, 0) | 39 | (11, 2) | (0, 3) | 132 | (30, 0) | (60, 0) | 131 | (25, 1) | (57, 3) |

42 | (6, 6) | (8, 0) | 41 | (9, 3) | (4, 0) | 134 | (34, 4) | (56, 8) | 133 | (32, 2) | (58, 4) |

44 | (2, 8) | (12, 2) | 43 | (4, 7) | (10, 1) | 136 | (40, 4) | (54, 12) | 135 | (37, 4) | (55, 10) |

46 | (0, 6) | (18, 0) | 45 | (1, 7) | (15, 1) | 138 | (48, 0) | (54, 12) | 137 | (44, 2) | (54, 12) |

48 | (0, 0) | (18, 4) | 47 | (0, 3) | (18, 2) | 140 | (48, 6) | (56, 8) | 139 | (48, 3) | (55, 10) |

50 | (0, 0) | (20, 4) | 49 | (0, 0) | (19, 4) | 142 | (50, 8) | (60, 0) | 141 | (49, 7) | (58, 4) |

52 | (8, 0) | (24, 0) | 51 | (4, 0) | (22, 2) | 144 | (54, 6) | (52, 10) | 143 | (52, 7) | (56, 5) |

54 | (12, 2) | (20, 6) | 53 | (10, 1) | (22, 3) | 146 | (60, 0) | (46, 16) | 145 | (57, 3) | (49, 13) |

56 | (18, 0) | (18, 8) | 55 | (15, 1) | (19, 7) | 148 | (56, 8) | (42, 18) | 147 | (58, 4) | (44, 17) |

58 | (18, 4) | (18, 6) | 57 | (18, 2) | (18, 7) | 150 | (54, 12) | (40, 16) | 149 | (55, 10) | (41, 17) |

60 | (20, 4) | (20, 0) | 59 | (19, 4) | (19, 3) | 152 | (54, 12) | (40, 10) | 151 | (54, 12) | (40, 13) |

62 | (24, 0) | (12, 8) | 61 | (22, 2) | (16, 4) | 154 | (56, 8) | (42, 0) | 153 | (55, 10) | (41, 5) |

64 | (20, 6) | (6, 12) | 63 | (22, 3) | (9, 10) | 156 | (60, 0) | (30, 12) | 155 | (58, 4) | (36, 6) |

66 | (18, 8) | (2, 12) | 65 | (19, 7) | (4, 12) | 158 | (52, 10) | (20, 20) | 157 | (56, 5) | (25, 16) |

68 | (18, 6) | (0, 8) | 67 | (18, 7) | (1, 10) | 160 | (46, 16) | (12, 24) | 159 | (49, 13) | (16, 22) |

70 | (20, 0) | (0, 0) | 69 | (19, 3) | (0, 4) | 162 | (42, 18) | (6, 24) | 161 | (44, 17) | (9, 24) |

72 | (12, 8) | (10, 0) | 71 | (16, 4) | (5, 0) | 164 | (40, 16) | (2, 20) | 163 | (41, 17) | (4, 22) |

74 | (6, 12) | (16, 2) | 73 | (9, 10) | (13, 1) | 166 | (40, 10) | (0, 12) | 165 | (40, 13) | (1, 16) |

76 | (2, 12) | (24, 0) | 75 | (4, 12) | (20, 1) | 168 | (42, 0) | (0, 0) | 167 | (41, 5) | (0, 6) |

78 | (0, 8) | (26, 4) | 77 | (1, 10) | (25, 2) | 170 | (30, 12) | (14, 0) | 169 | (36, 6) | (7, 0) |

80 | (0, 0) | (30, 4) | 79 | (0, 4) | (28, 4) | 172 | (20, 20) | (24, 2) | 171 | (25, 16) | (19, 1) |

82 | (0, 0) | (36, 0) | 81 | (0, 0) | (33, 2) | 174 | (12, 24) | (36, 0) | 173 | (16, 22) | (30, 1) |

84 | (10, 0) | (34, 6) | 83 | (5, 0) | (35, 3) | 176 | (6, 24) | (42, 4) | 175 | (9, 24) | (39, 2) |

86 | (16, 2) | (34, 8) | 85 | (13, 1) | (34, 7) | 178 | (2, 20) | (50, 4) | 177 | (4, 22) | (46, 4) |

88 | (24, 0) | (36, 6) | 87 | (20, 1) | (35, 7) | 180 | (0, 12) | (60, 0) | 179 | (1, 16) | (55, 2) |

90 | (26, 4) | (40, 0) | 89 | (25, 2) | (38, 3) | 182 | (0, 0) | (62, 6) | 181 | (0, 6) | (61, 3) |

92 | (30, 4) | (34, 8) | 91 | (28, 4) | (37, 4) | 184 | (0, 0) | (66, 8) | 183 | (0, 0) | (64, 7) |

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**MDPI and ACS Style**

Bonatsos, D.; Martinou, A.; Peroulis, S.K.; Mertzimekis, T.J.; Minkov, N.
The Proxy-SU(3) Symmetry in Atomic Nuclei. *Symmetry* **2023**, *15*, 169.
https://doi.org/10.3390/sym15010169

**AMA Style**

Bonatsos D, Martinou A, Peroulis SK, Mertzimekis TJ, Minkov N.
The Proxy-SU(3) Symmetry in Atomic Nuclei. *Symmetry*. 2023; 15(1):169.
https://doi.org/10.3390/sym15010169

**Chicago/Turabian Style**

Bonatsos, Dennis, Andriana Martinou, Spyridon Kosmas Peroulis, Theodoros John Mertzimekis, and Nikolay Minkov.
2023. "The Proxy-SU(3) Symmetry in Atomic Nuclei" *Symmetry* 15, no. 1: 169.
https://doi.org/10.3390/sym15010169