The Proxy-SU(3) Symmetry in Atomic Nuclei

Abstract

The microscopic origins and the current predictions of the proxy-SU(3) symmetry model of atomic nuclei were reviewed. Beginning with experimental evidence for the special roles played by nucleon pairs with maximal spatial overlap, the proxy-SU(3) approximation scheme is introduced; its validity is demonstrated through Nilsson model calculations and its connection to the spherical shell model. The major role played by the highest weight-irreducible representations of SU(3) in shaping up the nuclear properties is pointed out, resulting in parameter-free predictions of the collective variables $\beta$ and $\gamma$ for even–even nuclei in the explanation of the dominance of prolate over oblate shapes in the ground states of even–even nuclei, in the prediction of a shape/phase transition from prolate to oblate shapes below closed shells, and in the prediction of specific islands on the nuclear chart in which shape coexistence is confined. Further developments within the proxy-SU(3) scheme are outlined.

1. Introduction

The purpose of the present article is to discuss the ideas that led to the introduction of the proxy-SU(3) symmetry model, its microscopic justification, and its application to nuclear structural problems. Starting with a review of nuclear structure models based on the SU(3) symmetry in Section 2, we discuss nucleon pairs favoring deformation in Section 3, and show how these lead to the introduction of the proxy-SU(3) symmetry in Section 4. The approximation leading to the proxy-SU(3) symmetry is tested using Nilsson model calculations in Section 5 and is “translated” into spherical shell model language in Section 6. We point out (in Section 7) the crucial roles played by the highest weight-irreducible representations. Their consequences on providing parameter-free predictions for the collective variables $\beta$ and $\gamma$ of even–even nuclei, explaining the dominance of prolate over oblate shapes in the ground states of even–even nuclei and the prediction of a prolate to oblate shape/phase transition are examined in Section 8; in Section 9, a mechanism predicting the nuclear chart’s specific islands (in which shape coexistence can appear) is discussed. Possible further work on parameter-free predictions of B(E2) transition rates, energy spectra, binding energies, and nucleon separation energies is outlined in Section 10. The literature was searched up until October 2022.

2. SU(3) Symmetry in Nuclear Structure

Symmetries have played major roles in nuclear structures since 1937, when Wigner [1] suggested that first approximation nuclear forces should be independent of the orientations of the spins and isospins of the nuclei constituting a nucleus, in a framework known as the SU(4) supermultiplet model [2,3].

In 1949, a major step forward in deciphering the experimental observations was made by Mayer and Jensen through the introduction of the shell model [4,5,6,7]; they interpreted the appearance of nuclear magic numbers in terms of a three-dimensional (3D) isotropic harmonic oscillator (HO), to which a crucial spin–orbit term was added. The 3D isotropic HO is known to possess shells labeled by the number of excitation quanta n, characterized by the unitary symmetries U((n + 1)(n + 2)/2), having SU(3) subalgebras [8,9,10,11]. Mayer, Jensen, and Wigner shared the Nobel Prize in Physics in 1963 [12].

The shell model was considered adequate for describing near-spherical nuclei in the vicinity of the magic numbers, away from which, however, large quadrupole moments were observed. In order to explain their appearance, Rainwater in 1950 suggested [13] that deformed shapes are energetically favored away from closed shells. In 1952, the collective model of Bohr and Mottelson was introduced [14,15], in which departures from the spherical shape and axial symmetry are described by the collective variables, $\beta$ and $\gamma$, respectively. Bohr, Mottelson, and Rainwater shared the Nobel Prize in Physics in 1975 [16].

In 1955, Nilsson introduced [17,18,19] a modified version of the shell model, in which axial-deformed nuclei with prolate (rugby ball-like) or oblate (pancake-like) shapes can be described through the use of a 3D anisotropic HO with cylindrical symmetry [20,21,22,23,24,25].

In 1958, Elliott proved that deformation within the nuclear $sd$ shell with U(6) symmetry can be described in terms of its SU(3) subalgebra [26,27,28,29,30,31,32], thus building a bridge between the spherical shell model and the collective model for the case of light nuclei, in which the consequences of the spin–orbit interaction on the ordering of the single-particle nucleon levels are small. Beyond the $sd$ nuclear shell, the spin–orbit force [4,5,6,7] is known to break the SU(3) symmetry of the 3D isotropic HO, by pushing within each HO shell the orbital possessing the highest angular momentum j to the shell below. As a result, each shell contains the orbitals left back after this removal, called the normal parity orbitals, plus the orbitals invading from the shell above with the opposite parity, called the intruder orbitals.

Efforts of extending the SU(3) symmetry to heavy nuclei [33,34,35,36] started in 1972, evolved gradually into the introduction of the vector boson model [37,38,39], while at the same time, the group’s theoretical structure of the Bohr–Mottelson model (with an overall U(5) symmetry possessing an O(5) subalgebra) was understood [40].

A major step forward in the extension of the SU(3) symmetry to heavy nuclei was taken in 1973, with the introduction of the pseudo-SU(3) symmetry [41,42,43,44,45,46,47]. Within the pseudo-SU(3) framework, a unitary transformation is used [48,49,50], through which the incomplete set of normal parity orbitals left in a shell is mapped onto the complete set of orbitals of the shell below, thus recovering the SU(3) symmetry of the 3D isotropic HO for the normal parity orbitals. This becomes possible by assigning to each orbital a pseudo-orbital angular momentum and a pseudo-spin, while the total angular momentum remains intact. As a result, within the pseudo-SU(3) scheme, each shell consists of a normal parity part, which possesses a U(n) symmetry and SU(3) subalgebra, and an intruder part which does not possess any SU(3) structure and has to be treated separately by shell model techniques [45,46]. The relativistic mean field origins of the pseudospin symmetry were later understood [51,52].

In 1974, it was realized that the nuclear quadrupole degree of freedom can be described in terms of SU(6) algebra [53] formed by five generalized coordinates, conjugated momenta, and their commutators.

Next year, the interacting boson model (IBM) [54,55,56,57,58,59,60] was introduced by Arima and Iachello. In the IBM, the monopole degree of freedom is taken into account in addition to the quadrupole one. IBM is characterized by an overall U(6) symmetry built by s-bosons of zero angular momentum and d-bosons of angular momentum two, possessing three limiting symmetries, U(5) [55] for vibrational nuclei, which correspond to the Bohr–Mottelson collective model, O(6) [57] for $\gamma$-unstable nuclei (which are soft towards triaxial deformation), and SU(3) [56] for deformed nuclei.

In 1977, the symplectic model [61,62,63,64,65] was introduced by Rowe and Rosensteel. The symplectic model, having an overall symmetry (called Sp(3,R) by Rowe and Rosensteel and Sp(6,R) by other authors [66,67]) is a generalization of the fermionic Elliott SU(3) model and includes many major oscillator shells in addition to core excitations. Its overall symmetry is characterized by non-compact algebra Sp(6,R), which does possess a compact SU(3) subalgebra along with other ones. A proton–neutron extension of the model, called the proton–neutron symplectic model (PNSM), is also developed [68,69,70].

In 1982, the interacting two-vector boson model [71,72] was introduced, in which two vector bosons of angular momentum are used as the building blocks, forming non-compact algebra Sp(12,R), of which the maximal compact subalgebra is U(6).

In 1987, the fermion dynamical symmetry model [73] was introduced, in which the total angular momentum of the nucleons is assumed to be split into active and inactive parts instead of orbital angular momentum and spin parts, the k-active part contains SU(3) subalgebra.

In 2000, an ab initio approach to a no-core shell model was introduced [74,75] for light nuclei. It was then realized that symplectic symmetry underlies the ab initio no-core shell model results, thus paving the way toward the development of the symplectic no-core shell model [76,77,78], which was extended to intermediate-mass nuclei [79,80,81,82]. The realization that nuclei are made of only a few equilibrium shapes [83] led to the introduction of the ab initio symmetry-adapted no-core shell model [84,85]. An SU(3)-adapted basis [80] plays a key role in this approach, taking advantage of the Elliott SU(3) symmetry underpinning the Sp(3,R) [61,62,63,64,65] (alias Sp(6,R) [66,67]) symplectic model.

The use of the SU(3) symmetry in the nuclear structure was reviewed in 2020 by Kota [86]. A historical account similar to the present section was given by some of the present authors in Reference [87].

In 2017, the proxy-SU(3) symmetry was suggested [88,89,90], which will be the subject of the present review. However, before describing the proxy-SU(3) symmetry itself, we will briefly review the physical motivation behind its introduction. A review similar to the next section was given by some of the present authors in Reference [87].

3. Nucleon Pairs Favoring Deformation

In 1953, deShalit and Goldhaber [91], in their studies on $\beta$ transition probabilities, noticed that within the proton–neutron pairs of orbitals (1p3/2, 1d5/2), (1d5/2, 1f7/2), (1f7/2, 1g9/2), (1g9/2, 1h11/2), (1h11/2, 1i13/2), the nucleons of one kind (protons, for example) have stabilizing effects on pairs of nucleons of the other kind (neutrons in the example), thus favoring the development of nuclear deformation. In the standard shell model notation $|nlj{m}_j\rangle$, in which states are labeled by the number of oscillator quanta n, the orbital angular momentum l, the total angular momentum j, and its z-projection $m_j$, the orbitals forming pairs differed by $|\Delta n\Delta l\Delta j\Delta m_j\rangle =|0110\rangle$. We are going to call these pairs the spherical shell model $|0110\rangle$ pairs, or the $|0110\rangle$ pairs.

In 1962, a major step forward in our understanding of effective interactions and coupling schemes in nuclei was made by Talmi [92] through the introduction of seniority [92,93,94,95], representing the number of nucleon pairs coupled to non-zero angular momentum, which explained the linear dependence of neutron separation energies on the mass number within various series of isotopes.

In 1977, Federman and Pittel [96,97,98] realized that when adding valence protons and valence neutrons to a nucleus, the proton–neutron pairs (1d5/2, 1d3/2), (1g9/2, 1g7/2), (1h11/2, 1h9/2), and (1i13/2, 1i11/2) play major roles in the onset of deformation, while later on, deformation is reinforced by the proton–neutron pairs (1d5/2, 1f7/2), (1g9/2, 1h11/2), (1h11/2, 1i13/2), and (1i13/2, 1j15/2), as shown in

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${}_{15/2}$

. In the shell model notation, these sets correspond to $|\Delta n\Delta l\Delta j\Delta m_j\rangle =|0010\rangle$, and $|0110\rangle$, respectively, the latter set coinciding with the de Shalit–Goldhaber pairs.

In 1985, the decisive roles played by proton–neutron pairs were demonstrated by Casten through the introduction of the $N_p{N}_n$ scheme [99,100], by showing the systematic dependence of several observables on the quadrupole–quadrupole interaction, measured through $N_p{N}_n$, where $N_p$ ($N_n$) is the number of valence protons (neutrons) counted from the nearest closed shell. In 1987, the P-factor, $P=N_p{N}_n/(N_p+N_n)$ [101,102], was introduced, and the systematic dependence of several observables on it was demonstrated. The P-factor expresses the competition between the quadrupole deformation, “measured” by the quadrupole–quadrupole interaction through $N_p{N}_n$, and the pairing interaction, “measured” through $N_p+N_n$.

In 1995, the quasi-SU(3) symmetry [103,104] was introduced, based on the proton–neutron pairs (1g9/2, 2d5/2), (1h11/2, 2f7/2), (1i13/2, 2g9/2), which lead to enhanced quadrupole collectivity [105]. The quasi-SU(3) pairs are expressed as $|\Delta n\Delta l\Delta j\Delta m_j\rangle =|1220\rangle$ in the shell model notation.

Following detailed studies of double differences in experimental binding energies [106,107,108,109], in 2010 it was realized (in Reference [110]) that proton–neutron pairs differed by $\Delta K[\Delta N\Delta n_z\Delta \Lambda ]=0\left[110\right]$ in the Nilsson notation [17,18,19] $K[N{n}_z\Lambda ]$, where N is the total number of oscillator quanta, $n_z$ is the number of quanta along the z-axis, and $\Lambda$, K, are the projections along the z-axis of the orbital angular momentum and the total angular momentum, respectively, which play a major role in the development of nuclear deformation, due to the large spatial overlaps [111]. This effect was corroborated by nuclear density functional theory calculations [112]. We are going to call these pairs the Nilsson 0[110] pairs, or simply the 0[110] pairs. Notice the difference in the notation in comparison to the spherical shell model $|0110\rangle$ pairs.

As we shall see below, the Nilsson 0[110] pairs play crucial roles in the replacements made within the approximation leading to the proxy-SU(3) symmetry. Furthermore, the relation between the Nilsson 0[110] pairs and the spherical shell model $|0110\rangle$ pairs will be clarified through the connection of the proxy-SU(3) symmetry to the spherical shell model. Evidence supporting the formation of 0[110] pairs was found recently within Monte Carlo shell model calculations [113].

4. The Proxy-SU(3) Approximation

The proxy-SU(3) symmetry was born when the desire to reestablish the SU(3) symmetry of the 3D-HO in medium mass and heavy nuclei, described in Section 2, met the experimental hint of the 0[110] Nilsson pairs, described in Section 3. The 0[110] Nilsson pairs were discovered through experimental observation [110], where the proton–neutron pairs of this type maximize the proton–neutron interaction. This maximization was attributed to the large spatial overlaps of the two orbitals involved in such pairs [111]. In other words, 0[110] Nilsson pairs are very similar since they possess identical angular momentum and spin properties (identical projections of the orbital angular momentum, the spin, and the total angular momentum) and very similar spatial shapes since they differ only by one excitation quantum in the z-axis (and, therefore, also by one quantum in the total number of quanta). The similarity holds equally well if one considers a 0[110] pair of protons, or a 0[110] pair of neutrons. Such pairs will still have identical angular momentum and spin properties and very similar spatial shapes. Therefore, in the framework of an approximation needed for some reason, each member of a 0[110] pair could replace the other, i.e., acting as a proxy, with minimal changes inflicted in the physical system under study.

Such a situation appears in the shells of the shell model beyond 28 nucleons. The intruder orbitals, which come down from the shell above, form 0[110] pairs with the orbitals, which desert the shell by fleeing into the shell below. We can, therefore, think of replacing the intruder orbitals with their 0[110] counterparts, which are the deserting orbitals, expecting that the changes caused in the physical system under study would be minimal. In other words, the deserting orbitals can act as proxies of the intruder orbitals. However, in this way, the shell obtains the SU(3) symmetry of the 3D-HO, which it lost after the deserting of the orbitals, which fled to the shell below.

An example can be seen in Figure 1, in which the $sdg$ shell is depicted, consisting of the $3s_{1/2}$, $2d_{3/2}$, $2d_{5/2}$, $1g_{7/2}$, $1g_{9/2}$ orbitals, having an overall symmetry of U(15), which possesses SU(3) subalgebra. The spin–orbit interaction pushes the orbital with the highest j, i.e., $1g_{9/2}$, to the shell below, while it brings down the $1h_{11/2}$ orbital from the shell above. In this way, the 50–82 shell of the shell model is formed, in which the SU(3) symmetry is lost. Looking into the details of the orbitals, a sign of hope appears. The intruder $1h_{11/2}$ orbital consists of the Nilsson orbitals 1/2[550], 3/2[541], 5/2[532], 7/2[523], 9/2[514], 11/2[505], while the deserting $1g_{9/2}$ orbital consists of Nilsson orbitals 1/2[440], 3/2[431], 5/2[422], 7/2[413], 9/2[404]. We observe that the first five Nilsson orbitals of $1h_{11/2}$ are 0[110] partners with the five orbitals making up $1g_{9/2}$. Therefore, it is plausible to replace the 1/2[550], 3/2[541], 5/2[532], 7/2[523], 9/2[514] orbitals by the 1/2[440], 3/2[431], 5/2[422], 7/2[413], 9/2[404] orbitals, hoping that the changes inflicted in the physical system would be minimal. In other words, the 1/2[440], 3/2[431], 5/2[422], 7/2[413], and 9/2[404] orbitals will act as proxies of the 1/2[550], 3/2[541], 5/2[532], 7/2[523], and 9/2[514] orbitals.

It is now time to consider the gains and losses of this replacement. The gain is that the SU(3) symmetry is re-established in the 50–82 shell since the shell will now contain all orbitals composing the $sdg$ shell. However, a couple of losses are lurking. First, the 11/2[505] Nilsson orbital has no 0[110] partner; thus, it is left behind within the 50–82 shell, where it stays outside the SU(3) symmetry, and in principle, it has to be treated separately by shell model techniques [45,46]. The good news in this case is that the 11/2[505] Nilsson orbital lies at the very top of the 50–82 shell, as one can see in the standard Nilsson diagrams [19,114]. Thus, it should be empty for most nuclei and, therefore, have little influence on the structures of most of them. The second problem arises from the fact that the intruder and deserting orbitals have opposite parities since they belong to adjacent shells differing by one unit in N. For even–even nuclei, this should not cause any trouble, since, for example, a pair of 9/2[514] particles will be replaced by a pair of 9/2[404] particles; therefore, no differences caused by parity should be seen. For odd nuclei, one would probably have to use projection techniques [115], a problem that has not been considered yet.

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

The first test [88] of the accuracy of the proxy-SU(3) approximation described in the previous section was performed in the framework of the Nilsson model. In each shell of the shell model, two calculations were performed, one with the real orbitals composing the shell, and another one with the intruder orbitals replaced by their 0[110] counterparts. Numerical results for the 82–126 shell are shown in

Table 2. Nilsson Hamiltonian matrix elements (in units of $\hslash {\omega }_0$) with $ϵ=0.3$ 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

. In the upper part of the table, the matrix elements of the Nilsson Hamiltonian, using the standard set of parameters [15], are shown, while in the lower partm the matrix elements occurring within the $PFH$ shell (resulting after the proxy-SU(3) approximation) are reported. We remark that in the last seven columns of the upper part, corresponding to the $1i_{13/2}$ orbitals, all non-diagonal matrix elements vanish, due to the fact that they connect orbitals of the opposite parity. On the contrary, in the last six columns of the lower part of the table, in which the proxies of the $1i_{13/2}$ orbitals, i.e., the $1h_{11/2}$ orbitals appear, not all off-diagonal matrix elements vanish since (in this case) they connect to orbitals of the same parity. However, the number of these off-diagonal matrix elements is small, and in addition, their sizes are small in comparison to the diagonal matrix elements; thus, they are not expected to affect the single-particle energy levels substantially. The diagonal matrix elements in the last six columns of the lower part of the table are also slightly modified in comparison to the corresponding matrix elements in the upper half of the table, but again the changes are small and are not expected to affect the single-particle energy levels radically. The evolution of the single-particle energy levels with deformation, depicted in Figure 2, shows that the changes caused in the Nilsson diagrams by the proxy-SU(3) approximation are minimal. Neither the order of the orbitals nor their dependence on the deformation (upward-sloping or downward-sloping) is modified. Similar tables and figures for other shells can be found in the supplemental material for Reference [88]. It can be seen there that the quality of the proxy-SU(3) approximation becomes better in higher shells.

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

The proxy-SU(3) symmetry is established in the nuclear shells beyond the $sd$ shell in the framework of the Nilsson model by taking advantage of the 0[110] Nilsson pairs. The issue arises if such a process is possible within the framework of the spherical shell model, allowing for the replacement of certain orbitals by their proxies in order to reestablish the SU(3) symmetry.

In order to examine if such a possibility exists, we start with the Elliott model [26,27,28,30,31], in which the Cartesian basis of the 3D isotropic HO is used. This basis is labeled $\left[n_z{n}_x{n}_y{m}_s\right]$, in which the number of quanta along the z, x, y directions and the z-projection of the spin appear. The first step is to transform this basis into the spherical basis, $\left[nl{m}_l{m}_s\right]$ in the l-s coupling, labeled by the principal quantum number n, the orbital angular momentum l, its z-projection ($m_l$), and the z-projection of the spin ($m_s$). This can be achieved through a unitary transformation [116,117,118]

$$\left[n_z{n}_x{n}_y{m}_s\right]=R\left[nl{m}_l{m}_s\right],$$

(1)

the details of which can be found in Reference [119]. Using Clebsch–Gordan coefficients [120,121], the spherical basis can be recoupled from the l-s coupling to the j-j coupling in the following way

$$\left[nl{m}_l{m}_s\right]=C\left[nlj{m}_j\right],$$

(2)

in which the total angular momentum j and its z-projection appear. Combining these two transformations, one obtains

$$\left[n_z{n}_x{n}_y{m}_s\right]=RC\left[nlj{m}_j\right],$$

(3)

i.e., the connection between the Cartesian Elliott basis and the spherical shell model basis in j-j coupling. Details of the calculations and transformation tables up to the $n=3$ shell can be found in Reference [119], while a succinct discussion of the approach, similar to the present section, was given by some of the present authors in Reference [87].

Using the above transformation, one sees that the Nilsson 0[110] replacements made within the proxy-SU(3) scheme are “translated” into $|0110\rangle$ replacements within the spherical shell model basis. The resulting correspondences between original shell model orbitals and proxy-SU(3) orbitals are summarized in

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	$1p_{\pm 1/2}^{1/2}$	$1p_{\pm 1/2}^{1/2}$	$U\left(3\right)$	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	$2s_{\pm 1/2}^{1/2}$	$2s_{\pm 1/2}^{1/2}$	$U\left(6\right)$	14–26	8–20
	$1d_{\pm 1/2,\pm 3/2}^{3/2}$	$1d_{\pm 1/2,\pm 3/2}^{3/2}$
	$\mathbf{1}{\mathbf{f}}_{\pm \mathbf{1}/\mathbf{2},\pm \mathbf{3}/\mathbf{2},\pm \mathbf{5}/\mathbf{2}}^{\mathbf{7}/\mathbf{2}}$	$\mathbf{1}{\mathbf{d}}_{\pm \mathbf{1}/\mathbf{2},\pm \mathbf{3}/\mathbf{2},\pm \mathbf{5}/\mathbf{2}}^{\mathbf{5}/\mathbf{2}}$
	$\mathbf{1}{\mathbf{f}}_{\pm \mathbf{7}/\mathbf{2}}^{\mathbf{7}/\mathbf{2}}$	-
28–50	$2p_{\pm 1/2}^{1/2}$	$2p_{\pm 1/2}^{1/2}$	$U\left(10\right)$	28–48	20–40
	$2p_{\pm 1/2,\pm 3/2}^{3/2}$	$2p_{\pm 1/2,\pm 3/2}^{3/2}$
	$1f_{\pm 5/2,\pm 3/2,\pm 1/2}^{5/2}$	$1f_{\pm 5/2,\pm 3/2,\pm 1/2}^{5/2}$
	$\mathbf{1}{\mathbf{g}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{7}/\mathbf{2}}^{\mathbf{9}/\mathbf{2}}$	$\mathbf{1}{\mathbf{f}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{7}/\mathbf{2}}^{\mathbf{7}/\mathbf{2}}$
	$\mathbf{1}{\mathbf{g}}_{\pm \mathbf{9}/\mathbf{2}}^{\mathbf{9}/\mathbf{2}}$	-
50–82	$3s_{\pm 1/2}^{1/2}$	$3s_{\pm 1/2}^{1/2}$	$U\left(15\right)$	50–80	40–70
	$2d_{\pm 1/2,\pm 3/2}^{3/2}$	$2d_{\pm 1/2,\pm 3/2}^{3/2}$
	$2d_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$	$2d_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$
	$1g_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$	$1g_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$
	$\mathbf{1}{\mathbf{h}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{9}/\mathbf{2}}^{\mathbf{11}/\mathbf{2}}$	$\mathbf{1}{\mathbf{g}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{9}/\mathbf{2}}^{\mathbf{9}/\mathbf{2}}$
	$\mathbf{1}{\mathbf{h}}_{\pm \mathbf{11}/\mathbf{2}}^{\mathbf{11}/\mathbf{2}}$	-
82–126	$3p_{\pm 1/2}^{1/2}$	$3p_{\pm 1/2}^{1/2}$	$U\left(21\right)$	82–124	70–112
	$3p_{\pm 1/2,\pm 3/2}^{3/2}$	$3p_{\pm 1/2,\pm 3/2}^{3/2}$
	$2f_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$	$2f_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$
	$2f_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$	$2f_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$
	$1h_{\pm 1/2,\dots ,\pm 9/2}^{9/2}$	$1h_{\pm 1/2,\dots ,\pm 9/2}^{9/2}$
	$\mathbf{1}{\mathbf{i}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{11}/\mathbf{2}}^{\mathbf{13}/\mathbf{2}}$	$\mathbf{1}{\mathbf{h}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{11}/\mathbf{2}}^{\mathbf{11}/\mathbf{2}}$
	$\mathbf{1}{\mathbf{i}}_{\pm \mathbf{13}/\mathbf{2}}^{\mathbf{13}/\mathbf{2}}$	-
126–184	$4s_{\pm 1/2}^{1/2}$	$4s_{\pm 1/2}^{1/2}$	$U\left(28\right)$	126–182	112–168
	$3d_{\pm 1/2,\pm 3/2}^{3/2}$	$3d_{\pm 1/2,\pm 3/2}^{3/2}$
	$3d_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$	$3d_{\pm 1/2,\dots ,\pm 5/2}^{5/2}$
	$2g_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$	$2g_{\pm 1/2,\dots ,\pm 7/2}^{7/2}$
	$2g_{\pm 1/2,\dots ,\pm 9/2}^{9/2}$	$2g_{\pm 1/2,\dots ,\pm 9/2}^{9/2}$
	$1i_{\pm 1/2,\dots ,\pm 11/2}^{11/2}$	$1i_{\pm 1/2,\dots ,\pm 11/2}^{11/2}$
	$\mathbf{1}{\mathbf{j}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{13}/\mathbf{2}}^{\mathbf{15}/\mathbf{2}}$	$\mathbf{1}{\mathbf{i}}_{\pm \mathbf{1}/\mathbf{2},\dots ,\pm \mathbf{13}/\mathbf{2}}^{\mathbf{13}/\mathbf{2}}$
	$\mathbf{1}{\mathbf{j}}_{\pm \mathbf{15}/\mathbf{2}}^{\mathbf{15}/\mathbf{2}}$	-

. This correspondence paves the way for taking advantage of the proxy-SU(3) symmetry in shell model calculations for heavy nuclei, in a way similar to that of the symmetry-adapted no-core shell model approach [81,82] used in light nuclei.

A unitary transformation connecting the orbitals being replaced within the proxy-SU(3) scheme was found [119] within the shell model basis, as depicted in Figure 3. This is reminiscent of the unitary transformation occurring in the framework of the pseudo-SU(3) symmetry [48,49,50].

A by-product of the above transformation is that the 0[110] Nilsson pairs identified in Reference [110], and used within the proxy-SU(3) scheme [88,89,90], are identical to the de Shalit–Goldhaber pairs [91] and the Federman–Pittel pairs [96,97,98] within the spherical shell model basis, where they are expressed as $|0110\rangle$ pairs.

The correspondence between Nilsson pairs and shell model pairs was corroborated by calculations [123] within the Nilsson model, in which the first justification of the proxy-SU(3) scheme was found, as described in Section 5. As one can see in

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 $ϵ$. 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.

$\frac{3}{2}\left[541\right]$	$\mathit{ϵ}$	$|{\mathit{Nljm}}_{\mathit{j}}\rangle$	$\left|51\frac{3}{2}\frac{3}{2}\right.〉$	$\left|53\frac{5}{2}\frac{3}{2}\right.〉$	$\left|53\frac{7}{2}\frac{3}{2}\right.〉$	$\left|55\frac{9}{2}\frac{3}{2}\right.〉$	$\left|55\frac{11}{2}\frac{3}{2}\right.〉$
	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
$\frac{\mathbf{3}}{\mathbf{2}}\left[\mathbf{651}\right]$	$\mathbf{ϵ}$	$|{\mathit{Nljm}}_{\mathbf{j}}\rangle$	$\left|\mathbf{6}\mathbf{2}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.〉$	$\left|\mathbf{6}\mathbf{2}\frac{\mathbf{5}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.〉$	$\left|\mathbf{6}\mathbf{4}\frac{\mathbf{7}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.〉$	$\left|\mathbf{6}\mathbf{4}\frac{\mathbf{9}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.〉$	$\left|\mathbf{6}\mathbf{6}\frac{\mathbf{11}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.〉$	$\left|\mathbf{6}\mathbf{6}\frac{\mathbf{13}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\right.〉$
	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

and

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 $ϵ$. 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.

$\frac{1}{2}\left[431\right]$	$\mathit{ϵ}$	$|{\mathit{Nljm}}_{\mathit{j}}\rangle$	$\left|40\frac{1}{2}\frac{1}{2}\right.〉$	$\left|42\frac{3}{2}\frac{1}{2}\right.〉$	$\left|42\frac{5}{2}\frac{1}{2}\right.〉$	$\left|44\frac{7}{2}\frac{1}{2}\right.〉$	$\left|44\frac{9}{2}\frac{1}{2}\right.〉$
	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
$\frac{\mathbf{1}}{\mathbf{2}}\left[\mathbf{541}\right]$	$\mathbf{ϵ}$	$|{\mathit{Nljm}}_{\mathbf{j}}\rangle$	$\left|\mathbf{5}\mathbf{1}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.〉$	$\left|\mathbf{5}\mathbf{1}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.〉$	$\left|\mathbf{5}\mathbf{3}\frac{\mathbf{5}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.〉$	$\left|\mathbf{5}\mathbf{3}\frac{\mathbf{7}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.〉$	$\left|\mathbf{5}\mathbf{5}\frac{\mathbf{9}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.〉$	$\left|\mathbf{5}\mathbf{5}\frac{\mathbf{11}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\right.〉$
	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	$-0.4789$	$-0.3671$

, the correspondence used in proxy-SU(3) works only for the Nilsson orbitals, which possess the highest total angular momentum j within their shell, which are exactly the orbitals, which are replaced within the proxy-SU(3) scheme.

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

In order to start examining the consequences of the existence of the proxy-SU(3) symmetry in medium mass and heavy nuclei, we should first consider a few properties of the irreducible representations (irreps) of SU(3).

In Elliott’s notation [26,27,28,29,30,31], the irreps of SU(3) are labeled by $(\lambda ,\mu )$, where the Elliott quantum numbers $\lambda$ and $\mu$ are connected to the number of boxes in the corresponding Young diagram through the relations

$$\lambda =f_1-f_2,\mu =f_2,$$

(4)

where $f_1$ ($f_2$) is the number of boxes in the first (second) line of the relevant Young diagram. Irreps with $\lambda >\mu$ are known to correspond to prolate (rugby ball-like) shapes, while irreps with $\lambda <\mu$ are known to describe oblate (pancake-like) shapes, with $\lambda =\mu$ irreps corresponding to maximally triaxial shapes [124,125,126].

A quantity characterizing the SU(3) irreps is the eigenvalue of the second-order Casimir operator of SU(3), given by [8,10]

$$C_2(\lambda ,\mu )=\frac{2}{3}({\lambda }^2+{\mu }^2+\lambda \mu +3\lambda +3\mu ).$$

(5)

This quantity is known to be connected to the eigenvalues of the quadrupole–quadrupole interaction by [26,27,28,47]

$$QQ=4C_2-3L(L+1),$$

(6)

where L is the eigenvalue of the orbital angular momentum. For the ground state bands of even–even nuclei, in which $L=0$, it is clear that the eigenvalue of $C_2$ is proportional to the eigenvalue of the quadrupole–quadrupole interaction, $QQ=4C_2$. Since the quadrupole–quadrupole interaction is known to cause nuclear deformation, it is expected that the irrep with the highest value of $C_2$ should correspond to the preferred ground state with maximum deformation.

There is, however, another factor that should be taken into account. The nucleon–nucleon interaction is known to be attractive and of short-range [127], therefore favoring the maximal spatial overlap [111] among the nucleons, which can be achieved for the most symmetric SU(3) irrep. In the Young diagrams, it is known that boxes on the same line correspond to symmetrized particles, while boxes in the same column correspond to anti-symmetrized particles [8,10]. The degree of symmetrization of a given SU(3) irrep can, therefore, be measured by the ratio of the symmetrized boxes over the total number of boxes, which is

$$r=\frac{f_1}{f_1+f_2}=\frac{\lambda +\mu }{\lambda +2\mu }.$$

(7)

It was proved [128] that the irreps with the highest value of r, i.e., with the highest degree of symmetry, correspond to what is called the highest weight (hw) irreps of SU(3). As such, these irreps are favored by the nucleon–nucleon interaction and, therefore, dominate the related nuclear properties.

The irreps possessing the highest eigenvalues of the second-order Casimir operator, called the highest $C_2$ irreps, for brevity, and the highest hw irreps of SU(3) in the various nuclear shells, are shown in Table 6. It is clear that up to the mid-shell, the $C_2$ and hw irreps are identical, while in the upper half of each shell, the $C_2$ and hw irreps are different, with the exception of the last 5 particle numbers (which correspond to states with up to 4 holes at the top of the shell). Further mathematical details on the dominance of the hw irreps can be found in References [129,130].

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

8.1. Prolate over Oblate Dominance

One of the long-standing puzzles in a nuclear structure is the dominance of prolate (rugby ball-like) shapes over oblate (pancake-like) shapes in the ground state bands of even–even nuclei. In Elliott’s notation, prolate (oblate) irreps correspond to $\lambda >\mu$ ($\lambda <\mu$) [26,27,28,29,30,31,131], while irreps with $\lambda =\mu$ correspond to maximal triaxiality. Despite several attempts [102,132,133,134,135] to resolve this puzzle, the issue is still considered open [136].

The dominance of the hw irreps over the highest $C_2$ irreps offers a simple, parameter-free justification of the prolate over oblate dominance. As an example, in

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)	$\underline{(20,22)}$	$\underline{(12,28)}$	$\underline{(6,30)}$	$\underline{(2,28)}$	$\underline{(0,22)}$

, the hw irreps corresponding to the rare earth nuclei with valence protons in the 82–126 shell and valence neutrons in the 126–184 shell are shown. These hw irreps are obtained by adding up the hw irrep $({\lambda }_{\pi },{\mu }_{\pi })$ corresponding to the valence protons and the hw irrep $({\lambda }_{\nu },{\mu }_{\nu })$ and corresponding to the valence neutrons into the most stretched irrep [45]$({\lambda }_{\pi }+{\lambda }_{\nu },{\mu }_{\pi }+{\mu }_{\nu })$. We notice that prolate irreps are obtained over most of the table, with the exception of its lower right corner, near which, a few oblate irreps (underlined in Table 7) appear in nuclei lying below the tops of both the proton valence shell and the neutron valence shell. The same effect appears in other shells as well, as one can see using the irreps appearing in Table 6. For example, results for rare earth with valence protons in the 50–82 shell and valence neutrons within the 50–82 and the 82–126 shells are given in Reference [89]. The conclusion of this subsection was recently corroborated by a heuristic method in Reference [137].

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)

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 $\gamma$

Further consequences of the dominance of the hw irreps over the highest $C_2$ irreps become evident if one considers the connection between the Elliott quantum numbers $\lambda$, $\mu$ and the collective variables $\beta$, $\gamma$ of the Bohr–Mottelson model. This connection is obtained by employing a linear mapping between the invariant quantities of the two theories, which are the invariants ${\beta }^2$ and ${\beta }^3\cos 3\gamma$ of the collective model [14,15] and the Casimir operators of the second and third order of SU(3) [8,10]. This mapping provides the angle collective variable $\gamma$ of the expression [124,126]

$$\gamma =\arctan \left(\frac{\sqrt{3}(\mu +1)}{2\lambda +\mu +3}\right),$$

(8)

while for the square of the deformation parameter $\beta$, being proportional to the second-order Casimir operator of SU(3) [58], it gives [124,126]

$${\beta }^2=\frac{4\pi }{5}\frac{1}{{(A\overline{r^2})}^2}({\lambda }^2+\lambda \mu +{\mu }^2+3\lambda +3\mu +3),$$

(9)

where A is the mass number of the nucleus and $\overline{r^2}$ is related to the dimensionless mean square radius [115], $\sqrt{\overline{r^2}}=r_0{A}^{1/6}$. The dimensionless mean square radius is obtained by dividing the mean square radius, which is proportional to $A^{1/3}$, by the oscillator length, which grows as $A^{1/6}$ [115]. The constant $r_0$ is determined from a fit over a wide range of nuclei [144,145]. We stick to the value used in Reference [124], $r_0=0.87$, in agreement with Reference [145].

A word of warning: since only the valence nucleons are taken into account in proxy-SU(3), the values of $\beta$ obtained from Equation (9) should be rescaled by multiplying them by a factor $A/(S_{\pi }+S_{\nu })$, where $S_{\pi }$ ($S_{\nu }$) is the size of the proton (neutron) valence shell. The need for this rescaling is discussed in detail in Section 5 of Reference [89]. In practice, this rescaling means that Equation (9), when used in the proxy-SU(3) framework, should read

$${\beta }^2=\frac{4\pi }{5}\frac{1}{{\left((S_{\pi }+S_{\nu })\overline{r^2}\right)}^2}({\lambda }^2+\lambda \mu +{\mu }^2+3\lambda +3\mu +3).$$

(10)

Numerical results for the collective variable $\beta$ ($\gamma$) for several rare earth nuclei are shown in Figure 4 and Figure 5 and are compared to other theoretical approaches, such as the D1S–Gogny interaction [146] and relativistic mean field theory [147], as well as experimental values [148]. By “Gogny D1S min”, we label the values of the deformation variables, $\beta$ and $\gamma$, at the HFB minimum energy (entries 4 and 5 in Reference [146]); “Gogny D1S mean” involves the mean ground state values (entries 11 and 12 in Reference [146]). It is seen that the parameter-free proxy-SU(3) predictions are in good agreement with both the theoretical approaches and the empirical values. Additional numerical results for $\beta$ and $\gamma$ obtained within the proxy-SU(3) approach can be found in [149,150,151]. Additional comparisons of proxy-SU(3) predictions to covariant density functional theory can be found in References [152,153,154], while comparisons to recent experimental findings can be found in [155,156].

In Figure 4, it is clear that the $\beta$ curve is not symmetric around the mid-shell, but it appears to exhibit higher values in the first half of the shell. The origin of this discrepancy can be traced in Figure 6, in which the square root of $C_2$, which is proportional to $\beta$, according to Equation (10), is shown. We see that the breaking of the symmetry around the mid-shell is due to the fact that in the upper half of the shell the highest weight irreps enter in the place of the highest $C_2$ irreps, as indicated by Table 6.

8.3. Prolate to Oblate Shape/Phase Transition

A second important consequence of the hw irreps dominance is seen in Figure 7, in which the proxy-SU(3) predictions for the collective variable $\beta$ for the rare earth with valence protons in the 50–82 shell and valence neutrons in the 82–126 shell are collected. The dip seen at $n=116$ signifies the occurrence of a shape/phase transition [158,159,160,161,162,163,164,165,166] from prolate to oblate shapes, for which extended experimental evidence exists [167,168,169,170,171], along with microscopic theoretical considerations [172,173,174,175,176,177,178], and relevant searches within the interacting boson model (IBM) [179,180,181,182,183,184] and the Bohr–Mottelson collective model [185,186]. In the framework of the Bohr Hamiltonian, this shape/phase transition is referred to as Z(5) [185]. A posteriori corroboration of the findings of Reference [89] is found in References [187,188,189].

The robustness of this result is emphasized by the fact that it also appears in atomic clusters [190,191,192,193,194,195]. The valence electrons in alkali metal clusters, in particular, are supposed to be free, thus forming shells with major magic numbers 2, 8, 20, 40, 58, 92,… [196,197,198,199,200,201,202,203]. Prolate and oblate shapes in alkali metal clusters were observed experimentally through optical response measurements, finding oblate shapes below cluster sizes 20 and 40 [204,205,206,207], while prolate shapes were seen above cluster sizes 8, 20, 40 [204,205,206,207,208]. In other words, prolate (oblate) shapes are seen above (below) the magic numbers, exactly as in atomic nuclei.

It should be noticed that the shape/phase transition from prolate to oblate shapes is seen equally clearly in the framework of pseudo-SU(3), if the dominance of the hw irreps over the highest $C_2$ irreps is taken into account, as discussed in detail in Reference [209]. Although pseudo-SU(3) and proxy-SU(3) are based on different approximations, involving different unitary transformations, they lead to the same physical conclusion, a fact providing evidence for the compatibility of the two approaches.

The compatibility of the proxy-SU(3) and pseudo-SU(3) approaches has also been pointed out recently [210,211,212] in the framework of the semi-microscopic algebraic quartet model (SAQM) [213], which generates the excitation spectra of shell-like quartets, formed by two protons and two neutrons in a well-defined shell configuration. Based on the SU(3) symmetry, in its initial form [213] the model had been applicable only to light nuclei, based on the Elliott SU(3) symmetry. However, it has recently [210,211,212] been extended to heavy nuclei, based on the proxy-SU(3) symmetry and on the pseudo-SU(3) symmetry, with similar results provided by both approaches. The use of proxy-SU(3) could also be extended in a similar way to the semi-microscopic algebraic cluster model (SACM) [214,215,216], in which the internal structures of the clusters are described in terms of the shell model SU(3) symmetry, while the relative motion is described in terms of the phenomenological algebraic vibron model [60,217,218,219].

9. Islands of Shape Coexistence

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

The shape coexistence (SC) involves the appearance in a nucleus of two bands lying close in energy but having radically different structures; for example, one of them being spherical and the other deformed, or both of them being spherical, but one of them exhibiting prolate (rugby ball-like) deformation and the other oblate (pancake-like) deformation. Shape coexistence has first been suggested in 1956 by Morinaga [220], in relation to the spectrum of ${}^{16}$O. Since then many experimental examples were found in both odd and even nuclei, as summarized in the relevant review articles [221,222,223,224]. From the theoretical point of view, SC was attributed to the existence of particle–hole excitations across shell or subshell closures, and was believed to be able to appear all over the nuclear chart, although in Figure 8 of the authoritative review article by Heyde and Wood [223], the regions in which SC was experimentally observed appear to form certain islands on the nuclear chart.

As already mentioned, SC is supposed to be due to particle–hole excitations across shell or subshell closures. However, the magic numbers of the shell models, 2, 8, 20, 28, 50, 82, 126,…, are known to be valid only at zero deformations. These magic numbers originate from the 3D-HO magic numbers 2, 8, 20, 40, 70, 112, 168,…, [8,10] (to be called the HO magic numbers in what follows) because of the action of the spin–orbit interaction [4,5,6,7]. As deformation sets in, the energy gaps separating different shells soon disappear, as one can see in the standard Nilsson diagrams [17,18,19,114] (see also References [122,226] for the evolution of magic numbers away from stability). Furthermore, as seen in Table 3 (see also Reference [227]), in the framework of the proxy-SU(3) symmetry a set of magic numbers 6, 14, 28, 50, 82, 126, 184,… (to be called the SO magic numbers in what follows) appears, corresponding to the strong presence of the spin–orbit interaction everywhere. The standard shell model magic numbers follow the HO magic numbers up to 20, while they follow the SO magic numbers beyond this point.

9.2. A Dual Shell Mechanism for Shape Coexistence

It was suggested [225,228,229] that the interplay between HO and SO magic numbers offers a simple justification for the appearance of islands of SC on the nuclear chart. Let us see how this is occurring.

Because of the collapse of the shell model quantum numbers as deformation sets in, the protons and/or the neutrons of a nucleus can follow either the HO or the SO magic numbers. The same number of protons or neutrons will then correspond to a different irrep in the HO framework and another irrep in the SO framework. These irreps can be seen in

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)

. As an example, let us consider 60 nucleons. In the HO framework, 40 is a magic number; thus, there are 20 nucleons left in the 40–70 shell, which has the U(15) symmetry and, therefore, the 20 nucleons correspond to the (20,0) hw irrep according to Table 6. In the SO framework, 50 is a magic number; thus, there are 10 nucleons left in the 50–82 shell, which in the proxy-SU(3) approximation has the U(15) symmetry; therefore, the 10 nucleons correspond to the (20,4) hw irrep of SU(3) according to Table 6. Indeed, in Table 8 for 60 nucleons, the hw irreps given are (20,4) for the SO case and (20,0) for the HO case.

For the $L=0$ band heads of two coexisting bands, one can use the very simple Hamiltonian [225]

$$H=H_0-\frac{\kappa }{2}QQ,$$

(11)

where $H_0$ corresponds to the 3D-HO Hamiltonian and $QQ$ to the quadrupole–quadrupole interaction. Both bands should belong to the same U(n) algebra within the SO and HO schemes. From Table 3, we see that this is possible for the nucleon number intervals 6–8, 14–20, 28–40, 50–70, 82–112, 126–168, in which both bands belong to the U(3), U(6), U(10), U(15), U(21), and U(28) algebra, respectively. We remark that the right borders of these regions are the HO magic numbers. The successful parameter-free predictions of the $\beta$ and $\gamma$ collective variables for the ground states of nuclei seen in Section 8.2 imply that the ground-state band will belong to the SO irrep, thus the band head of the coexisting band should lie higher in energy. One can easily see that this requirement leads to the condition

$$Q{Q}_{SO}\ge Q{Q}_{HO};$$

(12)

the full details of the argument are given explicitly in Section 8 of Reference [225]. From Equation (6), one then sees that this condition is equivalent to the condition

$$C_2({\lambda }_{SO},{\mu }_{SO})\ge C_2({\lambda }_{HO},{\mu }_{HO}).$$

(13)

The eigenvalues of the Casimir operator $C_2$ in the SO and HO frameworks are shown for the various shells in Figure 8. We see that the condition of Equation (13) starts being fulfilled at the nucleon numbers 7, 17, 34, 59, 96, and 146, which, therefore, stand for the left borders of the regions in which SC could be possible, with the right borders given by the HO magic numbers mentioned above. Therefore, we conclude that SC can occur only within the nucleon intervals 7–8, 17–20, 34–40, 59–70, 96–112, and 146–168, bearing U(3), U(6), U(10), U(15), U(21), and U(28) symmetry, respectively. These intervals define horizontal and vertical stripes on the nuclear chart, shown in color in Figure 9. One can easily see that the islands of SC seen in Figure 8 of Reference [223] do lie within the stripes predicted by the dual shell mechanism within the proxy-SU(3) framework just found.

9.3. From Stripes to Islands of Shape Coexistence

The stripes 7–8, 17–20, 34–40, 59–70, 96–112, and 146–168 (determined in the previous subsection) represent necessary conditions for the appearance of SC, but not sufficient ones. Further work is needed in order to narrow down the stripes into islands. A step in this direction was taken within the covariant density functional theory [230,231,232,233,234,235,236], using the DDME2 functional [237] and the code of Reference [238]. A systematic search was made [239,240] for particle–hole excitations, which are believed to be the microscopic mechanisms behind SC. Indeed, specific islands of SC were located [239,240] around the proton shell closures Z = 82 and Z = 50, in which proton particle–hole excitations were caused by the neutrons, characterizing these cases as part of neutron-induced shape coexistence. In addition, specific islands of SC were located [239,240] around the neutron numbers n = 90 and n = 60, in which neutron particle–hole excitations were caused by the protons, therefore characterizing these cases as part of proton-induced shape coexistence. Furthermore, an island of SC was found around $Z=n=40$, in which both the proton-induced and neutron-induced mechanisms are present simultaneously. All of these islands are consistent with the stripes of the dual shell mechanism within the proxy-SU(3) symmetry, as well as with the empirical islands reviewed in Figure 8 of Reference [223].

Further corroboration of the predictions of the dual shell mechanism was provided by various relativistic microscopic calculations [241,242,243,244,245,246], as well as by calculations using the Bohr Hamiltonian [246,247] and the IBM [248].

The above predictions of specific islands of SC are based on the particle–hole excitation mechanism. Some other microscopic mechanisms may create SC in regions of the nuclear chart outside the islands predicted here. This can be the subject of further investigation.

9.4. Multiple-Shape Coexistence

Recent experimental evidence shows that multiple-shape coexistence (of up to four bands) can be seen in certain nuclei [249,250], while it is also predicted theoretically in others [245]. Multiple-shape coexistence can occur within the dual shell mechanism described above, since the protons can follow either the SO or the HO scheme, and so can (independently) the neutrons. As a result, four different irreps, based on the proton–neutron combinations SO-SO, SO-HO, HO-SO, and HO-HO can occur in general, giving rise to multiple coexistence of four bands, or three bands in the special case in which equal numbers of valence protons and valence neutrons occupy the same shell. This idea, however, still needs to be tested against experimental evidence.

10. Conclusions and Outlook

In this article, we discussed the following: the physical ideas that led to the introduction of the proxy-SU(3) symmetry, the calculations proving its validity and its connection to the shell model framework, the first successful applications in predicting (in a parameter-free way) the values of the collective variables ($\beta$ and $\gamma$) for even–even nuclei, the dominance of prolate over oblate shapes in the ground states of even–even nuclei, a prolate to oblate shape/phase transition, and the existence of islands on the nuclear chart in which shape coexistence can appear. Several directions for further investigations are briefly presented below.

The proxy-SU(3) symmetry offers the possibility of making parameter-free predictions for B(E2) transition rates. The value of $B(E2;2_1^{+}\to 0_1^{+})$ is known to be connected to the collective variable $\beta$ [148,251]; thus, it can be determined in a parameter-free way from the $\beta$ values calculated as described in Section 8.2. Since ratios of B(E2)s will only depend on angular momentum coupling coefficients of SO(3) [120,121] and SU(3) [252,253,254,255,256,257], parameter-free predictions for all B(E2)s could be obtained in principle. Some first steps in this direction were taken in [151,157].

Based on the experience acquired within the pseudo-SU(3) model [45,46], it is expected that the description of nuclear spectra within the proxy-SU(3) symmetry will require the use of third-order and fourth-order operators. In particular, the O(3) symmetry-preserving three-body operator $\Omega$ and four-body operator $\Lambda$ (their mathematical names being the $O_l^0$ and $Q_l^0$ shift operators, respectively) [258,259,260,261], will be needed in order to break the degeneracy between the ground state band (GSB) and the ${\gamma }_1$ band, which in the proxy-SU(3) approach lies within the same SU(3) irrep. This can be seen, for example, in Table 7, where almost all nuclei are characterized by SU(3) irreps with $\mu \ge 2$. Since K takes the values $K=0,2,\cdots ,\mu$ [10,26,58], the hw irrep will contain both the $K=0$ (GSB) and $K=2$ (${\gamma }_1$) bands [262]. The parameter-free reproduction of the empirical observation (where the energy differences between the ${\gamma }_1$ band and the ground state band decrease as functions of the angular momentum L in deformed nuclei [263], with an opposite trend seen in vibrational and $\gamma$-unstable nuclei) should be tested. Some first steps in this direction were taken in References [264,265]. The energy scale can be fixed in an even–even nucleus by determining the energy of the first excited state $2_1^{+}$ from the value of the $B(E2;2_1^{+}\to 0_1^{+})$, determined in a parameter-free way, as described in Section 8.2. This can be achieved through the microscopically derived [266,267] Grodzins relation [268], connecting the energy of the $2_1^{+}$ state and $B(E2;2_1^{+}\to 0_1^{+})$.

Nuclear binding energies and nucleon separation energies are basic nuclear structure quantities, for which extended experimental data [269] and theoretical predictions [147,270,271] exist. It would be an interesting project to examine the degree to which proxy-SU(3) is able to predict these quantities, preferably in a parameter-free way. Some first steps in this direction were taken in References [272,273]. The calculations of two-neutron separation energies are of particular interest due to the recently discovered [274,275] connections between them and the neutron capture cross sections, which are essential for understanding the astrophysical s and r processes.