Proxy-SU(3) Model: Symmetry in Heavy Nuclei

• Proxy-SU(3) is a symmetry-based analytic framework that reconstructs near-complete oscillator shells by replacing intruder orbitals with their 0[110] partners.
• It provides parameter-free, group-theoretical predictions of nuclear shape observables (β and γ) and explains prolate dominance and the prolate–oblate transition using SU(3) invariants.
• The model leverages SU(3) algebra to derive spectroscopic properties and transition rates, complementing microscopic approaches and offering insights into shape coexistence and triaxiality.

The Proxy-SU(3) model is a symmetry-based, analytic framework developed to restore and exploit an approximate SU(3) symmetry in medium-mass and heavy nuclei where strong spin–orbit interactions would otherwise destroy the exact SU(3) symmetry known from light shells. By systematically replacing most intruder (unique parity) orbitals with their 0[110] partners from the next-lower major shell, the model reconstructs a near-complete oscillator shell in which nucleon configurations can be classified by familiar SU(3) irreducible representations. This approach is parameter-free, relies on the Pauli principle and the short-range character of the nucleon–nucleon interaction, and permits direct calculations of nuclear shape observables and spectroscopic properties through group-theoretical arguments.

1. Analytic Foundations and the Proxy Approximation

The Proxy-SU(3) framework restores a near-SU(3) symmetry by proxying intruder orbital configurations. The prescription is as follows: in any shell where the strong spin–orbit term causes some high-$j$ or unique-parity Nilsson orbitals to invade the shell, these intruder orbitals (save for the highest in the intruder subshell) are replaced by their 0[110] partners—orbitals from the next shell below that match $K$, $\Lambda$, and angular momentum projections but differ by $N \to N-1$ and $n_z \to n_z-1$. This replacement reconstructs the full oscillator space (restoring, for instance, $U(15)$ for 50–82 protons or $U(21)$ for 82–126 neutrons), within which the Elliott SU(3) symmetry and its irreps (labeled by $(\lambda,\mu)$) re-emerge.

Mathematically, for each deformed nucleus, the valence nucleon space is described by a product of SU(3) irreps for protons and neutrons, with the total SU(3) irrep determined either by direct sum ($\lambda_{tot} = \lambda_p + \lambda_n$, $\mu_{tot} = \mu_p + \mu_n$) or via Young diagram methods maximizing spatial symmetry. The highest weight irrep dominates the structure and properties of the ground state and low-lying bands.

2. Shape Variables: Analytic Prediction of $\beta$ and $\gamma$

The model achieves a linear mapping between SU(3) invariants and the collective deformation variables of the Bohr–Mottelson model:

• The overall quadrupole deformation $\beta$ is related to the quadratic Casimir of SU(3):

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

where $A$ is the mass number and $\overline{r^2}$ is determined by a typical mean-square radius formula ($r_0 \sim 0.87$).

• The triaxiality parameter $\gamma$ is determined via

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

This formula directly links the SU(3) quantum numbers to geometric shape: $\gamma = 0^\circ$ is prolate (axial), $\gamma = 60^\circ$ is oblate, and $\gamma \approx 30^\circ$ is maximally triaxial.

Parameter-free predictions for these variables, assigned from the SU(3) quantum numbers by nucleon counting and proxy mapping, are in agreement with both empirical data (e.g., from $B(E2)$ values and bandhead ratios) and with more microscopic models such as RMF and Gogny D1S mean-field calculations (Bonatsos et al., 2017, Bonatsos et al., 2017, Martinou et al., 2017, Martinou et al., 2017).

3. Prolate Dominance and Prolate–Oblate Shape Transition

An essential result of the Proxy-SU(3) model is the natural emergence of prolate shape dominance in heavy deformed nuclei, directly traceable to the properties of highest weight SU(3) irreps:

• When filling a shell, the SU(3) irrep assignment overwhelmingly yields $\lambda \gg \mu$, so $\gamma$ is small (prolate).
• This dominance arises because the spatial part of the wavefunction is maximized (consistent with the short-range force) for the highest weight irrep subject to the antisymmetry required by the Pauli principle.
• Particle–hole symmetry is explicitly broken: the irrep with the largest Casimir value is not always of maximal spatial symmetry for holes. Past mid-shell, the filling order (and the highest weight irrep) no longer coincides with the Casimir-maximizing (oblate) irrep, and so prolate shapes remain prevalent until near shell closure (Bonatsos et al., 2017, Sarantopoulou et al., 2017, Bonatsos et al., 2017).

The model analytically predicts a prolate–oblate transition at the shell ends, such as at $N \approx 116$ in rare earth nuclei, where the irrep assignment inverts ($\lambda < \mu$), shifting $\gamma$ to oblate values. These predictions align with observed shifts in $E(2^+_2) - E(4^+_1)$, energy ratios, and spectroscopic quadrupole moments in W, Os, and Pt isotopes (Bonatsos et al., 2017, Bonatsos et al., 2017).

4. SU(3) Algebraic Structure and Relation to Other Models

The mathematical core of Proxy-SU(3) lies in the invariants and commutation relations of the SU(3) algebra:

• Quadrupole and angular momentum operators ($Q, L$) form the algebra via $[Q_i, Q_j] = i\epsilon_{ijk}L_k$ and $[L_i, Q_j] = i\epsilon_{ijk}Q_k$.
• The explicit Casimir:

$$C_2(\lambda, \mu) = (2/3)(\lambda^2 + \lambda \mu + \mu^2 + 3\lambda + 3\mu)$$

directly quantifies collective quadrupole energy and is pivotal for analytic shape predictions.

Proxy-SU(3) is distinct from (but analogous to) the pseudo-SU(3) scheme, which reassigns normal-parity levels within the same shell, while proxy-SU(3) employs 0[110] replacement across shells to reconstruct a full oscillator symmetry. The model is complementary to bosonic Interacting Boson Approximation approaches and provides a symmetry-based lens with no adjustable parameters (Bonatsos et al., 2017, Bonatsos et al., 2017).

5. Microscopic Justification: Nilsson and Shell Model Basis

The theoretical justification of the approximation is underpinned by the simple structure of Nilsson orbitals that form 0[110] “proxy” pairs (Bonatsos et al., 2020, Martinou et al., 2020, Bonatsos et al., 2021):

• Nilsson orbitals with highest $j$ in a shell are nearly pure in the shell-model basis, mapping one-to-one to their 0[110] partners.
• In the shell model basis, these correspond to a well-defined pair transformation (|1 1 1 0⟩ rule): a change of one quantum along the $z$ direction, preserving all orbital and angular momentum projections.
• The proxy substitution is formally equivalent to a unitary transformation in the $n_z$ coordinate, leaving $n_x$, $n_y$ unchanged, thus preserving the underlying symmetry of the wavefunction in the $x$–$y$ plane, which dictates collective properties.

This mapping has been validated by both direct Nilsson Hamiltonian calculations and explicit transformations between the Elliott cartesian and spherical shell model bases.

6. Shape Coexistence, Triaxial Shapes, and Next Highest Weight Irreps

Proxy-SU(3) not only explains collective ground-state properties but also accounts for more nuanced collective phenomena:

• Shape Coexistence: The dual-shell mechanism predicts that, in overlap regions of harmonic oscillator and spin–orbit–like shells (intervals of $Z$ or $N$: 7–8, 17–20, etc.), two configurations with distinct SU(3) irreps can coexist at similar energies, yielding shape coexistence “islands” (Martinou et al., 2021, Bonatsos et al., 2022).
• Triaxiality: Regions with oblate or maximally triaxial SU(3) irreps ($\lambda \leq \mu$) and small energy separation between hw and next-highest weight (nhw) irreps can induce substantial triaxial deformation ($\gamma \sim 15$–$45^\circ$) (Bonatsos et al., 19 Nov 2024, Kota, 1 Oct 2025). Accurate quantitative description in many nuclei requires mixing of both hw and nhw irreps, especially when the hw irrep is fully symmetric ($\mu=0$).
• Transition Rates: Analytic, parameter-free expressions for $B(E2)$ (and, via more elaborate expansions, $B(E1)$) transition rates are obtained within the SU(3) algebra using group-theoretical matrix elements, and these predictions show good agreement with experiment for rare earth and actinide nuclei (Martinou et al., 2017, Martinou et al., 2017, Restrepo et al., 7 May 2024).

Table: Central Proxy-SU(3) Shape Relations

Observable	Expression	Variables
Quadrupole Deformation, $\beta^2$	$$\frac{4\pi}{5} (1/(A \overline{r^2})^2)(\lambda^2 + \lambda\mu + \mu^2 + 3\lambda + 3\mu + 3)$$	$(\lambda, \mu), A$
Triaxiality, $\gamma$	$$\arctan( \frac{ \sqrt{3} (\mu + 1) }{ 2\lambda + \mu + 3 } )$$	$(\lambda, \mu)$
Casimir Invariant, $C_2$	$$(2/3)(\lambda^2 + \lambda\mu + \mu^2 + 3\lambda + 3\mu)$$	$(\lambda, \mu)$

7. Applications, Limitations, and Future Directions

• Predictive Power: Proxy-SU(3) generates deformation and transition-rate systematics for a wide range of medium-mass and heavy nuclei, providing shape evolution trends, the prolate–oblate transition locus, and collective band properties in good agreement with empirical and microscopic data.
• Triaxiality: Systematic studies have shown that triaxial shapes are predicted to occur in broad “stripes” of the nuclear chart, particularly for nucleon numbers 22–26, 34–48, 74–80, 116–124, and 172–182, matching regions with substantial oblate or triaxial SU(3) irreps and regions of shape coexistence (Bonatsos et al., 19 Nov 2024).
• Extensions: For nucleon numbers $32 \leq Z,N \leq 46$, proxy-SU(4) symmetry—coupling orbital (SU(3)) and spin-isospin (SU(4)) algebras—emerges and modifies irrep assignments and mixing relevant for the calculation of triaxiality and collective observables (Kota et al., 2023, Kota, 1 Oct 2025).
• Configuration Mixing: Accurate predictions, especially for nuclei with $\mu=0$, require inclusion of neighboring irreps (nhw, nnhw), as their mixing significantly affects the calculated $\gamma$ and, consequently, the predicted triaxial softness or rigidity.
• Limitations and Outlook: Proxy-SU(3) is best-suited for well-deformed (collective) nuclei; it requires refinement (e.g. inclusion of higher-order interactions or configuration-mixing schemes) to capture band splitting and fine spectroscopic details or for nuclei near closed shells. Ongoing research targets systematic incorporation of these effects, applications to more complex electromagnetic transitions (e.g., $M1$, $E1$), and the exploration of shape coexistence and triaxiality in new regions of the nuclear chart.

The Proxy-SU(3) model, by bridging group-theoretical symmetry principles and modern shell-model spectroscopic analysis, represents a robust analytic tool for understanding and predicting a range of collective phenomena in atomic nuclei without reliance on phenomenological parameter fitting (Bonatsos et al., 2017, Bonatsos et al., 2017, Bonatsos et al., 2017, Bonatsos et al., 2017, Sarantopoulou et al., 2017, Bonatsos et al., 2017, Martinou et al., 2017, Bonatsos et al., 2017, Martinou et al., 2017, Bonatsos et al., 2019, Bonatsos et al., 2020, Martinou et al., 2020, Martinou et al., 2021, Bonatsos et al., 2021, Bonatsos et al., 2022, Kota et al., 2023, Restrepo et al., 7 May 2024, Bonatsos et al., 6 Sep 2024, Bonatsos et al., 19 Nov 2024, Kota, 1 Oct 2025).