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Systematic study of E2 matrix elements in the framework of the Triaxial Projected Shell Model

This paper presents a systematic study of extended sets of E2 matrix elements for various nuclides using the Triaxial Projected Shell Model, successfully accounting for experimental energies and both intra- and inter-band transition properties.

Original authors: Stefan Frauendorf, Syed Rouoof, Gowhar Bhat, Sheikh Jehangir, Nazira Nazir, Kouser Qureshie, Niyaz Rather, Javid Sheikh

Published 2026-09-10

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CC BY 4.0

Original authors: Stefan Frauendorf, Syed Rouoof, Gowhar Bhat, Sheikh Jehangir, Nazira Nazir, Kouser Qureshie, Niyaz Rather, Javid Sheikh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

At the heart of every atom lies a nucleus, a dense cluster of protons and neutrons that defies the simple image of a perfect, round ball. For decades, physicists have understood that these nuclei can stretch into ellipses, much like a rugby ball, or flatten into disks. But a more subtle and complex possibility exists: the nucleus can twist into a three-sided shape, lacking a single axis of symmetry. This triaxial form is difficult to pin down because the nucleus is not a solid object but a quantum system, constantly fluctuating and vibrating. To understand these shapes, scientists measure how the nucleus interacts with electric fields, specifically looking at how it transitions between different energy levels. These measurements reveal the "stiffness" of the shape—whether it holds a rigid form or if it is "soft," wobbling freely between different configurations. Knowing the difference is crucial because the shape of the nucleus dictates how it spins, vibrates, and ultimately how it behaves in the universe.

A team of researchers, led by Stefan Frauendorf and colleagues, has taken a fresh, microscopic look at this problem using a powerful theoretical tool called the Triaxial Projected Shell Model. Instead of treating the nucleus as a smooth, collective fluid, this approach builds the nucleus from the bottom up, starting with the individual protons and neutrons and how they pair up and move. The team applied this method to a wide variety of atomic nuclei, including isotopes of Germanium, Ruthenium, Erbium, Osmium, and Platinum. Their goal was to see if this detailed, particle-based model could accurately reproduce the complex shapes and energy patterns observed in real experiments, particularly those involving the "gamma" band, a specific type of excited state that is highly sensitive to triaxiality.

The researchers found that their model works with remarkable precision. By adjusting the static triaxial deformation of the mean field, fixed by two parameters, as the only input to the experiment, they were able to calculate the energy levels and the electric transition strengths for all the studied nuclei. The results matched experimental data from Coulomb excitation experiments, a technique where nuclei are bombarded with heavy ions to probe their structure without breaking them apart. The model successfully described nuclei that behave as rigid triaxial shapes, those that are soft and fluctuating, and those that are strictly axial. A key discovery was that the model could explain the "softness" of certain nuclei not by assuming the shape is inherently unstable, but by including the effects of virtual excitations. These are fleeting, microscopic changes where pairs of protons or neutrons are briefly excited, and their mixing with the ground state creates the appearance of a wobbling, soft shape.

One of the most significant findings concerns how scientists determine whether a nucleus is rigid or soft. Traditionally, physicists have looked at the spacing of energy levels in the gamma band, using a pattern called "staggering" to make this judgment. If the energy levels follow a specific up-and-down pattern, the nucleus is considered rigid; if they follow a different pattern, it is considered soft. However, the researchers discovered a contradiction when they applied their microscopic model to the Osmium isotopes. While the energy staggering pattern suggested these nuclei were switching back and forth between soft and rigid as their mass changed, the direct calculation of their shape invariants—mathematical quantities derived from the electric transitions that measure the average shape and its fluctuations—showed them to be consistently rigid triaxial. This suggests that the traditional method of judging shape based solely on energy spacing can be misleading, as it may reflect complex interactions between different particle states rather than a simple change in the nucleus's physical shape.

To visualize what is happening, the researchers used a concept of shape invariants, which act like a statistical map of the nucleus's geometry. They calculated the average shape and the spread of possible shapes for each nucleus. For the Osmium isotopes, this map showed a stable, rigid triaxial form, confirming that the nucleus does not actually wobble between soft and rigid states despite what the energy levels might suggest. In contrast, for nuclei like Germanium, the model correctly identified a soft, fluctuating shape where the nucleus explores a wide range of deformations. The study demonstrates that the Triaxial Projected Shell Model is capable of capturing these subtle differences by accounting for the superposition of many different particle configurations. It shows that the complex behavior of the nucleus, including its apparent softness, emerges naturally from the interactions of its constituent particles, providing a unified and accurate description of nuclear structure that bridges the gap between simple collective models and the complex reality of the quantum world.

Technical Summary: Systematic Study of E2 Matrix Elements in the Triaxial Projected Shell Model

Problem Statement
The paper addresses the challenge of systematically describing nuclear shapes and collective excitations, specifically distinguishing between axial, rigid triaxial ( $\gamma$ -rigid), and soft triaxial ( $\gamma$ -soft) nuclei. While the Collective Model (CM) and the 5-Dimensional Bohr Hamiltonian (5DBH) provide a macroscopic framework for these phenomena, there is a need for a microscopic approach that can simultaneously account for rotational bands, $\gamma$ vibrations, and quasiparticle excitations. The study focuses on reproducing experimental electric quadrupole (E2) matrix elements and energy spectra for a wide range of nuclides (including Ge, Se, Mo, Ru, Pd, Er, Os, and Pt) obtained via Coulomb excitation (COULEX) experiments. A specific point of inquiry is the apparent contradiction between shape classifications derived from energy staggering patterns versus those derived from E2 matrix elements in certain isotopic chains (e.g., Osmium).

Methodology
The authors employ the Triaxial Projected Shell Model (TPSM), a microscopic framework that combines symmetry restoration with configuration mixing. The methodology proceeds as follows:

Basis Construction: The model constructs a basis from triaxially deformed quasiparticle states. This involves diagonalizing the triaxial Nilsson Hamiltonian (incorporating axial deformation $\varepsilon$ and non-axial deformation $\varepsilon'$ ) and incorporating pairing correlations via the BCS approximation.
Symmetry Restoration: Rotational symmetry is restored using a three-dimensional angular-momentum projection operator, $P^{I}_{MK}$ , which projects the triaxially deformed vacuum onto good angular momentum $I$ .
Configuration Mixing: The model space for even-even nuclei includes the projected quasiparticle vacuum, as well as two-proton, two-neutron, and four-quasiparticle configurations. These basis states are used to diagonalize a pairing-plus-quadrupole shell-model Hamiltonian.
Parameter Adjustment: The axial deformation is determined from experimental $B(E2)$ values or self-consistent mean-field calculations. The non-axial deformation is adjusted specifically so that the calculated $\gamma$ -band head reproduces the observed excitation energy. Effective charges ( $e_p = 1.5e$ , $e_n = 0.5e$ ) are used to account for core polarization.
Analysis via Shape Invariants: To classify nuclear shapes, the authors calculate Kumar-Cline shape invariants (second, third, fourth, and sixth order) derived from the reduced E2 matrix elements. These invariants provide mean values and fluctuations (dispersions) for the deformation parameters $\beta$ and $\gamma$ (or $\delta$ ).

Key Results
The TPSM calculations were applied to a systematic set of nuclides, including $^{70,72,74,76}$ Ge, $^{76,78,80,82}$ Se, $^{100}$ Mo, $^{104}$ Ru, $^{110}$ Pd, $^{168}$ Er, $^{186,188,190}$ Os, and $^{184}$ Pt.

Spectroscopic Agreement: The model successfully reproduces experimental energies for the ground band, quasi- $\gamma$ band, and excited $0^+$ bands, as well as individual intra-band and inter-band E2 matrix elements. The agreement is described as "remarkable" across nuclei classified as axial ( $^{168}$ Er), $\gamma$ -soft ( $^{72}$ Ge), and $\gamma$ -rigid (as exemplified by the Os isotopes and $^{184}$ Pt).
Role of Quasiparticles: A critical finding is that the vacuum configuration alone yields an "odd-I-down" staggering pattern characteristic of $\gamma$ -rigid nuclei. The inclusion of quasiparticle excitations is necessary to generate the "even-I-down" feature associated with $\gamma$ -softness.
Shape Invariants: The calculated shape invariants confirm that the TPSM describes the mean deformation and triaxiality of yrast and $\gamma$ bands accurately. For example, $^{104}$ Ru is classified as $\gamma$ -soft with a preference for prolate shape, while $^{168}$ Er is identified as rigid prolate.
Resolution of Contradictions: The study highlights a discrepancy in the classification of Osmium isotopes ( $^{186,188,190}$ Os). While the phenomenological 5DBH interprets the changing energy staggering pattern ( $S(I)$ ) as a transition between $\gamma$ -soft and $\gamma$ -rigid, the E2-derived shape invariants from the TPSM consistently classify all these isotopes as $\gamma$ -rigid. The authors attribute this to the different nature of excited $0^+$ bands in the microscopic model (superpositions of two-quasiparticle states) compared to the collective vibrational nature assumed in the 5DBH.

Significance and Claims
The paper claims that the TPSM provides a unified microscopic description of collective quadrupole excitations without requiring the nuclear shape to be a dynamical degree of freedom in the Hamiltonian. Instead, the model uses a static triaxial mean field as input.

The significance of the work lies in demonstrating that:

Microscopic Origin of Softness: Nuclei exhibiting $\gamma$ -soft characteristics can be described starting from a static triaxial mean field. The necessary shape fluctuations are generated microscopically through the superposition of multi-quasiparticle excitations (virtual excitations of quasiparticle pairs) rather than through explicit collective coordinate fluctuations.
Limitations of Phenomenological Models: The study suggests that the correlation between energy staggering patterns and $\gamma$ -softness, which holds in the phenomenological 5DBH, may not hold in microscopic approaches where excited bands have complex quasiparticle structures.
Predictive Power: The model achieves similar accuracy for nuclei not explicitly detailed in the figures (e.g., $^{70,74}$ Ge, $^{76,80,82}$ Se, $^{100}$ Mo, $^{110}$ Pd) as it does for the detailed examples, validating its systematic applicability.

The authors conclude modestly that while the current version of TPSM accounts well for the shape fluctuations encountered in the studied nuclei, it remains to be seen to what extent the model can account for more drastic shape changes.

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