⚛️ nuclear theory

B $_{4/2}$ anomaly related to the shape coexistence in $^{94}$ Zr

This paper utilizes the Bohr-Mottelson Hamiltonian with a sixth-order anharmonic oscillator potential to demonstrate that the $B_{4/2}$ anomaly and low-lying states of $^{94}$ Zr arise from a triple shape coexistence scenario involving spherical, axially-symmetric deformed, and $\gamma$ -unstable triaxial configurations.

Original authors: P. Buganu, R. Benjedi

Published 2026-10-22

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: P. Buganu, R. Benjedi

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

Inside the heart of every atom lies a nucleus, a dense cluster of protons and neutrons that behaves less like a rigid marble and more like a droplet of liquid. In the world of nuclear physics, these droplets can change their shape. Sometimes they are perfectly round, like a sphere. Other times, they stretch out into an oval, resembling a rugby ball, or flatten into a disk. For decades, physicists have built mathematical models to predict how these shapes vibrate and rotate, expecting the nucleus to settle into one stable form or transition smoothly between them. However, nature often surprises us with exceptions that defy these neat expectations. One such puzzle involves a specific measurement of how the nucleus emits energy, a value that sometimes drops to a surprisingly low number, breaking the patterns that standard theories predict. This anomaly, known as the B4/2 anomaly, has appeared in a handful of nuclei, leaving scientists to wonder what hidden complexity is driving this behavior.

A recent study focused on the nucleus of Zirconium-94, an isotope that exhibits this strange behavior, to uncover the source of the mystery. The researchers, working from institutions in Romania and Morocco, turned to a sophisticated framework known as the Bohr-Mottelson model, which treats the nucleus as a collective system of moving parts. Instead of using a simple, smooth potential to describe the forces holding the nucleus together, they employed a more complex, sixth-order potential. This mathematical tool allowed them to explore a wider range of possibilities, specifically looking for scenarios where the nucleus might not just vibrate in one shape, but actually host two or more distinct shapes at the same time. This phenomenon, called shape coexistence, suggests that a single nucleus can simultaneously support a spherical configuration and a deformed one, with the different shapes mixing together to create the observed energy levels.

The team applied their model to the lowest energy states of Zirconium-94, comparing their calculations against the most recent experimental data available. They found that a single, simple description of the nucleus was insufficient to explain the full picture. Instead, the data pointed toward a triple shape coexistence scenario. The lowest energy state, the ground state, behaves like a nearly perfect sphere. However, the first excited state with a spin of four, a specific type of angular momentum, clearly belongs to a different family: it is shaped like a stretched, axially symmetric oval. The transition of energy between these two specific states is what creates the anomaly. Because the nucleus has to jump from a spherical shape to a stretched oval shape, the probability of this transition is significantly suppressed, resulting in the unusually low value that puzzled scientists for years.

Yet, the story did not end there. When the researchers examined other excited states, specifically those with zero and two units of spin, the picture shifted again. These states did not fit the simple oval model. Instead, they behaved as if the nucleus was dynamically unstable, fluctuating in a way that suggests a triaxial deformation, where the shape is neither a perfect sphere nor a simple oval, but something more complex that wobbles in three dimensions. The study revealed that the second excited zero-spin state and the second excited two-spin state are dominated by this triaxial character, though they still retain a small mixture of the spherical shape. This mixing is crucial, as it explains why certain energy transitions between these states are so weak, matching the experimental observations that a single-shape model could not.

The researchers concluded that the key to understanding Zirconium-94 is not choosing between a spherical, an oval, or a triaxial shape, but recognizing that all three exist within the same nucleus. The anomaly in the energy transition is the direct result of the nucleus switching between the spherical ground state and the oval-shaped fourth state, separated by a significant energy barrier that makes the jump difficult. Meanwhile, the other states dance to the tune of a triaxial rhythm. This triple coexistence provides a complete explanation for the energy levels and the strength of the electromagnetic transitions observed in the lab. While the current model successfully reproduces most of the data, the researchers noted that some transitions remain difficult to describe perfectly, suggesting that a future, even more general model might be needed to treat all three shape configurations on equal footing. For now, however, the study offers a clear and compelling explanation for a long-standing nuclear puzzle, showing that the interior of an atom can be far more versatile and complex than previously imagined.

Technical Summary: B4/2 Anomaly and Triple Shape Coexistence in 94Zr

Problem Statement
The paper addresses the "B4/2 anomaly," a phenomenon observed in certain nuclei where the quadrupole transition strength $B(E2; 4^+_1 \to 2^+_1)$ is smaller than $B(E2; 2^+_1 \to 0^+_1)$ , resulting in a subunitary ratio $B_{4/2} < 1$ . This behavior contradicts predictions from traditional collective nuclear models. While recent studies suggest that triaxial deformation and band mixing contribute to this anomaly, they indicate that triaxiality alone is insufficient. The specific case of the $^{94}$ Zr nucleus presents a challenge: previous models (such as the Interacting Boson Model with Configuration Mixing) successfully predicted a spherical ground state and a triaxial-oblate $0^+_2$ state but could not calculate the $B_{4/2}$ anomaly due to model space limitations. The authors aim to reproduce the observed $B_{4/2}$ anomaly in $^{94}$ Zr (where the ratio $B_{4/2} = 0.181$ ) and elucidate its relationship with shape coexistence and triaxial deformation.

Methodology
The study employs solutions of the Bohr-Mottelson Hamiltonian utilizing a sixth-order anharmonic oscillator potential. Two distinct deformation scenarios are applied to the lowest states of $^{94}$ Zr:

Axially-symmetric deformation: Corresponding to a specific separation constant $\omega = L(L+1)/3$ .
$\gamma$ -unstable (dynamical triaxial) deformation: Corresponding to $\omega = \tau(\tau+3)$ , where $\tau$ is the seniority quantum number.

The potential $v(\beta)$ includes terms up to $\beta^6$ ( $v(\beta) = \beta^2 + b_1\beta^4 + b_2\beta^6$ ). The radial equation is solved numerically by diagonalizing the Hamiltonian in a basis of infinite square well solutions. The parameters $b_1$ and $b_2$ are fitted using a root-mean-square procedure to reproduce the reduced energies of the experimental states. Scaling factors ( $\hbar^2/2B$ and $\beta_M$ ) are subsequently fixed to match the experimental energy of the $2^+_1$ state and the $B(E2; 2^+_1 \to 0^+_1)$ transition strength, respectively. The study analyzes energy spectra, electromagnetic transition strengths ( $E2$ and $E0$ ), effective potentials, and probability density distributions of the deformation $\beta^2$ .

Key Results
The application of the two approaches yields a composite picture of the $^{94}$ Zr structure:

Axially-Symmetric Approach: This model successfully reproduces the energies and $B(E2)$ transitions for the $4^+_1$ and $4^+_2$ states. The analysis reveals a high potential barrier separating an approximately spherical minimum (associated with the $2^+_1$ state, $\beta^2 \approx 0.083$ ) and a well-deformed axially-symmetric minimum (associated with the $4^+_1$ state, $\beta^2 \approx 0.300$ ). This separation suppresses the $B(E2; 4^+_1 \to 2^+_1)$ transition strength, explaining the $B_{4/2}$ anomaly. The calculated value (0.42 W.u.) is close to the experimental value (0.87(5) W.u.).
$\gamma$ -Unstable Approach: This model provides a superior fit for the $0^+_2$ and $2^+_2$ states and their associated transitions. The probability density for the $0^+_2$ state shows a three-peak structure, indicating a mixing of spherical and $\gamma$ -unstable deformed components. This mixing explains the small monopole $E0$ transition and the strong $B(E2; 0^+_2 \to 2^+_1)$ transition. The $2^+_2$ state is predominantly $\gamma$ -unstable deformed.
Discrepancies: The models struggle to simultaneously describe the $4^+_2 \to 2^+_2$ transition, as the $4^+_2$ state is better described by the axially-symmetric approach while the $2^+_2$ state requires the $\gamma$ -unstable approach.

Significance and Claims
The paper proposes that the $B_{4/2}$ anomaly in $^{94}$ Zr is not solely a result of triaxiality but arises from a triple shape coexistence scenario involving:

An approximately spherical configuration (dominant in the $2^+_1$ state).
An axially-symmetric deformed configuration (dominant in the $4^+_1$ state).
A $\gamma$ -unstable (triaxial) deformed configuration (dominant in the $0^+_2$ and $2^+_2$ states).

The authors claim that this specific coexistence and mixing of three distinct shape configurations is essential for understanding the level structure and electromagnetic transition strengths in $^{94}$ Zr. While the current separate approaches successfully explain specific subsets of the data, the authors modestly conclude that a more general model capable of treating these three configurations on an equal footing is necessary to fully reproduce all inter- and intra-band transitions without overestimation.

More like this