Study of the shape coexistence in the 96Zr, 96Mo, 96Ru isobars

This study investigates shape coexistence and mixing in the stable $^{96}$Zr, $^{96}$Mo, and $^{96}$Ru isobars by combining Covariant Density Functional Theory for ground-state deformations with a Bohr-Mottelson Hamiltonian featuring an octic potential for excited states, revealing that these phenomena significantly influence the nuclear structure near the Z=40 and N=50 shell closures.

The Gist

Imagine the atomic nucleus not as a solid, unchanging marble, but as a drop of liquid that can squish, stretch, and twist into different shapes. Sometimes, a single nucleus can be "confused" about its shape, existing in two different forms at the same time. This strange phenomenon is called shape coexistence.

In this study, researchers looked at three specific "twins" of atoms—nuclei with the same total weight but different recipes of protons and neutrons: Zirconium-96, Molybdenum-96, and Ruthenium-96. They wanted to see if these twins were suffering from this shape confusion and, if so, how they behaved.

To solve this mystery, the team used two different "lenses" or tools:

1. The Microscopic Lens (CDFT): Think of this as a high-powered microscope that looks at the individual particles (protons and neutrons) inside the nucleus. It calculates the "energy landscape"—a map showing where the nucleus feels most comfortable resting.
2. The Phenomenological Lens (Bohr-Mottelson Hamiltonian): This is more like a mathematical model of a vibrating drum or a wobbling jelly. It describes how the nucleus moves and vibrates when it's excited, using a special "octic potential" (a fancy mathematical hill with two valleys) to see if the nucleus can sit in one valley or jump between them.

Here is what they found for each of the three atomic twins:

1. Zirconium-96: The "Split Personality"

• The Shape: The microscope showed this nucleus likes to be slightly flattened (oblate), like a pancake.
• The Behavior: When they looked at the excited states (when the nucleus is wiggling), they found two distinct "valleys" in the energy landscape. One valley is for a nearly round shape, and the other is for a more stretched shape.
• The Twist: The ground state (the calm, resting state) sits in the rounder valley, while the first excited state sits in the stretched valley. Crucially, there is a high "wall" between them. Because the wall is so high, the two shapes don't mix much; they stay separate. It's like having two people in the same house who live on different floors and never talk to each other. This is shape coexistence without mixing.

2. Molybdenum-96: The "Shape Shifter"

• The Shape: This nucleus is "triaxial," meaning it's not just a sphere or a simple pancake; it's a bit lopsided and unstable, like a wobbly spinning top.
• The Behavior: Here, the two valleys in the energy landscape are much closer together, and the wall between them is lower.
• The Twist: The nucleus doesn't just sit in one shape; it mixes them. The excited states are a blend of a round shape and a deformed shape. As the nucleus spins faster (gains more energy), it actually undergoes a "shape transition." It starts looking round, then wobbles through a critical point where it's undecided, and finally settles into a more deformed shape. It's like a dancer who starts with a slow, round movement and gradually transitions into a sharp, stretched pose.

3. Ruthenium-96: The "Confused Wobbler"

• The Shape: This one is tricky. It looks almost round (spherical) but acts a bit like a wobbly, unstable shape (gamma-unstable).
• The Behavior: The energy levels of this nucleus didn't follow the usual rules for a spinning top. Instead of getting harder to spin as it spins faster, the energy gaps actually shrank.
• The Twist: Like Molybdenum, this nucleus shows shape coexistence with mixing. The ground state is a mix of a round shape and a deformed one. The researchers found that the probability of the nucleus being in a certain shape changes as you look at higher energy levels, suggesting a dynamic dance between being round and being wobbly.

The Big Picture

The main takeaway is that these three nuclei, which are neighbors in the periodic table, all show evidence of shape coexistence, but they handle it differently:

• Zirconium keeps its shapes separate (no mixing).
• Molybdenum and Ruthenium blend their shapes together (mixing).

The study confirms that these nuclei aren't static balls; they are dynamic systems that can exist in multiple shapes simultaneously or transition between them as they gain energy. The researchers used their two mathematical tools to map out these "energy valleys" and "walls," proving that the complex dance of protons and neutrons creates these fascinating shape-shifting behaviors.

Technical Summary

Technical Summary: Study of the shape coexistence in the 96Zr, 96Mo, 96Ru isobars

Problem Statement
The paper investigates the phenomenon of shape coexistence and mixing in three stable isobars: $^{96}\text{Zr}$, $^{96}\text{Mo}$, and $^{96}\text{Ru}$. These nuclei are located near the harmonic oscillator proton shell closure ($Z=40$) and the spin-orbit neutron shell closure ($N=50$). This mass region is characterized by a competition between shell closures, which facilitates shape coexistence via a "dual-shell mechanism." While previous studies have suggested this scenario for these nuclei, the authors aim to provide a more accurate and comprehensive view by employing two complementary theoretical approaches to analyze ground state deformations and excited state properties.

Methodology
The study utilizes a dual-approach framework combining a microscopic model and a phenomenological model:

1. Covariant Density Functional Theory (CDFT):

   • Framework: Relativistic Hartree–Bogoliubov (RHB) approach.
   • Interaction: Density-dependent point-coupling interaction (DD-PCX).
   • Application: Used to determine the ground-state potential energy surfaces (PES) and extract ground-state deformation parameters ($\beta_2, \gamma$). Pairing correlations are handled self-consistently using a separable pairing force in momentum space.

2. Bohr-Mottelson Hamiltonian with Octic Potential (BMH-OP):

   • Framework: A phenomenological collective model using the Bohr-Mottelson Hamiltonian.
   • Potential: An octic potential ($V(\beta) \propto \beta^2 + b_1\beta^4 + b_2\beta^6 + b_3\beta^8$) is applied to the $\beta$ degree of freedom.
   • Symmetry: The model is applied to both axially symmetric (prolate/oblate) and $\gamma$-unstable deformations.
   • Solution Method: The Hamiltonian is solved numerically using a basis of Bessel functions of the first kind, allowing for the description of shape coexistence with and without mixing.
   • Observables: The model calculates energy spectra, $B(E2)$ electromagnetic transition probabilities, and monopole $E0$ transition strengths. Parameters are fitted to experimental data using a root-mean-square procedure.

Key Contributions and Results

• Ground State Deformations:

  • $^{96}\text{Zr}$: CDFT calculations indicate an approximately oblate ground state ($\beta_2 \approx 0.20, \gamma \approx 48^\circ$).
  • $^{96}\text{Mo}$: CDFT indicates a triaxial deformation with a wide potential well.
  • $^{96}\text{Ru}$: CDFT indicates a prolate shape ($\beta_2 \approx 0.10, \gamma = 0^\circ$), though the deformation is small, suggesting proximity to a spherical limit.

• $^{96}\text{Zr}$ Analysis (Axially Symmetric/Oblate):

  • The excited states are described using an axially symmetric (oblate) BMH-OP approach.
  • The effective potential exhibits two minima: a near-spherical one and a well-deformed one.
  • Shape Coexistence: The ground state ($0^+_1$) is positioned above the less-deformed minimum, while the first excited $0^+_2$ state corresponds to the well-deformed minimum. The high barrier between minima results in negligible mixing.
  • Observables: The monopole $E0$ transition is small due to the lack of mixing. The $2^+_1$ and $2^+_2$ states show probability density distributions with two peaks, indicating shape coexistence with mixing for these higher states.

• $^{96}\text{Mo}$ Analysis ($\gamma$-Unstable):

  • The nucleus is treated as $\gamma$-unstable. The energy difference $\Delta E = E(0^+_2) - E(2^+_1)$ is calculated as 459 keV (experimental: 370 keV), satisfying the criterion ($<800$ keV) for an "island of shape coexistence."
  • Shape Coexistence: The $0^+_2$ state probability density shows three peaks, suggesting coexistence with mixing between a less-deformed and a well-deformed shape.
  • Dynamical Shape Transition: The ground band exhibits a shape transition as spin increases. The $0^+_1$ and $2^+_1$ states are centered on the less-deformed minimum, while higher spin states ($4^+_1$ through $8^+_1$) transition to the well-deformed minimum.
  • Transitions: The model reproduces the contraction in $B(E2)$ strength for the $4^+_1 \to 2^+_1$ transition and the large $B(E2; 0^+_2 \to 2^+_1)$ value, attributing these to the mixing of shapes.

• $^{96}\text{Ru}$ Analysis ($\gamma$-Unstable):

  • Despite the CDFT prediction of a prolate shape, the prolate BMH-OP model failed to reproduce the experimental energy spectrum (which shows energy contraction rather than rotor-like expansion).
  • The $\gamma$-unstable BMH-OP model successfully reproduces the energy levels and the contraction trend in the ground band.
  • Shape Coexistence: The effective potential shows two minima (near-spherical and well-deformed). The ground state exhibits a two-peak probability density, while the $0^+_2$ state shows a mirror-image two-peak structure. This indicates shape coexistence with mixing between an approximately spherical shape and a $\gamma$-unstable one.
  • Discrepancies: The model predicts a forbidden $B(E2; 2^+_\gamma \to 0^+_1)$ transition, whereas experiment shows a large value (35 W.u.), which the authors note deviates from the systematic trend of the Ru isotopic chain.

Significance and Claims
The authors claim that the combined application of CDFT and BMH-OP provides a consistent description of the low-lying collective states in these isobars. The study highlights that shape coexistence and mixing phenomena significantly influence the nuclear structure in this region, leading to non-standard behaviors such as:

1. Contraction in Energy and $B(E2)$: Observed in the ground bands of $^{96}\text{Mo}$ and $^{96}\text{Ru}$, contrary to typical rotational behavior.
2. Dynamical Shape Transition: Specifically identified in $^{96}\text{Mo}$, where the ground band evolves from an approximately spherical shape to a more deformed shape as excitation energy increases, crossing a point of nearly degenerate minima.
3. Model Versatility: The octic potential within the BMH-OP framework is shown to be effective in describing both shape coexistence with and without mixing, as well as the transition between different shape regimes, offering a general solution compared to quasi-exact methods.

The paper concludes that these phenomena lead to a structure of states that differs from standard expectations, reinforcing the utility of the dual-approach methodology for elucidating shape coexistence in nuclei near shell closures.