Systematic Cranked Shell Model Calculations for $^{87, 89, 91}$Br

This paper employs systematic cranked shell model calculations to successfully reproduce experimental rotational properties and elucidate the shape evolution, including $\gamma$-softness and shape coexistence, of odd-mass neutron-rich bromine isotopes $^{87, 89, 91}$Br.

The Gist

Imagine an atomic nucleus not as a static, hard marble, but as a soft, squishy blob of dough made of tiny particles called protons and neutrons. Now, imagine you start spinning this blob of dough on your finger.

This paper is about what happens to a specific family of these "dough balls" (called Bromine isotopes) when you spin them faster and faster. The scientists used a super-computer to simulate this spinning and figured out how the shape of the dough changes, how it holds together, and what happens when the particles inside start to rearrange themselves.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: The Spinning Dough

The researchers looked at three specific types of Bromine atoms (numbers 87, 89, and 91). These are "odd" atoms because they have one extra unpaired proton, like a single sock in a drawer of pairs.

They used a mathematical model called the Cranked Shell Model. Think of this as a high-tech simulation where they:

• Crank the handle: They increase the spinning speed (rotational frequency).
• Watch the shape: They see if the dough stays round, stretches into a football (prolate), flattens into a pancake (oblate), or gets wobbly in the middle (triaxial).

2. The Shape-Shifting Drama

The most exciting part of the paper is how the shape of these atoms changes as they spin faster. It's like watching a gymnast change their pose mid-routine.

• The "Magic" One (85Br): This atom has a "magic number" of neutrons (50), which makes it very stable. It's like a perfectly round basketball. Even when you spin it, it tries to stay round because its internal structure is so strong.
• The Stretchers (87Br & 89Br): As you add a few more neutrons, the dough becomes a bit more flexible. At slow speeds, they are a bit wobbly (gamma-soft), meaning they can easily squish into different shapes. But as they spin faster, they stretch out into a football shape (prolate).
• The Pancakes (91Br & 93Br): Here is the twist! As you get to the heavier versions, the dough doesn't stretch; it flattens out into a pancake (oblate).
  • The Big Reveal: The scientists found a "tipping point" at a specific number of neutrons (N=56). Before this point, the atoms stretch like footballs. After this point, they flatten like pancakes. It's like a magic switch that flips the entire family's shape preference.

3. The Dance of the Particles (Quasiparticles)

Inside the spinning dough, the protons and neutrons are dancing.

• The "Blocking" Effect: Because these atoms have one extra unpaired proton, that proton acts like a dance partner who refuses to let go of their spot. It "blocks" the other particles from moving freely. This locks the overall shape of the atom.
• The Neutron Team: Even though the proton is the boss holding the shape, the neutrons are the ones doing the heavy lifting to generate the spin. As the atom spins faster, the neutrons start to line up (align) with the spin, kind of like a crowd of people turning to face the same direction in a stadium wave.
• The "Backbend": At very high speeds, the scientists predict a moment where the neutrons suddenly snap into a new alignment. Imagine a group of dancers suddenly breaking formation to do a synchronized flip. This causes a sudden jump in how much energy the atom can hold.

4. Why Does This Matter?

You might ask, "Why do we care about spinning dough balls?"

• Understanding the Rules: This helps scientists understand the fundamental rules of how matter behaves under extreme conditions. It's like learning the physics of a spinning top, but for the building blocks of the universe.
• Predicting the Future: The model they used is very accurate. It didn't just explain what we already knew; it predicted that at even higher speeds (which we haven't measured yet), the atoms will undergo a dramatic shape change.
• The "Missing" Data: The paper explains why we haven't seen these high-speed states in experiments yet. It's like trying to see a hummingbird's wings with the naked eye; the energy required to spin them that fast breaks the "pairing" of the neutrons, causing the atom to fall apart or scatter its energy in a way that is hard to catch.

The Bottom Line

This paper is a roadmap for the shape-shifting world of atomic nuclei. It tells us that as you spin these atoms faster, they don't just spin faster; they fundamentally change their identity from round to football-shaped, and then to pancake-shaped. The scientists successfully mapped out this journey, proving that their computer models can predict these complex dances of subatomic particles with great precision.

In short: Spin the atom, and watch it morph.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the complex evolution of nuclear shapes and rotational dynamics in neutron-rich odd-mass bromine isotopes ($^{87,89,91}\text{Br}$). While the interplay between collective rotation and single-particle structure is well-studied in stable nuclei, exotic neutron-rich regions present challenges regarding:

• Shape Evolution: The transition from spherical to deformed shapes, and specifically the shift from prolate to oblate deformation as neutron number ($N$) increases toward 56.
• Band Assignments: The difficulty in assigning specific quasiparticle configurations to observed rotational bands in odd-Z nuclei where an unpaired proton couples with a deformed core.
• High-Spin Behavior: Understanding the mechanisms behind angular momentum alignment (backbending) and the role of neutron pairing in odd-even nuclei.

Previous studies using Large Scale Shell Model (LSSM) and Discrete Nonorthogonal (DNO) models have suggested a prolate-to-oblate transition at $N=56$, but a microscopic description using the Cranked Shell Model (CSM) with configuration constraints was needed to explicitly link these shape changes to specific orbital occupations.

2. Methodology

The authors employed the Configuration-Constrained Cranked Shell Model (CSM) combined with the Strutinsky shell-correction method. The theoretical framework involves:

• Hamiltonian: The model uses a triaxial Woods-Saxon potential with universal parameters to generate the single-particle spectrum. The total Hamiltonian includes the Woods-Saxon potential, a pairing interaction (treated via BCS equations), and the cranking term ($-\omega J_x$) to simulate rotation.
• Total Energy Calculation: The total energy ($E_{TOT}$) is calculated as the sum of the rotating liquid-drop energy ($E_{RLDM}$) and the microscopic shell correction ($\delta E$).
  • $E_{RLDM}$ accounts for surface and Coulomb energies of a triaxially deformed shape.
  • $\delta E$ is the sum of the shell correction (Strutinsky smoothing of single-particle levels) and the pairing correction.
• Configuration Constraints: To study specific rotational bands, the occupancy of specific quasiparticle orbitals (blocking) is fixed. The Total Routhian Surface (TRS) is then minimized with respect to deformation parameters ($\beta_2, \gamma$) at fixed rotational frequencies ($\omega$).
• Observables: The study calculates:
  • Total Routhian Surfaces (TRS): To determine equilibrium shapes ($\beta_2, \gamma$) and shape coexistence.
  • Kinematic Moments of Inertia ($J^{(1)}$): To compare rotational response with experiment.
  • Aligned Angular Momentum ($\langle J_x \rangle$): To analyze the contribution of protons vs. neutrons to angular momentum generation.

3. Key Contributions

• Systematic CSM Analysis: Provides a comprehensive CSM study of the $^{87,89,91}\text{Br}$ isotopic chain, explicitly constraining the odd proton to the $\pi g_{9/2}$ orbital to isolate the effects of the even-N core.
• Shape Evolution Mapping: Successfully reproduces the transition from spherical ($N=50$) to prolate ($N=52, 54$) and finally to oblate ($N=56$) shapes, validating the pseudo-SU(3) regime suggested by other models.
• Mechanism of Alignment: Clarifies that in these odd-Z nuclei, the structural "skeleton" is defined by the blocked proton, but the angular momentum alignment is driven almost entirely by the even-N core neutrons.
• Prediction of High-Spin Features: Predicts a sharp up-bend in aligned angular momentum at high frequencies ($\hbar\omega > 0.45$ MeV) due to the alignment of a neutron pair, explaining why these states are difficult to populate experimentally.

4. Key Results

A. Shape Evolution (Total Routhian Surfaces)

• $^{85}\text{Br}$ ($N=50$): Remains spherical at low frequencies due to the robust $N=50$ shell closure.
• $^{87}\text{Br}$ ($N=52$): Exhibits a prolate shape ($\beta_2 \approx 0.17$) with pronounced $\gamma$-softness (flat energy surface regarding triaxiality) at low frequencies, stabilizing as rotation increases.
• $^{89}\text{Br}$ ($N=54$): Shows prolate deformation ($\beta_2 \approx 0.18$) at low frequencies but transitions to triaxial shapes ($\gamma \approx 45^\circ\text{--}60^\circ$) at higher frequencies ($\omega = 0.6$ MeV).
• $^{91}\text{Br}$ and $^{93}\text{Br}$ ($N=56, 58$): Display a clear prolate-to-oblate transition. At low frequencies, they favor oblate shapes ($\beta_2 \approx -0.2$). This aligns with recent DNO shell-model calculations identifying a pseudo-SU(3) driven oblate minimum.
• Comparison: The CSM results for ground-state shapes agree well with DNO calculations but differ from global macroscopic models (like FRDM), which predict persistent prolate shapes, highlighting the importance of microscopic single-particle effects.

B. Band Assignments and Alignment

• Configuration: The positive-parity bands in $^{87,89,91}\text{Br}$ are assigned to the one-quasiproton configuration 'a' ($\pi g_{9/2}[440]1/2^+$).
• Decoupled Behavior: The calculated aligned angular momentum ($\langle J_x \rangle$) and kinematic moment of inertia ($J^{(1)}$) match experimental data up to the current limits ($21/2^+$).
• Neutron-Driven Collectivity: The analysis reveals that the proton contribution to alignment is nearly constant (due to blocking), while the neutron contribution drives the increase in alignment. This confirms that the even-N core is responsible for the rotational collectivity.
• High-Spin Prediction: The model predicts a sharp increase (up-bend) in $\langle J_x \rangle$ and $J^{(1)}$ at $\hbar\omega \approx 0.5$ MeV. This is attributed to the alignment of a pair of $g_{9/2}$ neutrons (band crossing).
  • Significance: The lack of experimental data beyond this point is explained by the high energetic cost of breaking the neutron pair and the subsequent fragmentation of $\gamma$-decay intensity, making these states hard to observe in fission experiments.

5. Significance and Conclusion

This work demonstrates that the Configuration-Constrained CSM is a robust tool for describing rotational dynamics in neutron-rich odd-mass nuclei.

• It successfully explains the shape coexistence and the prolate-to-oblate transition near $N=56$, a critical region for understanding shell evolution away from stability.
• It provides a transparent microscopic interpretation of quasiparticle configurations, confirming that the odd proton dictates the band structure while the even neutron core drives the rotational alignment.
• The prediction of a neutron-pair alignment crossing offers a physical explanation for the current experimental limits in high-spin spectroscopy of these isotopes, motivating future high-statistics experiments to probe these high-spin frontiers.

The study bridges the gap between low-spin shell effects and high-spin collective behavior, offering a coherent picture of structural evolution in the $A \approx 80\text{--}90$ mass region.