The coexistence of possible magnetic and chiral rotation in $^{129}\mathrm{Cs}$ and $^{131}\mathrm{La}$: a microscopic investigation

Using three-dimensional tilted axis cranking covariant density functional theory, this study reveals the coexistence of magnetic and chiral rotational modes built on identical quasiparticle configurations in $^{129}\mathrm{Cs}$ and $^{131}\mathrm{La}$, establishing a new type of shape coexistence and uncovering a distinctive rotational mode transition from principal-axis to planar and finally to chiral rotation as frequency increases.

The Gist

Imagine the atomic nucleus not as a static ball of clay, but as a bustling, spinning dance floor where protons and neutrons are the dancers. For decades, physicists have watched these dancers and noticed two distinct ways they move together: Magnetic Rotation and Chiral Rotation.

This paper is like a high-tech surveillance report on two specific dance floors: Cesium-129 and Lanthanum-131. The researchers used a powerful computer simulation (a "microscopic investigation") to figure out exactly how these dancers are moving and, surprisingly, discovered that these two nuclei are doing both types of dances at the same time, depending on how fast they spin.

Here is the breakdown of the discovery using simple analogies:

1. The Two Dance Styles

To understand the paper, we first need to know the two "moves" the nucleus can do:

• Magnetic Rotation (The "Shear" Dance):
  Imagine a group of dancers holding hands in a circle. In a normal spin, they all lean outward together. But in this "magnetic" dance, the protons (positive dancers) lean one way, and the neutrons (neutral dancers) lean the opposite way. They create a "shear" force, like scissors closing. As they spin faster, they slowly close the scissors, generating energy without changing their shape much. This creates a specific pattern of light (radiation) that looks like a magnetic field.

  • The Paper's Finding: The researchers confirmed that the bands of energy labeled B8 in Cesium-129 and Band 13 in Lanthanum-131 are indeed performing this "Shear Dance."

• Chiral Rotation (The "3D Spiral" Dance):
  Now, imagine the dancers aren't just leaning left or right; they are twisting in 3D space. If you look at the nucleus from the front, it looks like a left-handed spiral. If you look at it from the back, it looks like a right-handed spiral. Because the universe treats left and right slightly differently in these quantum states, the nucleus creates a "twin" pair of energy bands that are almost identical but mirror images of each other. This is called Chirality (like your left and right hands).

  • The Paper's Finding: The researchers found evidence that these same nuclei might also be capable of this complex 3D spiral dance under different conditions.

2. The Big Discovery: "Shape Coexistence"

Usually, a nucleus picks one style of dance and sticks to it. But this paper reveals a fascinating phenomenon called Shape Coexistence.

Think of it like a person who can be a sprinter (fast, straight lines) and a figure skater (twisting, spinning) depending on the music.

• In Cesium-129 and Lanthanum-131, the nucleus has the exact same group of dancers (the same arrangement of protons and neutrons).
• However, depending on how fast the "music" (rotational frequency) plays, the nucleus can switch between the "Shear Dance" (Magnetic) and the "3D Spiral" (Chiral).

The researchers calculated the "shape" of the nucleus for both dances:

• For the Magnetic Dance: The nucleus is slightly flattened and tilted (like a pancake that's been pushed sideways).
• For the Chiral Dance: The nucleus is slightly more spherical but twisted (like a slightly squashed ball).

3. The Evolution: From Flat to Twisted

The most exciting part of the paper is the story of how the dance evolves as the nucleus spins faster. The researchers found a specific progression, like a story with three acts:

1. Act 1: The Principal Axis (The Standard Spin): At low speeds, the nucleus spins like a normal top, rotating around its main axis.
2. Act 2: The Planar Rotation (The Flat Spin): As it speeds up, it shifts into the "Shear" mode (Magnetic Rotation). The dancers are leaning, but they are still mostly in a flat plane.
3. Act 3: The Chiral Twist: If it spins even faster, it suddenly breaks out of that flat plane and starts twisting in 3D space, entering the "Chiral" mode.

It's as if the nucleus starts by spinning on a table, then leans over to slide across the table, and finally stands up to do a complex aerial twist.

4. How Did They Know? (The Tools)

The scientists didn't just guess; they used a super-advanced computer model called 3DTAC-CDFT.

• Think of this as a virtual reality simulator for the atomic nucleus.
• They programmed the simulator with the laws of physics (specifically Einstein's relativity and quantum mechanics) to see how the protons and neutrons would behave.
• They compared the simulator's output with real-world experiments (measuring how long the nucleus spins and what kind of light it emits).
• The Result: The simulation matched the real-world data almost perfectly. This confirmed that their theory about the "Shear Dance" and the potential "3D Spiral" is correct.

Summary: Why Does This Matter?

This paper is a breakthrough because it shows that the atomic nucleus is far more flexible and complex than we thought.

• It's a "Chameleon": The same nucleus can change its entire personality (shape and movement style) just by spinning faster.
• It connects two worlds: It proves that Magnetic Rotation and Chiral Rotation aren't separate, isolated phenomena; they can live side-by-side in the same atom.
• New Physics: It helps us understand the fundamental forces that hold the universe together, showing how tiny particles organize themselves into complex, spinning structures.

In short, Cesium-129 and Lanthanum-131 are the "chameleons of the atomic world," capable of performing two very different, exotic dances simultaneously, and this paper is the first to write down the choreography.

Technical Summary

1. Problem Statement

The study addresses the complex rotational phenomena in atomic nuclei, specifically focusing on the coexistence of magnetic rotation (MR) and chiral rotation (CR) within the same nucleus. While magnetic rotation (driven by the "shear mechanism" of proton and neutron angular momentum vectors) and chiral rotation (arising from triaxial deformation and the breaking of chiral symmetry) are distinct modes, their simultaneous existence in a single nucleus with identical quasiparticle configurations remains a challenging theoretical and experimental topic.

The authors focus on the isotones $^{129}\text{Cs}$ and $^{131}\text{La}$ (both with $N=74$). Previous experimental data suggested magnetic rotation in $^{129}\text{Cs}$ (Band B8) and $^{131}\text{La}$ (Band 13), while chiral candidates have been reported in neighboring isotopes. The specific problem is to theoretically verify:

1. Whether these specific bands exhibit magnetic rotation characteristics.
2. Whether chiral rotation can coexist with magnetic rotation in these nuclei based on the same underlying quasiparticle configuration ($\pi h_{11/2} \otimes \nu h_{11/2}^{-2}$).
3. The microscopic mechanisms governing the evolution of these rotational modes.

2. Methodology

The investigation employs the Three-Dimensional Tilted Axis Cranking Covariant Density Functional Theory (3DTAC-CDFT). This is a fully self-consistent, microscopic approach that treats the nucleus as a relativistic system.

• Theoretical Framework: The method solves the Dirac equation in an intrinsic frame rotating with a constant angular velocity vector $\vec{\omega}$. The orientation of $\vec{\omega}$ is determined self-consistently by minimizing the total Routhian, allowing for non-planar (tilted) rotation axes.
• Interaction: The PC-PK1 point-coupling density functional is used.
• Configurations: The study analyzes four specific valence nucleon configurations:
  • Config 1 & 2: Based on the experimental suggestion of $\pi h_{11/2} \otimes \nu h_{11/2}^{-2}$, but with different neutron hole distributions (some neutrons in $s_{1/2}/d_{3/2}$ orbitals vs. all in $h_{11/2}$).
  • Config 3 & 4: Self-consistent calculations yielding the same unpaired configuration ($\pi h_{11/2} \otimes \nu h_{11/2}^{-2}$) but with specific occupation patterns of the neutron core ($s_{1/2}/d_{3/2}$ orbitals filled).
• Observables Calculated:
  • Excitation energy spectra and spin vs. rotational frequency ($\hbar\omega$) relations.
  • Deformation parameters ($\beta, \gamma$).
  • Electromagnetic transition probabilities: $B(M1)$, $B(E2)$, and their ratios.
  • Angular momentum decomposition along the principal axes ($s, m, l$) to identify shear mechanisms and chiral geometry.
  • Orientation angles ($\theta, \phi$) of the total angular momentum vector.

3. Key Contributions

• Discovery of Shape Coexistence: The paper establishes a new type of shape coexistence where magnetic rotation and chiral rotation coexist in the same nucleus ($^{129}\text{Cs}$ and $^{131}\text{La}$) built upon identical quasiparticle configurations ($\pi h_{11/2} \otimes \nu h_{11/2}^{-2}$).
• Microscopic Mechanism Elucidation: It provides a detailed microscopic explanation of how the neutron occupancy in the $h_{11/2}$ shell versus the $s_{1/2}/d_{3/2}$ shell dictates the transition between different rotational modes.
• Dynamic Evolution of Rotational Modes: The study uncovers a distinctive rotational mode transition sequence as rotational frequency increases:
  1. Principal-axis rotation.
  2. Planar rotation.
  3. Chiral rotation (aplanar).
• Validation of Magnetic Bands: It confirms the magnetic nature of Band B8 in $^{129}\text{Cs}$ and Band 13 in $^{131}\text{La}$ through the reproduction of experimental energy spectra and transition probabilities without adjustable parameters.

4. Key Results

A. Magnetic Rotation Characteristics

• Deformation: The magnetic bands are characterized by oblate shapes with triaxiality.
  • $^{129}\text{Cs}$: $\beta \approx 0.23, \gamma \approx 41^\circ$.
  • $^{131}\text{La}$: $\beta \approx 0.25, \gamma \approx 42^\circ$.
• Shear Mechanism: The calculations reveal the "shear mechanism" where proton angular momentum ($J_\pi$) aligns along the short axis and neutron hole angular momentum ($J_\nu$) aligns along the long axis. As frequency increases, these vectors "close" like a shear pair, increasing total spin while maintaining the total angular momentum direction.
• Transition Probabilities: The calculated $B(M1)$ values are large and show a decreasing trend (or stability in $^{129}\text{Cs}$), while $B(E2)$ values are small. The $B(M1)/B(E2)$ ratios match experimental data well, confirming the magnetic nature of these bands.

B. Chiral Rotation Characteristics

• Deformation: The chiral candidates exhibit slightly different deformations:
  • $^{129}\text{Cs}$: $\beta \approx 0.20, \gamma \approx 29^\circ$.
  • $^{131}\text{La}$: $\beta \approx 0.20, \gamma \approx 27^\circ$.
• Critical Frequency & Geometry:
  • At low frequencies, the system undergoes principal-axis rotation ($\phi \approx 0, \theta \approx 90^\circ$).
  • A critical frequency ($\hbar\omega_{crit}$) is identified ($\approx 0.33$ MeV for $^{129}\text{Cs}$, $\approx 0.37$ MeV for $^{131}\text{La}$).
  • Beyond this frequency, the system transitions to aplanar rotation (non-zero $\phi$ and $\theta$), indicating the onset of static chirality.
• Angular Momentum: The emergence of non-zero angular momentum along the intermediate ($m$) axis is the signature of chiral geometry. The calculations show that the proton particle in $h_{11/2}$ contributes to the short axis, while the neutron holes contribute to the long axis, with collective effects populating the intermediate axis.

C. Comparison with Experiment

• The calculated excitation energies and spin-frequency relations for the magnetic bands agree excellently with experimental data.
• The theoretical $B(M1)/B(E2)$ ratios are slightly overestimated but considered acceptable given the absence of pairing correlations in the calculation (which are known to reduce $B(M1)$ values).

5. Significance

• Theoretical Advancement: This work demonstrates the power of 3DTAC-CDFT in describing complex rotational modes and shape coexistence without phenomenological adjustments. It bridges the gap between magnetic and chiral rotation, showing they are not mutually exclusive but can coexist in the same nuclear configuration.
• Nuclear Structure Insight: It clarifies the role of neutron occupancy (specifically the interplay between $h_{11/2}$ and $s_{1/2}/d_{3/2}$ orbitals) in driving the transition from planar to chiral rotation.
• Experimental Guidance: The study predicts the existence of chiral doublet bands in $^{129}\text{Cs}$ and $^{131}\text{La}$ (specifically suggesting a possible M$\chi$D or magnetic-chiral doublet in $^{129}\text{Cs}$), providing a clear target for future experimental verification.
• Mass Region Understanding: It contributes significantly to the understanding of the "chiral island" in the $A \approx 130$ mass region, a region known for rich structural phenomena.

In conclusion, the paper successfully identifies $^{129}\text{Cs}$ and $^{131}\text{La}$ as prime candidates for the coexistence of magnetic and chiral rotation, offering a comprehensive microscopic explanation of how these exotic modes evolve and interact within the same nuclear system.