Shape phase transition, coexistence and mixing in the $^{98-106}$Ru isotopes

This study investigates the deformation properties of the $^{98-106}$Ru isotopes using Covariant Density Functional Theory and a phenomenological Bohr-Mottelson Hamiltonian, revealing a shape phase transition, evidence of shape coexistence and mixing among various configurations, and explaining $\gamma$-band staggering through these mixing effects.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: Nuclear Shape-Shifting

Imagine an atomic nucleus not as a rigid, hard marble, but as a soft, squishy blob of dough. Depending on how much energy is in the blob and how many "ingredients" (protons and neutrons) it has, this dough can change its shape.

Sometimes it's a perfect sphere (like a ball). Sometimes it stretches into a football (prolate). Sometimes it gets a bit lopsided or wobbly (triaxial or $\gamma$-unstable).

This paper is a study of a specific family of these "dough blobs": the Ruthenium (Ru) isotopes, ranging from mass 98 to 106. The scientists wanted to answer three big questions:

1. Shape Phase Transition: As we add more neutrons (ingredients) to the dough, does it suddenly snap from a ball to a football?
2. Shape Coexistence: Can the nucleus be two different shapes at the same time?
3. Mixing: Do these shapes blend together, like mixing blue and yellow paint to get green?

The Tools: Two Different Lenses

To understand these shapes, the researchers used two different "lenses" or models to look at the data:

1. The Microscope (CDFT): This is a high-tech, microscopic view. It looks at the individual particles (nucleons) inside the nucleus and calculates how they interact. It's like looking at the dough under a microscope to see the individual flour and water molecules.
2. The Macro Lens (Bohr-Mottelson Hamiltonian with an Octic Potential): This is a mathematical model that treats the nucleus as a whole, vibrating object.
   • The "Octic" Part: Imagine the energy landscape of the nucleus as a bowl. Usually, scientists use simple bowls (like a parabola). Here, they used a very complex, 8th-degree polynomial bowl (an "octic" potential). Think of this as a bowl with a very specific, wiggly shape that can have flat bottoms, steep sides, or even two dips (double-well) to allow for shape mixing.
   • Two Modes: They ran this model in two modes:
     • $\gamma$-Unstable: The nucleus is wobbly and can spin around easily in different directions (like a spinning top that isn't perfectly balanced).
     • Prolate ($\gamma$-Stable): The nucleus is a stable, rigid football.

The Journey: What They Found

The researchers looked at the Ruthenium family from lightest (98) to heaviest (106). Here is what they discovered, step-by-step:

1. The "Wobbly" Start (98Ru and 100Ru)

The lighter isotopes (98 and 100) are like the dough just starting to stretch. They aren't perfect spheres, but they aren't rigid footballs either. They are wobbly.

• The Finding: The data fits best with the "wobbly" ($\gamma$-unstable) model. The nucleus is fluctuating between shapes.
• The Analogy: Imagine a jelly on a plate. It's mostly round, but if you poke it, it jiggles and changes shape easily.

2. The "Double-Identity" Crisis (102Ru and 104Ru)

As we get to the middle of the family (102 and 104), things get weird. The nucleus seems to be having an identity crisis.

• The Finding: Some states (energy levels) look like they are wobbly, while others look like they are rigid footballs.
• The Analogy: This is Shape Coexistence. Imagine a person who is a professional athlete in the morning (rigid football) but a clumsy dancer in the afternoon (wobbly). The nucleus is doing both.
• The Mixing: The scientists found that the "wobbly" shape and the "rigid" shape are mixing. They aren't just switching back and forth; they are blending. It's like the nucleus is a "shape-shifter" that exists in a superposition of being both a ball and a football simultaneously.

3. The "Rigid" End (106Ru)

By the time we reach the heaviest isotope (106), the dough has settled.

• The Finding: The nucleus has become a more stable, deformed football (prolate).
• The Analogy: The jelly has finally set into a firm, stretched-out shape. It's no longer wobbly; it's a stable rugby ball.

The "Staggering" Mystery

One of the coolest things they found involves the $\gamma$-band (a specific set of excited states).

• The Puzzle: In a perfect, rigid football, the energy levels of these states should follow a smooth, predictable pattern. But in Ruthenium, they "stagger" (go up and down like a zigzag).
• The Surprise: Usually, this zigzag pattern is a sign of a "wobbly" nucleus. However, the researchers found that even when the nucleus looks like a stable football (using the Prolate model), it still shows this zigzag pattern.
• The Explanation: This happens because of the mixing. Even though the nucleus is mostly a football, the "ghost" of the wobbly shape is still there, mixing with the football shape and causing the energy levels to zigzag. It's like a rigid drum that still hums with a wobbly vibration because it's made of mixed materials.

The Conclusion: It's Not Black and White

The main takeaway from this paper is that nature is messy and complex. You can't just say "This nucleus is a ball" or "That nucleus is a football."

• Complementary Views: To understand Ruthenium, you need to look at it through both lenses (the wobbly model and the rigid model) at the same time.
• The Spectrum: The Ruthenium chain shows a smooth transition from a wobbly sphere to a rigid football, but along the way, the nucleus spends a lot of time in a "gray area" where shapes coexist and mix.

In short: The nucleus is like a piece of clay being molded. Sometimes it's a ball, sometimes a football, and sometimes it's a weird, half-ball/half-football hybrid. The scientists used advanced math to prove that this "hybrid" state is real and is the key to understanding how atomic nuclei evolve.

Technical Summary

Here is a detailed technical summary of the paper "Shape phase transition, coexistence and mixing in the 98−106Ru isotopes".

1. Problem Statement

The study addresses the complex nuclear structure of the even-even Ruthenium (Ru) isotopic chain ($^{98-106}$Ru). These nuclei are known to exhibit intricate behaviors near critical points of nuclear shape phase transitions, specifically involving:

• Shape Phase Transitions: The evolution from spherical vibrators to $\gamma$-unstable (triaxial soft) or $\gamma$-stable prolate rotors as neutron number increases.
• Shape Coexistence and Mixing: The phenomenon where low-lying excited states (often $0^+$ states) possess shapes distinct from the ground state, leading to bands with different deformation characteristics coexisting within the same nucleus.
• Model Limitations: Previous studies often relied on single-limit descriptions (e.g., pure $\gamma$-unstable or pure prolate) or quasi-exactly solvable methods that restrict the generality of the potential. There is a need for a unified framework that can simultaneously describe phase transitions, coexistence, and mixing, particularly in the context of the octic (eighth-order) potential in the Bohr-Mottelson Hamiltonian.

2. Methodology

The authors employed a dual-approach strategy combining microscopic and phenomenological models:

A. Microscopic Approach: Covariant Density Functional Theory (CDFT)

• Framework: Relativistic Hartree-Bogoliubov (RHB) formalism.
• Interaction: Density-Dependent Point-Coupling (DD-PCX) parametrization.
• Purpose: To calculate the ground-state potential energy surfaces (PES) in the $(\beta_2, \gamma)$ plane. This provides a microscopic baseline for the deformation parameters ($\beta_2, \gamma$) of the isotopes, serving as a guide for the phenomenological models.

B. Phenomenological Approach: Bohr-Mottelson Hamiltonian (BMH) with Octic Potential

The study utilizes the Bohr-Mottelson Hamiltonian with an octic potential ($V(\beta) \sim \beta^2 + b_1\beta^4 + b_2\beta^6 + b_3\beta^8$) in the $\beta$ variable. The solution is split into two distinct symmetry cases:

1. Octic & $\gamma$-unstable:
   • Assumes a $\gamma$-independent potential (flat in $\gamma$).
   • Solves the $\gamma$-equation using the $SO(5)$ Casimir operator eigenvalues.
   • Solved numerically using a basis of Bessel functions of the first kind (derived from an Infinite Square Well potential in $\beta$).
2. Octic & $\gamma$-stable (Prolate):
   • Introduces a new solution for the prolate case where the potential has a minimum at $\gamma = 0^\circ$.
   • Uses a small-angle approximation for the $\gamma$ variable, reducing the problem to a 2D harmonic oscillator.
   • Also solved numerically using the Bessel function basis.

• Numerical Technique: Unlike previous quasi-exact methods, this numerical approach allows for a more general solution, enabling the proper treatment of shape coexistence, mixing, and anomalously small $B(E2)$ transition ratios.

3. Key Contributions

• Novel Solution for Prolate Octic Potential: The paper introduces the first application of the Bohr-Mottelson Hamiltonian with an octic potential specifically for the $\gamma$-stable prolate case, extending previous work limited to $\gamma$-unstable systems.
• Unified Analysis of Coexistence: Instead of forcing a single symmetry description on the isotopes, the authors apply both the $\gamma$-unstable and prolate models concurrently. They demonstrate that different bands (ground, $\beta$, $\gamma$) within the same nucleus may favor different symmetries, providing evidence for shape coexistence and mixing.
• Explanation of $\gamma$-Band Staggering: The study shows that the characteristic staggering of $\gamma$-band states, typically a signature of $\gamma$-unstable nuclei, can also appear in prolate systems when shape mixing and coexistence are accounted for.
• Comprehensive Dataset: The work provides a complete set of calculated energy spectra, $B(E2)$ transition rates, $B(E0)$ monopole transitions, and probability density distributions for the entire $^{98-106}$Ru chain.

4. Results

The analysis of the $^{98-106}$Ru isotopes yields the following specific findings:

• Ground State Deformations (CDFT):

  • $^{98,100}$Ru: Prolate ground states.
  • $^{102,104,106}$Ru: Increasing triaxial deformation ($\gamma \approx 18^\circ - 24^\circ$).
  • The CDFT results show very shallow prolate and triaxial minima, suggesting softness.

• Phase Transitions and Critical Points:

  • $^{98}$Ru: Fits well with the Octic & $\gamma$-unstable model. The $R_{4/2}$ ratio (2.14) places it between the spherical vibrator and the $E(5)$ critical point.
  • $^{100,102}$Ru: Exhibit characteristics of the critical point between spherical and $\gamma$-unstable shapes. The effective potentials are flat, and the $\gamma$-band staggering is well-reproduced by the $\gamma$-unstable model.
  • $^{104,106}$Ru: Show a transition toward $\gamma$-unstable symmetry (single deformed minimum in effective potentials), though the prolate model still captures certain features.

• Shape Coexistence and Mixing:

  • $^{100}$Ru: The ground band fits the $\gamma$-unstable model, while the $\beta$-band fits the prolate model better. The presence of two peaks in the probability density distribution of deformation for the $0^+_2$and$2^+_g$ states confirms shape coexistence with mixing.
  • $^{102}$Ru: The $\beta$-band states ($2^+, 4^+, 6^+$) show signatures of prolate structure, while the ground band is$\gamma$-unstable.
  • $^{104}$Ru: Exhibits dynamical shape transitions within bands. For example, the ground band states evolve from a less-deformed minimum to a well-deformed minimum as spin increases.
  • $^{106}$Ru: The $\beta$-band is distinctly prolate, while the ground and $\gamma$ bands are $\gamma$-unstable.

• Electromagnetic Transitions:

  • The models successfully reproduce experimental $B(E2)$ values.
  • Large $B(E2; 0^+_\beta \to 2^+_g)$ and $B(E2; 2^+_\gamma \to 2^+_g)$ values are attributed to the mixing of wavefunctions with different deformation shapes.
  • The monopole transition $\rho^2(E0)$ values are large, further supporting the presence of shape coexistence.

5. Significance

• Theoretical Advancement: The paper validates the Octic & Prolate solution as a powerful tool for describing nuclei that are not strictly in a limiting symmetry. It proves that the "staggering" of $\gamma$-bands is not exclusive to $\gamma$-unstable nuclei but can arise from shape mixing in prolate systems.
• Structural Insight: It resolves the ambiguity in the Ru isotopic chain by showing that these nuclei do not belong to a single dynamical symmetry. Instead, they represent a complex landscape where spherical, $\gamma$-unstable, and prolate configurations coexist and mix depending on the specific excitation band and spin.
• Methodological Impact: The use of a numerical Bessel-function basis for the octic potential offers a more flexible and general solution than quasi-exact methods, allowing for the accurate modeling of complex shape mixing phenomena and anomalous transition rates.
• Experimental Guidance: The calculated monopole transitions and specific $B(E2)$ values for unmeasured states provide crucial predictions for future experimental campaigns in the $A \approx 100$ mass region.

In conclusion, the study demonstrates that a complementary application of $\gamma$-unstable and prolate octic potential models, supported by microscopic CDFT calculations, is essential for a complete understanding of the nuclear structure in the $^{98-106}$Ru chain, revealing a rich interplay of shape phase transitions, coexistence, and dynamical shape evolution.