Microscopic investigation of $E2$ matrix elements in atomic nuclei -- II

This paper extends a microscopic triaxial projected shell model (TPSM) investigation to six additional nuclides ($^{70}$Ge, $^{76-82}$Se, and $^{100}$Mo), demonstrating that the approach successfully describes experimental $E2$ matrix elements and reveals $\gamma$-soft behavior in most cases while showing no clear correlation between $\gamma$-band energy staggering and shape invariants, contrary to phenomenological collective model predictions.

The Gist

Imagine the atomic nucleus not as a boring, solid marble, but as a squishy, dancing blob of dough. Sometimes this dough is perfectly round like a ball. Other times, it stretches out like a rugby ball (prolate) or flattens out like a pancake (oblate). But the most interesting shapes are the "triaxial" ones—imagine a potato that is stretched in three different directions, or a spinning top that wobbles as it spins.

This paper is a scientific investigation into the shapes of these nuclear "doughs," specifically looking at six new types of atomic nuclei (isotopes of Germanium, Selenium, and Molybdenum) that were recently measured in a lab.

Here is the breakdown of what the scientists did, using some everyday analogies:

1. The Experiment: The "Coulomb Excitation" Dance

To figure out the shape of these invisible nuclei, scientists use a technique called Coulomb Excitation.

• The Analogy: Imagine you have a mystery object in a dark room. You can't touch it, but you can throw a tennis ball at it. If the object is round, the ball bounces off one way. If it's a long stick, it bounces off differently.
• In the Lab: Scientists fire beams of heavy ions (like tiny cannonballs) at these nuclei. The electrical repulsion between the beam and the nucleus makes the nucleus "dance" (get excited). By watching how the nucleus dances and how it emits energy (light, essentially), they can deduce its shape.

2. The Tool: The "Microscopic Chef" (TPSM)

The scientists used a computer model called the Triaxial Projected Shell Model (TPSM).

• The Analogy: Think of the nucleus as a complex stew.
  • Old Models (Phenomenological): These were like guessing the taste of the stew just by looking at the pot. They assumed the whole thing moved as one big, smooth blob.
  • The TPSM (Microscopic): This model is like a chef who knows exactly what every single ingredient (proton and neutron) is doing. It calculates how every single particle interacts with every other particle to create the final shape. It's much more detailed and "from the ground up."

3. The Discovery: The "Wobbly Top" vs. The "Rigid Spinning Top"

The main goal was to see if these nuclei are "soft" (wobbly and changing shape easily) or "rigid" (stiff and holding a specific shape).

• The "Staggering" Clue: The scientists looked at the energy levels of the nuclei as they spun faster.
  • The Analogy: Imagine a spinning top. If it's perfectly balanced, it spins smoothly. If it's slightly unbalanced, it wobbles in a specific rhythm.
  • The Finding: For most of the nuclei they studied (like Germanium-70 and most Selenium isotopes), the "wobble" pattern suggested they are γ-soft. This means they are like a wobbly jelly; they can easily shift their shape as they spin.
  • The Surprise: For two specific nuclei (Selenium-76 and Molybdenum-100), the math suggested they should be γ-rigid (stiff like a rock). However, the actual energy data (the "dance moves") looked a bit messy and didn't fit the simple "stiff top" story perfectly.

4. The Conflict: The Map vs. The Terrain

This is the most exciting part of the paper.

• The Old Map (Collective Model): For decades, physicists had a rulebook (the Collective Model) that said: "If the nucleus is stiff, the dance moves will look like X. If it's wobbly, they will look like Y."
• The New Terrain (TPSM Results): The new, detailed "microscopic" calculations showed that for Selenium-76 and Molybdenum-100, the dance moves (energy patterns) didn't match the old rulebook, even though the shape calculations said they were stiff.
• The Takeaway: The old rulebook is too simple. It's like trying to predict the weather using only a thermometer. The new model shows that the nucleus is a complex system where individual particles (quasiparticles) mix and interact in ways that create a "wobble" even when the overall shape is supposed to be stiff.

Summary in Plain English

The scientists took a closer look at six atomic nuclei using a super-detailed computer simulation that tracks every single particle inside. They found that:

1. Most of them are "wobbly": They change shape easily as they spin.
2. Two of them are confusing: The computer says they are stiff, but their behavior looks complicated.
3. The lesson: The old, simple rules we used to describe how nuclei spin are sometimes wrong. We need to look at the tiny, individual particles inside to understand the whole picture.

Why does this matter?
Understanding these shapes helps us understand the fundamental forces of the universe. Just as knowing the shape of a building tells you how strong it is, knowing the shape of a nucleus helps us understand how matter holds together, which is crucial for everything from nuclear energy to understanding why the universe exists the way it does.

Technical Summary

1. Problem Statement

The determination of atomic nuclear shapes remains a central challenge in nuclear physics. While electromagnetic probes (specifically Coulomb excitation, or COULEX) allow for the measurement of electric quadrupole (E2) transition probabilities, interpreting these data to deduce nuclear deformation (specifically triaxiality and $\gamma$-softness) often relies on phenomenological collective models. These models, such as the Bohr-Mottelson model, assume a direct correlation between the energy staggering pattern of the $\gamma$-band and the rigidity of the triaxial deformation. However, recent experimental data for various nuclides have shown discrepancies with these phenomenological predictions. The authors aim to address these discrepancies by systematically calculating E2 matrix elements and shape invariants using a microscopic approach to test the validity of collective model assumptions.

2. Methodology

The study employs the Triaxial Projected Shell Model (TPSM), a microscopic framework that constructs nuclear states from angular-momentum projected deformed basis states rather than spherical ones.

• Hamiltonian: The model utilizes the Pairing Plus Quadrupole-Quadrupole (PPQ) Hamiltonian. The intrinsic basis is generated by solving the triaxially deformed Nilsson Hamiltonian within the BCS approximation for pairing correlations.
• Configuration Space: The model space includes the triaxial vacuum and configurations involving 2-quasiparticle (2qp) and 4-quasiparticle (4qp) excitations.
• Projection: Three-dimensional angular momentum projection is applied to restore good angular momentum quantum numbers, allowing for the description of rotational bands up to high spins.
• Calculations:
  • Input: Deformation parameters ($\epsilon$ and $\epsilon'$) were chosen to reproduce experimental $\gamma$-band head energies and $B(E2; 2^+_1 \to 0^+_1)$ values.
  • Pairing: Standard pairing gaps were used for ground-state bands. For excited $0^+$ bands, "reduced" pairing gaps were employed to account for blocking effects and better match experimental excitation energies.
  • Shape Invariants: Using the calculated E2 matrix elements, the authors derived shape-invariant quantities (Kumar-Cline sum rules) via the GOSIA code. These include the mean quadrupole moment $\langle Q^2 \rangle$, the triaxiality parameter $\langle \cos 3\delta \rangle$, and their dispersions ($\sigma$).
• Target Nuclides: The study extends previous work to six new nuclides: $^{70}$Ge, $^{76,78,80,82}$Se, and $^{100}$Mo.

3. Key Contributions

• Systematic TPSM Analysis: This is the first comprehensive microscopic calculation of E2 matrix elements and shape invariants for this specific set of six nuclides using the TPSM approach.
• Comprehensive Data Set: The authors calculated and tabulated 103 reduced E2 matrix elements for each nucleus (covering diagonal, in-band, and inter-band transitions up to spin $I=10$), providing a benchmark for future experimental verification.
• Challenge to Phenomenology: The work explicitly tests the correlation between $\gamma$-band energy staggering and shape rigidity, a cornerstone of the collective model, using a microscopic many-body approach.

4. Key Results

A. Excitation Energies

• The TPSM successfully reproduces the experimental yrast, $\gamma$, and excited $0^+$ band structures for all six nuclides.
• The use of reduced pairing gaps for excited $0^+$ bands was crucial for matching experimental energies, effectively simulating blocking effects.

B. E2 Matrix Elements

• Agreement: The calculated E2 matrix elements show good agreement with available experimental COULEX data.
• Diagonal Elements:
  • $^{70}$Ge shows slightly positive diagonal elements (tendency toward oblate shape).
  • Other isotopes ($^{76,78,80,82}$Se, $^{100}$Mo) show negative diagonal elements (tendency toward prolate shape), though $^{100}$Mo is near maximal triaxiality.
• Shape Changes: The model predicts shape changes (sign flips in diagonal matrix elements) at specific spins for $^{70}$Ge ($I=8$) and $^{76}$Se ($I=10$).

C. Shape Invariants and Triaxiality

• $\gamma$-Softness: For most nuclides ($^{70}$Ge, $^{78,80,82}$Se), the large fluctuations in the triaxiality parameter ($\sigma(\cos 3\delta)$) indicate $\gamma$-soft behavior.
• $\gamma$-Rigidity: For $^{76}$Se and $^{100}$Mo, the small fluctuations in $\cos 3\delta$ indicate $\gamma$-rigid behavior according to the sum rules.

D. The Staggering Paradox (Major Finding)

The study reveals a significant contradiction between the TPSM results and the phenomenological collective model:

• Collective Model Prediction: $\gamma$-rigid nuclei should exhibit an "odd-I-down" energy staggering pattern in the $\gamma$-band.
• TPSM/Experimental Reality:
  • For $^{76}$Se and $^{100}$Mo, the sum rules indicate $\gamma$-rigidity, yet the TPSM predicts complex staggering patterns (e.g., $^{100}$Mo shows an "even-I-down" pattern initially, and $^{76}$Se shows a phase change).
  • The TPSM attributes this complex staggering to the mixing of quasiparticle excitations with the vacuum configuration, a mechanism absent in simple collective models.
  • Consequently, the paper demonstrates that no clear correlation exists between the energy staggering pattern and the deduced shape invariants when using the microscopic TPSM approach.

5. Significance

• Limitations of Collective Models: The work provides strong evidence that phenomenological collective models, which rely on the quadrupole degree of freedom alone, fail to capture the complexity of nuclear structure in transitional nuclei. Specifically, the assumption that energy staggering is a direct proxy for $\gamma$-rigidity is shown to be unreliable in the presence of quasiparticle mixing.
• Microscopic Insight: The TPSM approach successfully explains experimental data by incorporating the microscopic degrees of freedom (quasiparticles), offering a more robust description of shape coexistence and softness.
• Future Directions: The authors note that while TPSM works well for deformed nuclei, it requires extension (via the Generator Coordinate Method) to handle near-spherical nuclei (like Pd and Cd isotopes mentioned in the conclusion). The extensive tabulation of E2 matrix elements serves as a critical resource for experimentalists to design future Coulomb excitation experiments to verify the predicted complex staggering patterns.

In summary, this paper validates the TPSM as a superior tool for analyzing nuclear shapes in transitional regions and highlights the necessity of microscopic treatments to resolve contradictions found in phenomenological interpretations of Coulomb excitation data.