Nuclear shape evolution of neutron-deficient Au and kink structure of Pb isotopes

This study utilizes the deformed relativistic Hartree-Bogoliubov theory in continuum to explain the abnormal charge radii evolution in neutron-deficient gold isotopes through prolate-to-oblate shape transitions and successfully reproduces the kink structure observed in lead isotopes near the $N=126$ shell.

The Gist

Imagine the atomic nucleus not as a rigid, unchanging marble, but as a squishy, dynamic blob of dough. Sometimes this dough is perfectly round like a basketball. Other times, it stretches out like a rugby ball (prolate) or flattens out like a pancake (oblate).

This paper is a detective story about two specific families of these "nuclear doughs": Gold (Au) and Lead (Pb). Scientists recently used high-tech lasers to measure the size of these nuclei and found some very strange behavior. The authors of this paper used a super-computer model to figure out why these nuclei are acting so weirdly.

Here is the breakdown of their findings using simple analogies:

1. The Gold Mystery: The "Shape-Shifter"

The Observation:
When scientists looked at Gold isotopes (atoms of gold with different numbers of neutrons), they saw two weird things:

• The Stagger: In a specific range, the size of the nucleus jumped back and forth. If you had an even number of neutrons, the nucleus was one size; if you added one more (making it odd), it suddenly got bigger or smaller. It was like a dance where partners kept switching sizes.
• The Sudden Jump: At a certain point (around 108 neutrons), the size of the nucleus changed abruptly, like a light switch being flipped.

The Explanation (The "Shape Coexistence" Analogy):
The authors explain this using the concept of Shape Coexistence. Imagine a ball sitting in a valley. Usually, a ball rolls to the very bottom (the most stable shape). But in these Gold nuclei, the "valley" has two dips: one deep dip for a stretched-out shape (rugby ball) and a slightly shallower dip for a flattened shape (pancake).

• The Tug-of-War: For some Gold atoms, the "rugby ball" shape is the winner. For others, the "pancake" shape wins.
• The Switch: As you add or remove neutrons (changing the recipe of the dough), the balance shifts. Suddenly, the nucleus decides it's easier to be a pancake than a rugby ball.
• The Result: Because the nucleus is constantly switching between these two shapes as you move across the periodic table, the size measurements look chaotic (the "stagger") or jump suddenly (the "kink"). The paper shows that if you account for this shape-shifting, the weird data makes perfect sense.

2. The Lead Mystery: The "Kink" at the Finish Line

The Observation:
Lead isotopes are famous for having a "magic number" of neutrons (126). Think of this like a finish line in a race where the runners (neutrons) are very organized and stable. Scientists noticed that right after crossing this finish line (adding neutrons beyond 126), the size of the Lead nucleus suddenly swelled up. This sharp bend in the size graph is called a "kink."

The Explanation (The "Balloon" Analogy):
The authors explain this using the behavior of the neutrons themselves.

• The Core: Imagine the nucleus has a tight, stable core of protons (the positive charge).
• The Outer Layer: When you add neutrons past the magic number of 126, they don't just sit quietly. They start to fill up specific "seats" (quantum states) that are very large and fluffy.
• The Swelling: It's like blowing air into a balloon. The protons stay mostly the same size, but the new neutrons are like extra air that pushes the outer skin of the balloon out. This "swelling" of the neutron cloud is what causes the sudden "kink" in the size measurement. The paper confirms that the protons aren't changing much; it's the neutrons expanding that does the work.

The Tool They Used: The "Super-Model"

To solve these mysteries, the scientists didn't just guess; they used a powerful computer model called DRHBc.

• Think of this model as a virtual physics lab. It simulates the nucleus by calculating the forces between every single proton and neutron, taking into account that the nucleus can be squashed, stretched, and that the outer edges of the nucleus are fuzzy (continuum effects).
• It's like a weather simulation, but instead of predicting rain, it predicts whether a nucleus will be round, flat, or stretched, and how big it will be.

The Big Takeaway

The paper concludes that nuclei are much more flexible than we thought.

• Gold is a shape-shifter that flips between rugby-ball and pancake forms, causing its size to dance and jump.
• Lead has a stable core, but once it passes a certain limit, the extra neutrons puff it up like a balloon, creating a sharp bend in its size curve.

By understanding these "shape coexistence" and "neutron swelling" effects, scientists can better understand the fundamental rules that hold the universe's matter together. It turns out that at the subatomic level, things are a lot more squishy and dynamic than they appear!

Technical Summary

1. Problem Statement

Recent experimental advancements in laser spectroscopy have revealed complex and anomalous behaviors in the nuclear charge radii of neutron-deficient isotopes, specifically Gold (Au) and Lead (Pb). Two primary phenomena require theoretical explanation:

• Gold (Au) Isotopes: Significant changes in ground-state deformation are observed, characterized by odd-even shape staggering (OES) in the neutron number ($N$) range of 98–100 and an abrupt change in charge radii starting at $N=108$.
• Lead (Pb) Isotopes: An abrupt "kink" structure is observed in the charge radii near the $N=126$ neutron shell closure.

While various nuclear models exist, few can simultaneously and accurately reproduce the strong OES and the abrupt kinks across neutron shell closures. The authors aim to elucidate the underlying nuclear structure mechanisms driving these phenomena, specifically investigating the role of shape coexistence and single-particle state evolution.

2. Methodology

The study employs the Deformed Relativistic Hartree–Bogoliubov theory in Continuum (DRHBc). This framework is chosen for its ability to simultaneously handle:

• Nuclear deformation (axial and triaxial possibilities).
• Pairing correlations.
• Continuum effects (crucial for weakly bound nuclei).
• Odd-mass and odd-odd nuclei via an automatic blocking method (utilizing the equal filling approximation to preserve time-reversal symmetry).

Key Computational Parameters:

• Basis: Dirac Woods-Saxon basis with energy cutoff $E_{cut}^+ = 300$ MeV and angular momentum cutoff $J_{max} = (23/2)\hbar$.
• Interaction: Meson-exchange or point-coupling density functionals.
• Pairing: Density-dependent zero-range force with pairing strength $V_0 = -325.0$ MeV fm$^3$ and a pairing window of 100 MeV.
• Expansion: Legendre polynomial expansion truncated at $\lambda_{max} = 8$.
• Validation: The model's binding energies were first validated against experimental data, showing root-mean-square deviations of < 1.43 MeV for Au and < 1.93 MeV for Pb.

3. Key Contributions and Results

A. Shape Coexistence in Gold (Au) Isotopes

• Deformation Transition: The DRHBc calculations reveal a competition between prolate (elongated) and oblate (flattened) shapes.
  • For $A \le 188$ ($N \le 109$), prolate deformation is generally favored.
  • At $N=109$ ($^{188}$Au), an abrupt transition to oblate deformation occurs.
• Mechanism of Anomalies: The study identifies shape coexistence as the root cause of the observed charge radius anomalies.
  • In the region $N=98 \sim 100$ and near $N=108$, the Total Binding Energy (TBE) curves show local minima for both prolate and oblate shapes with energy differences of approximately 1 MeV.
  • Odd-Even Staggering: The staggering in charge radii arises because odd-$N$ isotopes (e.g., $^{177,179}$Au) access different shape minima (often oblate) compared to their even-$N$ neighbors (often prolate), leading to significant radius variations.
  • Abrupt Change: The sudden change at $N=108$ is attributed to a transition from oblate to prolate deformation driven by the shifting balance of shape coexistence.
• Single-Particle Evolution: Analysis of occupation probabilities reveals that the surge in charge radii is driven by the increased occupation of the $\nu 1i_{13/2}$ neutron orbital and the $\pi 1h_{9/2}$ proton orbital in prolate shapes, coupled with a decrease in $\nu 1h_{9/2}$ and $\nu 2f_{7/2}$ occupations.

B. Kink Structure in Lead (Pb) Isotopes

• Reproduction of the Kink: The DRHBc model successfully reproduces the abrupt increase in charge radii (the "kink") observed at $N=126$ (near $^{208}$Pb).
• Neutron Swelling: The kink is primarily caused by the swelling of the neutron core rather than changes in the proton distribution.
  • Beyond the $N=126$ shell closure, pairing correlations lead to a significant increase in the occupation probabilities of the $\nu 2g_{9/2}$ and $\nu 1i_{11/2}$ neutron orbitals.
  • The occupation probabilities of proton states ($\pi 3s_{1/2}$ and $\pi 1h_{9/2}$) remain largely unchanged, confirming that the proton core ($Z=82$) remains symmetric and stable.
• Shape Evolution: While earlier models predicted large deformations for neutron-deficient Pb isotopes ($100 \le N \le 117$) that contradicted experimental radii, the DRHBc approach, by considering shape coexistence (including spherical, prolate, and oblate minima within ~1.5 MeV), aligns theoretical results with experimental data.

4. Significance

• Theoretical Validation: The study demonstrates that the DRHBc theory is a robust tool for describing complex nuclear phenomena, particularly where shape coexistence is prevalent. It successfully reproduces both the subtle odd-even staggering in Au and the sharp kinks in Pb, which many other models fail to do simultaneously.
• Mechanistic Insight: The work clarifies that:
  1. Shape coexistence is the primary driver for the irregular charge radius evolution in neutron-deficient Au isotopes.
  2. Neutron shell closure effects (specifically the occupation of high-$j$ orbitals like $\nu 1i_{11/2}$) drive the kink structure in Pb, highlighting the decoupling of neutron and proton core behaviors near magic numbers.
• Future Directions: The authors suggest that while axial deformation explains the current data, the potential existence of triaxial deformation ($\gamma$ deformation) in these isotopes warrants further investigation, as it could turn local minima into saddle points in the $(\beta, \gamma)$ plane.

In conclusion, the paper provides a comprehensive microscopic explanation for the nuclear shape evolution in Au and Pb isotopes, attributing experimental anomalies to the interplay of shape coexistence and specific single-particle orbital occupations within a relativistic mean-field framework.