Prolate-oblate shape competition and impact on charge radii in Bk isotopes

Using the deformed relativistic Hartree-Bogoliubov theory, this study reveals that in well-deformed odd-$A$ Bk isotopes, oblate shapes yield larger charge radii than prolate ones due to a central proton density depression caused by the non-occupation of the $3s_{1/2}$ orbital, thereby establishing a microscopic link between nuclear shape, single-particle occupancy, and nuclear size.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Picture: Measuring the "Belly Button" of Atoms

Imagine you are trying to measure the size of a giant, invisible balloon. In the world of physics, this balloon is an atomic nucleus. Scientists usually measure its size by looking at its charge radius—essentially, how far out the positive "proton" charge stretches.

For most atoms, this is straightforward. But for heavy, exotic atoms like Berkelium (Bk), things get weird. These atoms are like unstable, wobbly balloons that can change shape. Sometimes they stretch out like a rugby ball (prolate), and sometimes they flatten out like a pancake (oblate).

This paper is a deep dive into what happens when these heavy atoms switch between being "rugby balls" and "pancakes," and how that shape-shifting changes their size.

The Main Discovery: The "Pancake" is Bigger

The researchers used a super-advanced computer simulation (called DRHBc) to predict the shapes and sizes of Berkelium isotopes (different versions of the element with varying numbers of neutrons).

Here is the surprising twist they found:

The "Rugby Ball" vs. The "Pancake" Paradox
Usually, if you squish a ball into a pancake, you might think it gets smaller in one direction but bigger in another, keeping the total volume the same. However, the team discovered that for these heavy atoms, the flat "pancake" shape actually makes the atom larger overall than the stretched "rugby ball" shape, even if the amount of squishing is the same.

Think of it like this:

• The Rugby Ball (Prolate): The protons are stretched out along a long axis, but they stay relatively close to the center in the middle.
• The Pancake (Oblate): The protons spread out wide, but here's the kicker—they also evacuate the center.

The Secret Ingredient: The "Bubble" in the Middle

Why does the pancake shape make the atom bigger? It's because of a bubble.

In the "pancake" (oblate) version of these atoms, the protons don't just spread out; they actually leave a hollow space in the very center of the nucleus. It's like a donut or a hollowed-out ball.

• The Rugby Ball: The protons are packed fairly tightly in the middle, just stretched out.
• The Pancake: The protons are pushed so far to the edges that the middle becomes empty.

Because the protons are pushed further out to the edges to fill this "donut" shape, the overall radius of the atom increases. The "bubble" in the middle forces the outer shell to expand.

The Microscopic Reason: The Missing Seat

Why does this bubble happen? It comes down to a game of musical chairs played by subatomic particles.

Inside the nucleus, protons sit in specific "seats" (orbitals).

• In the Rugby Ball shape, there is a specific seat right in the middle called the 3s1/2 orbital. The protons are happy sitting there, filling up the center.
• In the Pancake shape, the rules of the game change. That specific middle seat becomes "too expensive" or energetically unfavorable to sit in. So, the protons leave that seat empty.

When the protons abandon the center seat, the center becomes empty (the bubble), and the remaining protons spread out to the edges, making the whole atom bigger.

Why Does This Matter?

1. Solving a Mystery: Scientists have noticed that some heavy elements (like Curium) seem smaller than their neighbors, which didn't make sense with old formulas. This paper suggests that if these elements are flattening out and developing "bubbles," it explains why their sizes behave strangely.
2. Better Maps: We can't easily measure these super-heavy atoms in a lab because they are hard to make and decay quickly. This study provides a theoretical "map" that predicts their sizes based on their shape, helping scientists understand the limits of the periodic table.
3. The Formula Breakdown: Old formulas assumed that shape only mattered as a simple square (squaring the deformation). This paper shows that shape matters more than that. A flat shape isn't just a scaled version of a round shape; it fundamentally changes the internal structure (creating bubbles) and increases the size in a way old math couldn't predict.

Summary in One Sentence

This paper reveals that when heavy atoms like Berkelium flatten into a pancake shape, they develop a hollow "bubble" in their center, which pushes their outer edges further out, making the atom larger than if it were stretched into a rugby ball.

Technical Summary

Here is a detailed technical summary of the paper "Prolate-oblate shape competition and impact on charge radii in Bk isotopes."

1. Problem Statement

The nuclear charge radius ($r_c$) is a fundamental observable for understanding nuclear structure, including shell evolution, shape coexistence, and halo structures. However, experimental data for charge radii in the actinide region (specifically for Berkelium, $Z=97$) are exceptionally scarce. While theoretical models exist, there is a lack of systematic understanding regarding how the competition between prolate (elongated) and oblate (flattened) nuclear shapes influences charge radii in heavy, odd-$A$ systems.

A specific puzzle exists in neighboring actinides (Uranium, Plutonium, Curium): Curium isotopes systematically exhibit smaller charge radii than their Uranium and Plutonium counterparts at similar neutron numbers, a discrepancy not fully explained by existing empirical formulas (which assume charge radius depends only on the square of the quadrupole deformation, $\beta_2^2$, regardless of shape sign). This study aims to resolve this by investigating the microscopic mechanisms linking nuclear shape, single-particle occupancy, and nuclear size in odd-$A$ Berkelium isotopes.

2. Methodology

The authors employed the Deformed Relativistic Hartree-Bogoliubov theory in Continuum (DRHBc) using the PC-PK1 density functional. This framework is chosen for its ability to self-consistently handle:

• Relativistic effects: Essential for heavy nuclei.
• Pairing correlations: Crucial for odd-$A$ systems.
• Continuum effects: Important for weakly bound nuclei near drip lines.
• Deformation: Explicitly treating quadrupole deformation ($\beta_2$).

Key Technical Details:

• Scope: Systematic investigation of odd-$A$ Bk isotopes from the proton drip line ($^{227}\text{Bk}$) to the neutron drip line ($^{355}\text{Bk}$).
• Calculations:
  • Constrained DRHBc calculations were performed to generate Potential Energy Curves (PECs) as functions of $\beta_2$.
  • Unconstrained calculations identified ground states and local minima.
  • The "equal filling approximation" was used to handle blocking effects in odd-$A$ nuclei.
  • Single-particle energy levels and density distributions were analyzed to understand microscopic origins.
• Basis: Solutions were obtained in the Dirac Woods-Saxon (DWS) basis with a box size of 20 fm and an energy cutoff of 300 MeV.

3. Key Contributions and Results

A. Shape Evolution and Coexistence

• Prolate Dominance in Ground States: The ground states of most Bk isotopes are predominantly prolate, with significant deformation ($|\beta_2| > 0.3$) in mid-shell regions.
• Oblate Local Minima: Corresponding local minima are predominantly oblate.
• Energy Landscape: The energy difference ($\Delta E$) between prolate ground states and oblate local minima is large (up to 9.3 MeV) in mid-shell regions but diminishes near neutron shell closures ($N=184, 258$).
• Shape Coexistence: The study confirms shape coexistence in $^{331}\text{Bk}$, where prolate and oblate minima appear at nearly degenerate energies (separated by only 0.57 MeV).

B. Anomalous Charge Radius Behavior

• Deformation Impact: DRHBc predictions yield systematically larger charge radii than spherical RCHB (Relativistic Continuum Hartree-Bogoliubov) calculations, confirming that deformation extends the charge density distribution.
• Shape Dependence: A critical finding is that for a given absolute deformation $|\beta_2|$, oblate shapes yield larger charge radii than prolate shapes in well-deformed nuclei.
• Failure of Empirical Formulas: The standard empirical formula $r_c(\beta_2) = (1 + \frac{5}{4\pi}\beta_2^2)r_c(0)$ fails to capture this behavior because it assumes charge radius depends only on $\beta_2^2$, ignoring the sign of deformation. The study shows that for large deformations, the relationship is better described as $\Delta r_c \propto \Delta \beta_2^2 + G$, where $G$ is a positive offset favoring oblate shapes.

C. Microscopic Origin: The "Bubble" Structure

The paper provides a microscopic explanation for why oblate shapes are larger:

• Central Depression: Oblate minima exhibit a pronounced central depression (a "bubble" structure) in the proton density distribution, whereas prolate minima do not.
• Orbital Occupancy: This bubble structure arises from the non-occupation of the spherical $3s_{1/2}$($\Omega=1/2$) orbital in the oblate configuration.
  • In prolate ground states, the $3s_{1/2}$ orbital lies below the Fermi level and is occupied, filling the center.
  • In oblate local minima, this orbital is pushed above the Fermi level and remains unoccupied, creating a hole in the center and pushing the remaining proton density outward, thereby increasing the charge radius.
• Weak Deformation Case: In isotopes with small deformations (near shell closures), the $3s_{1/2}$ orbital remains occupied in both shapes, the bubble disappears, and the charge radii of prolate and oblate states become nearly identical.

4. Significance

• Resolving the Actinide Anomaly: The findings offer a theoretical explanation for why Curium isotopes (which may favor oblate shapes or specific shell effects) have smaller radii than expected or smaller than neighbors, by highlighting the complex interplay between shape and single-particle occupancy.
• Microscopic-Macroscopic Link: The study establishes a clear, quantitative link between single-particle quantum numbers (specifically the $3s_{1/2}$ orbital), nuclear shape (prolate vs. oblate), and macroscopic observables (charge radii).
• Predictive Power: By demonstrating that oblate shapes can produce larger radii than prolate shapes of the same deformation magnitude, the study refines theoretical predictions for exotic nuclei where experimental data is unavailable. This is crucial for interpreting future laser spectroscopy experiments in the actinide region.
• Validation of DRHBc: The results reinforce the DRHBc framework as a superior tool for describing heavy, deformed, and exotic nuclei compared to spherical approximations.

In conclusion, the paper demonstrates that nuclear size is not merely a function of deformation magnitude but is critically dependent on the sign of the deformation and the resulting single-particle orbital occupancy, specifically the presence or absence of a central density depression (bubble) driven by the $3s_{1/2}$ orbital.