Spectroscopic factors as a probe of nuclear shape in $^{44}$S via one-neutron knockout reaction

This study utilizes antisymmetrized molecular dynamics combined with the generator coordinate method to demonstrate that spectroscopic factors and reaction cross sections from one-neutron knockout reactions on $^{44}$S serve as sensitive probes for investigating shape mixing and fluctuations in this neutron-rich nucleus.

The Gist

Imagine the atomic nucleus not as a rigid, solid marble, but as a drop of liquid that can stretch, squish, spin, and change its shape. In the world of nuclear physics, scientists are trying to understand why some of these "drops" stay round and stable, while others are wobbly, stretching into long ovals or flattening out like pancakes.

This paper focuses on a specific, exotic nucleus called Sulfur-44 ($^{44}\text{S}$). It's a "neutron-rich" version of sulfur, meaning it has way more neutrons than the stable sulfur we find on Earth. Because of this extra weight, the usual rules that keep atomic nuclei in a nice, round shape start to break down.

Here is the story of the paper, explained through simple analogies:

1. The Mystery: A Wobbly Nucleus

Think of the nucleus as a dancer. In most atoms, the dancer has a specific, rigid routine (a fixed shape). But in Sulfur-44, the "music" (the forces holding the nucleus together) is changing. The dancers are confused. They aren't sure if they should stand tall (prolate shape), lie flat (oblate shape), or spin in a weird, twisted way (triaxial shape).

Scientists call this Shape Coexistence. It's like a dancer who can instantly switch between a ballet pose, a breakdance move, and a yoga stretch, all at the same time. The big question is: Which shape does Sulfur-44 actually prefer?

2. The Problem: Two Different Maps

To figure this out, the researchers used a powerful computer simulation called AMD+GCM. Think of this as a high-tech weather forecast for the nucleus. However, there's a catch: the forecast depends on the "climate model" (the mathematical rules) you use.

The team ran the simulation twice using two different sets of rules (called D1S and D1M):

• The D1S Model: Predicts that Sulfur-44 is a chameleon. It says the nucleus is extremely wobbly, constantly fluctuating between different shapes. It's like a dancer who can't stop moving, mixing all styles together.
• The D1M Model: Predicts that Sulfur-44 is a stickler for rules. It says the nucleus is mostly rigid and prefers one specific shape (a stretched oval). It's like a dancer who only does one specific move and sticks to it.

Both models are mathematically valid, but they tell two completely different stories about the same nucleus. How do we know which one is right?

3. The Solution: The "Knockout" Test

You can't just look inside a nucleus with a microscope. So, the scientists proposed a clever experiment: The One-Neutron Knockout.

Imagine the nucleus is a tightly packed suitcase full of clothes (protons and neutrons).

• The Experiment: They shoot a high-speed proton at the Sulfur-44 nucleus. It's like throwing a tennis ball at the suitcase to knock one piece of clothing (a neutron) out.
• The Result: What's left inside the suitcase is now Sulfur-43.

The key insight is this: How the suitcase reacts depends on how the clothes were packed.

• If the original suitcase was a wobbly mix of shapes (D1S), knocking out a neutron might leave the remaining clothes in many different, messy arrangements.
• If the original suitcase was rigid and ordered (D1M), knocking out a neutron will leave the remaining clothes in a very specific, predictable arrangement.

4. The Clues: Spectroscopic Factors

The paper introduces a concept called Spectroscopic Factors. Think of this as a "probability score."

• It tells you how likely it is to find the remaining nucleus in a specific pose after the knockout.
• The researchers found that the 3/2⁻ and 7/2⁻ states (specific poses of the remaining Sulfur-43) are the "smoking guns."
  • If the D1S (Wobbly) model is right, you will see a lot of these specific states appearing.
  • If the D1M (Rigid) model is right, you will see very few of them.

It's like checking the aftermath of a shaken box of marbles. If the box was full of loose sand (wobbly), the marbles will scatter everywhere. If the box was full of glued-together blocks (rigid), they will stay in a neat pile.

5. The Verdict: What the Paper Says

The paper concludes that we don't need to guess anymore. We just need to do the experiment.

By measuring the results of this "knockout" reaction (specifically looking at the momentum and energy of the leftover nucleus), scientists can tell which model is correct:

• If the data matches the D1S predictions: We confirm that Sulfur-44 is a chaotic, shape-shifting chameleon, proving that the "magic" rules of nuclear stability have completely collapsed here.
• If the data matches the D1M predictions: We confirm that the nucleus is more stable and rigid than we thought.

Why Does This Matter?

This isn't just about Sulfur-44. It's about understanding the fundamental forces of nature.

• If nuclei can change shape so easily, it means our current understanding of how protons and neutrons stick together needs an update.
• It helps us understand how heavy elements are formed in the universe (like in exploding stars).
• It proves that "Shape Mixing" is a real, observable phenomenon, not just a theory.

In short: This paper is a recipe for a physics experiment. It says, "We have two theories about a wobbly nucleus. Here is exactly how to knock a piece off it, and here is exactly what to look for to see which theory is the winner." It turns a complex mathematical debate into a clear, testable question for experimentalists to answer.

Technical Summary

1. Problem Statement

The neutron-rich nucleus $^{44}$S lies in a region where the traditional $N=28$ shell closure weakens, leading to the breakdown of magic numbers and the emergence of shape coexistence and large-amplitude collective motion (LACM). While experimental evidence suggests the coexistence of prolate, spherical, and oblate shapes in $^{44}$S, the precise nature and degree of shape mixing in its ground state remain ambiguous.

A critical issue in theoretical nuclear physics is the dependence of structural predictions on the choice of effective interactions. Different parameterizations of the same interaction model can yield contradictory predictions regarding the magnitude of shape fluctuations. The authors aim to resolve this by identifying observables that are highly sensitive to these theoretical differences, specifically focusing on whether $^{44}$S exhibits strong shape mixing (LACM) or rigid deformation.

2. Methodology

The study employs a multi-faceted theoretical approach combining microscopic structure calculations with reaction theory:

• Structure Model (AMD+GCM):

  • The authors use Antisymmetrized Molecular Dynamics (AMD) combined with the Generator Coordinate Method (GCM).
  • The wave functions are constructed by superposing projected AMD states using $(\beta, \gamma)$ deformation parameters as generator coordinates.
  • Interaction Dependence: Calculations are performed using two different Gogny effective interactions, D1S and D1M, to explore how the choice of interaction affects the predicted shape mixing.
  • Observables Calculated: Monopole ($E0$) and quadrupole ($E2$) transition strengths, and spectroscopic factors ($C^2S$) for neutron removal.

• Reaction Model (DWIA):

  • The Distorted Wave Impulse Approximation (DWIA) is used to calculate cross-sections for the one-neutron knockout reaction $^{44}$S($p, pn$)$^{43}$S.
  • The code pikoe is utilized, employing the Franey-Love nucleon-nucleon effective interaction and Dirac phenomenological optical potentials (EDAD1 set).
  • The incident energy is set to 250 MeV/nucleon, typical for facilities like RIKEN.
  • Longitudinal Momentum Distributions (LGMD): The study analyzes the momentum distributions of the residual nucleus ($^{43}$S) to distinguish between different orbital angular momentum transfers ($l$-dependence).

3. Key Contributions

• Interaction Sensitivity Analysis: The paper demonstrates that the manifestation of LACM in $^{44}$S is not a robust feature across all Gogny parameter sets. D1S predicts strong shape mixing, while D1M predicts relatively rigid shapes.
• Identification of Probes: The authors identify specific electromagnetic transitions and, more importantly, spectroscopic factors for specific states in the residual nucleus ($^{43}$S) as direct probes to distinguish between these competing theoretical scenarios.
• Reaction Formalism Application: The study provides a comprehensive framework linking microscopic wave function overlaps (spectroscopic factors) to measurable reaction cross-sections and momentum distributions, specifically for the $^{44}$S($p, pn$)$^{43}$S channel.

4. Results

A. Structural Properties of $^{44}$S

• D1S Interaction: Predicts a soft energy surface along the triaxial ($\gamma$) degree of freedom. The $0^+$ states exhibit broad GCM overlaps, indicating strong shape mixing (LACM) involving prolate, oblate, and triaxial configurations.
• D1M Interaction: Predicts a rigid energy surface. The $0^+$ states are localized in either prolate or oblate minima with minimal mixing, suggesting a more static deformation.
• Excitation Spectra: Both interactions reproduce the general low-lying spectrum but underestimate the energy of the $0^+_2$ state compared to experiment, likely due to insufficient pairing correlation treatment.

B. Electromagnetic Transitions

• Monopole Transitions ($E0$): The transition strength $\rho^2(E0; 0^+_1 \to 0^+_2)$ is significantly larger for D1S (13.3 $\times 10^{-3}$) than for D1M (4.4 $\times 10^{-3}$). This is a direct signature of configuration mixing.
• Quadrupole Transitions ($E2$): Transitions between $2^+$ states ($2^+_1 \to 2^+_2$) show an opposite trend, being enhanced in D1M.
• Conclusion: Monopole transitions between $0^+$ states and quadrupole transitions between $2^+$ states are the most sensitive electromagnetic probes for distinguishing the shape mixing scenarios.

C. Spectroscopic Factors and Knockout Reaction

• Overlap Amplitudes: The spectroscopic factors ($C^2S$) for the $^{44}$S($p, pn$)$^{43}$S reaction depend heavily on the shape of the $^{44}$S ground state.
  • D1S (Strong Mixing): Predicts substantial $C^2S$ values for populating both prolate-band states (e.g., $3/2^-_1$) and oblate/triaxial states (e.g., $3/2^-_2$, $7/2^-_1$) in $^{43}$S.
  • D1M (Rigid Prolate): Predicts strong population of prolate-band states but suppressed population of oblate ($3/2^-_2$) and triaxial ($7/2^-_1$) states due to shape mismatch.
• Longitudinal Momentum Distributions (LGMD):
  • The calculated LGMDs for the $3/2^-$ and $7/2^-$ states show distinct differences between the two interactions.
  • For the D1S case, the cross-sections for oblate/triaxial states are comparable to prolate states. For D1M, they are drastically reduced.
  • The ratio of cross-sections ($R(\sigma_{AMD})$) between different states serves as a robust observable, differing by factors of ~2.4 between the two interaction models.

5. Significance and Conclusion

The paper concludes that spectroscopic factors and reaction cross-sections are superior probes for investigating shape mixing in $^{44}$S compared to static properties alone.

• Experimental Proposal: The measurement of the $^{44}$S($p, pn$)$^{43}$S reaction, specifically the population ratios of the $3/2^-$ and $7/2^-$ states in $^{43}$S, can directly test the theoretical predictions.
• Discrimination Power: If experiments observe significant population of the oblate ($3/2^-_2$) and triaxial ($7/2^-_1$) states, it would confirm the strong shape mixing (LACM) predicted by the D1S interaction. Conversely, a dominance of prolate states would support the rigid deformation picture of D1M.
• Broader Impact: This work establishes a methodology for using knockout reactions to probe the "softness" of nuclear energy surfaces in exotic nuclei, providing a critical link between microscopic nuclear forces (effective interactions) and macroscopic nuclear shapes.