Quasi-elastic scattering for the nuclear ground state structure: An intriguing case of $^{30}$Si

By combining quasi-elastic scattering measurements with coupled-channels and shell-model calculations, this study reveals that while $^{28}$Si has a distinct oblate ground state, the addition of two neutrons in $^{30}$Si leads to a sudden structural change where the nucleus lacks a well-defined intrinsic shape, suggesting the presence of ground-state shape fluctuations.

The Gist

Imagine the nucleus of an atom not as a hard, solid marble, but as a drop of liquid that can change its shape. Sometimes it's a perfect sphere, sometimes it stretches out like a rugby ball (prolate), and sometimes it flattens like a pancake (oblate). Scientists have long been trying to figure out exactly what shape these tiny drops take in their most stable, "ground state" condition.

This paper is a detective story about two specific atomic nuclei: Silicon-28 and Silicon-30. They are neighbors on the periodic table, differing only by two neutrons (tiny neutral particles inside the nucleus). You might expect them to look very similar, but the researchers found something surprising: they behave like completely different characters.

The Experiment: Bouncing Balls to See Shapes

To see these invisible shapes, the scientists didn't use a microscope. Instead, they used a technique called Quasi-Elastic (QEL) scattering.

Think of it like this: Imagine you are in a dark room trying to figure out the shape of a hidden object. You throw a bunch of soft rubber balls (the Silicon projectiles) at it and listen to how they bounce back.

• If the object is a perfect sphere, the balls bounce back in a predictable, smooth pattern.
• If the object is a flattened pancake or a stretched rugby ball, the balls bounce back in a specific, jagged way that reveals the object's "squishiness" and orientation.

The team fired beams of Silicon-28 and Silicon-30 at a target made of Zirconium-90. By measuring the energy of the bouncing particles at different angles, they could reconstruct the "shape" of the Silicon nuclei.

The Discovery: One is a Pancake, the Other is a Chameleon

1. Silicon-28: The Flat Pancake
When they analyzed Silicon-28, the data was very clear. It behaved exactly like a flattened pancake (an "oblate" shape). The "bounce-back" pattern was distinct and asymmetric, leaving no doubt about its shape. It's a rigid, well-defined shape.

2. Silicon-30: The Shape-Shifter
Then came Silicon-30. This is where it gets weird. Even though it only has two extra neutrons compared to Silicon-28, the data refused to pick a single shape.

• The scientists tried to fit the data to a pancake shape. It worked perfectly.
• They tried to fit it to a rugby ball (prolate) shape. It also worked perfectly.
• They even tried a perfect sphere that vibrates. That worked too!

It was as if the Silicon-30 nucleus was a chameleon that could be a pancake, a rugby ball, or a sphere, and the experiment couldn't tell which one it was because it seemed to be all of them at once.

The "Shape Fluctuation" Mystery

Why is Silicon-30 so confused? The paper suggests that this nucleus doesn't have a single, rigid shape. Instead, it suffers from "shape fluctuations."

Imagine a ball of jelly sitting on a table.

• Silicon-28 is like a firm gelatin mold; it holds its pancake shape firmly.
• Silicon-30 is like a very soft, wobbly jelly. It doesn't know if it wants to be flat or round. The energy required to be flat is almost the same as the energy to be round. So, it constantly wobbles and fluctuates between these shapes.

The researchers call this a "$\gamma$-soft" nucleus. In simple terms, it's "soft" and "fluid-like" rather than rigid.

The Microscopic Reason: A Tug-of-War

To understand why this happens, the scientists looked at the tiny particles inside (protons and neutrons) using a computer model called the "Shell Model."

• In Silicon-28, the protons and neutrons are all working together, pulling in the same direction to flatten the nucleus. It's a team effort.
• In Silicon-30, the two extra neutrons change the game. The protons want to pull one way (flattening), but the neutrons want to pull the other way (rounding or stretching). It's a tug-of-war where both sides are equally strong. Because they cancel each other out, the nucleus can't decide on a shape, leading to that wobbly, fluctuating state.

The Conclusion

The paper concludes that while Silicon-28 is a well-defined, flat pancake, Silicon-30 is a unique case of a nucleus that lacks a single, fixed shape. It is a "shape-fluctuating" system that constantly shifts between being flat, round, and stretched.

This is a big deal because it shows that adding just two tiny neutrons can completely change the fundamental nature of an atom's structure, turning a rigid object into a fluid, shape-shifting one. The study serves as a crucial test for future theories trying to predict how atomic nuclei behave.

Technical Summary

Technical Summary: Quasi-elastic Scattering and Ground State Structure of $^{30}\text{Si}$

Problem Statement
Determining the exact ground-state shapes of atomic nuclei is inherently challenging due to the complex interplay between collective liquid-drop behavior and single-particle levels near the Fermi surface. While nuclei near major shell closures typically exhibit spherical shapes, those in the $sd$-shell region (between magic numbers 8 and 20) often display shape coexistence or fluctuations. The nucleus $^{30}\text{Si}$ ($N=Z+2$) presents a specific case of interest where theoretical predictions and experimental measurements have yielded conflicting results. Relativistic mean-field calculations suggest an oblate ground state, while beyond-mean-field approximations and certain Skyrme-Hartree-Fock calculations predict a broad, flat potential energy surface (PES) centered around a spherical shape or exhibiting dual prolate/oblate minima. Experimentally, $\alpha$-scattering and deuteron scattering have suggested a prolate shape, whereas Coulomb excitation measurements have indicated spherical or oblate characteristics. This study aims to resolve these discrepancies by investigating the ground-state structure of $^{30}\text{Si}$ using quasi-elastic (QEL) scattering, a technique highly sensitive to ground-state deformation, and comparing it with its neighbor $^{28}\text{Si}$, which is known to possess a uniquely oblate ground state.

Methodology
The experimental investigation utilized $^{28}\text{Si}$ and $^{30}\text{Si}$ projectiles incident on a spherical, closed-shell $^{90}\text{Zr}$ target at energies surrounding the Coulomb barrier (70–102 MeV). The choice of $^{90}\text{Zr}$ was driven by the high charge product ($Z_P Z_T = 560$), which enhances the channel-coupling form factor and sensitivity to the projectile's ground-state structure.

• Experimental Setup: QEL events were detected using silicon surface barrier detectors at backward angles ($140^\circ$–$170^\circ$). Projectile-like fragments (PLFs) were distinguished from evaporated light charged particles (LCPs) via pulse-height analysis.
• Data Analysis: Differential cross-sections were normalized to Rutherford scattering. Quasi-elastic excitation functions were converted into barrier distributions ($D_{qel}$) by taking the energy derivative of the cross-section ratio.
• Theoretical Framework: The experimental data were analyzed using Coupled-Channels (CC) calculations (via the CCFULL code). The analysis explored a large parameter space of quadrupole ($\beta_2$) and hexadecapole ($\beta_4$) deformations for the projectile, while explicitly including vibrational couplings of the $^{90}\text{Zr}$ target ($2^+$ and $3^-$ states).
  • Rotational Models: Calculations tested rigid-rotor descriptions for prolate ($\beta_2 > 0$) and oblate ($\beta_2 < 0$) shapes.
  • Vibrational Models: Calculations tested a spherical description with vibrational coupling strengths ($\beta_{vib}$).
  • Statistical Analysis: A $\chi^2$ minimization and Markov-Chain Monte Carlo (MCMC) Bayesian analysis were employed to determine the probability distributions of deformation parameters.
• Microscopic Validation: Large-scale shell-model calculations using the USDB interaction and Hartree-Fock-Bogoliubov (HFB) calculations with the Gogny interaction were performed to interpret the macroscopic findings in terms of single-particle configurations and transition strengths.

Key Results

1. Contrast with $^{28}\text{Si}$: The QEL excitation function and barrier distribution for $^{28}\text{Si}$ exhibited a distinct asymmetric "shoulder" structure, consistent with a uniquely oblate ground state. In sharp contrast, the data for $^{30}\text{Si}$ displayed a smooth, symmetric barrier distribution, indicating a fundamentally different ground-state structure despite the addition of only two neutrons.
2. Ambiguity in $^{30}\text{Si}$ Deformation: CC calculations revealed that the experimental data for $^{30}\text{Si}$ could be equally well reproduced by three distinct configurations:
   • An oblate shape ($\beta_2 \approx -0.30, \beta_4 \approx 0.00$).
   • A prolate shape ($\beta_2 \approx +0.31, \beta_4 \approx -0.10$).
   • A spherical shape with vibrational coupling ($\beta_{vib} \approx 0.26$).
     All three configurations yielded reduced $\chi^2$ values in the range of 1.7 to 1.9, indicating that the QEL data alone cannot uniquely determine a single intrinsic shape for $^{30}\text{Si}$.
3. Microscopic Origins: Shell-model calculations explained the lack of a rigid shape through the interference of valence nucleons. In $^{30}\text{Si}$, the active $s_{1/2}$ neutrons interfere destructively with the $d_{5/2}$ protons, leading to a near-zero spectroscopic quadrupole moment ($Q_s \approx -0.05$ eb). This contrasts with $^{28}\text{Si}$ (constructive interference, sizable moment) and $^{32}\text{Si}$ (restored constructive interference).
4. $\gamma$-Softness: Theoretical HFB calculations showed a flat deformation-energy surface for $^{30}\text{Si}$ extending across oblate, prolate, and spherical configurations ($\gamma$ values from $0^\circ$ to $60^\circ$). This is supported by experimental $B(E2)$ ratios and excitation energies that suggest $\gamma$-softness rather than a rigid deformation.

Significance and Claims
The paper concludes that $^{30}\text{Si}$ does not possess a well-defined intrinsic shape (neither spherical nor rigidly deformed). Instead, the ground state is characterized by strong shape fluctuations, making it a candidate for a $\gamma$-soft nucleus. The study highlights that the addition of two neutrons beyond the $N=Z$ line in $^{30}\text{Si}$ disrupts the delicate balance of nucleonic forces that define the oblate shape of $^{28}\text{Si}$, leading to a fluid-like nuclear structure.

The authors assert that their findings provide a critical test-bed for advanced theoretical treatments, particularly those aiming to describe shape coexistence and fluctuations in the $sd$-shell. The work underscores the necessity of combining reaction data (QEL scattering) with microscopic structure calculations to resolve ambiguities that arise when different theoretical models or experimental probes yield conflicting results. The paper modestly suggests that future work could extend coupled-channels approaches to explicitly account for the softness of intrinsic shapes in both projectile and target.