⚛️ nuclear theory

Determining the dynamic deformation of $^{140}$ Ce by constraining coupled-channels parameters for fusion

This study systematically determines the dynamic deformation parameters of $^{140}$ Ce by combining experimental fusion data with Gaussian analytic-barrier and coupled-channels analyses across multiple projectile systems, revealing that the extracted quadrupole and octupole deformations are robust and effectively validated by their ability to reproduce fusion excitation functions and barrier distributions in the $^{28}$ Si+ $^{140}$ Ce reaction.

Original authors: Chandra Kumar, Rohan Biswas, J. Gehlot, Gonika, A. Parihari, N. Madhavan, A. Vinayak, Amritraj Mahato, S. Nath

Published 2026-07-03

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CC BY 4.0

Original authors: Chandra Kumar, Rohan Biswas, J. Gehlot, Gonika, A. Parihari, N. Madhavan, A. Vinayak, Amritraj Mahato, S. Nath

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine an atom's nucleus not as a tiny, hard marble, but as a squishy, vibrating blob of jelly. Sometimes this jelly is perfectly round, but often it wobbles, stretches, or bulges in specific ways. Physicists call these shapes "deformations." The big challenge is figuring out exactly how squishy and what shape a specific nucleus is, like the one in this study: Cerium-140 (140Ce).

Here is a simple breakdown of what the researchers did and found, using everyday analogies:

1. The Goal: Measuring the "Jelly" Shape

The scientists wanted to know the exact shape of the 140Ce nucleus. They knew it wasn't a perfect sphere, but they didn't agree on how deformed it was. Some previous studies said it was almost round; others said it was quite squashed.

To find the answer, they didn't just look at the nucleus; they bounced other nuclei into it. Think of it like trying to figure out the shape of a hidden object by throwing different balls at it and seeing how they bounce off or stick together.

2. The Experiment: The "Sticky Ball" Test

They used two different "projectiles" (the balls they threw):

Oxygen-16 (16O): A perfectly round, smooth ball.
Sulfur-36 (36S): Another round, smooth ball.

They fired these at the 140Ce target at different speeds. When the balls hit the target, sometimes they bounced off, and sometimes they fused (stuck together) to form a new, heavier nucleus.

The Key Insight: If the target nucleus (the jelly) is perfectly round, it's hard to stick. But if the jelly is wobbly or deformed, it's easier for the incoming ball to "grab on" and fuse, especially when the balls are moving slowly. By measuring exactly how often fusion happened at different speeds, the scientists could work backward to calculate the shape of the jelly.

3. The Problem with Old Tools

In the past, scientists used a method to analyze these results that was a bit like trying to draw a smooth curve by connecting dots with a shaky hand. It worked okay in the middle, but at the edges (higher energies), the lines got jagged and full of errors. This made it hard to be sure about the shape.

The New Tool: This team used a "Gaussian analytic recipe." Imagine instead of connecting dots with a shaky hand, you use a flexible ruler that automatically smooths out the wobbles. This new method gave them a much clearer, smoother picture of the "barrier distribution" (a map showing how easy or hard it was to fuse at different speeds).

4. The Results: Finding the Sweet Spot

By using this new smooth method and combining data from both the Oxygen and Sulfur experiments, they used a computer to find the "Goldilocks" shape for 140Ce.

They tested thousands of different shapes (some very round, some very squashed).
They found the specific shape that made the computer's prediction match the real-world experiment perfectly.

The Verdict: They determined that 140Ce has a specific "squishiness" (quadrupole deformation) and a specific "bulge" (octupole deformation). Their numbers were consistent whether they used the Oxygen ball or the Sulfur ball, which gave them high confidence that their answer was correct.

5. The Final Test: The "Complex" Scenario

To prove their findings were solid, they took their new shape measurements and applied them to a third, much more complicated scenario: Silicon-28 (28Si) hitting 140Ce.

This third scenario was tricky because:

The Silicon ball wasn't perfectly round; it was already a bit deformed.
There was a "bonus" effect where pairs of neutrons could be swapped between the nuclei (like swapping a pair of shoes between two dancers), which makes fusion easier.

When they ran the simulation with their newly discovered shape for 140Ce, it worked perfectly. The computer predicted exactly what happened in the real experiment.

If they used the old, disputed shape measurements, the prediction failed.
If they ignored the "neutron swap" (the shoe swap), the prediction failed.
But with their new shape + the neutron swap, the model matched reality.

Summary

This paper is like a detective story where the scientists:

Gathered clues by smashing round balls into a wobbly target.
Used a better magnifying glass (the Gaussian method) to see the clues clearly without the usual blur.
Solved the mystery of the target's shape (140Ce).
Proved they were right by using that solution to successfully predict what would happen in a much more complex, messy scenario involving a third type of ball.

They showed that by combining different experiments and using smarter math, they could finally agree on exactly how "wobbly" this specific atomic nucleus really is.

Technical Summary: Determining the Dynamic Deformation of $^{140}$ Ce by Constraining Coupled-Channels Parameters for Fusion

Problem Statement
The extraction of nuclear shape parameters (quadrupole $\beta_2$ , octupole $\beta_3$ , and hexadecapole $\beta_4$ ) remains a central challenge in nuclear physics. While fusion reactions serve as sensitive probes of nuclear deformation via coupled-channels (CC) analyses, current methodologies face significant limitations. Specifically, parameter tuning is often restricted to individual reaction systems, leading to inconsistent or unphysical results when applied to different projectile-target combinations. Furthermore, the conventional double-derivative (DD) method for deriving fusion barrier distributions ( $D$ ) is highly sensitive to step size and exhibits poor accuracy at higher energies, limiting the reliability of extracted structural information. For the nucleus $^{140}$ Ce, a known vibrational nucleus, previous studies have yielded markedly different deformation parameters (e.g., $\beta_2$ ranging from 0.003 to 0.15) depending on the experimental probe and theoretical model used.

Methodology
This study employs a systematic approach combining new experimental data, a robust analytic framework for barrier distribution, and rigorous statistical analysis across multiple reaction systems:

Experimental Data Acquisition:
- $^{16}$ O + $^{140}$ Ce: New fusion cross-section ( $\sigma_{fus}$ ) measurements were performed using the Heavy Ion Reaction Analyzer (HIRA) at the Inter-University Accelerator Centre (IUAC). Data were collected from $\simeq$ 12.4% below to $\simeq$ 17% above the Bass barrier (57–76 MeV).
- $^{36}$ S + $^{140}$ Ce: Existing fusion data from the literature were utilized.
- $^{28}$ Si + $^{140}$ Ce: Existing experimental excitation functions were used as a validation testbed.
Barrier Distribution Analysis:
- The fusion barrier distribution ( $D$ ) was derived using two methods: the conventional DD method and the Gaussian analytic recipe. The latter was chosen for its independence from step size and resilience to large uncertainties in $\sigma_{fus}$ , providing a complete, well-defined spectrum.
Coupled-Channels (CC) Calculations:
- Calculations were performed using the ccfull code with a Woods-Saxon nuclear potential.
- Target Modeling: $^{140}$ Ce was modeled as a purely vibrational nucleus incorporating one-phonon quadrupole ( $2^+$ ) and octupole ( $3^-$ ) excitations.
- Parameter Space: $\beta_2$ and $\beta_3$ were systematically varied from 0.00 to 0.60.
- Validation: The $^{28}$ Si + $^{140}$ Ce system was analyzed with a revised coupling scheme, including the projectile's intrinsic deformation (treated as both a vibrator and a rotor) and the positive Q-value two-neutron ( $2n$ ) transfer channel.
Statistical Analysis:
- $\chi^2$ Minimization: Deformation parameters were extracted by minimizing the difference between the analytic barrier distribution and theoretical CC results.
- Bayesian Analysis: Markov Chain Monte Carlo (MCMC) frameworks were used to determine posterior probability distributions for $\beta_2$ and $\beta_3$ in both the $^{16}$ O and $^{36}$ S systems.
- Model Averaging: Independent Bayesian Model Averaging (NBMA) was applied to combine the results from the two systems, yielding optimal values with associated uncertainties.

Key Results

Barrier Distribution: The Gaussian analytic recipe successfully produced a well-defined barrier distribution for the $^{36}$ S + $^{140}$ Ce system, revealing a major peak at $E \simeq 104$ MeV and a broad high-energy structure. This contrasts with the conventional DD method, which produced a spurious three-peak structure, highlighting the limitations of the DD approach at higher energies.
Extracted Deformation Parameters:
- For $^{16}$ O + $^{140}$ Ce: $\beta_2 = 0.06^{+0.07}_{-0.06}$ and $\beta_3 = 0.18^{+0.02}_{-0.04}$ .
- For $^{36}$ S + $^{140}$ Ce: $\beta_2 = 0.09^{+0.02}_{-0.02}$ and $\beta_3 = 0.18^{+0.01}_{-0.01}$ .
- NBMA Combined Result: The study determines the dynamic deformation parameters of $^{140}$ Ce to be $\beta_2 = 0.09 \pm 0.03$ and $\beta_3 = 0.18 \pm 0.02$ . These values are consistent across both projectile systems.
Validation in $^{28}$ Si + $^{140}$ Ce:
- When the extracted parameters ( $\beta_2=0.09, \beta_3=0.18$ ) were applied to the $^{28}$ Si + $^{140}$ Ce system, the CC calculations successfully reproduced both the fusion excitation function and the analytic barrier distribution.
- The inclusion of the positive Q-value $2n$ -pickup channel was found to be essential for reproducing the sub-barrier fusion enhancement and the shape of the barrier distribution.
- The treatment of $^{28}$ Si as either a vibrator or a rigid rotor resulted in negligible differences in the outcomes, suggesting the robustness of the extracted target parameters.
Comparison with Literature: The extracted $\beta_2$ value aligns closely with Pritychenko et al. ( $\beta_2 = 0.10$ ) but differs from earlier fusion-based studies suggesting $\beta_2 \approx 0.15$ . The extracted $\beta_3$ is larger than values reported in previous literature.

Significance and Claims
The paper claims that this work demonstrates the consistency and robustness of extracted deformation parameters when constrained by multiple reaction systems, addressing the ambiguity often found in single-system analyses. The study validates the Gaussian analytic recipe as an effective and sensitive tool for deriving fusion barrier distributions, proving superior to the conventional DD method in handling experimental uncertainties. Furthermore, the successful reproduction of data across three distinct projectile systems ( $^{16}$ O, $^{36}$ S, and $^{28}$ Si) using a single set of deformation parameters for $^{140}$ Ce underscores the reliability of the adopted methodology. The authors posit that such multi-system constraints are crucial for deriving physically justified nuclear structure parameters and that the Gaussian analytic approach serves as a powerful probe for intrinsic nuclear deformation.