⚛️ nuclear theory

Neutron radii of Mg isotopes and semiphenomenological treatment for neutron distributions

This study extracts neutron radii for $^{24-38}$ Mg isotopes by combining Glauber model reaction cross-section calculations with a semiphenomenological core+n description to validate a one-neutron halo structure in $^{37}$ Mg and predict a two-neutron halo-like structure in $^{40}$ Mg.

Original authors: Govind Kumar, M. Imran, Z. Hasan, Z. A. Khan

Published 2026-08-11

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CC BY 4.0

Original authors: Govind Kumar, M. Imran, Z. Hasan, Z. A. Khan

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 the atom not as a tiny, static solar system, but as a bustling, fuzzy cloud of energy. At the very center sits a dense, heavy nucleus, packed with protons and neutrons. For decades, scientists have been trying to map out exactly how these particles are arranged. We know the protons (which carry a positive charge) form a sort of "skin" that defines the atom's electrical size, but the neutrons (which are neutral) are trickier. They don't have a charge, so they are invisible to the tools we usually use to see atoms. However, in the wild, unstable world of "neutron-rich" atoms, these neutrons can sometimes stretch out far beyond the protons, forming a fuzzy, ghostly halo or a thick, puffy skin. Understanding this arrangement is like reading the blueprint of the universe; it helps us figure out how the most extreme objects in space, like neutron stars, are built and how they behave under crushing pressure.

Now, picture a team of physicists acting like cosmic detectives. They are investigating a specific family of atoms called Magnesium (Mg). Some of these Magnesium atoms are stable and common, but others are exotic, unstable, and packed with extra neutrons. The scientists wanted to know: How far do these extra neutrons stretch out? To find out, they couldn't just look at the atoms; they had to smash them. They fired these Magnesium atoms at a target of Carbon atoms at incredibly high speeds—240 million electron volts per nucleon. By watching how the Magnesium atoms scattered and interacted with the Carbon, they could work backward to figure out the size of the neutron cloud. They used two different mathematical "lenses" to interpret the data: one that treats the nucleus like a neat, organized grid of particles (SDHO), and another that treats it like a smooth, fuzzy ball (2pF).

The paper's main goal was to extract the "neutron radius" (how big the neutron cloud is) for Magnesium isotopes ranging from mass 24 to 38. They found that while both mathematical lenses gave similar trends, the "fuzzy ball" lens (2pF) and the "organized grid" lens (SDHO) sometimes disagreed on the exact size. However, by adjusting their models to match the experimental data, they successfully mapped out the neutron radii for these isotopes.

Here is where the story gets really interesting. The authors noticed that for some of these unstable Magnesium atoms, the neutrons weren't just sitting close to the core; they were stretching out into space, almost like a tail. To explain this, they proposed a clever, semi-phenomenological idea: treat the atom as a "core" (the main body) plus a "tail" (one or two loose neutrons). They called this the "core+n" (core plus one neutron) or "core+2n" (core plus two neutrons) approach.

When they applied this "core plus tail" idea to the data, something remarkable popped out. For the isotope Magnesium-37, the model showed a massive spread of neutrons, confirming that it has a "one-neutron halo"—a single neutron wandering far away from the rest of the nucleus. This matched previous suspicions but gave it a solid mathematical backing.

Encouraged by this success, the team took a leap of faith and applied the same logic to an even more exotic, unstable atom: Magnesium-40. Since Magnesium-40 has a very specific energy balance where it's easier to remove two neutrons at once than one, they treated it as a "core+2n" system (a Magnesium-38 core with two loose neutrons). They predicted what its neutron radius and reaction cross-section (how likely it is to hit the Carbon target) would be. Their calculations suggested that Magnesium-40 has a "two-neutron halo," meaning it has a puffy, extended cloud of two neutrons stretching far out.

The paper doesn't claim to have "solved" the mystery of all atomic nuclei, nor does it say this is the final word. Instead, it suggests that this "core plus tail" approach is a promising tool. It seems to work well for predicting the behavior of these exotic atoms, especially when direct measurements are hard to get. The findings support the idea that Magnesium-37 is a one-neutron halo and that Magnesium-40 likely has a two-neutron halo structure, offering a new way to visualize the fuzzy, stretched-out edges of the atomic world.

Technical Summary: Neutron Radii of Mg Isotopes and Semiphenomenological Treatment for Neutron Distributions

Problem Statement
Reliable information regarding the spatial distribution of protons and neutrons is essential for understanding exotic phenomena in neutron-rich nuclei, such as neutron halos and neutron skins, as well as the equation of state for nuclear matter. While electron scattering and isotope shift measurements effectively determine charge radii for stable and some unstable nuclei, they are limited for exotic species far from the $\beta$ -stability line. Although charge-changing cross-section (CCCS) measurements offer an alternative for proton distribution, they are unavailable for the isotopic chains of Magnesium (Mg), Neon, and Sodium. Consequently, researchers must rely on model-dependent estimates for charge radii and analyze reaction (interaction) cross-sections ( $\sigma_R$ ) to infer neutron radii and skin thickness. This work addresses the extraction of neutron radii for $^{24-38}\text{Mg}$ isotopes and the investigation of their asymptotic neutron distributions, specifically probing for halo structures in $^{37}\text{Mg}$ and $^{40}\text{Mg}$ .

Methodology
The study employs the Glauber model to analyze reaction cross-sections of Mg isotopes on a $^{12}\text{C}$ target at 240 MeV/nucleon. The theoretical framework incorporates:

Input Parameters: Proton radii are fixed using values calculated via the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc). Neutron radii are extracted by varying density distribution parameters to fit experimental $\sigma_R$ data.
Density Distributions: Two distinct nuclear density descriptions are utilized:
- SDHO: Slater determinants involving harmonic oscillator single-particle wave functions.
- 2pF: Two-parameter Fermi distributions.
Semiphenomenological Core+Tail Approach: To understand the asymptotic behavior (spread) of neutron distributions, the authors propose a "core+n" (for $S_n < S_{2n}$ $S_{n} < S_{2 n}$ ) or "core+2n" (for $S_n > S_{2n}$ $S_{n} > S_{2 n}$ ) description. In this model, the nucleus is treated as a core plus one or two valence neutrons. The density is formulated as $\rho_n(r) = \rho_{\text{core}}(r) + \rho_{\text{tail}}(r)$ $ρ_{n} (r) = ρ_{core} (r) + ρ_{tail} (r)$ , where the tail term accounts for the spread of loosely bound neutrons.
- Unlike previous approaches that fixed core parameters based on free-state charge radii, this work treats the core's density parameters (oscillator constant $\alpha_n^2$ for SDHO or diffuseness $a_n$ for 2pF) as variable parameters. These are adjusted to reproduce the neutron radii extracted from the $\sigma_R$ analysis, acknowledging that the interaction between the valence neutron and the core may alter the core's effective radius within the bound system.

Key Contributions and Results

Extraction of Neutron Radii: The authors successfully extracted neutron radii for $^{24-38}\text{Mg}$ isotopes. Comparisons between SDHO and 2pF densities revealed that while both reproduce experimental $\sigma_R$ reasonably well, the SDHO density generally yields smaller deviations from experimental data than the 2pF density.
Neutron Skin Thickness: The study assessed the density dependence of neutron skin thickness ( $r_n - r_p$ ). While the trend was consistent across both density models, absolute values differed. The results indicate that $^{37}\text{Mg}$ possesses the largest neutron skin thickness in the chain, coinciding with its low one-neutron separation energy ( $S_n = 0.240$ MeV) and a sharp rise in reaction cross-section.
Validation of Core+n Description: For isotopes $^{25-38}\text{Mg}$ (where $S_n < S_{2n}$ ), the core+n description was employed. The model successfully reproduced the experimental reaction cross-sections, validating the approach. The resulting neutron density distributions clearly demonstrated a one-neutron halo structure for $^{37}\text{Mg}$ .
Prediction for $^{40}\text{Mg}$ : Extending the methodology to $^{40}\text{Mg}$ $^{40} Mg$ (where $S_n > S_{2n}$ $S_{n} > S_{2 n}$ ), the authors utilized a core+2n description ( $^{38}\text{Mg} + 2n$ $^{38} Mg + 2 n$ ). Using the extracted neutron radius of $^{38}\text{Mg}$ $^{38} Mg$ as the core input, they predicted the neutron radius and reaction cross-sections for $^{40}\text{Mg}$ $^{40} Mg$ at 240 and 1000 MeV/nucleon.
- The predicted values showed a significant enhancement in both neutron radius and $\sigma_R$ compared to lighter Mg isotopes.
- These findings support the existence of a two-neutron halo-like structure in $^{40}\text{Mg}$ , consistent with previous three-body model predictions by Singh et al.

Significance and Claims
The paper claims that the proposed semiphenomenological core+n (and core+2n) approach is a promising tool for assessing the credibility of extracted neutron radii and predicting reaction cross-sections for isotopes where experimental data is scarce. By treating the core's density parameters as variables constrained by the extracted neutron radius of the full nucleus, the model accounts for the modification of the core by the valence nucleons. The work successfully identifies the one-neutron halo in $^{37}\text{Mg}$ and predicts a two-neutron halo structure in $^{40}\text{Mg}$ , providing a consistent theoretical framework for understanding neutron distributions in unstable nuclei using reaction cross-section data and DRHBc proton radii.

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