Predicting reaction observables for the two-neutron halo candidates $^{31}$F and $^{39}$Na

This paper presents the first microscopic prediction of reaction observables for the two-neutron halo candidates $^{31}$F and $^{39}$Na by combining the deformed relativistic Hartree-Bogoliubov theory in continuum with a validated Glauber reaction model, confirming their dilute halo structures through agreement with experimental cross sections and characteristic momentum distributions.

The Gist

Imagine the atomic nucleus not as a solid marble, but as a tiny solar system. Usually, the planets (protons and neutrons) huddle tightly around the sun (the core). But in some very rare, unstable atoms, the "planets" get so loose that they drift far away, forming a giant, fuzzy cloud around the core. Scientists call this a "halo."

This paper is about hunting for two new, very heavy "halo" atoms: Fluorine-31 and Sodium-39.

Here is the story of how the researchers found them, explained simply:

1. The Challenge: Seeing the Invisible

These atoms are so unstable that they fall apart almost instantly. You can't put them in a jar to look at them. To prove they have a "halo," scientists usually smash them into a target (like a wall of carbon atoms) and measure the debris.

• The Reaction Cross Section (RCS): Think of this as measuring how "big" the atom looks when it hits the wall. If the atom has a fluffy halo, it acts like a giant, fuzzy cloud. It will hit the wall more often than a tight, hard ball of the same weight.
• The Momentum Distribution: This is like measuring how fast the pieces fly off after the crash. If the halo is very loose and spread out, the pieces fly off slowly and in a narrow, focused beam. If the atom is tight, the pieces fly off wildly in all directions.

2. The Problem: The "Crystal Ball" was Cracked

The researchers wanted to predict what would happen in these crashes before the experiments were even done. To do this, they needed a "crystal ball" (a computer model).

They had a great crystal ball for lighter halo atoms (like Lithium-11), but they weren't sure if it would work for the heavier, stranger ones like Fluorine-31 and Sodium-39. The physics gets much more complicated when you add more neutrons.

3. The Solution: A New Detective Tool

The team combined two powerful tools:

1. DRHBc Theory: This is a super-advanced mapmaker. It calculates the internal structure of the atom, telling them exactly how the neutrons are arranged and how "fuzzy" the cloud is.
2. The Glauber Model: This is the crash simulator. It takes the map from the first tool and simulates the collision with the carbon wall to predict the results.

The Test Drive:
Before using their new tool on the mystery atoms, they tested it on the famous "Lithium-11" halo. They ran the simulation, and the results matched the real-world data perfectly. It was like driving a new car on a familiar track to make sure the brakes worked. They were ready to race.

4. The Discovery: The "Fuzzy" Neighbors

They ran the simulation for Fluorine and Sodium isotopes (versions of these elements with different numbers of neutrons).

• The Fluorine Surprise: As they added more neutrons to Fluorine, the "size" of the atom (Reaction Cross Section) started to jump up unexpectedly. It was like a balloon suddenly inflating much faster than expected. The simulation showed that Fluorine-31 has a massive, dilute halo.
• The Sodium Surprise: Similarly, for Sodium, the atom Sodium-39 showed a sudden jump in size. The simulation revealed that the neutrons had moved from a tight orbit to a very wide, diffuse orbit (a "p-wave halo"), creating a giant fuzzy cloud.

The Smoking Gun:
When they looked at the "momentum distribution" (how the pieces fly off), the results for Fluorine-31 and Sodium-39 were very narrow and focused. This confirmed that the neutrons are indeed drifting far away in a loose cloud, just like a halo.

5. Why Does This Matter?

Think of the periodic table as a map of the "island of stability." Most atoms live in the middle. As you go to the edges (very neutron-rich), the islands usually sink.

This paper suggests that Fluorine-31 and Sodium-39 are new, hidden islands in the middle of the ocean. They are heavier than any other known two-neutron halo atoms.

The Takeaway:
The researchers built a bridge between the invisible quantum world inside the atom and the visible crash data outside. By proving their "crash simulator" works, they have given experimentalists a roadmap. They are saying, "Don't just guess; go look at Fluorine-31 and Sodium-39. We are 99% sure they are the heaviest halo atoms we've ever found."

This helps us understand how matter behaves under extreme pressure and could change our understanding of how stars create heavy elements.

Technical Summary

1. Problem Statement

The identification of exotic nuclei, particularly two-neutron (2n) halo nuclei, relies heavily on experimental reaction observables such as Reaction Cross Sections (RCSs) and longitudinal momentum distributions of fragments. While the existence of 2n halo nuclei like $^{11}$Li and $^{29}$F has been established, the status of heavier neutron-rich isotopes, specifically $^{31}$F and $^{39}$Na, remains under investigation.

• The Gap: Previous theoretical studies on $^{31}$F and $^{39}$Na have focused primarily on nuclear structure (e.g., binding energies, deformation, shell closures) but lacked a direct bridge to reaction observables.
• The Challenge: Experimental data for momentum distributions of fragments after 2n knockout reactions are unavailable for these specific isotopes. Without these "direct and decisive" evidences, confirming their halo nature is difficult.
• Objective: To establish a microscopic framework connecting nuclear structure theory to reaction observables to predict the 2n halo nature of $^{31}$F and $^{39}$Na and guide future experiments.

2. Methodology

The authors developed a unified theoretical framework combining microscopic nuclear structure theory with a reaction model:

• Structure Input: Deformed Relativistic Hartree-Bogoliubov theory in Continuum (DRHBc)

  • Solves the relativistic Hartree–Bogoliubov (RHB) equations using a Dirac Woods–Saxon basis, which provides superior asymptotic wave functions crucial for weakly bound systems.
  • Uses the density functional PC-PK1.
  • Provides inputs for the reaction model, including valence neutron wave functions, density distributions, and separation energies ($S_{2n}$).
  • Specifics: For $^{31}$F, calculations predict a very small $S_{2n}$ ($\approx 0.41$ MeV). For $^{39}$Na, it predicts an oblate 2n halo surrounding a prolate core with $S_{2n} \approx 1$ MeV.

• Reaction Model: Glauber Model

  • Applied in the microscopic optical-limit approximation.
  • Calculates Reaction Cross Sections ($\sigma_R$) for projectiles bombarding a carbon target ($^{12}$C).
  • Calculates Longitudinal Momentum Distributions of the residues (core nucleus) after 2n knockout reactions.
  • Validation: The model was first validated against the well-established 2n halo nucleus $^{11}$Li + $^{12}$C. The authors successfully reproduced experimental RCSs and momentum distributions for $^{11}$Li using identical structural inputs from previous works, confirming the model's reliability for 2n systems.

• Simulation Parameters:

  • Bombarding energies: 240 MeV/A (matching available experimental data for F/Na isotopes) and 1000 MeV/A (for cross-checking $^{39}$Na).

3. Key Contributions

1. First Microscopic Bridge: This is the first study to connect DRHBc structural predictions directly to Glauber reaction observables for 2n halo candidates in the medium-mass region ($A \approx 40$).
2. Model Validation: Successfully demonstrated that the Glauber model, when fed with DRHBc inputs, accurately reproduces the benchmark data for $^{11}$Li, establishing confidence in its application to unknown nuclei.
3. Prediction for $^{31}$F and $^{39}$Na: Provided quantitative predictions for RCSs and momentum distributions for isotopes where experimental momentum data is currently missing.

4. Results

A. Fluorine Isotopes ($^{21-31}$F)

• Reaction Cross Sections (RCSs):
  • Calculated RCSs for $^{21-29}$F agree well with existing experimental data.
  • Anomaly: A pronounced increase in RCS is predicted for $^{29}$F and $^{31}$F, deviating significantly from the linear trend observed in lighter neighbors ($^{25}$F to $^{27}$F). This "kink" suggests a structural change consistent with halo formation.
• Momentum Distributions:
  • The longitudinal momentum distributions for the residues after 2n removal from $^{29}$F and $^{31}$F exhibit pronounced narrow peaks.
  • Narrow distributions are a hallmark of dilute spatial extension (halo structure).

B. Sodium Isotopes ($^{22-39}$Na)

• Reaction Cross Sections (RCSs):
  • Results up to $^{35}$Na match experimental data.
  • A significant jump in RCS is predicted for $^{39}$Na compared to the linear extrapolation of lighter isotopes.
  • Mechanism: This increase is attributed to a transition of valence neutrons from dominant $d$-orbitals to spatially diffuse $p$-orbitals, facilitating a $p$-wave halo structure.
  • The prediction at 1000 MeV/A for $^{39}$Na aligns with an independent theoretical study by Horiuchi et al., further validating the model.
• Momentum Distributions:
  • The distribution for $^{39}$Na is significantly narrower than that of $^{37}$Na.
  • Quantitative Metric: The Full Width at Half Maximum (FWHM) decreases from 182 MeV/c ($^{37}$Na) to 158 MeV/c ($^{39}$Na). This narrowing confirms the dilute 2n halo nature of $^{39}$Na.

5. Significance

• Confirmation of Halo Candidates: The study provides strong theoretical evidence that $^{31}$F and $^{39}$Na are promising candidates for two-neutron halo nuclei, characterized by dilute spatial distributions and narrow momentum widths.
• Guidance for Future Experiments: By predicting specific reaction observables (RCSs and momentum distributions), the paper offers concrete targets for future experiments at facilities like RIKEN (Radioactive Isotope Beam Factory).
• Methodological Advancement: The successful application of the DRHBc + Glauber approach establishes a robust tool for exploring the "island of inversion" and identifying new halo nuclei in the medium-mass region, extending beyond the traditional light-mass halo nuclei.
• Structural Insight: The work highlights the role of shell evolution (e.g., the collapse of the $N=28$ shell closure in $^{39}$Na) and orbital transitions ($d$ to $p$) in driving the formation of heavy halo structures.