Ab initio optical potentials for magnesium isotopes: from stability to the island of inversion

This paper presents the first *ab initio* nonlocal optical potential calculations for magnesium isotopes ($^{24,26,28,32}$Mg) using the symmetry-adapted no-core shell model and multiple-scattering theory, successfully reproducing experimental data for $^{24}$Mg and providing parameter-free predictions for heavier isotopes that validate global models near the N=20 island of inversion while highlighting their limitations.

The Gist

The Big Picture: Predicting How Tiny Billiard Balls Bounce

Imagine you are trying to predict how a billiard ball will bounce when it hits a cluster of other balls. In the world of physics, these "balls" are protons and neutrons (nucleons) that make up the nucleus of an atom.

Scientists often want to shoot a particle (like a neutron or proton) at an atomic nucleus to learn what's inside it. To do this, they need a "map" of how the particle will interact with the nucleus. This map is called an Optical Potential.

For decades, scientists have used "phenomenological" maps. Think of these like Google Maps built from traffic reports. They are incredibly useful because they are based on real data from millions of cars (experiments) on stable roads (stable atoms). However, if you try to use Google Maps to drive on a brand-new, uncharted road in a remote jungle (a rare, unstable atom), the map might be wrong because it's just guessing based on patterns from the city.

This paper is about building a new kind of map from scratch. Instead of guessing based on traffic reports, the authors built their map using the fundamental laws of physics (the "blueprints" of the universe) to calculate exactly how the atoms should behave, even for atoms that don't exist naturally on Earth.

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The Cast of Characters

1. The Magnesium Isotopes: The authors focused on a family of atoms called Magnesium.
   • 24Mg: The "stable" family member. It's like a well-behaved adult. We have lots of data on it.
   • 32Mg: The "wild child." It lives in a place physicists call the "Island of Inversion." It's unstable, rare, and behaves strangely. It's like a teenager who breaks all the rules of how atoms are supposed to act.
2. The SA-NCSM (The Architect): This is a super-smart computer program used to calculate the internal structure of these atoms. It's like a 3D printer that builds a perfect model of the atom's interior, showing exactly where every proton and neutron is sitting.
3. The Spectator Expansion (The Rulebook): This is the mathematical method used to figure out how the incoming particle bounces off the nucleus.
   • The Analogy: Imagine a crowded dance floor (the nucleus). A new dancer (the projectile) tries to cut in.
   • Leading Order: The simplest rule: The new dancer bumps into one person on the floor.
   • Higher Orders: The new dancer bumps into one person, who bumps into another, who bumps into a third (a chain reaction).
   • The authors used the "Leading Order" (just one bump) because it's the most important part, though they admit that at lower energies, the "chain reactions" start to matter more.

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What They Did (The Experiment)

The team did two main things:

1. They tested their new map on the "well-behaved" atom (24Mg).
They used their "from-scratch" physics calculations to predict how neutrons and protons would bounce off Magnesium-24. They compared their predictions to real-world data from experiments.

• The Result: Their map was spot on. It matched the real-world data almost perfectly for high-energy collisions. It was like their new GPS predicted the traffic in the city exactly as well as the old, data-heavy Google Maps.

2. They ventured into the "jungle" (26Mg, 28Mg, and 32Mg).
Since they couldn't easily get experimental data for the unstable, rare isotopes (like 32Mg), they used their new "from-scratch" map to predict what would happen.

• They compared their predictions to the "old" maps (the phenomenological ones like KDUQ) that scientists usually use for these rare atoms.
• The Surprise: The old maps were mostly right, but they were a bit too optimistic. They predicted that the particles would be absorbed (stopped) by the nucleus slightly less than the new physics-based map predicted.
• The "Island of Inversion" Check: For the wild child, 32Mg, the old maps didn't show much uncertainty. But the authors' new method showed that because 32Mg is so weird, we should actually be more unsure about the predictions. The old maps were hiding the fact that they were just guessing.

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Why This Matters (The Takeaway)

1. We don't need to guess anymore.
For rare atoms that are hard to study, scientists usually have to guess how they behave. This paper shows that we can calculate these behaviors using pure math and physics, without needing to tweak numbers to fit data. It's like calculating the trajectory of a rocket using physics equations rather than just looking at where previous rockets landed.

2. The "Old Maps" are good, but have limits.
The paper validates that the old, popular maps (KDUQ) are actually quite good at predicting what happens in these rare isotopes. This gives scientists confidence to keep using them for now. However, the paper also warns that these maps might be underestimating the "uncertainty" (the risk of being wrong) when we get to the very edge of the periodic table.

3. A Bridge to the Future.
This work acts as a bridge. It connects the "real world" of experimental data with the "theoretical world" of pure physics. It proves that our fundamental understanding of how atoms work is strong enough to predict the behavior of atoms we haven't even seen yet.

In a Nutshell

The authors built a physics-based GPS for atomic collisions. They tested it on a known city (stable Magnesium) and found it worked perfectly. Then, they used it to navigate a remote jungle (unstable Magnesium). They found that while the old, data-based maps were helpful, the new GPS gave a more honest picture of the risks and uncertainties involved in exploring the unknown frontiers of the atomic world.

Technical Summary

1. Problem Statement

The study of nuclear reactions involving rare isotopes (near the drip lines) relies heavily on optical potentials to model the interaction between a projectile and a target nucleus. Currently, the field depends on global phenomenological optical potentials (e.g., Koning-Delaroche, Weppner-Penney) which are fitted to experimental data from stable nuclei and then extrapolated to exotic systems.

• Limitations: These extrapolations introduce significant uncertainties, particularly for nuclei far from stability (e.g., the $N=20$ "island of inversion") where elastic scattering data is scarce or non-existent.
• Gap: There is a critical need for parameter-free, microscopic ($ab$ $initio$) optical potentials that treat nuclear structure and reaction dynamics on an equal footing, providing reliable predictions for unstable isotopes without adjustable parameters.

2. Methodology

The authors developed a framework to calculate nonlocal, translationally invariant optical potentials from first principles for Magnesium isotopes ($^{24,26,28,32}\text{Mg}$).

• Structure Input (SA-NCSM):

  • Used the Symmetry-Adapted No-Core Shell Model (SA-NCSM) to compute ground-state wavefunctions.
  • Employed the NNLOopt chiral nucleon-nucleon (NN) interaction, which is consistent with Quantum Chromodynamics (QCD) and calibrated to NN scattering data up to 135 MeV.
  • Key Advantage: The SA-NCSM leverages emergent symplectic symmetries ($Sp(3,R) \supset SU(3) \supset SO(3)$) to drastically reduce the basis space, allowing calculations for deformed, open-shell nuclei (up to $A \approx 48$) that are computationally intractable for standard NCSM.
  • Generated translationally invariant, off-shell scalar and spin-projected one-body densities and momentum distributions.

• Reaction Theory (Spectator Expansion):

  • Applied the leading-order (LO) term of the spectator expansion of multiple-scattering theory.
  • In this approximation, the projectile interacts with a single target nucleon (two active nucleons total).
  • The effective interaction is constructed by folding the NN amplitudes (Wolfenstein parameterization) with the microscopic densities derived from SA-NCSM.
  • Solved the Watson optical potential integral equation to account for the difference between proton and neutron densities and to generate the full optical potential ($U_p$).

• Comparative Models:

  • Results were benchmarked against the Koning-Delaroche Uncertainty Quantified (KDUQ) global potential (Bayesian calibrated) and the Weppner-Penney (WP) potential.
  • Comparisons were also made with ENDF nuclear data evaluations.

3. Key Contributions

• First $ab$ $initio$ Calculations for Mg Chain: This is the first application of leading-order spectator expansion optical potentials to a chain of Magnesium isotopes spanning from stable $^{24}\text{Mg}$ to the neutron-rich, deformed $^{32}\text{Mg}$ (located in the $N=20$ island of inversion).
• Parameter-Free Predictions: The approach utilizes no adjustable parameters; the only inputs are the chiral NN interaction and the many-body structure calculation.
• Validation of Extrapolation: The study provides a rigorous test of whether phenomenological potentials (like KDUQ) can be trusted when extrapolated to the drip line by comparing them against a microscopic baseline.
• Handling Deformation: Demonstrated that the SA-NCSM framework is uniquely suited to handle the strong deformation and collectivity present in these isotopes, which often challenge standard shell-model approaches.

4. Results

A. Validation on $^{24}\text{Mg}$ (Stable)

• Neutron Total Cross Sections (65–250 MeV): The $ab$ $initio$ predictions matched experimental data exceptionally well for energies $>100$ MeV. Slight deviations at lower energies (65–80 MeV) were attributed to the omission of higher-order rescattering terms (beyond leading order).
• Proton Scattering: Excellent agreement was found for differential cross sections and analyzing powers ($A_y$) across the energy range.
• Model Dependence: Results showed minimal sensitivity to the harmonic oscillator frequency ($\hbar\Omega$) and model space size ($N_{max}$) at higher energies, confirming convergence.
• Comparison with KDUQ: While KDUQ agreed well with data (as it was fitted to $^{24}\text{Mg}$), it began to diverge at higher angles and energies (250 MeV), highlighting the limits of its extrapolation range.

B. Predictions for Exotic Isotopes ($^{26,28,32}\text{Mg}$)

• Neutron Total Cross Sections: Showed a gradual increase from $^{24}\text{Mg}$ to $^{32}\text{Mg}$. The $ab$ $initio$ results agreed well with ENDF evaluations for $^{26,28}\text{Mg}$ above 65 MeV.
• Proton Reaction Cross Sections:
  • The $ab$ $initio$ model predicted a systematic rise in reaction cross sections across the isotopic chain, consistent with the increasing charge radius.
  • Discrepancy: The KDUQ and WP phenomenological potentials systematically underpredicted the reaction cross sections by approximately 10–15% compared to the $ab$ $initio$ results.
  • Volume Integrals: Analysis of the volume integrals revealed that the microscopic potential possesses a larger imaginary strength (absorption) than KDUQ, particularly at low energies. This explains the higher reaction cross sections in the $ab$ $initio$ model.
• Differential Scattering:
  • For $^{32}\text{Mg}$, the diffraction pattern shifted to smaller angles due to the larger nuclear radius.
  • KDUQ maintained good agreement with the $ab$ $initio$ results at forward angles but showed increasing uncertainty in the shape of the angular distribution at larger angles.
  • Notably, KDUQ's uncertainty bands did not significantly widen when moving from stable $^{24}\text{Mg}$ to neutron-rich $^{32}\text{Mg}$, suggesting the model may underestimate epistemic uncertainty when extrapolating to the drip line.

5. Significance and Outlook

• Validation of Phenomenology: Despite the quantitative differences, the study validates the use of KDUQ for analyzing experimental data on neutron-rich Magnesium isotopes, particularly for forward-angle scattering. This provides confidence for users of reaction codes (e.g., TALYS) working in the $A \approx 30$ mass region.
• Limitations of Global Models: The systematic underprediction of reaction cross sections and the lack of increased uncertainty in KDUQ for exotic nuclei highlight the limitations of linear extrapolation schemes based on $(N-Z)/A$. Future phenomenological models may need to include higher-order terms or model-discrepancy inference.
• Future Applications: The derived $ab$ $initio$ optical potentials can be generated on demand for any energy between 65–250 MeV. They serve as robust inputs for other reaction channels (e.g., knockout, charge-exchange) where experimental data is lacking, offering a path toward predictive nuclear physics for rare isotopes.

In conclusion, this work successfully bridges the gap between microscopic nuclear structure and reaction theory, providing a benchmark for global optical potentials and demonstrating that $ab$ $initio$ methods can reliably predict scattering observables for deformed, unstable nuclei near the drip line.