Nuclear Reaction Data for Fission Products Off Stability

This paper outlines a methodology combining machine learning, improved predictive modeling that accounts for nuclear deformation, and experimental data to address the lack of reliable neutron cross-section data for unstable fission products, with the goal of producing evaluated files for future ENDF/B releases.

The Gist

Imagine you are trying to predict the weather. If you are looking at a well-known city with thousands of weather stations, you have a very accurate forecast. But if you are trying to predict the weather in a remote, uncharted jungle where no one has ever stood, you have to guess based on what you know about similar jungles elsewhere.

This paper is about doing exactly that, but for atomic nuclei instead of weather.

The Problem: The "Uncharted Jungle" of Atoms

Scientists need to know how neutrons (tiny particles) interact with specific atoms created during nuclear fission (splitting atoms). This is crucial for things like managing nuclear waste, ensuring nuclear safety, and understanding how stars work.

For stable atoms (the ones that exist naturally on Earth), we have "weather stations"—lots of real experiments and data. We know exactly how they behave. But for unstable fission products (atoms created in reactors that don't last long), there are almost no experiments. It's like trying to forecast the weather in that remote jungle with zero data.

Currently, scientists have to use "simplified guesses" (theoretical models) to fill in the blanks. The problem is, these guesses often assume atoms are perfect spheres, like billiard balls. But many of these unstable atoms are actually squashed or stretched, like rugby balls or distorted blobs. Using a "billiard ball" model for a "rugby ball" leads to big errors.

The Solution: A Smarter Toolkit

The authors, a team from Brookhaven, Lawrence Livermore, and Ohio University, are building a new toolkit to get better answers. They call their project RREFPOS (Realistic Reaction Evaluations for Fission Products Off Stability).

Here is how they are fixing the problem, using three main tools:

1. The "Shape-Shifter" Model (Accounting for Deformation)
Instead of pretending all atoms are perfect spheres, they are using a new method that accounts for the atom's actual shape.

• The Analogy: Imagine throwing a ball at a wall. If the wall is flat (a sphere), the ball bounces back predictably. If the wall is curved or bumpy (a deformed nucleus), the ball bounces differently.
• The Fix: They are using a "coupled-channels" approach that treats these atoms like rugby balls. They feed the computer the specific "deformation parameters" (how squashed or stretched the atom is) so the math reflects reality, not a simplified fantasy.

2. The "AI Translator" (Machine Learning)
Since they can't measure every single unstable atom, they are using Artificial Intelligence to help.

• The Analogy: Think of a translator who knows how to speak "German" and "French." If you ask them to translate a sentence from a language they've never seen ("Swahili"), they might struggle. But if you give them a dictionary of how German and French relate, they can make a very educated guess about Swahili based on those patterns.
• The Fix: They are training a neural network (a type of AI) to learn the patterns of how neutron reactions work across the "map of atoms." The AI doesn't just guess; it uses advanced physics theories to look at a known neighbor atom and translate that knowledge to the unknown, unstable atom. This gives them a "best guess" that is much smarter than a random roll of the dice.

3. The "New Weather Station" (Experimental Measurements)
To make sure their guesses are right, they are building new "weather stations" in the lab.

• The Analogy: Instead of just guessing the weather in the jungle, they are sending a drone up to take a few direct measurements.
• The Fix: They are performing new experiments (using particle accelerators) to measure the "nuclear level density" (a fancy way of counting how many energy states an atom has) for specific atoms like Zirconium and Niobium. This provides real data to anchor their models, ensuring the AI and the shape-shifting math aren't drifting off course.

The Goal: A Better "User Manual" for Atoms

The ultimate goal is to create a new, high-quality "user manual" (called an evaluated file) for these unstable atoms.

• Current State: The manual is full of blank pages or rough scribbles because we lack data.
• Future State: They want to fill those pages with realistic data that accounts for the weird shapes of these atoms and uses AI to fill in the gaps.

They plan to submit these new manuals to the ENDF/B library, which is the global database that engineers and scientists use to design reactors and analyze nuclear events. By making this data more accurate, they hope to improve the safety and efficiency of nuclear energy and non-proliferation efforts.

In short: They are moving from "guessing the weather in a jungle" to "using drones, AI, and shape-shifting math to map the jungle accurately," so we can navigate it safely.

Technical Summary

Technical Summary: Realistic Reaction Evaluations for Fission Products Off Stability (RREFPOS)

Problem Statement
Accurate modeling of isotope generation and depletion is critical for applications ranging from nuclear nonproliferation and forensics to spent-fuel assay, reactor design, and astrophysics. These applications require complete descriptions of neutron cross sections for fission products. While evaluated nuclear data libraries (e.g., ENDF/B-VIII.1) provide reliable data for isotopes on or near stability, data for unstable fission products are often scarce or non-existent. Consequently, evaluators must rely on theoretical extrapolations. Current alternatives, such as the TENDL library, often employ simplified assumptions—such as stochastic resonance generation or spherical optical model potentials—that fail to account for nuclear deformation and lack quantified uncertainties. These limitations render existing data insufficient for high-fidelity applications involving unstable nuclei.

Methodology
The RREFPOS project aims to develop and disseminate complete neutron evaluations for off-stability fission products, specifically targeting nuclei produced in the spontaneous fission of $^{235}$U, with secondary goals for $^{239}$Pu and $^{252}$Cf. The methodology addresses three energy regimes:

1. Resonance Region:

   • Challenge: Unstable nuclei lack experimental resonance data for direct R-matrix fitting. Stochastic generation of resonance parameters (as in TENDL) often yields meaningless results unless averaged, and current libraries do not include uncertainties in average parameters.
   • Approach: The project leverages machine learning to establish the cross-section probability distribution, $P(\sigma|E, T)$, as a function of neutron energy ($E$) and target temperature ($T$). Rather than generating random resonance sets, the goal is to develop a model that predicts $P(\sigma|E, T)$ directly from average resonance parameters ($\Gamma_\gamma, \Gamma_n, D$), similar to approaches used for unresolved resonances in reactor calculations.

2. Fast-Range Reaction Calculations:

   • Reaction Modeling: In the absence of direct measurements, the project prioritizes microscopic predictive models. A critical component is the Optical Model Potential (OMP). Recognizing that most nuclei of interest are well-deformed (as indicated by $E_{4^+}/E_{2^+}$ ratios near 3.3), the project rejects spherical-nucleus OMPs as poor approximations. Instead, it employs an adiabatic model based on multipole deformation parameters. This model is extended to off-stability nuclei by combining it with nuclear deformations derived from microscopic models, such as HFB nuclear densities.
   • Machine Learning Priors: The project utilizes generative neural models trained on systematic trends across the nuclear chart (informed by density functional theory). These networks translate cross-sections from stable to neighboring unstable isotopes. The outputs are not used as final values but as informative priors to guide and constrain model parameters, ensuring the extrapolation remains grounded in theoretical trends.

3. Level Density Measurements:

   • Challenge: Current Nuclear Level Density (NLD) models are constrained only by discrete levels and specific resonance spacings, offering limited guidance for off-stability nuclei.
   • Approach: The project conducts targeted experimental measurements using the particle evaporation technique at the Edwards Accelerator Lab. This method provides NLD data over a wider excitation energy range (from ground state to neutron separation energy) and broader spin ranges, specifically focusing on mass regions near the stability line to improve parameter extrapolation reliability.

Key Contributions and Preliminary Results

• Experimental Campaigns: Two experimental campaigns have been completed, yielding NLD measurements for $^{92,96}$Nb and constraints for $^{94}$Nb via $(p,n)$ reactions. Analysis of $(d,n)$ reaction data for $^{95}$Nb is ongoing.
• Modeling Infrastructure: An infrastructure has been developed to integrate nuclear deformation databases with the EMPIRE reaction code, enabling the adiabatic coupled-channels approach for all relevant nuclei.
• Preliminary Calculations: Fast-region calculations have been formatted into ENDF-6 files. A comparison for the $^{98}$Zr(n,total) cross section demonstrates a substantial difference between the adiabatic coupled-channels approach (accounting for deformation) and the direct use of a spherical OMP. The spherical model closely follows TENDL predictions, whereas the deformation-inclusive model provides a significantly more realistic description for nuclei with strong dynamic quadrupole deformation (e.g., $\beta_2 \approx 0.33$).

Significance and Future Outlook
The primary significance of this work lies in moving beyond simplified assumptions to produce evaluated files that account for nuclear deformation and utilize machine learning to constrain model parameters. The project aims to submit these evaluated files to the ENDF/B library for consideration in the future ENDF/B-IX.0 release.

The authors emphasize that once the evaluation system is established for the primary goal (nuclei in the first fission yield hump of $^{235}$U), the effort required to extend the process to other unstable fission products (secondary and "stretch" goals) will be relatively small. The ultimate objective is to provide the nuclear data community with realistic, validated evaluations for off-stability fission products, enabling better testing, feedback, and fine-tuning for nonproliferation and reactor applications.