Westcott $g$ Factors Extended to Arbitrary Neutron Energy Spectra

This paper presents an updated methodology and open-source software for calculating Westcott $g$ factors using ENDF/B-VIII.1 cross-section data across both Maxwellian and arbitrary non-Maxwellian neutron spectra, specifically addressing the needs of guided thermal and cold-neutron beams at facilities like the Budapest Research Reactor and FRM II.

The Gist

Imagine you are a chef trying to bake a perfect cake. You have a recipe that says, "Add 1 cup of flour." But in your kitchen, the flour isn't sitting in a neat, uniform pile; it's a chaotic mix of fine dust, medium grains, and a few big clumps. If you just scoop a cup without thinking about the size of the grains, your cake might turn out dry or dense.

In the world of nuclear physics, scientists face a similar problem when they try to measure elements using Neutron Activation Analysis (NAA) or Prompt Gamma-ray Activation Analysis (PGAA). They shoot neutrons (tiny subatomic particles) at a sample to see what elements are inside. To do the math correctly, they need to know exactly how the neutrons are behaving.

For decades, scientists used a "standard recipe" called the Westcott $g$-factor. Think of this factor as a correction knob or a translation tool.

The Problem: The "Perfect" vs. The "Real"

Most elements play by the rules: the faster a neutron moves, the less likely it is to get caught by the nucleus. This is called the $1/v$ law (inverse velocity). For these "good students," the correction knob is set to 1. It's simple.

But some elements are "troublemakers" (called irregular nuclei). They have "resonances"—like a swing that only moves when pushed at a very specific rhythm. If the neutrons hit them at just the right speed, these nuclei gobble them up much more aggressively than the standard rules predict.

Historically, scientists assumed the neutrons in their experiments were like a perfectly mixed pot of soup (a Maxwellian distribution), where the temperature is uniform. They calculated their correction knob ($g$-factor) based on this perfect soup.

The Discovery: The Soup is Actually a Smoothie

The authors of this paper, D.A. Matters and colleagues, realized that in many modern labs (like the Budapest Research Reactor and FRM II), the neutrons aren't a uniform soup. They are more like a specialized smoothie or a guided beam.

Because of the way these labs use "neutron guides" (mirrors that reflect neutrons) and cold sources, the neutrons arrive with a very specific, weird energy distribution. It's not a smooth, predictable curve anymore.

The Analogy:
Imagine you are trying to catch fish.

• Old Method: You assume the fish are swimming in a calm, uniform lake. You use a standard net size (the Westcott $g$-factor) to estimate how many you'll catch.
• New Reality: The fish are actually swimming in a fast-moving, narrow river with a weird current. If you use the "calm lake" net size, you will either catch too few or too many fish, and your data will be wrong.

The Solution: A Custom Calculator

The paper introduces a new, open-source software tool called WestcottFactors (and updates to an existing tool called DeCE).

Think of this software as a smart, custom-fit net. Instead of guessing the water conditions, you can feed the software the actual map of the river (the measured neutron energy spectrum). It then calculates the exact correction factor needed for that specific situation.

Why This Matters

1. Accuracy: For "troublemaker" elements (like Gadolinium or Lutetium), using the old "perfect soup" assumption could lead to errors of 20% or more. That's a huge mistake in science.
2. Flexibility: The new tools work for any neutron beam, not just the theoretical ones. Whether you are at a standard reactor or a high-tech facility with guided beams, you can get the right answer.
3. Open Source: The authors didn't keep this secret. They built the software and put it on GitHub (like a public library of code) so anyone can use it to check their own experiments.

The Bottom Line

This paper is like telling the scientific community: "Stop using the generic map for a city you've never visited. We've built a GPS that works for the actual streets you're driving on."

By using these new tools, scientists can ensure their "cakes" (experimental results) are baked perfectly, regardless of how chaotic the "flour" (neutron spectrum) actually is.

Technical Summary

1. Problem Statement

In Neutron Activation Analysis (NAA) and Prompt Gamma-ray Activation Analysis (PGAA), accurate quantification of reaction rates requires correcting for deviations from the ideal $1/v$ law (where neutron capture cross-section $\sigma \propto 1/v$). Many nuclei, termed "irregular" or "non-$1/v$" absorbers, possess neutron resonances in the low-energy range (< 5 eV) that distort this behavior.

To address this, the Westcott $g$-factor is used as a correction factor to relate the effective cross-section ($\sigma_{eff}$) to the thermal cross-section ($\sigma_0$) at 2200 m/s. Historically, these factors have been calculated assuming a Maxwellian neutron flux distribution defined by a single temperature ($T$).

The Core Issue:
Modern experimental facilities (e.g., Budapest Research Reactor and FRM II) utilize guided thermal and cold-neutron beams. These beams undergo spectral shaping via supermirror guides and cryogenic moderators, resulting in non-Maxwellian energy spectra.

• Using a standard Maxwellian approximation for these complex spectra introduces significant errors in the calculated $g$-factors.
• Existing tabulated $g$-factors often rely on simplified resonance approximations (using only the lowest-energy resonance) or outdated cross-section libraries, which fail to capture the complexity of multiple overlapping resonances in irregular nuclei.

2. Methodology

The authors developed a comprehensive framework to calculate Westcott $g$-factors for arbitrary neutron energy spectra, moving beyond the Maxwellian assumption.

A. Theoretical Framework

The paper defines the $g$-factor generally as:
$$g = \frac{\sigma_{eff}}{\sigma_0} = \frac{1}{\sigma_0 v_0} \frac{\int_0^\infty \sigma(v)\phi(v) dv}{\int_0^\infty \phi(v)/v \, dv}$$
where $\sigma(v)$ is the neutron-capture cross-section and $\phi(v)$ is the neutron flux distribution.

B. Computational Approaches

Two distinct methods were implemented and compared:

1. Irregularity Function Method: An approximation using the Breit-Wigner resonance formula (Lorentzian lineshape) based on the lowest-energy resonance. This is the traditional method found in many handbooks.
2. Full Cross-Section Integration: A direct numerical integration of the full neutron-capture cross-section data from the Evaluated Nuclear Data File (ENDF/B-VIII.1). This method accounts for all resonances, including those with negative energies relative to the neutron separation energy and multiple overlapping resonances.

C. Software Implementation

To make these calculations accessible, the authors developed and released two open-source tools:

• WestcottFactors (Python): A toolkit designed to ingest data from the Generalized Nuclear Database Structure (GNDS) format (used in ENDF/B-VIII.1). It utilizes scipy for numerical integration and allows users to input custom CSV-formatted neutron spectra.
• Modified DeCE (C++): A fork of the existing DeCE (Descriptive Correction of ENDF-6 format) code, adapted to handle the legacy ENDF-6 format and perform similar $g$-factor calculations.

D. Validation Cases

The methods were tested on six representative irregular nuclei: $^{83}$Kr, $^{115}$In, $^{149}$Sm, $^{151}$Eu, $^{157}$Gd, and $^{176}$Lu.

• Spectra Used: Maxwellian distributions (various $T$), Ideal Guide spectra, and experimentally measured spectra from the Budapest Research Reactor (BRR) and Forschungsreaktor München II (FRM II).

3. Key Contributions

1. Extension to Arbitrary Spectra: The primary contribution is the methodology and software to calculate $g$-factors for any user-specified neutron spectrum, not just Maxwellian distributions. This is critical for facilities using guided beams where the spectrum deviates significantly from thermal equilibrium.
2. Open-Source Tools: The release of WestcottFactors and the modified DeCE fork provides the community with robust, traceable tools to calculate $g$-factors using the latest ENDF/B-VIII.1 data.
3. Comprehensive Tabulation: The paper provides extensive tables (Appendix A1 and A2) of $g$-factors for a wide range of isotopes across various Maxwellian temperatures and Ideal Guide distributions, calculated using the superior full cross-section integration method.
4. Resolution of Discrepancies: The work identifies and explains discrepancies between historical $g$-factor tables (e.g., IAEA, PGAA Handbook) and new calculations, often attributing them to the use of incomplete resonance data or incorrect sign conventions in older parameters.

4. Key Results

• Accuracy of Integration vs. Approximation: For nuclei with multiple low-energy resonances (e.g., $^{151}$Eu, $^{176}$Lu), the traditional "irregularity function" method (using only the lowest resonance) produces significant errors.
  • Example: For $^{151}$Eu at 500 K, the irregularity function method yielded a $g$-factor 13.2% lower than the full cross-section integration method.
  • For $^{176}$Lu, differences reached 4.5% at 600 K compared to IAEA database values.
• Impact of Non-Maxwellian Spectra: Comparing calculated $g$-factors for BRR and FRM II beams against Maxwellian approximations revealed differences of 20% or more for highly irregular nuclei like $^{149}$Sm, $^{151}$Eu, and $^{176}$Lu.
  • Example: For $^{149}$Sm at the BRR cold-neutron beam, the $g$-factor was 0.721, whereas a Maxwellian approximation at the equivalent average temperature (140 K) yielded 0.942.
• Consistency of Tools: The WestcottFactors (Python/GNDS) and DeCE (C++/ENDF-6) tools produced nearly identical results (differences < 1%), validating the numerical integration approach regardless of the data format used.
• Data Source Sensitivity: The choice of resonance parameters (ENDF/B-VIII.1 vs. Atlas of Neutron Resonances) had minimal impact on results for most nuclei, but the method of calculation (full integration vs. single-resonance approximation) was the dominant source of error.

5. Significance

This work fundamentally updates the practice of Neutron Activation Analysis for modern facilities.

• Precision: It demonstrates that relying on tabulated Maxwellian $g$-factors for guided-beam experiments can lead to substantial systematic errors in quantitative analysis.
• Best Practice: The authors recommend that whenever an experimental neutron spectrum is available, it should be used with the provided software tools to calculate specific $g$-factors, rather than relying on theoretical approximations.
• Community Resource: By providing open-source software and updated tables based on ENDF/B-VIII.1, the authors enable the PGAA and NAA communities to achieve higher accuracy in elemental analysis, particularly for the challenging "irregular" nuclei that are critical in various scientific and industrial applications.