MuDirac 1.3.0: A Sustainable Software Tool for Calculating Ground State Nuclear Properties Using Muonic X-Ray Measurements

This paper introduces MuDirac 1.3.0, a sustainable and efficient open-source software tool that enables the negative muon community to accurately calculate nuclear properties, such as the charge radius, by modeling muonic X-ray transition energies under a two-parameter Fermi distribution.

The Gist

Imagine the nucleus of an atom as a tiny, dense city. For a long time, scientists have tried to map out exactly how the "citizens" (protons and neutrons) are arranged inside this city. One of the most important things to know about this city is its size, specifically its "charge radius."

For decades, scientists have used a special tool to measure this: muons. You can think of a muon as a "heavy electron." It's about 200 times heavier than a regular electron. When you drop a muon into an atom, it doesn't just hang out on the outside; it crashes right into the inner rings, replacing a regular electron. As it settles down to its lowest energy level, it emits a flash of light called an X-ray.

The tricky part is that the color (energy) of this X-ray flash depends entirely on the shape and size of the nuclear city it's orbiting. If the city is slightly bigger or has a fuzzy edge, the X-ray changes.

The Problem: A One-Way Street

Until now, the software used to analyze these X-rays (called MuDirac) worked like a one-way street.

• The Old Way: You had to guess the size and shape of the nuclear city first. You'd plug those guesses into the computer, and it would tell you, "Based on your guess, the X-ray should look like this."
• The Limitation: If your guess was slightly off, the computer's prediction wouldn't match the real X-ray you measured in the lab. To find the real size, scientists had to play a tedious game of "guess and check," trying thousands of different city shapes until one finally matched the data. It was slow and computationally expensive.

The Solution: MuDirac 1.3.0 (The Reverse Engineer)

The authors of this paper have upgraded MuDirac to version 1.3.0. Think of this new version as a reverse engineer or a detective.

Instead of guessing the city size and checking the X-ray, the new software starts with the real X-ray measurement and works backward to figure out exactly what the city must look like to produce that specific flash of light.

Here is how they made it work, using some simple analogies:

1. The "Fuzzy Ball" Model (The 2pF Model)
To describe the nuclear city, scientists use a mathematical shape called the "2-parameter Fermi distribution." Imagine a ball of clay.

• Parameter 'c': This is the radius of the hard core of the ball.
• Parameter 't': This is the thickness of the fuzzy, soft skin on the outside of the ball.
  The old software just picked a standard skin thickness and looked up the core size in a table. The new software asks: "What specific combination of core size and skin thickness creates the exact X-ray we measured?"

2. The Map and the Compass (Polar Coordinates)
Finding the right combination of core size and skin thickness is like trying to find a specific spot on a map.

• The Old Way (Brute Force): Imagine walking every single square inch of a huge field, checking if you found the spot. It takes forever.
• The New Way (Polar Coordinates): The authors realized that the "correct" answers for the core and skin thickness always line up in a specific pattern, like a curved path on a map. They changed the software's "compass" to polar coordinates. Instead of walking a grid, the software now walks along the curved path directly. This is like switching from a slow, grid-like search to a high-speed train that only travels on the tracks where the answer actually exists.

3. The Best Detective (The Optimization Algorithm)
Even with the new compass, you need a smart detective to find the exact spot. The authors tested many different "detectives" (mathematical algorithms) to see which one could find the answer fastest and most accurately. They found that a specific method called Levenberg-Marquardt (powered by a tool called Ceres Solver) was the champion. It found the perfect match between the theory and the experiment much faster than the old methods.

What Did They Find?

The team tested this new "detective" on a variety of atoms, from light ones like Zinc to heavy ones like Gold and Lead.

• The Result: In every case, the new MuDirac 1.3.0 was able to pinpoint the nuclear size (the charge radius) with much higher precision than the old method.
• The Proof: When they compared their results to the "gold standard" reference values that scientists have trusted for years, the new software matched them almost perfectly.

The Bottom Line

MuDirac 1.3.0 is a free, open-source tool that allows scientists to stop guessing and start deducing. By reversing the math, it takes the X-ray flashes captured in experiments and instantly calculates the precise size and shape of the atomic nucleus that created them. It's a faster, more efficient way to understand the fundamental building blocks of our universe.

Technical Summary

1. Problem Statement

The nuclear charge radius ($R_{rms}$) is a fundamental property of the atomic nucleus, crucial for understanding nuclear structure and deformation. While $R_{rms}$ can be deduced from muonic X-ray transition energies, the relationship between these energies and the nuclear charge distribution ($\rho$) is complex.

• Current Limitation: Previous versions of the MuDirac software (e.g., v1.2.4) were designed primarily for elemental analysis. They utilized a 2-parameter Fermi (2pF) model for the nuclear charge distribution where the parameters—half-density radius ($c$) and skin thickness ($t$)—were fixed a priori based on nuclear data tables (specifically, $t$ was fixed at 2.3 fm).
• The Gap: There is a growing need in the negative muon community to reverse-engineer the Dirac equation. Researchers require a tool that can take experimentally measured muonic X-ray transition energies and efficiently calculate the optimal 2pF parameters ($c$ and $t$) to determine the nuclear charge distribution and $R_{rms}$ with high precision, rather than relying on pre-tabulated values.

2. Methodology

The authors developed MuDirac 1.3.0, an open-source, sustainable software tool that optimizes the 2pF model parameters to fit experimental data. The methodology involves three core components:

A. Theoretical Framework

• Dirac Equation Solver: The software solves the radial Dirac equation for a muon orbiting a nucleus with a potential $V(r)$ defined by the 2pF model.
• 2pF Model: The nuclear charge density $\rho(r)$ is modeled as:
  $$\rho(r) = \rho_0 \left[ 1 + \exp\left( \frac{4 \ln(3)(r - c)}{t} \right) \right]^{-1}$$
  where $c$ is the half-density radius and $t$ is the skin thickness.
• Objective Function: The goal is to minimize the difference between experimental transition energies ($E_k^{exp}$) and simulated energies ($E_k^{sim}$) using a least-squares approach. The cost function is defined as:
  $$\chi^2(c, t) = \sum_k \frac{(E_k^{exp} - E_k^{sim}(c, t))^2}{\sigma_k^2}$$
  where $\sigma_k$ is the experimental uncertainty.

B. Optimization Strategy

To solve the inverse problem (finding $c$ and $t$ that minimize $\chi^2$), the authors introduced two major algorithmic improvements over brute-force scanning:

1. Polar Coordinate Parametrization:
   • Analysis showed that valid bands of $(c, t)$ parameters are roughly parallel to contours of constant $R_{rms}$.
   • The authors transformed the search space from Cartesian $(c, t)$ to polar coordinates $(R_{rms}, \theta)$. This allows the algorithm to scan along the relevant $R_{rms}$ contours, significantly reducing the search domain and computational cost.
   • The transformation is defined by:
     $$c = r(\theta, R_{rms}) \cos \theta, \quad t = r(\theta, R_{rms}) \sin \theta$$
2. Efficient Optimization Algorithms:
   • The authors benchmarked various minimizers using the FitBenchmarking tool against 24 muonic atom datasets.
   • The Levenberg-Marquardt (lm) algorithm, implemented via the Ceres Solver (a C++ library), was identified as the most efficient and accurate solver.
   • This approach replaced the computationally expensive brute-force grid search with a gradient-based optimization that converges rapidly to the global minimum.

C. Software Implementation

• Input/Output: MuDirac 1.3.0 accepts two input files: a main configuration file (.in) and a new experimental data file (.xr.in) containing transition energies and uncertainties.
• Output: It generates a file containing the optimized $c$ and $t$ values, the resulting $R_{rms}$, the $\chi^2$ value, and the execution time.
• Availability: The tool is open-source, hosted on GitHub, and available via the Conda environment.

3. Key Contributions

1. MuDirac 1.3.0 Release: The first version of the software capable of reverse-engineering the Dirac equation to extract 2pF parameters ($c$ and $t$) directly from experimental muonic X-ray data.
2. Algorithmic Efficiency: The implementation of a polar coordinate system combined with the Levenberg-Marquardt algorithm reduces the computational time from hours (required for high-resolution brute-force scans) to approximately one minute per atom, while maintaining or improving accuracy.
3. Improved Precision: The tool allows for the determination of nuclear charge radii with uncertainties conditioned on accurate simulations, providing a more robust estimation of ground-state nuclear properties.
4. Community Resource: The software is designed to be sustainable, publicly available, and accessible to the negative muon community for both light and heavy nuclei.

4. Results

The authors validated MuDirac 1.3.0 using 24 muonic atom datasets (ranging from light elements like Zinc to heavy elements like Lead and Bismuth) sourced from Engfer et al.

• Reduction in $\chi^2$: In all tested cases, the fitted 2pF parameters (MuDirac 1.3.0) significantly reduced the $\chi^2$ value compared to the default parameters (MuDirac 1.2.4). The simulated energies fell well within the experimental uncertainty bounds.
• Accuracy of $R_{rms}$: The calculated root-mean-square radii for elements such as $^{89}\text{Y}$, $^{114}\text{Cd}$, $^{197}\text{Au}$, and $^{206}\text{Pb}$ showed excellent agreement with reference values from Engfer et al.
  • Example: For $^{197}\text{Au}$, MuDirac calculated $R_{rms} = 5.4207 \pm 0.02$ fm, compared to the reference $5.415 \pm 0.007$ fm.
• Robustness: The tool successfully handled cases where multiple experimental datasets with different uncertainties were available (e.g., $^{206}\text{Pb}$ data from three different sources), finding an optimal parameter set that satisfied all constraints.

5. Significance

• Scientific Impact: MuDirac 1.3.0 bridges the gap between experimental muonic spectroscopy and nuclear theory. It enables researchers to extract precise nuclear charge distributions and radii for both stable and unstable nuclei without relying on pre-existing tabulated values, which may not exist for exotic isotopes.
• Computational Sustainability: By optimizing the search space and using efficient solvers, the tool makes high-precision nuclear property calculations accessible and computationally feasible for a broader range of researchers.
• Future Work: The authors note that the current uncertainty estimation relies on the gradient of the chi-squared error. Future work aims to improve the modeling of the muon cascading process and provide more robust uncertainty quantification methods that account for simulation limitations.

In summary, MuDirac 1.3.0 represents a significant advancement in computational nuclear physics, providing a fast, accurate, and open tool for deriving fundamental nuclear properties from muonic X-ray measurements.