Improving the Accuracy of Muonic X-ray Simulations: New Nuclear Models and Cascading Processes for MuDirac

by Dr Leandro Liborio (Scientific Computing Department, STFC, UKRI, Harwell, UK), Dr Philip Jones (Department of Physics, University of Warwick, Coventry, UK)

Monday 23 Mar 2026, 15:00 → 16:00 Europe/Zurich

SZ-WHGA/121

MuDirac is an open, sustainable, software tool that is being developed in the UK as a collaboration between the ISIS Muon Group, the Scientific Computing Department at STFC, and colleagues from the University of Warwick.

The current version of MuDirac can compute most muonic atom transition energies up to precisions of KeV, and it can take into account things like the eGects of finite nuclear size, vacuum polarizability and electronic shielding.

In this talk, we will present the key design ideas behind the new version of MuDirac, the work that we have been doing to add new nuclear capabilities to the code, and some of the work and ideas we have for improving the modelling of the cascade process.

Nuclear capabilities:

The nuclear charge radius is a fundamental property of the atomic nucleus, and its value depends on features of the nuclear structure such as the nuclear charge distribution, ρ. The nuclear charge radius can be mathematically defined using ρ, which in turn can be deduced from the X-ray transition energies of a muonic atom. The binding energy of the muonic atom is sensitive to ρ and, in principle, the nuclear charge radius can be determined by the muonic X-ray measurements. This is, however, a difficult problem, as the relationship between muonic X-ray transition energies and ρ is quite complex.

One way of treating this problem is assuming a functional form for ρ, such as the 2-parameter Fermi distribution (2pF), which assumes that the nucleus has a homogenously charged spherical core with radius c, and a skin of thickness t along which the charge exponentially decreases in a radial direction.

The current version of MuDirac uses the most common implementation of the 2pF, where the values of c are taken from tables and the value of t is set at 2.3fm. MuDirac then numerically solves the Dirac equation and calculate the muonic X-ray transition energies in a muonic atom, which can then be compared to experimental values.

In the new version of MuDirac, it will be possible to use experimental X-ray transition energies to eGiciently estimate nuclear properties, such as the nuclear charge radius. The new version of MuDirac assumes a 2-parameter Fermi distribution of the nuclear charge and uses experimental X-ray transition energies to reverse-engineer the Dirac equation, optimise the values of the c and t, obtain a new nuclear charge distribution ρ, and then use it to calculate the nuclear charge radius.

Cascade process:

The cascade process is a key part of correctly modelling muonic x-ray intensities. There are several physical processes involved in this, which include: radiative transitions, Auger transitions, electron refilling, and initial capture distribution. All of these are treated in the well-known Akylas-Vogel cascade code, but not without a significant level of approximation. In particular, the use of hydrogen-like orbitals for all rates is clearly not strong enough for studying high-Z elements. Through solving the Dirac equation, this approximation can be lifted to bring the method into a more modern framework.

In this part of the talk, we will present the theory, implementation, and verification of both relativistic radiative and Auger transition rates, as a means of improving the cascade process. Furthermore, we will outline how the cascade process may be further improved to obtain more accurate x-ray intensities.