🔬 atomic physics

CDW-EIS calculations of atomic ionization by bare ions and antiprotons

This paper presents *ab initio* CDW-EIS calculations of atomic ionization cross sections for a wide range of targets and projectiles (including bare ions and antiprotons), provides an openly accessible dataset with systematic trend analysis, and introduces a neural-network surrogate (CDWEIS-NN) to interpolate and extend these results for practical applications.

Original authors: Darío M. Mitnik, Jorge E. Miraglia, Claudia. C Montanari

Published 2026-10-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Darío M. Mitnik, Jorge E. Miraglia, Claudia. C Montanari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

When a fast-moving particle, like a stripped atom or an antiproton, smashes into another atom, it can knock electrons loose. This process, called ionization, is the engine behind a powerful tool used by scientists and art historians alike: particle-induced X-ray emission. By firing a beam of particles at a sample, researchers can cause the atoms to emit X-rays that act as a unique fingerprint, revealing exactly what elements are present. This technique allows for the non-destructive analysis of everything from ancient pigments in a Renaissance painting to the steel alloys in a modern skyscraper. To read these fingerprints accurately, however, scientists need to know the precise probability that a collision will knock an electron out of a specific shell around the atom's nucleus. These probabilities, known as cross sections, are difficult to calculate because atoms are complex systems where electrons interact with one another in intricate ways, and the outcome changes depending on how fast the incoming particle is moving and whether it carries a positive or negative charge.

A team of researchers from the Institute of Astronomy and Space Physics in Buenos Aires has now created a massive, open library of these probabilities, filling in gaps for a wide range of materials that are crucial for both cultural heritage and industrial technology. Using a sophisticated computer model called the continuum distorted-wave eikonal initial-state approach, they simulated thousands of collisions between bare ions and antiprotons and atoms ranging from hydrogen to krypton, resulting in a database of 6,239 tabulated cross sections. The study focused heavily on elements found in the third and fourth rows of the periodic table, such as aluminum, silicon, and the transition metals like iron and copper, which are common in pigments and advanced materials. The researchers calculated the likelihood of ionization for every occupied electron shell in these atoms across a broad spectrum of speeds. They found that while the details of the atom's outer structure matter, the energy holding the innermost electrons in place is the dominant factor that dictates how the atom responds to a collision. This discovery allowed them to organize thousands of complex data points into a few clear, predictable patterns.

One of the most striking findings involves the difference between hitting an atom with a positively charged proton versus a negatively charged antiproton. At high speeds, the two particles behave almost identically, stripping electrons with similar efficiency. However, as the speed drops, a distinct difference emerges: the antiproton becomes more effective at ionizing the atom than the proton. This reversal happens because the negative charge of the antiproton interacts differently with the atom's electrons at lower energies, a phenomenon that had been predicted but is now clearly mapped out in this new dataset. The researchers also discovered that this charge-sign effect is superimposed on a robust rule: if you scale the speed of the incoming particle by the binding energy of the electron shell being hit, the results for different atoms collapse onto a single, universal curve. This means that the behavior of a heavy atom like copper can be understood through the same fundamental scaling laws that apply to a light atom like aluminum, provided the energy is adjusted correctly.

To make this vast amount of data useful for everyday applications, the team developed a neural network, a type of artificial intelligence trained on their simulation results. This digital tool, named CDWEIS-NN, can instantly predict the ionization probabilities for any combination of projectile and target within the studied range, even for systems the researchers did not explicitly calculate. It acts as a fast, accurate bridge between the heavy, time-consuming physics simulations and the practical needs of experimentalists. The network was tested against both the researchers' own simulations and real-world experimental data, showing that it can reproduce the complex physics with an error rate of less than five percent in most cases. It successfully predicted the behavior of systems involving carbon and oxygen ions striking chlorine, potassium, titanium, iron, and copper—combinations that were not part of the original training set.

The result is a dual resource for the scientific community: a comprehensive database of raw, high-precision calculations and a fast, intelligent tool to access them. This work ensures that when scientists analyze a historical artifact or a new material, they have access to the most accurate theoretical data available to interpret their measurements. By confirming that the underlying physics holds up against experimental benchmarks and by providing a way to interpolate between known data points, the study removes a significant barrier to precise elemental analysis. The researchers have made all their data and the neural network tool freely available online, allowing anyone to query the system and obtain the necessary cross sections without needing to run complex simulations themselves. This accessibility ensures that the insights gained from these deep theoretical calculations can be immediately applied to real-world problems in materials science and cultural heritage preservation.

Technical Summary: CDW-EIS Calculations of Atomic Ionization by Bare Ions and Antiprotons

Problem Statement
Accurate ionization cross sections for K, L, and M shells are essential for reliable Particle-Induced X-ray Emission (PIXE) analysis, particularly in art, cultural heritage, and materials science applications involving elements from periods 3 and 4 of the periodic table (Na to Kr). While semi-empirical models like ECPSSR exist, they often lack the precision required for complex multielectron systems. Furthermore, a systematic ab initio treatment for a broad range of projectile charges (including antiprotons) and target subshells has been computationally demanding. There is a specific need for data covering the charge-sign dependence (proton vs. antiproton) and the behavior of inner-shell ionization in metallic targets, where valence electrons are better described as a conduction gas.

Methodology
The authors employed the Continuum Distorted-Wave Eikonal Initial-State (CDW-EIS) model to perform ab initio calculations of atomic ionization.

Theoretical Framework: The model treats the collision between a bare projectile (charge $Z_P$ ) and an atomic target (charge $Z_T$ ) nonperturbatively, including projectile distortion in both initial and final channels. The total ionization cross section is derived from a four-dimensional integral of the transition matrix element.
Computational Implementation: Bound and continuum wave functions were obtained as numerical solutions to the same Hamiltonian using the radialf code, ensuring orthogonality. The central potentials were derived from Hartree–Fock–Slater orbitals using the Depurated Inversion Method (DIM). Radial integrations were performed numerically, and angular variables were integrated using vector-coupling techniques.
Database Construction: Calculations were performed for approximately 130 projectile–target systems (bare ions with $|Z_P| \le 48$ and antiprotons) impacting atomic targets from H to Kr. Cross sections were calculated for all occupied $nl$ subshells across projectile velocities of $v = 1–20$ a.u., resulting in 6,239 tabulated values.
Neural Network Surrogate (CDWEIS-NN): To address the computational cost of direct CDW-EIS calculations for uncalculated systems, a neural network surrogate was developed. Trained on a subset of the database ( $|Z_P| \le 8$ , $1 \le Z_T \le 36$ ), the model uses a 9-dimensional input vector (projectile charge magnitude, antiproton flag, target $Z$ , subshell $n, l$ , occupation, binding energy, velocity, and reduced velocity). The network employs an ensemble of five feed-forward neural networks trained with a repeated group-held-out strategy to prevent data leakage and ensure generalization to unseen projectile–target pairs.

Key Contributions

Open-Access Database: The authors released a comprehensive, open-access repository containing subshell-resolved total ionization cross sections for a wide variety of projectile–target systems, including antiproton impacts.
Systematic Analysis: The study identified and analyzed systematic trends, including:
- Binding-Energy Scaling: A reduced velocity ( $u = v/\sqrt{2I_{nl}}$ ) and reduced cross section ( $S_{nl}$ ) were shown to produce an almost universal behavior for K-shell ionization across different targets for both protons and antiprotons.
- Charge-Sign Effects: The calculations confirmed that while proton and antiproton cross sections converge at high velocities (scaling with $Z_P^2$ ), antiproton-induced ionization exceeds proton-induced ionization at lower velocities, a manifestation of higher-order collision effects (Barkas-type effect).
CDWEIS-NN Model: A validated neural network surrogate capable of interpolating the CDW-EIS database and predicting cross sections for projectile–target combinations not explicitly calculated.

Results

Validation against Experiment: The CDW-EIS calculations were benchmarked against experimental data for total ionization of H and He, and K- and L-shell ionization for various targets (Al, Ar, Si, P, S, Cl, Cu, etc.). The calculations reproduced the absolute scale, velocity dependence, and charge-sign effects (including the low-velocity crossing of proton/antiproton curves) without empirical normalization.
Neural Network Performance:
- Held-Out Validation: On a held-out dataset of 2,371 cross sections (47 unseen projectile–target pairs), the CDWEIS-NN ensemble achieved a median relative error of 3.1% and a log-space mean absolute error of 0.0192. 99.5% of predictions differed from the CDW-EIS reference by less than 30%.
- External Validation: When applied to K-shell ionization by $C^{6+}$ and $O^{8+}$ on Cl, K, Ti, Fe, and Cu (systems not in the training set), the CDWEIS-NN predictions showed satisfactory agreement with experimental measurements by Kadhane et al., reproducing both magnitude and energy dependence.
Scaling Laws: The reduced representation confirmed that the K-shell binding energy determines the principal target-dependent scale for both proton and antiproton impacts, whereas 3p subshells retain a residual dependence on outer-shell electronic structure.

Significance
The paper establishes a reliable, open database of ab initio ionization cross sections that addresses the limitations of semi-empirical models for complex atomic targets. By demonstrating that binding-energy scaling can unify cross-section behaviors across different targets and projectile charges, the work provides a robust physical framework for understanding ionization processes. The development of the CDWEIS-NN surrogate significantly extends the practical utility of these calculations, enabling rapid, accurate predictions for experimental configurations where direct ab initio computation is currently unavailable. The authors emphasize that the database and surrogate serve as complementary resources: the former provides the physical systematics and ground truth, while the latter facilitates interpolation and prediction for uncalculated systems, supporting applications in PIXE analysis and stopping-power descriptions in metallic targets.

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