🔬 atomic physics

HIR: A collisional-radiative model for hydrogen ionization in plasma with anisotropic fast ions

This paper introduces the HIR software package, a collisional-radiative model for calculating hydrogen ionization in hot plasmas with anisotropic fast ions, demonstrating that stepwise ionization significantly impacts total rates and mean free paths while revealing that fast ion anisotropy has a negligible effect on ionization dynamics.

Original authors: Sergey Polosatkin

Published 2026-07-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sergey Polosatkin

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

The Great Escape: Why Atoms Don't Always Play Fair

Imagine you are trying to keep a crowd of energetic people inside a giant, invisible stadium made of magnetic fields. This is the dream of nuclear fusion: trapping super-hot plasma (a soup of charged atoms) to create clean, limitless energy. But there's a problem. To keep the stadium full, you have to keep feeding it new fuel—hydrogen atoms. However, these new atoms are like shy guests who don't want to join the party. If they stay neutral, they can slip right through the magnetic walls and escape. To get stuck in the game, they need to get "zapped" by the heat and lose an electron, turning into ions that the magnetic fields can hold tight.

The tricky part is figuring out exactly how long it takes for these atoms to get zapped. It's not just a simple game of "hit and miss." Sometimes, an atom gets hit, jumps to a slightly higher energy level (like climbing a single step on a ladder), and then gets hit again to finish the job. This "step-by-step" climbing is called stepwise ionization. Scientists have known this happens, but they've been guessing how much it matters, especially when the plasma contains "fast ions"—super-speedy particles moving in weird, lopsided patterns. If we get the math wrong, we might think our fuel is staying put when it's actually leaking out, or vice versa. This is where a new piece of software called HIR comes in, acting like a super-accurate calculator to track every single step these atoms take before they become part of the plasma.

The Paper: HIR and the Ladder of Ionization

In this paper, S.V. Polosatkin introduces a new software package called HIR (Hydrogen Ionization Rates). Think of HIR as a high-tech traffic cop for hydrogen atoms in a plasma. Its job is to count exactly how many atoms are sitting on each "rung" of the energy ladder (the different quantum levels) and calculate how fast they get knocked off the ladder entirely (ionized) to become part of the plasma soup.

The researchers built HIR using a "steady-state" model. Imagine a busy highway where cars (atoms) are constantly entering and leaving, but the total number of cars on the road stays the same because the flow is balanced. HIR tracks these flows, accounting for atoms bumping into electrons, bumping into other ions, swapping electrons (charge exchange), and even glowing as they drop down energy levels.

The Big Discovery: The "Step-Up" Effect
The most exciting finding is that the "step-by-step" route is a major player. The paper shows that stepwise ionization contributes up to 20% of the total rate at which atoms get ionized. This means if you ignore the intermediate steps and only look at the direct hits, you are missing a huge chunk of the action. To get this right, the model had to include energy levels up to the principal quantum number N=8. If you stop counting at level 7, your math is off. It's like trying to count the steps to a roof but stopping at the 7th step when the roof is actually on the 8th; you'd think you're almost there, but you're actually still falling short.

Fast Ions and Weird Shapes
The paper also tackles a specific headache: what happens when the plasma contains "fast ions" that aren't moving in a nice, round, uniform ball? In some fusion devices, like the GDML facility, these fast ions are "anisotropic," meaning they are moving in a lopsided, stretched-out shape (like a rugby ball instead of a soccer ball). Some scientists worried that this weird shape might change how atoms get ionized.

However, the HIR simulations found that for the GDML facility, this anisotropy has a negligible effect—less than 1% difference in the ionization dynamics. So, even though the fast ions are dancing in a weird pattern, the hydrogen atoms don't really care; they get ionized almost the same way as if the ions were moving in a perfect circle.

The "Non-Additive" Surprise
Here is where it gets counter-intuitive. You might think that if an atom has a 10% chance of getting hit by an electron and a 10% chance of getting hit by an ion, the total chance is just 20%. The paper shows that in the world of stepwise ionization, 1 + 1 does not equal 2.

Because atoms can get stuck in excited states and then get hit by different particles to finish the job, the effects of electrons and ions mix together in a complex way. They are "non-additive." In the GDML facility, this mixing actually reduces the "mean free path" (the average distance an atom travels before getting ionized) by up to 7% at 40 keV. It's like a game of tag where the players work together to catch the runner faster than they could individually.

The Cost of Joining the Party
Finally, the paper looks at the "ionization cost"—the energy it takes to turn a neutral atom into an ion, including the energy lost as light (glowing photons). In the GOL-NB facility, where the plasma is cooler, this cost depends heavily on how crowded the room is (plasma density). At higher densities, atoms bump into each other so often that they lose energy through collisions rather than glowing, which changes the total energy cost. The paper calculates that this cost varies by about 10% across the density range of 10¹² to 10¹⁴ cm⁻³.

The Verdict
The authors have made their tool, HIR, available to everyone on GitHub. It proves that to understand how hydrogen behaves in fusion plasmas, especially with fast ions, you can't just use simple math. You have to count every step up the ladder (up to N=8) and realize that different particles in the plasma team up in surprising ways. While the weird shape of fast ions turns out to be a minor detail, the step-by-step ionization is a giant factor that changes how far atoms can travel and how much energy we need to keep the fusion fire burning.

Technical Summary: HIR – A Collisional-Radiative Model for Hydrogen Ionization in Plasma with Anisotropic Fast Ions

Problem Statement
In magnetic confinement fusion, particularly within linear mirror trap systems, understanding the interaction between hot plasma and hydrogen (or its isotopes) is critical. Neutral hydrogen atoms, introduced via fueling, wall desorption, or neutral beam injection, undergo ionization processes that determine their penetration depth and the associated energy costs (ionization cost). While stepwise ionization—where atoms are excited to upper levels before ionizing—is a known phenomenon, existing models often assume Maxwellian, isotropic ion distributions. This assumption fails to capture the physics of facilities like the GDML (Gas-Dynamic Mirror Laboratory) and GOL-NB, where fast ion populations exhibit significant anisotropy due to loss cones in velocity space. Furthermore, the non-additive nature of collisional processes involving multiple plasma species (electrons, warm ions, and fast ions) in the presence of stepwise ionization requires a more rigorous modeling approach.

Methodology
The paper presents the HIR (Hydrogen Ionization Rates) software package, which employs a steady-state collisional-radiative model to calculate atomic hydrogen level populations and ionization dynamics. The model assumes that external plasma parameters (temperature, density) change on timescales longer than the lifetimes of excited states.

Physical Model: The code solves a system of balance equations for hydrogen atom concentrations across principal quantum numbers ( $N$ ). It accounts for electron and ion impact excitation, de-excitation, ionization, charge exchange, spontaneous radiative transitions, and the incoming flux of atoms.
Atomic Data: Cross-sections for ionization, excitation, and charge exchange are derived from reference [5], with corrections applied for known inaccuracies in the source formulas (e.g., missing $n^2$ factors for $n>3$ ). De-excitation cross-sections are calculated using the principle of detailed balance and the Klein–Rosseland relation. Transition probabilities are computed via the Johnson formula.
Anisotropy Handling: The package includes specific functions to calculate elementary process rates for collisions with plasma particles having two-dimensional axisymmetric (anisotropic) distribution functions, as well as isotropic distributions.
Validation: The code was verified against the OPEN ADAS library. While discrepancies of approximately 10% were observed (attributed to differences in underlying cross-section data), the HIR package successfully reproduces trends and provides a self-consistent framework using the specific cross-section data from [5].

Key Contributions and Results

Significance of Stepwise Ionization:
- Stepwise ionization contributes up to 20% to the total ionization rate.
- Accurate modeling requires including energy levels up to the principal quantum number $N=8$ . Levels beyond this make negligible contributions in the studied regimes.
- This effect is most pronounced at low temperatures and high densities but remains significant even in high-temperature plasma ( $>10^{13}$ cm $^{-3}$ ).
Non-Additivity of Collisional Processes:
- The paper demonstrates that collisions with different plasma species are non-additive due to stepwise ionization. The total inverse mean free path is not equal to the sum of the inverse mean free paths calculated for individual species (electrons, warm ions, fast ions) separately.
- In the GDML facility at 40 keV, this non-additivity reduces the mean free path by up to 7% compared to additive approximations.
- In the GOL-NB facility, the non-additivity of effective capture cross-sections increases with density, reaching approximately 4% at $10^{14}$ cm $^{-3}$ .
Impact of Anisotropy:
- Despite the highly anisotropic nature of fast ions in the GDML facility, the anisotropy of the ion distribution function has a negligible effect (<1%) on the ionization dynamics and mean free path of fast hydrogen atoms. Consequently, isotropic distribution functions may be used for such calculations in this specific context without significant loss of accuracy.
Ionization Cost and Density Dependence:
- The ionization cost (energy expended to ionize an atom, including line radiation) in the GOL-NB plasma exhibits a strong dependence on plasma density.
- As density increases, the ionization cost decreases. This is attributed to the competition between collisional de-excitation (which quenches radiative losses) and radiative de-excitation.
Facility-Specific Analysis:
- GDML: At 40 keV, collisions with thermal ions contribute ~50% to ionization, while electrons and fast ions contribute ~25% each.
- GOL-NB: Capture occurs predominantly via collisions with plasma ions (~80%), with electron collisions contributing ~20%. The effective capture cross-section varies by 10% across the facility's characteristic density range ( $10^{12}$ – $10^{14}$ cm $^{-3}$ ).

Significance
The HIR package provides an openly available tool for modeling neutral beam physics in both mirror and toroidal magnetic confinement systems. Its primary significance lies in its ability to rigorously account for stepwise ionization and the resulting non-additivity of collisional processes, which are often overlooked in simplified models. While the study found that ion anisotropy has a minimal impact on ionization rates in the specific GDML regime, the software's capability to handle anisotropic distributions ensures its applicability to a broader range of fusion scenarios. The code is designed to improve the accuracy of penetration depth predictions and energy balance calculations in fusion plasma research.

The software is available on GitHub ([4]), and the work is supported by the Russian Science Foundation (grant No. 25-22-00117).

The Great Escape: Why Atoms Don't Always Play Fair

The Paper: HIR and the Ladder of Ionization

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