Partial ionisation cross sections for the binary-encounter Bethe model

This paper evaluates the performance of the Binary-Encounter Bethe (BEB) model when utilizing experimental ionisation thresholds instead of theoretical ones, demonstrating that this adjustment improves the accuracy of partial ionisation cross sections for predicting subsequent plasma processes.

The Gist

Imagine you are trying to understand how a billiard ball (an electron) hitting a cluster of marbles (an atom or molecule) breaks the cluster apart. This is the core of electron impact ionisation: a fast electron smashes into a gas particle, knocking one of its own electrons out, leaving the particle charged (ionised).

Scientists need to predict exactly how often this happens and what kind of "broken pieces" (ions) are created. To do this, they use a mathematical recipe called the Binary-Encounter Bethe (BEB) model.

Here is the story of this paper, told simply:

1. The Old Recipe: "Theoretical Guessing"

For decades, scientists used the BEB model like a cookbook. The recipe said: "To calculate the breakage, look at the theoretical map of the atom's electrons. Assume every electron sits in a specific, perfect seat (an orbital) with a specific energy cost to remove it."

The problem? The map was wrong.
The theoretical map (based on the Hartree-Fock method) was like a blueprint drawn by an architect who had never actually visited the construction site. It guessed the energy needed to remove an electron was higher than it really was.

• The Result: The model worked surprisingly well for the total amount of breakage (like guessing the total weight of the rubble), but it failed miserably when trying to identify which specific pieces fell off. It was like guessing the total weight of a shattered vase but getting the shape of the shards completely wrong.

2. The New Insight: "Check the Receipts"

The authors of this paper, Anthony, Alejandro, and Nikolai, decided to stop trusting the blueprint and start reading the receipts (experimental data).

They looked at Photoelectron Spectroscopy. Imagine shining a light on the gas and watching exactly how much energy it takes to knock an electron out. This gives the real energy thresholds.

• The Change: They rewrote the BEB recipe. Instead of using the theoretical "guess" for the energy cost, they plugged in the actual experimental numbers.

3. The Analogy: The "Broken Egg"

Think of an atom as a delicate egg.

• The Old Model: You assume the eggshell is thick and uniform. You predict that if you hit it, it will crack in a specific, symmetrical way.
• The Reality: The eggshell is actually thinner in some spots and thicker in others. When you hit it, it might crack into a jagged shard, a tiny crumb, or a whole half-shell, depending on exactly where you hit.
• The Paper's Contribution: By using the real "thin spots" (experimental thresholds), the authors can now predict not just that the egg broke, but exactly what kind of broken piece you get. This is crucial because different broken pieces (ions) glow with different colors or behave differently in a plasma (like the atmosphere or a fusion reactor).

4. Why This Matters: The "Plasma Weather"

Why do we care about these broken pieces?
In the upper atmosphere (and in space), there is a soup of charged particles called plasma. When solar storms hit Earth, they send a flood of electrons into the air.

• If we want to predict how the atmosphere reacts (like the aurora borealis or radio blackouts), we need to know exactly which ions are created.
• Some ions might glow blue; others might be invisible. Some might break apart further; others might stay stable.
• The old model was a blurry photo. The new model is a high-definition video. It allows scientists to simulate "plasma weather" with much greater accuracy.

5. The Catch: "It's Complicated"

The authors found a twist. When they used the real numbers, the model didn't always predict the total amount of breakage as well as the old, wrong model did.

• Why? It turns out the old model had a "happy accident." The errors in the energy guesses accidentally cancelled out other errors in the math, making the total result look right by chance.
• The Lesson: Just because a model works for the "big picture" doesn't mean it understands the details. The authors argue we must accept that the model needs a little "tuning" (a scaling factor) to work perfectly with the new, accurate data.

Summary

This paper is a call to stop using "theoretical guesses" for the energy levels of electrons and start using real-world measurements.

• Old Way: "Let's assume the electron is here." -> Good for totals, bad for details.
• New Way: "Let's measure where the electron actually is." -> Good for details, requires a little math tweaking for totals.

By doing this, the authors have given scientists a better tool to understand the complex dance of particles in our atmosphere and in space, helping us predict how our technology and environment react to the constant bombardment of space radiation.

Technical Summary

Here is a detailed technical summary of the paper "Partial ionisation cross sections for the binary-encounter Bethe model" by Jeseněk, Luque, and Lehtinen.

1. Problem Statement

The Binary-Encounter Bethe (BEB) model, originally developed by Kim and Rudd, is a widely used semi-empirical formula for calculating total electron impact ionisation cross sections. While highly accurate for total ionisation, the model faces significant limitations when applied to partial ionisation cross sections (i.e., ionisation leading to specific excited ionic states).

The core issues identified are:

• Theoretical vs. Experimental Thresholds: The standard BEB model relies on Hartree-Fock (HF) orbital binding energies ($B_j$) to define ionisation thresholds. These theoretical values systematically overestimate experimental ionisation potentials ($I_j$) derived from photoelectron spectroscopy.
• The "Frozen Orbital" Approximation: The model assumes ionisation is simply the removal of an electron from a specific orbital while the remaining electrons remain "frozen." In reality, ionisation often leads to complex ionic states involving electron correlation, spin-flips, configuration mixing, and vibrational excitations that a single-orbital HF picture cannot capture.
• Accidental Compensation: The paper argues that the BEB model's success in predicting total cross sections using theoretical thresholds is partly due to a fortuitous cancellation of errors: the overestimation of thresholds (which lowers cross sections) compensates for the model's inclusion of optically allowed excitations that do not actually lead to ionisation (which would otherwise overestimate the cross section).

2. Methodology

The authors propose a revised application of the BEB model to generate partial cross sections by shifting the paradigm from orbital-based decomposition to ionisation channel-based decomposition.

• Input Parameter Replacement:
  • Thresholds ($I_j$): Instead of using theoretical HF binding energies ($B_j$), the authors use experimental vertical ionisation thresholds obtained from photoelectron spectroscopy.
  • Scaling Parameter ($K_j$): The scaling factor $K_j$ (representing the kinetic energy gain of the incident electron) is calculated using two distinct semi-empirical formulas depending on the shell:
    • Valence Shells: The Burgess denominator formula (Eq. 2), using experimental $I_j$ as the binding energy.
    • Core (1s) Shells: The screening correction formula (Eq. 3) by Guerra et al., which accounts for effective nuclear charge.
• Weighting Factor ($N_j$): The number of electrons in the channel is redefined. Instead of the simple orbital occupation number, $N_j$ is weighted by the statistical multiplicity of the final ionic state, calculated as $(2S+1)(2L+1)$, where $S$ is total spin and $L$ is total orbital angular momentum.
• Exclusion of Forbidden Channels: Spin-forbidden ionisation channels (requiring electron exchange) are identified but assigned negligible weights ($N \approx 0$) based on their low probability and steep asymptotic decay ($\sim 1/\epsilon^3$).
• Target Scope: The methodology is applied to 12 targets relevant to atmospheric physics: 4 atoms (C, N, O, F), 4 diatomic molecules (CO, N$_2$, O$_2$, NO), and 4 triatomic molecules (CO$_2$, H$_2$O, NO$_2$, O$_3$).

3. Key Contributions

1. Channel-Based Decomposition: The paper provides a framework to decompose the BEB model into specific ionisation channels corresponding to experimental ionic states rather than theoretical orbitals. This allows for the prediction of partial cross sections for excited ionic states.
2. Quantification of Error Compensation: The authors demonstrate that the original BEB model's accuracy for total cross sections is partly "accidental." When experimental thresholds are used, the total cross sections often deviate from experimental data, revealing that the theoretical thresholds were masking the model's inherent overestimation of the dipole oscillator strength sum.
3. Introduction of the BEQ Model Context: The paper clarifies the relationship between the BEB model and the BEQ model (where $Q < 1$). It argues that if one wishes to model partial cross sections accurately using experimental thresholds, the dipole oscillator strength sum must be adjusted (via parameter $Q$) to account for discrete excitations that do not lead to ionisation (e.g., autoionisation).
4. Comprehensive Dataset: The authors provide a complete set of input parameters ($I$, $K$, $N$) for 12 atmospheric species in Table 1, along with Python code (BEB.py) to reproduce the results.

4. Results

• Partial Cross Sections: The revised model successfully generates partial cross sections for various ionic states (singlets, triplets, quartets) for atoms and molecules. For example, in atomic Oxygen, it distinguishes between $^4S^o$, $^2D^o$, and $^2P^o$ states; in NO, it separates singlet and triplet manifolds.
• Total Cross Section Discrepancies:
  • Atoms: For Carbon and Nitrogen, using experimental thresholds increases the total cross section compared to the original BEB, bringing it closer to B-spline R-matrix calculations (though discrepancies remain regarding autoionisation). For Fluorine, the model overestimates the cross section, suggesting the need for a $Q$-factor correction.
  • Molecules: For diatomics like N$_2$ and CO, the difference between the original and revised models is minor. However, for O$_2$, NO$_2$, and CO$_2$, the use of experimental thresholds reveals significant overestimations in the total cross section, confirming that the original model's success relied on the cancellation of errors (overestimated thresholds vs. un-subtracted excitation strength).
• Core Shell Ionisation: The model performs reasonably well for K-shell ionisation when using the screening-based $K$ parameter, though discrepancies near the threshold are attributed to experimental uncertainties in fluorescence yields.
• Complex Molecules: For Ozone (O$_3$), the single-orbital picture breaks down completely due to strong configuration mixing. The authors note that the BEB model struggles here, highlighting the need for a more generalized approach based on integrated dipole oscillator strengths over spectral bands rather than discrete orbital holes.

5. Significance and Outlook

• Plasma Modeling: The ability to predict partial ionisation cross sections is crucial for Monte Carlo and kinetic modeling of electron swarms in atmospheric plasmas. It allows researchers to track specific excited ionic species and predict subsequent optical radiations and non-radiative transitions.
• Model Refinement: The paper serves as a critical correction to the community's understanding of the BEB model. It warns against relying on the "fortuitous" agreement of the original model and advocates for the use of experimental thresholds combined with a scaling factor ($Q$) (transitioning to the BEQ model) to achieve physical consistency.
• Future Directions: The authors propose that future improvements should:
  1. Replace theoretical binding energies with vertical ionisation thresholds.
  2. Use the asymptotic overestimation of the total cross section to determine the appropriate $Q$ factor (accounting for non-ionising excitations).
  3. Treat the scaling parameter $K$ as an adjustable parameter or derive it from ab initio distorted-wave calculations rather than simple semi-empirical formulas.
  4. Develop models for complex molecules (like O$_3$) that rely on integrated band intensities rather than single-hole approximations.

In conclusion, this work transforms the BEB model from a "black box" for total ionisation into a more transparent, physically grounded tool for partial ionisation, provided that experimental data is used for thresholds and the model is adjusted to account for the dipole oscillator strength sum rule.