[Paper Review] A novel numerical tool to study electron energy distribution functions of spatially-anisotropic and non-homogeneous ECR plasmas

Bharat Mishra, Angelo Pidatella+6 more|arXiv (Cornell University)|Jul 3, 2021

Plasma Diagnostics and Applications35 references16 citations

TL;DR

This paper presents a novel numerical tool to reconstruct spatially resolved, continuous electron energy distribution functions (EEDFs) in electron cyclotron resonance (ECR) plasmas with anisotropic and non-homogeneous electron populations. Using a trial-and-error fitting method on self-consistently simulated electron data and benchmarking against experimental X-ray spectra from an argon plasma, the method successfully retrieves warm electron temperature and density, revealing competing warm and hot electron components and enabling improved modeling of electron-dependent reactions like Kα emission.

ABSTRACT

A numerical tool for analysing spatially anisotropic electron populations in electron cyclotron resonance (ECR) plasmas has been developed, using a trial-and-error electron energy distribution function (EEDF) fitting method. The method has been tested on space-resolved warm electrons in the energy range $2-20\,\mathrm{keV}$, obtained from self-consistent simulations modelling only electron dynamics in ECR devices, but lacked real-world validation. For experimentally benchmarking the method, we attempted to numerically reproduce the experimental X-ray emission spectrum measured from an argon plasma. Results of this analysis have provided crucial information about density and temperature of warm electrons, and competing distributions of warm and hot electron components. This information can be fed back to simulation models to generate more realistic data. Subsequent application of the numerical tool as described to the improved simulation data can result in continuous EEDFs that reflect the nature of charge distributions in anisotropic ECR plasmas. These functions can be also applied to electron dependent reactions, in order to reproduce experimental results, like those concerning space-dependent K$\alpha$ emissions.

Motivation & Objective

• To develop a numerical method for reconstructing continuous, space-resolved electron energy distribution functions (EEDFs) in spatially anisotropic and non-homogeneous ECR plasmas.
• To address the lack of reliable experimental validation for self-consistent simulation-based EEDFs, particularly for the warm electron component (1–10 keV).
• To enable accurate modeling of electron-dependent processes such as Kα emission by providing realistic EEDFs derived from experimental X-ray spectra.
• To improve simulation fidelity by feeding back experimentally constrained EEDF parameters to kinetic models.
• To demonstrate the method’s utility through benchmarking against measured X-ray emission spectra from an argon ECR plasma.

Proposed method

• A trial-and-error fitting approach is used to determine continuous EEDFs by matching simulated X-ray emission spectra to measured experimental spectra from an argon ECR plasma.
• The plasma is spatially segmented into seven regions-of-interest (ROIs), each analyzed independently to derive position-dependent EEDFs.
• EEDFs are modeled as multi-component distributions (2- or 3-component) with free parameters for electron temperature (kBT), relative population (Cj, Sj), and energy range.
• The method uses a least-squares minimization of mean squared error (MSE) and coefficient of determination (r²) to evaluate fit quality between simulated and experimental X-ray emissivity spectra.
• The simulation data—generated via a self-consistent kinetic code coupled with a 3D electromagnetic solver (COMSOL) and particle tracking (MATLAB)—provides initial 3D electron energy and density maps.
• Experimental X-ray emission data from a 2016 campaign at INFN-LNS and ATOMKI are used as a benchmark for EEDF validation and refinement.

Experimental results

Research questions

• RQ1 Can a numerical fitting method accurately reconstruct continuous, spatially resolved EEDFs for warm electrons (1–10 keV) in anisotropic ECR plasmas using experimental X-ray spectra?
• RQ2 What are the spatially varying temperature and density of warm electron populations in an argon ECR plasma, as inferred from X-ray spectral fitting?
• RQ3 How do competing contributions from warm and hot electron components affect the measured X-ray emission spectrum in ECR plasmas?
• RQ4 To what extent can experimentally constrained EEDFs improve the realism of self-consistent kinetic simulations of ECRIS plasmas?
• RQ5 Can the reconstructed EEDFs be used to quantitatively reproduce spatially resolved Kα emission maps in ECR plasmas?

Key findings

• The method successfully reproduced the experimental X-ray emission spectrum of an argon ECR plasma using a 2-component EEDF model, with a coefficient of determination (r²) of 0.9904 in ROI j=1 and improving to 0.9945 in ROI j=7.
• The warm electron temperature (kBT) in the plasma ranged from 8.9 to 54.7 eV (0.0086–30 keV in the fit parameters), with the highest values found in the central region (ROI j=7).
• The volumetric density of warm electrons was estimated to be in the range of 1.15×10¹⁴ to 9.63×10¹⁴ cm⁻³ across ROIs, with the highest density in the central region.
• The 3-component EEDF models (EEDF3 and EEDF4) showed improved fit quality (r² > 0.97) and lower mean squared error (MSE) compared to 2-component models, especially in outer ROIs.
• The analysis revealed a competing distribution between warm and hot electron components, with the hot component (kBT ~ 1–11 keV) dominating in outer regions, while warm electrons (kBT ~ 1–5 keV) were most prominent in the central zone.
• The reconstructed EEDFs, particularly from EEDF4, showed consistent trends across ROIs with a decreasing contribution of the cold component (Cj) and increasing hot component (S2j) toward the periphery, indicating spatial evolution of electron populations.