Influence of Cathode Boundary and Initial Electron Swarm Width on Electron Swarm Parameter Determination with the Pulsed Townsend Experiment

TL;DR

The paper addresses the bias in swarm-parameter extraction from Pulsed Townsend measurements caused by neglecting finite initial pulse width and electrode boundaries.It develops a finite-domain drift-diffusion model with a Gaussian temporal initial condition and derives a corresponding current expression that can be convolved with the initial pulse to fit data.The key contributions include a new analytical current expression for finite domains, demonstrated improvements in extracting $W^b$, $D_L^b$, and $R_{net}$ (notably a $\sim$40% enhancement in $R_{net}$), and a publicly available curve-fitting implementation.The work shows that these improvements enable more accurate swarm parameter measurements across gases and pressures, with practical impact for gas discharge modeling and related fields.

Abstract

The Pulsed Townsend experiment enables the extraction of relevant electron transport properties in different gases such as the electron drift velocity $W$ (or equivalently the mobility $μ$), the longitudinal diffusion coefficient $D_{\mathrm{L}}$, and the effective ionization rate $R_{\mathrm{net}}$ (or equivalently the effective ionization coefficient $α$). Existing analysis techniques lack an accurate representation of the experimental initial and boundary conditions. This work aims to provide an improved evaluation approach by appropriately considering both initial and boundary conditions in order to extract more accurate swarm parameters from measurement data. Simulative and experimental measurement results verify an increased evaluation accuracy. Furthermore, the longitudinal diffusion coefficient $D_{\mathrm{L}}$ can now be accurately extracted from Pulsed Townsend measurements. The developed curve fitting code is made publicly available.

Figures (9)

• Figure 1: Electrode arrangement and initial electron swarm, which is assumed to be distributed as a Gaussian pulse with $\sigma$ denoting the Gaussian RMS width in longitudinal direction, and $z_{\mathrm{0}}$ the initial position of released electrons (/position of the peak). The electrodes have a total diameter of $d_{\mathrm{total}} = 165mm$ with the flat surface being around $d_{\mathrm{flat}} = 114mm$ wide, and a gap spacing of around $L = 10\dots35mm$Haefliger_thesis. The photocathode has a PdZn nanocrystalline coating Photocathode and is $25mm$ wide. A custom-made electrode mounted transimpedance amplifier is included for illustrative purposes.
• Figure 2: Example electron number density over space and time within an electrode gap ($L = 20mm$) for net ionizing conditions. The resulting externally measurable current $I_{\mathrm{e}}(t)$ (in arbitrary units) is also illustrated.
• Figure 3: Example experimental curve fitting results for CO$_{\mathrm{2}}$ gas under different conditions using the state-of-the-art approach (see equation \ref{['eq:casey-eq']}). The black curve represents the raw experimental waveform, the blue curve is used for fitting and results from the raw waveform by cutting and downsampling, the pink curve shows the obtained fit. (Please note that some of the shown raw experimental waveforms have already been made publicly available and can be found at Zenodo-PT-Paper-HVL, and some related swarm data over LXCat CO2_LXCat).
• Figure 4: Example experimental curve fitting results for CO$_{\mathrm{2}}$ gas under different conditions using the proposed method. The black curve represents the raw experimental waveform, the blue curve is used for fitting and results from the raw waveform by cutting and downsampling, the pink curve shows the obtained fit. (Please note that some of the shown raw experimental waveforms have already been made publicly available and can be found at Zenodo-PT-Paper-HVL, and some related swarm data over LXCat CO2_LXCat).
• Figure 5: Electron swarm parameters for simulated waveforms of CO$_{2}$ gas that are evaluated with the state-of-the-art evaluation method in equation \ref{['eq:casey-eq']}. The initial pulse width is chosen as $\sigma_{\mathrm{t}} = 4.5ns$ and the initial swarm position around $z_{\mathrm{0}} = 100um$. The underlying swarm parameter values are calculated with MultiBolt Stephens_2018Flynn_2022 using Biagi's cross section set for CO$_{\mathrm{2}}$Biagi_CO2_LXCat. Simulated electron current waveforms (see section \ref{['sec:fluid-code']}) for the three pressures $p = 50Pa$ ($N \approx 1.23e22\per m\cubed$), $p = 500Pa$ ($N \approx 1.23e23\per m\cubed$) and $p = 5kPa$ ($N \approx 1.23e24\per m\cubed$) over a field range of $E/N = 10\dots2000Td$ are fitted using the existing model (equation \ref{['eq:casey-eq']}). The top row shows the evaluated swarm parameters and the underlying Boltzmann solver values, and the bottom row displays the relative error percentages between the two.