Electron Energy Distribution Function Research Articles

Electron kinetics in a positive column of AC discharges in a dynamic regime

We have performed hybrid kinetic-fluid simulations of a positive column in alternating current (AC) argon discharges over a range of driving frequencies f and gas pressure p for the conditions when the spatial nonlocality of the electron energy distribution function (EEDF) is substantial. Our simulations confirmed that the most efficient conditions of plasma maintenance are observed in the dynamic regime when time modulations of mean electron energy (temperature) are substantial. The minimal values of the root mean square electric field and the electron temperature have been observed at f/p values of about 3 kHz Torr−1 in a tube of radius R = 1 cm. The ionization rate and plasma density reached maximal values under these conditions. The numerical solution of a kinetic equation allowed accounting for the kinetic effects associated with spatial and temporal nonlocality of the EEDF. Using the kinetic energy of electrons as an independent variable, we solved an anisotropic tensor diffusion equation in phase space. We clarified the role of different flux components during electron diffusion in phase space over surfaces of constant total energy. We have shown that the kinetic theory uncovers a more exciting and rich physics than the classical ambipolar diffusion (Schottky) model. Non-monotonic radial distributions of excitation rates, metastable densities, and plasma density have been observed in our simulations at pR > 6 Torr cm. The predicted off-axis plasma density peak in the dynamic regime has never been observed in experiments so far. We hope our results stimulate further experimental studies of the AC positive column. The kinetic analysis could help uncover new physics even for such a well-known plasma object as a positive column in noble gases.

Accounting for plasma resistance in the interpretation of probe measurements in gas discharge plasma

The influence of the finite plasma resistance between the Langmuir probe and the reference electrode on the results of measurements of the characteristics of low-temperature plasma: the plasma potential, the average electron energy, the electron concentration, and the electron energy distribution function is analyzed. Specific calculations were made for glow discharge conditions supported by a hollow cathode in helium at a pressure of several millibars.

Reconstruction algorithm for the runaway electron energy distribution function of the ITER hard x-ray monitor

Hard x-ray (HXR) spectrometry in ITER can provide information about Runaway Electrons (REs) in tokamak plasmas. Non-trivial reconstruction techniques must be applied to study the Energy distribution of REs in tokamaks since the diagnostic signals are convoluted with the emission of bremsstrahlung radiation from REs reaching the detector and the detector response function. A developed tool, coupled with the PREDICT code, has been described in this report for reconstructing the Runaway Electron energy distribution function (REDF) from HXR spectrum. Bremsstrahlung emission spectra and the detector response function are utilized in a forward modelling process to generate synthetic HXR spectra for different test REDF to which artificial noise is added. These HXR spectra are utilized to reconstruct the REDFs that can provide information about the REs in the plasma. The reconstruction process has been applied to the ITER HXR Monitor configuration for the first time. The effect of reduced optical transmission efficiency is studied on the reconstruction process and the accuracy of the extracted RE parameters. The performance of the reconstruction process is also tested for different amount of photon counts to identify the minimum number of photon counts required for optimal reconstruction. Preliminary results of RE-current estimation using the reconstruction process are also presented.

Effects of high energy electrons on electron temperature measurements of asymmetric double Langmuir probes

In this work, electron temperature was measured with both the asymmetric double Langmuir probe (ADLP) and the single Langmuir probe to investigate the reliability of the ADLP on the electron temperature measurement in multi-temperature Maxwellian plasmas. A series of I–V traces of the ADLP were obtained at various plasma conditions with different area ratios and analyzed with different methods including conventional ADLP analysis and two-temperature Maxwellian fitting with results measured by a single planar Langmuir probe analyzed with three-temperature Maxwellian fitting as reference. The measured Te of the ADLP is found to reflect that of the temperature of the degraded primary electrons when the area ratio of the probe tips is close to ∼16 and approaches the real effective electron temperature as the area ratio increases to a value of ∼30% higher than that measured by a single Langmuir probe, which occurs even when the area ratio is higher than the flux ratio of electrons and ions entering their respective sheaths. This effect is consistent with the distortion effect of Langmuir probe I–V traces caused by the presence of hotter electron species, which was computationally reconstructed and agreed well with the experimental observations. This result implies that an area ratio, possibly ∼20 times much larger than what was conventionally assumed, is needed for an ADLP to be reliably treated as a single Langmuir probe in practical settings, where electron energy distribution functions of plasmas are generally expected to be multi-temperature Maxwellian. This effect is also analogous to the current balance between a single Langmuir probe and the device wall, implying that this effect would also affect the application of the single Langmuir probe in plasmas, where the ion loss to the device wall can be reduced, such as plasmas in miniaturized devices, strong magnetic fields, or a highly ion-neutral collisional environment.

Investigation of Characteristics and Parameters of Plasma of an Overstressed Nanosecond Discharge in Krypton between Electrodes Made of a Superionic Conductor – Silver Sulfide

The results of a study of the electrical and optical characteristics of an overvoltage nanosecond atmospheric pressure discharge between electrodes made of a superionic conductor (Ag2S) in krypton are presented. The destruction of electrodes in the discharge and the introduction of Ag2S vapor into the interelectrode gap occurred due to microexplosions of inhomogeneities on the working surfaces of polycrystalline electrodes (the formation of ectons) in order to synthesize thin films based on this compound on the surface of a dielectric substrate which was installed near the electrodes. By numerically solving the Boltzmann kinetic equation for the electron energy distribution function, the temperature and the electron density in the discharge, the specific discharge power losses for the main electronic processes, and the rate constants of the electronic processes depending on the parameter E/N for the plasma of vapor-gas mixtures based on krypton and silver sulfide have been calculated. Homogeneous thin films based on silver sulfide were synthesized on the quartz substrates by the gas-discharge method under conditions of ultraviolet assisted discharge plasma.

Basic metal oxide integrated DBD packed bed reactor for the decomposition of CO2

A coaxial dielectric barrier discharge (DBD) reactor was used to perform catalytic non-thermal plasma for CO2 conversion under ambient conditions. A systematic study was made to understand the role of basic oxides in activating weakly acidic CO2. For this purpose, MO/γ-Al2O3 (M = Mg, Ca, Sr, and Ba) catalysts were synthesized and integrated with the DBD reactor. The applied voltage was varied from 16 kV to 22 kV to determine the effect of applied voltage on CO2 conversion. Plasma discharge generates high-energy electrons that activate the basic oxides. Integration of basic metal oxides with the non-thermal plasma reactor resulted in a higher CO2 conversion. The adsorption of weakly acidic CO2 on the basic sites is responsible for the higher conversion of catalytic plasma reactor over plasma reactor alone. Among the basic metal oxides studied, SrO loaded γ-Al2O3 resulted in the best conversion, where CO2 conversion of ∼ 12% and energy efficiency of ∼ 1.46 mmol kJ−1 was attained at a power of 1.8 W. The concentration of O2 and O3 was measured during the reaction. The hybridized system's superior performance may be due to increased charge deposition and altered gas-phase chemistry because of catalyst integration. BOLSIG + software was used to compute the mean electron energies and electron energy distribution function for various packing conditions.

A new hard x-ray spectrometer for runaway electron measurements in tokamaks

Runaway electron gamma-ray detection system, a novel hard x-ray (HXR) spectrometer optimized for bremsstrahlung radiation measurement from runaway electrons in fusion plasmas, has been developed. The detector is based on a 1‘×1’ LaBr3:Ce scintillator crystal coupled with a photomultiplier tube. The system has an energy dynamic range exceeding 20 MeV with an energy resolution of 3% at 661.7 keV. The detector gain is stable even under severe loads, with a gain shift that stays below 3% at HXR counting rates in excess of 1 MCps. The performance of the system enables unprecedented studies of the time-dependent runaway electron energy distribution function, as shown in recent runaway electron physics experiments at the ASDEX Upgrade and COMPASS tokamaks.

Numerical strategy for solving the Boltzmann equation with variable E/N using physics-informed neural networks

A novel strategy to solve the Boltzmann equation with variable reduced electric field by using an artificial neural network (ANN) is introduced, where is the electric field and is the gas number density. In this method, the ANN learns the electron velocity distribution function (EVDF) for arbitrary in the Boltzmann equation. Thus, the ANN can calculate the EVDFs in the training range of without additional training. For validation of the ANN, the EVDFs in each Ar and SF6 gas were calculated with the trained ANN. The electron energy distribution function and electron transport coefficients calculated from the EVDF quantitatively agree with those from another ANN for a single and those from a Monte Carlo simulation, proving the validity of the present method.

Partial EEDF analysis and electron diagnostics of atmospheric-pressure argon and argon–helium DBD plasma

Partial arbitrary electron energy distribution function (EEDF) results for cold atmospheric-pressure plasma are reported. The EEDF is obtained using the visible bremsstrahlung inversion (VBI) method. This machine learning method requires only optical emission spectroscopy (OES) measurement and a momentum transfer cross section to determine a partial EEDF. Numerical EEDF of a pure-argon dielectric barrier discharge dataset with changing peak-to-peak voltage and a helium–argon discharge with changing mixture ratio are reported. Resemblance between the numerical EEDF and a two-temperature Maxwell distribution is observed and a simplified three-point numerical EEDF is obtained. The electron temperature and relative electron number density for the bulk and high-energy electron populations are measured. The bulk electron temperature was consistently 0.3 eV. For pure argon, the high-energy electron temperature decreased exponentially from 3 to 2.2 eV with increasing peak-to-peak voltage from 3.6 to 6.3 kV. For the helium–argon dataset, the high-energy electron temperature decreased linearly from 4.2 to 2.2 eV with increasing argon fraction 25%–100%. From an OES measurement, the arbitrary EEDF can be observed by utilization of the VBI method. Based on this numerical EEDF, appropriate assumptions can be applied to simplify the quantification of electron diagnostics.

Specificity of the electron energy distribution function in a low-pressure nitrogen plasma

In this work, self-consistent simulations of a glow discharge positive column in low-pressure nitrogen plasma are presented. The electron energy distribution function (EEDF) was calculated by solving the complete (i.e. nonlocal) Boltzmann kinetic equation for electrons in both the energy and coordinate (space) variables. The accuracy and range of applicability of the two limiting cases are considered when the kinetic equation is reduced to a dependence on only one variable: kinetic energy in the local approximation or total energy in the so-called totally nonlocal (Holstein–Tsendin) approximation. It is shown that the Holstein–Tsendin approximation, when electron spatial transport dominates the electron energy relaxation (i.e. the ratio between the typical plasma inhomogeneity characteristic length and the electron energy relaxation length ), can be used at low nitrogen pressures (when the parameter pR ⩽ 0.05 Torr). With a further increase in pressure, there is a large intermediate region where the EEDF is nonlocal in the sense that the full kinetic equation must be solved. So, even at pR ≈ 0.5 Torr, the EEDF in nitrogen plasma is nonlocal. Therefore, the use of a local approximation to find the EEDF, when all terms with spatial gradients and an ambipolar field can be discarded in the kinetic equation, requires careful verification. Particular attention should be paid to the periphery of the discharge volume, where the self-consistent ambipolar field inevitably exceeds the external electric field that heats the electrons and produces plasma.

Benchmarking and validation of a hybrid model for electropositive and electronegative capacitively coupled plasmas

In this work, a fluid/Monte Carlo collision (fluid/MCC) hybrid model is developed based on the framework of multi-physics analysis of plasma sources. This hybrid model could be highly accurate in predicting the nonequilibrium phenomena in capacitively coupled plasmas and meanwhile avoid the limitation caused by the computational cost. Benchmarking against the well-established particle-in-cell/MCC (PIC/MCC) method and comparison with experimental data have been presented both in electropositive N2 discharges and electronegative O2 discharges. The results indicate that in N2 discharges, the ion density evolves from a uniform distribution to an edge-high profile as power increases. Besides, the electron energy distribution function (EEDF) at the bulk center exhibits a ‘hole’ at about 3 eV, and the ‘hole’ becomes less obvious at the radial edge, because more low energy electrons are generated there. In O2 discharges, the EEDF exhibits a Druyvesteyn-like distribution in the bulk region, and it evolves to a Maxwellian distribution in the sheath, indicating the dominant influence of the electric field heating there. The results obtained by the hybrid model agree well with those calculated by the PIC/MCC method, as well as those measured by double probe, except for a slight discrepancy in absolute values. The qualitative agreement achieved in this work validates the potential of this hybrid model as an effective tool in the deep understanding of plasma properties, as well as in the improvement of plasma processing.

Numerical Study on Kinetic Effects of Driving Frequency in Atmospheric Radio Frequency Discharges Using Deep Neural Network

The particle-in-cell Monte Carlo collision (PIC-MCC) model is an essential way to investigate the kinetic behaviors of low-temperature plasmas, but usually, it is hugely time consuming in simulating atmospheric radio frequency (RF) plasmas. In this study, a deep neural network (DNN) with multiple hidden layers is developed to predict the kinetic discharge characteristics of atmospheric RF plasmas. The results obtained from the PIC-MCC model are used as the training dataset for the DNN, and the well-trained DNN is able to efficiently yield various kinetic behaviors of atmospheric RF discharges with very high precision. The validation of the results predicted by the DNN algorithm is performed by comparing them with the simulation results directly from the PIC-MCC model. Compared with the time-consuming PIC-MCC simulations, the well-trained DNN takes only 0.01 s to yield the essential kinetic characteristics of atmospheric RF discharges, which saves seven orders of magnitude of computation time compared with the traditional PIC-MCC simulation. The predicted data show that the discharge current of atmospheric RF discharges increases monotonically with the driving frequency at a given applied voltage, and as the driving frequency increases, the electric field in the sheath region is strongly enhanced with the sheath region shrinks and the bulk plasma region expanding. Additionally, the electron energy distribution function (EEDF) can be accurately predicted by DNN; moreover, as the driving frequency increases, the low-energy electrons can be transformed into medium-energy electrons, leading to a transition of the EEDF from a three-temperature distribution to a Maxwellian distribution from the DNN prediction. The study indicates that the well-trained DNN is a promising tool for plasma simulation with very high efficiency and accuracy compared with the PIC-MCC model with huge computational costs, which could provide enough kinetic results to further understand the discharge behaviors in atmospheric RF discharges.

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ 0/x p , where is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.

The comparison of two-term Boltzmann approximation and Monte-Carlo solutions for e+Ar0 plasma

In this paper, we examine the energy distribution function of electrons in the case of a very weakly ionized argon plasma at sub-atmospheric pressure and external electric field using Boltzmann kinetic equation. We consider a spatially uniform model because our primary interest is the behavior of the collisional part of the equation. The purpose of the study is to compare two different numerical approaches: a deterministic one (using two-term local non-stationary approximation) and a stochastic approach (using the Monte Carlo method) over a wide range of reduced electric fields. We compare steady-state and time-dependent solutions, isotropic and anisotropic parts of the electron energy distribution function, and reaction constants. The research will also help to identify any limitations and challenges of these methods.

Data processing techniques for ion and electron-energy distribution functions

Retarding field energy analyzers and Langmuir probes are routinely used to obtain ion and electron-energy distribution functions (IEDF and EEDF). These typically require knowledge of the first and second derivatives of the current–voltage characteristics, both of which can be obtained using analog and numerical techniques. A frequent problem with electric-probe plasma diagnostics is the noise from the plasma environment and measurement circuits. This poses challenges inherent to differentiating noisy signals, which often require prior filtering of the raw current–voltage data before evaluating the distribution functions. A review of commonly used filtering and differentiation techniques is presented. It covers analog differentiator circuits, polynomial fitting (Savitzky–Golay filter and B-spline fitting), window filtering (Gaussian and Blackman windows) methods as well as the AC superimposition and Gaussian deconvolution routines. The application of each method on experimental datasets with signal-to-noise ratios ranging from 44 to 66 dB is evaluated with regard to the dynamic range, energy resolution, and signal distortion of the obtained IEDF and EEDF as well as to the deduced plasma parameters.

Discussion on the transport processes in electrons with non-Maxwellian energy distribution function in partially-ionized plasmas

In a previous work (Alvarez Laguna et al 2022 Phys. Plasmas 29 083507), we have developed a non-linear moment model for electrons that self-consistently captures non-Maxwellian electron energy distribution function effects. The model does not rely in the local approximation and the transport coefficients are calculated by expanding the distribution function into Hermite polynomials and by taking moments of the Boltzmann equation, including the collision operator for elastic and inelastic collisions with arbitrary cross sections. This model captures the classical Fick’s, Fourier’s, and Ohm’s law as well as Soret, Dufour, and Peltier effects. In addition, novel non-local transport phenomena appear as a result of spatial gradients of the kurtosis of the distribution function. In this paper, we discuss on the transport effects by analyzing two collisional models: constant collision frequency and constant cross section. We estimate the order of magnitude of the transport processes in non-equilibrium electrons by analyzing the Langmuir probe measurements of a low-pressure argon inductively-coupled discharge. The results show that, under these conditions, the transport produced by the spatial gradients in the kurtosis of the distribution function produces a heat-flux contribution that is of the same order of magnitude as the Fourier and Dufour’s effects. These transport effects are beyond the local field or the electron gradient expansions, commonly used in the low-temperature plasma modeling.

Effects of oxygen concentration on streamer propagation and ozone production in a single-filament streamer discharge at atmospheric pressure

Limited research has been conducted on the formation mechanism of chemically active species in streamer discharges with respect to the oxygen concentration, which is critical to various applications such as ozone generation, air purification, and plasma-assisted combustion, among others. Herein, the oxygen concentration in an N2/O2 gas atmosphere is varied from 1% to 99% under atmospheric pressure and room temperature to investigate changes in the characteristics of streamer discharge propagation and generation of chemically active species. As the oxygen concentration increases from 10% to 90%, the decay rate of the discharge current, propagation velocity of the primary streamer, and ozone production efficiency increase. These phenomena are qualitatively explained by the electron attachment reaction to oxygen molecules and changes in the electron energy distribution function caused by the change in the oxygen concentration. However, the amount of discharge emission from N2(C) cannot be explained by changes in the fraction of electron energy lost in excitation of N2(C) and its quantum yield, implying that changes in the production of N2(C) in the primary and secondary streamers must be considered in a spatiotemporal manner. This study demonstrates that the ozone and N2(C) production characteristics in streamer discharges vary nonlinearly with respect to the oxygen concentration.

Design of an optimised movable electrostatic diagnostic for the investigation of plasma properties in a large negative ion source

This contribution describes the design of a movable Langmuir probe system to be permanently installed in the SPIDER negative ion source. This diagnostic aims at measuring the axial profiles of plasma density, electron temperature and floating potential in both the driver and the expansion region, with a view to further investigate the positive ion dynamics. Furthermore, a deeper understanding of the plasma formation mechanism and of the RF coupling can be provided, e,g. via the estimation of the Electron Energy Distribution Function (EEDF) inside the driver. Thermal analyses were performed to define both the allowable duration of the probe exposure to the plasma and the extent of the voltage sweep. The maximum exposure time for the probe shaft is 10 s, thus a fast manipulation system is included in the design. The integration in the SPIDER source is described, considering all the limitations due to the space availability, the presence of the RF generators and the vacuum-compatibility requirements. To comply with all the constraints, several solutions are proposed for the probe head design.

Activation of vibrational-induced CO2 dissociation in cold non-equilibrium plasma

The activation of vibrational-induced dissociation of CO2 in cold non-equilibrium plasma discharges is investigated by means of a 0D self-consistent kinetic model, which, with a state-to-state approach, is able to calculate the CO2 vibrational distribution function (vdf) of the asymmetric mode levels, the electron energy distribution function and the corresponding vibrational-induced and electron impact CO2 dissociation rates. The conditions for the onset of such activation are linked to the achievement of a sufficiently high CO2 vibrational excitation characterized by the presence of a non-equilibrium plateau in the CO2 vdf, resulting from the combined effect of electron–vibrational and vibrational–vibrational collisions, which, by overpopulating the higher vibrational levels, enhances dissociation. Such non-equilibrium conditions are maximized at lower gas temperature, lower pressure and higher power density values. In particular, for the power density, an activation threshold value can be obtained from simulations and its dependence on the gas temperature and pressure can be investigated. The dependence of the maximum vibrational temperature reached at the end of the discharge as a function of the gas temperature and pressure is also analyzed. A satisfactory agreement from our simulation results with the Kotov’s criterion for vibrational activation has been found.

Gas Discharge Properties of Inverse-Brush-Electrodes Driven by Alternating Current

The inverse-brush-electrodes (IBE) structure with a large area is designed to investigate the discharge mechanism driven by alternating current (ac). It is found that the IBE structure increases the number of secondary electrons and reduces the breakdown voltage by comparing the discharge processes of IBE and plate electrodes. Meanwhile, the breakdown mechanism of gas discharge driven by ac is analyzed in detail. Additionally, the designed IBE structure has a negative differential conductivity enhancement effect than that of plate electrodes when plasmas are generated in the heavy noble gas. Finally, the negative differential conductivity enhancement phenomenon of the IBE structure is analyzed by experiments and simulations. This effect originates from the non-equilibrium electron energy distribution function (EEDF), which is due to the more suprathermal electrons generated by the IBE structure.