bi-Maxwellian Distribution Research Articles

Experimental studies of H2/Ar plasma in a cylindrical inductive discharge with an expansion region

Abstract The electrical parameters of H2/Ar plasma in a cylindrical inductive discharge with an expansion region are investigated by a Langmuir probe, where Ar fractions range from 0% to 100%. The influence of gas composition and pressure on electron density, the effective electron temperature and the electron energy probability functions (EEPFs) at different spatial positions are present. In driver region, with the introduction of a small amount of Ar at 0.3 Pa, there is a rapid increase in electron density accompanied by a decrease in the effective electron temperature. Additionally, the shape of the EEPF transitions from a three-temperature distribution to a bi-Maxwellian distribution due to an increase in electron–electron collision. However, this phenomenon resulting from the changes in gas composition vanishes at 5 Pa due to the prior depletion of energetic electrons caused by the increase in pressure during hydrogen discharge. The EEPFs for the total energy in expansion region is coincident to these in the driver region at 0.3 Pa, as do the patterns of electron density variation between these two regions for differing Ar fractions. At 5 Pa, as the discharge transitions from H2 to Ar, the EEPFs evolved from a bi-Maxwellian distribution with pronounced low energy electrons to a Maxwellian distribution in expansion region. This evolve may be attributed to a reduction in molecular vibrational excitation reactions of electrons during transport and the transition from localized electron dynamics in hydrogen discharge to non-localized electron dynamics in argon discharge. In order to validate the experimental results, we use the COMSOL simulation software to calculate electrical parameters under the same conditions. The evolution and spatial distribution of the electrical parameters of the simulation results agree well with the trend of the experimental results.

Preliminary Exploration of Low Frequency Low-Pressure Capacitively Coupled Ar-O2 Plasma

Non-thermal plasma as an emergent technology has received considerable attention for its wide range of applications in agriculture, material synthesis, and the biomedical field due to its low cost and portability. It has promising antimicrobial properties, making it a powerful tool for bacterial decontamination. However, traditional techniques for producing non-thermal plasma frequently rely on radiofrequency (RF) devices, despite their effectiveness, are intricate and expensive. This study focuses on generating Ar-O2 capacitively coupled plasma under vacuum conditions, utilizing a low-frequency alternating current (AC) power supply, to evaluate the system’s antimicrobial efficacy. A single Langmuir probe diagnostic was used to assess the key plasma parameters such as electron density (ne), electron temperature (Te), and electron energy distribution function (EEDF). Experimental results showed that ne increases (7 × 1015 m−3 to 1.5 × 1016 m−3) with a rise in pressure and AC power. Similarly, the EEDF modified into a bi-Maxwellian distribution with an increase in AC power, showing a higher population of low-energy electrons at higher power. Finally, the generated plasma was tested for antimicrobial treatment of Xanthomonas campestris pv. Vesicatoria. It is noted that the plasma generated by the AC power supply, at a pressure of 0.5 mbar and power of 400 W for 180 s, has 75% killing efficiency. This promising result highlights the capability of the suggested approach, which may be a budget-friendly and effective technique for eliminating microbes with promising applications in agriculture, biomedicine, and food processing.

Plasma properties in the vicinity of the last closed flux surface in hydrogen and helium fusion plasma discharges

The origin of the bi-Maxwellian electron energy distribution function (EEDF) observed in the scrape-off layer (SOL) of tokamak plasmas by means of Langmuir probes is still under discussion. It has been assumed that the ionization of hydrogen and deuterium neutrals by thermal electrons penetrating the SOL from the bulk plasma is the main reason for the appearance of a second Maxwellian. To validate this assumption, radial measurements of the electron temperatures and densities, or the plasma properties in helium plasmas in the GOLEM tokamak and the TJ-II stellarator were performed. The radial profiles of the low-temperature electron group densities follow the trend of the calculated radial profiles of the electron sources arising from the ionization of neutrals in both deuterium and helium plasmas in TJ-II. The difference in the radial location where the bi-Maxwellian EEDF appears can be explained by the difference in the rate coefficients for ionization of deuterium and helium. The results of probe measurements in GOLEM and the WEST tokamak divertor, at one radial location in the SOL, are compatible with the hypothesis concerning the ionization of neutral atoms and the type of the EEDF.

Numerical characterization of dual-frequency capacitively coupled plasmas modulated by electron beam injection

The modulated approach of electron beam (EB) injection can achieve favorable parameters for capacitive coupled plasmas (CCP). In this work, a one-dimensional particle-in-cell/Monte Carlo collision (PIC/MCC) model is used to simulate the stable dual-frequency CCP with EB injection. First, when the parameters of EB are kept constant at 0.01 A and 30 eV, the results demonstrate significant enhancements in electron density, self-bias voltage, and ion flux. Furthermore, the electron energy probability function (EEPF) appears to have a transition from a typical bi-Maxwellian distribution to a Maxwellian distribution, and the dominant heating mode shifts from the α-mode to the α-γ-mode. Secondly, when the EB current and energy are all changed, the basic parameters of DF-CCP can be achieved by different modulations. Furthermore, we also discuss the transition of the electron heating mode as the current increases from 0.001 A to 1 A and the energy increases from 10 eV to 490 eV. In particular, we conduct a comparative study among different cases of EB injection. According to these results, the modulation capability of EB injection in DF-CCP is thoroughly investigated, which can greatly benefit atom-scale etching in practical applications.

Analysis of a second peak of electron density observed in high-power impulse magnetron sputtering plasma using a Langmuir probe

Plasma characteristics of a high-power impulse magnetron sputtering (HIPIMS) for copper deposition were investigated using a time-resolved Langmuir probe to explore HIPIMS discharge physics. Various discharge frequencies and pulse widths were employed while operating the HIPIMS in a constant current mode. Waveforms of the HIPIMS cathode current remained constant throughout the HIPIMS voltage pulse. It was found that electrons exhibited a bi-Maxwellian energy distribution both during and after the HIPIMS pulse. After the HIPIMS pulse, plasma density built up to a second peak while the bulk electron temperature quickly decreased. By examining the effect of pulse width and discharge frequency on the temperature of hot electrons through Langmuir I–V curves, it is suggested that the hot electron ionization contributed to the occurrence of the second peak.

Parametrization of coefficients for sub-grid modeling of pitch-angle diffusion in global magnetospheric hybrid-Vlasov simulations

Sub-grid models are key tools to accurately describe the physical processes at play in a system when high-resolution simulations are not feasible. We previously developed a sub-grid model for pitch-angle diffusion in hybrid-Vlasov simulations of Earth's magnetosphere. However, a more precise description of the pitch-angle diffusion coefficient is required to apply this model to global simulations. In this study, we use an existing method to parametrize pitch-angle diffusion coefficients from monotonic distribution functions and adapt it to bi-Maxwellian distributions. We determine these coefficients for various values of the ion temperature anisotropy and plasma β∥. We use these newly parametrized coefficients in our sub-grid model and show that it accurately models reduction of temperature anisotropy in both local simulations and global simulations of the Earth's magnetosphere, while using minimal computational resources.

Excitation of proton firehose instability in magnetospheric cold and hot proton plasma: a quasilinear approach

Abstract Unstable states of different charged species in the solar wind and Earth’s magnetosphere are governed with the collective and collisional processes. For these dilute plasmas, the contribution of microinstabilities driven by the anisotropic particle distribution and heat flux becomes important in defining the stable/equilibrium states of electrons and ions/protons. The present paper highlights the key role of proton firehose instability to regulate an unchecked rise in the temperature anisotropy in these solar wind and magnetospheric environments. Right-handed circularly polarized proton firehose mode becomes unstable when the temperature condition of T ‖p > T ⊥p is satisfied, where the directional subscripts denote directions with respect to the ambient magnetic field. Based on the observations of magnetospheric multi-scale (MMS) space mission, we assume the bi-Maxwellian nature of the model distribution for the multi-component proton plasma. To study the time evolution of the unstable mode, we further allow the time variation in the cold and hot proton temperatures. For the choice of the initial conditions related with observations, we unveil the wave properties (growth and unstable wave number domain) corresponding to the cold/hot proton temperature anisotropy and the plasma betas of constituents proton components. In the back action of proton firehose instability, we highlight the time-scale modifications and saturation of initial bi-Maxwellian distributions and resulting wave-energy densities for various choices of initial cold-hot temperature anisotropy and plasma betas.

Investigation of spatial distribution of EEPFs and neutral species in nitrogen inductively coupled plasmas by 2D hybrid simulation

Neutral species in nitrogen plasmas play a crucial role in many applications related to semiconductor fabrication. In this research, a two-dimensional fluid/electron Monte Carlo hybrid model is employed to simulate nitrogen inductively coupled plasmas, and the spatial distributions of electron energy probability distributions (EEPFs), as well as their influence on the neutral species, are discussed under various pressures. It is found that the EEPF in the bulk region is relatively uniform, and it exhibits a bi-Maxwellian distribution at 3 mTorr. As pressure increases, the high energy tail declines due to the more frequent collisions. Moreover, a hole appears at around 3 eV in the EEPF above the substrate, and it becomes less obvious toward the skin layer below the dielectric window. Moreover, the maxima of metastable species densities, i.e., N2(A3Σu+), N(2D), and N(2P), are located at the center of the chamber at low pressure, and they gradually move to the skin layer under the coils as pressure increases. The behaviors of neutral species can be understood by examining the reactant densities of the main generation and loss mechanisms, as well as the corresponding rate coefficients which are calculated according to EEPFs. In addition, since the ground state N(4S) is mainly produced by the quenching of metastable atoms and neutralization of ions at the walls, the maximum of the N(4S) density appears below the dielectric window and above the substrate at 3 mTorr, and the peak under the dielectric window becomes more obvious at higher pressure due to the stronger locality.

Statistical Study of Anisotropic Proton Heating in Interplanetary Magnetic Switchbacks Measured by Parker Solar Probe

Magnetic switchbacks, which are large angular deflections of the interplanetary magnetic field, are frequently observed by Parker Solar Probe (PSP) in the inner heliosphere. Magnetic switchbacks are believed to play an important role in the heating of the solar corona and the solar wind as well as the acceleration of the solar wind in the inner heliosphere. Here, we analyze magnetic field data and plasma data measured by PSP during its second and fourth encounters, and select 71 switchback events with reversals of the radial component of the magnetic field at times of unchanged electron-strahl pitch angles. We investigate the anisotropic thermal kinetic properties of plasma during switchbacks in a statistical study of the measured proton temperatures in the parallel and perpendicular directions as well as proton density and specific proton fluid entropy. We apply the “genetic algorithm” method to directly fit the measured velocity distribution functions in field-aligned coordinates using a two-component bi-Maxwellian distribution function. We find that the protons in most switchback events are hotter than the ambient plasma outside the switchbacks, with characteristics of parallel and perpendicular heating. Specifically, significant parallel and perpendicular temperature increases are seen for 45 and 62 of the 71 events, respectively. We find that the density of most switchback events decreases rather than increases, which indicates that proton heating inside the switchbacks is not caused by adiabatic compression, but is probably generated by nonadiabatic heating caused by field–particle interactions. Accordingly, the proton fluid entropy is greater inside the switchbacks than in the ambient solar wind.

A simple and fast approach for computing the fusion reactivities with arbitrary ion velocity distributions

Calculating fusion reactivity involves a complex six-dimensional integral of the fusion cross section and ion velocity distributions of two reactants. We demonstrate a simple Monte Carlo approach that efficiently computes this integral for arbitrary ion velocity distributions with a time complexity of O(N), where N is the number of samples. This approach generates random numbers that satisfy the reactant velocity distributions. In cases where these numbers are not readily available, we propose using Gaussian random numbers with weighted factors. For cases where only a small number of N samples are available, a O(N2) method can be used. We benchmarked this approach against analytical results for drift bi-Maxwellian distributions and provided examples of drift ring beam and slowing down distributions. Our results show that the error can be less than 1% with N∼104 samples for our standard approach.

Fusion reactivities with drift bi-Maxwellian ion velocity distributions

The calculation of fusion reactivity involves a complex six-dimensional integral that takes into account the fusion cross section and velocity distributions of two reactants. However, a more simplified one-dimensional integral form can be useful in certain cases, such as for studying fusion yield or diagnosing ion energy spectra. This simpler form has been derived in a few special cases, such as for a combination of two Maxwellian distributions, a beam-Maxwellian combination, and a beam-target combination, and can greatly reduce computational costs. In this study, it is shown that the reactivity for two drift bi-Maxwellian reactants with different drift velocities, temperatures, and anisotropies can also be reduced to a one-dimensional form, unifying existing derivations into a single expression. This result is used to investigate the potential enhancement of fusion reactivity due to the combination of beam and temperature anisotropies. For relevant parameters in fusion energy, the enhancement factor can be larger than 20%, which is particularly significant for proton-boron (p–B11) fusion, as this factor can have a significant impact on the Lawson fusion gain criteria.

Toward developing a comprehensive algorithm for solving kinetic plasma dispersion relations for parallel propagation with a kappa distribution

In general, it is challenging to numerically solve all the roots of plasma wave dispersion relations. The velocity distributions of multi-component particles in an anisotropic high-energy plasma can be better described by a drift loss-cone bi-Kappa distribution or a mixed drift loss-cone distribution containing bi-Kappa and bi-Maxwellian plasma in space and laboratories. In this work, we have developed a code with a new numerical algorithm to solve all roots of the kinetic dispersion relation for parallel propagation in hot magnetized plasmas with drift loss-cone bi-Kappa distribution. Solving all roots of the rational expansions of the kinetic dispersion relation is equivalent to a matrix eigenvalue problem of a linear system. We have performed detailed numerical solutions for three kinds of plasmas: bi-Maxwellian, bi-Kappa, and cold plasmas. We have also proposed a unified numerical method to solve the mixed dispersion relation based on the bi-Kappa and bi-Maxwellian distributions. The numerical results and benchmark studies demonstrate that the new algorithm is in agreement with the data from previous studies. This is a crucial step toward revealing a full picture of kinetic plasma waves and instabilities.

Electron distribution-controlled electron cyclotron harmonic wave generation in the magnetosphere

Electron cyclotron harmonic (ECH) waves, contributing significantly to magnetospheric dynamics, usually appear as a series of harmonics between the multiples of electron gyrofrequency. Previous studies have demonstrated that ECH waves are the electron Bernstein mode excited by the electron loss cone distribution. To investigate how the electron distribution controls the frequency and wave normal angle of excited ECH waves, we derive a concise analytic formula for ECH growth rates using the bi-Maxwellian distribution. Parametric studies show that the expansion of the loss cone size could effectively enlarge ECH growth rates and move the peak growth rates to a smaller wave number k (higher frequency) region, while the deepening of the loss cone can only increase the growth rates. It is also found that the growth rate pattern in the k space exhibits different trends on the two sides of 1.5 fce because the dominant resonance orders on the two sides are different. This study further develops our understanding of the generation mechanism of ECH waves.

Effect of parallel resonance on the electron energy distribution function in a 60 MHz capacitively coupled plasma

In general, as the radio frequency (RF) power increases in a capacitively coupled plasma (CCP), the power transfer efficiency decreases because the resistance of the CCP decreases. In this work, a parallel resonance circuit is applied to improve the power transfer efficiency at high RF power, and the effect of the parallel resonance on the electron energy distribution function (EEDF) is investigated in a 60 MHz CCP. The CCP consists of a power feed line, the electrodes, and plasma. The reactance of the CCP is positive at 60 MHz and acts like an inductive load. A vacuum variable capacitor (VVC) is connected in parallel with the inductive load, and then the parallel resonance between the VVC and the inductive load can be achieved. As the capacitance of the VVC approaches the parallel resonance condition, the equivalent resistance of the parallel circuit is considerably larger than that without the VVC, and the current flowing through the matching network is greatly reduced. Therefore, the power transfer efficiency of the discharge is improved from 76%, 70%, and 68% to 81%, 77%, and 76% at RF powers of 100 W, 150 W, and 200 W, respectively. At parallel resonance conditions, the electron heating in bulk plasma is enhanced, which cannot be achieved without the VVC even at the higher RF powers. This enhancement of electron heating results in the evolution of the shape of the EEDF from a bi-Maxwellian distribution to a distribution with the smaller temperature difference between high-energy electrons and low-energy electrons. Due to the parallel resonance effect, the electron density increases by approximately 4%, 18%, and 21% at RF powers of 100 W, 150 W, and 200 W, respectively.

Effect of temperature anisotropy on the dynamics of geodesic acoustic modes

In this work, we revisit the linear gyro-kinetic theory of geodesic acoustic modes (GAMs) and derive a general dispersion relation for an arbitrary equilibrium distribution function of ions. A bi-Maxwellian distribution of ions is then used to study the effects of ion temperature anisotropy on GAM frequency and growth rate. We find that ion temperature anisotropy yields sensible modifications to both the GAM frequency and growth rate as both tend to increase with anisotropy and these results are strongly affected by the electron to ion temperature ratio.

Experimental investigation on the hysteresis in low-pressure inductively coupled neon discharge

A hysteresis phenomenon observed in neon inductive discharge at low gas pressure is investigated in terms of the evolution of the electron energy distribution function (EEDF). Generally, the hysteresis phenomenon has been reported at high-pressure Ramsauer gas discharges. However, in neon plasma, we found that the hysteresis phenomenon occurs even at low gas pressure (5 mTorr). Furthermore, the hysteresis vanishes with an increase in the gas pressure (10 and 25 mTorr). To analyze this hysteresis, the EEDF is measured depending on the radio frequency power. The EEDF at 10 mTorr sustains the bi-Maxwellian distribution during an E–H transition. On the other hand, the EEDF at 5 mTorr changes dramatically between discharge modes. At 5 mTorr, the measured EEDF for the E mode has the Maxwellian distribution due to high collisional heating in the bulk plasma. The EEDF for the H mode has the bi-Maxwellian distribution because collisionless heating in the skin depth is dominant. This apparent evolution of the EEDF causes a nonlinear energy loss due to collisions during the discharge mode transition. Therefore, the plasma can maintain the H mode discharge with high ionization efficiency, even at a lower applied power, which results in the hysteresis.

Saturating Magnetic Field of Weibel Instability in Plasmas with Bi-Maxwellian and Bikappa Particle Distributions

Saturating Magnetic Field of Weibel Instability in Plasmas with Bi-Maxwellian and Bikappa Particle Distributions

Model of Thomson scattering from z-pinch plasma: Application in experimental design for Plasma Focus

The present work develops a model of Thomson scattering (TS) for z-pinch plasmas. Sustained on the phenomenology observed in dynamical-pinch discharges of interest in fusion studies, the plasma dynamics is modeled by axisymmetric bi-Maxwellian velocity distribution with axial and radial drift velocities. Expressions for TS form factor and screening integrals are deduced, and TS spectra are reconstructed. A characteristic temperature of the spectrum is identified, which is determined by a weighted-sum of the axial and radial temperatures, whose coefficients are given by the square of the respective axial and radial components of k→ over the square of the magnitude of k→. It is shown that it is not possible to determine the velocity distribution function of the plasma from just one direction of measurement. Additionally, an experimental setup, which requires two complementary observation directions for a complete determination of the proposed distribution function, is analyzed and its capacity to measure thermal anisotropy and drift velocities is studied for plasma conditions expected in the pinch phase of a plasma focus discharge.

Influence of electron energy distribution on fluid models of a low-pressure inductively coupled plasma discharge

The aim of the present paper is to examine the influence of assumption on the electron energy distribution function on the relation between the plasma potential and the electron temperature for both electropositive (argon) and electronegative (chlorine) plasmas. A one-dimensional fluid model is used for simplicity although similar results were obtained using a self-consistent two-dimensional fluid model coupled with the Maxwell's equations for inductively coupled plasmas. We find that for electropositive plasma only a bi-Maxwellian electron energy distribution function provides reasonable results compared to measurements in low-pressure inductively coupled plasmas, namely, the increasing plasma potential for increasing electron temperature. For electronegative plasma, the plasma potential is an increasing function of the electron temperature for all electron distributions considered in the model. However, the scaling factors do not agree with the conventional plasma theory. We explain these results by the deviation of electrons from a Boltzmann distribution, which is due to non-equilibrium and non-local nature of plasma at the low-pressure conditions.

Effects of Landau damping and collision on stimulated Raman scattering with various phase-space distributions

Stimulated Raman scattering (SRS) is one of the main instabilities affecting success of fusion ignition. Here, we study the relationship between Raman growth and Landau damping with various distribution functions combining the analytic formulas and Vlasov simulations. The Landau damping obtained by Vlasov–Poisson simulation and Raman growth rate obtained by Vlasov–Maxwell simulation are anti-correlated, which is consistent with our theoretical analysis quantitatively. Maxwellian distribution, flattened distribution, and bi-Maxwellian distribution are studied in detail, which represent three typical stages of SRS. We also demonstrate the effects of plateau width, hot-electron fraction, hot-to-cold electron temperature ratio, and collisional damping on the Landau damping and growth rate. They gives us a deep understanding of SRS and possible ways to mitigate SRS through manipulating distribution functions to a high Landau damping regime.