non-Maxwellian Distribution Function Research Articles

Parallel expansion of a fuel pellet plasmoid

The problem of the assimilation of a cryogenic fuel pellet injected into a hot plasma is considered. Due to the transparency to ambient particles of the plasmoid, the localised region of high-density plasma created by ionisation of the ablated pellet material, electrons reach a ‘quasiequilibrium’ (QE) state which is characterised by a steady-state on the fastest collisional time scale. The simplified electron kinetic equation of the QE state is solved. Taking a velocity moment of the higher-order electron kinetic equation, which is valid on the expansion time scale, permits a fluid closure, yielding an evolution equation for the macroscopic parameters describing the QE distribution function. In contrast to the Braginskii equations, the closure does not require that electrons have a short mean free path compared with the size of density perturbations, and permits an anisotropic and highly non-Maxwellian distribution function. As the QE distribution function accounts for both trapped and passing electrons, the self-consistent electric potential that causes the expansion can be properly described, in contrast to earlier models of pellet plasmoid expansion with an unbounded potential. The plasmoid expansion is simulated using both a Vlasov model and a cold-fluid model for the ions. During the expansion plasmoid ions and electrons obtain nearly equal amounts of energy; as hot ambient electrons provide this energy in the form of collisional heating of plasmoid electrons, the expansion of a pellet plasmoid is expected to be a potent mechanism for the transfer of energy from electrons to ions on a time scale shorter than that of ion–electron thermalisation.

Ten-moment fluid model with heat flux closure for gasdynamic flows

The gasdynamic 10-moment equations are revisited, applying a Chapman-Enskog type expansion to achieve the closure for heat flux. The resulting gradient-based closure performs favorably against previously employed closures in regions of shocks when compared to the kinetic results obtained from Direct Simulation Monte Carlo simulations. The 10-moment model can capture finite kinetic effects due to non-Maxwellian velocity distribution functions, as compared to the 5-moment Navier-Stokes equations, and remains hyperbolic-parabolic with real wave speeds under all initial conditions. Self-consistent kinetic boundary conditions are derived without consideration of a distinct Knudsen layer. The model is applied to a number of canonical gasdynamics problems in one and two dimensions such as steady normal and oblique shocks, as well as the Sod shock tube problem.

Local Proton Heating at Magnetic Discontinuities in Alfvénic and Non-Alfvénic Solar Wind

We investigate the local proton energization at magnetic discontinuities/intermittent structures and the corresponding kinetic signatures in velocity phase space in Alfvénic (high cross helicity) and non-Alfvénic (low cross helicity) wind streams observed by Parker Solar Probe. By means of the partial variance of increments method, we find that the hottest proton populations are localized around compressible, coherent magnetic structures in both types of wind. Analysis of parallel and perpendicular temperature distributions suggest that the Alfvénic wind undergoes preferential enhancements of T ∥ at such structures, whereas the non-Alfvénic wind experiences preferential T ⊥ enhancements. Although proton beams are present in both types of wind, the proton velocity distribution function displays distinct features. Hot beams, i.e., beams with beam-to-core perpendicular temperature T ⊥,b /T ⊥,c up to three times larger than the total distribution anisotropy, are found in the non-Alfvénic wind, whereas colder beams are in the Alfvénic wind. Our data analysis is complemented by 2.5D hybrid simulations in different geometrical setups, which support the idea that proton beams in Alfvénic and non-Alfvénic wind have different kinetic properties and different origins. The development of a perpendicular nonlinear cascade, favored in balanced turbulence, allows a preferential relative enhancement of the perpendicular plasma temperature and the formation of hot beams. Cold field-aligned beams are instead favored by Alfvén wave steepening. Non-Maxwellian distribution functions are found near discontinuities and intermittent structures, pointing to the fact that the nonlinear formation of small-scale structures is intrinsically related to the development of highly nonthermal features in collisionless plasmas. Our results contribute to understanding the role of different coherent structures in proton energization and their implication in collisionless energy dissipation processes in space plasmas.

Effect of nonthermal electron distributions on dust acoustic solitons in cometary plasmas

We investigate the effect of nonthermal electrons modeled by two non-Maxwellian distribution functions, i.e., the (r, q) and Cairn’s distributions on the formation of dust acoustic (DA) solitons in an un-magnetized dusty plasma by incorporating the effect of dust streaming. We adopt the pseudopotential technique to obtain solitary wave solutions from fluid equations. It is seen that only rarefactive soliton can be obtained in such plasmas where ions are considered Boltzmannian and electrons non-Maxwellian. We find that soliton characteristics are strongly dependent on the nonthermal spectral indices r, q, and α and dust temperature Td. For (r, q) distribution, it is found that soliton amplitude increases but width decreases when the positive (negative) value of r decreases (increases). For Cairn’s distribution, we find that with the increase in α, soliton amplitude decreases. In space environments, such as cometary tails, solar wind, and Earth’s magnetosphere, where non-Maxwellian populations of electrons are present, our theoretical results show that the amplitude of soliton remains smaller than the Maxwellian case. Thus, Maxwellian distribution overestimates the soliton amplitude in such space environments. Therefore, we feel that our results will better interpret the results of observations, from cometary tails, and other space plasmas where nonlinear DA structures are likely to be observed.

Non-Maxwellian electron effects on the macroscopic response of a Hall thruster discharge from an axial–radial kinetic model

A 2D axial–radial particle-in-cell (PIC) model of a Hall thruster discharge has been developed to analyze (mainly) the fluid equations satisfied by the azimuthally-averaged slow dynamics of electrons. Their weak collisionality together with a strong interaction with the thruster walls lead to a non-Maxwellian velocity distribution function (VDF). Consequently, the resulting macroscopic response differs from a conventional collisional fluid. First, the gyrotropic (diagonal) part of the pressure tensor is anisotropic. Second, its gyroviscous part, although small, is relevant in the azimuthal momentum balance, where the dominant contributions are orders of magnitude lower than in the axial momentum balance. Third, the heat flux vector does not satisfy simple laws, although convective and conductive behaviors can be identified for the parallel and perpendicular components, respectively. And fourth, the electron wall interaction parameters can differ largely from the classical sheath theory, based on near Maxwellian VDF. Furthermore, these effects behave differently in the near-anode and near-exit regions of the channel. Still, the profiles of basic plasma magnitudes agree well with those of 1D axial fluid models. To facilitate the interpretation of the plasma response, a quasiplanar geometry, a purely-radial magnetic field, and a simple empirical model of cross-field transport were used; but realistic configurations and a more elaborate anomalous diffusion formulation can be incorporated. Computational time was controlled by using an augmented vacuum permittivity and a stationary depletion law for neutrals.

Gyrokinetic modelling of non-linear interaction of Alfvén waves and EGAMs in ASDEX-Upgrade

Abstract Energetic particle (EP) dynamics and excitation of EP driven instabilities is an important topic of study for the physics of fusion reactors. In this paper we consider EPs injected in the plasma by neutral beams at high energies to heat it. EP species exist far from thermal equilibrium in the form of anisotropic non-Maxwellian distribution functions. EP driven modes, such as Alfvén waves (AWs) and EP driven geodesic acoustic modes (EGAMs), can redistribute EPs in phase-space and harm confinement. We examine the effects of experimental-like anisotropic EP distribution functions on the excitation and the non-linear coupling of such instabilities with the gyrokinetic particle-in-cell code ORB5. The growth rate of EGAMs is found to be sensitively dependent on the phase-space shape of the distribution function as well as on the non-linear wave-wave coupling with AWs. Experimental findings are compared with numerical results.

Inertial and anisotropic pressure effects on cross-field electron transport in low-temperature magnetized plasmas

In this paper, a one-dimensional (1D) particle-in-cell Monte Carlo collision (PIC-MCC) model is developed to investigate the effects of anisotropic pressure and inertial terms due to non-Maxwellian velocity distribution functions on cross-field electron transport. The conservation of momentum is evaluated by taking the moments of the first-principles gas-kinetic equation. A steady-state discharge is obtained without any low-frequency ionization oscillations by considering an anomalous electron scattering profile. The results obtained from the 1D PIC-MCC model are compared with fluid models, including the quasi-neutral drift-diffusion (DD), non-neutral DD, and full fluid moment models. The discharge current obtained from the PIC-MCC model is in good agreement with the fluid models. The cross-field electron transport due to the inertial terms, i.e. the gradient of axial and azimuthal drift, is evaluated. Moreover, PIC-MCC simulation results show non-zero, anisotropic, off-diagonal pressure tensor terms due to asymmetric non-Maxwellian electron velocity distribution function, potentially contributing to cross-field electron transport.

The nonlocal electron heat transport under the non-Maxwellian distribution in laser plasmas and its influence on laser ablation

The electron heat transport plays an important role in laser driven inertial confinement fusion. For the plasmas created by intense laser, the traditional Spitzer–Härm theory cannot accurately describe the electron heat transport process mainly due to two physical effects. First, the electron distribution function would significantly differ from the Maxwellian distribution because of the inverse bremsstrahlung heating. Second, the long mean free paths of heat carrying electrons relative to the temperature scale length indicate that the electron heat flux has the nonlocal feature. In 2020, we have developed a nonlocal electron heat transport model based on the non-Maxwellian electron distribution function (NM-NL model) to describe the electron heat flux in laser plasmas. Recently, this model is successfully incorporated into our radiation hydrodynamical code RDMG. In this article, we numerically investigated the electron heat flux in laser plasmas, especially the nonlocal feature of heat flux and the influence of the non-Maxwellian distribution. The influence of electron heat transport on laser ablation is also discussed. The simulated plasma conditions based on different electron heat transport models are presented and compared with experiments. Our results show that the nonlocal feature of heat flux and the influence of non-Maxwellian distribution function are considerable in plasmas heated by intense lasers.

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.

Discussion on Electron Temperature of Gas-Discharge Plasma with Non-Maxwellian Electron Energy Distribution Function Based on Entropy and Statistical Physics

Electron temperature is reconsidered for weakly-ionized oxygen and nitrogen plasmas with its discharge pressure of a few hundred Pa, with its electron density of the order of 1017m-3 and in a state of non-equilibrium, based on thermodynamics and statistical physics. The relationship between entropy and electron mean energy is focused on based on the electron energy distribution function (EEDF) calculated with the integro-differential Boltzmann equation for a given reduced electric field E/N. When the Boltzmann equation is solved, chemical kinetic equations are also simultaneously solved to determine essential excited species for the oxygen plasma, while vibrationally excited populations are solved for the nitrogen plasma, since the EEDF should be self-consistently found with the densities of collision counterparts of electrons. Next, the electron mean energy U and entropy S are calculated with the self-consistent EEDF obtained, where the entropy is calculated with the Gibbs's formula. Then, the "statistical" electron temperature Test is calculated as Test=[∂S/∂U]-1. The difference between Test and the electron kinetic temperature Tekin is discussed, which is defined as [2/(3k)] times of the mean electron energy U=⟨ϵ⟩, as well as the temperature given as a slope of the EEDF for each value of E/N from the viewpoint of statistical physics as well as of elementary processes in the oxygen or nitrogen plasma.

Comparison of Langmuir probe and optical emission spectroscopy on a volume-cusp filament ion source using helium.

D-Pace has a Penning ion source that runs with helium for studies of α-particle production. We want to study its plasma properties as a function of varying operational parameters, which results in varied output ion beam characteristics. In order to diagnose the ion source plasma, a collisional-radiative code for helium to be used with optical emission spectroscopy measurements is developed. This has the advantage of being non-invasive, which allows for measurements using the emitted light from the plasma. This collisional-radiative code is shown to compare well with the Yacora on the Web model developed at IPP-Garching, and improves upon it with the addition of radiation trapping. Furthermore, the sensitivity of this model to the inclusion of additional excited state populations and non-Maxwellian electron energy distribution functions is investigated. It is shown that non-Maxwellian distributions can significantly affect spectroscopy measurements. This diagnostic is benchmarked to Langmuir probe measurements on the TRIUMF-licensed volume-cusp ion source to determine whether it can replicate the measured electron density and electron temperature as a function of varied operational parameters. The operational parameters are helium gas flow (10-40 SCCM), arc voltage (100-200V), and arc current (1-5A). The measured plasma properties, while different in absolute value, have similar trends for each operational parameter except when varying arc voltage. It is shown that this mismatch as a function of arc voltage is likely due to high energy non-Maxwellian electrons from the cathode sheath, which are not included within the collisional-radiative model.

A regularized high-order moment model to capture non-Maxwellian electron energy distribution function effects in partially ionized plasmas

A model for electrons in partially ionized plasmas that self-consistently captures non-Maxwellian electron energy distribution function (EEDF) effects is presented. The model is based on the solution of scalar and vectorial velocity moments up to the contracted fourth-order moment. The set of fluid (macroscopic) equations is obtained with Grad's method and exact expressions for the collision production terms are derived, considering the electron–electron, electron–gas, and electron–ion elastic collisions as well as for electron–gas excitation and ionization collisions. A regularization of the equations is proposed in order to avoid spurious discontinuities, existing in the original Grad's moment model, by using a generalized Chapman–Enskog expansion that exploits the disparity of mass between the electrons and the heavy particles (ions and atoms) as well as the disparity of plasma and gas densities, typical of gas discharges. The transport model includes non-local effects due to spatial gradients in the EEDF as well as the impact of the EEDF in the calculation of the elastic and inelastic collision rates. Solutions of the moment model under spatially homogeneous conditions are compared to direct simulation Monte Carlo and a two-term Boltzmann solver under conditions that are representative of high plasma density discharges at low-pressure. The moment model is able to self-consistently capture the evolution of the EEDF, in good quantitative agreement with the kinetic solutions. The calculation of transport coefficients and collision rates of an argon plasma in thermal non-equilibrium under the effect of an electric field is in good agreement with the solutions of a two-term Boltzmann solver, largely improving models with a simplified Bhatnagar–Gross–Krook collisional operator.

Two-dimensional simulation of non-equilibrium silicon nitride plasma discharge for thin film solar cell deposition

In this work, we model a Two-dimensional (2D) radiofrequency (RF) plasma discharge of hydrogenated silicon nitride (SiH4/NH3/H2) in a capacitive coupled plasma reactor (CCP), using non-Maxwellian electron energy distribution function (EEDF). The purpose is to investigate whether there is a real advantage and a significant contribution in 2D simulation using non-Maxwellian EEDF rather than Maxwellian one for determining the fundamental characteristics of radiofrequency plasma discharge. The results show a comparative study of the 2D evolution of the fundamental characteristics of the RF plasma using Maxwellian and non-Maxwellian EEDF. By considering the criterion of the density and the electronic temperature, the analysis of the results shows that the adoption of the non-equilibrium 2D EEDF makes it possible to obtain an approximation closer to the real conditions and thus a better understanding of the plasma discharge.

On the electron sheath theory and its applications in plasma–surface interactions

In this work, an improved understanding of electron sheath theory is provided using both fluid and kinetic approaches while elaborating on their implications for plasma–surface interactions. A fluid model is proposed considering the electron presheath structure, avoiding the singularity in electron sheath Child–Langmuir law which overestimates the sheath potential. Subsequently, a kinetic model of electron sheath is established, showing considerably different sheath profiles in respect to the fluid model due to non-Maxwellian electron velocity distribution function and finite ion temperature. The kinetic model is then further generalized and involves a more realistic truncated ion velocity distribution function. It is demonstrated that such a distribution function yields a super-thermal electron sheath whose entering velocity at the sheath edge is greater than the Bohm criterion prediction. Furthermore, an attempt is made to describe the electron presheath–sheath coupling within the kinetic framework, showing a necessary compromise between a realistic sheath entrance and the inclusion of kinetic effects. Finally, the secondary electron emissions induced by sheath-accelerated plasma electrons in an electron sheath are analysed and the influence of backscattering is discussed.

Influence of the electron kinetics on Ar/NF3 inductively coupled plasma

A self-consistent two-dimensional axisymmetric fluid plasma model coupled with Maxwell's equations is used to analyze the plasma kinetics of inductively coupled Ar/NF3 low-pressure plasma. We have developed a plasma chemical reaction mechanism for Ar/NF3 plasma and validated the same against the experimental data found in the literature. We analyze two sets of the electron-neutral reaction rate coefficients obtained for assumed Maxwellian and non-Maxwellian electron energy distribution functions. We find that the electron density in the discharge is sensitive to the choice of the electron energy distribution, while the fluorine atoms density is less sensitive. Since the density of atoms is controlled by a balance between production by dissociation reactions and diffusion to the walls, their density is very sensitive to the wall sticking probability.

On the calculation of superstatistics ionization rates of neutral Hélium

Electron-impact ionization (EII) can be important in dynamic systems where atoms are suddenly exposed to higher electron temperatures. EII can also have a significant effect on the charge state distribution for plasmas with a non-thermal electron energy distribution. In this paper, we present the effects of hot electrons on the calculation of ionization rates for neutral Helium using superstatistics. We also exchange the distribution function with the effective Boltzmann factor of superstatistics. The ionization rates of neutral Helium are obtained from cross sections obtained by FAC code, are weighted by a non-maxwellian distribution function. The results are compared to those reported by Kato. et. al. We note that non-maxwellian energy distribution for different fractions of hot electrons showed the effects of these rates on the fractions of hot electrons and the forms of ionization rates.

The power loss requirement in the non-thermal ³He-³He fusion plasma

The control of power loss is the most important issue in the plasma and it should be minimised by suitable selection of design parameters. This research is done with the aim of determining power loss and the parameters affecting the ³He-³He plasma with isotropic non-Maxwellian electron velocity distribution function (EVDF). In order to investigate the power loss requirement, bi-Maxwellian EVDF is considered. The results show that the rate of entropy production decreases as the electron density decreases and collision time increases. The ratio of the minimum power loss to the fusion power decreases as the electron temperature, and electron and helium-3 densities decrease, and the ion temperature increases. It is favourable to decrease the electron temperature lower than the thermal limit. Moreover, decreasing the electron temperature increases the fusion power.

Characterization of thermal transport and evolution of Au plasma in ICF experiments by Thomson scattering

This paper demonstrates the capability of optical Thomson scattering (OTS) to measure thermal transport, and provides support to radiation hydrodynamic and kinetic simulations of electron thermal transport and plasma evolution. OTS theory and plasma simulations are applied to the interpretation of experimental measurements of laser-produced plasma from spherical gold targets on the OMEGA laser facility. The dynamical form factor, S(k→,ω), of electron density fluctuations that is used in the fitting of Thomson scattering spectra includes ion–ion collisions and effects of non-Maxwellian distribution functions. OTS measurements and their interpretation are consistent with the nonlocal transport model in radiation hydrodynamic simulations as well as with kinetic simulations in the second half of the probe pulse duration. In particular, the reversal of heat transport during cooling is observed to be consistent with simulations, while some discrepancies are noted during the initial heating of the Au targets.

Macroscopic plasma analysis from 1D-radial kinetic results of a Hall thruster discharge

A radial particle-in-cell model of the weakly-collisional plasma discharge in a Hall thruster, provides the non-Maxwellian velocity distribution functions (VDF) of ions and electrons. The model considers a radial magnetic field, secondary electron emission from the two walls, and phenomenological models of anomalous electron scattering. The electron VDF is used to assess the different terms in the macroscopic momentum and energy equations, identifying those differing from the standard fluid model for a near-Maxwellian VDF. The pressure tensor consists of an anisotropic gyrotropic part and a small gyroviscous part. Nonetheless, the gradient of this last one affects the cross-field electron current density, generating radial undulations that resemble those reported for near-wall conductivity. A gyroviscous energy flux is identified too. The heat flux parallel to the magnetic lines does not follow a conductive-type law but a convective-type one, already found in other weakly-collisional plasmas. The tails of the electron velocity distribution function are partially depleted due to wall collection, leading to reduced electron fluxes of particles and energy, which are characterized with parameters useful for fluid models. Differences in the plasma response for annular and planar channel geometries are highlighted. The levels of replenishment of the electron VDF and of the asymmetries in radial profiles differ for isotropic and anisotropic anomalous scattering models.

Free Energy Sources in Current Sheets Formed in Collisionless Plasma Turbulence

Abstract Collisionless dissipation of macroscopic energy into heat is an unsolved problem of space and astrophysical plasmas, e.g., solar wind and Earth’s magnetosheath. The most viable process under consideration is the turbulent cascade of macroscopic energy to kinetic scales where collisionless plasma processes dissipate the energy. Space observations and numerical simulations show the formation of kinetic scale current sheets in turbulent plasmas. Instabilities in these current sheets (CS) can provide collisionless dissipation and influence the turbulence. Spatial gradients of physical quantities and non-Maxwellian velocity distribution functions provide the free energy sources for CS plasma instabilities. To determine the free energy sources provided by the spatial gradients of plasma density and electron/ion bulk velocities in CS formed in collisionless turbulent plasmas with an external magnetic field B 0, we carried out two-dimensional particle-in-cell-hybrid simulations and interpret the results within the limitations of the simulation model. We found that ion-scale CS in a collisionless turbulent plasma are formed primarily by electron shear flows, i.e., electron bulk velocity inside CS is much larger than ion bulk velocity while the density variations through the CS are relatively small (<10%). The electron bulk velocity and, thus, the current density inside the sheets are directed mainly parallel to B 0. The shear in the perpendicular electron and ion bulk velocities generates parallel electron and ion flow vorticities. Inside CS, parallel electron flow vorticity exceeds the parallel ion flow vorticity, changes sign around the CS centers, and peaks near the CS edges. An ion temperature anisotropy develops near CS during the CS formation. It has a positive correlation with the parallel ion and electron flow vorticities. Theoretical estimates support the simulation results.