Nonlocal electron kinetics in collisional gas discharge plasmas

Nonlocal phenomena in electron kinetics of collisional gas discharge plasmas, their kinetic treatment by a nonlocal approach, and relevant experimental results are reviewed in this paper. Using the traditional two-term approximation for the electron distribution function, a general method to analyze electron kinetics in nonuniform plasmas in DC and RF fields for atomic gases is presented for the nonlocal case, when the electron energy relaxation length exceeds the characteristic spatial scale of bounded plasmas. The nonlocal method, which is based on the great difference between the electron mean free path for the momentum transfer and the electron energy relaxation length, considerably simplifies the solution of the kinetic equation and, in a number of cases, allows one to obtain analytical and semi-analytical solutions. The main simplification is achieved for trapped electrons by averaging the Boltzmann equation over space and fast electron motion. Numerous examples of spatial nonlocality are considered in the positive column and near the electrodes of DC glow discharges, in spatial relaxation of the electron distribution and in striations, and in capacitively and inductively coupled low-pressure RF discharges. The modeling of fast beam-like electrons is based on a continuous-energy-loss approximation with the assumption of forward scattering. Simple analytic expressions for the fast electron spectrum are obtained in cathode regions of DC discharges with planar and hollow cathodes     

Substantiation of the two-temperature kinetic model by comparing calculations within the kinetic and fluid models of the positive column plasma of a dc oxygen discharge

Results from kinetic and fluid simulations of the positive column plasma of a dc oxygen discharge are compared using commercial CFDRC software (), which enables one to perform numerical simulations in an arbitrary 3D geometry with the use of both the fluid equations for all the components (fluid model) and the kinetic equation for the electron energy distribution function (kinetic model). It is shown that, for both the local and nonlocal regimes of the formation of the electron energy distribution function (EEDF), the non-Maxwellian EEDF can satisfactorily be approximated by two groups of electrons. This allows one to take into account kinetic effects within the conventional fluid model in the simplest way by using the proposed two-temperature approximation of the nonequilibrium and nonlocal EEDF (2T fluid model).     

Self-consistent hybrid model for a stratified positive column of a low pressure glow discharge

The stratification of a positive column of a low-pressure glow discharge in inert gases has been studied with the help of a self-consistent hybrid model. The model is based on the solution of a nonlocal kinetic Boltzmann equation for the electron distribution function, a nonstationary drift-diffusion equation for the ions, and the Poisson equation for the electric field. Spatial electron and ion density distributions and the electric field distribution in the positive column were obtained. The converged solution of the model gives a self-consistent resonant strata length L and the value and the form of the modulated plasma parameters. An unexpected surprising result was obtained: for a given potential drop in the positive column of a low-pressure glow discharge, a self-consistent spatially modulated striation-like electric field does not lead to the resonant increasing of the ionization frequencies in the discharge as compared with a constant electric field with the same potential drop. Usually, it was assumed that, in spatially modulated field distributions, all the parameters in a striated plasma will be more pronounced and have a resonant form. The text was submitted by the authors in English.     

Self-consistent mechanism for sustaining ionization waves in a low-pressure discharge

The possibility of constructing a self-consistent model for the sustaining of ionization waves is demonstrated for a low-pressure discharge in an inert gas. The model is based on the combined solution of the kinetic equation of the electrons and the equation of motion of the ions in a spatially periodic field. The distribution function is constructed in an experimentally measured field and then used to calculate the spatial distributions of the plasma density and the ionization rate. The solution of the equation of motion of the ions makes it possible to reconstruct a field similar to the original one. One specific feature of the mechanism considered is the pronounced nonlocal character of the formation of the electron distribution function by the entire nonuniform potential profile of the ionization wave. Zh. Tekh. Fiz. 67, 24–30 (February 1997)     

Viscous effects on motion and heating of electrons in inductivelycoupled plasma reactors

A transport model is developed for nonlocal effects on motion and heating of electrons in inductively coupled plasma reactors. The model is based on the electron momentum equation derived from the Boltzmann equation, retaining anisotropic stress components which in fact are viscous stresses. The resulting model consists of transport equations for the magnitude of electron velocity oscillation and terms representing energy dissipation due to viscous stresses in the electron energy equation. In this model, electrical current is obtained in a nonlocal manner due to viscous effects, instead of Ohm's law or the electron momentum equation without viscous effects, while nonlocal heating of electrons is represented by the viscous dissipation. Computational results obtained by two-dimensional numerical simulations show that nonlocal determination of electrical current indeed is important, and viscous dissipation becomes an important electron heating mechanism at low pressures. It is suspected that viscous dissipation in inductively coupled plasma reactors in fact represents stochastic heating of electrons, and this possibility is exploited by discussing physical similarities between stochastic heating and energy dissipation due to the stress tensor     

Self-consistent multidimensional electron kinetic analysis forinductively coupled plasma sources

Based upon the kinetic equations coupled with electromagnetic analysis for the recently developed inductively coupled plasma sources (ICPS), a self-consistent electron kinetic model is presented for 2-D (r, z) in a cylindrically symmetric configuration space and 2-D (ν _∂, ν_z) in the velocity space, The EM model is based on the mode analysis, while the kinetic analysis gives the perturbed Maxwellian distribution of electrons by solving the Boltzmann-Vlasov equation. The kinetic analysis shows that the RF energy in an ICPS is extracted by a collisionless dissipation mechanism, once the electron thermovelocity is close to the RF phase velocities determined by the reactor height and mode indexes. In this context, the effect of varying the reactor geometry is reported in terms of the electron energy distribution function. The analytical results are compared to the experimental data of Barnes et al. (see Appl. Phys. Lett., vol.62, no.21, p.2622-4 (1993)), which shows qualitative agreements in many aspects     

Low-pressure RF discharge in the free-flight regime

The self-consistent equations system for low-pressure RF discharge in the free-flight regime is formulated. The expressions for the electron energy diffusion coefficient due to electron-neutral collisions and to the electron collisions with the plasma-space charge moving boundary (stochastic heating) are derived. If the electron-neutral elastic collisions frequency exceeds the inelastic one, the conventional two-term approximation for the electron distribution function (EDF) can be generalized, and the space-time-averaged electron kinetic equation can be reduced to the one-dimensional energy diffusion one. The fast electrons attached to the electrode surface can also be accounted for in this equation. It is shown that in the cases of (a) spatially uniform ion profile, (b) for frequencies that are small compared with the electron bounce frequency, and (c) for frequencies exceeding the electron plasma one in the sheath, the stochastic heating vanishes     

Diffusion-path approximation in nonlocal electron kinetics

For the positive column of a discharge, nonlocal distribution functions obtained by averaging over radial diffusion paths are compared with the exact solution to the kinetic elliptic equation. For a discharge in argon, as an example, the limits of applicability of the approximate solution for various macroscopic characteristics of the plasma were identified. It was previously believed that the approximation based on averaging has the limits of applicability determined by the condition that the plasma inhomogeneity size be smaller than the energy relaxation length. This condition restricts the applicability of the approximation to low pressures. In the present work, it is shown that, for determining a number of macroscopic parameters, such as the concentration, mean energy, mobility, diffusion coefficient, and thermal conductivity of electrons, the pathaveraging approximation works well over a pressure range of up to a few Torr. A number of subtle characteristics, such as the excitation rate, ionization rate, and others, largely influenced by fast electrons, cannot be calculated from the averaged distribution functions at pressures above a few tenths of a Torr.     

Kinetic Theory of a Weakly‐Nonideal Spatially Nonuniform Polarizable Plasma

A nonmarkovian and nonlocal kinetic equation, which is a generalization of the (nonlinear) Balescu‐Lenard equation, is derived for a weakly nonuniform multicomponent polarizable plasma. A specific expansion of the plasma resolvent, which allow to separate the dymamic, kinetic and hydrodynamic scales, is proposed. Explicit expressions of the collision integral and of the nonequilibrium pair correlation function are given in an approximation which take into account the effect of patial and temporal nonlocality a well as of the polarization. Balance equation for the momentum and energy densitie are calculated in the first order in nonlocality, and potential contributions to the fluxes due to polarization are obtained.     

Self-consistent modelling of X-ray preionized XeCl-laser discharges

In this theoretical work a 0-D model for a self-sustained X-ray preionized XeCl-laser discharge is presented. The model is self-consistent in the sense that it simultaneously solves, contrarily to the usual decoupling procedure, the Boltzmann equation for electrons, the kinetic equations for excited and ionic species, the equations for the electrical circuit and the laser photon density. It includes a rather complete kinetics of HCl(v) vibrational excitation, dissociation and dissociative attachment. The influence of electron collisions with excited species and of e-e Coulomb collisions on the plasma parameters and transport coefficients is discussed. Some evidence of the non-stationary equilibrium between the electron distribution and the reduced electric field E/N is given. Results of the model are compared with experimental ones corresponding to a XeCl-laser discharge driven by a L-C inversion circuit. The model predicts well the main trends for the variation of the laser energy in a large range of experimental conditions. The discrepancy between experiment and model for absolute values of the laser energy is discussed.     

A phase-space particles motion scheme for electron kineticsimulation

A phase-space particle motion scheme (PSPMS) for electron kinetic simulation is proposed. This scheme is based on the convenient representation of electron motion in phase space using the particles method. PSPMS offers the possibility of calculating the electron energy distribution function (EEDF) much faster than by conventional methods. PSPMS electron calculation is shown to match convective scheme and Monte Carlo simulations of swarms in the uniform electric field and the cathode fall (CF). PSPMS may be used for a self-consistent model of low-pressure glow discharge construction     

Numerical Study on Characteristics of Argon Radio-Frequency Glow Discharge with Varying gas Pressure

A one-dimensional fluid simulation on argon rf glow discharge with varying linearly gas pressure from 1 Torr to 100 Tort is performed. The model based on mass conservation equations for electron and ion under diffusion and mobility approximation, and the electron energy conservation equation is solved numerically by finite volume method. The numerical results show that a uniform plasma with high density can be obtained from rf glow discharge with varying gas pressure, and the density of plasma becomes higher as the gas pressure varies from 1 Tort to 100 Tort. It is also shown that in the range of the gas pressure from 1 Tort to 100 Tort with the slower rate of varying gas pressure, higher density of plasma can be obtained.     

Radial Distributions of Dusty Plasma Parameters in a DC Glow Discharge

A non‐stationary non‐local kinetic model for radial distributions of dusty plasma parameters based on the solution of Boltzmann equation for electron energy distribution function is presented. Electrons and ions production in ionizing collisions and their recombination on dust particle surface were taken into account. The drift‐diffusion approximation for ions was used. To obtain the self‐consistent radial distribution of electric potential the Poisson equation was used. It is shown that at high dust particle density the recombination of electrons and ions can exceed their production in ionization collisions in the region of dusty cloud. In this case the non‐monotonous radial distribution of the electric field is formed, the radial electric field becomes reversed and the radial electron and ion fluxes change their direction toward the center of the tube (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

RF electric field penetration and power deposition into nonequilibrium planar-type inductively coupled plasmas

The RF electric field penetration and the power deposition into planar-type inductively coupled plasmas in low-pressure discharges have been studied by means of a self-consistent model which consists of Maxwell equations combined with the kinetic equation of electrons. The Maxwell equations are solved based on the expansion of the Fourier--Bessel series for determining the RF electric field. Numerical results show the influence of a non-Maxwellian electron energy distribution on the RF electric field penetration and the power deposition for different coil currents. Moreover, the two-dimensional spatial profiles of RF electric field and power density are also shown for different numbers of RF coil turns.     

DC column plasma kinetics in a longitudinal magnetic field

The cylindrical column plasma of a neon dc glow discharge under the influence of a weak longitudinal magnetic field is studied. An extended, fully self-consistent model of the column plasma has been used to determine the kinetic quantities of electrons, ions and excited atoms, the radial space charge field, and the axial electric field for given discharge conditions. The model includes a nonlocal kinetic treatment of the electrons by solving their spatially inhomogeneous kinetic equation, taking into account the radial space charge field and the axial magnetic field. The treatment is based on the two-term expansion of the velocity distribution and comprises the determination of its isotropic and anisotropic components in the axial, radial, and azimuthal direction. A transition from a distinctly nonlocal kinetic behavior of the electrons in the magnetic-field-free case to an almost local kinetic behavior has been found by increasing the magnetic field. The establishment of the electron cyclotron motion around the column axis increasingly restricts the radial electron energy transport and reduces the radial ambipolar current. The complex interaction of these transport phenomena with the alterations in the charge carrier production leads finally to a specific variation of the electric field components. The axial field increases by applying weak magnetic fields, however, decreases with increasingly higher magnetic fields. At higher magnetic fields, the radial space-charge field is considerably reduced     

On the self-energy of electrons in metals

The electron self-energy of unoccupied states is investigated taking into account dynamical screening and nonlocal exchange. To obtain agreement with experiment it is crucial to go beyond the framework of the homogeneous electron gas and include the nonlocal exchange with electrons in valence and core shells. This contribution of atomic origin gives rise to a considerable, almost linear energy dependence of the self-energy over a wide energy range in agreement with experimental findings for many substances and in disagreement with the local density approximation. Quantitative results are presented for Ag.     

Dissociative attachment of low-energy electrons to vibrationally excited hydrogen molecules

Dissociative electron attachment to hot hydrogen molecules is studied in the framework of nonlocal resonance model. The method based on the use of the Bateman approximation, well known in nuclear physics, is adapted for solving the Lippmann-Schwinger integral equation of the nonlocal resonance model and applied to the calculation of cross sections of inelastic resonant electron-molecule collisions. The proposed method is compared with the Schwinger-Lanczos algorithm used extensively for the treatment of these processes. It is shown that the Bateman approximation is very useful and efficient for treating the non-separable nonlocal potentials appearing in the integral kernels of the nonlocal resonance models. The calculated cross sections for the dissociative attachment of electrons to vibrationally excited hydrogen molecules are of importance for astrophysics. This paper is dedicated to Prof. J. Bičák on the occasion of his 60th birthday.