Nonstationary and Quasi-stationary Treatment of Distribution Anisotropy in the Temporal Relaxation of an Energetic Electron Group

A relaxation study of an electron group in collision dominated weakly ionized plasmas has been performed. The study is based on the two-term approximation of the Legendre polynomial expansion of the electron velocity distribution in the nonstationary Boltzmann equation. To overcome the limitation of the conventional quasi-stationary treatment of the distribution anisotropy, a very efficient solution approach of the nonstationary kinetic equation in two-term approximation has been developed which allows for a strict nonstationary treatment of the distribution anisotropy. By using this approach the temporal evolution of the isotropic and anisotropic distribution of the electrons has been investigated for a model plasma, which involves typical features of an inert gas plasma. A comparison of the results with corresponding ones obtained by applying the conventional approach under various parameter conditions clearly indicates a pronounced falsification of the real relaxation course by the latter approach.     

Temporal relaxation of plasma electrons acted upon by directcurrent electric and magnetic fields

On the basis of the time-dependent electron Boltzmann equation the temporal relaxation of the electrons in the presence of electric and magnetic fields in weakly ionized, collision dominated plasmas has been studied. The relaxation process is treated by using a strict time-dependent two-term approximation of the velocity distribution function expansion in spherical harmonics. A new technique for solving the time-dependent electron kinetic equation in this two-term approximation for arbitrary angles between the electric and magnetic fields has been developed and the main aspects of the efficient solution method are presented. Using this new approach and starting from steady-state plasmas under the action of time-independent electric fields only, the impact of superimposed DC magnetic fields on the electron relaxation is analyzed with regard to the control of a neon plasma. The investigations reveal an important effect of the magnetic field on the temporal relaxation process. In particular, it has been found that the relaxation time of the electron component with respect to the establishment of steady-state can be enlarged by some orders of magnitude when increasing the magnetic field strength     

The Electron Relaxation to Stationary States in Collision Dominated Plasmas in Molecular Gases

The temporal collision dominated relaxation of electrons to new stationary states, starting from initial stationary states and due to jump-like changes of the electric field, was studied in the plasmas of the molecular gases N₂ and CO. Numerical solving of the time dependent Boltzmann equation for the electrons yields the temporal evolution of their energy distribution function and of resulting macroscopic quantities. The varying relaxation due to different values of the field strength in the final stationary state has been investigated considering the molecules of the plasma only as vibrationally non-excited and, in another case, including the additional impact of collisions with vibrationally excited molecules. The results obtained are discussed and, in particular, the relaxation times found for the transitions to the new stationary states are analysed on the basis of the energy transfer effectiveness by the collision processes. An approximative microphysical basis for the understanding of the main features of the relaxation in such complex molecular gas plasmas could be obtained.     

Comparison of monte carlo and boltzmann two-term calculations of time-dependent velocity distribution functions of electrons in an electric field in a gas

Summary An extended comparison has been made between Boltzmann two-term and rigorous Monte Carlo calculations of time-dependent velocity distribution functions of electrons in a d.c. electric field in a gas to assess the limits of the conventional approximations. It is shown that under various conditions the two-term approximation is unable to represent the velocity distribution correctly even if the conventional quasi-stationary approximation of the velocity distribution anisotropy, usually adopted in this kind of calculations, is abandoned. The conditions which permit to use the two-term theory are discussed.     

Time - Dependent Multi - Term Treatment of Plasma Electrons Acted upon by RF Electric Fields

Using a new method for solving the time-dependent electron Boltzmann equation in higher order accuracy, studies of the temporal behaviour of electrons in weakly ionized, collision-dominated plasmas under the action of rf fields have been performed. The method is based upon a multi-term approximation of the Legendre polynomial expansion of electron velocity distribution function and is applied to investigate the established periodic behaviour of the electron velocity distribution in helium, argon and CO plasmas. The results obtained in various approximation orders are discussed. The analysis has shown that the simplified treatment using only two terms of the velocity distribution expansion can fail in several conditions. In general, the four-term approximation gives already a good representation of the convergent solution of the electron Boltzmann equation at each instant of the rf period. The discrepancies between two-term and convergent results are found to depend sensitively on the specific atomic data, in particular on the magnitude of the various electron collision cross sections involved. Furthermore, the results obtained in the multi-term approximation are compared with corresponding ones obtained by accurate Monte Carlo simulations. Very good agreement between convergent eight-term Boltzmann and Monte Carlo calculations is found.     

Field Free Relaxation of the Electron Component in the Temporally Decaying Molecular Gas Plasmas Hydrogen and Nitrogen

This paper deals with the temporal relaxation of the electron component of weakly ionized, anisothermal and collision dominated plasma in the molecular gases hydrogen and nitrogen after jump-like switching off of the electric field starting from stationary states. The investigation is based upon a solution of the non-stationary Boltzmann equation using a finite difference approach of the resulting partial differential equation. Besides the temporal development of the energy distribution and of some macroscopic quantities of the electrons especially the characteristic relaxation times and their physical nature are discussed and compared with former results on the relaxation in an inert gas plasma.     

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     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials. 2. Comprehensive Investigation of a Model Plasma

Applying the new technique for finding the converged solution of the Boltzmann equation in a weakly ionized plasma, which was developed in the first part of this paper, a comprehensive study of the electron velocity distribution function for a model plasma with elastic and exciting collisions is performed by solving the Boltzmann equation with increasing order of approximation. The purpose of this investigation is that of calculating the isotropic distribution f₀, the first contribution f₁ to the anisotropy of the velocity distribution, the important macroscopic quantities and, more generally, that of studying the total anisotropy as well as the changes of all these quantities when the approximation degree is enlarged beyond the 2 terms of the conventional Lorentz approximation. By varying some parameters of the model plasma, that is the electric field strength, the magnitude of the excitation cross section and the excitation threshold, the main features of plasmas in inert as well as molecular gases are modelled and the impact of these parameters on the mentioned quantities is analysed. Some of the converged results are compared with results of corresponding Monte Carlo simulations. The approximation degree required to find the converged values of isotropic distribution, main macroscopic quantities and electron distribution in the velocity space (and thus its real anisotropy) is estimated by solving the Boltzmann equation over wide parameter ranges.     

Stoßbestimmte Relaxation des Elektronenensembles im Plasma bei zusätzlicher Aufheizung durch ein elektrisches Feld I. Die charakteristischen Zeiten für den Übergang in stationäre Zustände

Collision Dominated Relaxation of the Electron Ensemble in a Plasma with Additional Heating by an Electric Field. I. Characteristic Times for the Transition to Stationary States Starting from time dependent Boltzmann equation for electrons the time development of the isotropic part of the distribution function and of macroscopic quantities as mean energy, mobility, excitation frequency and energy transfer quotients during transition between two stationary states are determined. The computation is referred to weak ionized neon plasmas which are typical for low pressure and for medium pressure discharges. As a result of this investigations we get informations about the characteristic relaxation times which are different in order of magnitude, and about their dependence of the processes of energy transfer. The energy transfer quotients which determine the energy loss by different collision processes in consideration are found to be suitable quantities to characterize the relaxation times.     

Response of the Electron Kinetics on Spatial Disturbances of the Electric Field in Nonisothermal Plasmas

An efficient method for solving the inhomogeneous electron Boltzmann equation for a weakly ionized collision dominated plasma is represented. As a first application this method is used to investigate in a helium plasma the response of the electron velocity distribution function and of the relevant macroscopic quantities to the impact of spatially limited disturbances in the electric field. In addition to the field action elastic and (conservative) inelastic collisions of electrons with gas atoms are taken into account in the kinetic treatment. In this way the spatial relaxation behaviour of the electrons and their establishment into homogeneous plasma states could be studied on a strict kinetic basis. Unexpectedly large relaxation lengths in electron acceleration direction have been found at medium electric fields.     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials 1. Theory

On the basis of our recent investigations concerning the mathematical structure of the hierarchy which results from the Legendre polynomial expansion of the electron velocity distribution function in Boltzmann's equation a new technique for solving this equation in multi-term even-order approximation is presented. This method is, even if more complex, the logical generalization of the well known technique for solving Boltzmann's equation by backward integration in the conventional two-term approximation. A weakly ionized, spatially homogeneous and stationary plasma with elastic and exciting electron-atom collisions is considered acted upon by a dc electric field. The technique, presented in detail, determines the distribution function in even order 2l of the expansion at the end by l-fold backward and 2l-fold forward integration of the hierarchy and by continuous connection of the resulting non-singular parts of the general solutions at low and high energies at an appropriate connection point. A first application of this method is made on a model gas for the even orders from 2 to 10 and under conditions with distinct anisotropy in the velocity space due to intensive exciting collisions. The converged macroscopic quantities and the corresponding first coefficients of the distribution expansion itself are compared with very accurate Monte Carlo simulations under the same conditions where a perfect agreement between the results obtained with both techniques was found confirming the high accuracy of the new technique to be presented.     

Kinetics of the Electron Component in the Turbulent Electron-Beam Discharge Plasma in Diatomic Gases and Comparison of some Macroscopic Properties between the Plasmas of the Beam and the Glow Discharge

The present paper is devoted to the investigation of the kinetics of the electron component in the stationary beam discharge plasma in molecular nitrogen. Using the Boltzmann equation with the inclusion of the elastic and the main inelastic binary collisions and also the Coulomb interaction between the charged particles we have calculated the energy distribution function and some macroscopic quantities of the electrons within a large range of parameters. Using our earlier results for the beam discharge plasma in hydrogen, also the dependence of the macroscopic quantities on the kind of molecular gas is discussed. Finally, the comparison of some macroscopic properties of the beam and the glow discharge plasma was performed under the condition of equal power input per volume unit in both types of plasmas in nitrogen.     

On the Relaxation of Energetic Electrons in Plasmas

The impact of individual collision processes on the relaxation of the velocity distribution function of a group of electrons, initially localized in a narrow region at relatively high energies, has been studied. By having recourse to solutions of the non-stationary Boltzmann equation and to corresponding Monte-Carlo simulations, the temporal behaviour of electrons in CO₂ plasmas, both in the absence and the presence of an external dc field, has been investigated. A microphysical interpretation of observed relaxation phenomena, based on the data relevant to the individual collision processes, is also given.     

ISSDE: A Monte Carlo implicit simulation code based on Stratonovich SDE approach of Coulomb collision

A Monte Carlo implicit simulation program, Implicit Stratonovich Stochastic Differential Equations(ISSDE), is developed for solving stochastic differential equations(SDEs) that describe plasmas with Coulomb collision. The basic idea of the program is the stochastic equivalence between the Fokker–Planck equation and the Stratonovich SDEs. The splitting method is used to increase the numerical stability of the algorithm for dynamics of charged particles with Coulomb collision. The cases of Lorentzian plasma, Maxwellian plasma and arbitrary distribution function of background plasma have been considered. The adoption of the implicit midpoint method guarantees exactly the energy conservation for the diffusion term and thus improves the numerical stability compared with conventional Runge–Kutta methods. ISSDE is built with C++ and has standard interfaces and extensible modules. The slowing down processes of electron beams in unmagnetized plasma and relaxation process in magnetized plasma are studied using the ISSDE, which shows its correctness and reliability.     

A deterministic solver for the transport of the AlGaN/GaN 2D electron gas including hot-phonon and degeneracy effects

The transport of the two-dimensional electron gas formed at an AlGaN/GaN heterostructure in the presence of strain polarization fields is investigated. For this purpose, we develop a deterministic multigroup model to the Boltzmann transport equations. The envelope wave functions for the confined electrons are calculated using a self-consistent Poisson–Schrödinger solver. The electron gas degeneracy and hot phonons are included in our transport equations. Numerical results are given for the dependence of macroscopic quantities on the electric field strength and on time and for the electron and phonon distribution functions. We compare our results to those of Monte Carlo simulations and with experiments.     

Strict Kinetic Treatment of the Electron Response to Spatial Field Disturbances in Nonequilibrium Plasmas

In generalization of former approaches for the simplified solution of the inhomogeneous electron Boltzmann equation a higher order solution technique has been developed. This technique is based on a multi-term expansion of the electron velocity distribution function and allows a strict study of the electron kinetics in plasmas acted upon by space-dependent electric fields. This solution technique is used to investigate the response of the plasma electrons to spatially limited disturbances of the electric field in weakly ionized plasmas of helium and mercury. By solving the kinetic equation with increasing order of the multi-term expansion the convergent solution of the kinetic problem and thus the strict spatial behaviour of the velocity distribution and of significant macroscopic properties of the electrons has been determined and analysed. Furthermore, the impact of higher order terms of the expansion has been revealed and the falsification of the velocity distribution and of related macroscopic properties has been evaluated when instead of the multi-term solution the simpler two-term solution of the kinetic equation is used.