Electron Kinetics in Nonisothermal Plasmas with Spatially Two-Dimensional Structures

A method for the study of the steady-state nonlocal electron kinetics in radially and axially inhomogeneous cylindrical plasmas is presented. The method is based on the solution of the relevant space-dependent kinetic equation for the electron velocity distribution using the two-term expansion. A three-dimensional initial boundary value problem for the isotropic distribution component with the total energy as the evolution variable of the kinetic problem has to be treated. The technique is applied to analyze the electron kinetics in a cylindrical DC column plasma under impact of a space-independent axial field and a radially increasing radial field. Particularly, the spatial relaxation of the electron gas in axial direction in response to a disturbance of the axial homogeneity is investigated. A detailed analysis of the spatially resolved energy balance is given. Considerable modification of the results with respect to those obtained by earlier studies of axially homogeneous DC column plasmas, as well as of relaxation processes in one spatial dimension, has been found.     

On the Nonlocal Electron Kinetics in s- and p-Striations of DC Glow Discharge Plasmas: I. Electron Establishment in Striation-like Fields

The nonlocal behavior of the electrons in strongly modulated and period-averaged electric fields typical of s- and p-striations in neon glow discharge plasmas is investigated by numerically solving the axially inhomogeneous electron Boltzmann equation. A good agreement between the period lengths measured in the striations and those obtained from the spatially periodic electron relaxation in the period-averaged field of the striations is found confirming the close relation of both phenomena. The s- and p-striations represent the fundamental and first harmonics of the inherent periodic electron relaxation. Furthermore, starting from different boundary conditions the establishment of the velocity distribution function and of selected macroscopic quantities of the electrons into unique periodic states under the action of strongly modulated striation-like fields is investigated. It is shown that the same damping processes that cause in homogeneous fields a relaxation into homogeneous states lead to unique periodic states in strongly modulated fields.     

Two-Term and multi-term approximation of the nonstationary electron velocity distribution in an electric field in a gas

A very efficient, strict nonstationary multi-term approach has been developed as a generalization of the conventional and the strict nonstationary two-term approximations used for solving the nonstationary , electron Boltzmann equation. As a first application the temporal relaxation of electrons in a model plasma acted upon by a de electric field has been investigated. The results are compared with corresponding ones obtained by the conventional and the strict nonstationary two-term approach as well as with very accurate Monte Carlo simulations. Perfect agreement between nonstationary eight-term Boltzmann and Monte Carlo calculations is found.     

A New Method of the Microwave Diagnostics for Determining the Electron Density

A improved method of the microwave diagnostics for the determination of the electron density in the column plasma of dc glow discharges is presented. This method overcomes significant limitations involved in the conventional method. The latter is mainly based on the measurement of the resonance frequency shift of the resonator cavity caused by the presence of the dc column plasma and is strictly valid only in the so-called high field frequency limit of the resonator operation. Of significant importance with respect to the improved method is the additional inclusion of the measurable phase delay between the electric hf field, superimposed on the dc plasma by the microwave field of the resonator, and the induced hf electron current. This new method has been critically evaluated with respect to its extended applicability and has been successfully applied to the experimental determination of the electron density in dc plasmas of neon and molecular nitrogen.     

Spatial Electron Relaxation: Comparison of Monte Carlo and Boltzmann Equation Results

In former investigations on the spatial relaxation of the electron component of weakly ionized plasmas a considerable spectrum of distinct spatial structures has been found in the velocity distribution function as well as in various macroscopic quantities of the electrons. These results have been mainly obtained by solving the spatially inhomogeneous electron Boltzmann equation considering the action of uniform electric fields and the impact of elastic and inelastic collisions of the electrons with the gas atoms. To verify these partly unexpected results on the complex structure formation, analogous Monte Carlo simulations were performed now for a helium plasma at various reduced electric field strengths. Detailed comparisons were made between the results of the two independent kinetic approaches with respect to the spatial evolution of the velocity distribution function as well as of associated macroscopic quantities. Good agreement was generally found, thus confirming the earlier results on the complex spatial relaxation structures.     

On the Nonlocal Electron Kinetics in s and p Striations of DC Glow Discharge Plasmas: II. Electron Properties in Periodic States

The spatial relaxation behavior of the electron gas acted upon by stronglymodulated striation-like electric fields, typically occurring in s and pstriations of dc neon plasmas, has been investigated on a kinetic basis inpart I of this paper. These studies have clearly shown that the electronrelaxation in the striationlike field leads to the establishment of anunambiguous spatially periodic state, which is characterized by largemodulation depths and phase delays of the microscopic and macroscopicquantities with respect to the striationlike field. The present part givesa detailed analysis of these quantities in the established striationlikestates. These investigations clearly reveal the distinctly nonlocal electronproperties, which are especially pronounced in the s-type of striations. Inaddition, the kinetic studies are completed by analyzing the impact of thefield modulation degree and the sensitive destruction of the resonancebehavior of the electron gas when imposing slightly nonresonant electric fields.     

Electron kinetics of weakly ionized collision-dominated RF plasmas in CO

Based on the nonstationary electron Boltzmann equation this paper deals with the time-resolved electron kinetics in the rf plasma in CO, i.e., with the calculation of the temporal evolution of the energy distribution and of the resultant macroscopic quantities for the established steady state. A particular aspect of this plasma is the distinctly resonance-like behavior of the vibrational excitation of the CO molecules by electron collisions. This causes the lumped frequencies for energy and impulse dissipation in collisions, recently introduced in the study of the rf kinetics in Ne and H₂, to become extremely dependent on the electron energy. Despite this fact, it could be verified that the field frequency dependence of the temporal evolution of the electron kinetics can be interpreted by means of these two dissipation frequencies even under such complicated conditions as given by the atomic data in CO.     

On the mechanisms of spatial electron relaxation in nonisothermal plasmas

The impact of different collision processes of the electrons with the gas atoms on their spatial relaxation under the action of space-independent electric fields is analyzed in a weakly ionized, collision-dominated helium plasma. Based on the numerical solution of the one-dimensional inhomogeneous electron Boltzmann equation, the spatial evolution of the electron velocity distribution function and of the related macroscopic quantities is investigated for different models concerning the treatment of the elastic, exciting, and ionizing collision processes. The spatial relaxation into homogeneous states is initiated by a disturbance which is directly imposed on the electron velocity distribution as a boundary condition. At moderate and higher electric fields this disturbance excites spatially periodic structures in the distribution function which are damped out by special mechanisms inherent in the exciting and ionizing collision processes. With decreasing field strength the damping due to the energy loss in elastic collisions becomes more effective and causes at lower fields an aperiodic establishment into homogeneous states.     

Spatial relaxation of electrons in nonisothermal plasmas

The spatial relaxation of electrons to homogeneous states under the action of space-independent electric fields is investigated in helium, krypton, and N₂ plasmas for various electric field strengths. These investigations are based on a new method recently developed for solving the one-dimensional inhomogeneous electron Boltzmann equation in weakly ionized, collision-dominated plasmas. Elastic as well as conservative inelastic collisions of electrons with gas atoms have been included in the kinetic treatment. The spatial relaxation is caused by an imposed direct disturbance in the velocity distribution of the electrons on a spatial boundary. A pronounced dependence of the relaxation structure and the resultant relaxation length on the atomic data of the electron collision processes in different gases has been found. Furthermore the relaxation process sensitively depends on the electric field strength in the region of medium field values.     

New developments in theory of fast electron scattering in solids: Applications to microbeam analysis

The study of fast electron interaction with solids in the energy range from 100 eV to several tens of keV is prompted by quickly developing microbeam analysis techniques such as electron probe microanalysis, scanning electron microscopy, electron energy loss spectroscopy and so on. It turned out that for random solids the electron transport problem might be solved on the basis of the generalized radiative field similarity principle. The latter states that the exact differential elastic cross section in the kinetic equation may be replaced by an approximate one provided the conditions of radiative field similarity are fulfilled. Application of the generalized similarity principle to electron scattering in solids has revealed many interesting features of electron transport. Easy to use and effective formulae have been obtained for the angular and energy distribution of electrons leaving a target, total yields of characteristic photons and slow electrons escaping from a sample bombarded by fast primaries, escape probability of Auger electrons as a function of depth etc. The analytical results have been compared with Monte Carlo calculations and experiments in a broad range of electron energies and scattering properties of solids and good agreement has been observed.     

Microwave Diagnostics in Diffusive and Constricted Medium-Pressure Discharges

The high-frequency (HF) electron current induced in a dc discharge plasma bysuperimposing a HF electric field presents a useful tool for the diagnosticsof the time-dependent electron behavior of the plasma. This response to theHF field has been recently studied in diffusive discharge plasmas at lowergas pressures and discharge currents. These studies are extended tomedium-pressure plasmas operating in the diffusive as well as in theconstricted mode. In particular, the impact of the electron–electroninteraction on the phase delay between the HF field and electron current inconstricted column plasmas has been experimentally and theoreticallyanalyzed. Furthermore, the problem has been studied if, under the conditionsof pronounced electron–electron interaction, the determination of theelectron density will further on be possible by using the phase delay. Themeasurements of the delay have been performed by means of the microwaveresonator method in a medium-pressure krypton glow discharge operating inthe diffusive as well as the strongly constricted mode. In addition, thedelay has been theoretically determined by treating the appropriatetime-dependent electron kinetic equation at high frequencies of thesuperimposed microwave field.     

Spatially Dependent Kinetics of Electrons and Excited Atoms in the Column Plasma of a He–Xe dc Discharge

The radially varying kinetics of electrons and excited atoms in the cylindrical axially homogeneous positive column of a dc glow discharge in a gas mixture of helium and 2% xenon was studied. The experimental investigations comprise the radially resolved measurements of the isotropic part of the electron velocity distribution function (EVDF) using a single-probe technique and of the densities of atoms in the lower excited states by using a laser diode absorption method. The theoretical investigations are based on the solution of the space-dependent kinetic equation for the EVDF and the balance equations of excited gas atoms. Besides a strict solution, various simplified treatments of the electron kinetics as the conventional homogeneous approach and the nonlocal approach have been applied. The electron kinetic behavior in the helium–xenon column plasma changes remarkably with increasing helium gas pressure from a distinctly nonlocal behavior at a low pressure of 100 Pa to a nearly local behavior at a medium pressure of 600 Pa.     

Comparison of reaction rates for helium plasmas calculated using nonequilibrium and equilibrium electron energy distribution functions

The electron energy distribution functions for helium plasmas have been calculated using the Boltzmann equation. Three characteristic temperatures of these distribution functions have been determined (from mean energy, the Einstein formula, and the logarithmic slope). The reaction rates for nonequilibrium and three equilibrium (corresponding to these three characteristic temperatures) distribution functions have been calculated and compared We have found that the use of equilibrium values for reaction rates of processes going from the ground state can lead to great errors in results, the use of equilibrium values for processes going from higher levels is possible for higher reduced electric fields, and there is no problem with using equilibrium values. for superelastic processes.     

Model of Confined Atoms in Arbitrary Static Electric Fields: Relevance to Non-degenerate Plasmas

Abstract

The inhomogeneous electron cloud in atomic ions ‘confined’ in hot plasmas and subjected to high static electric fields is studied, because of a body of experimental data on multiphoton ionization. In particular, the canonical (Bloch) density matrix is obtained in closed form for independent electrons moving in a static electric field of arbitrary strength and confined by a harmonic oscillator potential. To bring the model into contact with atoms in plasmas, the oscillator force constant is connected with the plasma density. For non-degenerate electrons an ‘atomic’ potential is included, by means of the Thomas—Fermi (TF) method. In an Appendix, a fully non-local theory is then developed which transcends this TF approximation. Simple numerical examples are presented for realistic values of field, temperature and plasma density.     

Attachment and ionization in a self-consistent treatment of the electron kinetics of a collision-dominated rf plasma

This paper deals with the self-consistent determination of the rf field amplitude for sustaining the steady-state collision-dominated weakley ionized plasmas in the bulk of the rf discharge and of the time-resolved behavior of the isotropic part of the distribution function as well as of relevant macroscopic quantities in plasmas whose particle loss is dominantly determined by electron attachment. The strict timeresolved treatment is based on the nonstationary Boltzmann equation of the electrons and its numerical solution including, apart from electron number conservative collision processes, the electron attachment and ionization. The investigations are related to an rf plasma in a model gas and in SF₆ and are performed for reduced rf field frequencies around 10 MHz Torr^–1 which are of particular interest from the point of application of rf discharges for plasma processing. The numerical results show that a large field amplitude of around 160 V cm^–1 Torr^–1 is necessary to maintain the discharge and that the isotropic distribution, the relevant collision frequencies for attachment and ionization, and the electron density undergo a large modulation during a period of the rf field.     

Kinetics of the Electrons in Striations of Spherical Glow Discharges

Recent observations of spherical striations in large-volume nitrogen dcdischarges with a central anode have stimulated investigations of thenonlocal electron kinetics in these striations by solving the spatiallyinhomogeneous Boltzmann equation adapted to spherical geometry. Becausethe radial course of the electric potential is largely unknown in thisdischarge, different models concerning its radial course have been developedand used. These models are based on the measured radii of the striationsand the assumption that the potential drop between successive striationsdoes not change. As a consequence, with decreasing distance between thestriations the electric field strongly increases toward the centralanode. It has been found that spherical striations are only obtained ifthe electric field is strongly modulated. In this case, a highly nonlocalbehavior of the velocity distribution function and strongly modulatedradial courses of the macroscopic quantities have been obtained.     

Impact of Electron–Electron Collisions on the Spatial Electron Relaxation in Non-Isothermal Plasmas

The impact of electron–electron collisions on the spatial relaxation of electrons in the column-anode plasma of a glow discharge, acted upon by a space-independent electric field and initiated by a constant influx at the cathode side of the plasma, is investigated in inert gas plasmas. The investigations are based on a new method for numerically solving the one-dimensional inhomogeneous Boltzmann equation of the electrons including electron–electron interaction in weakly ionized, collision-dominated plasmas. A detailed analysis of the spatial behaviour of the velocity distribution function and relevant macroscopic properties of the electrons is given for various degrees of ionization and electric field strengths. A significant impact of the electron–electron collisions on the relaxation structure and the resultant relaxation length already at relatively low ionization degrees has been found for low to medium electric fields.     

Experimental check of the high-frequency behavior of the electrons in molecular and atomic gas plasmas

The high-frequency behavior of the electron component in collision-dominated nitrogen plasmas of dc glow discharges, acted upon by an additional microwave field, has been studied on an adequate kinetic basis for field frequencies exceeding the characteristic frequency for energy dissipation in electron collisions with nitrogen molecules. In particular, the phase delay of the electron current density with respect to the driving microwave field has been calculated. To check the validity of the results obtained by the electron kinetic approach, the phase delay has been experimentally determined adapting an appropriate microwave resonator method to the dc plasma. The comparison of the theoretically and experimentally determined phase delay of the ac electron current in the nitrogen plasma leads to a good agreement in the entire range of high-field frequencies and confirms the conclusions on the high-frequency behavior of the electrons deduced from the electron kinetic approach. Using previous results for a neon plasma, the remarkable impact of the atomic data of the collision processes in different gases on the high-frequency behavior of the electron component in these gas plasmas is additionally evaluated.