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.     

Microwave Diagnostics of the Electron Density in Molecular Mixture Plasmas

The electron density in the positive column of DC glow discharges has been determined by means of an improved method of the microwave diagnostics. This method is based on the measurement of both the resonance frequency shift of the resonator cavity and the change of the inverse resonator quality. It makes explicit use of the associated phase delay between the microwave field and the induced HF electric current. To evaluate the applicability of the method and of a simplified version of it to complex molecular mixture plasmas, the electron density in CO₂ CO/O₂ mixture plasmas has been measured for larger ranges of gas pressure and discharge current. The applicability check is made by a comparison between measured and calculated results for the phase delay. The good agreement between experimental and theoretical results supplies evidence for a far‐reaching applicability of the simplified version of the improved microwave diagnostics method that allows the determination of the electron density without detailed information about the plasma composition and the electron energy distribution. The theoretical determination of the phase delay has been done by solving the time‐dependent electron Boltzmann equation. Two different techniques have been employed to solve this kinetic equation. In particular, the conventional two‐term treatment has been analysed with respect to its suitability to molecular gas mixtures and has been found to be sufficient for the determination of the phase delay.     

Zum Verhalten des Elektronenensembles im anisothermen schwachionisierten Plasma bei Einwirkung eines hochfrequenten elektrischen Feldes

Investigations Concerning the Electron-ensemble in a Nonisothermal Low Ionized Plasma under the Influence of a High Frequency Electric Field The adjustment and the periodic behaviour of the electron-ensemble under the influence of a high frequency electric field were investigated in a nonisothermal low ionized plasma. By direct numerical solution of the nonstationary Boltzmann equation the time behaviour of some macroscopic quantities of the electrons was determined via the calculation of the isotropic distribution function for a neon-plasma. An interpretation of the obtained results is given. The investigations were performed within a range of frequencies of the electric field, where the time behaviour of the isotropic distribution function is essential effected as well as by the electric a.c. field and the collision processes, whilst the alteration of the anisotropic part of the velocity distribution function takes place in a quasistationary way.     

The Impact of Different CO Admixtures on the Electron Kinetics in Collision Dominated RF He/CO Bulk Plasmas

Electron energy distribution functions and mean collision frequencles for the uniform bulk plasma of established rf discharges in He/CO have been obtained by solving the time dependent Boltzmann equation. The results, which are of importance for plasma processing and laser technology, show that already small CO admixtures to He remarkably changes the periodic behaviour of the energy distribution of the electrons by drastical enhancement of the collisional energy dissipation effectiveness in the mixture. These changes can be interpreted on the basis of a lumped energy dissipation frequency in all electron collisions which is determined by the atomic data of the electron collision processes as well as by the admixture fractions and which drastically increases with increasing CO admixture. The degree of modulation of the energy distribution in different parts of the electron energy space and its population on the time average at higher energies are determined by the relation between the rf field frequency and the mentioned lumped energy dissipation frequency.     

Glow discharge in rare-gas and metal vapour mixture

Applying the Boltzmann equation to a He-Cd mixture discharge the electron energy distribution functions, kinetic coefficients and collision frequencies are numerically calculated. Calculations are made for a homogeneous and stationary discharge plasma subjected to an externally applied electric field. The collision processes which have been taken into account are elastic and inelastic collisions of electrons with He and Cd atoms as well as mutual encounters of electrons. In this case the electron energy distribution and all the quantities calculated from it are dependent on the reduced electric field, the ionization degree and the relative cadmium concentration.     

Harmonics Analysis of the Electron Component of rf Bulk Plasmas. I. Theoretical Base

rf discharges are increasingly used in low pressure plasma processing, i.e. for etching, film deposition and sputtering. The modelling of such discharges is a very complex task, especially dependent on discharge conditions, however of large importance for the insight into the main physical processes and thus for their control to improve the final results. One main important aspect is the determination of the electron velocity distribution function and of relevant rate and transport coefficients. The paper contributes to the treatment of this problem. In the first part a systematic Fourier expansion of the kinetic equation and of the consistent particle, energy and momentum balance equation is described. Then, a mathematical analysis of the resulting ordinary differential equation system for the coefficients of the Fourier expansion is performed. Based upon this we succeeded to develop a numerical approach to calculate the physical relevant solution of this system. By this approach in addition to the harmonics of the distribution function that of relevant macroscopic quantities, as transport coefficients and collision frequencies, can be determined. In the second part of this paper this method will be applied to investigate the bulk plasma of a rf discharge in molecular hydrogen.     

Theoretical modeling of microwave-pumped high-pressure gas lasers

The electron avalanche and laser excitation processes in high-pressure discharges at microwave frequencies are investigated. In our model, the applied electromagnetic field is treated classically and assumed to be monochromatic. The Boltzmann equation for the electron velocity distribution function under the influence of an alternating electric field is numerically solved for a typical XeCl laser gas mixture. All relevant elastic, inelastic and electron-electron collisions are included in solving the Boltzmann equation. The theoretical modeling of microwwave-pumped high-pressure gas lasers are developed based on the first law of thermodynamics in order to determineE _rms/n (root-mean-square field strength/total number density of gas molecules) which is required by the Boltzmann equation to calculate the electron kinetics rates and microwave-power absorption by the plasma. A sample calculation of the microwave-pumped XeCl laser is presented, and a fair agreement between theory and experiment is seen.Paper partially presented at the 10th Int. Conf. on Lasers and Applications, Lake Tahoe, Nevada, USA (1987)     

Die Relaxation der Geschwindigkeitsverteilungsfunktion der Elektronen im homogenen elektrischen Feld mit elastischen und anregenden Stößen. II. Das Verhalten des Elektronenensembles bei aperiodischer Feldänderung zwischen zwei stationären Zuständen

Recently the relaxation of the electron component was investigated under field-free conditions after sudden switch-off the electric field and otherwise after sudden changing the electric field to a new value of field strength. To continue these papers we consider now the relaxation process of the electron ensemble in the presence of a monotonous time variable electric field. The investigations are based on a computation of time change in the isotropic part of the velocity distribution function of the electrons and of the macroscopic parameters determined by the distribution function. The start-ing-point is the non-stationary Boltzmann equation with stationary initial states taking into consideration elastic and exciting collisions. Besides the representation of the received numerical results a physical interpretation is obtained for the duration of the whole relaxation process, for its initial stage as well as for the momentary stage of the relaxation by introduction of normalized characteristic time quantities. Further characterizing conditions are found related to the relaxation after quasi-jumplike change of the electric field and in the case of quasistationary field alteration respectively. With the introduced characteristic time quantities statements about the degree of realization of one of this limiting cases are possible for any given monotonous field. The investigations are per-formed in a low ionized non-thermal neon plasma.     

Electron Energy Distribution Functions in RF Molecular Plasmas: The Role of Second Kind Collisions

Electron energy distribution functions in rf molecular plasmas have been calculated by solving the time dependent Boltzmann equation in the presence as well as the absence of vibrationally and electronically excited molecules and thus of first kind and second kind (superelastic) collisions with them. The results, which refer to a model plasma composed by three components (the ground state, a lumped vibrational state and a lumped electronic state), show that these collisions with vibrationally and electronically excited molecules strongly affect the modulation of the electron energy distribution function and related quantities.     

Allgemeine Berechnung der Elektronenverteilungsfunktion von schwach ionisierten Plasmen in beliebig zeitabhängigen elektromagnetischen Feldern

The behaviour of a weakly ionized plasma in external, arbitrarily time-dependent, electromagnetic fields is treated within the framework of kinetic theory. The Boltzmann kinetic equation is solved using the Lorentz ansatz, taking into account elastic collisions between electrons and neutral particles and assuming that the collision frequency is independent of the electron velocity. The drift velocity of electrons enters into the isotropic part f₀ and into the direction-dependent part f₁ of the electron distribution function. A method is given for the calculation of the drift velocity, which is calculated explicitly for the important but difficult case of a sinusoidal electric field in the presence of a magnetic switching field. f₀ and f₁ are calculated; f₀ is investigated generally. f₀ consists of an expansion in generalized Laguerre polynomials. The influence of the electromagnetic fields on the distribution function and its time variation is discussed and the relaxation behaviour is shown. The following two special cases are calculated explicitly: a linear rising electric field and a sinusoidal electric field, both in the presence of a constant magnetic field.     

ELectron Collision Rates and Transport Coefficients of a Weakly Ionized dc Plasma in Ar/Sih4-Mixtures

The paper deals with electron kinetics of a Ar/SiH₄ dc plasma, using the stationary and spatially uniform Boltzmann equation. The solution of this kinetic equation has been obtained by applying higher order Legendre polynomial expansion of the electron velocity distribution. For varying mixture composition the energy distribution and relevant macroscopic quantities as mean energy, drift velocity, rate coefficients for excitation, dissociation and ionization and relevant energy transfer rates for these processes have been calculated. In particular, it has been found that a most effective activation of the feed gas occurs for SiH₄ admixtures in the range from 5 to 10 per cents.     

Non-equilibrium vibrational,attachment and dissociation kinetics of HCl in XeCl selfsustained laser discharges

The vibrational kinetics of HCl in a Ne-buffered XeCl selfsustained laser discharge is self-consistently calculated by coupling the vibrational master equation with the chemical one, the Boltzmann equation for electron transport and the circuit equations. The results show the limits of the classical simplified schemes of HCl vibrational kinetics used in the literature. Problems connected with the neglection of HCl depletion instabilities in the calculations are discussed using a parallel resistor network model.     

Zur allgemeinen kinetischen Theorie von partiell ionisierten Plasmen in zeitabhängigen Magnetfeldern und elektromagnetischen Zusatzfeldern

A general method is given for the calculation of the electron distribution function of a weakly ionized plasma in external time-dependent magnetic fields and additional electromagnetic fields. The Boltzmann equation of kinetic theory is solved taking into account elastic collisions between electrons and neutrals. The isotropic part f⁰ of the distribution function follows from a general linear integro-differential equation and contains all known standard distributions (Druyvesteyn, Davydov, Margenau and others) as special cases. The direction-dependent part f¹ gives the transport tensors.     

弱电离大气等离子体电子能量分布函数的理论研究

使用球谐展开的方法求解玻尔兹曼方程,得到了弱电离大气等离子体(79％氮气和21％的氧气)的电子能量分布函数(EEDF).发现当约化电场较小时(E/N＜100 Td),EEDF在2-3 eV急剧下降,在此情况下,高能尾部比麦氏分布要小;当约化电场增加,E/N＞ 400 Td,分布函数趋近于麦氏分布;当约化电场进一步增加,E/N＞ 2000 Td,EEDF的高能尾部(超过200 eV)相对于麦氏分布增加,在高频场作用下,EEDF更倾向于麦氏分布.当ω》vm时,有效电子温度只依赖于E/ω,而与碰撞频率无关;当ω《vm时,有效电子温度只依赖于E/N,与微波频率无关.与一些单原子分子等离子体中电子-电子碰撞在电离度大于10-6时就会影响EEDF不同,空气等离子体中,只有当电离度大于0.1％时,电子-电子碰撞才会对EEDF有明显影响.     

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.