The multiterm spherical-harmonic representation of the velocity distribution function of ``reacting'' charged-particle swarms in a gaseous medium is discussed from a general viewpoint, using spherical tensors throughout, in contrast to the traditional mixed spherical-Cartesian notation usually employed in analysis of the hydrodynamic regime. The resulting hierarchy of kinetic equations generated from the Boltzmann equation has a universal validity, applicable to all experimental arrangements, as do the associated transport and reaction-rate coefficients. The structure of these equations and the nature of the eigenvalue problem associated with them are discussed generally, independently of any numerical technique adopted for their solution, of which the moment method of Lin, Robson, and Mason [J. Chem. Phys. 71, 3483 (1979)] is just one possibility.

Velocity distribution function and transport coefficients of electron swarms in gases: Spherical-harmonics decomposition of Boltzmann's equation.

The 'two-term' approximation (representation of the electron distribution by the first two terms of an expansion in spherical harmonics in velocity space) continues to occupy a central role in the low-temperature plasma physics literature, in spite of the mass of evidence illustrating its inadequacy in the swarm (free diffusion) limit for many molecular gases. Part of the problem lies in the failure of many authors to specify quantitatively what they mean when they say that the two-term approximation is 'acceptable'. Thus for example, an error of 10% in transport coefficients may well be acceptable in many plasma applications, but for analysis of highly accurate swarm experiments to compare with ab initio and beam-derived cross-sections, 0.1% or less is required, making 'multi-term' analysis mandatory. While reconciliation of the swarm and plasma literature along the lines of two different accuracy regimes may thus be possible, we dispute claims that the two-term approximation is generally satisfactory for inversion of swarm experiment data to obtain electron impact cross-sections. The unsatisfactory nature of other assumptions implicit in much of the modern plasma kinetic theory literature is also discussed.

https://iopscience.iop.org/article/10.1088/0022-3727/36/24/006/pdf

Is the classical two-term approximation of electron kinetic theory satisfactory for swarms and plasmas?

Plasma modelling is an exciting subject in which virtually all physical disciplines are represented. Plasma models combine the electromagnetic, statistical and fluid dynamical theories that have their roots in the 19th century with the modern insights concerning the structure of matter that were developed throughout the 20th century. The present cluster issue consists of 20 invited contributions, which are representative of the state of the art in plasma modelling and numerical simulation. These contributions provide an in-depth discussion of the major theories and modelling and simulation strategies, and their applications to contemporary plasma-based technologies. In this editorial review, we introduce and complement those papers by providing a bird’s eye perspective on plasma modelling and discussing the historical context in which it has surfaced.

https://iopscience.iop.org/article/10.1088/0022-3727/42/19/190301/pdf

Plasma modelling and numerical simulation

This paper reports about systematic investigations of the velocity distribution function and resulting macroscopic quantities of the electrons in a weakly ionized and spatially homogeneous rf plasma in the molecular gas H2 and about a comparison with an analogous study of the rf plasma in the inert gas Ne for a wide field frequency range. These results are obtained by a strict treatment of the non-stationary electron Boltzmann equation considering the action of the rf field as well as that of elastic and inelastic electron - neutral gas particle collision processes. Furthermore, such investigations could be extended to very high field frequencies for the rf plasma in neon. This provides the basis for a general characterization of the complex temporal behaviour of the isotropic and anisotropic part of the velocity distribution as well as of the several macroscopic quantities in the range from low to very high field frequencies. The ration of the field frequency to two dissipation frequencies which describe the lumped dissipation of energy and impulse due to elastic and all inelastic collision processes and which are only determined by the atomic data of these processes governs the temporal behaviour of the electron kinetics in the rf plasma in any gas.



Wesentliche Grundzuge der Elektronen-Kinetik in Stosbestimmten, Eingeschwungenen rf Plasmen



Diese Arbeit berichtet uber systematische Untersuchungen der Geschwindigkeits-Verteilungsfunktion und hierdurch bestimmter makroskopischer Grosen der Elektronen in einem schwach ionisierten und raumlich homogenen rf Plasma im molekularen Gas H2 und uber einen Vergleich mit einer analogen Untersuchung des rf Plasmas im Edelgas Ne fur einen weiten Bereich der Feldfrequenz. Diese Ergebnisse werden vermittels einer konsequenten Behandlung der nichtstationaren Elektronen Boltzmann-Gleichung erhalten, wobei sowohl die Wirkung des rf Feldes als auch diejenige von elastischen und unelastischen Stosprozessen zwischen Elektronen und Neutralgas-Teilchen berucksichtigt werden. Daruber hinaus konnten derartige Untersuchungen fur das rf Plasma in Neon bis zu sehr hohen Feldfrequenzen ausgedehnt werden. Dies bildet die Grundlage fur eine allgemeine Charakterisierung des komplexen zeitlichen Verhaltens sowohl des isotropen und anisotropen Teils der Geschwindigkeitsverteilung als auch der verschiedenen makroskopischen Grosen im Bereich von kleinen bis hin zu sehr grosen Feldfrequenzen. Das Verhaltnis der Feldfrequenz zu zwei Dissipations-Frequenzen, welche die summarische Dissipation der Energie bzw. des Impulses infolge von elastischen und allen unelastischen Stosprozessen beschreiben und die allein durch die atomaren Daten dieser Prozesse bestimmt werden, beherrscht das zeitliche Verhalten der Elektronen-Kinetik im rf Plasma in jedem Gas.

Main Features of the Electron Kinetics in Collision Dominated Steady-State rf Plasmas

A new multi-term solution technique for the electron Boltzmann equation of weakly ionized steady-state plasmas

The conventional two-term solution of Boltzmann’s equation for electrons in a gas in an electric field is extended to orders higher than the first. Numerical solutions are obtained under the assumption that the Legendre polynomial expansion of the velocity distribution function can be truncated after four terms. Monte Carlo simulations are used to assess the accuracy of the new theoretical approach and the limits of the four-term approximation itself in a simple model gas with a single inelastic process. Substantial improvements are observed when using the four-term approximation for the calculation of energy distributions and transport or rate coefficients in comparison with analogous calculations performed with the conventional approach. Conditions under which the four-term approximation becomes inadequate because of the high rate of inelastic collisions are also analysed.

Multiterm solution of Boltzmann’s equation for electrons in a gas in an electric field

Recently it has been shown that in steady-state situations under certain parameter conditions the conventional Boltzmann equation solution approach using the well known two-term Legendre polynomial expansion fails to provide a sufficiently accurate description of the energy distribution and resulting macroscopic quantities of the electrons. Over the past few years, detailed studies of the temporal relaxation of electrons in a homogeneous, collision dominated, weakly ionized plasma have been performed using always the same approximation technique. Thus it is of particular interest to check the applicability of such a two-term approximation to temporal relaxation processes by an independent approach without such limitation. An important improvement of Monte Carlo methods recently obtained allows to determine the energy distribution function far into the high energy range for steady-state situations; further simulations of the non-stationary evolution are now possible with high accuracy for a limited time period. Using these Monte Carlo techniques and the conventional approach we compared the energy distribution functions and some macroscopic quantities, in particular the excitation collision frequencies, in the inert gas plasmas of Ne and Ar for three selected cases of temporal relaxation. The results obtained with these two methods for different initial situations and for both inert gas plasmas with their very different atomic data show a very good agreement and thus confirm the validity of the two-term approximation over the whole relaxation process for the considered cases.



Vergleich zweier Methoden zur Untersuchung der stosbestimmten Relaxation der Elektronen in schwachionisierten Plasmen bei Einwirkung eines elektrischen Feldes



Kurzlich hat sich erwiesen, das das ubliche Losungsverfahren der Boltzmann-Gleichung mittels der bekannten 2-Term-Legendre-Polynom-Entwicklung in stationaren Zustanden unter gewissen Parameterbedingungen die Energieverteilungsfunktion und hiermit folgende makroskopische Grosen der Elektronen nicht ausreichend genau beschreibt. Unter Verwendung der gleichen Naherung wurden in den letzten Jahren umfangreiche Untersuchungen der zeitlichen Relaxation der Elektronen in einem homogenen stosbestimmten, schwachionisierten Plasma durchgefuhrt. Somit ist es von besonderem Interesse, die Gultigkeit einer solchen 2-Term-Naherung wahrend des Ablaufs von Relaxationsprozessen mittels einer unabhangigen Methode, welche keine solche Beschrankung enthalt, zu uberprufen. Eine neuerdings erreichte wesentliche Verbesserung der Monte Carlo Methode erlaubt die Bestimmung der Energieverteilungsfunktion fur stationare Bedingungen bis weit in den Hochenergiebereich; ferner sind jetzt Simulationen der nichtstationaren Entwicklung mit groser Genauigkeit fur ein begrenztes Zeitintervall moglich. Unter Verwendung solcher Monte Carlo Verfahren und der konventionellen Losungsmethode wurde ein Vergleich zwischen den Energieverteilungsfunktionen und einigen makroskopischen Grosen, insbesondere zwischen den Anregungs-Stosfrequenzen, in den Plasmen der Edelgase Ne und Ar fur drei ausgewahlte Falle der zeitlichen Relaxation durchgefuhrt. Die mit ihren sehr verschiedenartigen atomaren Daten erhaltenen Resultate zeigen eine sehr gute Ubereinstimmung und bestatigen somit die Gultigkeit der 2-Term-Naherung wahrend des ganzen Relaxationsprozesses in den betrachteten Fallen.

Comparison between Two Approaches to the Study of the Collision Dominated Electron Relaxation in Weakly Ionized Plasmas under the Action of an Electric Field

An appropriate approach is presented for solving the Boltzmann equation for electron swarms and nonstationary weakly ionized plasmas in the hydrodynamic stage, including ionization and attachment processes. Using a Legendre-polynomial expansion of the electron velocity distribution function the resulting eigenvalue problem has been solved at any even truncation-order. The technique has been used to study velocity distribution, mean collision frequencies, energy transfer rates, nonstationary behaviour and power balance in hydrodynamic stage, of electrons in a model plasma and a plasma of pure SF6.

The calculations have been performed for increasing approximation-orders, up to the converged solution of the problem.

In particular, the transition from dominant attachment to prevailing ionization when increasing the field strength has been studied.

Finally the establishment of the hydrodynamic stage for a selected case in the model plasma has been investigated by solving the nonstationary, spatially homogeneous Boltzmann equation in twoterm approximation.

J. Wilhelm

Papers

Multiterm solution of Boltzmann’s equation for electrons in a gas in an electric field

Comparison between Two Approaches to the Study of the Collision Dominated Electron Relaxation in Weakly Ionized Plasmas under the Action of an Electric Field

Electron Kinetics with Attachment and Ionization from Higher Order Solutions of Boltzmann's Equation