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.     

On the Solution of the Electron Boltzmann Equation in Higher Order Legendre Polynomial Expansion

The paper deals with the mathematical structure of the general solution and the selection of the physical relevant solution of the hierarchy resulting from the electron Boltzmann equation by the Legendre polynomial expansion for a weakly ionized plasma with elastic and exciting collisions and under the action of an electric field. At first the properties of the general solution for the two and four term approximation of the distribution function are analyzed and then the study is extended to arbitrarily even order of the hierarchy using the theory of weakly and strongly singular differential equation systems for the investigation of the solution behaviour of the truncated hierarchy at small and at large energies, respectively. In this way, especially the free parameters of the non-singular part of each general solution, which is obtained for small and large energies and is of interest for the construction of the desired solution, could be found, and the procedure for their final determination is explained. A first illustrative example is given of the application of these studies for the determination of the velocity distribution function of the electrons in four term approximation for a model plasma.     

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

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.     

A stochastic approach to the kinetic theory of gases

A theory of fluctuations in non-equilibrium diluted gases is presented. The velocity distribution function is treated as a stochastic variable and a master equation for its probability is derived. This evolution equation is based on two processes: binary hard sphere collisions and free flow. A mean-field approximation leads to a non-linear master equation containing explicitly a parameter which represents the spatial correlation length of the fluctuations. An infinite hierarchy of equations for the successive moments is found. If the correlation length is sufficiently short a truncation after the first equation is possible and this leads to the Boltzmann kinetic equation. The associated probability distribution is Poissonian. As to the fluctuation of the macroscopic quantities, an approximation scheme permits to recover the Langevin approach of fluctuating hydrodynamics near equilibrium and its fluctuation-dissipation relations.     

Generalized Boltzmann equation for lattice gas automata

In this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for then-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlnear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard-sphere systems, describing the time evolution of pair correlations. The ring equation is solved to determine the (nonvanishing) pair correlation functions in equilibrium for two models that violate semidetailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid-type model on a triangular lattice. The numerical predictions agree very well with computer simulations.     

Response of a Lorentzian gas to an a. c. pulse

The response of a “Lorentz gas” to a pulsed a.c. electric field and its relaxation after the cessation of this field has been studied by solving the Boltzmann's transfer equation. Explicit expressions for the electron distribution function and the current density are obtained under the assumption that the collision frequency is independent of the electron velocity.     

A stochastic master equation representation of statistical mechanics

A functional master equation for a probability density of the velocity distribution function is presented. This constitutes a representation of statistical mechanics which is equivalent to the BBGKY hierarchy but makes alternative approximation schemes available.     

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.     

Tensorielle Kugelflächenfunktionsentwicklung des Stoßintegrals für ein schwach ionisiertes Plasma II

A spherical harmonic expansion of the deduced approximation [1] of the collision integral is obtained by expanding the electron distribution in spherical harmonics of the electron velocity. General spherical harmonic tensors are used. From the expansion a hierarchy of partial differential equations for the expansion tensors follows. This hierarchy is equivalent to the Boltzmann-equation. In this paper the calculations are performed up to linear terms of the approximation.     

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     

Theory of open systems I

A new method of treating open systems is presented. The normal treatment using the generalized master equation with the projection of Argyres and Kelley is meaningful only if the state of the reservoir never deviates appreciably from the reference state which appears in the projection. Otherwise, one must make at least a partial resummation of the perturbative expansion of the kernel of the generalized master equation. The present method avoids the introduction of a projection operator and allows us to overcome such resummation difficulties. It is based on an integrodifferential equation for the statistical operator of the composite system, which naturally provides a hierarchy of equations involving the statistical operator ?(t) of the open system and suitable quantities describing higher and higher order bath-system correlations. Treating the deviations of the bath from its initial equilibrium or stationary state as expansion parameters, one gets an approximation scheme, each step of which gives a closed system of equations for ?(t) and a suitable set of correlation quantities.Eliminating such quantities one obtains a closed linear integrodifferential equation for ?(t). The zeroth approximation in the deviations coincides with the Born approximation of the generalized master equation which uses the projection of Argyres and Kelley.On the other hand, even the first approximation is equivalent to the resummation of infinite contribution of the Born series of such a generalised master equation. When it is suitable, the concentration of the bath can also be used as an expansion parameter to handle the hierarchy.     

Relativistic Single Particle Kinetic Theory

Single particle kinetic theory is the study of the dynamics of a single particle moving through a medium. The only mechanism for change in this theory is through two-body collisions between the single particle and the particles of the medium given by the collision term of Boltzmann's equation in kinetic theory. This article contains a summery of relativistic dynamics and a method of projecting the relativistic dynamical system into phase space. This enables us to express relativistic kinetic theory in terms of phase space variables and also to apply the techniques of approximation in the non-relativistic theory to the relativistic domain.     

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.     

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)     

Self-consistent theory of population,ionization and electron distribution

We derive analytical formulae for population of excited levels and degree of ionization of atoms not necessarily restricted to the Maxwellian distribution of the free electrons. The results are obtained from solving Boltzmann's equation with a simplified form of the inelastic collision term. Good agreement with numerical results of other authors is obtained.     

Quantum statistical hierarchy equation in nonequilibrium systems

An extension is given for the Fourier expansion method with the contraction technique, which was introduced by Balescu for quantum statistical systems. This is attained by introducing a diagrammatic method with a concept of moving contraction. Then the hierarchy equation for the Contracted Fourier coefficient of the Wigner distribution function is obtained. As an application, a generalized master equation involvingn-body collision effects and quantum statistical effects is also derived.     

Semiclassical approximation and 1/n expansion in quantum-mechanical problems

The semiclassical approximation and the technique of 1/n expansion are used to calculate the eigenenergies and the wave functions for the radial Schrödinger equation. It is shown that the expressions that are asymptotically exact in the limit n=n _r+l+1 → ∞ and which describe the above eigenenergies and the asymptotic coefficients at the origin and at infinity ensure a satisfactory precision even for states characterized by modest values of the quantum numbers n _r and l, including the ground state.     

Asymmetric Bethe-Salpeter Equation for Pairing and Condensation

The Martin-Schwinger hierarchy of correlations are reexamined and the three-particle correlations are investigated under various partial summations. Besides the known approximations of screened, ladder and maximally crossed diagrams the pair-pair correlations are considered. It is shown that the recently proposed asymmetric Bethe-Salpeter equation to avoid unphysical repeated collisions is derived as a result of the hierarchical dependencies of correlations. Exceeding the parquet approximation we show that an asymmetry appears in the selfconsistent propagators. This form is superior over the symmetric selfconsistent one since it provides the Nambu-Gorkov equations and gap equation for fermions and the Beliaev equations for bosons while from the symmetric form no gap equation results. The selfenergy diagrams which account for the subtraction of unphysical repeated collisions are derived from the pair-pair correlation in the three-particle Green’s function. It is suggested to distinguish between two types of selfconsistency, the channel-dressed propagators and the completely dressed propagators, with the help of which the asymmetric expansion completes the Ward identity and is Φ-derivable.     

On the Kinetics of the Active Zone of the Stationary SF6 Beam Discharge Plasma

The paper deals with the impact of intensive electron attachment on the kinetics of the electrons in the active zone of the stationary band-like beam discharge plasma in SF₆ which is an alternative useful plasma medium for “dry etching”. The energy distribution of the electrons in this plasma was obtained by numerically solving the Boltzmann equation which includes apart from elastic collisions, different exciting collision processes, attachment in electron collisions, direct ionization, the ambipolar loss of electrons, Coulomb interaction between electrons and of electrons with ions and the power input to the electrons by the turbulent electric field. In particular, due to the needed fulfilment of the consistent electron particle balance, for an extended region of the turbulence energy density in this plasma a large impact on the electron kinetics of the intensive electron attachment, which is the prevailing electron loss process, was found enforcing independent of the turbulence energy density always a large power input to the electrons, smooth and only slowly decreasing energy distributions even in the energy region of direct ionization.     

Research on the ionospheric ion velocity distribution and the incoherent scattering spectra of the radar with the 16-moments approximation for the relaxation collision model

The distribution function which given by Grad cannot be used to describe the ionospheric ion velocity distribution due to the larger anisotropic temperature appears in the high-latitude ionosphere. In this article, based on the Boltzmann equation, the relaxation collision model (RCM) was used to substitute the Boltzmann collision integration, and a non-Maxwell ion velocity distribution function with the 16-moments approximation for the bi-Maxwell distribution was given, and the ion transport equation with the 16-moments approximation was also derived and solved. Moreover, the ion velocity distribution, the solution of transport equation and the incoherent scattering spectra with the 13-moments and 16-moments approximation for the relaxation collision model were simulated, analyzed and compared. The research shows that compared with the 13-moments approximation, the 16-moments approximation with the bi-Maxwell distribution is more suitable to describe the characteristics of the anisotropic temperature ion distribution in the high-latitude ionosphere.