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.     

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.     

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.     

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.     

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.     

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.     

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)     

Zur mikrophysikalischen Beschreibung des schwachionisierten Säulenplasmas von Glimmentladungen in Stickstoff-Neon-Gemischen. I. Vergleich zwischen gemessenen und berechneten Geschwindigkeitsverteilungsfunktionen der Elektronen

The isotropic part of the velocity distribution function of electrons is determined experimentally and theoretically for the plasma of the positive column of glow discharges in N₂-Ne mixtures. The distribution functions are measured with electrical probes using the Druyvesteyn method. The calculation of the isotropic part of the distribution functions for homogeneous and stationary conditions from the Boltzmann equation takes into consideration the essential elastical and unelastical collision processes. The character of the obtained distribution functions is fixed by the reduced field strength E/p₀ and the mixture ratio x = N₂/N₂ + Ne), which is varied in this paper in the range between pure nitrogen and pure neon. The distribution functions are found to be influenced by elastic and unelastic collisions of electrons with nitrogen molecules even at smal admixtures of N₂. Comparing the experimental results for the distribution functions with those received by the calculations a good agreement is observed.     

A variational derivation of the velocity distribution functions for nonequilibrium,multispecies, weakly interacting,spherically symmetric many-body systems

The most probable velocity distribution function of each component,f _a , of a nonequilibrium multispecies spherically symmetric system of particles (stellar plasma atmospheres and winds, stellar systems, pellet-fusion systems) is analytically derived forthe case in which each component is described by the first six moments of f _a . This is achieved by the aid of a variational approach based on the requirement that the BoltzmannH function for the system be a minimum, subject to the constraints provided by the sets of six macroscopic parameters describing the nonequilibrium state. The use of the so-obtained velocity distribution functions for the closure of the moment equations as well as for the calculation of their collisional terms (via the Fokker-Planck equation) is discussed. The limitations on the maximum deviations from the equilibrium state which are consistent with the assumptions used are also indicated.     

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 BGK Model with External Confining Potential: Existence,Long-Time Behaviour and Time-Periodic Maxwellian Equilibria

We study global existence and long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential. For an initial datum f ₀≥0 with bounded mass, entropy and total energy we prove existence and strong convergence in L ¹ to a Maxwellian equilibrium state, by compactness arguments and multipliers techniques. Of particular interest is the case with an isotropic harmonic potential, in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states. This behaviour is shared with the Boltzmann equation and other kinetic models. For all these systems we study the multistability of the time-periodic Maxwellians and provide necessary conditions on f ₀ to identify the equilibrium state, both in L ¹ and in Lyapunov sense. Under further assumptions on f, these conditions become also sufficient for the identification of the equilibrium in L ¹.     

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.     

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.     

An approximate expression for the galvanomagnetic tensor

Using the iterative solution to the Boltzmann equation for electrons in d.c. electric and magnetic fields, an expression for the resistivity tensor can be obtained in the form of an infinite series. This series can be approximated by retaining only the first two terms. In the cases where relaxation times exist — in the sense that the collision term in the Boltzmann equation can be written asg(k)/τ(k), whereτ(k) is the relaxation time, andf (k) = f _E(ɛ k) + [∂f _E(εk)/∂ε]·g(k) the distribution function for electrons with wavevectork — this approximation is exact. For polyvalent metals in the one-OPW approximation, the complete galvanomagnetic tensor can be obtained using this approximation and the result differs from that obtained by using a time of relaxation given by an expression suggested byZiman. A calculation for a simple model Fermi surface, with screened Coulomb scattering, is carried out and the results compared with those of the relaxation time approximation.     

Emission characteristics and electron kinetic coefficients of the plasma of a transverse volume discharge initiated in a mixture of heavy inert gases with chlorine molecules

The results of studying the radiation due to argon, krypton, and xenon monochloride bands, as well as to the bands of chlorine molecules, from the plasma of a transverse Ar-Kr-Xe-Cl₂ volume discharge are reported. The working mixture of a pulse radiation source is optimized with regard to its pressure and elemental composition and parameters of an excitation system. By numerically solving the Boltzmann kinetic equation for the electron energy distribution function, the transport characteristics of plasma electrons and discharge power specific losses are found for different values of the reduced electric field strength. The plasma parameters are simulated for the quaternary mixture, which is most appropriate for a multiwave UV-VUV source. Qualitative analysis is conducted for the most important electron processes in the multicomponent plasma that govern the joint formation of argon, krypton, and xenon monochlorides in the transverse discharge.     

Nonlocal phenomena in the positive column of a medium-pressure glow discharge

Commercial CFDRC software () is used to self-consistently simulate the plasma of the positive column (PC) of a medium-pressure dc discharge in argon. The software allows simulations in an arbitrary 3D geometry by using Poisson’s equation for the electric potential and fluid equations for the heavy components and by solving a nonlocal kinetic equation for electrons. It is shown that, in calculating the electron distribution function, the local approximation is almost always inapplicable not only at relatively low pressures (pR<1 cm Torr), but also at relatively high pressures (pR<10 cm Torr), i.e., under the real conditions of a diffuse PC usually met in practice. The use of the local approximation in solving the kinetic equation for electrons leads to significant errors in determining the main parameters of the PC. A paradoxical effect has been revealed: the peaks of the profiles of the excitation rates shift from the discharge axis toward the periphery as the pressure increases from low to medium values (1 cm Torr<pR<10 cm Torr). It is shown that the effect is related to the nonlocal character of the electron distribution function.     

Electron kinetics in dc CF4 + O2 mixture discharge

The electron distribution function, transport coefficients and rate coefficients for the dissociation of CF₄ and the excitation of a¹ _g and b¹ _g ⁺ states of oxygen have been calculated for a CF₄ + O₂ mixture discharge. The two-term approximation was used in solving the electron Boltzmann equation. The influence of the dissociation was ignored, so that the results are parametric in the density reduced electric fieldE/N and the ratio of the mixture components. Only slight dependence of the calculated quantities on the composition of the gas mixture has been observed. The critical electric field for the onset of the discharge was found to beE/N=1.1–1.25 × 10^–15 Vcm² and it is weakly decreasing with growth of the oxygen content. With a small amount of oxygen added to CF₄ the electron distribution function and the mean kinetic energy remain practically unchanged.