On the Relaxation of Energetic Electrons in Plasmas

The impact of individual collision processes on the relaxation of the velocity distribution function of a group of electrons, initially localized in a narrow region at relatively high energies, has been studied. By having recourse to solutions of the non-stationary Boltzmann equation and to corresponding Monte-Carlo simulations, the temporal behaviour of electrons in CO₂ plasmas, both in the absence and the presence of an external dc field, has been investigated. A microphysical interpretation of observed relaxation phenomena, based on the data relevant to the individual collision processes, is also given.     

A convenient expansion of the master equation

We present a simple and very convenient expansion of the master equation.     

Montecarlo and boltzmann calculations of electron energy distributions in RF fields

Summary Detailed comparisons between Montecarlo and Boltzmann calculations of electron energy distributions in gases acted upon by RF fields are presented. Attention is turned to model gases of special theoretical interest but various calculations have also been made for real gases such as pure CO and He−CO mixtures. The analysis has shown that large discrepancies exist between energy distributions obtained with the two mentioned techniques under conditions of particular physical interest. The discrepancies are found to be the consequence of the two-term approximation and are expected to disappear if an appropriate multiterm solution of the Boltzmann equation is adopted.     

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.     

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.     

Modification of the electron velocity distribution function in weakly ionized plasmas by nonisotropic elastic-scattering processes

Summary After a brief presentation of a technique for solving Boltzmann's equation in multiterm approximation in the presence of nonisotropic elastic and (conservative) inelastic collisions of the electrons with gas atoms or molecules, we report the results of a systematic study of the impact of a nonisotropic scattering in elastic collisions. By using an appropriate distribution for the scattering angle, the effects on isotropic (i.e. energy) distribution, total anisotropy of the velocity distribution and some transport and rate coefficients are investigated for a model plasma, from mainly forward to mainly backward scattering. The results are then compared with those found for isotropic scattering. A particularly large impact has been found, for instance, on energy distribution and some macroscopic quantities for narrow forward scattering, while the classical two-term approximation especially is found to fail for backward scattering. The results are qualitatively explained by considering the balance between energy gain by the field and collisional dissipation and the modifications produced when changing the anisotropy of elastic scattering.

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Riassunto Dopo una breve presentazione di una nuova tecnica per la risoluzione dell'equazione di Boltzmann in approssimazione ?multi-term? in presenza di urti elastici non isotropi e urti anelastici conservativi da parte di elettroni con atomi o molecole del gas, vengono riportati i risultati di uno studio sistematico dell'effetto dell'anisotropia dello scattering elastico. Facendo ricorso ad appropriate distribuzioni dell'angolo di scattering, si studia l'effetto su distribuzione energetica, anisotropia della distribuzione delle velocità ed alcuni coefficienti del trasporto utilizzando un conveniente gas modello e variando lo scattering da ?prevalentemente in avanti? a ?prevelentemente all'indietro?. I risultati sono poi confrontati con quelli ottenuti con la comune assunzione di scattering isotropo. Si è potuto così mettere in luce che esiste un effetto particolarmente pronunciato dello scattering in avanti su distribuzione energetica ed alcune importanti grandezze macroscopiche e, contemporaneamente, anche osservare che la convenzionale approssimazione a due termini tende a divenire inadeguata particolarmente in presenza di scattering all'indietro. Tutti i risultati ottenuti sono qualitativamente spiegati considerando il bilanciamento tra guadagno di energia dovuto al campo elettrico e dissipazione dovuta agli urti e, in particolare, analizzando gli effetti che sono prodotti quando l'anisotropia dello scattering elastico viene alterata.

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Резюме Сначала обсуждается техника решения уравнения Больщмана в многочленном приближении в присутствии неизотропных упругих и (консервативных) неупругих соударений электронов с атомами или молекулами газа. Затем мы приводим результаты систематического исследования влияния неизотропного рассеяния в упругих соударениях. Используя соответсвующее распределение для угла рассеяния, для модельной плазмы исследуются влияние на изотропное (т.е. энергетическое) распределение, полная анизотропия распределения по скоростям и некоторые транспортные и вероятностные коэффициенты. Полученные результаты сравниваются с результатами вычислений для изотропного рассеяния. Обнаружено особенно большое влияние, например, на энергетическое распределение и на некоторые макроскопические величины для узкого угла рассеяния вперед, хотя классическое двух-членное приближение не обнаруживает изменения для рассеяния назад. Полученные результаты качественно обьясняются с помощью рассмотрения баланса между приращением энергии из-за полевой и столкновительной диссипации и изменениями, связанными с изменением анизотропии упругого рассеяния.

     

A humidity-sensitive nanocomposite solid ion conductor: sulfonated poly-ether-ether-ketone in nanotubular TiO2 or ZrO2 matrix

Journal of Solid State Electrochemistry - A nanocomposite solid ion conductor was prepared by infiltrating zirconia or titania nanotube arrays, made by electrochemical anodization of Zr or Ti...