Plasma Boltzmann equation

In a plasma, the constituent atoms, ions, and electrons are made to move faster by an electromagnetic field and not by application of heat externally or through combustion processes. Nevertheless, the result is the same as if the plasma had been heated externally the constituent atoms, ions, and electrons are made to move faster and faster, eventually reaching a distribution of kinetic energies that would be characteristic of the Boltzmann equation applied to a gas that had been... 

Plasmas typical of C02 laser discharges operate over a pressure range from 1 Torr to several atmospheres with degrees of ionization, that is, nJN (the ratio of electron density to neutral density) in the range from 10-8 to 10-8. Under these conditions the electron energy distribution function is highly non-Maxwellian. As a consequence it is necessary to solve the Boltzmann transport equation based on a detailed knowledge of the electron collisional channels in order to establish the electron distribution function as a function of the ratio of the electric field to the neutral gas density, E/N, and species concentration. Development of the fundamental techniques for solution of the Boltzmann equation are presented in detail by Shkarofsky, Johnston, and Bachynski [44] and Holstein [45]. 

As noted above, solving the Boltzmann equation is problematic because of the multidimensionality of the problem. A promising approach to calculating the electron distribution function in low pressure plasmas is the so-called non-local approach to electron kinetics. This was proposed by Bernstein and Holstein [56] and popularized by Tsendin [57], who initially suggested this approach for the positive column of a DC discharge. Since then, the non-local approach has been applied to a variety of low pressure gas discharge systems [58]. 

A radio-fiequency (rf) discharge generates a bulk plasma containing nearly the same number of positive and negative particles and a sheath containing predominantly positive ions both of them are periodically modulated by the rf field. The electron distribution/(x, v, r) as a function of position x, velocity v, and time f in a gas imder an oscillating applied electric field F may be determined by the Boltzmann equation (Winkler, 1999)... 

THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS OF ELECTRONS IN WEAKLY IONIZED PLASMAS... 

By the preceding representations, an attempt has been made to give, on the basis of the electron Boltzmann equation, an introduction to the kinetic treatment of the electron component in steady-state, time-dependent, and space-dependent plasmas and to illustrate by selected examples the large variety of electron kinetics in anisothermal weakly ionized plasmas. 

The Boltzmann Equation and Transport Coefficients of Electrons in Weakly Ionized Plasmas, R. Winkler... 

On the other hand, it is also quite important to study reaction kinetics in nitrogen plasmas to understand quantitative amount of various excited species including reactive radicals. Many theoretical models have been proposed to describe the number densities of excited states in the plasmas. Excellent models involve simultaneous solvers of the Boltzmann equation to determine the electron energy distribution function (EEDF) and the vibrational distribution function (VDF) of nitrogen molecules in the electronic ground state. Consequently, we have found noteworthy characteristics of the number densities of excited species including dissociated atoms in plasmas as functions of plasma parameters such as electron density, reduced electric field, and electron temperature (Guerra et al, 2004 Shakhatov Lebedev, 2008). 

In the present analysis, the EEDF is determined by solving the Boltzmann equation as a fimction of the reduced electric field E/N so that the electron mean energy equals 3/2 times the electron temperature experimentally measured by the probe. The Boltzmann equation is simultaneously solved with the master equations for the vibrational distribution function (VDF) of the N2 X iZg+ state, since the EEDF of N2-based plasma is strongly affected by the VDF of N2 molecules owing to superelastic collisions with vibrationally excited N2 molecules. A more detailed account of obtaining the EEDF is given in the next section. 

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