The various analytic approaches to the solution of the electron Boltzmann equation in non-uniform glow discharge plasmas in atomic gases are presented. For slow electrons with kinetic energy w < t1 (e1 is the first excitation potential), for which the traditional two-term approximation for the distribution function is valid, the situation can be radically simplified in the non-local case, for which the energy relaxation length exceeds the characteristic scale. Numerous manifestations of non-locality in the positive column, the anode and the cathode regions of the direct current glow discharge, in striations, in capacitively and inductively coupled low-pressure radiofrequency discharges and so on are discussed. For fast electrons of w >? E , , the approximation of continuous energy losses can be used. The simplest approximation corresponds to the neglect of scattering. In this case, simple analytic expressions for the distribution of the fast electrons, for the multiplication factor and so on can be derived, and the structure of normal and abnormal cathode regions for plane and hollow cathodes can be analysed.

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With the continuing progress of plasma modelling in recent years the desire arose to develop simplified approaches to the kinetic description of the electron component in weakly ionized plasmas. Methods which are based on the direct solution of the Boltzmann equation in some limiting situations, namely the 'nonlocal' approximation in a weakly collisional plasma and the 'local' approximation in a highly collisional plasma, may be much more efficient than conventional simulation techniques. In this paper the foundation of these approaches is reviewed. Their quantitative accuracy and their applicable range is examined on the basis of a comparison to the numerical solution of the complete, space-dependent Boltzmann equation for a positive column plasma.

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Macroscopic models for the equilibrium of a three-component electronegative gas discharge are developed. Assuming the electrons and the negative ions to be in Boltzmann equilibrium, a positive ion ambipolar diffusion equation is derived. Such a discharge can consist of an electronegative core and may have electropositive edge regions, but the electropositive regions become small for the highly electronegative plasma considered here. In the parameter range for which the negative ions are Boltzmann, the electron density in the core is nearly uniform, allowing the nonlinear diffusion equation to be solved in terms of elliptic integrals. If the loss of positive ions to the walls dominates the recombination loss, a simpler parabolic solution can be obtained. If recombination loss dominates the loss to the walls, the assumption that the negative ions are in Boltzmann equilibrium is not justified, requiring coupled differential equations for positive and negative ions. Three parameter ranges are distinguished corresponding to a range in which a parabolic approximation is appropriate, a range for which the recombination significantly modifies the ion profiles, but the electron profile is essentially flat, and a range where the electron density variation influences the solution. The more complete solution of the coupled ion equations with the electrons in Boltzmann equilibrium, but not at constant density, is numerically obtained and compared with the more approximate solutions. The theoretical considerations are illustrated using a plane parallel discharge with chlorine feedstock gas of p = 30, 300 and 2000 mTorr and n e0 = 10 10 cm −3 , corresponding to the three parameter regimes. A heuristic model is constructed which gives reasonably accurate values of the plasma parameters in regimes for which the parabolic profile is not adequate.