SELF-CONTAINED SOLUTION TO THE SPATIALLY INHOMOGENEOUS ELECTRON BOLTZMANN-EQUATION IN A CYLINDRICAL PLASMA POSITIVE-COLUMN

In this paper we develop a self-contained formulation to solve the ste ady-state spatially inhomogeneous electron Boltzmann equation (EBE) in a plasma positive column, taking into account the spatial gradient an d the space-charge field terms. The problem is solved in cylindrical g eometry using the classical two-term approximation, with appropriate b oundary conditions for the electron velocity distribution function, es pecially at the tube wall. A condition for the microscopic radial flux of electrons at the wall is deduced, and a detailed analysis of some Limiting situations is carried out. The present formulation is self-co ntained in the sense that the electron particle balance equation is ex actly satisfied, that is, the ionization rate exactly compensates for the electron loss rate to the wall. This condition yields a relationsh ip between the applied maintaining field and the gas pressure, termed the discharge characteristic, which is obtained as an eigenvalue solut ion to the problem. By solving the EBE we directly obtain the isotropi c and the anisotropic components of the electron distribution function (EDF), from which we deduce the radial distributions of all relevant macroscopic quantities: electron density, electron transport parameter s and rate coefficients for excitation and ionization, and electron po wer transfer. The results show that the values of these quantities acr oss the discharge are lower than those calculated for a homogeneous si tuation, due to the loss of electrons to the wall. The solutions for t he EDF reveal that, for sufficiently low maintaining fields, the radia l anisotropy at some radial positions can be negative, that is, direct ed toward the discharge axis, for energies above a collisional barrier around the inelastic thresholds. However, at the wall, the radial ani sotropy always points to the wall, due to the strong electron drain oc curing in this region. We further present pertinent comparisons with o ther formulations recently proposed in the literature to model the pre sent inhomogeneous problem.