Sheath structure in front of an electron emitting electrode immersed in a two-electron temperature plasma: a fluid model and numerical solutions of the Poisson equation

Abstract

A one-dimensional fluid model of sheath formation in front of a large, planar, floating, electron emitting electrode (collector) immersed in a two-electron temperature plasma is presented. For certain values of the parameters the Bohm criterion is triple valued. Three methods to select the correct Bohm velocity are compared. The simplest is to find the parameters, where the floating potentials that correspond to the high and to the low Bohm velocity are equal. But sometimes—when the electron emission is small—such parameters cannot be found. The second method is the ‘maximum positive ion flux criterion’, proposed recently by Fernández Palop et al (2002 J. Appl. Phys. 91 2587), for electronegative plasmas. In this work it is modified for a two-electron temperature plasma with emitted electrons and applied successfully. This method is sometimes—when the electron emission from the collector is increased—not applicable because the positive ion flux as a function of potential has only 1 maximum. The third method is a numerical solution of the Poisson equation. The transition from the low to the high Bohm velocity occurs at the parameters where the numerically found space charge density profile indicates the formation of a double layer in the sheath. These parameters are always identical to the parameters found with the maximum positive ion flux method, when the latter can be applied. The ‘shooting method’ for the correct determination of electric field at the collector is presented. This method reveals the existence of regular and irregular numerical solutions of the Poisson equation. The irregular solutions of the Poisson equation fulfill only one of the two sheath edge conditions and they correspond to the solutions of the fluid model, that are not physically acceptable. The parameters, where the numerical solutions of the Poisson equation become irregular, are identical to those found with the ‘positive square of electric field’ test, which is also presented.