Approximate sheath solutions for a planar plasma anode

Abstract

Solutions to the anode sheath problem have been studied for the case where the electron and plasma temperature is constant and equal. The planar geometry description of steady, low-temperature collisional plasmas permits certain simplifications to the species continuity equations whereby a single, highly nonlinear equation is obtained for the entire region disturbed by the electrode. This equation is given in terms of the electric field and a variable representing the currents. Our formulation includes charged-particle production with a net current flow. Solutions are generated with a nontrivial analytic expression for the electric field of the form: E ( y ) = E,exp{A/(y+a)'}, where a, A, and E, are shape factors. Defects in the approximation can be minimized by identifying the coefficients for the net production of charges U posteriori, thereby yielding largely non-numerical solutions of some validity. Generically, this approach is not restricted to the form of the electric field utilized and other more exact approaches, while expected to refine the numerical values given here, should retain the relationships between parameters basically intact. Meaningful results have been found for nitrogen at moderately high densities [NE loz4 m-'], which at sufficiently IOW currents [JE 100 mA/cm'] are realistic for electron-impact ionization and two-body recombination. This case is completely described by two nondimensional groupings (q and z), which are sufficient to define the sheath and ambipolar regions. The constants A and a are evaluated using information for nitrogen at 6000 K, a case with well documented values, corresponding to degrees of ionization above loa. Other practical discharge gases are also represented in the example. Scaling laws result from our formulation, which allow for relatively quick answers to many single-temperature collisional plasmas at positive probes and anodes.

Plasma-sheath approximate solutions for planar and cylindrical anodes and probes

“…&,=E, @f(a), (30) where (31) Note that the infinite series converges for all values of q, and that the y/r0 term in the denominator of Eq. (29) is only relevant for the wire-thin (and smaller) anodes.…”