On universal properties of the plasma-sheath transition and large-size sheath structures

摘要

We investigate properties of the plasma-sheath transition in the Bissell and Johnson (B&J) extension [1] of the archetypal Tonks and Langmuir (T&L) collision-free (CF) plasma and sheath equation [2] (assuming a cold ion source) to discharges with Maxwellian sources of arbitrary temperatures. Our work is based on highly accurate numerical and analytic solutions and simulations of the one-dimensional time-independent kinetic equations for the ion and electron velocity distribution functions (VDFs) f_(i,e)(x,v) coupled with the self-consistent electric field. We consider symmetric boundary conditions at two perfectly absorbing co-planar plates, located at positions x= ±L and characterized by the electric potential {formula}, assuming that starting from the symmetry plane (x= 0, Φ= 0) the electrostatic potential Φ(x) decreases monotonically in directions positive and negative directions. The basic equations of the problem are {formula} where the ion and electron densities are defined by n_(i,e) = f fi,_edv and quantities related to higher velocity moments {formula}dv, are, e.g. the directional velocities {formula} temperatures {formula}, and other quantities of interest read, in their alternative normalized and un-normalized forms: {formula} Here, n_0= n_i(0)= n_e(0), and T_e,_(i,0)= T_(e,i)(0), are the density and temperature in the center of the discharge, e is the positive elementary charge, k Boltzmann's constant, ε_0 is the vacuum permeability, {formula} is the cold-ion sound velocity, with m_(i,e) the ion/electron mass, and {formula} is the electric field. The ion source {formula} is a function of potential, i.e., {formula} with β= 0,1 here. The frequency v_i= Rn_n= c_(s0)/L_i is either due volume ionization or to an external ion source originating from the perpendicular direction, e.g., in the cases when the model is applied to scrape-off-layer [3] (SOL) plasma, with the source temperature T_n different from the self-consistently established ion temperature