Comparison of Monte Carlo simulations and nonlocal calculations of the electron distribution function in a positive column plasma

Recently the nonlocal approximation for solving the Boltzmann equation to determine the electron distribution function (EDF) in modeling of low-pressure discharges has attracted great interest. The nonlocal approximation is strictly applicable only to electrons which are confined in the plasma volume by the space charge electric field. The unconfined electrons which have a sufficiently high total energy to overcome the space charge potential barrier in front of the walls, and which can therefore be lost from the plasma to the walls, are not consistently addressed by the nonlocal approximation. We compare EDF's from nonlocal calculations in positive column plasmas with and without inclusion of the wall losses in different approximations to results of an efficient, accurate Monte Carlo benchmark method. The expected range of (column radius)×(gas density) for applicability of the nonlocal approach with wall losses is confirmed. The anisotropy of the EDF caused by wall losses of unconfined electrons and by the axial electric field is studied using Monte Carlo simulations. The impact of the anisotropy on the applicability of the nonlocal approximation is discussed. The importance of the appropriate inclusion of the wall losses of unconfined electrons in the nonlocal approximation is demonstrated. An approximation of the treatment of the wall losses in nonlocal calculations is studied, which yields good agreement with the Monte Carlo results in the entire applicable range of the nonlocal approximation.