Ion distribution function in a weakly collisional sheath

The ion distribution function in a weakly collisional sheath is obtained analytically for arbitrary electric-field configurations, based on kinetic equations with elastic collisions. In the light of application to plasma etching and deposition, the distribution function is integrated to derive the angular and energy distributions of the ion flux incident at the electrodes. Under the assumption of a constant-sheath electric field, the simple analytic expressions ${\mathrm{\Gamma }}_{\mathrm{\theta }}$∝2 sinθ cosθ(-4 ln sinθ+${\sin }^4$θ-1)/(1+${\sin }^2$θ${)}^2$ and ${\mathrm{\Gamma }}_{\mathrm{en}}$∝-ln(1-η) for the angular and energy distributions of the nonballistic ion flux are obtained, where η denotes the normalized ion kinetic energy. For the more realistic case of a self-consistent electric field, integral formulas for ${\mathrm{\Gamma }}_{\mathrm{\theta }}$ and ${\mathrm{\Gamma }}_{\mathrm{en}}$, which are amenable to numerical quadrature, are also obtained from the Boltzmann-Poisson system. These analytic expressions are compared with Monte Carlo sheath simulations under various conditions and are found to be in excellent agreement.