The positive column and wall sheath in a gas discharge are studied with allowance for ion collisions in a plasma and ion reflection from a solid surface under conditions of incomplete ion neutralization. The kinetic equation for ions in a positive column is reduced to a Fredholm equation of the second kind. This makes it possible to solve the kinetic equation using a resolvent and thereby derive a single integrodifferential equation for the potential, which is referred to as a generalized plasma-sheath equation. Specific versions of the plasma-sheath equation are obtained that take into account charge exchange of the ions in a plasma and the thermal spread in velocities of the ionization-produced ions.
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Dvinin et al. (2000) studied this question.
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