On the importance of slow ions in the kinetic Bohm criterion

Between a plasma and a solid target lies a positively charged sheath of several Debye lengths λD width, typically much smaller than the characteristic length scale L of the main plasma. This scale separation implies that the asymptotic limit ε = λD/L → 0 is useful to solve for the plasma-sheath system. In this limit, the Bohm criterion must be satisfied at the sheath entrance. A new derivation of the kinetic criterion, admitting a general ion velocity distribution, is presented. It is proven that, for ε → 0, the distribution of the velocity component normal to the target, vx, and its first derivative must vanish for |vx| → 0 at the sheath entrance. These two conditions can be subsumed into a third integral one after it is integrated by parts twice. A subsequent interchange of the limits ε → 0 and |vx| → 0 is invalid, leading to a divergence which underlies the misconception that the criterion gives undue importance to slow ions.