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 in 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. The kinetic Bohm criterion emerges by combining these two conditions with a third integral one upon integrating by parts twice. A subsequent interchange of the limits ε → 0 and |vₓ| → 0 is shown to be invalid, causing a divergence which underlies a common misconception that the criterion gives undue importance to slow ions.