On the importance of slow ions in the kinetic Bohm criterion

TL;DR

This paper derives a new kinetic Bohm criterion considering general ion velocity distributions, clarifying the role of slow ions and correcting misconceptions about their importance in plasma sheath formation.

Contribution

It presents a novel derivation of the kinetic Bohm criterion that accounts for arbitrary ion velocity distributions and clarifies the significance of slow ions.

Findings

The distribution of normal ion velocity component must vanish at the sheath entrance.

The first derivative of this distribution must also vanish at zero velocity.

Incorrect limit interchange leads to divergence, explaining misconceptions about slow ions.

Abstract

Between a plasma and a solid target lies a positively charged sheath of several Debye lengths $\lambda_{\rm D}$ in width, typically much smaller than the characteristic length scale $L$ of the main plasma. This scale separation implies that the asymptotic limit $\epsilon = \lambda_{\rm D} / L \rightarrow 0$ 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 $\epsilon \rightarrow 0$ the distribution of the velocity component normal to the target, $v_x$, and its first derivative must vanish for $|v_x| \rightarrow 0$ 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 $\epsilon \rightarrow 0$ and $|v_x| \rightarrow 0$ is invalid, leading to a divergence which underlies the misconception that the criterion gives undue importance to slow ions.