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 $\lambda _{D}$ width, typically much smaller than the characteristic length scale $L$ of the main plasma. This scale separation implies that the asymptotic limit $\epsilon = \lambda _{D} / L \rightarrow 0$...

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Bibliographic Details
Published in:Journal of plasma physics 2024-11, Vol.90 (6), Article 905900604
Main Authors: Geraldini, Alessandro, Brunner, Stephan
Format: Article
Language:English
Subjects:
Bohm criterion
Derivatives
Flow velocity
Integrals
Ion velocity
Ions
Magnetic fields
Plasma
Sheaths
Velocity distribution
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Summary:Between a plasma and a solid target lies a positively charged sheath of several Debye lengths $\lambda _{D}$ width, typically much smaller than the characteristic length scale $L$ of the main plasma. This scale separation implies that the asymptotic limit $\epsilon = \lambda _{D} / L \rightarrow 0$ 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.
ISSN:0022-3778
1469-7807
DOI:10.1017/S0022377824001387