IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 35, NO. 5, OCTOBER 2007 1341 The Bohm Plasma-Sheath Model and the Bohm Criterion Revisited Natalia Sternberg and Valery Godyak, Fellow, IEEE Abstract—The plasma–sheath model and the Bohm criterion introduced by Bohm are among the earliest attempts to separately model the plasma and the sheath and to find a way to join the plasma and the sheath solutions. Although there is hardly a paper on plasma–sheath modeling that does not quote the Bohm criterion, Bohm’s paper and his results are widely misunderstood. The reason for this is that, in his paper, Bohm himself misin- terpreted his result by concluding that the sheath edge coincides with the reference point of his plasma–sheath model. As a result, the criterion for the reference point obtained by Bohm to ensure monotonicity of his sheath solution (i.e., the Bohm criterion) was erroneously applied to the sheath edge and was used in literature as a criterion for sheath formation. In this paper, we show that the Bohm criterion when applied to the sheath edge contradicts Bohm’s own definition of the sheath and cannot be obtained from Bohm’s plasma–sheath model. Index Terms—Bohm criterion, plasma–sheath model. I. I NTRODUCTION I T IS WELL known that a bounded plasma consists of bulk plasma and space-charge sheath. There is a transition region between the plasma and the sheath [1], [2]. The bulk plasma is characterized by quasi-neutrality, while the sheath is the region where no ionization occurs. Ionization is caused by electron collisions with gas atoms wherever electrons are present. Therefore, the lack of ionization in the sheath could be caused only by the absence of electrons. Indeed, it has been shown in [3], using asymptotic matching techniques, that the assumption of no ionization in the sheath implies that the electron density in the sheath is negligible. Thus, one can say that the sheath is an electron-free region, which is precisely Langmuir’s [4, p. 970] view of the sheath, accepted by Bohm in [1, p. 79]. In literature, the plasma and the sheath are often modeled separately. No matter which model is used to describe the sheath, one has to address three separate issues: 1) what is the point of reference for the potential in the sheath model; 2) which point represents the sheath edge; and 3) which point can be used Manuscript received March 1, 2007; revised May 14, 2007. This work was supported in part by the Air Vehicle Directorate, Air Force Research Laboratory, Wright-Patterson AFB, OH and in part by the Air Force Office of Scientific Research Award FA9550-07-1-0415. N. Sternberg is with the Department of Mathematics and Computer Science, Clark University, Worcester, MA 01610 USA (e-mail: nsternberg@clarku.edu). V. Godyak is with the OSRAM Sylvania, Beverly, MA 01915USA (e-mail: valery.godyak@sylvania.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPS.2007.905944 as a patching point in order to patch the plasma and the sheath solutions. In general, those are the three different points. In [1], Bohm proposed a model, which he called the plasma–sheath model. That model can be derived from the stan- dard fluid plasma-wall equations [5] by neglecting ionization but keeping the electron density. Bohm’s goal was to find a solution that describes the sheath. He first specified a point of reference for the potential where the plasma is still quasi- neutral. Choosing a zero electric field at the reference point, he integrated the sheath model starting at that point. He then found that the solution he obtained is monotone only if the ion velocity at the reference point is greater than or equal to the ion sound speed. Bohm’s result is purely mathematical and ensures existence of a monotone solution of his model. It is his misinterpretation of that mathematical result as a condition for the sheath formation that has been causing confusion in the literature. Bohm justified his choice of a zero field as an initial condition for his solution by his claim that the electric field in the plasma is negligible. He was obviously trying to patch the plasma with the sheath solution, interpreting the reference point as the patch- ing point. Thinking that his patching was successful, he further interpreted his solution as the sheath solution and the patching point as the sheath edge. Based on those interpretations, Bohm concluded that the sheath begins where the ions reach the ion sound speed. His conclusion is known in the literature as the Bohm criterion. It is known today that Bohm’s solution yields an infinite sheath [6]. It is known that Bohm’s solution cannot be used for patching [2], [3], [7]. It is known that the Bohm criterion does not specify the sheath edge [3]. Nevertheless, the Bohm crite- rion has remained at the center of controversy about the position of the sheath edge. The present paper is an attempt to resolve this controversy by providing an insight into Bohm’s model, his solution, and his conclusions that led to the formulation of the Bohm criterion. In the present paper, using the geometric theory of differ- ential equations [8], we give a mathematical analysis of the Bohm plasma–sheath model. Our analysis shows that Bohm’s physical interpretation of his mathematical result, which led to the Bohm criterion, was based on his erroneous assumption that the reference point coincides with the patching point and the sheath edge. We show that Bohm’s model and methods do not provide sufficient information about the position of the sheath edge. We show that a criterion for existence of a monotone solution of Bohm’s plasma–sheath model is not sufficient for providing a criterion for the position of the sheath edge or for the sheath formation. 0093-3813/$25.00 © 2007 IEEE