In the limit of a small Debye length ( lambda D to 0) the analysis of the plasma boundary layer leads to a two-scale problem of a collision free sheath and of a quasi-neutral presheath. Bohm's criterion expresses a necessary condition for the formation of a stationary sheath in front of a negative absorbing wall. The basic features of the plasma-sheath transition and their relation to the Bohm criterion are discussed and illustrated from a simple cold-ion fluid model. A rigorous kinetic analysis of the vicinity of the sheath edge allows one to generalize Bohm's criterion accounting not only for arbitrary ion and electron distributions, but also for general boundary conditions at the wall. It is shown that the generalized sheath condition is (apart from special exceptions) marginally fulfilled and related to a sheath edge field singularity. Due to this singularity, smooth matching of the presheath and sheath solutions requires an additional transition layer. Previous investigations concerning particular problems of the plasma-sheath transition are reviewed in the light of the general relations.

https://iopscience.iop.org/article/10.1088/0022-3727/24/4/001/pdf

The Bohm criterion and sheath formation

The sheaths that occur at surfaces in laboratory and space plasmas are reviewed with an emphasis on numerical models that can be solved with modest computational resources. The surfaces in plasma may be the interior walls of confinement devices or inserted probes. Fluid and kinetic models are presented in some detail, and particle-in-cell models are discussed briefly. The numerical methods find the spatial profile of the potential, the particle densities near the surfaces and the current to the surfaces. Maxwellian electrons and cold ions are assumed at the outset and subsequently the models are expanded to encompass (1) multiple electron populations, (2) multiple ion species, (3) finite ion temperature, (4) surfaces that emit electrons such as heated cathodes or emissive probes and (5) surfaces that emit plasma as in the Q-machine. These complications may produce nonmonotonic sheaths in which the first derivative of the potential changes sign or double layers in which the second derivative changes sign. The effect of charge-exchange collisions on ion losses to the wall and on ion current to probes is discussed, but models with collisions of electron are omitted. Some recent advances are discussed, including experiments that measure the ion distribution function in sheaths using laser-induced fluorescence, experiments and numerical models on sheaths with multiple ion species and computational models of sheaths surrounding objects in flowing plasma.

https://iopscience.iop.org/article/10.1088/0741-3335/55/9/093001/pdf

Sheaths in laboratory and space plasmas

The spatial distribution of collisionless ions collected by a spherical object of radius much larger than the Debye length, in a flowing plasma, is calculated using a particle-in-cell code. The results provide the first rigorous theoretical calibration of a `Mach probe' in a plasma with negligible magnetic field. They are also applicable, for example, to spacecraft-plasma interactions. Ion to electron temperature ratios 0.1<Ti/ZTe<10 are explored. Qualitative differences are observed between the results for low and high Ti/ZTe, arising from ion dynamics in the wake. However, a practically universal form for the upstream to downstream current density ratio dependence on flow velocity is observed for Ti/ZTe3.

http://iopscience.iop.org/article/10.1088/0741-3335/44/9/313/pdf

Ion collection by a sphere in a flowing plasma: I. Quasineutral

Accurate measurements of the plasma potential is a critical challenge especially for complex plasmas such as magnetized and flowing. We compare various emissive probe techniques for measurements of the plasma potential. The measurements were conducted in a low-pressure magnetized discharge of the Hall thruster. The thruster was operated with xenon gas in subkilowatt power range and the discharge voltage range of 200–450 V. The probe was placed at the channel exit where, the electron temperature is in the range of 10 to 60 eV and the plasma potential is in the range of 50 to 250 V. The floating point method is expected to give a value ∼Te/e below the plasma potential. The experimental results are consistent with these expectations. Specifically, it is shown that the floating potential of the emissive probe is ∼2Te/e below the plasma potential. It is observed that the separation technique varies wildly and does not give a good measure of the plasma potential

A Comparison of Emissive Probe Techniques for Electric Potential Measurements in a Complex Plasma

The major emissive probe techniques are compared to better understand the floating potential of an electron emitting surface in a plasma. An overview of the separation point technique, floating point technique, and inflection point in the limit of zero emission technique is given, addressing how each method works as well as the theoretical basis and limitations of each. It is shown that while the floating point method is the most popular, it is expected to yield a value ∼1.5Te/e below the plasma potential due to a virtual cathode forming around the probe. The theoretical predictions were checked with experiments performed in a 2 kW annular Hall thruster plasma (ne ∼ 109−1010 cm−3and Te ∼ 10−50 eV). The authors find that the floating point method gives a value around 2Te/e below the inflection point method, which is shown to be a more accurate emissive probe technique than other techniques used in this work for measurements of the plasma potential.

A comparison of emissive probe techniques for electric potential measurements in a complex plasma

A theory has been developed for cylindrical and spherical probes and other collectors in collisionless plasmas, in the limit where the ratio of Debye length to probe radius (the Debye number λD) vanishes. Results are presented for the case of equal electron and ion temperatures. On the scale of the probe radius, the distributions of potential and density in the presheath appear to have infinite slope at the probe surface. The dimensionless current–voltage characteristic is the same for the cylinder as for the sphere, within the limits of error of the numerical results, although no physical reason for this is evident. As the magnitude of probe potential (relative to space) increases, the current does not saturate abruptly but only asymptotically; its limiting value is about 45% larger than at space potential. Probe currents for small nonzero λD approach those for zero λD only very slowly, showing power‐law behavior as function of λD in the limit as λD → 0, with power‐law exponents less than unity, resulting in infinite limiting derivatives with respect to λD.

J. G. Laframboise

Papers

Theory of cylindrical and spherical Langmuir probes in the limit of vanishing Debye number