One‐dimensional fluid model for a planar sheath in front of a negative ion‐emitting electrode surface immersed in a collision‐less, non‐magnetized, electronegative plasma is presented. It was found ...that the positive ion speed at the plasma–sheath boundary (PSB) increases linearly with negative ion emission from the electrode but attains a saturation value as soon as a virtual cathode is formed near the electrode surface. The effect of negative ion emission on the pre‐sheath region shows that the potential drop increases across the pre‐sheath in accordance with the rise in positive ion speed at the PSB. The sheath width obtained using the present model shows a similar trend as the Child‐Langmuir law, but its magnitude is found to be consistently higher compared with a non‐emitting electrode. A plausible explanation has been given to explain these effects.
Abstract
The magnetic pre-sheath (MPS) length,
L
MPS
, is a critical parameter to define the sheath potential, which controls the ion trajectory of low-Z species (D, T, He, and C), as well as the ...prompt re-deposition of high-Z species. To determine
L
MPS
, we fabricated micro-trenches (30 × 30 × 4
μ
m) via focused ion beam milling on a silicon surface and exposed them to L-mode deuterium plasmas in DIII-D via the divertor material evaluation system (DiMES) removable sample exposure probe. The areal distribution of impurity depositions, mainly consisting of carbon, was measured by energy-dispersive x-ray spectroscopy (EDS) to reveal the deuterium ion shadowing effect on the trench floors. The carbon deposition profiles showed that the erosion was maximized for the azimuthal direction of
φ
= −40° (referenced to the toroidal magnetic field direction) as well as the polar angle of
θ
= 80°. A Monte Carlo equation-of-motion (EOM) model, based on a collisionless MPS, was used to calculate the azimuthal and polar deuterium ion angle distributions (IADs) at the surface for a range of
L
MPS
=
k
×
ρ
i
, where
ρ
i
is the ion gyro radius and
k
= 0.5–4. Then, gross erosion profiles were calculated by a Monte Carlo micro-patterning and roughness (MPR) code for ion sputtering using as input the calculated azimuthal and polar IADs for each value of
k
. Good agreement with the experimental C deposition profiles was obtained for the case
k
= 2.5–3.5. This result is consistent with a previous kinetic modeling prediction of
k
∼ 3, as well as previous analytical investigations that predicted the
L
MPS
to be several ion gyro radii. A validation of theoretical sheath models supports its applicability to ITER and pilot plant divertors to successfully predict plasma–materials interactions.
An analytic solution is presented in this paper for the electric potential near a wall in a confined plasma. This is well fitted for both the sheath and pre-sheath regions. In the sheath region, the ...potential is well adapted to the differential equation proposed by Bohm. In the pre-sheath region, the potential is also well suited, decaying to zero electric field in the plasma, which is a physical condition. The potential is also valid for any value of the parameter K measuring the dimensionless Bohm velocity.
Edge-to-center plasma density ratios-so-called h factors-are important parameters for global models of plasma discharges as they are used to calculate the plasma losses at the reactor walls. There ...are well-established theories for h factors in the one-dimensional (1D) case. The purpose of this paper is to establish h factors in two-dimensional (2D) systems, with guidance from a 2D particle-in-cell (PIC) simulation. We derive analytical solutions of a 2D fluid theory that includes the effect of ion inertia, but assumes a constant (independent of space) ion collision frequency (using an average ion velocity) across the discharge. Predicted h factors from this 2D fluid theory have the same order of magnitude and the same trends as the PIC simulations when the average ion velocity used in the collision frequency is set equal to the ion thermal velocity. The best agreement is obtained when the average ion velocity varies with pressure (but remains independent of space), going from half the Bohm velocity at low pressure, to the thermal velocity at high pressure. The analysis also shows that a simple correction of the widely-used 1D heuristic formula may be proposed to accurately incorporate 2D effects.
The energy distribution of particles in a gaseous system is well understood through the implementation of a statistical tool, namely, the Maxwell–Boltzmann distribution function in the velocity–space ...coordinate system. The Maxwell–Boltzmann distribution function is utilized to investigate the velocity distribution of plasma particles like electrons, assuming that their collision frequency does not depend on the velocity. However, there is a swift transition in converting the Maxwell–Boltzmann distribution function to the Druyvesteyn distribution function for the case where a collision frequency is directly proportional to the velocity. Our aim is to incorporate the frequency components to investigate the Maxwell–Boltzmann and Druyvesteyn distribution functions. Employing the equation of motion, we observe that the collisional electron velocity is equal to the equilibrium electron velocity
∼eE/m
e
ω
multiplied by the collisional frequency over the external source frequency
β
=
ν/ω
corresponding to the externally applied electric field. We investigate the difference in the Druyvesteyn distribution function between sheath and pre-sheath regions, when a stream of electrons is traversing or effusing through the part of a pre-sheath region corresponding to the dimension of the order of mean free path. Velocity and corresponding energy distribution functions are compared for non-effusion and effusion cases in the collisional and non-collisional regimes. The Maxwell–Boltzmann and Druyvesteyn velocity and energy distributions are competitive when the collisional frequency is twice the frequency of the applied electric field.
This paper solves and analyses the complete characterization of the plasma-sheath transition and the characteristic I-V curves of an active and collisional plasma close to a cylindrical or spherical ...wall considering a wide range of the parameter which describe the model to be useful for experimental measures. Despite the difficulty of including the three possible pre-sheath mechanisms, this characterization is obtained from a self-consistent model using three easily measurable parameters, namely the electric potential of the wall, the positive ion current collected by the wall, and the radius of the wall. These parameters are easy to measure and facilitate the diagnosis of plasmas from an experimental point of view.
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