Shown below is another animated gif result from a simulation of ion beam formation at the end of a plasma column. The plasma forms in the chamber on the left. A very short cylindrical opening in the conducting wall (at +25 kV) of the plasma chamber is formed by a truncated double cone with 54 degree half-angle. Beam is extracted by the strong electric field of an adjacent "puller" conical electrode at -5 kV. With growing positive ion density, the plasma surface is seen to advance slightly toward the beam hole and flatten, so the beam waist moves downstream and becomes wider. The highest extracted current is about 40 mA.
The extraction potential is made visible by four equipotential surfaces (+0.1, 0.0, -0.1, -0.2 kV relative to the extraction cone). The plasma potential is indicated above the plot, referenced to the empty system potential on axis at launch surface left (24.798 kV). In the sequence the proton charge density triples but the plasma potential increases only gradually, to a maximum of about 200 V.
The total (e + p) charge density is displayed by a false color plot, with logarithmic scale (a factor e between colors: magenta = 100% of positive ion density in plasma chamber, blue = 37%, cyan = 13.5%, green, yellow, red). Below 0.7% is transparent. The density in a tight waist on axis may be higher, so color here is clamped to maintain useful color contrast elsewhere.
The plot shows electrons in thermal equilibrium, with electron temperature adjusting itself slightly for each frame of the sequence in the range 16+-1eV. The adjustment damps numerical oscillations in the plasma potential which is sensitive to the Debye length via the sheath thickness. The electron density near the wall is chosen to approximately equilize ion and electron flows to the wall of the chamber. Electron density in the plasma interior is set at 99% of the proton density (to satisfy the Bohm stability criterion). Between these limits, electron density depends on potential through the Boltzmann factor.
The sheath is visible as a band of high total charge density near the leftmost cone, which continues along the extraction equipotential surface.
In this plot, ions have zero temperature and are launched from the left at 10 eV kinetic energy. The 9 frames show a sequence of increasing proton beam densities, (rho = 16000, 20000, 4000*n; n = 4, 5, ..12; units of microCoul/m3). The ion paths are shown by tracking 16 rays from left to right, with equal radial spacing of 0.5 mm. Many hit the chamber wall to form the plasma sheath, with 5 or 6 extracting and reaching the cutoff z.
An earlier animated gif at lower proton density is available
here. The electron treatment in this version of the simulation is less satisfactory with higher temperature and a tendency to break away from the chamber wall, allowing the plasma potential to grow unrealistically large.