Shown below is another pair of animated gif results from a simulation of ion beam formation at the end of a plasma column. The plasma will form in a chamber on the left. A short cylindrical opening in the conducting wall ends in a cone with 45 degree half-angle. Beam is extracted by the strong electric field of an adjacent grounded "puller" electrode. The first animation shows growing proton beam density with no electrons. The second shows the added electron cloud with growing density and dropping temperature, as the plasma surface advances into the beam hole.
For both cases, the extraction potential is made visible by a family of equipotential surfaces (10.0, 9.5, 9.0 ..) kV spacing. The wall is at 10 keV. The highest potential location and value is indicated above the plot, rising from 20.9 V to 1129.7V above the wall potential without electrons then dropping to as low as 69.2 V for the coldest plasma kT = 13.84 eV.
The total charge density is displayed by a false color plot, with logarithmic scale (a factor e between colors: magenta = 100% of maximum density, blue = 37%, cyan = 13.5%, ..). Below 0.7% is white. When the plasma forms, grey shading at 8% is added to the spectrum to call attention to the shape of the plasma surface.
The first plot has the plasma electrons switched off. Ions have zero temperature and are launched from the left at 10 eV kinetic energy. A sequence of increasing proton beam densities is shown, (rho = 2, 4, 8,16*n; n = 1, 2..16). The ion paths are shown by tracking 32 rays from left to right, with equal radial spacing. Most hit the wall, with 11 extracting at low density and 8 at high density. A few rays are deflected upward into the tube wall by space charge repulsion.
Note the rapid drop in density as ions accelerate in the extraction field. The beam is sharply focussed by the extraction field for low initial density, so the peak density which defines the magenta color is found at the waist. Also the density change in initial acceleration is reduced by the low space charge potential.
The conical exit has a well-known beneficial effect on beam optics with strong space charge (see J. R. Pierce "Theory and Design of Electron Beams" D. Van Nostrand 1949). Here we see the absence of ray crossover for outer rays in the beam bundle.
The second plot shows electrons in thermal equilibrium, with electron temperature set on each iteration to 20% of the plasma (space charge) potential. This number is chosen to approximately satisfy the sheath equilibrium condition for equilizing ion and electron flows to the wall of the chamber.
The early iterations are not shown as the plasma shape changes only slightly while the potential falls from over 1 kV to 236 V. Then steps of -40, -20, -10 V in potential are shown. The control parameter is the ratio of electron to proton density at the point of maximum potential. Stability becomes more difficult to maintain as the electrons cool, and control is lost in the final frame when electron density exceeds the proton density (which violates the Bohm criterion).
Note that the exit beam properties are not very sensitive to the plasma surface position. The position and shape of the 10 kV equipotential also change little. The sheath thickness falls by a factor of about 2 during the sequence, as expected since the Debye length depends on the square root of the electron temperature.