xmpl3d09, 9th 'example' data file for CPO3D
Space-charge limitation of current in a strong magnetic field.
Electrons of energy 1 eV in a beam of radius 1 mm are in a field of 0.1 Tesla. Theoretically, the maximum current is 0.0324 mA and the potential depression at the centre of the beam is then such that the electrons there have a kinetic energy of 0.174 eV. Since it is impossible to simulate the idealised system, for which the beam has an infinite length and the potential at the outer surface of the beam is a constant, the beam is enclosed between 2 discs and a cylinder, all of potential zero. The current is put at 95% of the theoretical maximum value, and after several iterations the kinetic energy at the centre of the beam converges to 0.306 eV. If the full current is used the calculation fails to converge.
The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.
The maximum perveance is known, but this cannot be a benchmark test because a beam of infinite length cannot be simulated, and electrodes have to used.
A finite beam is enclosed between 2 discs and a cylinder, all of radius 1 mm and potential zero. The electrons start from one of the discs.
A set of parallel rays is defined as coming from a disc of radius 1 mm and going through a 2nd disc at a very large distance away. The current is given the initial value 0.0324*1.333, which is reduced to 0.0324*0.95 for the next 2 iterations.
The helical motion of the electrons has a small pitch length, approximately 0.2 mm, and so the ray tracings are necessarily rather slow. To make them faster for this demonstration a large mesh spacing, 0.3 mm, has been used for the ray tracings and the space-charge cells, and the ray inaccuracy has been given the large value of 0.03. But the essential features of the rays are still reproduced by the program.
The final potential at the centre of the beam is -0.685, as can be seen by zooming in on the central volume and looking at the potential contours. After several more iterations this potential converges to -0.694. The corresponding kinetic energy is therefore 0.306 eV, which is somewhat different from the value 0.174 eV mentioned above. But only 95% of the theoretical has been taken in this example. If the full current (0.0324 mA) is used the calculation fails to converge. The discrepancies are no doubt partly due to the fact that an infinite length of beam is not present to contribute to the potential at the centre, but are also due to the inaccuracies in the present example.