xmpl3d21, 21st 'example' data file for CPO3D

Space-charge limited cylindrical diode with thermal emission energies.

The file was written 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.

An inner cathode of radius 1 mm and length 8 mm is coaxial with an anode of radius 4 and potential 100V. Fringe field correcting hoops are placed at the ends of the diode. The thermal energy of emission from the cathode is 0.1 eV.

To obtain convergent iterations in this example it seems to be necessary to make the value of j0 (in the last line of the data) not more than about 20% larger than the space-charge limited j found by the program (in fact j0 is 1.1 mA/mm**2, while the value of j obtained by the program, see below, is 0.946 mA/mm**2). Otherwise there tend to be oscillations in the individual ray currents -for example in one run the currents in the interior of the diode might be large while those in the outer region are small and then in the next run this situation might reverse.

There is no theoretical treatment with which the results of this test can be compared (as far as the author of this program knows), but when the thermal energy is zero the theoretical space-charge limited current density at the cathode surface is 0.875 mA/mm**2 (see I Langmuir and K B Blodgett, Phys. Rev. vol 22, 347, 1923). The current that appears on the screen and in the output data file is the total cathode current divided by 16 (because all 4 planes of reflection symmetry are used in this example) and so should be 2.749 mA (for kT = 0). In fact in this example it is 2.972 mA (for kT = 0.1 eV). The innermost 9 rays give a current equivalent to a total current of 2.871 mA -which is 4.4% higher than the expected value for kT = 0.

To explore the depth and position of the potential minimum (that is, the 'virtual cathode') in front of the surface of the cathode, the number of cathode interpolation points can be increased from 4 to 12 (to obtain a better simulation of this region) and the 'contour' option can be used at the end of the iterations. The potential minimum is then seen to have a depth of approximately 0.079 V at a distance of approximately 0.021 mm from the surface of the cathode.

The above file is easily modified to give a diode with kT = 0, and the current obtained is then approximately 2.844 mA, which is 3.5% higher than the theoretical value.

The present simulation is however imperfect in that the diode does not have a length much larger than the diameter of the diode (which would be necessary to simulate the infinite length assumed in the theoretical derivation), the number of segments is not large (in order to give a quick demonstration) and the mesh size and depth of the cathode region are not small. All these aspects can of course be improved (and the computing time will then be longer).