Test3d10, 10th 'benchmark test' data file for CPO3D
Space-charge limited current of a spherical diode (concave cathode)
The concentric spherical cathode and anode have radii of 1.0 and 0.1 respectively, and a voltage difference of 1V. The total space-charge limited current (which is independent of the scale length) should be 1.005 A, and is given by the program as 1.000 A, an error of 0.5%, in a total computing time of less than about a minute.
(This is a 2-dimensional simulation and so of course can be solved with higher accuracy by CPO2D).
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 cathode is a sphere of radius 1 and voltage 0 outside a spherical anode of radius 0.1 and voltage 1.
An unsupported spherical anode is not a practical proposition, but this example has a known analytical solution (see below) and the further important advantage that extra artificial boundaries do not have to be added. It also represents a precursor to a form of Pierce gun.
The number of subdivisions of each sphere is put at 10 in this data file, but the program changes the number to 12 (as can be seen from the information put on the screen and in the processed data file), and so the number of cathode segments is entered as 12.
All 4 planes of reflection symmetry are used, and so the minimum sector is 1/16 of the complete sphere. The plane x+0.414*y+z = 0.5 (which is parallel to the plane through the 3 extremities of the part of the cathode that is in the minimum sector) is used as the test plane (so that rays which fail to reach this plane do not contribute to the space-charges -although there are no failures in this example).
The depth of the cathode region and the number of interpolation points are put at 0.1 and 3 respectively. The program therefore creates a region between radii of 1.0 and 0.9 in which a form of Childs Law is used that has been adapted to the curvature of the cathode surface.
The mesh spacing for the ray space charge cells is put at 0.1. These cells come into effect outside the 'Childs Law' cathode region.
2 iterations are called, with a maximum total current density much higher than the theoretical value (because at this stage the program successfully limits the current). The fastest convergence seems to be obtained, in this example, with a damping factor of about 0.3 and an initial cathode current density of about the theoretical value.
The theoretical space-charge limited total current is 1.005 micro-Amp, see G R Brewer, Focussing of Charged Particles, Vol 2, p1 (Ed A Septier, Academic Press, New York 1967) or G A Nagy and M Szilagyi, Introduction to the Theory of Space-Charge Optics (Macmillan, 1974).
The current that appears on the screen and in the output data file is the total current divided by 16 (because there are 4 planes of reflection symmetry) and so should be 0.0000628 mA. In fact in this example it is 0.0000610 (giving an error of 2.9%) in the last iteration, with a short total computing time. Later iterations converge to 0.000613 mA (giving an error of 2.4%).
Doubling the number of segments and the number of interpolation points in the cathode region (and also halving the ray inaccuracy and the maximum step length) gives the current 0.0000630 mA, that is, an error of 0.3%, but with an increase in computing time of approximately 6.
The option to choose the colours of the rays is demonstrated here. The option is activated by putting 'col' in spaces 21 to 23 of the line that specifies the type of rays. This is then followed by an extra line that gives the repetition number (in this case 4) and the numbers of the colours that are to be repeated (in this case the colour numbers are 1, 2, 4 and 14, corresponding to blue, green, red and yellow respectively).
(Added June 1999: When simulating the geometry used in test2d10.dat, with radii 1 and 2, V=10, test-plane d=2, it is found that many more iterations are required, eg 10, to obtain approximately the correct result.)