Test3d11, 11th 'benchmark test' data file for CPO3D

Space-charge limited current of a spherical diode (convex cathode)

The cathode is a complete sphere of radius 0.1 mm, inside a concentric anode of radius 10 mm. The space-charge limited current should be 8.0304 A, and is reproduced by the program with an error of 0.1% -a notable achievement.

(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 0.01 and voltage 0 inside a spherical anode of radius 1 and voltage 1.

An unsupported spherical cathode 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. The simulation is however more difficult than that of the concave cathode treated in test3d10.dat, and so an accurate solution takes longer.

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 outer corners of the minimum sector are at

(1,0,0,), (0.707,0.707,0), (0,0,1),

which are in the plane x+0.414*y+z = 1, which has been used above 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).

Because the cathode is convex, the program automatically creates two regions that surround the cathode. The first region has the cathode surface as a boundary, and is called the 'inner cathode' region. Here Childs Law is used, appropriately corrected for the curvature of the cathode. The second region is called the 'outer cathode' region, and is created only for convex cathodes. Outside the 'outer cathode' region is what can be called the 'mesh cell' region, in which the space charges of the rays are assigned to mesh cells, as described in the relevant note. The purpose of the 'outer cathode' region is to bridge the gap between the 'inner cathode' and 'mesh cell' regions. This region is necessary when the spacing of the mesh cells is comparable to, or larger than, the dimensions of the cathode, as will frequently happen for convex cathodes. For example, in the present file the mesh spacing is equal to the cathode radius. If the 'outer cathode' region did not exist it would be necessary to use a much smaller mesh spacing, with a consequent large increase in computing time. The creation of the outer region allows the study of cathodes that are much smaller than the dimensions of the rest of the system.

The depth of the overall cathode region (that is, the 'inner' plus 'outer' cathode regions) and the number of interpolation points are put at 0.05 and 6 respectively. The overall radius of the cathode region is therefore 0.06.

As explained in the relevant note, the 'inner' region will lie between radii of 0.01 and 0.01625 (that is, its thickness is 1/8th of that of the combined 'inner' plus 'outer' regions), and 2 interpolation points will be used there, and the 'outer' region will lie between radii of 0.01625 and 0.06, and here the remaining 4 interpolation points will be used.

The mesh spacing for the ray space charge cells is put at 0.05. As explained above, these cells come into effect outside the outer cathode region.

The 'direct' method of ray tracing is used (and is necessary) and the maximum step length is 0.125. The program determines the step lengths in the cathode regions.

4 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 0.8 and an initial cathode current density of about the theoretical value.

The theoretical space-charge limited total current is 8.031 microamp, see 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 0.5019 microamp. In fact in this example it is 0.5092 (giving an error of 1.4%) in the last iteration, with a total computing time of a few minutes. In further iterations the error increases to 7.2% and then the result converges to 0.5013 microamp, giving an error of 1.5%.

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 0.5089 microamp, that is, an error of 1.4%, which is only a small improvement, but with an increase in computing time of approximately 6.

To illustrate the use of the 'zero initial kinetic energy' option, the lines above that specify the thermionic cathode can be replaced by:

zero initial kinetic energy

12 0. number of segments, current density

i 0. 0. 0. 'i' for 'inside, and xyz of reference point

0.005 distance from electrode for starting rays

n calculate space-charges?

1 1 1 1 symmetries of rays in yz, zx, xy and x=y planes

The 10 rays then start at a distance of 0.005mm from the surface of the inner electrode, with the kinetic that they would have had if they had started from the inner electrode with zero kinetic energy.

To illustrate the use of this option with space charge, these lines can be tried:

zero initial kinetic energy

12 6.391 number of segments, current density

i 0. 0. 0. 'i' for 'inside, and xyz of reference point

0.005 distance from electrode for starting rays

Y calculate space-charges?

0.01 distance parameter for space-charges

repeat of previous rays*****************************

1.00 = factor to multiply ORIGINAL currents

repeat of previous rays*****************************

1.00 = factor to multiply ORIGINAL currents

finish of space-charge iterations***************************

The 'space-charge tube' method is called here for the space-charge. In this example there is no space-charge between the inner electrode and the starting point of the rays (unlike the situation with the cathode options).