Test2d12, 12th 'benchmark test' data file for CPO2D
Planar Pierce gun
The cathode has a width of 4mm in the x direction and is infinite in the y direction, the cathode-anode spacing is 10mm in the z direction and the cathode-anode voltage difference is 10V. The focus electrode that surrounds the cathode is inclined at the well-known 'Pierce angle' of 67.5 degrees. Childs Law is used in the immediate vicinity of the cathode surface, assuming a zero cathode temperature (thermal emission can be treated with CPO3DS). The current density should be 0.73808 microamp/mm**2, and is given by the program with an error of 0.5% after 7 iterations, in a total computing time of much less than a minute.
The x = 0 plane is used as a plane of voltage reflection symmetry. The anode surround is made up of 4 circular sections (see below) and the whole is enclosed by 2 plates separated by 10mm.
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.
For simplicity, and also to try to accurately reproduce the theoretical behaviour, the anode hole is replaced by a plate.
The potential outside the beam is given by (for example see P. H. Hawkes and E. Kasper, Principles of Electron Optics, 1989, Academic Press):
V = C*r**(4/3)*cos(4*phi/3),
x = a + r*sin(phi), z = r*cos(phi),
C = (2.25*j/epsilon0)^(2/3)*(0.5*m/e)^(1/3),
and where 2*a is the width of the beam.
These equations have been used in the present example to determine the shape of the anode surround (which is accurately reproduced by 4 circular sections, each of the appropriate radius and centre of curvature) and to determine the potential distribution along the enclosing plate (which is reproduced by 10 sections in each of which the voltage varies linearly).
The theoretical Childs Law current density is w*(V^1.5)/L^2, where w = .0023340 mA/mm^2. In the present simulation, V = 10, L = 10, so the current density should be .73808 microamp/mm**2 and the current should be 1.4762 microamp per linear mm in the y direction, for the part of the cathode that has x>0.
The present example is intended as a quick test of the accuracy of CPO2D, and so the cathode and anode are given only 10 subdivisions each. The 'test plane' is put at the anode, z = 10 (so that rays which fail to reach this plane do not contribute to the space-charges -in fact some failures occur in the second iteration).
As explained elsewhere the depth 'd' of the cathode region should be significantly greater than the mesh spacing 's'. In fact, for a quick calculation we have put 'd' = 0.5 and 's' = 0.2.
7 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 0.5. The initial cathode current density is set at 0.001 mA, about 35% above the theoretical value.
The current that should be given by the program is 1.4762 microamp (see above). In fact in this example it is converges to 1.4830 microamp, giving an error of 0.5%, with a total computing time of much less than a minute.
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 1.4792 mA, that is, an error of 0.2%, but with an increase in computing time of approximately 4.