xmpl2d10, 10th 'example' data file for CPO2D
Space-charge limited diode, with zero initial energies, in planar symmetry (that is, flat, parallel cathode and anode, infinite in one direction).
This example is similar to the 9th example, given in file xmpl2d09.dat.
The answer is of course well known, but this cannot be a benchmark test because the beam is not infinite in the transverse direction.
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.
Here 'planar' symmetry is used, so that the system extends an infinite distance in the +/- y directions.
An infinite planar diode cannot be simulated and so it is necessary to add boundary electrodes in this simulation. These are plates that are given the theoretical potentials, proportional to z**(4/3).
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 cathode and anode both have a length of 10 mm in the x direction, are infinite in the y direction, are separated by 10 mm in the z direction, and have voltages 0 and 100 respectively. The x=0 plane is made a plane of positive voltage reflection symmetry. 10 rays start from the part of the cathode that has x>0. The enclosing plates are separated by 10 mm.
As explained elsewhere the depth 'd' of the cathode region should in principle be significantly greater than the mesh spacing 's'. In fact we have put 'd' = 0.5 and 's' = 0.2.
The theoretical Childs Law current density is w*(V**1.5)/L**2, where w=.0023340 mA/mm**2. In the present simulation, V=100, L=10, so the current density should be .023340 mA/mm**2 and the current should be 0.11670 mA per mm in the y direction, for the part of the cathode that has x>0.
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 0.5 and an initial cathode current density approximately equal to the theoretical value.
The current given by this program is 0.11003 mA. The error would therefore seem to be 6% However the rays nearest to the artificial boundaries can be ignored because they cross the boundary (mostly due to the use of the 'mesh' method of ray tracing, for speed, and the fact that some of the mesh points lie outside the boundary). Therefore averaging over the 5 innermost rays (which cover 0.5 of the cathode area) gives a current equivalent to 0.11784 mA, and hence an error of 1% (in a total computing time of 86 sec, with a 66 MHz 486 laptop PC). The errors are similar in later iterations.