Test2d10, 10th 'benchmark test' data file for CPO2D

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


The concentric spherical cathode and anode have radii of 2mm and 1mm respectively, and the cathode-anode voltage difference is 10V. The space-charge limited current is given by the program with an error of 1.8% in the last (that is, the fourth) iteration, with a total computing time of much less than a minute. The error remains at this level in later iterations.


Usable only with the full, space-charge version of CPO2D.

Detailed description:

The spherical diode is the precursor, for cylindrical symmetry, of the type of electron gun design known as the Pierce gun.


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.


Childs Law is used in the immediate vicinity of the cathode surface, assuming a zero cathode temperature (thermal emission can be treated with CPO3DS).

According to I. Langmuir and K. Blodget, Phys. Rev. 24, 49 (1924) (see G. R. Brewer, Focussing of Charged Particles, Editor A. Septier, Academic Press, 1967), the total current I should be 29.329 micro-Amps/(alpha**2*V**1.5), where alpha**2 = 0.74985 for r_cathode/r_anode = 2. The current density at the cathode should therefore be 0.024607 mA/square mm (approximately one third of that given by Child's law for the same spacing and voltage difference), and the current in the hemisphere at z>0 should be 0.6184 mA.


The present example is intended as a quick test of the accuracy of CPO2D, for cathode systems in cylindrical symmetry. The z = 0 plane is used as a plane of reflection symmetry and the cathode and anode are given 10 and 5 subdivisions respectively.


The outer corners of the minimum sector are at (r,z) = (2,0) and (0,2), which are in the plane rx + z = 2, 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).

As explained elsewhere, we should have the radius 'r' (of the cathode) much greater than the depth 'd' of the cathode region, which in turn should be significantly greater than the mesh spacing 's'. In fact, for a quick calculation we have put 'r' = 2, 'd' = 0.1 and 's' = 0.1 (a smaller 's' would increase the computing time, and a larger 'd' would give a larger error in the total current).

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 of about 30% above the theoretical value.


The current that should be given by the program is 0.6184 mA (see above). In fact in this example it is 0.6296 mA (giving an error of 1.8%) in the last (that is, the fourth) iteration, with a total computing time of much less than a minute. The error remains at this level in later iterations.