Section 2.12 of the User's Guide for CPO2D and CPO3D
(or proceed to section 2.13)
Introduction to thermionic cathodes.
For detailed information see the links given in the general note on cathodes.
Here we give a brief introduction and a step-by-step description of the steps that the program takes in an example simulation.
When a thermionic cathode is called for, a finely-divided 'cathode space-charge region' is created by the program in front of the cathode. The appropriate space-charges are deposited in this region, and are limited, if necessary, according to Childs Law (or Langmuir's relationship when kT is non-zero), and these are automatically modified to take account of any curvature that the cathode has.
The starting point of the rays is the outer edge of the cathode space-charge region.
For a convex cathode, such as one in which electrons are emitted outwards from a spherical surface, the program uses an additional refinement, to match the cathode to the space-charge cells or tubes that hold the ray space-charges.
A step-by-step description of the actions taken by the program, to help in understanding
-for the simplest case, a flat cathode that follows Child’s Law.
All this information is already elsewhere in Help (and the details are given in the publication Simulation of Thermionic Cathodes, by F. H. Read and N. J. Bowring, Nucl. Instrum. Meth. A531, 407-415, 2004).
(1) The program (CPO2DS or CPO3DS) starts by setting up a series of N ‘virtual’ cathode segments in front of each real cathode segment. The value of N is chosen by the user. The virtual segments are parallel to the real segments and occupy a ‘cathode region’ of depth d in front of the cathode (and again d is chosen by the user).
(2) The program calculates the surface charges on the real boundary segments in the usual way, including the effects of the charges on the virtual segments (which are zero at the beginning) and the trajectory space-charges in the rest of the system (which are also zero at the beginning).
(3) The program calculates the potential difference Vd across the cathode region.
(4) Vd is inserted into Child’s Law to give the space-charge limited current density j from the cathode segment. In the first iteration (see below) the value of j is compared with the ‘initial current density’ ji supplied by the user. The lower of the two values is used to determine the current I of the trajectory from this segment (I is the product of j and the segment area). In later iterations j is compared in the same way with the value of the ‘maximum current density’ jm, supplied by the user.
(5) The trajectories are all traced through the system, starting at the outer edge of the cathode region (that is, at the distance d). The ‘trajectory space-charges’ that they leave in the system are stored for use in the next iteration. The current I of each trajectory is used to calculate the cathode space-charges on the virtual segments and these are also stored for use in the next iteration.
(6) The program then repeats steps (2) to (5), after applying a ‘damping factor’ (supplied by the user) to the individual trajectory currents and also to the overall space-charges. These iterations continue until the specified convergence is reached or until the specified maximum number of iterations is exceeded.
How should the user choose the various parameters?
(1) The value of N is usually not critical and a suitable value is usually 2 or 3, but higher values can be used if very high accuracy is required (more details are given in the publication referred to above).
(2) The value of d is also usually not critical and a suitable value is usually in the region of the average width of the cathode segments. The program will not allow it to be so small that the virtual segments become too crowded. If it is too large the cathode region will extend to regions where the field is not uniform and where Child’s Law will not apply in its simplest form.
(3) To help with convergence the initial current density ji should usually be significantly smaller (say 10 times smaller) than the expected final current density. This mirrors a practical situation in which the cathode temperature is increased gradually from zero to a final value. If ji is chosen to be too high then the trajectory space-charges in the first run might be so large that the potential difference Vd becomes repulsive, which terminates the iterations.
(4) The value chosen for the ‘maximum current density’ jm is usually less important, unless the cathode is intended to be temperature limited. Otherwise an arbitrarily high value can be used.
(5) Finally the damping factor can be anywhere from 0 (no damping) to 0.98 (very heavy damping). If the damping is too light then violent oscillations might occur in the iterations, while if it is too heavy the iterations will converge too slowly. For an ordinary simple harmonic oscillator such as a pendulum there is an optimum ‘critical’ damping that gives only one excursion before the oscillator is stopped in the minimum time. For cathode iterations the user has to experiment with the damping factor to find a suitable value, which is usually in the range 0.3 to 0.7. In fact the user should experiment with all the parameters.
The parameters of the rays also have to be carefully chosen.
The step lengths should be small at the start of a ray -use the cathode depth as a guide.
The later step lengths should be larger, using the advanced option to change the step lengths.
If space-charge tubes are being used then the advaced option to change their diameters should be used.
Are there too many parameters for the user to choose? It might seem like this when the program is first used. But cathode simulations are notoriously difficult and often show signs of instability. As far as we know at present the CPO programs offer the most careful and accurate method.
Could some of the parameters be pre-set or could the damping factor be adjusted by the program? Yes, perhaps for some problems, but there is such a wide variety of different cathode systems that this cannot be done in general. And we have not yet mentioned curved cathodes and other thermionic types, such as those that follow the Langmuir relationship!. It is hard to escape from the fact that cathode simulations need some feedback from the user.
(proceed to next section)