General notes on space-charge calculations.

The CPO programs are extremely accurate, particularly for problems involving space-charge and/or cathodes. But these simulations can be difficult, so here is some general advice:

When you first start to set up a new simulation choose the numbers of segments and rays, the inaccuracies and step lengths so that the running tim is only a few minutes. The set-up might then look too simplistic, but you will be changing it before anyone else sees it.

Then, after you have optimised the parameters and have nunderstood how your system behaves, -BUT NOT BEFORE- nincrease the numbers of segments and rays, decrease the nrequested inaccuracies and use shorter step lengths (but not too short) to obtain the final answers.

This note deals with:

Iterative use of space-charge.

Surface charges.

Step lengths.

Region in front of cathode.

Reflection symmetries of voltages.

Reflection symmetries of rays.

Using minimum sector.

Iterative use of space-charge:

The space charges in a beam or ion cloud are established iteratively. In the first run the charged particles move in a zero space-charge environment and leave behind them their space charges in the form of charge-filled cells or cylinders that surround each step in the ray tracing. In the second iteration the ions are traced in that space-charge environment, again leaving their new charges along the new rays. And so on until an acceptable consistency (chosen by the user) has been reached.

The inaccuracy of the calculation of the potential due to all the ray space-charges is given (approximately) by the 'ray inaccuracy' that has been entered.

Surface charges:

The presence of the space-charges of the rays causes changes to the charges on the boundary elements (that is, the electrodes and their segments), which are therefore re-calculated at each iteration.

Step lengths:

WARNING. Large ray step lengths can give inaccurate space-charges. This is because in depositing the charge into the mesh cells that are crossed by a ray step, or the space-charge tubes along the step, the step is taken to be linear (that is, a straight line between the end points) and the velocity is assumed to be constant. The total charge in the step is the known current times the known time for the step.

Region in front of a thermionic cathode:

When a thermionic cathode is called for, a 'cathode space-charge region' is created by the program in front of the cathode -see note on setting up a thermionic cathode.

Reflection symmetries of voltages (see also general note on symmetries and reflections and symmetries of cathode rays and segments).

The symmetries in the reflection planes cannot be negative, because the space-charge cloud cannot have negative symmetry.

Reflection symmetries of rays (see also general note on symmetries and reflections and symmetries of cathode rays and segments).

:

The rays are automatically reflected in the relevant planes when space-charge is present.

It is not possible to incorporate all the relevant safeguards into the program, so the user must be careful, and should CAREFULLY INSPECT THE RAYS PLOTS to check on the symmetries.

Using minimum sector:

ADVICE: Confine the first set of rays (before reflections) to the smallest possible sector (for example if x = 0 and z = 0 are reflection planes in a 2D planar simulation, confine to the region of positive x and z sector, or if x = 0 and y = 0 are reflection planes in a 3D simulation, confine to the region of positive x and y). Otherwise unnecessary mesh points will be created, slowing down the calculation.