The cell method for ray space charges

(Note that the 'tube' method is usually the more suitable for general use, especially when the beams are long and narrow).

The ray space-charge 'cell' method.

Cells are created in the region of a beam, and when a ray passes through one of these cells a space-charge is deposited in the cell. The amount of this space-charge is

q = i*t,

where i is the current carried by the ray and t is the time spent inside the cell. (Remember that each ray represents a model particle that moves in the total electric field as if it were a single electron but carries the charge and current of many adjacent electrons.)

In CPO3DS the cells are cubic (in principle they could have different lengths in the 3 different directions, but in practice it would then be difficult to calculate or approximate the potentials due to them). In CPO2DS the cells are square; for axial symmetry the square cells are rotated around the axis to give a ring of charge, while for planar symmetry they are extended to large distances in the +/- y directions.

The distance parameter s is the cell size (the length of the sides of the cubic or square cell).

These cells are created only where they are needed, in the volume traversed by the rays.

The space-charge cells are completely independent of the ray cells (if these are used), and so the mesh spacing of the space-charge cells does not have to be the same as the mesh spacing used for the ray mesh points when rays are traced by the mesh method. In fact a space-charge calculation can be carried out when the 'direct' method of ray tracing is used and when there are therefore no ray mesh points. In fact calculations can sometimes be done more quickly this way, for a given inaccuracy, when the total number of rays is small.

When a ray passes through a space-charge cell the appropriate space-charge, with the appropriate centre-of-gravity, is added to the cell. In a later stage of the calculation the total charges in the cells (each with its weighted centre-of-gravity) are used to calculate space-charge potentials. For this purpose the charge in each cell (of side s, say) is represented by a uniformly charged sphere (in CPO3DS) or circle (in CPO2DS), in both cases centred at the centre-of-gravity of the charge.

In CPO3DS the sphere has a radius 0.6343*s (and with this choice of radius the potential at the centre of an isolated sphere is equal to that at the centre of an isolated uniformly charged cube of the same total charge). In CPO2DS the circle has a radius 0.5705*s in planar symmetry or 0.5766*s in cylindrical symmetry (and then the potential at the centre of the circular section is equal to that at the centre of the true cell of square section).

The distance s (that is, the side of the cell) must be smaller than the radius of the space-charge beam. This is a condition that is often difficult to satisfy, but the tube method can then be used.

If the 'mesh' method of ray tracing is used then it is important that the ray mesh spacing should be smaller than the radius of the beam -otherwise the variations of field strength due to the space-charges will not be correctly reproduced in the fields at the ray mesh points. But the ray mesh spacing should not be too small because this occasionally causes the rays nearest to the axis to be incorrectly accelerated towards the axis. Otherwise the 'direct' method of ray tracing should be used.

Reflection symmetries:

The program assumes that the rays all start in the minimum sector, so the user must be careful to ensure this.

In the cell method the program puts all the cells into the minimum sector.

The program automatically reflects the rays and their space charges in all the symmetry planes of the segments and surface charges.