The tube method for ray space charges.

The tube method is usually the more suitable for general use, especially when the beams are long and narrow.

In the ray tracing procedure a ray is advanced in steps, the maximum length of which is specified by the user (see note on ray tracing details). The tube method makes use of these ray steps. Each step is treated as a straight line between its end points (that is, the intermediate points that are created by the program - see note on ray tracing details- are not used). The program assigns a charge q uniformly distributed along the step line, where

q = i*t

and where i is the current carried by the ray and t is the time difference between the end points of the step. (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 the CPO3DS program the charge q is put in a cylindrical tube that surrounds the step. In the CPO2DS program, for a system that has axial symmetry the line of charge is rotated around the axis to give a conical or tubular sheet of charge, while for planar symmetry the line of charge becomes a flat sheet.

A radius d is required. This is used only occasionally by the program, when it needs to calculate the potential or field at a point that is very near to a line of charge. The CPO3DS program then treats the line of charge as a cylinder or 'tube' of radius d, with a uniform charge density inside the tube. This is done to avoid the very high field strengths that exist near to a charge line of zero thickness. The CPO2DS program treats the sheet of charge as having a finite thickness d, thus avoiding the very high field strengths that exist near to the ends of charge sheets of zero thickness.

When a space-charge contour option is called the programs average the charge density over a radius 3*d.

The user should experiment with a range of values of d.

When d is too small an occasional ray might experience an unwanted large deflection because it has experienced a strong field near to an end of a ray step. This might happen for example at the starting point of a ray, because then the point is certainly at the first end of the first step (and to help here the program automatically improves the accuracy level for the first step).

When d is too large the fields might be effectively averaged over a distance that is too large.

Here is a procedure for choosing the initial (experimental) value of d:

(1) Usually the critical part of the beam, as far as space-charge repulsion is concerned, is the part that has the smallest radius r. Try to make d less than about r/4.

(2) Also look at the part of the beam that has the largest radius, ignoring regions where space-charge repulsion is not important, and find the mean spacing s between the rays. Then try to make d greater than about s/3.

In practice, particularly for initial approximate simulations with a small number of rays, it might not be possible to satisfy the above two conditions simultaneously, d ≤. r/4, d ≥ s/3. It becomes easier to do this after the number of rays has been increased. As stated above, the user should experiment with a range of values of d.

The 'tube' method is primarily intended for simulations involving long thin beams. For example a beam might have an initial and final diameter of 0.1mm, a maximum diameter of 1mm and an overall length of 100mm. With the 'cell' method an accurate treatment might require that the cell dimension be 0.01mm, which would result in the creation of more than 10 million cells. On the other hand with the 'tube' method there might be 20 trajectories with step lengths that are typically 2mm, giving of the order of 1000 tubes.

An ‘advanced option’ is available for varying the tube radius during trajectory integrations

Examples of the use of the 'tube' method are given in xmpl2d11, test3d11 and xmpl3d07.

The other method of assigning space charge is the 'cell' method.

Reflection symmetries:

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

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