46th 'example' file for CPO2D.

Simulation of the space-charge repulsion of an isolated beam that initially converges to a point, using the 'ray space-charge tube' method and the option

to vary the tube radius.

This simulation is similar to that of xmpl2d11, in which the current was the maximum current that would give a beam waist at the original point of convergence of the beam. In the present simulation the current is smaller, giving a beam waist beyond the original point of convergence. The geometry is the same as for xmpl2d11.

Here the beam starts from a plate at a finite distance from the cross-over, and is surrounded by a cylinder at ground potential to try to reduce any external lensing action of the beam on itself. The energy of the electrons is 1eV.

There are 25 rays, all starting from a disc of effective radius 3 at z = -10, and all directed towards the point (r,z) = (0,0). The 'beam' option is used, with a 'uniform' distribution of current across the starting disc. Each ray therefore carries the same current and each represents the same area of the disc (and so they are not uniformly spaced in the radial direction).

The initial current is multiplied in 14 stages of iteration, with a final total current 0.0012mA.

The 'mesh' method of ray tracing is used, with a mesh spacing of 0.05 and 0.5 in the r and z directions respectively. The 'space-charge tube' method of assigning space-charge is used, and the distance parameter for the tubes is 0.03. However, the option is used to vary this distance parameter r_tube, together with the step length dl for ray tracing. r_tube and dl are varied according to the z position, as follows:

z |
dl |
r_tube |

-11 to -3 |
1 |
0.1 |

-3.1 to -1.5 |
0.5 |
0.05 |

-1.5 to 5 |
0.2 |
0.03 |

5 to 11 |
0.5 |
0.05 |

The last few iterations show some consistency. The beam waist is approximately at z = 2.218 and the radius of the waist is approximately 0.296.

The flow is not exactly laminar, since the rays that are outermost at the source penetrate the beam after the waist. This behaviour continues to be observed for other values of the variables (dl, r_tube, etc) and so seems to be genuine. One possible explanation is that the beam does not have an infinite length. Therefore some negative space-charge at z < -10 is missing and positive charges are present on the plate at z = -10. This is more likely to affect the outer rays than the inner ones. It seems difficult to investigate this by starting the beam at lower values of z, since although the incorrect charges would be smaller they would have an effect over a longer distance. A similar non-laminar behaviour can be seen in xmpl2d11.

It was noted in xmpl2d11 that the field at the edge of the beam is not radial, thus invalidating the theoretical treatments. This is also true for the present simulation.