xmpl2d11, 11th 'example' data 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.

This cannot be a benchmark test because the beam is not infinite in extent and the assumptions made in deriving the theoretical expressions are not valid. A beam is directed towards a point on the axis. The size and position of the beam waist are approximately correct.

The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.

Theoretically, the maximum current for a isolated beam that is directed towards a point is (0.0386mA)*(V**1.5)*(tan theta)**2, where theta is the initial convergence half angle. The assumptions made when this is derived in text-books are that the beam is infinite in extent, that there is no external lens action on the beam, and that the forces on the electrons are essentially radial (for example Space-Charge Flow, by P T Kirstein, G S Kino and W E Waters, McGraw Hill, 1967). These conditions can never be met, and are perhaps not usually (perhaps not ever) well approximated in practice.

At the maximum current the waist should be at the initial convergence point and should have a diameter that is a factor 2.35 smaller than the initial diameter.

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 10 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, as described in the relevant note, 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 total initial current is one quarter that given by the above formula for tan theta = 0.3, namely one quarter of 0.003474 mA. The current is multiplied in stages until the value 0.003474 is reached.

The 'mesh' method of ray tracing is used, with a mesh spacing of 0.25. The 'ray space-charge tube' method of assigning space-charge is used, and the distance parameter for the tubes is also chosen to be 0.25.

There are 8 space-charge iterations, the last 2 of which show some consistency. The beam waist is approximately at z = -0.27 and the radius of the waist is approximately 1.168. The values for an ideal beam would be 0 and 1.244 respectively (where the expected waist radius has been adjusted to allow for the fact that the outermost ray starts from a radius 2.924 rather than 3). The flow is approximately laminar.

But remember that this is a quick demonstration and that the present simulation of an isolated beam is only approximate. In particular, you will see from contour plots of potentials and fields that the assumptions made in the theoretical derivations are not valid here (for example the field at the edge of the beam is not radial). It would of course be possible to modify this example file by using confining electrodes that have the expected boundary potentials for an ideal beam, which would presumably result in a closer approximation to the ideal result. But where should the equipotential surface be put, at the surface of the beam or along the axis or somewhere else? Another problem is that the potential at the centre of the beam is lower than that at the surface and so the electrons are slower there, forming a larger space-charge. For a cylindrical beam the difference in potential is I/(4*pi*epsilon_0*vel), which is 0.052 volt in the present example. This is not taken into account in the usual theoretical derivations.

The 'space-charge cell' method can also be used. In this method cubic cells are created in the region of the beam and the ray space-charges are put in these cells. Using the same distance parameter for the cells as for the ray space-charge tubes, a higher ray inaccuracy can be used (0.001 instead of 0.0003) to give results of approximately the same overall accuracy, but the computing time is approximately doubled in this example.

The following table summarises the results obtained using the 'space-charge tube' and 'space-charge cell' methods:

method |
accuracy |
tan(theta) |
N |
r_i/r_w |
r_f/r_i |
minimum potl |

tube |
normal |
0.3 |
8 |
2.503 |
1.130 |
-0.192 |

cell |
normal |
0.3 |
8 |
2.445 |
1.146 |
-0.192 |

tube |
high |
0.3 |
14 |
2.271 |
1.280 |
-0.203 |

tube |
high |
0.075 |
14 |
2.528 |
0.954 |
-0.0207 |

Here r_i = initial outer radius of the beam, r_w = radius of the waist, r_f = final outer radius. The minimum potential exists on the axis, in the vicinity of the waist. In the results labelled 'normal accuracy' the parameters are as described above. In the results labelled 'high accuracy' the ray inaccuracy is 0.00003, the ray mesh size is 0.05, the ray step length is 0.5, the space charge tube minimum distance is 0.05, and 20 rays are used.