xmpl3d07, 7th 'example' data file for CPO3D
Space-charge repulsion of an isolated beam that initially converges to a point.
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.
This simulation has axial symmetry and so a more accurate simulation can of course be obtained using CPO2D, with the data file xmpl2d11. The present file is the 3D analogue of xmpl2d11..
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 20 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 total initial current is half that given by the above formula for tan theta = 0.3, namely half 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 0.5.
The 'trajectory charge-tube' method of assigning the space-charges of the trajectories is used, and the tubes are given the diameter 0.2.
There are 5 space-charge iterations, the last 2 of which show some consistency. The beam waist is approximately at z = -0.2 and the radius of the waist is approximately 1.11. The values for an ideal beam would be 0 and 1.209 respectively (where the expected waist radius has been adjusted to allow for the fact that the outermost ray starts from a radius 2.842 rather than 3). The flow is approximately laminar. The potential minimum at the centre of the beam is approximately -0.19. The sizes of the ray tracing mesh and the trajectory space-charge tubes are both large in the present example (for an accurate calculation they would ideally both be much smaller than the beam radius).
This example is intended only as a quick demonstration and so 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? And the electrons at the centre of the beam move in a potential that is lower than at the surface and so have a lower velocity, which is not taken into account in the usual theoretical derivations.