Test2d04, 4th 'benchmark test' data file for CPO2D

Parabolic motion in uniform field (planar symmetry)

This test is almost identical to that in test2d03.dat, except that it is for a system of planar rather than cylindrical symmetry.

Detailed description:

The number of segments used in the present example is small enough for the example to be run with the demo' version of CPO2D. Higher accuracy could of course be obtained with more segments, using the standard or full versions of CPO2D.

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.

A uniform electric field is set up between 2 flat parallel plates. The volume between the plates is enclosed by 2 further plates. The left-hand plate is at z = -1 with V = 1, while the right-hand plate is at z = 1 with V = -1. The potential of the enclosing plates varies linearly from V = 1 at z = -1 to V = -1 at z = 1. The z = 0 plane is taken as a plane of negative reflection symmetry for the voltages, and the x = 0 plane as a plane of positive reflection symmetry. A total of 40 segments is used (before the reflection in the z = 0 plane).

The first information that appears on the screen consists of potentials and fields along the lines specified in the data file. The potentials and fields are correct to within the requested inaccuracy of 0.1%, except for the field at the faces of the electrodes (which are indefinite). The potentials and fields in the region of the corners have larger errors, for the reasons explained in the note potentials, fields and contours near to boundaries. As explained in there, these errors are easily reduced either by making the segments near the corners shorter or by adding extra short lengths of electrodes that touch the corners and lie outside the square, but in practice this would be necessary only if (exceptionally) electrons travel in this region.

An electron starts with total energy 1 eV at x = -0.5, z = 0, a point that has zero potential, so the initial kinetic energy is 1 eV. It starts at 45 degrees to the x and z axes. The path of the electron should be the parabola z = 0.5 - 0.5*(x-0.5)**2. Therefore at the test plane x = 0.5, z should maximise at 0.5 and the energy should be 0.5 eV.

In fact the calculated maximum z (see the end of the output file tmp4a.dat) is 0.49982 and the energy is 0.5000. The errors in z is therefore approximately 0.04%, and so is consistent with the requested ray inaccuracy of 0.1%. The number N of segments (40 in the present example) also affects the accuracy of the final value of z. We find empirically that for the present simulation the fractional error of z is of the order of 0.5/N**2 (giving an error of the order of 0.03% for N = 40).

The direct method is used here. Using the mesh method gives the same accuracy in a shorter time, with a mesh spacing as high as 0.4, but this is not a realistic comparison for this example because the field is uniform.