Section 2.3 of the User's Guide for CPO2D and CPO3D

(or proceed to section 2.4)

Electrodes and segments

(and the charges on them)

The elements to which voltages are applied are referred to as electrodes, and the parts into which they are subdivided are referred to as segments. Cross-sections of the electrodes, in a plane through the axis of cylindrical symmetry, are referred to as sections.

When a new electrode geometry is entered for the first time, the electrodes are subdivided into an initial set of basic segments. In CPO2D these have a simple straight cross-section, while in CPO3D they are either flat triangles or rectangles, as appropriate (for example, a sphere is divided into triangles). The program then further subdivides these segments, if required, to give the final set of segments, and the corners of these segments continue to lie on the curved surface of the parent electrode and continue to follow the same boundaries as the parent electrode.

Examples of the subdivision of electrodes into segments are shown in some of the figures in the Users Guide, namely figures 1.1 (perspective view of a quadrupole-type system), 1.3 (schematic of a double cylinder lens), 1.5 (a sphere automatically subdivided into triangles) and several others.

The program sometimes further subdivides these segments, if requested by the user, to give the final set of segments, and the corners of these segments continue to lie on the curved surface of the parent electrode and continue to follow the same boundaries as the parent electrode.

The maximum number of segments for the present program is given in the information box on the screen (but be very cautious about using a large number -see chapter 3 of the User`s Guide or HELP!!).

In CPO2D the electrodes can have either axial or planar symmetry. They can be straight or arcs of circles. When the symmetry is axial (sometimes also called 'cylindrical') the electrodes are symmetric with respect to rotations about the z axis and the coordinates used to define a position on an electrode are (r,z), where r is the perpendicular distance from the axis and z is the distance along the axis (and where a third coordinate, the azimuthal angle phi around the axis is not relevant). When the symmetry is planar the system is assumed to extend to plus and minus infinity in the y direction and so the coordinates are (x,z).. In either case their are only two active coordinates -hence the description 2-dimensional- and electrodes are represented by their cross-section in the plane (r,z) or (x,z). The program allows the cross-section through an electrode to be either a straight line or an arc of a circle.

A CPO2D systems with axial symmetry can also be used for non-meridional rays (that is, rays that have non-zero angular momentum about the axis and so cannot pass through the axis). The option cannot be combined with the options for entering total energy, or using variable mass, time dependence, planar symmetry, use saved trajectories or focus iterations. Examples are given in test2d22.dat and test2d23.dat.

In CPO3D the coordinates are (x,y,z) and the electrodes can be of several types, including the following:

(a) whole spheres or sections of spheres (bounded by two parallel circles)

(b) whole cylinders, cones or discs

(c) triangles that are part of a spherical or conical surface

(d) rectangles that are part of a cylindrical surface

(e) triangles or rectangles that are part of a flat surface.

Most electrodes are catered for by (a) and (b), while (c), (d) and (e) allow the user to construct arbitrary shapes. The electrodes listed in (a) and (b) can also be stretched. Detailed information on the electrodes can be found in note on 2D eletrode specification and note on 3D electrode types.

In more detail, in CPO3D the electrodes can be of the following types:

(4) disc electrode

(5) evenly-divided disc electrode

(7) rectangle that is on a cylindrical surface

(9) triangle that lies on a spherical surface

(10) triangle that lies on a cylindrical surface

(12) triangle that lies on a conical surface

3 further types of interest to the more experienced user are:

(14) end conical or cylindrical triangle

(15) end disc triangle

Other shapes of electrodes can also be created.

After all the segments have been established, the charges on them are evaluated. More precisely, the unit charges are evaluated, which are the charges on the segments when each of the applied voltages is put equal to unity in turn while the remaining voltages are zero. These sets of charges are stored (in the binary processed data file) and used later to synthesise charges for any set of applied voltages.

The synthesis is carried out as follows. Suppose that the applied potentials Vi is unity while all the other potentials Vj, where j = 1 to (i-1) or (i+1) to m, are zero, and suppose that the resulting charges on the segments are then qni, where n = 1 to N. The charges qni are the unit charges. Since the equations that connect charges and potentials are linear equations, the charge qn on segment n is given by

for a general set of potentials Vi.

The electrodes are treated by the program as the sites of infinitesimally thin sheets of charge (see description of surface charge method). The electrodes themselves therefore cease to exist, as far as the program is concerned. The charges are those that would exist on the surfaces of the electrodes. For a thin metal electrode, the charges reside on both sides of the electrode, but these charges can be added together and treated as a single sheet of charge, as far as the calculation of potentials is concerned (if the electrode is thin enough).

Care must be taken if there are rays that lie outside a set of electrodes, because at long distances D from all the electrodes the magnitude of the potential is proportional to Q/D, where Q is the total charge on all the electrodes. The potential in the outer region is therefore not constant. In regions near but outside the electrodes there is a similar fall-off in the potential. Therefore if rays need to be traced in such regions it is necessary to enclose these regions with extra electrodes at the appropriate voltages.

The program allows the voltages applied to an electrode to vary linearly with distance, but in the z direction only.

There are two important restrictions on the electrodes, for CPO2D and CPO3D:

(a) Electrodes at different potentials MUST NOT be allowed to touch. There is an increased danger of electrodes touching when reflection symmetries are used (see the next section).

(b) When there are planes of reflection symmetry electrodes MUST NOT lie in or across those planes (except for electrode geometries of types (a) and (b) in CPO3D, which are dealt with automatically by the program).

If the program crashes without an explanation being given, or if some of the results are obviously wrong, it is usually because one of these two conditions has been violated. The program tries to detect and prevent these conditions being violated, but it has not been possible to build in a full set of tests in the program to trap this type of error on all occasions.

(proceed to next section)