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Outline of the Boundary Element Method The Boundary Element Method (BEM) is used by the program to find electrostatic potentials and fields for systems of conducting electrodes. This method is more accurate than the older, simpler methods used in most other programs. The first group to use the BEMto
obtain systematic lens data was the Electrostatic Cylinder Lenses I: Two element lenses, by F H Read, A Adams and J R Soto-Montiel, J.Phys.E (Sci.Instrum.) 4, 625-32 (1971), and the ‘standard’ data book Electrostatic Lenses, by Since then we have published over 50 papers on charged particle optics and instrumentation and have been involved in many consultancies. The principle of the BEM is simple. The method is based on the fact that in a system of conducting electrodes, real charges appear on the surfaces of the electrodes when potentials are applied to them. These surface charges are then the sources of all the potentials and fields in the system. In the BEM the electrodes are effectively replaced by these charges. Consider for example an isolated conducting sphere. If it has a non-zero potential then there is a uniformly distributed charge on its surface. The external potential is proportional to 1/r and the field is proportional to 1/r2. This potential and field are due to the surface charges on the sphere. If the sphere could be removed without disturbing the surface charges then the potential and field would remain unchanged. As stated above, the surface charges are the sources of all the potentials and fields in any electrostatic system. If all the surface charges are known then Coulomb’s Law can be used to calculate the potentials and fields in any part of the system. The only parts of the system that the User has to model are the surfaces of the electrodes. It is not necessary to create an artificial set of points in the space enclosed by the electrodes (as is needed in other methods). Nor is it necessary to enclose the system of electrodes. There are no restraints on the relative sizes of the electrodes. Cathodes can be of almost any shape and are easy to deal with.
The technique used to obtain the surface charges The first step is to subdivide the electrode surfaces into segments, which are either triangles or rectangles (depending on the shape of the electrode). The surface charge density on each segment is taken to be uniformly distributed over its surface, on the assumption that the segments have been chosen to be sufficiently small. The question that has to be answered is: What are the charges qi on the segments i (where i = 1 to n and n is the total number of segments) when the potentials Vi are applied to the segments (or more precisely, to the parent electrodes)? In the BEM the problem is looked at from the reverse direction, and the question is asked: What are the potentials Vi that would result from a given set of charges qi? To answer this we note that the potential Vj of segment j is related, essentially through Coulomb’s Law, to the charges qi on all the segments i, including the self‑potential due to the charge qj on the segment j itself. There is therefore an equation that connects Vj linearly to all the charges qi. There are n of these equations, for j = 1 to n. There are also n unknowns, qi, for i = 1 to n. Therefore the equations can be solved for a given set of Vj’s to give the segment charges qi. The electrostatic problem is then completely solved. The system of electrodes has become a set of segments each of which carries a known charge uniformly distributed over its surface. Potentials and fields can now be calculated anywhere in space, again essentially by using Coulomb’s Law. Laplace’s equation, the equation for the electrostatic field, has therefore been solved.
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