SIMION’s Refine function calculates the potentials of all non-electrode points in some volume when given some “boundary condition”, which is typically defined by voltages on electrode points but sometimes the location of mirror planes or PA edges too. This Refine is performed using an optimized numerical solution to the so-called Laplace Equation. Even if you’re not familiar with the Laplace equation, it’s important to note that this equation, upon which Refine is based, provides valid solutions only if certain conditions are met. The First Uniqueness Theorem given below is one sufficient condition, and the most typical one:

First Uniqueness Theorem: “The solution to the Laplace Equation in some volume is uniquely determined if [the voltage] is specified on the boundary surface.” [so-called “Dirichlet” boundary conditions] (Griffiths, 1999)

This theorem implies that SIMION’s calculated non-electrode point potentials are valid for any point inside some “closed” surface of electrode points By “closed” we mean that if you fill the boundary with water, it won’t leak.

The First Uniqueness Theorem is not the only condition that can ensure correctness. The Laplace equation still has a unique solution if part of this boundary is instead defined by “Neumann” boundary conditions, in which the normal derivative of potential on the boundary (not the potential itself) is specified. Nuemann boundary conditions are defined as 0 on mirror planes (SIMION mirror X/Y/Z). PA edges of non-electrode points, even if non-mirrored, also define some type of Neumann-like boundary condition of 0 (see SIMION User Group discussion 346 below). Therefore, every SIMION PA has a closed boundary consisting of some combination of Dirichlet and Neumann-like boundary conditions, and SIMION will go on to calculate it (see p. 6-3 of the SIMION 8.0 or 7.0 manuals), but whether these Neumann or Neumann-like boundary conditions assumed here are valid approximations of your system is something you’ll need to determine. To avoid ambiguities sometimes present in Neumann boundaries, it is often suggested to avoid them by enclosing your model in a “ground can” whenever possible (p. 6-3). Note that neither the potential nor the shape of the so-called “ground can” is important. In some cases it may be sufficient to put enough empty space around your system so that the Neumann-like conditions become more valid.

Figure 1: 2V circular electrode enclosed entirely with Neumann-like boundaries on the PA edges. All potentials are calculated as 2V, which is probably not desired.

Figure 2: 2V circular electrode enclosed with 1V circular Dirichlet boundary. All potentials are calculate between 1-2V as expected. Potentials in the ignored outside area are 1V due to Neumann-like boundaries on the PA edges.

For illustration, Figure 1 on the right shows a worst-case scenario. We have a centered 2V electrode with no other electrode points. That is, the external boundary on the PA edges is entirely Neumann-like. The Laplace Equation (and SIMION) calculates 2V on all non-electrode points. This is probably not what you intended. You probably expected potentials to decrease to zero by the 1/r relationship of Coulomb’s Law. In general, the Laplace equation can be thought of as an averaging process, and the potentials calculated on the non-electrode points will always be somewhere inside the interval defined by the minimum and maximum potentials on the boundary points (which is [2V,2V] here). Figure 2 shows the correction of this, where the electrode is fully enclosed by a 1V circular electrode (defining a Dirichlet boundary), and the potentials are as we would expect.

Figure 3 shows a milder but still potentially problematic scenario. This is two parallel plates offset to one corner inside a PA whose edges are entirely non-electrode points. Here, the parallel plates are Dirichlet boundaries and the PA edges are Neumann-like boundaries. SIMION does calculate this, and the calculated potentials between the plates and even around the openings are reasonably good, so it may be fine if particles are flown through the plates. Outside the plates, the results are more questionable because Neumann-like boundaries are assumed, but they may be ok along the beam line. In general, the contour lines tend to be more perpendicular to the PA edges (Neumann-like condition) than probably exist in your actual system without these Neumann boundaries. Since the plates are offset with repect to the PA edges, one sees a clear asymmetry in the equipotential contour lines–this shows that the amount of empty space around your system can affect the contours.

Figure 3: Parallel plates offset to one corner in a PA enclosed entirely with Nuemann-like boundaries on the PA edges. Equipotential contour lines are shown.

SIMION’s einzel\einzel.pa# and magnet\mag90.pa# examples demonstrate partly open systems (Neumann-like boundaries). These are examples where things like Figure 3 can be ok.

See also SIMION User Group discussion 346 comments by Bud and others.