By D.W.O. Heddle

This e-book allows readers to layout lens platforms having worthwhile features. The textual content covers the fundamental thought of the movement of charged debris in electrostatic fields and describes numerous tools for the calculation of the capability and box distribution for varied electrode geometries. tools, the Bessel functionality growth procedure which was once constructed by way of the writer and his scholars and the nine-point implementation of the finite distinction strategy, are given precise emphasis simply because they're fairly applicable for implementation via the green person. different equipment are mentioned in much less element and reference is made to web pages from which demonstration courses for those equipment might be received.

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**Example text**

The abscissa is the aspect ratio, X/Y , which allows values for a wide range to be shown compactly. 1) is ∂ 2V 1 ∂V ∂ 2V = 0. + + 2 2 ∂z ∂r r ∂r Consider a rectangular array of points separated by distances h with the polar axis forming the lower boundary. 19b) with similar expressions for the two other neighbouring points. Taking appropriate sums and differences and ignoring terms of fourth order and above we can write The determination of the axial potential 39 ∂ 2V ∂z2 ∂ 2V V (z, r + h) + V (z, r − h) − 2V (z, r) = h2 2 ∂r h2 ∂V h V (z, r + h) − V (z, r − h) = .

There are commercial programs available. 6, but they all use the five point method which as we have seen has limitations which may be significant. 1) V (z, r) = (Ak ekz + Bk e−kz )J0 (kr) + V0 . 1, we choose to treat positive and negative values of kz explicitly. 17 shows the geometry of a lens formed by two coaxial cylinders of equal diameter, D, separated by an axial distance, g, and having potentials V1 and V2 . We show the walls of the cylinder to be of finite thickness as we shall later use a result from a finite difference calculation to resolve a problem inherent in this method.

Such an approach has a long history, but with the amount of computing power readily available nowadays, there is little point in using this sort of method though we give a brief account in the next section. Results of much greater accuracy can be obtained by approximations based on the laws of electrostatics. In later sections we consider three such approaches: the direct application of Coulomb’s law, numerical solutions of Laplace’s equation on discrete networks of points and the analytical solution of Laplace’s equation subject to boundary conditions which closely approximate the true situation.

### Electrostatic Lens Systems by D.W.O. Heddle

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