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# Download Advances in Electronics and Electron Physics, Vol. 76 by Peter W. Hawkes (Ed.) PDF By Peter W. Hawkes (Ed.)

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Extra info for Advances in Electronics and Electron Physics, Vol. 76

Sample text

YAVOR Another numerical technique to solve the field problems is the method of finite elements. Its application turns out to be rational to calculate systems with a complex electrode geometry and also when the finite magnetic permeability in magnetic lenses is to be taken into account. The method is based on the postulation of a variational extremal principle, valid for the whole area. The solution minimizes a certain functional, which is defined as an integral of unknown functions over the whole area.

24 L. A. BARANOVA A N D S. YA. YAVOR Eqs. (40) provide better accuracy in finding trajectories, especially when the fields are defined by numerical methods. B. The Paraxial Characteristics of Electron Lenses The general solution of each paraxial equation is a linear combination of two partial independent solutions. We shall select the pairs of independent solution x u , x D , and y a , y,, in such a way that they will satisfy the initial conditions in the plane z = zo Note that x a ,x,, y,, and y, are not paraxial trajectories per se, because they satisfy the nonparaxial initial conditions and, moreover, x , and y, are dimensionless values.

A symmetry of potential distribution can also facilitate the solution of the problem. Although the integral equations describing electrostatic potentials have long been known, their wide use started only with the introduction of fastoperating computers, which can provide sufficiently high accuracy and short computation time. The development of large-memory and high-speed computers and computational mathematics has stimulated a widespread of numerical methods for field solution. We shall discuss briefly two of them-the finite-difference method and the finite-element method.