Electron ray tracing by means of resistor network and digital computer

Abstract

The problem of calculating electron trajectories in an electrostatic field is solved in two steps. First, the potential distribution for a given electrode configuration is found by means of a high precision resistor network. The second step requires a series of numerical calculations using the above potential values to determine the electron trajectories. All mathematical operations have been programmed for an IBM 709 digital computer. Three different mathematical methods are presented and compared. To check their accuracy test problems involving simple electrode configurations, for which the trajectories can be calculated analytically, were used. The error of a trajectory is defined by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta/l</tex> , where δ is the maximum transverse deviation of a trajectory from its correct position and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> is the length of the trajectory. Using the most accurate of the three methods for an electron in a homogeneous electric field between parallel planes the average error was <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.3 \times 10^{-4}</tex> . With the electric field between concentric spheres the average error was <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4 \times 10^{-3}</tex> . The method of ray tracing described can be used for two-dimensional <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(x, y)</tex> and axially symmetric <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(r, z)</tex> fields. Space charge and magnetic fields can be taken into account. The time for calculating one trajectory over the whole length of the resistor network (50 units) is approximately 25 s.