Computer simulation of an electrostatic spherical analyzer used as an energy spectrograph

Exit positions and velocities of charged particles passing through an electrostatic spherical analyzer were generated by a computer program which utilizes transformations to the trajectory plane of the particle to obtain an exact solution in the central force field of the analyzer. Entrance and exit planes were treated as a potential jump to zero potential, thus simulating grounded wire meshes at the entrance and exit. It was found that particle input energies could be determined as a function of exit position. A wide variety of designs were tried, varying plate radii, plate voltages, and incoming collimation of the particles. A power-law relationship between exit radius and input energy was found for designs in which the mean radial position between the plates corresponded to zero potential. However, the total input energy spread ΔE of the particles passed by an analyzer could never be made to approach twice the mean input energy E of the particles passed. Since a parabolic analyzer can be made to have a ΔE/E=1.92, we conclude that the parabolic analyzer makes a better spectrograph.