Multipole Expansion of the Stationary Electromagnetic Field

Abstract

Constant currents form stationary magnetic fields. In the static limit, the field-producing charges are at rest and the currents are zero. In this case, the magnetic field vanishes. Therefore, within the frame of our definition, static fields are purely electrostatic. We rarely encounter time-dependent fields in charged-particle optics because in most cases the reciprocal transition time of the particle through the system is significantly smaller than the maximum frequencies of the fields. Hence, we can consider these fields as stationary with a sufficient degree of accuracy. In most cases, charged-particle optics is concerned with the propagation of a confined ensemble of particles through a system. Examples are the electron microscope, accelerators, spectrometers, and beam-guiding systems. For these systems, it is advantageous to choose the central trajectory as the z-axis of an orthogonal coordinate system, as schematically illustrated in Fig. 3.1. In order that we can develop the curved sections into a plane, the torsion of the curved axis must be zero. In this case, all sections, which contain the centers of curvature of the optic axis, are plane sections.Charged particles must propagate in vacuo. The beam-guiding electro magnetic fields are formed by the voltages applied to the electrodes and the currents within the coils of the magnets. The spatial distribution of the elec tric and magnetic potentials is determined by their boundary values on the surfaces of the electrodes and pole pieces, respectively. The task of electron optics is an inverse problem because we must determine the geometry of the electrodes and pole pieces, which will provide the required imaging or propa gation. Unfortunately, we cannot directly solve this delicate problem.