Chapter Seven - Examples of Applications

Abstract

In this section, the general approach developed in Sections 1 through 6 is illustrated by some specific problems. A complete set of exact solutions with additive and multiplicative separation of variables is used to validate the axisymmetric geometrized models. It is shown that the remarking transformation of the transversal coordinate significantly affects the accuracy of the approximate solution. A comparison between the first approximation of the geometrized theory of tubular beams and the traditional asymptotic theory is performed. A formal coincidence of the relation on the stream tube within the framework of the geometrized model with the equation of the asymptotic theory allows us to indicate the analytical solutions of the latter equation, corresponding to the exact solutions of the beam equations. The recommendations of the theory of antiparaxial expansions, which are given in this section, refer to the construction of a more precise model of the near-cathode region to be used in trajectory analysis software. This model is based on solving the boundary-value problem in the near-cathode lens – the domain, the longitudinal size of which is comparable with the curvature radius of the cathode or with the cathode-anode distance. It is shown that using a non-Pierce value for the angle of zero-potential forming electrode in order to optimize the characteristics of a high-compression electron beam, or a random choice of the expansion gap shape, brings the numerical model beyond the hydrodynamic description. In this case, the error may exceed by the order the accepted level for the beams with the linear compression about 20. A theory is constructed and the calculation examples are given for (1) a multibeam multi-layer electron-optical system of the relativistic injector with a triode-type potential distribution and (2) a single-layer electron-optical system with the non-relativistic beam traveling in a guiding magnetic field. The multibeam interaction is described using the model of an effective macro-beam whose parameters are equivalent to the set of micro-jets. The evolution of a micro-beam with curved axis and elliptical cross section in the field of a macro-flow is described within the framework of 3D paraxial theory. The macro-beam consists of either a single tubular layer or multiple layers with the near-axis and near-surface asymptotics, which should be analytically coupled on the layers' boundary. The geometrized theory is applied to construct a combined model of the magnetron-injection gun for plasma-beam microwave devices. In contrast to the electron gun used in gyrotrons, in this case the cathode has a wide emitting belt. The model allows design of a cathode with a technologically realizable conical shape, as well as control over the emission current density. The model's algorithm may also be used to design a classical variant of magnetron-injection guns. The geometrized theory of higher approximations is used to study the problem of landing the ribbon beam onto the collector. It is shown that, for orthogonal landing, the collector's surface cannot simultaneously (1) have a beforehand-prescribed shape, (2) be equipotential, and (3) operate in the mode of a uniformly distributed heat load.