Geometrized theory of relativistic electron beams

Abstract

The geometrized theory of dense electron beams is the newest part of contemporary corpuscular optics which includes the problems of non-paraxial relativistic beams synthesis, the questions of approximate solution constmction and geometrized beam equations numerical integration.If we try to calculate the flow with some a priori needed parameters than the using of coordinate system connected with the flow geometry is natural. We may demand that the coordinate lines coincide with the particle trajectories or one of the coordinate surfaces family describes the flow tubes. In first case the velocity has only one component, in second case two tangential to the flow tube velocity components don't equal to zero. Because the flow geometry a priori is unknown we must write down the beam equations in tensor form for arbitrary non-orthogonal coordinate system x' with metric tensor &k, supplement the connection equations for Cartesian and curvilinear coordinates and the Euclidean conditions for space. These conditions are the second order non-hear partial differential equations for gik.It is clean that the properties of xi-system and the physical parameters of medium become no more independent from each other. In limit we may talk about full reduction of physical problem (electron beam calculation) to the geometrical task of corresponding coordinate system determination (full geometrization). An analogous problem we find in classical variants of single field theory.The geometrized theory pre-hystory is the single-component space-charge-flow theory [l-4) which main idea is the probability to describe 3-space potential electrostatic beam by an ordinary differential equation. The results and mistakes of this theory are discussed in [ 5 ] .The first example of successive geometrized theory is [6] where the problem of syntliesis for non-paraxial non-relativis tic tubular electrostatic flow in orthogonal coordinates is interpreted. The higher order paraxial beam theory for thin tubular beams is constructed in 173.The general ideas of geometrized theory firstly developed for super-sonic viscous gas jets [8] are reformulated for dense electron beams in [9-121. It is interesting to point out that on this way we must go to the general relativity results: the Einstein's equations tend to the Euclidean conditions: for space when gravitating masses and fields tend to zero.The geometrized theory equations analysis shows that in general we can't include the trajectories as coordinate lines in orthogonal system. For space-charge-flow with axial symmetry the local system non-orthogonality near the emitting surface is connected with the fulfilling of space-charge-limited conditions under external magnetic field arbitrary orientation, In [ 12,131 for axially symmetric flows it is shown that the beam equations may be transformed to the expression on flow tube which has the form of ordinary differential equation for g22 with longitudinal x1 coordinate as argument and evolutional fwst order equations system. The last gives expressions for tramver...

Optimization of a ribbon diode with magnetic insulation for increasing the current density in a high-current relativistic electron beam

“…The situation with the scale difference is complicated by the fact that, for diodes with a strong magnetic field, the use of this model requires a small grid step near the cathode (d ζ 0 ) to satisfy the condition h d. Therefore, the solution of the problem using the model of a starting layer without a magnetic field can require considerable computing resources (operative, disk storage and computing time) to calculate the angular characteristics of the electrons beam with the required accuracy. To overcome these difficulties, special algorithms have been developed (see, for example, [6,7]) in which analytical expansions take into account factors such as the external magnetic field, surface curvature, and emission nonuniformity on the cathode surface. As a rule, these algorithms are designed for particular classes of problems and are not universal.…”