Solution of regular beam equations in arbitrary emission conditions on a curvilinear surface

Abstract

The regular beam equations are solved analytically for the case of emission from an arbitrary surface in conditions of total space charge (ρ-mode) and in a given external magnetic field H ≠ (§2) for temperature-limited emission (T-mode), in an external magnetic field H (§3); and for emission with nonzero initial velocity (§4). The emitter is taken as the coordinate surface x1=0 in an orthogonal system x1 (i = =1,2,3), while the current density J and field ɛ on it are given functions j(x2, x3), ɛ (x2, x3. The solution is written as series in (x1)α with coefficients dependent on x2, x3, determined from recurrence relations. For emission in the ρ-mode and H ≠ 0, α=1/3; for temperature-limited emission, α=1/2; with nonzero initial velocity, α=1. The results are extended to the case of a beam in the presence of a moving background of uniform density (5).