Calculating the parameters of an electron beam that drifts across a strong homogeneous magnetic field

Abstract

Electron drift in specified fields has been examined in [1] and, as applied to a magnetron, in [2–4] with the averaging method. In [1,2], a first- and in [3,4] in a second-order approximation of the small parameter ν∼ η) E/Ω2L was used. Here and below, E and H=(c/η)Ω are the field strengths, L is the characteristic dimension of the field heterogeneity, η is the charge-mass ratio of an electron (η>0), and c is the velocity of light. An attempt to construct similar approximations for a drifting electron beam with allowance for the space-charge field, within the framework of the averaging method, involves considerable mathematical difficulties. This paper describes an attempt to solve the latter problem for a stationary monoenergetic beam that drifts under the influence of a plane electric field with potential ϕ(x,y) across a strong homogeneous magnetic field Hz ≡ H=const. Solutions are constructed by the method of successive approximations, in powers of the parameter ɛ=h/L, where h is the Larmor electron radius for narrow beams with a width on the order of 2h.