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

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 ) E/²L was used. Here and below, E and H=(c/) are the field strengths, L is the characteristic dimension of the field heterogeneity, is the charge-mass ratio of an electron (>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 (x,y) across a strong homogeneous magnetic field H_z H=const. Solutions are constructed by the method of successive approximations, in powers of the parameter =h/L, where h is the Larmor electron radius for narrow beams with a width on the order of 2h.I thank A. N. Ievlevu for assistance in the computational and graphical work, V. Ya. Kislov for a discussion of the results, and L. A. Vainshtein for suggesting the problem examined in §3 and for critical comments.