Exact Solutions for the Electron Distribution in Plane Magnetrons. II. Parabolic Potential with a Minimum
Proceedings of the Royal Society A
Electron space charge and tangential current density distributions in crossed electric and magnetic fields are obtained for plane systems characterized by a parabolic potential distribution with a potential minimum in front of the cathode. Complete distribution of initial electron velocities in all three directions is taken into account; both medium and low magnetic fields are considered. This work completes some earlier calculations which have already been presented (Lindsay 1965). . I n t r o
... 1965). . I n t r o d u c t i o n In earlier publications the problem of electron velocity distribution in crossed electric and magnetic fields has been discussed both for the magnetic-field-limited regime of operation (Lindsay i960), when the magnetic field is relatively strong and for the space-charge-limited regime (Lindsay 1964), when the magnetic field is relatively weak (zero in the limit of a thermionic diode). Self-consistent solutions of the Poisson equation have been obtained for the first regime of operation (Lindsay 1962; Lindsay & Goodell 1965) , but not for the second. However, there are reasons to believe th a t in the second case the potential distribution does not depart greatly from the parabolic distribution (van Duzer & Whinnery 1961; Yankina 1959), and it is thus of interest to calculate the space charge and tangential current distributions for such systems, the normal or anode current distributions having been already calculated by others (see, for example, Ho & van Duzer 1968). These calculations will also serve to close a gap left by earlier work, where linear and parabolic potential distributions, the latter however without a potential minimum, have been con sidered (Lindsay 1965; to be referred to as I). 2 . P o t e n t ia l d i s t r i b u t i o n , p h a s e -space d e n s i t y AND THE EQUATIONS OF MOTION The geometry of the systems discussed in this paper is shown in figure 1 , where the magnetic field B = (0,0, B z) is pointing in the positive z direction and is thus parallel to the two plane-parallel electrodes of infinite extent. Let us assume th a t 31 [ 479 ] Vol. 315. A.