Nullification of Space-Charge Effects in a Converging Electron Beam by a Magnetic Field

This paper presents the conditions necessary for maintaining a uniformly converging conical electron beam in the presence of space charge. It is an extension of the Brillouin focusing condition to conical flow, requiring a converging rather than a uniform magnetic field. In this type of electron flow, the diverging effects of space charge are balanced against magnetic reaction forces for reasonably small cone angles of convergence. Though the balance of forces is exact only for infinitesimal angles, it is reasonably accurate for cones of half-angle as great as 10 degrees. The minimum beam size will be limited only by the effects of thermal velocities, by gun aberrations, and by the magnetic field obtainable.