Stability of Thin Electron Layers

An instability that consists of a bunching transverse to the directed motion of an infinitely thin layer of electrons is investigated theoretically. The initial model is a plane sheet, infinite in extent, and neutralized by positive ions. This configuration is found to be unstable if all electrons travel with the same relativistic velocity. If the electrons are monoenergetic, but have a small angular spread in velocities, a criterion for stability is obtained that places a lower limit on the spread. This limit is directly proportional to the wavelength of the perturbation. The theory is also applied to a thin cylindrical sheet inside a perfectly conducting cylinder. The system is infinite in the axial direction, and the electrons move in the azimuthal direction. In the absence of axial velocity, the instability is present as in the plane sheet. Considering a spread in axial velocity leads to a stability criterion independent of perturbation wavelength. A numerical example, employing the proposed parameters of the Astron device, indicates that an angular spread of about 60° is necessary for stability.