Periodic focusing of beams from partially shielded cathodes

Using both an analog computer and an approximate analytical method, the equation describing the beam shapes which result from a uniform pencil beam of electrons entering a periodic magnetic field has been studied. The investigation has been primarily directed toward beams emergent from ideal cathodes only partially shielded from the magnetic field. Cases of both small and large scalloping have been considered. Curves relating the magnetic field coefficient α and the space-charge coefficient β are presented. A previously proposed relationship between α and β for small scalloping is shown to be inaccurate for large scalloping. Stable solutions for the motion of an electron beam having no thermal velocities are obtained for values of α less than 0.66 and in addition they are also found for some values of α greater than 0.66.