Instability of hollow beams

This paper gives a linearized theory for the breakup of magnetically focused hollow beams of electrons. A zero-thickness developed beam is assumed. Growing waves are found to be possible both at zero frequency and at finite frequencies. When there are electrodes close to the beam inside and out, the waves grow more rapidly as the electrodes are moved further from the beam. If the current is increased and the magnetic field is increased just enough to keep the beam together, the rate of growth is increased. The theory predicts a greater rate of growth for a higher number n of cycles of variation around the circumference of the beam, but for actual beams of finite thickness the theory is inaccurate for large values of n. In a simple case, frequency becomes important only when the wavelength of waves along the beam becomes comparable with the wavelength measured around the circumference of the beam. Increasing waves are also found in a zero-thickness beam in crossed electric and magnetic fields, as in a magnetron amplifier or carcinotron, but not at zero frequency.