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A cylinder of electrons infinitely long and infinitesimally thin in the radial direction is used to show the equivalence of the diochotron and negative mass instabilities. The electrons have a unique relativistic velocity in the azimuthal direction. The external magnetic field has a radial gradient, and a criterion for stability is derived that places a lower limit on the field gradient. This criterion shows that the instability can occur at energies below the transition energy of the magnetic field.
Neil et al. (1965) studied this question.