Relation between Diochotron and Negative Mass Instabilities

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.