The Negative Mass Instability in High Current Modified Betatrons at Low Energies

We have developed a new linearized, rigid-disk model of negative mass instabilities in high current betatrons. Taking the beam and accelerator cavity cross sections to be rectangular permits the electromagnetic fields to be evaluated exactly in toroidal geometry. Growth rates from the model agree well with results of three-dimensional numerical simulations for beams and cavities with rectangular or cylindrical cross sections. Generally, negative mass instability growth rates are greatest for beam energies within a factor of two of the so-called transition energy and in that energy regime scale inversely with the square root of the toroidal magnetic field strength. At much higher energies, growth rates are nearly independent of toroidal field strength. Growth rates increase with beam current and toroidal mode number, but the scaling laws tend to be complicated by competition between capacitive and inductive components of the toroidal cavity fields.