Rotational Instability of the Axisymmetric Trivelpiece–Gould Modes in the Inhomogeneous Electron Beam

Instability of axisymmetric electrostatic oscillations in an inhomogeneous beam of electrons confined by an axial magnetic field is studied within the framework of an ideal hydrodynamic model. The equation of perturbation of the beam electrostatic potential is derived in the linear approximation. It is demonstrated that the effects of differential beam rotation in self inhomogeneous electric field destabilize the Trivelpiece–Gould modes. The spatial structure of the unstable eigenmodes is calculated analytically and numerically for model profiles of the beam density, and their spectrum is analyzed.