Shear-driven instabilities of annular relativistic electron beams in vacuum

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Abstract
The study of instabilities driven by azimuthal shear (diocotron) and axial shear is extended to annular relativistic beams. The analysis is done nonrelativistically in the beam frame where the instability is assumed to be electrostatic, and the dispersion relation is then transformed back to the laboratory frame where magnetic perturbations are non‐negligible. This requires keeping finite axial wavenumber kz in the analysis. The axial and azimuthal shears are related through the self‐consistent equilibrium, assuming emission from an equipotential cathode. Axial shear destabilizes the diocotron modes at higher azimuthal wavenumbers l. It also produces a set of modes including l=1 modes, that are unstable as the result of wave–particle interactions. The annular configuration introduces an important second pole in the differential equation.

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