Multi-Stream Instability of a Single Long Electron Bunch in a Storage Ring

A multi-stream instability is observed experimentally in a longitudinally expanding electron beam in a storage ring. The instability is observed when the beam expands such that its length is several times the circumference of the ring, and portions of the beam overlap. While portions of the beam overlap in physical space, due to the nature of the expansion process, the portions form multiple streams and remain separate in velocity space. The streams become unstable as their number increases and their separation in velocity decreases. An analytical theory predicts the onset of the instability, consistent with simulations and measurements, over a wide range of peak line-charge densities and bunch lengths. This work extends previous calculations to include the dynamic non-linear elongation of the bunch, with a given initial length, and defines an onset criterion for the filament velocity separation.