On the beam break-up instability in storage rings

Numerous coherent instabilities of bunched beams in storage rings can be associated with the interaction of particles with the low Q-value elements of the vacuum chamber. Traditionally this scope of problems is solved by the calculation of the eigenfrequency spectra for the linearized system of Vlasov's equation. In this report, only the main features of fast transverse coherent single-turn instability and the main stabilizing factors for these oscillations are briefly discussed. For the sake of simplicity, it is assumed that only vertical coherent oscillations are excited in beam. The formula for the unperturbed oscillations of particles is described. The mathematical procedure is explained. The damping of considered instability is discussed. By the detuning of head-on and tail-on particles, the oscillations can be damped. The action of beam cooling and Landau damping on fast coherent oscillations is considered. The BNS damping is discussed. The simple calculation of the BNS damping can be done. It sufficiently uses the resonant nature of instability, and assumes suppression by the introduction of the deviation of particle oscillation frequency along a bunch. (K.I.)