Analysis of the high frequency longitudinal instability of bunched beams using a computer model

Abstract

The effects of high frequency longitudinal forces on bunched beams are investigated using a computer model. These forces are thought to arise from the transfer of energy between the beam and various structures in the vacuum chamber, this coupling being characterized by a longitudinal impedance function. The simulation is performed with a passive cavity-like element. It is found that the instability can be generated if three conditions are fulfilled: (1) the impedance must be sufficiently large, (2) the induced field must have a fast wake, and (3) the frequency of the induced field must be high enough. In particular, it is shown that the coasting beam threshold criterion for the longitudinal impedance accurately describes the onset of instability, if local values along the bunch of energy spread and current are used. It is also found that the very fast initial growth rate is in good agreement with linear theory and that the coasting beam overshoot expression may be used as a rough guide of the limiting growth for unstable bunches. Concerning the wake field, it is shown how the instability tends to disappear as the fields persist longer. It is furthermore demonstrated that as the wavelength of the unstable mode is increased, initially unstable conditions begin to weaken and vanish. This, it should be emphasized, is primarily a result of the strong correlation between the unstable mode frequency and the time rate of attenuation of the induced fields. ISR parameters are used throughout and a correspondence between the microwave instability observed in the ISR bunches and the simulated instability is suggested.