Saw-tooth instability studies at the Stanford Linear Collider damping rings

Saw-tooth instability occurs during high current operation in the Stanford Linear Collider (SLC) damping rings. This instability is single bunch and it can be cast as a longitudinal microwave instability. It is caused by the beam interaction with short range wakefields in the ring vacuum chamber. The saw-tooth instability manifests itself in the periodic blowup in quadrupole or higher moments in the longitudinal beam distribution.; Most of our instability studies have been experimental. Since the measurements of coherent particle motion within a short ultrarelativistic beam are largely unconventional we had to develop some original diagnostics. These includes, for example, the down-conversion of the high frequency (10 GHz) broad-band beam position monitor (BPM) signals. We have also employed the state-of the art Hamamatsu streak camera that is capable of resolving the longitudinal beam distribution with sub-picosecond accuracy.; As a result of our streak camera experiments we have quantitatively described the phase space of unstable bunches. We have found the radial structure of the instability mode and established that it only displaces a few percent of the beam particles. In another series of experiments we have correlated the instability signals from the beams before the extraction from the damping rings with their trajectories in the linac downstream. This showed that the instability results in a significant transverse beam jitter in the linac which compromises the damping ring performance as an injector. In addition, we have studied the instability behavior under the broad range of stored beam parameters using both passive and driven excitation. These measurements revealed unexpected beam behavior significantly above the instability threshold. Finally we performed several low current experiments to estimate the damping ring vacuum chamber impedance.; We also present some analytical results regarding the instability and compare them to the observations. In particular, these include the explanation of unequal sidebands in the spectrum of the BPM signal coming from unstable bunches. In addition we have obtained several results regarding the instability onset criteria and proposed a new method of estimating the instability threshold based on the steady-state solution of the Fokker-Planck equation.