A Simulation Study of the Sawtooth Behavior

The fact that bunch lengthening sometimes occurs with a sawtooth behavior has received some attention recently.[ 1-6] Various possible mechanisms which might explain the sawtooth behavior have been suggested. In particular, in Ref.6, Baartman and D'Yachkov proposed a mechanism that involves an interplay of synchrotron oscillation, potential well distortion (which at some moment of bunch oscillation creates a double-humped longitudinal beam distribution), quantum diffusion, and radiation damping and performed computer simulations to demonstrate this mechanism. Although this BD mechanism is not the only possible explanation of a sawtooth behavior, this note is an attempt to follow up on this trend of thought by yet another simulation study, and to draw a few tentative conclusions from this study. The collective effect is presumably caused by some wake function W(z). We assume the wake function is short-ranged and only single-turn wake needs to be considered. To enhance the BD mechanism in our simulation, a wake function model has been chosen which (a) has a range approximately equal to a few times the natural bunch length {sigma}{sub z}, and (b) flips sign once in this range (to make it easier to produce a second hump in potential well). In fact, we have chosenmore » a wake function W(z) = W{sub 0} when 0 > {sigma}{sub z}, the quantum diffusion (needed to transport particles from one hump to the other) is too slow to give a clear sawtooth behavior. One also expects that when z{sub 0} << {sigma}{sub z}, the diffusion is too fast and only a chaotic behavior appears. In the simulation, parameters W{sub 0} and z{sub 0} are varied.« less