Interesting Mathematical Sequence and Its Relation to the Threshold for Turbulent Bunch Lengthening

摘要

We may state that by considering the sequence, a/sub n+1/ = C/sup a/sub n//, some physical insight into the so-called turbulent bunch instability can be gained. When the constant C is too small, the sequence acts as if it is over-focused. That is, in going from term a/sub n/ to a/sub n+1/ the sequence heads toward the limiting solution alpha but overshoots the mark. Likewise, in a storage ring it takes a whole turn (or perhaps a synchrotron oscillation period) for the bunch to gain complete information about the shape and strength of the potential well. If there is a perturbation away from the limiting solution, the distribution will attempt an adjustment. If the ''focusing'' is too strong, the adjustment will be too large and will overshoot the equilibrium value. Analogous behavior takes place in, for example, a system of periodic focusing elements when the phase shift between elements exceeds pi . 4 figs.