Anomalous electron bunch length due to fast instability

Abstract

A theory of bunch lengthening in electron storage rings is proposed. The equilibrium bunch length is determined by a balancing of beam induced fields tending to cause coherent instability and bunch frequency spread tending to prevent the buildup of coherent modes (Landau damping). The mechanism is related to the existence of a single bunch, single mode “fast” longitudinal instability. For such an instability to develop, the growth rate must be sufficiently faster than the synchrotron frequency. It is suggested that the source of the anomalous electron bunch length is a broad band resistive impedance at high frequency which could be produced by the combination of many closely spaced high frequency resonators, such as vacuum flanges. It follows that bunch lengthening and the observed higher mode heating are directly correlated. A scaling law for the equilibrium bunch length is derived. The scaling parameter is g=( hV cos ϕ s)/ I, with h, V and ϕ s the usual rf parameters and I the average beam current. Data taken in SPEAR I and II, data in which g extends in value by more than three orders of magnitude, can be fit with an appropriate choice of coupling impedance. The impedance functions for SPEAR I and II are taken to be the same, a reflection of the fact that the high frequency sources are chamber discontinuities rather than structures connected with the rf systems. A parameter search leads to an impedance characterized by a central frequency ∼5 GHz, a width (fwhm) ∼1.8 GHz and a peak value ∼0.2 MΩ . In addition to the agreement obtained between predicted and experimentally observed bunch lengths, the measured higher mode resistance (i.e., heat dissipated) in SPEAR is also found to be in agreement with the theoretical predictions.