Longitudinal Stability of Recycler Bunches; Part I: Thresholds for Loss of Landau Damping

We examine the stability of intense flat bunches in barrier buckets used in the Recycler. We consider some common stationary distributions and show that they would be unstable against rigid dipole oscillations. We then discuss an analytical model for the line density that best describes measured bunch profiles. We include space charge in this model to predict the bunch intensity at which Landau damping would be lost. The dependence of this threshold on the bunch length is studied and related to the results of an experimental study with shorter bunch lengths. The threshold for the microwave instability is estimated. These studies will be followed by more detailed numerical studies.