Stability of Flat Bunches in the Recycler Barrier Bucket

We examine the stability of intense flat bunches in barrier buckets used in the Fermilab Recycler. We consider some common stationary distributions and show that they would be unstable against rigid dipole oscillations. We discuss the measurements which identify stable distributions. We also report on experimental studies on the impact of creating a local extremum of the incoherent frequency within the rf bucket. We considered two typical stationary distributions and found they were not adequate descriptions of the Recycler bunches. From the measured line density distribution we find (a) the tanh function is a good fit to the line density, and (b) the coherent frequency of the rigid dipole mode for this distribution is within the incoherent spread at nominal intensities. Stability diagrams when the beam couples to space charge and external impedances will be discussed elsewhere. Our initial experimental investigations indicate that longitudinal stability in the Recycler is, consistent with expectations, influenced by the ratio T2/(4T1) which determines the location of the extremum of the incoherent tune. The coherent tune depends strongly on the distribution in the bunch tails which is difficult to measure. Numerical studies using both a conventional tracking code and a Vlasov solver are in progress and should provide more insight into conditions that may lead to unstable behavior