Stability of barrier buckets with zero RF-barrier separations

A barrier bucket with very small separation between the rf barriers (relative to the barrier widths) or even zero separation has its synchrotron tune decreasing rather slowly from a large value towards the boundary of the bucket. As a result, large area at the bucket edges can become unstable under the modulation of rf voltage and/or rf phase. In addition, chaotic regions may form near the bucket center and extend outward under increasing modulation. Application is made to those barrier buckets used in the process of momentum mining at the Fermilab Recycler Ring