Stability Of Barrier Buckets With Short Or Zero Barrier Separations

A barrier bucket with very short or zero rf-barrier separation (relative to the barrier widths) has its synchrotron tune decreasing from a very large value towards the bucket boundary. As a result, chaotic region may form near the bucket center and extends outward under increasing modulation of rf voltage and/or rf phase. Application is made to those barrier buckets used in momentum mining at the Fermilab Recycler Ring.