Preservation of the distribution of beam particles with respect to longitudinal oscillation amplitude in a 3 to 1 bunch merge

In the work presented here we show in detail how the 3 to 1 bunch merge can be set up so that it preserves the distribution of beam particles with respect to longitudinal oscillation amplitude. This is an elaboration based on the work presented in reference 1. The desired merge is achieved by slowly reducing the area inside the three-lobed separatrix that contains the beam while keeping the areas of the three lobes equal to one another. A sequence of RF voltages that does the job is found by employing a numerical integration routine and a search algorithm. The integration routine calculates the areas of the three lobes for given voltages. This is used by the search algorithm to nd voltages that give three lobes of equal area. The resulting voltages are listed in Section 12 in order of decreasing lobe area. These may be multiplied by a common scaling factor to make the scheme work for any given longitudinal emittance in any given machine. The analytic basis of the scheme is presented in Sections 2 through 13. (These sections may be treated as appendices.) Demonstration of the merge is done by simulation. The results are shown in Figures 1 through 39 and are discussed in Section 1. References [1] through [10] and those cited therein provide a history of bunch merging (and splitting) relevant to the present work. Refs. [1] and [2] record the early work done at CERN. Refs. [3] through [10] record the setup and study of the 2 to 1 and 3 to 1 merges used in Booster and AGS. In particular, Refs. [9] and [10] record the recent work of K. Zeno who proposed and developed the 3 to 1 bunch merge in Booster. It is this merge that is studied in the present work.