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Analytical Model of Halo Formation

An analytical model for halo formation has been constructed based on a beam in a focusing channel which is “breathing” due to a mismatch with the channel. For a 2-D KV beam, an integral of motion can be obtained assuming the dominance of the parametric resonance (breathing mode frequency is twice the frequency of individual particles within the beam). These results correspond to the “peanut diagram” in particle phase space in all respects. The model is then extended to other 2-D distributions, as well as 3-D distributions involving both longitudinal and transverse breathing modes. Numerical simulations are then used to determine the behavior of the longitudinal and transverse halos which occur, and their dependence on the initial phase space distribution, the bunch charge and shape, and the amplitude of the mismatch.