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At present the super-conducting option of linear accelerators is considered for low energy, and new type of RF cavities is considered for this purpose. However, together with electrodynamics problems we should solve the transverse stability problem, since in structures with external focusing elements the focusing period is longer, and in higher accelerating field the defocusing factor increases as well. In this paper we consider the transverse stability problem, taking into account the non-linear space charge problem. The fundamental mechanism of hallo creation in super-conducting linear accelerators is investigated to minimize the particle losses. The theoretical results are supported by numerical simulation.
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