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The ion beam focusing and acceleration in an axisymmetric periodic RF undulator structure is considered. There is suggested that RF field has no a synchronous wave and accelerating force is to be driven by a combination of two non-synchronous waves. The influence of non-synchronous harmonics on ion beam dynamics is studied by means of a smooth approximation. Choice and optimization of RF field harmonics are made to obtain maximal transmission coefficient. The result is verified by a numerical simulation. The comparison with a conventional RF linac, where a synchronous harmonic accelerates a beam and non-synchronous one is focusing a beam, is described. This comparison is suitable for demonstration of the capabilities of an undulator linac.
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