Particle-core analysis of mismatched beams in a periodic focusing channel

A method is derived for applying particle-core analysis to mismatched beams in a periodic focusing channel. By carefully choosing the parameters to yield a favorable core frequency, a Poincaré surface of section plot is obtained. The plots for a periodic solenoid channel exhibit a striking resemblance to those in continuous focusing cases, while those for an alternating-gradient channel exhibit a strong chaoticity which is not seen in the corresponding continuous focusing situation. Only the breathing mode oscillation of a core has been considered in solenoid focusing cases. On the other hand, the quadrupole mode oscillation of a core has also been considered in the case of an alternating gradient channel. We examine the effects of both modes on the test particle stability, and find that the quadrupole mode oscillation can also drive the particle-core resonance and cause beam halo formation. The halo extent is also examined. The maximum halo width is found to be about twice as large as the maximum core width in breathing oscillation cases in both periodic solenoid and alternating-gradient channels. In quadrupole oscillation cases, the halo width exhibits density dependence, and ranges from 1.2 to 2 times the maximum core width. These results give us a practical criterion to determine the bore radius in designing high-intensity accelerators.