6 It is discovered that the sixth order 6σ=720° (or 6:2) resonance is manifested for high intensity 7 beams of linear accelerators through the space charge potential when the depressed phase 8 advance per cell σ is close to and below 120° but no resonance effect is observed for σ above 9 120°. Simulation studies show a clear emittance growth by this resonance and a characteristic 10 six-fold resonance structure in phase space. To verify that this is a resonance, a frequency 11 analysis was

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... cted and a study was performed of crossing the resonance from above and 12 from below the resonance. Canonical perturbation is carried out to show that this resonance 13 arises through perturbation of strong 2σ=360° (2:1) and 4σ=360° (4:1) space charge resonances. 14 Simulations also show that the space charge 6σ=360° (or 6:1) resonance is very weak. 15 16 PACS: 29.27.Bd, 41.75.-i 17