Sixth-Order Resonance of High-Intensity Linear Accelerators

It is shown that the sixth-order 6 sigma = 720 degrees (or 6:2) resonance is manifested for high-intensity beams of linear accelerators through the space charge potential when the depressed phase advance per cell sigma is close to and below 120 degrees but no resonance effect is observed for sigma above 120 degrees. Simulation studies show a clear emittance growth by this resonance and a characteristic sixfold resonance structure in phase space. To verify that this is a resonance, a frequency analysis was conducted and a study was performed of crossing the resonance from above and from below the resonance. Canonical perturbation is carried out to show that this resonance arises through perturbation of strong 2 sigma = 360 degrees (2:1) and 4 sigma = 360 degrees (4:1) space charge resonances. Simulations also show that the space charge 6 sigma = 360 degrees (or 6:1) resonance is very weak.