Sixth-Order Resonance of High-Intensity Linear Accelerators

Abstract

It is shown that the sixth-order $6\ensuremath{\sigma}=720\ifmmode^\circ\else\textdegree\fi{}$ (or $6\ensuremath{\mathbin:}2$) resonance is manifested for high-intensity beams of linear accelerators through the space charge potential when the depressed phase advance per cell $\ensuremath{\sigma}$ is close to and below 120\ifmmode^\circ\else\textdegree\fi{} but no resonance effect is observed for $\ensuremath{\sigma}$ above 120\ifmmode^\circ\else\textdegree\fi{}. 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 $2\ensuremath{\sigma}=360\ifmmode^\circ\else\textdegree\fi{}$ ($2\ensuremath{\mathbin:}1$) and $4\ensuremath{\sigma}=360\ifmmode^\circ\else\textdegree\fi{}$ ($4\ensuremath{\mathbin:}1$) space charge resonances. Simulations also show that the space charge $6\ensuremath{\sigma}=360\ifmmode^\circ\else\textdegree\fi{}$ (or $6\ensuremath{\mathbin:}1$) resonance is very weak.