Fourth order resonance of a high intensity linear accelerator

Abstract

It is discovered that, for a high intensity beam, the $4\ensuremath{\sigma}=360\ifmmode^\circ\else\textdegree\fi{}$ (or $4\ensuremath{\nu}=1$) resonance of a linear accelerator is manifested through the octupolar term of space charge potential when the depressed phase advance per cell $\ensuremath{\sigma}$ is close to and below 90\ifmmode^\circ\else\textdegree\fi{} but no resonance effect is observed when $\ensuremath{\sigma}$ is just above 90\ifmmode^\circ\else\textdegree\fi{}. To verify that this is a resonance, a frequency analysis is performed and a study of resonance crossing from above and from below the resonance is conducted. It is observed that this fourth order resonance is dominating over the better known envelope instability and practically replacing it. The simulation study shows a clear emittance growth by this resonance and its stop band. A proposal to GSI was made to perform an experiment to measure the stop band of this resonance using the UNILAC. The experiment confirmed this resonance and will be published in a separate paper.