A study of coherent and incoherent resonances in high intensity beams using a linear Paul trap

In this paper we present a quantitative measurement of the change in frequency (tune) with intensity of four transverse resonances in a high intensity Gaussian beam. Due to the non-linear space charge forces present in high intensity beams, particle motion cannot be analytically described. Instead we use the Simulator of Particle Orbit Dynamics (S-POD) and the Intense Beam Experiment (IBEX), two linear Paul traps, to experimentally replicate the system. In high intensity beams a coherent resonant response to both space charge and external field driven perturbations is possible, these coherent resonances are excited at a tune that differs by a factor $C_{m}$ from that of the incoherent resonance. By increasing the number of ions stored in the linear Paul trap and studying the location of four different resonances we extract provisional values describing the change in tune of the resonance with intensity. These values are then compared to the $C_{m}$ factors for coherent resonances. We find that the $C_{m}$ factors do not accurately predict the location of resonances in high intensity Gaussian beams. Further insight into the experiment is gained through simulation using Warp, a particle-in-cell code.