Analysis of single particle and collective beam effects in high intensity beams in a periodic quadrupole channel

In high intensity proton accelerators, there are two main mechanisms that can cause beam degradation: incoherent and coherent effects due to nonlinear space charge forces. The incoherent effects represent the single particle dynamics while the coherent effects represent the collective response of the beam. Of particular interest is the region above a zero current phase advance (${\sigma }_0$) of 90° where the (coherent) second-order envelope instability and the (incoherent) fourth-order particle resonance are seen to lead to emittance growth. Large emittance growth is also seen below the envelope instability region as the full current beam phase advance ($\sigma$) decreases. In the present study, we have studied the nonlinear effects in a high intensity beam propagating through a focusing-defocusing (FD) quadrupole channel for ${\sigma }_0$ greater than 90°, both analytically, by studying the solutions of Kapchinsky-Vladimirsky (KV) equation and the particle core model, and through detailed particle-in-cell (PIC) simulations using the tracewin code. The KV envelope equation gives the collective response of the beam while the particle core model gives the contribution of the single particle effects. With pic simulations, which resemble the behavior of the real beams more closely as compared to envelope calculations or the particle core model, it is possible to study the evolution of the beam in a self-consistent manner. Our studies show, that it is possible to identify the specific process responsible for beam degradation in high intensity beams. It is also possible to identify which process dominates under different conditions. We further show that the width of the emittance increase stop band calculated from PIC simulations is wider than that calculated by the envelope equations and that the width depends on the length of the channel being studied.