Radial mode evolution in longitudinal bunched beam instability

An indispensable aspect of the bunched beam instability mechanism is the variation of the particle distribution with respect to the beam intensity. This density variation can be shown as the evolution of radial modes. The radial modes, which are determined by the stationary particle distribution and the impedance, represent the coherence of the particle density variation governed by the Vlasov equation. Using this coherence in the beam instability analysis gives rise to not only the computational efficiency but also the physical insight into the instability mechanism. The evolution of the radial modes displays several interesting properties for the cases without and with synchrotron frequency spread. If the azimuthal mode coupling cannot be neglected, then corresponding to each coherent frequency shift there exists an extended radial mode which includes the interactions from other azimuthal modes. In this article, the radial mode evolution and the related physical implications are discussed, which are useful for the understanding of the beam instabilities, and also useful for the beam diagnostics