Microwave instability at transition-Stability diagram approach

A simple model of a beam at transition driven by a storage ring impedance is formulated in the framework of the nonlinear Vlasov equation. This yields a set of coupled equations of motion describing time evolution of a single coherent mode and the overall equilibrium density distribution function. At transition, contour integration in the dispersion relation can be carried out analytically and a simple closed formula for the coherent frequency is obtained. From the resulting stability diagram further conclusions about the growth time of the microwave instability and the longitudinal emittance blowup at transition are derived.<>