Analytical approach to the overshoot phenomenon for a coasting beam in particle accelerators

A nonlinear perturbation theory is developed for longitudinal instabilities of a coasting beam in particle accelerators. In contrast to the linear theory, the present perturbation approach to the Vlasov equation demonstrates that the part of the particle distribution function averaged over the azimuthal angle around the ring has the time dependence of second order in perturbed quantities. A set of differential equations in time is derived for the average distribution (or the energy spread) and the perturbation amplitude. A mapping of the time derivative of the energy spread onto the complex impedance plane is obtained and it shows that the region where the energy spread increases exists even in the stable area. The energy spread for an initially unstable beam is shown to increase beyond the threshold of the stability and eventually to converge to a final value determined by the initial energy spread and the threshold value. Quantitative agreements are obtained between the experimental results at the CERN ISR and the theoretical overshoot formula in the case where the initial energy spread is close to the threshold