An analytic model for the electrostatic contribution of the electron cloud to the vertical tune-shift

An analytic quasi-static model is developed for the analysis of the tune-shift associated with the presence of an electron cloud in a damping ring. The essential assumption is that in its direction of motion, a bunch experiences a uniform cloud density but the latter varies from one bunch to another. A second important component of the model is the life-time since it controls the build-up, the equilibrium as well as the decay of the cloud. It is demonstrated analytically that in case of a train of positron bunches, electrons may be trapped in the vertical direction for the entire train duration. Assuming that the ring is dominated by vertical magnetic fields due to either bends or wigglers, we found excellent agreement between the theoretical predictions and the experimental results reported at Cornell Electron/Positron Storage Ring. The ratio between the vertical and horizontal tune-shifts is shown to be indicative of the distribution of the cloud in the beam-chamber.