Beam-Beam Limit as a Function of the Damping Time in Electron Rings

Observations on the interactions between beams colliding head-on have shown the importance of the nonlinear resonances and the existence of a limiting beam strength. This limit is often associated with stochastic phenomena due to overlapping resonances. In this paper, a perturbation treatment for isolated sum resonances is used. The forces are deduced from the spacecharge potential of colliding beams. Single resonance crossings are the consequence of tune modulations. Different crossing regimes, associated with trapping, statistical blow-up and mere beating are considered. For the parameters associated with the LEP project and the PETRA ring, an estimate of the speed of resonance crossing is made, in order to determine the crossing regime and the blow-up rate as functions of the linear beam-beam tune shift, ΔQbb, for each order N. The corresponding life time is compared to the damping time and the order of the safe resonances is given for different ΔQbb's. Assuming a given value for the maximum order of resonances, which can be avoided in a certain region of the tune diagram, this allows a prediction of the beam-beam limit as a function of the damping time.