Longitudinal effects of colliding beam space charge forces in electron- positron storage rings with crossing angles

In electron- positron storage rings in which the orbits of the colliding beams cross at a substantial angle, a phase- dependent longitudinal force is exerted on the particles of each beam by the bunches of the other. The effects of this force on both incoherent and coherent longitudinal particles motion are investigated in this paper. These effects impose a limit on the number of particles which can be stored in a bunch in such storage rings and the limit is independent of the crossing angle (provided the crossing angle is sufficiently large) and of the rf accelerating system parameters. The limit is imposed by the requirement for stability of coherent particle motions.