LONGITbDINAL EFFECTS OF COLLIDING BEAM SPACE CHARGE FORCES IN ELECTRON-POSITRON STORkE RINGS WITH CROSkNG 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 particle motion are investigated. 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 radiofrequency accelerating sys tern parameters. The limit is imposed by the requirement for stability of coherent particle motions. (Submitted to Particle Accelerators) * Work support& 1)~. the U: S. At,omic Energy Commission. Introduction In electron-positron storage rings the electromagnetic forces exerted by the colliding beams on each other are very strong by comparison with the selfforces of each beam, and these colliding-beam forces give rise to a variety of effects which limit the performance of the storage rings. The forces,are highly nonlinear as functions of the particle coordinates, and characteristically they lead to avoidance of one beam by the other when either or both beams are too strong, with consequent reduction of the storage ring. luminosity. (The luminosity is defined as the interaction rate per unit reaction cross section.) Generally speaking the particles congregate in traveling bunches, each one of which is spread around a position of synchronous phase on the radiofrequency accelerating wave. Wren the beams are not colliding so that the strong colliding-beam forces are absent, the unperturbed beams assume gaussian density distributions in both transverse and longitudinal coordinates under the influence of radiation fluctuations and damping. When the beams collide these equilibrium distributions may be disturbed, and as a result, there may occur an optimum beam current at which maximum luminosity occurs. In this paper, we study these effects as they apply to the longitudinal or phase coordinate. As we shall see, such effects arise in the longitudinal coordinate only if the colliding beams cross at an angle, and in that case there is a limit on the number of electrons or positrons which can be stored in a bunch. In the past in storage rings using head-on collisions both incoherent and coherent transverse beam instabilities have been. observed and studied extensively because of their importance to achievable reaction rates. In the incoherent transverse instability, the weaker beam is disrupted by the stronger beam so that the central density of the weaker beam is diminished and the luminosity accordingly