NONLINEAR PHENOMENA IN CIRCULAR PARTICLE ACCELERATORS II: BEAM-BEAM INTERACTION

We calculate the Poincaré map for the equation of horizontal transverse motion of a test particle in an accelerator under the action of generic nonlinear forces. We particularize to beam-beam type interactions and we study the dynamics of the map. Beam instabilities due to beam-beam interaction are caused by chaotic behavior and not by amplitude growth, as is the case of other nonlinearities due to special purpose machine magnets. There exists a closed region in phase space where the motion is chaotic, even if the central design orbit is unstable. For large amplitudes in phase space the motion is regular and no chaotic behavior or amplitude growth exists. The beam-beam tune as a function of the beam position x is calculated and predictions agree with the numerical values within the maximum relative error of 10%, for beam positions |x| ≤ σ, where σ is the r.m.s. size of the beam. When beams cross at high distances the variation in the tune due to beam-beam effect disappears and the tune value approaches the betatron tune.