Overlap Knock-Out Resonances with Colliding Bunched Beams in the CERN ISR

A bunched beam can subject a `perturbed beam´ to overlap knock-out (OKO) resonances when the frequencies contained within the longitudinal spectrum of the bunches are equal to any of the transverse betatron frequencies of the perturbed beam. Usually with proton machines this condition can only be attained when the revolution frequencies of the two beams are different. With electron machines and their associated high synchrotron frequency, the betatron frequencies of the perturbed beam can overlap with longitudinal sidebands related to the synchrotron motion of the bunched beam. The worst case occurs when the first harmonic of the bunch frequency overlaps with the betatron frequencies of the perturbed beam. This situation was set up in the ISR for dipolar and quadrupolar (order 1 and 2) OKO resonances by operating the rings at largely asymmetric energies (26 and 11 GeV) and at tune values of the perturbed beam close to the integer (dipolar) and the half integer (quadrupolar). In this mode the half life of the perturbed bunched beam was reduced from several months to a few seconds by maximizing the strength of the OKO resonances.