To compensate the beam-beam tune spread and beam-beam resonance driving terms in the polarized proton operation in the Relativistic Heavy Ion Collider (RHIC), we will introduce a low energy DC electron beam into each ring to collide head-on with the opposing proton beam. The device to provide the electron beam is called an electron lens. In this article, using a 6D weak-strong beam-beam interaction simulation model, we will investigate the effects of head-on beam-beam compensation with electron lenses on the proton beam dynamics for the RHIC 250 GeV polarized proton operation. Frequency maps, dynamic apertures, and proton beam loss rates are calculated for this study. Key beam parameters involved in this scheme are varied to search for the optimum compensation condition. The sensitivities of head-on beam-beam compensation to beam imperfections and beam offsets are also studied.

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In the Brazilian synchrotron light source (LNLS-Laboratório Nacional de Luz Síncrotron), we observed that modulating the phase of the accelerating fields at approximately twice the synchrotron frequency suppressed remarkably well a longitudinal coupled-bunch mode of the beam driven by a higher order mode in one of the radiofrequency (rf) cavities. In this work, we present the results of a set of systematic measurements, in single and multibunch mode, aimed at characterizing the effects of rf phase modulation on the beam. We compare those experiments with the results of tracking simulations and of a theoretical model in which Landau damping is the stabilizing mechanism that explains the suppression of the longitudinal coupled-bunch instability. We also measure the frequency of the stable islands created in longitudinal phase space by phase modulation and the longitudinal beam transfer function as a function of the modulation frequency and amplitude. The experimental results are in good agreement with theoretical expectations.