Longitudinal beam instabilities in a double RF system

Operation with a double RF system is essential for many accelerators in order to increase beam stability, to change the bunch shape or to perform various RF manipulations. This is also the case for the operation of the CERN SPS as the LHC proton injector, where in addition to the main RF system, a fourth harmonic RF system is used in bunch shortening mode in order to increase the synchrotron frequency spread inside the bunch and thus to enhance Landau damping of the collective instabilities. In fact the double RF system operation in the SPS is one of the essential means, together with the controlled longitudinal emittance blow-up to significantly increase the longitudinal instability thresholds (single and multi-bunch) and deliver a good quality beam for the LHC. However, for the HiLumi-LHC (HL-LHC) and LHC injector upgrade (LIU) projects higher beam intensities are required. After all upgrades are in place, the main performance limitations of the LHC injector complex are beam instabilities and high intensity effects in the SPS. This thesis elaborates the benefits and the limitations of the operation in a double RF system. The study is primarily based on the beam and machine parameters of the SPS but most of the results can be generalized and used for other accelerators as well. In particular, the single-bunch longitudinal instability threshold is found from measurements, simulations and analytical calculations for the case of a purely reactive impedance and for the realistic case of the SPS impedance model. The effect of the relative phase and the voltage ratio between the two RF systems on beam stability is studied as well. Finally, the measured variation of the longitudinal emittance along the batch is explained by the modification of the synchrotron frequency distribution due to the residual effect of beam loading in the SPS double RF system.