Longitudinal stability of the hollow ion bunches after momentum slip-stacking in the CERN SPS

Momentum slip-stacking is planned to be used for the lead ion beams in the CERN SPS to double the beam intensity for the High-Luminosity LHC project. During this RF manipulation, two SPS batches, controlled by two independent RF systems, are going to be interleaved on an intermediate energy plateau, reducing the bunch spacing from 100 ns to 50 ns. However, there are limitations how close the frequencies of the two RF systems can approach each other, resulting in a hole in the longitudinal bunch particle distribution due to the offset in energy of the recaptured bunches. After filamentation, these bunches should be further accelerated to the SPS top energy, before extraction to the LHC. Macro-particle simulations have shown that Landau damping is lost for the bunches with the smallest longitudinal emittances in the batch, causing un-damped oscillations of the bunch core after recapture. The standard application of an additional, fourth harmonic RF system, successfully used in proton operation, was not able to damp the oscillations at top energy, and it is necessary to switch it on from the moment of recapture. In this paper the longitudinal stability of the bunches after slip-stacking is studied using both macro-particle simulations and analytical calculations.