Mechanism of Longitudinal Single-Bunch Instability in the CERN SPS

Understanding the origin of beam instabilities is essential for reaching the highest performance of proton synchrotrons. In the present work, the Oide-Yokoya eigenvalue method of solving the linearised Vlasov equation was used to shed light on the mechanism of longitudinal single-bunch instability in the CERN SPS. In particular, semi-analytical calculations were done for the full longitudinal impedance model, taking into account the RF nonlinearity. The obtained results agree well with macro-particle simulations and are consistent with available beam measurements. For the first time, the instability has been interpreted as a coupling of radial modes within a single azimuthal mode, due to a strong potential-well distortion of the synchrotron-frequency distribution. To avoid this instability, a higher RF voltage is required at a given emittance. Thus, the instability mechanism is very different from the loss of Landau damping, which, in contrast, is mitigated by a lower RF voltage. This understanding also allowed us to optimise the RF voltage programmes during the acceleration cycle for high-intensity bunches used in the AWAKE experiment at CERN.