Thesis title: Loss of Landau damping in double harmonic RF systems
Landau damping is a natural stabilization mechanism to mitigate coherent beam instabilities. In the absence of synchrotron radiation, it plays a crucial role in hadron accelerators, representing an effective way to maintain the beam stable. A double harmonic RF system is a common technique employed in several accelerators to increase beam stability as well as to perform RF manipulations. This method also finds application in the Proton Synchrotron~(PS) and Super Proton Synchrotron~(SPS) at CERN, which both utilize higher harmonic RF systems to reach the beam parameters for the High-Luminosity Large Hadron Collider~(HL-LHC).
In the longitudinal plane, loss of Landau damping~(LLD) occurs when a coherent mode of oscillation moves out of the incoherent band of the bunch synchrotron frequencies. This thesis expands the recent LLD studies to the important case of double harmonic RF systems. It is shown that in the bunch shortening mode (both RF systems in phase at the bunch position for a non-accelerating bucket), inductive impedance above transition energy leads to a vanishing LLD threshold for a binominal particle distribution, similar to the single harmonic RF case.
In this configuration, refined analytical estimates for the synchrotron frequency distribution allowed the derivation of an analytical equation for the LLD threshold by introducing an upper cutoff frequency. The LLD threshold is extensively studied using the concept of van Kampen modes and taking into account the effect of the voltage ratio and the relative phase between the two RF systems for reactive impedance below and above transition energy. The results are supported by self-consistent numerical calculations based on the Oide-Yokoya method implemented in the code MELODY, as well as macroparticle tacking simulation in BLonD. This validity of the theoretical studies is demonstrated with extensive beam measurements under different bucket-filling conditions in two synchrotrons, the PS and the SPS. The measurements consist of observing single-bunch oscillations after a rigid-dipole perturbation at the high-energy plateau to minimize space-charge contributions. Beyond the analytical estimates, the observations are moreover compared in detail to the results in the semi-analytical code MELODY and macroparticle simulation in BLonD.