Longitudinal loss of Landau damping in double harmonic RF systems below transition energy

Landau damping plays a crucial role in ensuring single-bunch stability in hadron synchrotrons. In the longitudinal plane, loss of Landau damping (LLD) occurs when a coherent mode of oscillation moves out of the incoherent synchrotron frequency band. The LLD threshold is studied for a purely inductive impedance below transition energy, specifically considering the common case of double harmonic RF systems operating in counter-phase at the bunch position. The additional focusing force due to beam-induced voltage distorts the potential well, ultimately collapsing the bucket. The limiting conditions for a binomial particle distribution are calculated. Furthermore, the contribution focuses on the configuration of the higher-harmonic RF system at four times the fundamental RF frequency operating in phase. In this case, the LLD threshold shows a non-monotonic behavior with a zero threshold where the derivative of the synchrotron frequency distribution is positive. The findings are obtained employing semi-analytical calculations using the MELODY code.