Landau damping in the longitudinal plane

Loss of Landau damping in the longitudinal plane can limit the performance of an accelerator and lead to particle losses via undamped bunch oscillations or to single- and multi-bunch instabilities. The threshold for loss of Landau damping for a single bunch is usually defined by comparing the position of the coherent bunch oscillation frequency with respect to the incoherent synchrotron frequency spread. Different ways of calculating this threshold are presented and compared, using the LHC as an example. Loss of Landau damping in longitudinal plane can be cured by increasing the synchrotron frequency spread, either through controlled emittance blow-up or the installation of an additional, higher- harmonic RF system.