Thresholds for loss of Landau damping in longitudinal plane

The Landau damping mechanism plays a crucial role in providing single-bunch stability in the LHC, high-luminosity LHC, and other existing as well as previous and future circular hadron accelerators. In this paper, the thresholds for the loss of Landau damping (LLD) in the longitudinal plane are derived analytically using the Lebedev matrix equation (1968) and the concept of the emerged van Kampen modes (1983). We have found that for the commonly used particle distribution functions from a binomial family, the LLD threshold vanishes in the presence of the constant inductive impedance ImZ/k above transition energy. Thus, the effect of the cutoff frequency or the resonant frequency of a broadband impedance on beam dynamics is studied in detail. The findings are confirmed by direct numerical solutions of the Lebedev equation as well as using the Oide-Yokoya method (1990). Moreover, the characteristics, which are important for beam operation, as the amplitude of residual oscillations and the damping time after a kick (or injection errors) are considered both above and below the threshold. Dependence of the threshold on particle distribution in the longitudinal phase space is also analyzed, including some special cases with a nonzero threshold for ImZ/k=const. All main results are confirmed by macroparticle simulations and consistent with available beam measurements in the LHC.