Thresholds for loss of Landau damping in longitudinal plane

TL;DR

This paper analytically derives the thresholds for the loss of Landau damping in the longitudinal plane of circular hadron accelerators, revealing conditions under which damping vanishes and confirming results with simulations and measurements.

Contribution

It provides a new analytical framework for determining Landau damping thresholds using Lebedev matrix equations and van Kampen modes, including effects of impedance and particle distribution.

Findings

LLD threshold vanishes with constant inductive impedance for common distributions

Cutoff frequency and resonant impedance significantly affect beam stability

Results are validated by numerical solutions, simulations, and LHC measurements

Abstract

Landau damping mechanism plays a crucial role in providing single-bunch stability in LHC, High-Luminosity LHC, other existing as well as previous and future (like FCC) 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 Im$Z/k$ above transition energy. Thus, the effect of the cutoff frequency or the resonant frequency of a broad-band 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 non-zero threshold for Im$Z/k = const$. All main results are confirmed by macro-particle simulations and consistent with available beam measurements in the LHC.