Enhancement of space-charge induced damping due to reactive impedances for head-tail modes

TL;DR

This paper investigates how reactive impedances influence Landau damping of head-tail modes, revealing that negative reactive impedances can enhance damping and alter stability thresholds, supported by an analytical model and simulations.

Contribution

It introduces an analytical model to predict damping thresholds considering reactive impedances and space-charge effects, validated by particle tracking simulations.

Findings

Negative reactive impedances enhance Landau damping.

Reactive impedances can change stability thresholds.

Damping rate depends on mode position in spectrum.

Abstract

Landau damping of head-tail modes in bunches due to spreads in the tune shift can be a deciding factor for beam stability. We demonstrate that the coherent tune shifts due to reactive impedances can enhance the space-charge induced damping and change the stability thresholds (here, a reactive impedance implies the imaginary part of the impedance of both signs). For example, high damping rates at strong space-charge, or damping of the $k=0$ mode, can be possible. It is shown and explained, how the negative reactive impedances (causing negative coherent tune shifts similarly to the effect of space-charge) can enhance the Landau damping, while the positive coherent tune shifts have an opposite effect. It is shown that the damping rate is a function of the coherent mode position in the incoherent spectrum, in accordance with the concept of the interaction of a collective mode with resonant particles. We present an analytical model, which allows for quantitative predictions of damping thresholds for different head-tail modes, for arbitrary space-charge and coherent tune-shift conditions, as it is verified using particle tracking simulations.