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Landau damping of head-tail modes in bunches due to spreads in the tune shiftcan be a deciding factor for beam stability. We demonstrate that the coherenttune shifts due to reactive impedances can enhance the space-charge induceddamping and change the stability thresholds (here, a reactive impedance impliesthe imaginary part of the impedance of both signs). For example, high dampingrates at strong space-charge, or damping of the $k=0$ mode, can be possible. Itis shown and explained, how the negative reactive impedances (causing negativecoherent tune shifts similarly to the effect of space-charge) can enhance theLandau damping, while the positive coherent tune shifts have an oppositeeffect. It is shown that the damping rate is a function of the coherent modeposition in the incoherent spectrum, in accordance with the concept of theinteraction of a collective mode with resonant particles. We present ananalytical model, which allows for quantitative predictions of dampingthresholds for different head-tail modes, for arbitrary space-charge andcoherent tune-shift conditions, as it is verified using particle trackingsimulations.

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