Stability Criterion for the Longitudinal Mode-Coupling Instability in the Presence of Both Space-Charge and Resonator Impedances

The equivalence between longitudinal mode-coupling and microwave instabilities has been pointed out a long time ago by Sacherer and Laclare in the case of broad-band driving resonator impedances. The purpose of this paper is to give a simple analytical derivation of the stability criterion for the longitudinal mode-coupling instability driven by a resonator with arbitrary bandwidth, taking into account the potential-well distortion due to both space-charge and resonator impedances. The formula can be expressed in the form of the Keil-Schnell-Boussard criterion with a general form factor. This factor is equal to one for a long bunch interacting with a broad-band resonator impedance, when the potential-well distortion is neglected, as expected.