Beam Instabilities in Circular Particle Accelerators

The theory of impedance-induced bunched-beam coherent instabilities is reviewed following Laclare's formalism, adding the effect of an electronic damper in the transverse plane. Both single-bunch and coupled-bunch instabilities are discussed, both low-intensity and high-intensity regimes are analyzed, both longitudinal and transverse planes are studied, and both short-bunch and long-bunch regimes are considered. Two similar approaches for coherent instabilities using the linearised Vlasov equation (and Garnier-Laclare’s formalism) are presented, leading to two Vlasov solvers: for transverse instabilities, GALACTIC (GArnier-LAclare Coherent Transverse Instabilities Code) and for longitudinal instabilities, GALACLIC (GArnier-LAclare Coherent Longitudinal Instabilities Code). The effect of the detuning impedance in the transverse planes (due to axially asymmetric structures) is also discussed, as well as the intra-bunch motion below-at-above the transverse mode-coupling instability and some mitigation measures.