Intra-Bunch Motion

Impedance-driven coherent beam instabilities are usually studied analytically with the linearised Vlasov equation, ending up with an eigenvalue system to solve. The eigenvalues describe the mode-frequency shifts, leading in particular to the Transverse Mode-Coupling Instability (TMCI) intensity threshold in the absence of chromaticity. This can be directly compared to measurements in particular for the lowest modes and in the absence of tune spread. Another important observable is the intra-bunch motion, which can be also accessed analytically thanks to the eigenvectors. The different regimes, below-at-above TMCI, are described and represented using a simple analytical model, which helps to really understand what happens at each step.