Longitudinal single-bunch instabilities

After introducing the concepts of longitudinal wakefield and coupling impedance, we review the theory of longitudinal single-bunch collective effects in storage rings. From the Fokker-Planck equation we first derive the stationary solution describing the natural single-bunch regime, and then treat the problem of microwave instability, showing the different approaches used for estimating the threshold current. We end the lecture with the semi-empirical laws that allow us to obtain the single-bunch behavior above threshold, and with a description of the simulation codes that are now reliable tools for investigating all these effects