Introduction to Non-linear Longitudinal Beam Dynamics

The interaction of a charged particle beam with radio-frequency (RF) systems in most linear or circular accelerators is an non-linear process. The large longitudinal electric fields for acceleration and longitudinal beam manipulations can only be generated thanks to the resonant build-up of the field in a high quality oscillator, the RF cavity, driven by a sinusoidal and hence inherently non-linear excitation. The course gives an introduction to linear and non-linear longitudinal beam dynamics, deriving the equations of motion, as well as the RF potential and the Hamiltonian of the longitudinal beam dynamics. Profiting from the non-linear dynamics, longitudinal beam manipulations to control bunch length, distance with multiple RF systems are shown as examples. Additionally, the distribution of the synchrotron frequencies of the particles in a bunch can be modelled thanks to the non-linearity of the applied RF voltage. Double- or even multi-harmonic RF systems are a powerful technique to improve the longitudinal stability of high-intensity beams.