Abstract
The first mention of the impedance concept appeared on November 1966 in the CERN internal report Longitudinal instability of a coasting beam above transition, due to the action of lumped discontinuities by V.G. Vaccaro. Then, a more general treatment of it appeared in February 1967 in the CERN yellow report Longitudinal instabilities of azimuthally uniform beams in circular vacuum chambers of arbitrary electrical properties by A.M. Sessler and V.G. Vaccaro. The concept of wake field came two years later, in 1969, in the paper The wake field of an oscillating particle in the presence of conducting plates with resistive terminations at both ends by A.G. Ruggiero and V.G. Vaccaro. This was the beginning of many studies, which took place over the last five decades, and today, impedances and wake fields continue to be an important field of activity, as concerns theory, simulation, bench and beam-based measurements. Building a reliable impedance or wake field model of a machine is the first necessary step to be able to evaluate the machine performance limitations, identify the main contributors in case an impedance reduction is required, and study the interaction with other mechanisms such as optics nonlinearities, transverse damper, noise, space charge, electron cloud, beam-beam (in a collider), etc. Beam collective instabilities, and their mitigation, cover a wide range of effects in particle accelerators and they have been the subjects of intense research. As the machines performance was pushed new mechanisms were revealed and nowadays the challenge consists in studying the interplays between all these intricate phenomena, as it is very often not possible to treat the different effects separately. With the increasing power of our computers this becomes easier but the need to continue and develop theories remains, to have a better understanding of the interplays between all these effects: the subject of impedance and beam instabilities in particle accelerators is far from being exhausted, as testified by the many new instability and stabilizing mechanisms which have been recently explained or discovered. Furthermore, in the context of the studies for possible future accelerators, some uncharted territories remain such as, for instance, the collective instabilities during the necessary ionization cooling for a muon collider.
Highlights
The concept of wake field came two years later, in 1969, in the paper “The wake field of an oscillating particle in the presence of conducting plates with resistive terminations at both ends”
This was the beginning of many studies, which took place over the last five decades, and today, impedances and wake fields continue to be an important field of activity, as concerns theory, simulation, bench and beam-based measurements
The methods based on beam excitation in Frequency Domain (FD) are well suited at low frequencies, where the CFL criterion poses a strong requirement on the time step
Summary
Impedance-driven (but ) coherent beam instabilities are usually studied analytically with the linearized Vlasov equation, ending up with an eigenvalue system to solve. 3.2 Impedance-induced instabilities Several Vlasov solvers have been developed over the years in both transverse and longitudinal planes such as GALACTIC and GALACLIC, which have been benchmarked in some regimes as discussed in [20] (see Figs. and 21). A resistive transverse damper destabilizes the single-bunch motion below the TMCI intensity threshold (for zero chromaticity), introducing a new kind of instability, which has been called ITSR instability (for imaginary tune split and repulsion) [30] (see Fig.). A resistive transverse damper destabilizes the single-bunch motion below the TMCI intensity threshold (for zero chromaticity), introducing a new kind of instability, which has been called ITSR instability (for imaginary tune split and repulsion) [30] (see Fig.22) Another important effect which needs to be carefully taken into account to correctly describe. Many other instability mechanisms have been identified over the past years by studying the interplays with other mechanisms, such as the coupled head-tail instability due to the destabilising effect of linear coupling [36] (and references therein), the destabilising effect of mode-coupling of colliding beams (beam-beam and impedance) [37], the destabilising effect of the interplay between Landau octupoles and beam-beam [38], the destabilising effect of noise [39], the destabilising effect of space charge [40] (and references therein), etc
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