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Abstract-Beam instabilities cover a wide range of effects in particle accelerators and they have been the subject of intense research for several decades. As the machines performance was pushed new mechanisms were revealed and nowadays the challenge consists in studying the interplays between all this intricate phenomena, as it is very often not possible to treat the different effects separately. The aim of this paper is to review the main mechanisms, discussing in particular the recent developments of beam instability theories and simulations.

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The resistive wall impedance of a vacuum chamber with elliptic cross section is of particular interest for circular particle accelerators as well as for undulators in free electron lasers. By using the electric field of a point charge and of a small dipole moving at arbitrary speed in an elliptical vacuum chamber, expressed in terms of Mathieu functions, in this paper we take into account the finite conductivity of the beam pipe walls by means of the surface impedance, and evaluate the longitudinal and transverse driving and detuning impedances for any beam velocity. We also extend the definition of the Yokoya form factors, valid in the thick wall regime, at any beam energy, and show that, in the ultra-relativistic limit, they coincide with the ones that are found in literature. The method is also extended to the multilayer vacuum chamber case. Under conditions generally satisfied with particle accelerator beam pipes, the classical transmission line theory can be used to modelling the impedance seen by a bunch in a vacuum chamber with several layers as an equivalent circuit with the same number of load impedances, giving, as result, a surface impedance that can be used in combination with the fields of the elliptic geometry to obtain the resistive wall impedance in an elliptical multilayer vacuum chamber. The results are also compared with a more time consuming 3D electromagnetic code and with solutions for known cases of circular and flat beam pipe.

Due to its effective pumping ability, nonevaporable getter (NEG) coating is considered for the vacuum chambers of the Compact Linear Collider (CLIC) electron damping rings (EDR). The aim is to suppress fast beam ion instabilities. The electromagnetic (EM) characterization of the NEG properties up to ultrahigh frequencies is required for the correct impedance modeling of the damping ring (DR) components. The properties are determined using rectangular waveguides which are coated with NEG. The method is based on a combination of complex transmission coefficient S 21 measurements with a vector network analyzer (VNA) and 3D simulations using CST Microwave Studio® (CST MWS). The frequency ranges discussed in this paper are 220-330 and 500-750 GHz.

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