This presentation discusses the generalization of the two-dimensional impedance model in the presence of an electron cloud. It will be discussed the implementation of a linear model of the e-cloud forces including both dipolar and quadrupolar forces to improve the modeling of the electron cloud instabilities. The linear model is included in the Vlasov equation, which allows for finding unstable modes. Benchmarking with conventional macro-particle tracking codes by also implementing the same linear model is discussed for negative, low, as well as large chromaticity. It is found that the instability modes by Vlasov agree well with those of the macro-particle simulations, using the same linear model for negative and low chromaticity. For large-chromaticity, the mode visible in the macro-particle simulations is among the unstable Vlasov modes. The present status of the checks with impedance-driven instabilities is being discussed also including recent benchmarking against tracking simulations and measurements.

The secondary emission of electrons and their interaction with the electromagnetic fields of charged particle beams can lead to the build-up of electron clouds in accelerator beam chambers. The interaction of the electrons with both the beam and the chamber walls leads to detrimental effects, such as transverse instabilities and emittance growth, beam loss, pressure rise and heat load. Such effects are systematically observed in the Large Hadron Collider (LHC) during operation with proton beams with the nominal bunch spacing of 25 ns. Furthermore, the severity of electron cloud effects has increased after each long shutdown period of the machine, due to a degradation of the beam screen surfaces with air-exposure. Consequently, electron cloud is already limiting the total intensity in the collider and is one of the main concerns for the performance of the HL-LHC upgrade. In this contribution, the present understanding of electron cloud in hadron accelerators is reviewed. Measurements and observations at the LHC are presented, the impact on performance is evaluated and mitigation measures are discussed along with lessons for future machines.

The proton complex is the first piece in the Muon Collider, it comprises a high power acceleration section, a compressor and a target delivery system. For the International Muon Collider Collaboration we are investigating the possibility of having a full energy 5-GeV linac followed by an accumulator and a compressor ring and finally a target delivery system. In this paper we present the initial studies for the complex and derived initial beam parameters at each interface.