A Vlasov approach to study transverse instabilities driven by electron clouds

We present a linearized method to study transverse instabilities due to electron clouds. It is based on a compact characterization of the cloud dipolar and quadrupolar forces that can be easily obtained from quick single-pass numerical simulations. The long-term stability properties of the bunch are predicted by solving the linearized Vlasov equation taking into account the dipolar forces introduced by the e-cloud along the bunch as well as the betatron tune modulation with the longitudinal coordinate due to the e-cloud quadrupolar forces. The identification of the beam coherent eigenmodes is achieved by solving an eigenvalue problem. An expression for the tune shift of the rigid-bunch mode is also derived.