Modelling the interaction of a relativistic beam particle with an electron cloud

We study the transverse and longitudinal forces acting on a relativistic beam particle due to the interaction with an electron cloud present in the beam pipe. It is found to be convenient to compute the electromagnetic field in a boosted reference frame, moving rigidly with the beam. In such a reference frame, charge and current densities are stationary, therefore the electric and magnetic fields are solutions of an electrostatic and a magnetostatic problem respectively. It is possible to show that the force acting on the bunch (in the lab frame) is simply proportional tot he gradient of the scalar potential and is therefore irrotational. This happens since the non-irrotational part of the electric field force is cancelled exactly by the force due to the magnetic field. For a relativistic beam the scalar potential can be calculated with good approximation as the solution of a 2D Poisson problem. The Hamiltonian of the resulting transformation can be written as a function of the position coordinates, showing that the map is symplectic and can be modelled as a "thin" element in tracking codes.