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Calculation of the transverse kicks generated by the bends of a hollow electron lens

Electron lenses are pulsed, magnetically confined electron beams whose current-density profile is shaped to obtain the desired effect on the circulating beam in high-energy accelerators. They were used in the Fermilab Tevatron collider for abort-gap clearing, beam-beam compensation, and halo scraping. A beam-beam compensation scheme based upon electron lenses is currently being implemented in the Relativistic Heavy Ion Collider at Brookhaven National Laboratory. This work is in support of a conceptual design of hollow electron beam scraper for the Large Hadron Collider. It also applies to the implementation of nonlinear integrable optics with electron lenses in the Integrable Optics Test Accelerator at Fermilab. We consider the axial asymmetries of the electron beam caused by the bends that are used to inject electrons into the interaction region and to extract them. A distribution of electron macroparticles is deposited on a discrete grid enclosed in a conducting pipe. The electrostatic potential and electric fields are calculated using numerical Poisson solvers. The kicks experienced by the circulating beam are estimated by integrating the electric fields over straight trajectories. These kicks are also provided in the form of interpolated analytical symplectic maps for numerical tracking simulations, which are needed to estimate the effects of the electron lens imperfections on proton lifetimes, emittance growth, and dynamic aperture. We outline a general procedure to calculate the magnitude of the transverse proton kicks, which can then be generalized, if needed, to include further refinements such as the space-charge evolution of the electron beam, magnetic fields generated by the electron current, and longitudinal proton dynamics.