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A new system used for monitoring energetic Coulomb-scattered electrons as themain diagnostic for accurately aligning the electron and ion beams in the newRelativistic Heavy Ion Collider (RHIC) electron lenses is described in detail.The theory of electron scattering from relativistic ions is developed andapplied to the design and implementation of the system used to achieve andmaintain the alignment. Commissioning with gold and 3He beams is then describedas well as the successful utilization of the new system during the 2015 RHICpolarized proton run. Systematic errors of the new method are then estimated.Finally, some possible future applications of Coulomb-scattered electrons forbeam diagnostics are briefly discussed.

High energy Coulomb-scattered electrons for relativistic particle beam diagnostics