Influences of the Transverse Motions of the Particles to the Recombination Rate of a Co-propagating Electron-ion System

For a system with the ion beam co-propagating with the electron beam, such as a traditional electron cooler or a Coherent electron Cooler (CeC), the recombination rate is an important observable for matching the energy of the electrons with the ions. In this work, we have developed the analytical expressions to investigate how the recombination rate depends on the energy difference of the two beams, with the influences from the transverse motions of the particles being considered. The analytical results are then used to analyze the measured recombination data collected during the CeC experiment in run 21.