Optimization of electron cooling for a medium energy accumulator ring

The binary collision model of electron cooling is used to discuss the principal issues in the use of electron cooling for accumulating ions at medium energies, where “medium energy” is defined by a beam velocity >0.8c. Some new general results are developed which are important in this context. Longitudinal and transverse cooling rates are calculated analytically for a particle executing betatron oscillations in a storage ring. The formulas obtained are compared with numerical results. Particular attention is paid to the case in which the transverse components of the relative velocities substantially exceed the longitudinal components. The exact analytical results for finite electron temperatures are presented. The time for longitudinal cooling of a Gaussian beam is calculated as a function of the acceptable fraction of uncooled particles; the optimum electron beam size is calculated and an optimum electron density distribution is found. The results obtained are applied to an electron cooling system for the Fermilab Recycler (Jackson, FERMILAB-TM-1991, Unpublished Internal Note, November 1999); the cooling time, optimum parameters and tolerances are calculated.