New directions in the theory of electron cooling

The theory of electron cooling of ions and positrons is reviewed. Formulas describing the retarding force of ions in an electron beam with an 'oblate' velocity distribution, which is typical for electron cooling, are considered for arbitrary intensities of a magnetic field. Considered for positrons are the cases of intermediate and strong magnetic fields, which are of the greatest practical interest. The friction force and the components of the positron velocity diffusion tensor are calculated. Also discussed is the relaxation of positrons in their electron cooling in positron storage rings and their transition to the stationary distribution. The stationary velocity distribution function for positrons is shown to practically coincide in this case with that for electrons. The feasibility of lowering the transverse electron temperature is analyzed, which is required for decreasing the positron spread in momentum.