Theory of the friction force using electron cooling as an intrabeam scattering process

Using the results found previously for the cooling rates of the emittances, due to collisions between the electrons and the ions, a result is found for the friction force acting on the ions. It is shown that the friction force found here when used to track the ion bunch will give the same emittance cooling rates as those found using the intrabeam scattering theory for electron cooling >.For the case of the uniform in space electron bunch distribution, the friction force found here agrees with the friction force result found with the usual theory of electron cooling.