Longitudinal energy-spread evolution due to radiation cooling in a compact electron recirculator

The analytical solution for electron energy in a compact radiation-cooling recirculator is obtained. Based on the analytical solution, the evolution of a longitudinal energy distribution function is calculated. From the obtained energy distribution function, a longitudinal energy spread is obtained. An example for a Gaussian energy distribution is shown in the race-track-type cooling recirculator with a bending radius of 33.4 cm at both ends. In this example, where the initial mean beam energy and the current are 100 MeV and 100 A, respectively, it is shown that the initial relative energy spread of 2.5% is reduced to less than 1.0% after 2 million turns in the cooling recirculator.