Evolution of hadron beams under intrabeam scattering

Based on assumptions applicable to many circular accelerators, we simplify into analytical forms the growth rates of hadron beams under Coulomb intrabeam scattering (IBS). Because of the dispersion that correlates the horizontal closed orbit to the momentum, the scaling behavior of the growth rates are drastically different at energies low and high compared with the transition energy. At high energies, the rates are approximately independent of the energy. Asymptotically, the horizontal and longitudinal beam amplitudes are linearly related by the average dispersion. At low energies, the beam evolves such that the velocity distribution in the rest frame becomes isotropic in all directions