Scaling laws with current for equilibrium momentum spread and emittances from intrabeam scattering and electron cooling

Based on the theories of Piwinski, Bjorken–Mtingawa and Martini of Coulomb scattering, expressions for the heating rates due to intrabeam scattering were known since a long time. Simplifications by Wei–Parzen and Rao and Piwinski led to analytic approximations which are easily applicable to existing lattices. We use these approximations and also the formulae from thermal equilibration of Struckmeier and equate them to either constant cooling rates from electron cooling or to the Novosibirsk cooling rates for electron cooling to calculate the equilibrium values of the horizontal and vertical emittances and the momentum spread (longitudinal emittance) for typical beams in the ESR or in the HESR. nstant cooling and all approximation formulae the ratio of current to the product of the three emittances remains almost constant. This yields a slope of the momentum spread with current between 0.2 and 0.3, in agreement with experimental data. Using the Novosibirsk cooling rates this slope is much larger.