An exact expression for the momentum dependence of the space charge tune shift in a Gaussian bunch [particle beam]

An analytical approach to calculate the incoherent tune shift in the presence of non-linear space charge forces is described in this paper. Closed form expressions to evaluate the dependence of the space charge detuning on betatron amplitude have been derived previously for a beam of elliptic cross section with Gaussian distribution in transverse dimensions, under the condition that each particle has the same momentum. The present computation considers in addition the dependence of the space charge detuning on momentum, for a beam with Gaussian momentum distribution. Two effects are taken into account for an exact calculation: the widening of the beam due to momentum spread and the variation of the tune shift with the longitudinal position in the bunch. Application to the present high intensity beams in the CERN PS machine and to the foreseen beam for LHC is discussed