The Q-shift of off-center particle beams in elliptic vacuum chambers

The change of the betatron frequency of a relativistic particle beam- usually called Q-shift-due to the presence of material boundaries is calculated for elliptical geometries, including the limiting cases of circular and parallel plate geometries. The potential of a line charge anywhere in the elliptic chamber is found by conformal mapping, and the coherent and incoherent image coefficients are calculated from the second derivatives of the potential after subtraction of the direct space charge terms. The values of the image coefficients at the beam location itself are obtained by series development. The incoherent Q- shifts found from these formulae are used in the CERN Intersecting Storage Rings (ISR) for dynamic compensation of the working line during stacking of large beams. (7 refs).