Longitudinal Space-Charge Geometric Factor for an Elliptical Beam

The longitudinal space-charge impedance for a particle beam, for which the azimuthal extent is much longer than the transverse, is typically expressed[3] in terms of a geo-metrical factor, g0, which depends on the transverse charge distribution and the geometry of the vacuum chamber. Ex-pressions for g0 for a round beam in a concentric round pipe are well known[3, 7]; and values may be given for the on-axis impedance or for the impedance ensemble-averaged over the beam cross-section. We have obtained analo-gous expressions for the on-axis and ensemble-average factors g0 for the case of an elliptical charge distribution ρ(m,x, y) = [1 − (x/a)2 − (y/b)2]m inside a confocal elliptic metallic pipe for the indices m = 0, 1, 2. How-ever, our results are not completely general; for example, the pipe cannot have the same eccentricity as the beam.