Longitudinal space-charge geometric factor for an elliptical beam

Abstract

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 geometrical factor, g 0 , which depends on the transverse charge distribution and the geometry of the vacuum chamber. Expressions for g 0 for a round beam in a concentric round pipe are well known [3,7]; and values may be given for the onaxis impedance or for the impedance ensemble-averaged over the beam cross-section. We have obtained analogous expressions for the on-axis and ensemble-average factors g 0 for the case of an elliptical charge distributionm inside a confocal elliptic metallic pipe for the indices m = 0, 1, 2. However, our results are not completely general; for example, the pipe cannot have the same eccentricity as the beam.

Generalizing the Method of Images for Complex Boundary Conditions : Application on the LHC Beam Screen

“…The transverse electric field can therefore be computed as if the proton charge distribution was a rest, uniformly distributed over the LHC ring. Also (from [3]), in the 2D electrostatic representation of our problem, we must distinguish the contributions of charged filaments of infinite length and linear charge density λ, for which (from Gauss' law) :…”