RMS envelope matching of electron beams from “zero” current to extreme space charge in a fixed lattice of short magnets

We present detailed calculations of RMS-envelope matching over a broad range of beam intensities for the University of Maryland Electron Ring (UMER). Containment of beams from zero current to extreme space charge, all without changing the strength of external focusing in the periodic lattice, is possible thanks to the high density of quadrupoles in UMER. In turn, the small-aspect ratio of the UMER magnets results in gradient or field profiles that are “all edges,” thus requiring special treatment when constructing accurate hard-edge models. Further, the results of matching calculations, for both symmetric and asymmetric FODO (alternating gradient) schemes, are compared with calculations from simple general expressions valid in the uniform-focusing approximation of the periodic lattice. Finally, some aspects of the source-to-FODO matching calculation/optimization problem are discussed, together with sensitivity studies of the matching solutions under realistic conditions. The examples from the UMER project, which include experimental results, emphasize the practical aspects of beam envelope matching.