Space Charge in Circular Machines

Abstract

Direct Coulomb forces play an important role in beam transport and storage. In this paper, the effect of space charge in beams stored in circular machines is reviewed. Starting from the concept of a matched beam, the most common particle distributions are introduced. Space charge forces are first addressed as ‘external frozen’ forces and, in this approximation, the modification of the machine optics is discussed, as well as the matching of high-intensity beams. Envelope equations and r.m.s. equivalence are presented as relevant tools for matching high-intensity beams. The space charge limit, space charge tuneshift, tune-spread and their relation with resonances are covered. The more
general beam response to Coulomb forces is discussed for mismatched beams, and a derivation of the coherent frequencies is presented for a Kapchinsky–Vladimirsky beam. The concept of free energy is examined in detail, as a source of emittance increase for high-intensity beams. More complex and relevant examples are presented with the fully coupled envelope equations, and the interplay of coherent and incoherent effects is addressed in the Montague resonance. Space charge as driving incoherent resonances is discussed. Longitudinal dynamics in the presence of space charge are briefly presented as well.