Coherent and incoherent bucket for a beam loaded RF system

For the case of a beam-loaded radio-frequency acceleration system, we have found the coherent oscillation equation by comparing the bunch centroid with the synchronous particle; and we have found the incoherent oscillation equation by comparing motion of a single particle with the bunch centroid. For the case of single particle motion, stability depends on the synchronous phase φ2,. Stability of coherent motion depends upon the generator induced partial voltage, whereas stability of incoherent motion depends upon the total cavity voltage including the beam induced voltage component. With beam loading, the stability of coherent motion depends upon φ2 + ψ + φg, with tuning angle ψ and generator current phase φg, and the coherent bucket is deformed. With beam loading, the stability of incoherent motion depends upon φb + φc, with beam current phase φb and perturbed cavity phase φc. For the case that the centroid is coincident with the synchronous particle, the incoherent bucket is found to be identical with the case of no beam loading. (author). 3 refs., 4 figs