Analytic model for the breakup of a coasting beam with space charge in isochronous accelerators

Abstract

An analytical model based on the negative mass instability is introduced in this paper to explain the formation of the breakup of a coasting beam into small clusters in isochronous machines such as the case observed by Pozdeyev and Rodriguez in a small isochronous ring. Solving Poisson’s equation in both charge and vacuum regions with the longitudinal beam density perturbation, the coherent radial space charge force which decreases the transition gamma is obtained. It is found that the modified transition gamma depends on the wave number of the density perturbation, longitudinal beam density distribution, beam intensity, and beam size. By combining the longitudinal space charge force caused by the perturbation and the modified transition gamma, a dispersion relation for a monoenergetic beam is derived and evaluated for the fastest-growing instability mode in terms of the beam parameters, such as energy, bunch length, intensity, and emittance. The fastest-growing negative mass mode number, which determines not only the cluster number but also the growth rate of the instability, is proportional to the orbit radius and inversely proportional to the initial beam size. With the growth of the instability, the particles at the points of local minimum density move to the ones of local maximum density, with the transition gamma increasing. Since the growth rate depends on the longitudinal density distribution, therefore, instead of a constant growth rate, our model shows that the growth rate decreases with time. The results above can be applied to both short and long wavelength limits. As an important application of this theory, the beam breakup effect in the isochronous cyclotron CYCIAE-100 is predicted.