UBC Theses and Dissertations
Single-particle and collective effects of cubic nonlinearity in the beam dynamics of proton synchrotrons Tran, Hy J.
This thesis describes some new studies of the effects of cubic nonlinearities arising from image-charge forces and octupole magnets on the transverse beam dynamics of proton synchrotrons and storage rings, and also a study of the damping of coherent oscillations using a feed-back damper. In the latter case, various corrective algorithms were modeled using linear one-turn maps. Kicks of fixed amplitude but appropriate sign were shown to provide linear damping and no coherent tune shift, though the rate predicted analytically was somewhat higher than that observed in simulations. This algorithm gave much faster damping (for equal power) than conventional proportional kicks, which damp exponentially. Two single-particle effects of the image-charge force were investigated: distortion of the momentum dispersion function and amplitudedependence of the betatron tunes (resulting in tune spread). The former is calculated using transfer maps and the method of undetermined coefficients, the latter by solving the cubic nonlinear equation of motion with the smooth approximation and time averaging. In the case of the CERN Large Hadron Collider, neither effect was found to be of serious concern. Two collective effects caused by tune spread, decoherence and Landau damping, were studied, using bounded binomial (rather than gaussian) density distributions to make the results valid for proton beams. The spread arises from a cubic term in the restoring force. To study decoherence, the motion of a transversely displaced beam bunch was determined using the Vlasov equation, and its centroid found by ensemble averaging; the displaced density distribution was Taylor-expanded in terms of the original one to simplify the integration boundary. The decoherence rate was found to depend primarily on the average density gradient. The centroid motion is also amplitude modulated at the synchrotron tune. To study Landau damping of the weak head-tail instability, a perturbation technique was applied to the Vlasov equation, and the dispersion-relation concept was augmented to handle many coupled radial modes by equating the inverse of the beam transfer function to the set of eigenvalues of the interaction matrix. The matrix elements are the overlaps of the impedance and the mode spatial spectra, while the eigenvalues are the mode frequencies. A new and rigorous derivation is given for the head-tail mode spectra for binomial distributions. In the case of an LHC-type bunch in the CERN PS, several modes were predicted to be unstable, with growth rates compatible with the rather imprecise measurements. The octupole strength needed to Landau damp these unstable modes was estimated by mapping out the stability region in the complex frequency plane.