Linear and nonlinear beam dynamics of vertical fixed-field accelerators

Abstract

Vertical fixed-field accelerators (vFFAs) feature a magnetic field that increases exponentially in the vertical direction, resulting in vertically stacked nonplanar orbits. Their magnetic field is highly nonlinear, and their solenoid and quadrupolar components induce strongly coupled optics. The detailed study of their beam dynamics must account for the transverse motion linear and nonlinear coupling. Specifically, the study of linear beam dynamics requires adequate coupling parametrizations, and the study of nonlinear beam dynamics, including the characterization of the dynamic aperture (DA), must be performed in the full 4D phase space. The zgoubi ray-tracing code is ideally suited to study the vFFA transverse dynamics, as it can perform step-by-step particle tracking in vFFA complex geometry and magnetic fields. This paper provides an in-depth study of the vFFA prototype ring designed under the ISIS-II proton driver prototype project to accelerate proton beams from 3 to 12 MeV. The magnets of this vFFA ring exhibit slow magnetic field falloffs, resulting in a significant influence of the neighboring cells on the optical lattice parameters. The determination of stable orbits and tunes requires superimposing 3D magnetic field maps in zgoubi to account for the neighboring cell residual fields. The study of nonlinear beam dynamics revealed the appearance of fourth-order stability islands. A complete characterization of the DA in the 4D phase space was conducted to give a measure of the stability domain and characterize the performance and limitations of this lattice. This study paves the way for further validation studies with experimental data and field maps.