Tracking protons in an FFAG and a synchrotron with tiny arcs

We study a ray tracing method that uses tiny arcs to track a proton through a magnetic field. We show how to apply this to an FFAG and a synchrotron that are designed for cancer therapy. Assuming that the field traversed by a proton is uniform over each time step, the required equations and a simple scheme for treating a lattice are developed. We compare the computation time with the code Zgoubi for a FODO lattice. To test its usefulness, it is applied to the spiral FFAG, designed in the RACCAM project, to find the horizontal phase space trajectories. It is also applied to a compact synchrotron, designed in Japan, to determine the phase space trajectories during slow extraction of the proton beam.