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The Poincaré Section Method for Accelerator Element Maps

Abstract

The computation of particle trajectories in particle optical elements is a task that has been continuously improved upon since the advent of accelerators, electron microscopes, spectrographs, and other particle optical systems. In the past transfer maps for the simplest of elements would be computed by hand to a few orders, codes such as COSY INFINITY now compute numerous elements to arbitrary order. The next logical step is in the direction of the computation of an element of an arbitrary electromagnetic field to an arbitrary order. The demand for this step is imminent in the requirements for higher precision and higher orders in various types of particle optical systems, including the simulation of the storage ring of the cooler synchrotron in Jülich for electric dipole moment (EDM) searches. In addition, conventional map methods are not natively suitable for the treatment of the radio frequency Wien filter which is an essential piece of equipment for studying the EDM at this facility in the frozen-spin method.

Convention and convenience in the beam physics community ask that for a given apparatus we are looking for a map that takes the initial state of particles as they enter the device to that of a final state as they exit. However, in general, we aren’t looking to provide a relationship between the initial state of particles and their final states at a given time; rather, we are looking for the relationship between the two at two different planes in space. Namely, we are looking for a map that relates the state of particles z_i on an initial plane s₀ = 0 to that of the state z_f on the final plane s_f as they travel through a specified electromagnetic field. This map can be obtained to arbitrary order within the framework of the code COSY INFINITY through the use of differential algebras (DA).

Often in dynamical systems, we ask for a mathematical construct called a Poincaré section in which we keep track of locations on a given plane that an integrated orbit passes through. This generally makes it easier for one of the requirements of performing a Poincaré section in a differential algebraic framework...

[This is an extract from the beginning page of the paper.]

R. Jagasia, M. Berz, B. Loseth, Microscopy and Microanalysis 21 Suppl. 4 (2015) 14

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