A Comparison of Storage Ring Modeling with COSY INFINITY, ZGOUBI, and MAD8

Robert Hippie, Martin Berz
2015 Microscopy and Microanalysis  
Currently there is significant interest in the use of storage rings to search for an electric dipole moment (EDM) in hadrons [1] . This requires utilizing the storage ring as a precision measuring device [2] . Part of understanding the detailed behavior of storage rings comes from careful analysis of fringe fields [3], but the various tracking codes available differ in their ability to model such behavior. It is the purpose of this paper to investigate these differences. A major storage ring
more » ... jor storage ring facility actively engaged in the search for hadron EDMs is the COoler SYnchrotron (COSY) at Forschungszentrum Jülich [4]. We modeled a simplified version of this storage ring using three well-known simulation codes -MAD8 [5], ZGOUBI [6] and COSY INFINITY [7]. MAD8 is a "transfer map" code of order 2, which means that the state of the particle in phase space is maintained as a vector, and the differential equations governing the motion of particle through the storage ring elements are represented by transfer maps. To track a particle through a system, one merely needs to perform map composition. MAD8 also has the capability to track particles symplectically using generating functions of third order [8]. ZGOUBI does not use the transfer map technique, but rather integrates the Lorentz equation by time stepping based on a Taylor series in path length. The coefficients of the Taylor series in time are determined by an additional Taylor expansion of the magnetic field, to fifth order maximum, if the fields are given analytically, and by an out of plane expansion based on numerical differentiation otherwise. ZGOUBI has few programming capabilities beyond a simple looping mechanism to provide multiple passes through an optical system. ZGOUBI provides support for fringe fields via Enge coefficients [9] . The software distribution also includes a powerful post processing module called ZPOP for plotting and data visualization. COSY INFINITY is a combination of the advantages of the transfer map approach and integration codes. It is primarily a transfer map code, but utilizes integration internally to create highly accurate maps for fringe fields [10] . Built into COSY INFINITY is an interpreter for the specialized COSYScript programming language [11] , which allows the researcher to simulate charged particle optics systems to a high degree of accuracy using the techniques of Differential Algebra. Fringe fields are specified by Enge coefficients which can be input by the user to model actual field measurements, or a default set of typical values can be chosen. To establish a baseline, we begin with a simplified hard-edge model of the COSY storage ring ( Figure 1 ). The ring is highly symmetric, incorporating two 40 m telescope regions, and two 52 m arcs. Each arc is composed of three identical bending segments, each with mirror symmetry. The bending elements are rectangular dipoles. There are 16 sextupoles (not indicated on the figure) at various locations around the lattice. Our model incorporates only the bending and focusing elements -the sextupoles in the actual lattice are not modeled. After implementing the storage ring elements into the three codes, we confirm that the first order transfer matrices are essentially identical (Table 1) . 2
doi:10.1017/s1431927615013045 fatcat:fgttve5emjgllbuuhh4fc6jcue