The Facility for Antiproton and Ion Research (FAIR) will build a set of accelerators and storage rings at the Gesellschaft fur Schwerionforschung (GSI). Nearly all of them transport beams of elliptical shape (SIS 100, CR, NESR, RESR, SuperFRS). Magnetic field calculations as well as magnetic measurements provide precise field information, which is used to improve the properties of the machines using numerical simulations. In this report we develop elliptical multipoles fulfilling the Laplace equation which enable us to describe the field within the whole aperture consistently. Further we show how circular multipoles valid within the elliptic aperture can be derived. We illustrate the advantage of this data representation on the FEM calculations performed during the SIS 100 R&D along with an approach to measure such fields using rotating coils and estimate the influence of measurement artifacts.
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