Theory and application of plane elliptic multipoles for static magnetic fields

Standard textbooks on beam dynamics study the impact of the magnetic field quality on the beam using field representations based on circular multipoles. Iron dominated magnets, however, typically provide a good field region with a non-circular aspect ratio (i.e. an ellipse whose axis a is significantly larger than the axis b  ); a boundary not ideal for circular multipoles. The development of superconductors, originally driven to reach fields above ≈2T, allows using them today in completely different fields: iron dominated DC magnets, to save the energy for coil powering as well as repeatedly fast ramped magnets. The cold mass of magnets, housed in common cryostats sectors, makes it tedious to implement additional correction magnets at a later stage, as it requires to warm up the sections where the magnets should be installed as well as unwelding the cryostat. Thus the field homogeneity of the magnets and its influence on the beam has to be thoroughly studied during the project planning phase.Elliptic multipoles, a new type of field expansion for static or quasi-static (here magnetic) two-dimensional fields, are proposed and investigated, which are particular solutions of the potential equation in plane elliptic coordinates obtained by the method of separation. The proper subsets of these particular solutions appropriate for representing static real or complex fields regular within an ellipse are identified. Formulas are given for computing expansion coefficients from given fields. The advantage of this new approach is that the expansion is valid, convergent and accurate in a larger domain, namely in an ellipse circumscribed to the reference circle of the common circular multipoles in polar coordinates. Formulas are derived for calculating the circular multipoles from the elliptical ones. The effectiveness of the approach was tested on many different magnet designs and is illustrated here on the dipole design chosen for the core synchrotron (SIS 100) of the FAIR project as well as on measurement data obtained by rotating coil probes.