Main and Fringe Field Computations for the Electrostatic Quadrupoles of the Muon $g$-2 Experiment Storage Ring

Abstract

We consider semi-infinite electrostatic deflectors with plates of different thickness, including plates with rounded edges, and we calculate their electrostatic potential and field using conformal mappings. To validate the calculations, we compare the fringe fields of these electrostatic deflectors with fringe fields of finite electrostatic capacitors, and we extend the study to fringe fields of adjacent electrostatic deflectors with consideration of electrostatic induction, where field falloffs of semi-infinite electrostatic deflectors are slower than exponential and thus behave differently from most magnetic fringe fields. Building on the success with electrostatic deflectors, we develop a highly accurate and fully Maxwellian conformal mappings method for calculation of main fields of electrostatic particle optical elements. A remarkable advantage of this method is the possibility of rapid recalculations with geometric asymmetries and mispowered plates. We use this conformal mappings method to calculate the multipole terms of the high voltage quadrupole used in the storage ring of the Muon g-2 Experiment (FNALE- 0989). Completing the methodological framework, we present a method for extracting multipole strength falloffs of a particle optical element from a set of Fourier mode falloffs. We calculate the quadrupole strength falloff and its effective field boundary (EFB) for the Muon g-2 quadrupole, which has explained the experimentally measured tunes, while simple estimates based on a linear model exhibited discrepancies up to 2%.