Abstract
The Laplace's equations for the scalar and vector potentials describing electric or magnetic fields in cylindrical coordinates with translational invariance along azimuthal coordinate are considered. The series of special functions which, when expanded in power series in radial and vertical coordinates, in lowest order replicate the harmonic homogeneous polynomials of two variables are found. These functions are based on radial harmonics found by Edwin~M.~McMillan in his more-than-40-years "forgotten" article, which will be discussed. In addition to McMillan's harmonics, second family of adjoint radial harmonics is introduced, in order to provide symmetric description between electric and magnetic fields and to describe fields and potentials in terms of same special functions. Formulas to relate any transverse fields specified by the coefficients in the power series expansion in radial or vertical planes in cylindrical coordinates with the set of new functions are provided. This result is no doubt important for potential theory while also critical for theoretical studies, design and proper modeling of sector dipoles, combined function dipoles and any general sector element for accelerator physics. All results are presented in connection with these problems.
Highlights
The description of sector magnets, any curved magnet symmetric along its azimuthal cylindrical coordinate is an important issue
In addition to his results, adjoint McMillan radial harmonics, Gn, are introduced in order to provide the symmetry in the description between electric and magnetic fields
The set of solutions in cylindrical coordinates, named sector harmonics, should not be confused with cylindrical harmonics where ρ-dependent terms are given by Bessel functions
Summary
The description of sector magnets, any curved magnet symmetric along its azimuthal cylindrical coordinate (longitudinal coordinate in accelerator physics) is an important issue. A widely used method, which goes back to Karl Brown’s 1968 paper [1], is based on a solution of Laplace’s equation for a scalar potential using a power series in cylindrical coordinates. In every new order of recurrence one has to assign an arbitrary constant, which will affect all other higher order terms This ambiguity leads to the fact that there is no preferred, unique choice of basis functions; it makes it difficult to compare accelerator codes, since. For higher order multipoles in cylindrical coordinates truncation without violation of Laplace’s equation is not possible. Joining my results to McMillan’s, I would like to present a new representation for multipole expansions in cylindrical coordinates. Appendixes D–F are supplementary materials with harmonics, fields, potentials and Taylor series
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