Analysis of 3D data for superconducting quadrupole magnets

The magnetic field in free space may be described by solving the equation B = ∇ φ in the appropriate coordinate system, where φ satisfied Laplace's equation. In the circular cylindrical coordinate system the radial component of B, Br, may be written in terms of a doubly infinite sum over n and m. n describes the azimuthal variation, and m describes the radial variation, with m = O being labeled the fundamental mode for any n. The local physical effect of the non-fundamental modes on the trajectories of charged particles may be significant in the design of beam lines. Therefore, accurate numerical simulation of quadrupole (n = 2) magnets requires that the ratio of the m ≠ O terms to the fundamental be known as a function of z. Analysis of measured data for a magnet at Los Alamos National Lab is presented via calculation of these ratios. Results show that m = 1 and 2, at least, must be considered in the design of quadrupoles such as those planned for PILAC at LAMPF