Modeling the fringe field of permanent magnet multipoles using numerical simulations

Permanent magnet multipoles (PMMs) are widely used in accelerators to either focus particle beams or confine plasma in ion sources. The real magnetic field created by PMMs is calculated by magnetic field simulation software and then used in particle tracking codes by means of a three dimensional magnetic field map. A common alternative is to use the so-called "hard-edge" model, which gives an approximation of the magnetic field inside the PMM assuming a null fringe field. This work proposes an investigation of the PMM fringe field properties. An analytical model of the PMM magnetic field is developed using the Fourier multipole expansion. A general axial potential function with a unique parameter λ, able to reproduce the actual PMM magnetic field (including its two fringe fields) with an explicit dependence on the PMM length, is proposed. An analytical first order model including the axial fringe field is derived. This simple model complies with the Maxwell equations [curl(B) = 0 and div(B) = 0] and can replace advantageously the "hard-edge" model when fast analytical calculations are required. The higher order analytical multiple expansion model quality is assessed by means of χ² estimators. The general dependence of the potential function parameter λ is given as a function of the PMM geometry for quadrupole, hexapole, and multipole, allowing one to use the developed model in simulation programs where the multipole geometry is an input parameter.