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Description
When exploring the design of constant-tune (isochronous) cyclotron magnets, we found it useful to employ dipole magnets with a variable gap height, allowing fine control of the radial field profile. To determine the optimal gap-shape profile that produces a prescribed magnetic field distribution, we employ an iterative search using 3D finite-element simulations (e.g., OPERA). To minimize the number of iterations, we use a Newton–Raphson scheme, which requires knowledge of the Jacobian matrix relating field errors to gap corrections. In this work, we derive exact analytical expressions for this Jacobian in two limiting cases: the non-saturated regime, obtained using conformal mapping, and the fully saturated regime, modeled with current sheets. We then describe a method to interpolate between these two limits to obtain a general Jacobian valid across the full range of magnetization. This approach serves as a practical building block for designing variable-gap magnets with only a few iterations; it can also readily be applied to the design of fixed-field alternating-gradient (FFA) magnets.
