Easy-to-implement passive shimming approach of Halbach magnet for low-field NMR measurement

Halbach magnets have been widely employed to NMR instruments due to their low weight, low cost, and minimal leakage of magnetic field. However, field inhomogeneity remains challenge due to discrete magnet rings and manufacturing deviations of the magnetic elements. This paper aims to address this limitation through an effective passive shimming approach, which is considered the first step toward constructing high-homogeneity magnets because of its non-powered and inherently stable characteristics. We focus on the transverse dipole field generated by Halbach magnets and develop an easily implementable linear programming-genetic algorithm (LP-GA) hybrid optimization approach for passive shimming. Our methodology first employs an equivalent magnetic dipole model to calculate the sensitivity matrix of the shim pieces in the Region of Interest (ROI). Then, the LP-GA hybrid optimization algorithm determines the optimal position, number, and thickness of the shim pieces. By combining shim pieces of three different thicknesses (1 mm, 1.5 mm, and 2 mm), we significantly reduce the field inhomogeneity of a 48 mT Halbach magnet system. The effectiveness of our approach is validated through NMR measurements using water samples with copper sulfate at different concentrations, demonstrating an improvement in field homogeneity from approximately 1229 ppm to 320 ppm. The experimental results confirm that the proposed approach effectively enhances magnetic field homogeneity of low-field Halbach magnet systems and could be applied to shimming various Halbach-like magnet arrays.