This paper presents a passive shimming design approach for a magic-angle-spinning (MAS) NMR magnet. In order to achieve a 1.5-T magic-angle field in NMR samples, we created two independent orthogonal magnetic vector fields by two separate coils: the dipole and the solenoid. These two coils create a combined 1.5-T magnetic field vector directed at the magic angle (54.74° from the spinning axis). Additionally, the stringent magnetic field homogeneity requirement of the MAS magnet is the same as that of a solenoidal NMR magnet. The challenge for the magic-angle passive shimming design is to correct both the dipole and solenoid magnetic field spherical harmonics with one set of iron pieces, the so-called ferromagnetic shimming. Furthermore, the magnetization of the iron pieces is produced by both the dipole and solenoid coils. In our design approach, a matrix of 2 mm × 5 mm iron pieces with different thicknesses was attached to a thin-walled tube, of 90-mm diameter and 40 mm high. Two sets of spherical harmonic coefficients were calculated for both the dipole and solenoid coil windings. By using the multiple-objective linear programming optimization technique and coordinate transformations, we have designed a passive shimming set that can theoretically reduce 22 lower order spherical harmonics and improve the homogeneity of our MAS NMR magnet.

中文翻译：

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一种魔角自旋核磁共振磁体被动匀场的理论设计方法

本文提出了一种用于魔角旋转 (MAS) NMR 磁体的被动匀场设计方法。为了在 NMR 样品中实现 1.5-T 魔角场，我们通过两个独立的线圈创建了两个独立的正交磁场矢量场：偶极子和螺线管。这两个线圈产生一个组合的 1.5-T 磁场矢量，指向魔角（距旋转轴 54.74°）。此外，MAS 磁体对磁场均匀性的严格要求与螺线管 NMR 磁体相同。魔角无源匀场设计的挑战是用一组铁片校正偶极子和螺线管磁场球谐函数，即所谓的铁磁匀场。此外，铁片的磁化是由偶极线圈和螺线管线圈产生的。在我们的设计方法中，一个由 2 mm × 5 mm 不同厚度的铁片组成的矩阵被连接到一个直径为 90 毫米、高为 40 毫米的薄壁管上。计算偶极子和螺线管线圈绕组的两组球谐系数。通过使用多目标线性规划优化技术和坐标变换，我们设计了一个被动匀场组，理论上可以减少 22 个低阶球谐函数并提高我们的 MAS NMR 磁体的均匀性。