For the 2018 version of our NMR Mandhala (working towards higher homogeneity along the cylindrical axis), I did a close reading of Soltner and Blumler’s 2010 article “Dipolar Halbach Magnet Stacks Made from Identically Shaped Permanent Magnets for Magnetic Resonance”. Below are some particularly useful findings.
Useful Information Regarding Comparing Finite Sized Magnets with Theory:
Magnetic field of final magnet is in very good approximation the sum of the field of its pieces. (Approximation still fairly good when you substitute each of its pieces with an ideal magnetic dipole, which is used for getting analytic results.)
For n=8 N52 1⁄2” cubes with remanent magnetization, Br = 1.48 T, analytic estimate of flux density at center would be $0.337613 \cdot 1.48 $ T $\approx 0.49967 $ T (for cubes as close to each other as possible).
Typically analytic and numeric results (using FEMM) differ by about 10% since magnetization inside magnets is reduced by magnetic interactions of neighboring magnetic material.
Analytic results also over-estimate because it used a dipole approximation where size of magnets is neglected.
Regard NMR Mandhala Stacks
For NMR Mandhala stack of 2, ideal placement of stacks similar to construction of a Helmholtz coil: $s_1/r = \pm 1/\sqrt{6} \approx \pm 0.408$ where $s_1$ is the distance between the stacks (measured from the center of each stack) and $r$ is the radius of the ring measured to the center of the magnets in each stack.
For our stack of 2 n=8 N52 1⁄2” cube Mandhalas, $r = 3$ cm, and optimal distance would be 0.48 inches, which is smaller than the actual physical size of the magnets.
Flux density at the center of two Mandhalas $\approx 1.36 \cdot$ flux density of single Mandhala (we got closer to $1.46 \cdot$ flux density of single stack).
For multiple stacked Mandhalas, normalized distance between stacks $0.44 \cdot r$ for the inner stacks. The outer stacks need to be made a bit closer to make up for them being at the ends.
Width of $\Delta B/B = 10^{-4}$ homogeneous region is $\pm 0.12 \cdot r$.
“Ultimate homogeneity value for given magnet shape is reached when two magnets touch each other. Their distance as measured from their centers cannot be smaller than their size”
For the reasons above, it made sense to try to move to longer magnets to both boost the field of the NMR Mandhala, as well as improve homogeneity along the cylindrical axis. We found 2” long 1⁄2” x 1⁄2” N52 magnets and switch to those for the second version of our NMR Mandhala (and kept the center Mandhala the same, since those magnets do not come in any longer sizes than 1”.)