Designing and Shimming Compact Iron-Core Dipole Magnets for NMR Spectroscopy

Abstract

Methods, equations, and software tools for designing iron-core dipole magnets, developed by the accelerator physics community but not well known in the compact NMR community, are introduced here. Many of these tools are due to Klaus Halbach, including the equations accounting for the “excess flux” at the edges of the poles and the software for calculating the fields throughout the magnet structure. The advantages of using magnetic ink for passive shimming of compact magnets are discussed, and the details of the implementation of ink-based shims are presented. Examples of combining ink shimming with space-efficient high-order active shims in compact magnets demonstrate the possibilities of iron-core dipole magnets as platforms for inexpensive, manufacturable NMR-based device development.

Appendix: Using Period-Correct Software

As stated in the main article, we make use of period-correct software developed in the 1960s and 1970s. This code, developed by Ron Holsinger with theoretical assistance from Halbach [20, 21], addressed a wide range of topics in accelerator physics. We are interested in the module PANDIRA which handles permanent magnet materials, as well as some supporting modules for plotting and producing output files. (For many decades, the POISSON/SUPERFISH code has been available for Windows via the Accelerator Code Group at Los Alamos National Laboratory, which maintained the code and made it available as a public service. At the time of the writing of this article, it appears to be unavailable at the LANL website. Interested readers should contact the authors for a copy of the software [22].) The documentation, while voluminous, is fragmented and suffers from poor version control. That said, support for those seeking to learn to use the code is available, the code runs quickly, we (as well as many others) have found it accurate, and it requires no fees. Here, we give a basic introduction to using the code and describe the workflow.

By far the best way to learn PANDIRA is via examples. A detailed narrative of how to run the code is provided by Tanabe [13], especially in his chapter 6. The magnet structure to be calculated is specified in a text file defining the physical regions of the 2D magnet design and the materials therein. In most cases, only one quadrant of the magnet structure is defined with the rest determined by symmetry through the specification of appropriate boundary conditions. A facile method for producing this text file is via a spreadsheet program, as detailed by Tanabe. This allows for easy modification to the magnet structure as one searches for the optimal design or responds to evolving user requirements.

Once the text file is established, a mesh on which the problem will be solved is created by running the module AUTOMESH. If the extension of the text file is “.am”, double clicking on the file will automatically invoke AUTOMESH, which then creates an intermediate file with the extensions “.T35” (which originally referenced a tape drive). At this point, opening the T35 file will display the magnet structure and (optionally) the mesh to be used. Now running PANDIRA on the T35 file (e.g., using a right mouse click) will calculate the solution and add it to the T35 file. Opening the T35 file again will show the field lines (see Fig. 3 for an example), and moving the mouse cursor to locations of interest will reveal the local values of the magnetic field and other parameters. Plots and text files containing the field values on user-specified lines or grids may be produced via the module INTERPOLATE. The user interface for this code is decidedly old-fashioned. However, once the production of the problem-defining text file is mastered, the calculations flow quickly.

In specifying the materials in the text file, one can choose between a few included types of low-carbon steel. User specified materials are also supported. The magnet material is specified by defining the values of ${B}_{cept}$ and ${H}_{cept}$ These parameters are the Y and X intercepts of the linearized $B-H$ curve (see Fig. 4). ${B}_{cept}$ is the remanent magnetization ${B}_{r}$. ${H}_{cept}$ is used by PANDIRA to calculate the value of $\mu$, so ${H}_{cept}$ will be slightly larger in magnitude than the ${H}_{c}$ specified by the manufacturer. $\mu \approx 1.05$ for most magnet materials. The orientation of each magnet block is specified by the parameter ${a}_{easy}$, the orientation of the easy axis of magnetization for that piece.

With a PANDIRA calculation result displayed, it is easy to check the field strength at various points by moving the mouse cursor to the location of interest. The most critical regions to inspect are discussed in the main article.

When setting the yoke and return thicknesses to limit the field strength to 10 kG, one needs to keep in mind that the PANDIRA model is axisymmetric. If the actual magnet will have lower symmetry, then the limit of the field strength calculated for the steel may need to be lowered by an amount given by the ratio of the flux-carrying areas of steel in the PANDIRA model and the actual magnet structure. In general, equivalence of areas is the most important guiding principle in matching the PANDIRA model to the actual magnet, for example in modeling the square-poled box magnet in Fig. 1. The impact of the actual magnet’s lack of structural symmetry on the magnet field in the gap, both strength and homogeneity, will be surprisingly weak.

When checking the homogeneity in the PANDIRA model, it may be helpful to use the INTERPOLATE module to calculate the field along the midplane of the magnet. This is useful for dialing in the Rose ring dimensions.