Development of an improved multilayer Halbach assembly magnet for PMR relaxometry based on optimization of magnet parameters

Relevance. A magnet is an important block in the design of measuring and analytical equipment based on nuclear (proton) magnetic resonance methods. A large volume of interpolar space and homogeneity of the magnetic field are desirable qualities of a magnet, directly affecting the accuracy of measurements. The Halbach magnetic assembly (HMA) is a promising type of magnet and is a priority for research for use in new generations of portable PMR relaxometers. They have smaller dimensions and weight but create a significantly greater magnetic field in the gap compared to dipole magnets of the same mass and distance between the poles. Therefore, the study and search for solutions to optimize the HMA parameters is an urgent task in the development of improved portable PMR relaxometers.

Objective. The aim of the work is to study and determine the factors influencing such important magnet parameters as: mass, dimensions, magnetic flux density and homogeneity of the magnetic field in the magnet gap and to develop mathematical equations to describe the influence of these factors on the magnet parameters. Based on the results of calculations and software modeling, the practical goal was to create an experimental prototype of HMA for evaluating and calibrating the quantities in mathematical equations.

Method. The applied method was the theory of magnetic combinations composed of identical magnets proposed by Klaus Halbach and other researchers in the field of magnets, based on which the corresponding mathematical equations are developed. ANSYS Maxwell magnetic field modeling software was used for preliminary evaluation of the theoretical calculations. An experimental study method was used in combination with statistical analysis to collect and process data during tests on the HMA prototype.

Results. The result is the development of a method for optimizing the HMA parameters by determining the influence of factors on the HMA parameters based on calculations and software modeling of the HMA prototype to test the optimization. As part of the study, a device for measuring magnetic field values in the range of 0–1.25 T with a resolution of 4 mV/mT based on a Hall sensor and an Arduino board was also developed, which complemented the research.

Conclusion. Based on the calculations of optimized parameters using a mathematical model, a prototype of the HMA was developed. It weighs 4.5 kg, dimensions 140×140×120 mm, and consists of 10 layers of cubic magnets stacked on top of each other. The developed HMA has a low price, is easy to manufacture and assemble, and has the flexibility of changing the design to adjust the magnetic field values in the gap. With a design of 10 layers, the magnetic flux density in the gap reaches B₀ = 0.344 T with inhomogeneity ΔB/B₀ = 2000 ppm in a volume of 6 cm³ in the center of the gap with a diameter of d = 30 mm. The value of the magnetic flux density in the HMA gap is like the value in the dipole magnets used in the PMR-NP1 and PMR-NP2 relaxometers, while the mass and dimensions are significantly less.

1. M. A. Yurchik, N. V. Dukmasova. Tsifrovizatsiya: novaya era dlya nefti i gaza // Sistema upravleniya ekologicheskoi bezopasnost'yu: sbornik trudov XIV mezhdunarodnoi nauchno-prakticheskoi konferentsii (Ekaterinburg, 20-21 maya 2020 g.). — Ekaterinburg: UrFU, 2020 g. - рр. 281-286. (In Russ).

2. Azieva R.Kh., Taimaskhanov Kh.E.. Neobkhodimost' I Vozmozhnosti Ispol'zovaniya Tsifrovykh Tekhnologii V Neftegazovoi Otrasli V Usloviyakh Tsifrovoi Transformatsii Ekonomiki.// Izvestiya SanktPeterburgskogo Gosudarstvennogo Ekonomicheskogo Universiteta. Nomer: 5 (125) God: 2020 рр. 178- 185.(In Russ).

3. Vorob'ev A.E., Tcharo Kh., Vorob'ev K.A. Tsifrovizatsiya neftyanoi promyshlennosti: «intellektual'nyi» neftepromysel // Vestnik Evraziiskoi nauki, 2018, №3. рр. 71-87.(In Russ).

4. Safiullin B.R., Kozelkova V.O., Kashaev R.S.,i dr. Ochistka nefti ot asfal'teno-smol i parafinov // Izvestiya vysshikh uchebnykh zavedenii. PROBLEMY ENERGETIKI. 2022. T.24. № 5. рр. 166-178.(In Russ).

5. Kozelkov Oleg Vladimirovich. Metody i sredstva ekspress-kontrolya kharakteristik skvazhinnoi zhidkosti i nefti na baze protonnoi magnitnoi rezonansnoi relaksometrii: Dis.dokt. tekhn. nauk. Kazan'; 2022.(In Russ).

6. Kashaev R.S., Svinin A. Yu., Kozelkov O.V.. Minimizatsiya oshibok eksperimenta v metode PMR i vozmozhnosti polucheniya spektra vremen relaksatsii. // Izvestiya vysshikh uchebnykh zavedenii. PROBLEMY ENERGETIKI. 2018, №11-12, t.20. рр.152-160.(In Russ).

7. K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nucl. Instrum. Methods. Feb. 1980, vol. 169, no. 1, pp. 1–10.

8. H. Raich and P. Blümler, “Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas,” Concepts Magn. Reson. Part B Magn. Reson. Eng., Oct. 2004, vol. 23B, no. 1, pp. 16–25.

9. Nguen Dyk An', Kashaev R.S. Usovershenstvovannie modeli magnita khal'bakha dlya relaksometra protonnogo magnitnogo rezoansa/ VIII Natsional'noi n/pr. konf. Priborostroenie i avtomatizirovannyi elekroprivod v TEK i ZhKKh. (Kazan' 8-9 dekabrya 2022g.). – Kazan', 2022. – T.1. – рр.82-87.(In Russ).

10. Amir D. Arslanov, Rustem S. Kashaev, Oleg V. Kozelkov. System of Oil Express Flow Control on the Basis of Proton Magnetic Resonance Relaxometry. 2024 International Russian Smart Industry Conference (SmartIndustryCon 2024), 24-30 March 2024. Sochi, Russia . pp. 214-219.

11. G. Moresi, R. Magin. Miniature permanent magnet for table – top NMR. Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering 19B (1). 2003. pp: 35-43. https://doi.org/10.1002/cmr.b.10082.

12. Soltner, H. and Blümler, P., Dipolar Halbach magnet stacks made from identically shaped permanent magnets for magnetic resonance. Concepts Magn. Reson., 36A(4), pp: 211-222, 2010 2010 https://doi.org/10.1002/cmr.a.20165.

13. Bernhard Blümich, Ernesto Danieli, Federico Casanova, et al. Mobile sensor for high resolution NMR spectroscopy and imaging. Journal of Magnetic Resonance, Volume 198, Issue 1, 2009, Pages 80-87. https://doi.org/10.1016/j.jmr.2009.01.022.

14. Cooley, C.Z., Stockmann, J.P., Armstrong, B.D., et al. Two – dimensional imaging in a lightweight portable MRI scanner without gradient coils. Magn. Reson. Med. 73(2), Pages 872–883 (2015). .https://doi.org/10.1002/mrm.25147.

15. A. Bogaychuk, V. Kuzmin; Accounting for material imperfections in the design and optimization of low cost Halbach magnets. Rev. Sci. Instrum. 1 October 2020; 91 (10): 103904. https://doi.org/10.1063/5.0013274.

16. Svinin A.Yu., Kashaev R.S., Kozelkov O.V. Razrabotka magnitnoi sistemy datchika dlya PMRanalizatora. // Izvestiya vysshikh uchebnykh zavedenii. PROBLEMY ENERGETIKI. 2020. T. 22. № 4 S. 115-122. doi:10.30724/1998-9903-2020-22-4-115-122.(In Russ).

17. O'Reilly, T., et al., In vivo 3D brain and extremity MRI at 50 mT using a permanent magnet Halbach array. Magnetic Resonance in Medicine, 2021. 85(1): p. 495-505.

18. C. Z. Cooley et al., "Design of Sparse Halbach Magnet Arrays for Portable MRI Using a Genetic Algorithm," in IEEE Transactions on Magnetics, vol. 54, no. 1, pp. 1-12, Jan. 2018, Art no. 5100112, doi: 10.1109/TMAG.2017.2751001.