Fig 1.
Schematic presentation of ULF relaxometry measurement.
(a) Pre-polarisation coil is switched on to generate Bp. (b) The net magnetization M reaches its maximum right before the pre-polarisation coil is switched off. (c) The measurement field Bm, perpendicular to Bp is switched on. (d) The net magnetisation vector M precesses about Bm and decays, the sample demagnetises. The localised magnetic field sensor (S) detects the sample signal (FID) during demagnetisation
Fig 2.
Setup of the dynamic SPMA model for ULF relaxometry.
(a) The SPMA model consists of three concentric cylindrical arrays with transversely (x-y plane) magnetised rods. Array A, required for pre-polarisation, consists of 24 magnets; Arrays B and C, required for generating the variable measurement field, consist of 12 magnets each. The origin of the coordinate system was positioned at the centre of the arrays and the axes of the arrays were aligned along the z-axis. In the Halbach configuration, each magnet has a specific orientation different to the neighbouring magnets. To study the magnetic forces, the magnets of array A were numbered counter-clockwise starting at the right-hand-side. (b) Side view indicating the concentric SPMA setup and sizes. Array A is fixed but each magnet rotates individually along its own axis in the z-coordinate direction (small red circular arrow). Arrays B and C (with fixed magnet orientation) rotate about the z-axis, indicated by the large red circular arrow
Fig 3.
Definition and visualisation of the magnetisation patterns considered in this study.
(a) Halbach, (b) reverse Halbach, (c) tangential and (d) radial. Shown as vector plots are the magnet remanent magnetisation (thick white arrows) and the normalized magnetic field distribution (thin white arrows) in the centre of the SPMA. The Halbach pattern leads to a strong, highly directional magnetic field, while the other patterns lead to non-directional, weak fields.
Fig 4.
Section of an array with radius RA.
Each cylindrical magnet, numbered in a counter-clockwise direction and with diameter dm, is evenly arranged along the circumference to ensure equidistant air gaps. The fill factor is defined as the ratio between dm and da. In this example dm equals da and the fill factor is 0.5
Fig 5.
Principle of generating Bm simulated with COMSOL.
(a) Two concentric arrays B and C (RB = 9 cm and RC = 8 cm), each with Halbach magnetisation pattern (see Fig 3A), generate opposing magnetic fields BB and BC. If their magnitude is matched, the field is nearly cancelled in the centre. (b) By rotating array B and C respectively clockwise and counter-clockwise about the axis of symmetry of the SPMA (see red arrows), the x-component of the magnetic field remains cancelled but a y-component, Bm, is generated
Fig 6.
(a) Elements of the SPMA prototype shown separately: Array D with Halbach pattern (I) and tangential pattern (II), array E (III) array F (IV) each with Halbach pattern. Array D magnets are fitted in the MDF frame I or II to achieve Bp ON or Bp OFF configurations, respectively. (b) Arrays E (III) and F (IV) fitted inside array D with tangential pattern (II). Bm magnitude control is achieved by rotating arrays E (III) and F (IV) in opposite directions with prescribed angles α. The white arrows indicate the magnetisation direction of each magnet
Fig 7.
Magnitude and direction plots of magnetic flux density of a SPMA.
Magnetic flux (a) during pre-polarisation (Bp) and (d) measurement state (Bm). Cross-sectional plots through the point of origin along the x-axis (b) and y-axis (c) of the ratio Bpy/Bp (red) and Bpz/Bp (blue). Bpy and Bpz are the y- and z-components of the pre-polarisation field Bp (= Bpx). Bpy and Bpz are at least six orders of magnitude smaller in all directions within the FOV. Plots along the z-axis were omitted since all ratios are well below 10−9. (e) Cross-sections of Bmx/Bm and Bmz/Bm along the x-axis (blue), y-axis (red) and z-axis (black) demonstrating the x and z-component of the resultant magnetic field, generated by arrays A, B and C, are at least three orders of magnitude smaller than Bm (f). Percent deviation from measurement field magnitude of Bm, plotted along the x-axis (curve 1, blue), y-axis (curve 2, red) and z-axis (curve 3, black). Arrays B and C were rotated by ~4.5° to achieve a magnitude of 40 μT
Fig 8.
2D cross-sectional plots of pre-polarisation field Bp along the x-axis (switched on).
For array A with constant fill factor, curve 1 (solid line) corresponds to 12 magnets, curve 2 (dashed line) to 16 and curve 3 (dash-dotted line) to 24 magnets. For array A with constant magnet dimensions (L = 70 cm, dm = 2.16 cm), curve 4 (dash-dotted line) corresponds to 24 magnets, curve 5 (dashed line) to 16 magnets and curve 6 (solid line) to 12 magnets. (a) In array A with constant fill factor 0.75, the field strength within the field of view (FOV) decreases with the number of magnets, since magnet volume and surface area increase. (b) For array A with constant magnet size, the field strength decreases with decreasing numbers of magnets. (c) Within the FOV the field inhomogeneity decreases slightly with decreasing numbers of magnets for both constant fill factor (c) and magnet size (d). In all cases, the field inhomogeneity within the FOV was well below 0.02% (200 ppm)
Fig 9.
Relative magnitude variation of the pre-polarisation field, Bp, generated by array A with 24 magnets.
Field inhomogeneity shown as line plots in z = 2 cm steps along the x axis (a) and y axis (b). Plotted are percent magnitude deviations from the magnitude of Bp at the centre of the array. Within the chosen FOV of 5 x 5 x 5 cm3, the inhomogeneity is less than 0.02% in all cases
Table 1.
Achievable magnetic field strength at the centre of array A and field inhomogeneity within the field of view (FOV) during pre-polarisation for varying number of magnets and fill factors calculated with COMSOL.
Fig 10.
Magnetic flux density plots of array A along the x-axis for different magnetisation patterns, corresponding to Fig 4.
In each array, 12 (solid line 1), 16 (dashed line 2) and 24 (dash-dotted line 3) magnets (L = 70 cm, dm = 2.16 cm, Br = 1 T, fill factor = 0.75) are considered for the (a) reverse Halbach, (b) radial and (c) tangential patterns. The FOV is indicated by the shaded area. Only the tangential pattern (c) is able to cancel the magnetic field within the FOV to magnitudes below 1 μT
Table 2.
Total magnetic energy contained within array A for different magnetisation pattern and varying numbers of magnets.
Fig 11.
Total stored magnetic energy for each magnet in array A with different number of magnets.
The solid line (curve 1) corresponds to 12 magnets, the dashed line (curve 2) to 16 magnets and the dash-dotted line (curve 3) to 24 magnets in array A. Magnet numbering follows Fig 2A. (a) Magnetic energy for each magnet in array A during pre-polarisation with Halbach magnetization pattern (see Fig 3A). (b) Magnetic energy difference between pre-polarisation and measurement with tangential magnetisation pattern. Negative values in Fig 11B indicate that all the magnets move to a lower magnetic energy state.
Fig 12.
Comparison of Bp and Bm generated by a SPMA prototype (left hand side) with numerical simulation (right hand side).
(a) Field direction of Bp indicated by array of needles (top inset, red circle) and surface plot of COMSOL (bottom inset, blue circle). (b) Field direction of Bm measured (top inset, red circle) and simulated (bottom inset, blue circle).
Table 3.
Comparison of simulated and measured magnetic fields generated by the SPMA prototype.
Fig 13.
Comparison of (a) Bp and (b) Bm generated by a SPMA prototype with results of numerical simulation. (a) The measured magnitude of Bp (yellow) shows an offset of 200 μT compared to the simulated value (blue), due to a higher remanent magnetisation than specifications provided by the manufacturer. (b) Arrow plot of Bm for two angular positions of arrays E and F. Dashed lines represent the measured field and continuous lines represent the simulated field for α = 00 and α = 100. The mismatch between simulation and measurement is produced predominantly by the leftover field produced by array D in the tangential configuration.