Conceptual design of a compact high gradient quadrupole magnet of varying strength using permanent magnets

A concept is presented to design magnets using cylindrical-shaped permanent-magnet blocks, where various types of magnetic fields can be produced by either rotating or varying the size of the magnetic blocks within a given mechanical structure. A general method is introduced to calculate the 3D magnetic field produced by a set of permanent magnets. An analytical expression of the 2D field and the condition to generate various magnetic fields like dipole, quadrupole, and sextupole are derived. Using the 2D result as a starting point, a computer code is developed to get the optimum orientation of the magnets to obtain the user-specific target field profile over a given volume in 3D. Designs of two quadrupole magnets are presented, one using 12 and the other using 24 permanent-magnet blocks. Variation of the quadrupole strength is achieved using tuning coils of a suitable current density and specially designed end tubes. A new concept is introduced to reduce the integrated quadrupole field strength by inserting two hollow cylindrical tubes made of iron, one at each end. This will not affect the field gradient at the center but reduce the integrated field strength by shielding the magnetic field near the ends where the tubes are inserted. The advantages of this scheme are that it is easy to implement, the magnetic axis will not shift, and it will prevent interference with nearby devices. Around 40% integrated field variation is achieved using this method in the present example. To get a realistic estimation of the field quality, a complete 3D model using a nonlinear $B\text{−}H$ curve is also studied using a finite-element-based computer code. An example to generate around an $80\mathrm{T}/\mathrm{m}$ quadrupole field gradient is also presented.

Figure 1

Position of the reference magnet at $(r_c,0)$ with magnetization along the $X$ axis and another magnet placed at an arbitrary angle at $(r_c,\alpha )$ whose magnetization makes an angle $\beta$ with the $X$ axis.

Figure 2

Schematic view of the base mechanical structure to accommodate 12 cylindrical-shaped permanent magnets and the position of the current-carrying conductors needed for field tuning.

Figure 3

Quadrupole field strength at the center of the magnet, when the radius of each PM is 5, 6, and 7 mm, respectively (a). The gradient uniformity of 0.1% is extended up to a radius of 1.2 cm, which is called the radius of good field zone (b).

Figure 4

Variation of the gradient in the good field zone for only 12 PMs each of 6 mm radius and after the addition of agradient tune coil of $10\mathrm{A}/{\mathrm{mm}}^2$ current density (a). In both cases, the values of the various multipole normalized with quadrupole strength are very low as desired (b).

Figure 5

Values of normalized multipole components for the model having 12 PMs each having a 6 mm radius for the cases where the direction of magnetization of each magnet randomly varies by $\pm 0.25°$, $\pm 0.50°$, $\pm 0.75°$, and $\pm 1.0°$, respectively, from its ideal orientation.

Figure 6

Values of normalized multipole components for the model having 12 PMs, each having a 6 mm radius, for the cases where the position of each magnet randomly varies by $\pm 25$, $\pm 35$, and $\pm 50\mu \mathrm{m}$, respectively, from its ideal position.

Figure 7

Arrangement of 12 PMs and the field in the good field radius at the center of the magnet ($Z=0$).

Figure 8

Variation of the field gradient in $\mathrm{T}/\mathrm{m}$ along the length of the magnet.

Figure 9

Position of 2-cm-long end tubes, one at each end of the 12 PM block assembly. The integrated field strength can be varied by varying the distance of the end tubes from the edge of the magnet.

Figure 10

A schematic diagram of the magnet assembly along with the movement mechanism of the end tube.

Figure 11

Field lines obtained from the 2D simulation in the presence of an iron tube at the bore of a quadrupole magnet are shown in the figure. Shielding of the magnetic field in the good field zone is evident.

Figure 12

Variation of the gradient along the length of the magnet of 20 cm length in the presence of end tubes. The distances of each end tube from the edge of the magnet are inf, 1, 0, $-1$, and $-2\mathrm{cm}$, respectively. The gradient for 18 cm magnet length and each end tube placed at the edge is also plotted.

Figure 13

Variation of integrated quadrupole gradient $\int g.dl$ in T with the distance of the end tubes from the edge of the magnet. The joining line is a guide to the eyes.

Figure 14

Multipole components, integrated over the magnet, normalized with the quadrupole component in the case of a magnet without an end tube and when the tubes are placed at 0, $-1$, and $-2\mathrm{cm}$ from the edge of the magnet.

Figure 15

Variation of the octupole component in arbitrary units along the length of the magnet in the case of a magnet without end tubes and when two end tubes are placed at $-2\mathrm{cm}$ from the edge of the magnet.

Figure 16

Arrangement of magnets in two layers for producing an $80.83\mathrm{T}/\mathrm{m}$ quadrupole gradient. The radius of each magnet for the inner and outer layers is 3.5 and 6 mm, respectively.

Figure 17

Variation of the quadrupole field gradient at the center of the magnet in the case of two layers (a). Values of the various harmonics normalized with quadrupole strength are also presented (b).

Figure 18

Variation of the gradient in the good field zone at the center of the magnet for the assembly of 24 PMs each of 3.4 mm radius and after the addition of a gradient tune coil of $10\mathrm{A}/{\mathrm{mm}}^2$ current density as obtained from the 2D calculation.

Figure 19

Gradient uniformity of 0.1% is extended up to a radius of 1.78 cm, for the assembly of 24 PMs, which is called the radius of good field zone.

Figure 20

Multipole components, as obtained from the 2D calculation, normalized with the quadrupole component are shown for the assembly of 24 PMs.

Figure 21

The 3D arrangement of 24 PMs in a quadrupole magnet and the field in Gauss (G) in the good field zone at the center of the magnet ($Z=0$). Field patterns in Gauss in the PM blocks are also shown. Half symmetry of the model is used for the calculation.

Figure 22

Variation of the field gradient in T/m along the length of the magnet for the 24 PMs assembly QP.