The flat Halbach array¹, which consists of a planar array of permanent magnets of thickness, $$$t$$$ , in which the magnetization in the x-y plane varies as $$$\textbf M=m\left(\textbf i \cos cx+\textbf k \sin cx \right)$$$ (Figure 1a), generates a spatially varying magnetic field $$\textbf B = \mu_{0} mct\left(\textbf i\cos cx+\textbf k \sin cx \right)e^{cx} \: \: \: Eq. 1 $$ on one side of the array,where z<-t/2, and zero field on the other, where z>t/2. This interesting behaviour has led to a diverse range of applications of planar Halbach arrays which include uses in fridge magnets, free electron lasers and magnetic levitation.

Planar arrangements that produce a one-sided local magnetic field variation are also of potential value in MRI where they could for example be used for selectively eliminating signals^2,3 from surface structures (e.g. subcutaneous fat). Here we show that a Halbach array can be formed by exposing appropriately oriented strips of material with anisotropic magnetic susceptibility to a strong static field, and also validate the predicted behaviour in experiments carried at 3T using a 40-element structure formed from pieces of pyrolytic graphite sheet (PGS).

A planar Halbach array can be formed by arranging long, thin pieces of material with anisotropic magnetic susceptibility, characterised by susceptibility tensor $$ \begin{bmatrix}\ \chi_{a}& 0 & 0 \\0 & -\frac{\chi_{a}}{2} & 0\\0 & 0 & -\frac{\chi_{a}}{2}\ \end{bmatrix}$$

in a static magnetic field ,so that the angle, $$$\theta $$$ , that the principal axis of the susceptibility tensor makes with the field varies linearly with $$$x$$$-position (Figure 1b) . When a magnetic field, $$$B_{0}$$$, is applied along the $$$x$$$-direction and $$$\theta = c x/2$$$, the magnetization of the array follows Eq. 1 with $$ m=\frac{3 \chi_{a}B_{0} }{2 \mu_{0}}\: \: \: Eq. 2$$thus generating a one-sided field perturbation of the form shown in Eq. 1. Similar behaviour obtains if the magnetic field is directed along $$$z$$$..

A planar array comprising 40 rotatable Perspex elements (each 4 x120x1 mm³ in size) was constructed (Fig.2), with one of the 4 mm faces of each element covered with a 70μm-thick layer of pyrolytic graphite sheet (EYGS121807–Panasonic), which has a highly anisotropic magnetic susceptibility. The array was immersed in a 27x19x9 cm³, saline-filled container and imaged at 3T using a 32 channel head Rx array. Field maps were acquired using a dual-GE sequence (1 mm isotropic resolution, TE = 4.8 ms, $$$ \Delta$$$TE = 2 ms), along with coarser resolution (2 mm) gradient echo images. Data were acquired with the spatial period of element rotation by angle π set equal to (16, 32 and 64 mm), giving c-values of 393, 196 and 98 m^-1, and also with all the elements parallel with one another (infinite period). Most images were acquired with the array lying in a coronal plane with the element orientation changing along the $$$B_{0}$$$-field direction, but data were also acquired with the array lying in an axial plane. Image data were processed in Matlab. Linear fits to the field maps were subtracted to remove the effect of large length-scale field variation.Figure 3 shows maps of the field variation in coronal planes located 6mm above (and below) the centre of the array for the different array modulations. A sinusoidally varying field following the array modulation period is evident below the array, but not above. Figure 4 shows that the amplitude of field modulation decreases exponentially with distance below the array in agreement with Eq. 1 with the rate of decay varying inversely with the period of modulation. Fitting the measured data to Eqs. 1 and 2 gave $$$ \chi_{a}$$$≈175ppm which is consistent with previous measurements of PGS anisotropy⁴. Additional measurements indicated that field modulation was produced above the array and zero field below when it was turned through 180^o –about the L-R axis, so that the sense of rotation of magnetization was reversed and we also found that a one-sided spatially varying field was also produced when it lay in an axial plane. Figure 5 shows axial magnitude images acquired with short (5 ms) and long (50 ms) TE with the array flipped in the coronal plane. It shows that the signal is more significantly attenuated above the array as a result of the greater signal dephasing due to field inhomogeneity. In general the magnitude of the gradient of the field produced by the array structure scales as $$$ \frac{3 }{2}\chi_{a}B_{0} c^2e^{cx}$$$ so that a > 1 mTm^-1 gradient was generated at 3T a distance of 5 mm from the array using less than 8 mg of PGS per cm².