Figure 1.

Schematic cross-sections and architecture of a Halbach magnet compatible for microscope systems. (A) An ideal Halbach dipole cylinder (black) providing a homogeneous magnetic field, B₀ (magnetic field amplitude is sketched by shades of blue and yellow field lines); (B) An ideal Halbach quadrupole (red) generating a constant radial magnetic field gradient. (C) In a combined Halbach dipole and quadrupole, only the gradient component along B₀ is relevant to generate the magnetic force (F_mag), on particles inside the innermost cylinder (F_mag is homogeneous in strength and direction in a distance r from the center if B₀ ≫ Gr is fulfilled). The angle of F_mag changes by 2α if the quadrupole is rotated by α relative to the dipole. For details see.²² (D–G) Effect on some magnetic particles in the inner volume. Here it is assumed that the particles have some remnant magnetization, hence they always possess a magnetic moment. (D) In the absence of any magnetic field the particles point in arbitrary directions. (E) In a homogeneous field they align with the field, but in the absence of a gradient there is no force on them. (F) In the gradient field of a quadrupole, the force (blue arrows) is radial, ie, zero in the centre and increasing with the distance from it. (G) the combination of homogeneous and gradient field results in a constant force in one direction. (H–J) Magnetic guiding system for microscopes. (H) Disassembled system: 1) Support to fit microscope stages using different adaptors. 2) Halbach dipole generating B₀= 0.097 T. Red line indicates poles. 3) Halbach quadrupole completely filled with magnets, producing G = 1.3 T/m. 4) Weaker quadrupole with only every forth magnet present, producing G = 0.33 T/m. (I) Assembled system with a weight of 273 g. (J) System mounted on a Leica TCS SP5 confocal microscope.