page_title: Halbach array
section_title: Uniform fields
hierarchical_section_title: Halbach array / Halbach cylinder / Uniform fields
caption_attribution_description: English: Cylindrical Halbach Arrach modelled in 2D FEM (MagNet, Infolytica). Inside the Halbach array visible uniform magnetic field.
context_page_description: A Halbach array is a special arrangement of permanent magnets that augments the magnetic field on one side of the array while cancelling the field to near zero on the other side. This is achieved by having a spatially rotating pattern of magnetisation.
  The rotating pattern of permanent magnets can be continued indefinitely and have the same effect. The effect of this arrangement is roughly similar to many horseshoe magnets placed adjacent to each other, with similar poles touching.
  The effect was discovered by John C. Mallinson in 1973, and these "one-sided flux" structures were initially described by him as a "curiosity", although at the time he recognized from this discovery the potential for significant improvements in magnetic tape technology.
  Physicist Klaus Halbach, while at the Lawrence Berkeley National Laboratory during the 1980s, independently invented the Halbach array to focus particle accelerator beams.
context_section_description: For the special case of k = 2, the field inside the bore is uniform and is given by
  where the inner and outer cylinder radii are Rᵢ and Rₒ respectively. H is in the y direction. This is the simplest form of the Halbach cylinder, and it can be seen that if the ratio of outer to inner radii is greater than e, the flux inside the bore actually exceeds the remanence of the magnetic material used to create the cylinder. However, care has to be taken not to produce a field that exceed the coercivity of the permanent magnets used, as this can result in demagnetization of the cylinder and the production of a much lower field than intended.
  This cylindrical design is only one class of designs that produce a uniform field inside a cavity within an array of permanent magnets. Other classes of design include wedge designs, proposed by Abele and Jensen, in which wedges of magnetized material are arranged to provide uniform field within cavities inside the design as shown.
  The direction of magnetization of the wedges in (A) can be calculated using a set of rules given by Abele and allows for great freedom in the shape of the cavity. Another class of design is the magnetic mangle (B), proposed by Coey and Cugat, in which uniformly magnetized rods are arranged such that their magnetization matches that of a Halbach cylinder, as shown for a 6-rod design. This design greatly increases access to the region of uniform field, at the expense of the volume of uniform field being smaller than in the cylindrical designs (although this area can be made larger by increasing the number of component rods). Rotating the rods relative to each other results in many possibilities including a dynamically variable field and various dipolar configurations. It can be seen that the designs shown in (A) and (B) are closely related to the k = 2 Halbach cylinder. Other very simple designs for a uniform field include separated magnets with soft iron return paths, as shown in figure (C).
  In recent years, these Halbach dipoles have been used to conduct low-field NMR experiments. Compared to commercially available (Bruker Minispec) standard plate geometries (C) of permanent magnets, they, as explained above, offer a huge bore diameter, while still having a reasonably homogeneous field.
caption: Uniform field inside Halbach cylinder
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