Halbach array

The rotating pattern of permanent magnets (on the front face; on the left, up, right, down) can be continued indefinitely and have the same effect.

The principle was further described by James (Jim) M. Winey of Magnepan in 1970, for the ideal case of continuously rotating magnetization, induced by a one-sided stripe-shaped coil.

[3] The effect was also 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.

is called a Hilbert transform; the components of the magnetization vector can therefore be any Hilbert-transform pair (the simplest of which is simply

The field on the non-cancelling side of the ideal, continuously varying, infinite array is of the form[6] where The advantages of one-sided flux distributions are twofold: Thus they have a surprising number of applications, ranging from flat refrigerator magnets through industrial applications such as the brushless DC motor, voice coils,[7] magnetic drug targeting[8] to high-tech applications such as wiggler magnets used in particle accelerators and free-electron lasers.

Scaling up this design and adding a top sheet gives a wiggler magnet, used in synchrotrons and free-electron lasers.

As the electrons are undergoing acceleration, they radiate electromagnetic energy in their flight direction, and as they interact with the light already emitted, photons along its line are emitted in phase, resulting in a "laser-like" monochromatic and coherent beam.

Such a device makes an efficient mechanical magnetic latch requiring no power.

[12][13] The difficulty of manufacturing a cylinder with a continuously varying magnetization also usually leads to the design being broken into segments.

These cylindrical structures are used in devices such as brushless AC motors, magnetic couplings and high-field cylinders.

Both brushless motors and coupling devices use multipole field arrangements: 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.

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.

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.

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.

[22] 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.

The method used to find the field created by the cylinder is mathematically very similar to that used to investigate a uniformly magnetised sphere.

, this is equivalently Since the problem is static there are no free currents and all time derivatives vanish, so Ampère's law additionally requires

over a small loop straddling the boundary and applying Stokes' theorem requires that the parallel component of

To obtain a second set of conditions, integrate Equation 1 across a small volume straddling the boundary and apply the divergence theorem to find where the notation

The sign difference is due to the relative orientation of the magnetisation and the surface normal to the part of the integration volume inside the cylinder walls being opposite at the inner and outer boundaries.

Through the method of separation of variables, it can be shown that the general homogeneous solution whose gradient is periodic in

If we can choose the constants such that the boundary conditions are satisfied, then by the uniqueness theorem for Poisson's equation, we must have found the solution.

So indeed the field is uniform inside and zero outside the ideal Halbach cylinder, with a magnitude depending on its physical dimensions.

[24] As the outside field of a cylinder is quite low, the relative rotation does not require strong forces.

The magnitude of the uniform field for a sphere also increases to 4/3 the amount for the ideal cylindrical design with the same inner and outer radii.

However, for a spherical struction, access to the region of uniform field is usually restricted to a narrow hole at the top and bottom of the design.

This results in the stretching of the sphere to an elliptical shape and having a non-uniform distribution of magnetization over the component parts.

As hard materials are temperature-dependent, refrigeration of the entire magnet array can increase the field within the working area further.