Halbach array

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A Halbach array, showing the orientation of each piece's magnetic field.

A Halbach array is a special arrangement of permanent magnets which augments the magnetic field on one side of the device while cancelling the field to near zero on the other side. In the diagram, the magnetic field is enhanced on the bottom side and cancelled on the top side (a one-sided flux).

The pattern (on the front face; left, up, right, down) of permanent magnets can be continued indefinitely and have the same effect. It is roughly similar to many horseshoe magnets placed adjacent to each other, with alternating polarity.

The effect was discovered by Mallinson in 1973, and these 'one-sided flux' structures were initially described by him as a 'curiosity', although he recognised at the time the potential for significant improvements in magnetic tape technology.^[1]

"In the 1980s, the late Klaus Halbach, a physicist at Lawrence Berkeley National Laboratory, invented the Halbach array to focus accelerator particle beams."^[2]

1 Magnetization
2 Applications
3 References
4 External links

[edit] Magnetization

Although this magnetic flux distribution seems somewhat counter-intuitive to those familiar with, for example, bar magnets or solenoids, the reason for this flux distribution can be intuitively visualised using Mallinson’s original diagram (note this uses the negative y-component unlike the diagram in Mallinson’s paper). The diagram shows the field from a strip of ferromagnetic material with alternating magnetization in the y direction (top left) and in the x direction (top right). Note that the field above the plane is in the same direction for both structures, but the field below the plane is in opposite directions. The effect of superimposing both of these structures is shown in the figure at the bottom:

Canellation of magnetic components resulting in a one-sided flux

The crucial point is that the flux will cancel below the plane and reinforce itself above the plane. In fact, any magnetization pattern where the components of magnetization are $π / 2$ out of phase with each other will result in a one-sided flux. The mathematical transform which shifts the phase of all components of some function by $π / 2$ is called a Hilbert transform; the components of the magnetization vector can therefore be any Hilbert transform pair (the simplest of which is simply $sin(x)cos(y)$ , as shown in the diagram above).

The advantages of one sided flux distributions are twofold:

The field is twice as large on the side on which the flux is confined (in the idealised case).
No stray field is produced (in the ideal, infinite length case) on the opposite side. This helps with field confinement, usually a problem in the design of magnetic structures.

Halbach arrays can be rolled into a cylindrical shape, known as a Halbach cylinder.

[edit] Applications

Although one sided flux distributions may seem somewhat abstract, they have a surprising number of applications ranging from the humble fridge magnet through industrial applications such as the brushless AC motor and magnetic coupling, to high-tech applications such as wiggler magnets used in particle accelerators and free electron lasers. This device is also the fundamental principle behind the Inductrack maglev system, a levitating train that requires no power to levitate; power is only used to create forward motion. The Halbach arrays repel buried loops of wire after they have been accelerated to speed, lifting the train.

Furthermore, one-sided flux theory could also, surprisingly, explain the lack of magnetic field on celestial bodies such as the moon and Mars, due to the cooling process after the formation of such planets ‘freezing in’ a spherical internal one-sided flux distribution, giving rise to zero field external to the planet.^[3]

The simplest example of a one sided flux magnet is a fridge magnet. These are usually composed of powdered ferrite in a binder such as plastic or rubber. The extruded magnet is exposed to a rotating field giving the ferrite particles in the magnetic compound a magnetization resulting in a one-sided flux distribution. This distribution increases the holding force of the magnet when placed on a permeable surface, compared to the holding force from, say, a uniform magnetization of the magnetic compound.

Scaling up this design and adding a top sheet gives a wiggler magnet, used in synchrotrons and free electron lasers. Wiggler magnets ‘wiggle’ or oscillate an electron beam perpendicular to the magnetic field. As the electrons are undergoing acceleration they radiate electromagnetic energy and these photons can be trapped between two parallel mirrors that form a resonant cavity similar to than of a conventional laser.

Schematic diagram of a free electron laser

The design shown above is usually known as a Halbach wiggler. The magnetization vectors in the magnetized sheets rotate in the opposite senses to each other; above, the top sheet’s magnetization vector rotates clockwise and the bottom sheet’s magnetization vector rotates counter-clockwise. This design is chosen so that the x-components of the magnetic fields from the sheets cancel and the y-components reinforce so that the field is given by

$H_y \approx \cos(kx)$

where k is the ‘wavenumber’ of the magnetic sheet given by the spacing between magnetic blocks with the same magnetization vector.

[edit] References

^ J.C. Mallinson, "One-Sided Fluxes — A Magnetic Curiosity?", IEEE Transactions on Magnetics, 9, 678-682, 1973
^ "Magnetically levitated train takes flight"
^ J.C.Mallinson, H.Shute and D. Wilton, "One-Sided Fluxes in Planar, Cylindrical and Spherical Magnetized Structures" IEEE Transactions on Magnetics,36, 2,Mar 2000