A Halbach array (German: [ˈhalbax]) 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 (on the front face; on the left, up, right, down) 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 one-sided flux configuration was first described in 1957 by W. K. Westmijze in US patent US2981871A. It also appears, in 1970, in US patent US3674946A Fig 29 by James (Jim) M. Winey of Magnepan. The effect was also discovered by John C. Mallinson in 1973 and later generalized. 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.
Linear arrays
Magnetization
The magnetic flux distribution of a linear Halbach array may seem somewhat counter-intuitive to those familiar with simple magnets or solenoids. The reason for this flux distribution can be visualised using Mallinson's original diagram (note that it uses the negative y component, unlike the diagram in Mallinson's article). 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. 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 {\displaystyle \pi /2} out of phase with each other will result in a one-sided flux. The mathematical transform that shifts the phase of all components of some function by π / 2 {\displaystyle \pi /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 ) {\displaystyle \sin(x)\cos(y)} , as shown in the diagram above).
The field on the non-cancelling side of the ideal, continuously varying, infinite array is of the form
F ( x , y ) = F 0 e i k x e − k y , {\displaystyle F(x,y)=F_{0}e^{ikx}e^{-ky},}
where
F ( x , y ) {\displaystyle F(x,y)} is the field in the form F x + i F y {\displaystyle F_{x}+iF_{y}} ,
F 0 {\displaystyle F_{0}} is the magnitude of the field at the surface of the array,
k {\displaystyle k} is the wavenumber (i.e., the spatial frequency) 2 π / λ . {\displaystyle 2\pi /\lambda .}
Applications 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 idealized case, missing references.) There is no stray field produced (in the ideal case) on the opposite side. This helps with field confinement, usually a problem in the design of magnetic structures. Thus they have a number of applications, ranging from flat refrigerator magnets through industrial applications such as the brushless DC motor, voice coils, magnetic drug targeting to high-tech applications such as wiggler magnets used in particle accelerators and free-electron lasers. The Inductrack maglev train and Inductrack rocket-launch system utilize the Halbach array to lift the train by repelling loops of wire in the track.
Flat flexible (not hard ceramic ferrite) refrigerator magnets are created with a Halbach magnetization pattern for a stronger holding force when attached to a flat ferromagnetic surface (e.g. a fridge door) than the holding force from a uniform magnetization. They're made from powdered ferrite mixed in a flexible binder (e.g. plastic or rubber) that is exposed to a Halbach magnetization field pattern as it is extruded, permanently giving the ferrite particles in the magnetic compound this one-sided flux distribution (which can be viewed with magnetic viewing film).
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 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. 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
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