Halbach sphere

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 (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.

Physicist Klaus Halbach, while at the Lawrence Berkeley National Laboratory during the 1980s, independently invented the Halbach array to focus particle accelerator beams.

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.

Although this magnetic flux distribution seems somewhat counter-intuitive to those familiar with simple 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:

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 ${\displaystyle \scriptstyle {\frac {\pi }{2}}}$ 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 ${\displaystyle \scriptstyle {\frac {\pi }{2}}}$ is called a Hilbert transform; the components of the magnetization vector can therefore be any Hilbert transform pair (the simplest of which is simply ${\displaystyle \scriptstyle \sin(x)\cos(y)}$, as shown in the diagram above).

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