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[[Image:Magnetic quadrupole moment.svg|right|thumb|250px|Four bar magnets configured to produce a quadrupole]]
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'''Quadrupole magnets''' consist of groups of four [[magnet]]s laid out so that in the planar [[multipole expansion]] of the field the dipole terms cancel and where the lowest significant terms in the field equations are [[quadrupole]]. Quadrupole magnets are useful as they create a [[magnetic field]] whose magnitude [[Linear function|grows rapidly]] with the [[radius|radial]] distance from its longitudinal [[Coordinate axis|axis]]. This is used in [[particle beam]] focusing.
 
The simplest magnetic quadrupole is two identical bar magnets parallel to each other such that the north pole of one is next to the south of the other and vice versa. Such a configuration would have no dipole moment, and its field will decrease at large distances faster than that of a dipole. A stronger version with very little external field involves using a ''k''=3 [[Halbach cylinder]].
 
In some designs of quadrupoles using [[electromagnet]]s there are four [[steel]] pole tips: two opposing [[magnetic north pole]]s and two opposing magnetic south poles. The steel is magnetized by a large [[electric current]] that flows in the coils of tubing wrapped around the poles. Another design is a [[Helmholtz coil]] layout but with the current in one of the coils reversed.<ref>[http://hyperphysics.phy-astr.gsu.edu/Hbase/magnetic/magquad.html Quadrupole Magnetic Field<!-- Bot generated title -->]</ref>
 
{{clear}}
 
==Quadrupoles in particle accelerators==
{{main|Strong focusing}}
 
{{Gallery
|title=Examples of quadrupole magnets
|width=250
|lines=4
|align=center
|File:Aust.-Synchrotron,-Quadrupole-Focusing-Magnet,-14.06.2007.jpg|A quadrupole electromagnet as used in the [[storage ring]] of the [[Australian Synchrotron]]
|File:Aust.-Synchrotron,-Quadrupole-Magnets-of-Linac,-14.06.2007.jpg|Quadrupole [[electromagnets]] (in blue), surrounding the [[Linear particle accelerator|linac]] of the [[Australian Synchrotron]], are used to [[Focus (optics)|focus]] the [[electron]] beam
}}
 
At the speeds reached in high energy [[particle accelerator]]s, magnetic deflection is more powerful than electrostatic, and use of the magnetic term of the [[Lorentz force]]:
 
: <math>\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),</math>
 
is enabled with various magnets that make up 'the lattice' required to bend, steer and [[strong focusing|focus]] a charged particle beam.
 
[[File:Magnetic field of an idealized quadrupole with forces.svg|thumb|300px|Magnetic field lines of an idealized quadrupole field in the plane transverse to the nominal beam direction. The red arrows show the direction of the magnetic field while the blue arrows indicate the direction of the [[Lorentz force]] on a positive particle going into the image plane (away from the reader)]]
 
The quadrupoles in the lattice are of two types: 'F quadrupoles' (which are horizontally focusing but vertically defocusing) and 'D quadrupoles' (which are vertically focusing but horizontally defocusing). This situation is due to the laws of [[electromagnetism]] (the [[Maxwell equations]]) which show that it is impossible for a quadrupole to focus in both planes at the same time. The image on the right shows an example of a quadrupole focusing in the vertical direction for a positively charged particle going into the image plane (forces above and below the center point towards the center) while defocusing in the horizontal direction (forces left and right of the center point away from the center).  
 
If an F quadrupole and a D quadrupole are placed immediately next to each other, their fields completely cancel out (in accordance with [[Earnshaw's theorem]]). But if there is a space between them (and the length of this has been correctly chosen), the overall effect is focusing in both horizontal and vertical planes. A lattice can then be built up enabling the transport of the beam over long distances—for example round an entire ring. A common lattice is a FODO lattice consisting of a basis of a focusing quadrupole, 'nothing' (often a bending magnet), a defocusing quadrupole and another length of 'nothing'.
 
==Mathematical description of the ideal field==
 
The components of the ideal magnetic field in the plane transverse to the beam is given by (see also [[multipole magnet]]):
 
: <math> B_y = K \cdot x  \qquad B_x = K \cdot y </math>
 
if the magnetic poles are arranged with an angle of 45 degrees to the horizontal and vertical planes. <math>K</math> is the field gradient
of the vertical component in the horizontal direction (or equivalently, the field gradient of the horizontal component in the vertical direction). Its [[SI unit]] is <math>\mathrm{T}/\mathrm{m}</math>. The sign of <math>K</math> determines whether (for a fixed particle charge and direction) the quadrupole focuses or defocuses particles in the horizontal plane.
 
{{clear}}
 
==See also==
*[[Charged particle beam]]
*[[Dipole magnet]]
*[[Electron optics]]
*[[Halbach cylinder]]
*[[Sextupole magnet]]
*[[Multipole magnet]]
 
==References==
{{reflist}}
 
[[Category:Accelerator physics]]
[[Category:Types of magnets]]

Latest revision as of 18:22, 24 December 2014

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