Multipole magnet

Multipole magnets are magnets built from multiple individual magnets, typically used to control beams of charged particles. Each type of magnet serves a particular purpose.

Dipole magnets are used to bend the trajectory of particles
Quadrupole magnets are used to focus particle beams
Sextupole magnets are used to correct for chromaticity introduced by quadrupole magnets ^[1]

Magnetic field equations

The magnetic field of an ideal multipole magnet in an accelerator is typically modeled as having no (or a constant) component parallel to the nominal beam direction ( $z$ direction) and for the transverse components can be written as complex numbers:^[2]

$B_{y}+iB_{x}=C_{n}\cdot (x+iy)^{{n-1}}$

where $x$ and $y$ are the coordinates in the plane transverse to the nominal beam direction. $C_{n}$ is a complex number specifying the orientation and strength of the magnetic field. $B_x$ and $B_{y}$ are the components of the magnetic field in the corresponding directions. Fields with a real $C_{n}$ are called 'normal' while fields with $C_{n}$ purely imaginary are called 'skewed'.

First few multipole fields
n	name	magnetic field lines	example device
1	dipole
2	quadrupole
3	sextupole

References

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[1] ttp://cas.web.cern.ch/cas/Bulgaria-2010/Talks-web/Brandt-1-web.pdf

[2] ttp://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf