Quadrupole magnets, abbreviated as Q-magnets, consist of groups of four magnets 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 grows rapidly with the radial distance from its longitudinal 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 will 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 electromagnets, there are four steel pole tips: two opposing magnetic north poles and two opposing magnetic south poles. The steel is magnetized by a large electric current 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.
At the particle speeds reached in high energy particle accelerators, the magnetic force term is larger than the electric term in the Lorentz force: and thus magnetic deflection is more effective than electrostatic deflection. Therefore a 'lattice' of electromagnets is used to bend, steer and focus a charged particle beam. on a positive particle going into the image plane The quadrupoles in the lattice are of two types: 'F quadrupoles' and 'D quadrupoles'. This situation is due to the laws of electromagnetism 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 while defocusing in the horizontal direction. If an F quadrupole and a D quadrupole are placed immediately next to each other, their fields completely cancel out. But if there is a space between them, 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', 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 : if the magnetic poles are arranged with an angle of 45 degrees to the horizontal and vertical planes. is the field gradient of the vertical component in the horizontal direction. Its SI unit is. The sign of determines whether the quadrupole focuses or defocuses particles in the horizontal plane.
