Quadrupole magnet
Quadrupole magnets, abbreviated as Q-magnets, consist of groups of four
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
In some designs of quadrupoles using
Quadrupoles in particle accelerators
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
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
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'.
Equations of motion and focal length for charged particles
A charged particle beam in a quadrupole magnetic field will experience a focusing / defocusing force in the transverse direction. This focusing effect is summed up by a focusing strength which depends on the quadrupole gradient as well as the beam's rigidity , where is the electric charge of the particle and
is the relativistic momentum. The focusing strength is given by
- ,
and particles in the magnetic will behave according to the ODE[2]
- .
The same equation will be true for the y direction, but with a minus sign in front of the focusing strength to account for the field changing directions.
Quadrupole ideal field
The components of the ideal magnetic field in the plane transverse to the beam are given by the following[3] (see also multipole magnet).
where is the field gradient of the normal quadrupole component and is the field gradient of the skew quadrupole component. The
See also
- Charged particle beam
- Dipole magnet
- Electron optics
- Halbach cylinder
- Sextupole magnet
- Multipole magnet
- Accelerator physics
References
- ^ Quadrupole Magnetic Field
- ^ Steven M. Lund, Transverse Particle Dynamics, US Particle Accelerator School (USPAS) Lectures on “Beam Physics with Intense SpaceCharge” https://people.nscl.msu.edu/~lund/uspas/bpisc_2017/lec_set_02/tpd.pdf
- ^ Shepard, Ben. "Conventional Magnets for Accelerators" (PDF).
External links
- Media related to Quadrupole magnet at Wikimedia Commons