Quadrupole magnet

A quadrupole magnet as used in accelerator physics

A quadrupole magnet in the DELTA (Dortmund)

A quadrupole magnet is an arrangement of four magnetic poles in which the north and south poles are opposite each other. Mathematically, the magnetic quadrupole field appears as a second non-vanishing term in the multipole expansion of the magnetic field . For a basic sketch see also the electric quadrupole .

In accelerator physics , quadrupole magnets are used to focus the particle beam in accelerators and beam guides . This makes use of the fact that their magnetic field has a gradient transverse to the direction of the beam, i.e. a location-dependent field strength and flux density , so that the Lorentz force acting on the particles also depends on the location.

Shape of the magnetic field

Let it be the direction of the beam axis and the two transverse directions. On the nominal beam axis (the orbit ), i.e. for , the flux density of the quadrupole field is zero. It grows outwards with a constant gradient, that is ${\ displaystyle y}$ ${\ displaystyle x}$ ${\ displaystyle z}$ ${\ displaystyle x = z = 0}$

{\ displaystyle B_ {x} (z) = gz}

and ,

{\ displaystyle B_ {z} (x) = gx \,}

if the equipotential surfaces , i.e. the surfaces of the iron yoke, are shaped hyperbolically . The proportionality factor is called the quadrupole strength. ${\ displaystyle g}$

Application in particle accelerators

Focus

A quadrupole magnet always has a focusing effect in one of the directions transverse to the particle beam, while it defocuses in the other transverse direction; that is, a horizontally focusing quadrupole magnet defocuses vertically and vice versa. In order to achieve overall focusing, i.e. to hold the beam together, an arrangement of quadrupoles must therefore be built up which alternately focus (F) and defocus (D) in a given direction; Abbreviations for these arrangements are z. B. FDFD, FFDD or FODO (O denotes a field-free so-called drift path). It is the same principle of strong focus that z. B. is used in the dipole magnets of modern cyclotrons and synchrotrons .

The focusing is necessary because the particles practically never fly parallel to the beam axis; Among other things, the space charge , ie the mutual repulsion of the particles among one another, contributes to this. Without focusing, the particles sooner or later ran against the wall of the vacuum chamber and were thus lost.

The quadrupole magnets cause the particles to oscillate around the target path, the so-called betatron oscillations . When building and configuring an accelerator, care must be taken that the amplitude of these vibrations does not become so large that the particles run into the wall. Among other things, this is the task of the so-called jet optics .

Sources of Polarized Ions

In some sources for polarized ions, a quadrupole magnet is used to separate atoms in different spin states from one another, see Stern-Gerlach experiment .

Application in physical analysis

By means of special arrangements of quadrupole magnets, complicated, superimposed fields can be generated, which generate sub-areas with a field strength of zero at defined points, which is advantageous when examining objects with magnetic resonance tomographs .

literature

F. Hinterberger: Physics of Particle Accelerators and Ion Optics . 2nd edition, Springer, 2008, ISBN 978-3-540-75281-3 .