2.3.2 Quadrupole Magnets
The particles traveling in bunches around the storage ring all have slightly different
positions and trajectories, and without additional restoring forces, the beam would
become more and more diffuse. Without focusing, particles would rapidly leave their
stable orbits and be lost from the ring. The key components for focusing charged
particles are quadrupole magnets.
Quadrupole fields are achieved by juxtaposing two electromagnets, each with a
pair of hyperbolic pole pieces, as illustrated by Fig. 2.6. They are usually arranged to
give either “upright” or “skew” (rotated) fields (Appendix C). The focusing action of
quadrupoles stems from the fact that they possess a field gradient, so that the angular
deflection of a particle is proportional to its distance from the beam center. Obviously, a larger field gradient results in a stronger “lens.” Quantitatively, the focusing
strength k of a quadrupole magnet is defined by k ¼ (e/βE)g, where g is the field
gradient, or in practical units:
k m
À2
Â
à ¼ 0:2998
g T=m
½
βE GeV
½
ð2:6Þ
The focal length f of a quadrupole magnet with path length l is then given by:
f m
½ ¼
1
kl
¼
3:3E GeV
½
g T=m
½
l m
½
ð2:7Þ
Since practical field gradients and path lengths are on the order of 10 T/m and
0.5 m, respectively, a typical focal length for 2 GeV electrons is ~1.3 m.
While optical lenses focus visible light in both horizontal and vertical planes, a
minor hitch for focusing charged particles with quadrupole magnets is that they can
only focus in one plane at a time—they are actually defocusing in the perpendicular
plane. To achieve net focusing in both horizontal and vertical planes, pairs or triplets
of quadrupoles are arranged together. Two magnetic lenses can be treated just like
glass lenses in geometrical optics in the thin lens approximation, where the focal
Fig. 2.6 Left to right: (a) quadrupole magnetic fields, (b) schematic of a quadrupole magnet for
SESAME, (c) magnetic field in a vertical sextupole, (d) schematic of a sextupole magnet for
SESAME
2.3 The Storage Ring: Inside the Shield Walls
17
The particles traveling in bunches around the storage ring all have slightly different
positions and trajectories, and without additional restoring forces, the beam would
become more and more diffuse. Without focusing, particles would rapidly leave their
stable orbits and be lost from the ring. The key components for focusing charged
particles are quadrupole magnets.
Quadrupole fields are achieved by juxtaposing two electromagnets, each with a
pair of hyperbolic pole pieces, as illustrated by Fig. 2.6. They are usually arranged to
give either “upright” or “skew” (rotated) fields (Appendix C). The focusing action of
quadrupoles stems from the fact that they possess a field gradient, so that the angular
deflection of a particle is proportional to its distance from the beam center. Obviously, a larger field gradient results in a stronger “lens.” Quantitatively, the focusing
strength k of a quadrupole magnet is defined by k ¼ (e/βE)g, where g is the field
gradient, or in practical units:
k m
À2
Â
à ¼ 0:2998
g T=m
½
βE GeV
½
ð2:6Þ
The focal length f of a quadrupole magnet with path length l is then given by:
f m
½ ¼
1
kl
¼
3:3E GeV
½
g T=m
½
l m
½
ð2:7Þ
Since practical field gradients and path lengths are on the order of 10 T/m and
0.5 m, respectively, a typical focal length for 2 GeV electrons is ~1.3 m.
While optical lenses focus visible light in both horizontal and vertical planes, a
minor hitch for focusing charged particles with quadrupole magnets is that they can
only focus in one plane at a time—they are actually defocusing in the perpendicular
plane. To achieve net focusing in both horizontal and vertical planes, pairs or triplets
of quadrupoles are arranged together. Two magnetic lenses can be treated just like
glass lenses in geometrical optics in the thin lens approximation, where the focal
Fig. 2.6 Left to right: (a) quadrupole magnetic fields, (b) schematic of a quadrupole magnet for
SESAME, (c) magnetic field in a vertical sextupole, (d) schematic of a sextupole magnet for
SESAME
2.3 The Storage Ring: Inside the Shield Walls
17