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6.1.1 Geometry of the Ring
For the bending force as well as for the focusing of a particle beam, magnetic fields
are applied in an accelerator. In principle, electrostatic fields would also be possible
but at high momenta (i.e. if the particle velocity is close to the speed of light) the
usage of magnetic fields is much more efficient. In its most general form, the force
acting on the particles is given by the Lorentz-force
F = q (E + v × B)
(6.1)
In high energy accelerators, the velocity v is close to the speed of light and
so represents a nice amplification factor whenever we apply a magnetic field. As
a consequence, it is much more convenient to use magnetic fields for bending
and focusing the particles. Neglecting the E component therefore in Eq. (6.1), the
condition for a circular orbit is defined as the equality of the Lorentz force and the
centrifugal force:
qvB =
mv 2
(6.2)
In a constant transverse magnetic field B, the particle will see a constant
deflecting force and the trajectory will be a part of a circle, whose bending radius ρ
is determined by the particle momentum p = mv and the external B field.
ρ =
p
qB
(6.3)
The term Bρ is called beam rigidity. Inside each dipole magnet in a storage
ring the bending angle—sketched out in Fig. 6.1—is given by the integrated field
strength via
α =
Bds
Bρ
(6.4)
Requiring a bending angle of 2π for a full circle, we get the condition for the
magnetic dipole fields in the ring. In the case of the LHC e.g. for a momentum of
p = 7000 GeV/c a number of 1232 dipole magnets are needed each having a length
of ~15 m with a B-field of 8.3 T. As a general rule in high energy rings, about
66% (2/3) of the circumference of the machine should be foreseen to install dipole
magnets, as they define the maximum particle momentum that can be carried by
the machine. This basic dipole structure is completed with focusing elements, beam
diagnostic tools etc. and forms the arcs of the ring. They are connected by long
B. J. Holzer et al.
6.1.1 Geometry of the Ring
For the bending force as well as for the focusing of a particle beam, magnetic fields
are applied in an accelerator. In principle, electrostatic fields would also be possible
but at high momenta (i.e. if the particle velocity is close to the speed of light) the
usage of magnetic fields is much more efficient. In its most general form, the force
acting on the particles is given by the Lorentz-force
F = q (E + v × B)
(6.1)
In high energy accelerators, the velocity v is close to the speed of light and
so represents a nice amplification factor whenever we apply a magnetic field. As
a consequence, it is much more convenient to use magnetic fields for bending
and focusing the particles. Neglecting the E component therefore in Eq. (6.1), the
condition for a circular orbit is defined as the equality of the Lorentz force and the
centrifugal force:
qvB =
mv 2
(6.2)
In a constant transverse magnetic field B, the particle will see a constant
deflecting force and the trajectory will be a part of a circle, whose bending radius ρ
is determined by the particle momentum p = mv and the external B field.
ρ =
p
qB
(6.3)
The term Bρ is called beam rigidity. Inside each dipole magnet in a storage
ring the bending angle—sketched out in Fig. 6.1—is given by the integrated field
strength via
α =
Bds
Bρ
(6.4)
Requiring a bending angle of 2π for a full circle, we get the condition for the
magnetic dipole fields in the ring. In the case of the LHC e.g. for a momentum of
p = 7000 GeV/c a number of 1232 dipole magnets are needed each having a length
of ~15 m with a B-field of 8.3 T. As a general rule in high energy rings, about
66% (2/3) of the circumference of the machine should be foreseen to install dipole
magnets, as they define the maximum particle momentum that can be carried by
the machine. This basic dipole structure is completed with focusing elements, beam
diagnostic tools etc. and forms the arcs of the ring. They are connected by long