338
F. Bordry et al.
velocity and on the space distribution of the field. The simplest case is that of a
uniform magnetic field with a single component and velocity v normal to it, in which
case the particle trajectory is a circle. A uniform field has thus a pure bending effect
on a charged particle, and the magnet that generates it is generally referred to as a
dipole.
By equating the Lorentz force to the centripetal force, we obtain the bending
radius ρ of the motion of a particle of charge q under the action of a magnetic field
B perpendicular to the motion:
1
ρ
=
qB
pc
,
(8.2)
By expressing the momentum p in practical units [GeV/c], we can write:
Bρ [Tm] =
10 9
c
p
Z
= 3.3356
p [GeV/c]
Z
,
(8.3)
where Z is the charge number of the particle, with q = Ze.
The product Bρ is known as magnetic rigidity and provides the link between
dipole strength and length based on the momentum of a charged particle in a circular
accelerator. Note how the formula shows clearly the trade-off between the bending
magnetic field B and the size of the machine (related to ρ).
Besides bending magnets, a number of other field shapes are required to focus
and control the beam. Most important are magnets that generate a pure gradient
field, i.e. a field that is zero on the axis of the magnet and grows linearly with
distance. This type of magnet is referred to as a quadrupole and is used to focus the
particles on the central trajectory of the accelerator. The strength of a quadrupoles is
customarily quoted in terms of the field gradient G, in units of [T/m]. A normalised
quadrupole strength for a quadrupole of length l is defined as the ratio of the
integrated quadrupole gradient to the beam rigidity, or: K = Gl/(Bρ). The angular
deflection α (in radians) of a particle passing at a distance x from the centre of a
quadrupole can be computed using the normalised quadrupole strength as:
α [rad] = Kx,
(8.4)
which shows that a particle on the quadrupole axis (x = 0) has a straight trajectory,
while a particle off-axis receives a kick proportional to its distance from the centre,
i.e. the expected focussing effect. Higher order gradient fields, such as sextupoles, or
octupoles, behave similarly and provide further non-linear means to control, correct
and stabilize the dynamics of the motion of the particles, as described elsewhere in
this handbook.
The accelerator magnets considered here have typically slender, long apertures
(the space available for the beam), where the magnetic field has components only
in the plane of the magnet cross section. In this plane, 2-D configuration, the
most compact representation of the magnetic field shape in the magnet aperture
F. Bordry et al.
velocity and on the space distribution of the field. The simplest case is that of a
uniform magnetic field with a single component and velocity v normal to it, in which
case the particle trajectory is a circle. A uniform field has thus a pure bending effect
on a charged particle, and the magnet that generates it is generally referred to as a
dipole.
By equating the Lorentz force to the centripetal force, we obtain the bending
radius ρ of the motion of a particle of charge q under the action of a magnetic field
B perpendicular to the motion:
1
ρ
=
qB
pc
,
(8.2)
By expressing the momentum p in practical units [GeV/c], we can write:
Bρ [Tm] =
10 9
c
p
Z
= 3.3356
p [GeV/c]
Z
,
(8.3)
where Z is the charge number of the particle, with q = Ze.
The product Bρ is known as magnetic rigidity and provides the link between
dipole strength and length based on the momentum of a charged particle in a circular
accelerator. Note how the formula shows clearly the trade-off between the bending
magnetic field B and the size of the machine (related to ρ).
Besides bending magnets, a number of other field shapes are required to focus
and control the beam. Most important are magnets that generate a pure gradient
field, i.e. a field that is zero on the axis of the magnet and grows linearly with
distance. This type of magnet is referred to as a quadrupole and is used to focus the
particles on the central trajectory of the accelerator. The strength of a quadrupoles is
customarily quoted in terms of the field gradient G, in units of [T/m]. A normalised
quadrupole strength for a quadrupole of length l is defined as the ratio of the
integrated quadrupole gradient to the beam rigidity, or: K = Gl/(Bρ). The angular
deflection α (in radians) of a particle passing at a distance x from the centre of a
quadrupole can be computed using the normalised quadrupole strength as:
α [rad] = Kx,
(8.4)
which shows that a particle on the quadrupole axis (x = 0) has a straight trajectory,
while a particle off-axis receives a kick proportional to its distance from the centre,
i.e. the expected focussing effect. Higher order gradient fields, such as sextupoles, or
octupoles, behave similarly and provide further non-linear means to control, correct
and stabilize the dynamics of the motion of the particles, as described elsewhere in
this handbook.
The accelerator magnets considered here have typically slender, long apertures
(the space available for the beam), where the magnetic field has components only
in the plane of the magnet cross section. In this plane, 2-D configuration, the
most compact representation of the magnetic field shape in the magnet aperture