22
E. Wilson and B. J. Holzer
2.1.7 Quadrupole Magnets
The first alternating gradient synchrotorns used alternating magnetic lenses formed
by bending magnets having the same vertical guide field but a radial gradient of
alternating sign. In a modern synchrotron the functions of guiding and focussing
the beam are separated. The dipole magnets which do the guiding have no gradient.
The principal focusing elements are quite a different kind of magnet with four poles
which produce gradient but no bending. The poles of these quadrupole magnets are
truncated rectangular hyperbolae and alternate in polarity around the aperture circle
which just touches the poles.
Figure 2.7 shows a particle’s view of the fields and forces in the aperture of a
quadrupole as it passes through normal to the plane of the paper. The field shape
is such that it is zero on the axis of the device but its strength rises linearly with
distance from the axis. This can be seen from a superficial examination of Fig. 2.7 if
we remember that the product of field and length of any field line joining the poles
is a constant. Symmetry tells us that the field is vertical in the median plane (and
purely horizontal in the vertical plane of asymmetry). The field must be downwards
on the left of the axis if it is upwards on the right.
The horizontal focusing force, −evB z , has an inward direction on both sides
and, like the restoring force on a weight suspended from a spring, rises linearly
with displacement, x. The strength of the quadrupole is characterised by its gradient
dB z /dx normalised with respect to magnetic rigidity:
k =
1
(BB)
dB z
dx
.
(2.10)
Fig. 2.7 Components of field and force in a magnetic quadrupole. Positive ions approach the
reader on paths parallel to the s axis (orthogonal to x and z) [2]
E. Wilson and B. J. Holzer
2.1.7 Quadrupole Magnets
The first alternating gradient synchrotorns used alternating magnetic lenses formed
by bending magnets having the same vertical guide field but a radial gradient of
alternating sign. In a modern synchrotron the functions of guiding and focussing
the beam are separated. The dipole magnets which do the guiding have no gradient.
The principal focusing elements are quite a different kind of magnet with four poles
which produce gradient but no bending. The poles of these quadrupole magnets are
truncated rectangular hyperbolae and alternate in polarity around the aperture circle
which just touches the poles.
Figure 2.7 shows a particle’s view of the fields and forces in the aperture of a
quadrupole as it passes through normal to the plane of the paper. The field shape
is such that it is zero on the axis of the device but its strength rises linearly with
distance from the axis. This can be seen from a superficial examination of Fig. 2.7 if
we remember that the product of field and length of any field line joining the poles
is a constant. Symmetry tells us that the field is vertical in the median plane (and
purely horizontal in the vertical plane of asymmetry). The field must be downwards
on the left of the axis if it is upwards on the right.
The horizontal focusing force, −evB z , has an inward direction on both sides
and, like the restoring force on a weight suspended from a spring, rises linearly
with displacement, x. The strength of the quadrupole is characterised by its gradient
dB z /dx normalised with respect to magnetic rigidity:
k =
1
(BB)
dB z
dx
.
(2.10)
Fig. 2.7 Components of field and force in a magnetic quadrupole. Positive ions approach the
reader on paths parallel to the s axis (orthogonal to x and z) [2]