4 Beam-based Correction and Optimization for Accelerators
The transverse motion of a beam in accelerators is determined by the
magnetic fields in the various types of magnets. The magnets along the beam
path in the accelerator constitute its lattice. Dipole magnets in the lattice
determine the orbit geometry. Transverse motion is described by the deviation
of particle motion from the reference orbit, which is typically the design orbit.
Imperfections in the accelerator often cause the beam in an actual machine to
travel on an orbit different from the reference orbit. Correction of the beam
orbit toward the ideal orbit is called orbit steering or orbit correction.
Particles in a beam tend to diverge from each other as they travel along
the orbit due to their slightly different directions of motion. If no intervention
is taken, the transverse beam size will indefinitely grow, causing beam loss
on the vacuum chamber. Quadrupole magnets provide a magnetic field that
varies linearly with transverse position. Such a field bends the stray particles
back toward the design orbit as particles with larger excursions receive larger
correcting kicks. This is called focusing. However, a quadrupole magnet that
focuses the beam in the horizontal direction necessarily defocuses it in the
vertical plane and vice versa. To keep the beam focused in both transverse
directions, quadrupole magnets with opposite polarities are placed alternately
along the beam path. This is the alternating gradient focusing scheme, also
called the strong focusing scheme. A properly designed strong focusing scheme
maintains the orbit stability and keeps a compact beam size.
Steering and focusing are two basic requirements for the transverse beam
motion. Magnets responsible for steering and focusing, namely dipole and
quadrupole magnets, have magnetic fields that are constant or linear with
the transverse position coordinates. The beam motion under such fields is
linear. With a proper focusing scheme, the periodic linear motion in a circular
accelerator is stable. However, synchrotrons and, in particular, storage rings,
typically need sextupole magnets to correct the focusing errors for particles
with energy errors (referred to as chromatic errors). The magnetic fields in
sextupoles have nonlinear dependence on transverse position coordinates and
hence the periodic motion becomes nonlinear. The nonlinearity causes the
motion to be unstable for particles with sufficiently large offsets from the
design orbit. Ensuring a large stable area is another critical requirement for
the transverse motion in circular accelerators.
In this chapter we will briefly introduce the theory of transverse and longitudinal beam motion in accelerators.
1.1 BEAM MOTION IN MAGNET LATTICES
1.1.1 Hamiltonian and the equations of motion
In an accelerator the particles in a beam are expected to closely follow the
design path. It is the deviations of the particles from the design path that are
of our concern. Therefore, typically we adopt a moving curvilinear coordinate
system to describe the particle motion, using the design orbit as the reference
The transverse motion of a beam in accelerators is determined by the
magnetic fields in the various types of magnets. The magnets along the beam
path in the accelerator constitute its lattice. Dipole magnets in the lattice
determine the orbit geometry. Transverse motion is described by the deviation
of particle motion from the reference orbit, which is typically the design orbit.
Imperfections in the accelerator often cause the beam in an actual machine to
travel on an orbit different from the reference orbit. Correction of the beam
orbit toward the ideal orbit is called orbit steering or orbit correction.
Particles in a beam tend to diverge from each other as they travel along
the orbit due to their slightly different directions of motion. If no intervention
is taken, the transverse beam size will indefinitely grow, causing beam loss
on the vacuum chamber. Quadrupole magnets provide a magnetic field that
varies linearly with transverse position. Such a field bends the stray particles
back toward the design orbit as particles with larger excursions receive larger
correcting kicks. This is called focusing. However, a quadrupole magnet that
focuses the beam in the horizontal direction necessarily defocuses it in the
vertical plane and vice versa. To keep the beam focused in both transverse
directions, quadrupole magnets with opposite polarities are placed alternately
along the beam path. This is the alternating gradient focusing scheme, also
called the strong focusing scheme. A properly designed strong focusing scheme
maintains the orbit stability and keeps a compact beam size.
Steering and focusing are two basic requirements for the transverse beam
motion. Magnets responsible for steering and focusing, namely dipole and
quadrupole magnets, have magnetic fields that are constant or linear with
the transverse position coordinates. The beam motion under such fields is
linear. With a proper focusing scheme, the periodic linear motion in a circular
accelerator is stable. However, synchrotrons and, in particular, storage rings,
typically need sextupole magnets to correct the focusing errors for particles
with energy errors (referred to as chromatic errors). The magnetic fields in
sextupoles have nonlinear dependence on transverse position coordinates and
hence the periodic motion becomes nonlinear. The nonlinearity causes the
motion to be unstable for particles with sufficiently large offsets from the
design orbit. Ensuring a large stable area is another critical requirement for
the transverse motion in circular accelerators.
In this chapter we will briefly introduce the theory of transverse and longitudinal beam motion in accelerators.
1.1 BEAM MOTION IN MAGNET LATTICES
1.1.1 Hamiltonian and the equations of motion
In an accelerator the particles in a beam are expected to closely follow the
design path. It is the deviations of the particles from the design path that are
of our concern. Therefore, typically we adopt a moving curvilinear coordinate
system to describe the particle motion, using the design orbit as the reference