Beam dynamics topics 51
s
x
f
δ > 0
δ = 0
δ < 0
s
x
f
δ > 0
δ = 0
δ < 0
Figure 2.6 Illustration of chromaticity correction with sextupole magnets. Top: focusing error of a quadrupole for off-momentum particles. Bottom: the feed-down
quadrupole field from a sextupole magnet provides correction to the focusing errors.
be corrected exactly at the locations of the error sources, correction of the
chromaticities usually does not completely eliminate chromatic beta beating.
The chromatic beta beating due to quadrupoles and sextupoles in the lattice
is a systematic error. It is important to properly arrange these magnets in the
lattice design in order to avoid excessive chromatic beta beating.
2.6 NONLINEAR BEAM DYNAMICS
Sextupole magnets are introduced into circular accelerators to correct chromaticities. The magnetic fields in sextupoles are nonlinear with respect to
transverse positions of the beam particles. The nonlinear forces can lead to
unstable beam motion and beam loss when the transverse offsets are sufficiently large as the particle motion can be driven onto nonlinear resonances
and become unstable under large oscillation amplitudes. A large stability region is critical for storage ring lattice designs.
The nonlinear beam motion in circular accelerator lattices can be analyzed
with the Hamiltonian dynamics approach or the Lie map approach.
2.6.1 Hamiltonian dynamics approach
In general, the Hamiltonian that describes the beam motion in an accelerator
lattice can be split into two parts, one representing the ideal linear motion by
s
x
f
δ > 0
δ = 0
δ < 0
s
x
f
δ > 0
δ = 0
δ < 0
Figure 2.6 Illustration of chromaticity correction with sextupole magnets. Top: focusing error of a quadrupole for off-momentum particles. Bottom: the feed-down
quadrupole field from a sextupole magnet provides correction to the focusing errors.
be corrected exactly at the locations of the error sources, correction of the
chromaticities usually does not completely eliminate chromatic beta beating.
The chromatic beta beating due to quadrupoles and sextupoles in the lattice
is a systematic error. It is important to properly arrange these magnets in the
lattice design in order to avoid excessive chromatic beta beating.
2.6 NONLINEAR BEAM DYNAMICS
Sextupole magnets are introduced into circular accelerators to correct chromaticities. The magnetic fields in sextupoles are nonlinear with respect to
transverse positions of the beam particles. The nonlinear forces can lead to
unstable beam motion and beam loss when the transverse offsets are sufficiently large as the particle motion can be driven onto nonlinear resonances
and become unstable under large oscillation amplitudes. A large stability region is critical for storage ring lattice designs.
The nonlinear beam motion in circular accelerator lattices can be analyzed
with the Hamiltonian dynamics approach or the Lie map approach.
2.6.1 Hamiltonian dynamics approach
In general, the Hamiltonian that describes the beam motion in an accelerator
lattice can be split into two parts, one representing the ideal linear motion by