Beam dynamics topics 55
which vanish, unless p is either 0 or a multiple of N . Therefore, it is beneficial
to retain high periodicity in the design of a storage ring lattice.
In a typical storage ring, the main sources of nonlinearity in the beam motion are the sextupole magnets. From Eq. (2.84), it is straightforward to show
that in the resonance basis the sextupole Hamiltonian consists of the following geometric terms: h 3000 , h 0300 , h 2100 , h 1200 , h 1011 , h 0111 , h 1020 , h 0102 , h 1002 ,
and h 0120 (here the fifth index, 0, is suppressed in the subscript). Each of these
terms contributes to driving a corresponding resonance. These resonances include 3ν x = p, ν x = p, ν x +2ν y = p, and ν x −2ν y = p. The above resonances are
driven by sextupoles through their direct impact on the linear motion, which
corresponds to the first order perturbation to the linear motion. The effect of
sextupole fields on the nonlinear beam motion perturbed by other sextupoles
or themselves on previous passes gives rise to additional resonances, which correspond to terms from the second or higher order perturbations. Resonances
driven by sextupoles through the second order perturbation include 4ν x = p,
2ν x = p, 2ν y = p, 4ν x ± 2ν y = p, and 2ν x ± 2ν y = p.
Aside from the systematic nonlinear resonances driven by sextupoles in the
design, a lattice always has field errors that are systematic or random deviations from the design. The field errors will drive many nonlinear resonances.
When the tunes of oscillating particles are shifted onto certain nonlinear resonances due to nonlinear detuning or linear and nonlinear chromaticities in
a storage ring, the particles can get lost. Particle loss from the nonlinear
beam motion limits the dynamic aperture and the local momentum aperture.
Sufficiently large dynamic aperture and local momentum aperture are basic
requirements for the operation of a storage ring. The lattice design of a storage
ring often relies on extensive optimization of the linear and nonlinear optics
to achieve the desired nonlinear dynamics performance. However, linear and
nonlinear errors in the real machine cause the operation conditions to deviate
from the design. During the commissioning phase, it is necessary to correct the
errors in the machine in order to restore the lattice performance. Beam-based
optimization may be used to compensate the effects of the errors when direct
correction methods are not available.
2.6.2 Lie map approach
The Hamiltonian dynamics approach gives a continuous description of the
beam motion. In the accelerator context, it often suffices to know the transfer
map between two locations. The transfer map can be given by a Taylor expansion of the coordinates at location 2 in terms of the coordinates at location
1, as shown in Eq. (1.48). The Taylor map is easy to evaluate, but it quickly
becomes large and cumbersome when it is extended to higher orders. More
importantly, the truncated Taylor map is generally not symplectic and hence
not ideal for the study of long term stability of beam motion. The Lie map is
an alternative representation of the transfer map, which is not only symplectic
but also compact.
which vanish, unless p is either 0 or a multiple of N . Therefore, it is beneficial
to retain high periodicity in the design of a storage ring lattice.
In a typical storage ring, the main sources of nonlinearity in the beam motion are the sextupole magnets. From Eq. (2.84), it is straightforward to show
that in the resonance basis the sextupole Hamiltonian consists of the following geometric terms: h 3000 , h 0300 , h 2100 , h 1200 , h 1011 , h 0111 , h 1020 , h 0102 , h 1002 ,
and h 0120 (here the fifth index, 0, is suppressed in the subscript). Each of these
terms contributes to driving a corresponding resonance. These resonances include 3ν x = p, ν x = p, ν x +2ν y = p, and ν x −2ν y = p. The above resonances are
driven by sextupoles through their direct impact on the linear motion, which
corresponds to the first order perturbation to the linear motion. The effect of
sextupole fields on the nonlinear beam motion perturbed by other sextupoles
or themselves on previous passes gives rise to additional resonances, which correspond to terms from the second or higher order perturbations. Resonances
driven by sextupoles through the second order perturbation include 4ν x = p,
2ν x = p, 2ν y = p, 4ν x ± 2ν y = p, and 2ν x ± 2ν y = p.
Aside from the systematic nonlinear resonances driven by sextupoles in the
design, a lattice always has field errors that are systematic or random deviations from the design. The field errors will drive many nonlinear resonances.
When the tunes of oscillating particles are shifted onto certain nonlinear resonances due to nonlinear detuning or linear and nonlinear chromaticities in
a storage ring, the particles can get lost. Particle loss from the nonlinear
beam motion limits the dynamic aperture and the local momentum aperture.
Sufficiently large dynamic aperture and local momentum aperture are basic
requirements for the operation of a storage ring. The lattice design of a storage
ring often relies on extensive optimization of the linear and nonlinear optics
to achieve the desired nonlinear dynamics performance. However, linear and
nonlinear errors in the real machine cause the operation conditions to deviate
from the design. During the commissioning phase, it is necessary to correct the
errors in the machine in order to restore the lattice performance. Beam-based
optimization may be used to compensate the effects of the errors when direct
correction methods are not available.
2.6.2 Lie map approach
The Hamiltonian dynamics approach gives a continuous description of the
beam motion. In the accelerator context, it often suffices to know the transfer
map between two locations. The transfer map can be given by a Taylor expansion of the coordinates at location 2 in terms of the coordinates at location
1, as shown in Eq. (1.48). The Taylor map is easy to evaluate, but it quickly
becomes large and cumbersome when it is extended to higher orders. More
importantly, the truncated Taylor map is generally not symplectic and hence
not ideal for the study of long term stability of beam motion. The Lie map is
an alternative representation of the transfer map, which is not only symplectic
but also compact.