DOI: 10.1201/9780429434358-2
C H A P T E R 2
Beam dynamics topics
CONTENTS
2.1
Beam orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2
Linear optics errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4
Linear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5
Chromatic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6
Nonlinear beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1 Hamiltonian dynamics approach . . . . . . . . . . . . . . . . . . . 51
2.6.2 Lie map approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7
Lattice modeling and particle tracking . . . . . . . . . . . . . . . . . . . . 59
2.7.1 Tracking different types of accelerator elements . . . 59
2.7.2 Calculation of lattice functions and beam
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
In Chapter 1 we studied the linear beam motion in an ideal lattice. A real
accelerator has all sorts of errors, which cause the beam motion to deviate
from the design, in terms of the beam orbit, linear optics, and linear coupling.
The intrinsic field errors in dipole and quadrupole magnets for off-energy particles introduce dispersion and chromatic aberration. The use of sextupole
magnets to correct chromaticity in storage rings makes the beam motion nonlinear. These effects are discussed in this chapter. The accurate modeling of
accelerators with simulation codes is also discussed.
2.1 BEAM ORBIT
In the ideal scenario, the magnetic fields along the beam path are identical
to what is specified in the design, and hence a beam launched on the design
orbit will travel on the design orbit. However, the actual field distribution on
the beam path will always be different from the design. The differences are
the magnetic field errors, which can be due to magnet imperfections, calibration errors, power supply regulation errors, power grid ripples, and magnet
misalignment. A major impact of the field errors to the beam is to cause the
31
C H A P T E R 2
Beam dynamics topics
CONTENTS
2.1
Beam orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2
Linear optics errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4
Linear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5
Chromatic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6
Nonlinear beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1 Hamiltonian dynamics approach . . . . . . . . . . . . . . . . . . . 51
2.6.2 Lie map approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7
Lattice modeling and particle tracking . . . . . . . . . . . . . . . . . . . . 59
2.7.1 Tracking different types of accelerator elements . . . 59
2.7.2 Calculation of lattice functions and beam
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
In Chapter 1 we studied the linear beam motion in an ideal lattice. A real
accelerator has all sorts of errors, which cause the beam motion to deviate
from the design, in terms of the beam orbit, linear optics, and linear coupling.
The intrinsic field errors in dipole and quadrupole magnets for off-energy particles introduce dispersion and chromatic aberration. The use of sextupole
magnets to correct chromaticity in storage rings makes the beam motion nonlinear. These effects are discussed in this chapter. The accurate modeling of
accelerators with simulation codes is also discussed.
2.1 BEAM ORBIT
In the ideal scenario, the magnetic fields along the beam path are identical
to what is specified in the design, and hence a beam launched on the design
orbit will travel on the design orbit. However, the actual field distribution on
the beam path will always be different from the design. The differences are
the magnetic field errors, which can be due to magnet imperfections, calibration errors, power supply regulation errors, power grid ripples, and magnet
misalignment. A major impact of the field errors to the beam is to cause the
31
C H A P T E R 2
Beam dynamics topics
CONTENTS
2.1
Beam orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2
Linear optics errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4
Linear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5
Chromatic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6
Nonlinear beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1 Hamiltonian dynamics approach . . . . . . . . . . . . . . . . . . . 51
2.6.2 Lie map approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7
Lattice modeling and particle tracking . . . . . . . . . . . . . . . . . . . . 59
2.7.1 Tracking different types of accelerator elements . . . 59
2.7.2 Calculation of lattice functions and beam
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
In Chapter 1 we studied the linear beam motion in an ideal lattice. A real
accelerator has all sorts of errors, which cause the beam motion to deviate
from the design, in terms of the beam orbit, linear optics, and linear coupling.
The intrinsic field errors in dipole and quadrupole magnets for off-energy particles introduce dispersion and chromatic aberration. The use of sextupole
magnets to correct chromaticity in storage rings makes the beam motion nonlinear. These effects are discussed in this chapter. The accurate modeling of
accelerators with simulation codes is also discussed.
2.1 BEAM ORBIT
In the ideal scenario, the magnetic fields along the beam path are identical
to what is specified in the design, and hence a beam launched on the design
orbit will travel on the design orbit. However, the actual field distribution on
the beam path will always be different from the design. The differences are
the magnetic field errors, which can be due to magnet imperfections, calibration errors, power supply regulation errors, power grid ripples, and magnet
misalignment. A major impact of the field errors to the beam is to cause the
31
C H A P T E R 2
Beam dynamics topics
CONTENTS
2.1
Beam orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2
Linear optics errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4
Linear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5
Chromatic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6
Nonlinear beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1 Hamiltonian dynamics approach . . . . . . . . . . . . . . . . . . . 51
2.6.2 Lie map approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7
Lattice modeling and particle tracking . . . . . . . . . . . . . . . . . . . . 59
2.7.1 Tracking different types of accelerator elements . . . 59
2.7.2 Calculation of lattice functions and beam
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
In Chapter 1 we studied the linear beam motion in an ideal lattice. A real
accelerator has all sorts of errors, which cause the beam motion to deviate
from the design, in terms of the beam orbit, linear optics, and linear coupling.
The intrinsic field errors in dipole and quadrupole magnets for off-energy particles introduce dispersion and chromatic aberration. The use of sextupole
magnets to correct chromaticity in storage rings makes the beam motion nonlinear. These effects are discussed in this chapter. The accurate modeling of
accelerators with simulation codes is also discussed.
2.1 BEAM ORBIT
In the ideal scenario, the magnetic fields along the beam path are identical
to what is specified in the design, and hence a beam launched on the design
orbit will travel on the design orbit. However, the actual field distribution on
the beam path will always be different from the design. The differences are
the magnetic field errors, which can be due to magnet imperfections, calibration errors, power supply regulation errors, power grid ripples, and magnet
misalignment. A major impact of the field errors to the beam is to cause the
31
C H A P T E R 2
Beam dynamics topics
CONTENTS
2.1
Beam orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2
Linear optics errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4
Linear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5
Chromatic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6
Nonlinear beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1 Hamiltonian dynamics approach . . . . . . . . . . . . . . . . . . . 51
2.6.2 Lie map approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7
Lattice modeling and particle tracking . . . . . . . . . . . . . . . . . . . . 59
2.7.1 Tracking different types of accelerator elements . . . 59
2.7.2 Calculation of lattice functions and beam
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
In Chapter 1 we studied the linear beam motion in an ideal lattice. A real
accelerator has all sorts of errors, which cause the beam motion to deviate
from the design, in terms of the beam orbit, linear optics, and linear coupling.
The intrinsic field errors in dipole and quadrupole magnets for off-energy particles introduce dispersion and chromatic aberration. The use of sextupole
magnets to correct chromaticity in storage rings makes the beam motion nonlinear. These effects are discussed in this chapter. The accurate modeling of
accelerators with simulation codes is also discussed.
2.1 BEAM ORBIT
In the ideal scenario, the magnetic fields along the beam path are identical
to what is specified in the design, and hence a beam launched on the design
orbit will travel on the design orbit. However, the actual field distribution on
the beam path will always be different from the design. The differences are
the magnetic field errors, which can be due to magnet imperfections, calibration errors, power supply regulation errors, power grid ripples, and magnet
misalignment. A major impact of the field errors to the beam is to cause the
31
C H A P T E R 2
Beam dynamics topics
CONTENTS
2.1
Beam orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2
Linear optics errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4
Linear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5
Chromatic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6
Nonlinear beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1 Hamiltonian dynamics approach . . . . . . . . . . . . . . . . . . . 51
2.6.2 Lie map approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7
Lattice modeling and particle tracking . . . . . . . . . . . . . . . . . . . . 59
2.7.1 Tracking different types of accelerator elements . . . 59
2.7.2 Calculation of lattice functions and beam
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
In Chapter 1 we studied the linear beam motion in an ideal lattice. A real
accelerator has all sorts of errors, which cause the beam motion to deviate
from the design, in terms of the beam orbit, linear optics, and linear coupling.
The intrinsic field errors in dipole and quadrupole magnets for off-energy particles introduce dispersion and chromatic aberration. The use of sextupole
magnets to correct chromaticity in storage rings makes the beam motion nonlinear. These effects are discussed in this chapter. The accurate modeling of
accelerators with simulation codes is also discussed.
2.1 BEAM ORBIT
In the ideal scenario, the magnetic fields along the beam path are identical
to what is specified in the design, and hence a beam launched on the design
orbit will travel on the design orbit. However, the actual field distribution on
the beam path will always be different from the design. The differences are
the magnetic field errors, which can be due to magnet imperfections, calibration errors, power supply regulation errors, power grid ripples, and magnet
misalignment. A major impact of the field errors to the beam is to cause the
31