C H A P T E R 1
Basics of beam
dynamics
CONTENTS
1.1
Beam motion in magnet lattices . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1 Hamiltonian and the equations of motion . . . . . . . . .
4
1.1.2 Magnets and magnetic fields . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Transverse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.1 Beam motion in linear components . . . . . . . . . . . . . . . .
9
1.2.2 Transfer matrix and transfer map . . . . . . . . . . . . . . . . . 13
1.2.3 Strong focusing principle and orbit stability . . . . . . 15
1.2.4 Courant-Snyder parametrization . . . . . . . . . . . . . . . . . . 17
1.2.5 Propagation of linear optics functions . . . . . . . . . . . . . 18
1.2.6 Beam distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3
Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A particle beam consists of particles that move roughly with the same speed
and in the same direction within a finite cross-section. In an accelerator, electromagnetic fields of various distributions in space and time are placed along
the path of the beam through the accelerator components to guide the beam
motion, change the beam energy, and provide focusing. Beam motion is also
affected by the electromagnetic fields generated by the beam itself, directly
or through the interactions with the environment. The study of beam motion
under the influence of electromagnetic fields in accelerators is called beam
dynamics.
It is necessary to understand the basics of beam dynamics in linacs and
synchrotrons as it is the foundation for the discussions of the operation requirements and the methods of fulfilling the requirements through beam-based
methods. Only single particle dynamics is covered here. The transverse dynamics describes beam motion in its deviation from the design orbit in the
plane perpendicular to the design orbit. The longitudinal dynamics describes
the motion in the direction along the design orbit, which involves oscillations
of beam energy and arrival time.
3
C H A P T E R 1
Basics of beam
dynamics
CONTENTS
1.1
Beam motion in magnet lattices . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1 Hamiltonian and the equations of motion . . . . . . . . .
4
1.1.2 Magnets and magnetic fields . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Transverse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.1 Beam motion in linear components . . . . . . . . . . . . . . . .
9
1.2.2 Transfer matrix and transfer map . . . . . . . . . . . . . . . . . 13
1.2.3 Strong focusing principle and orbit stability . . . . . . 15
1.2.4 Courant-Snyder parametrization . . . . . . . . . . . . . . . . . . 17
1.2.5 Propagation of linear optics functions . . . . . . . . . . . . . 18
1.2.6 Beam distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3
Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A particle beam consists of particles that move roughly with the same speed
and in the same direction within a finite cross-section. In an accelerator, electromagnetic fields of various distributions in space and time are placed along
the path of the beam through the accelerator components to guide the beam
motion, change the beam energy, and provide focusing. Beam motion is also
affected by the electromagnetic fields generated by the beam itself, directly
or through the interactions with the environment. The study of beam motion
under the influence of electromagnetic fields in accelerators is called beam
dynamics.
It is necessary to understand the basics of beam dynamics in linacs and
synchrotrons as it is the foundation for the discussions of the operation requirements and the methods of fulfilling the requirements through beam-based
methods. Only single particle dynamics is covered here. The transverse dynamics describes beam motion in its deviation from the design orbit in the
plane perpendicular to the design orbit. The longitudinal dynamics describes
the motion in the direction along the design orbit, which involves oscillations
of beam energy and arrival time.
3
C H A P T E R 1
Basics of beam
dynamics
CONTENTS
1.1
Beam motion in magnet lattices . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1 Hamiltonian and the equations of motion . . . . . . . . .
4
1.1.2 Magnets and magnetic fields . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Transverse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.1 Beam motion in linear components . . . . . . . . . . . . . . . .
9
1.2.2 Transfer matrix and transfer map . . . . . . . . . . . . . . . . . 13
1.2.3 Strong focusing principle and orbit stability . . . . . . 15
1.2.4 Courant-Snyder parametrization . . . . . . . . . . . . . . . . . . 17
1.2.5 Propagation of linear optics functions . . . . . . . . . . . . . 18
1.2.6 Beam distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3
Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A particle beam consists of particles that move roughly with the same speed
and in the same direction within a finite cross-section. In an accelerator, electromagnetic fields of various distributions in space and time are placed along
the path of the beam through the accelerator components to guide the beam
motion, change the beam energy, and provide focusing. Beam motion is also
affected by the electromagnetic fields generated by the beam itself, directly
or through the interactions with the environment. The study of beam motion
under the influence of electromagnetic fields in accelerators is called beam
dynamics.
It is necessary to understand the basics of beam dynamics in linacs and
synchrotrons as it is the foundation for the discussions of the operation requirements and the methods of fulfilling the requirements through beam-based
methods. Only single particle dynamics is covered here. The transverse dynamics describes beam motion in its deviation from the design orbit in the
plane perpendicular to the design orbit. The longitudinal dynamics describes
the motion in the direction along the design orbit, which involves oscillations
of beam energy and arrival time.
3
DOI: 10.1201/9780429434358-1
Basics of beam
dynamics
CONTENTS
1.1
Beam motion in magnet lattices . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1 Hamiltonian and the equations of motion . . . . . . . . .
4
1.1.2 Magnets and magnetic fields . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Transverse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.1 Beam motion in linear components . . . . . . . . . . . . . . . .
9
1.2.2 Transfer matrix and transfer map . . . . . . . . . . . . . . . . . 13
1.2.3 Strong focusing principle and orbit stability . . . . . . 15
1.2.4 Courant-Snyder parametrization . . . . . . . . . . . . . . . . . . 17
1.2.5 Propagation of linear optics functions . . . . . . . . . . . . . 18
1.2.6 Beam distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3
Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A particle beam consists of particles that move roughly with the same speed
and in the same direction within a finite cross-section. In an accelerator, electromagnetic fields of various distributions in space and time are placed along
the path of the beam through the accelerator components to guide the beam
motion, change the beam energy, and provide focusing. Beam motion is also
affected by the electromagnetic fields generated by the beam itself, directly
or through the interactions with the environment. The study of beam motion
under the influence of electromagnetic fields in accelerators is called beam
dynamics.
It is necessary to understand the basics of beam dynamics in linacs and
synchrotrons as it is the foundation for the discussions of the operation requirements and the methods of fulfilling the requirements through beam-based
methods. Only single particle dynamics is covered here. The transverse dynamics describes beam motion in its deviation from the design orbit in the
plane perpendicular to the design orbit. The longitudinal dynamics describes
the motion in the direction along the design orbit, which involves oscillations
of beam energy and arrival time.
3
C H A P T E R 1
Basics of beam
dynamics
CONTENTS
1.1
Beam motion in magnet lattices . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1 Hamiltonian and the equations of motion . . . . . . . . .
4
1.1.2 Magnets and magnetic fields . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Transverse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.1 Beam motion in linear components . . . . . . . . . . . . . . . .
9
1.2.2 Transfer matrix and transfer map . . . . . . . . . . . . . . . . . 13
1.2.3 Strong focusing principle and orbit stability . . . . . . 15
1.2.4 Courant-Snyder parametrization . . . . . . . . . . . . . . . . . . 17
1.2.5 Propagation of linear optics functions . . . . . . . . . . . . . 18
1.2.6 Beam distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3
Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A particle beam consists of particles that move roughly with the same speed
and in the same direction within a finite cross-section. In an accelerator, electromagnetic fields of various distributions in space and time are placed along
the path of the beam through the accelerator components to guide the beam
motion, change the beam energy, and provide focusing. Beam motion is also
affected by the electromagnetic fields generated by the beam itself, directly
or through the interactions with the environment. The study of beam motion
under the influence of electromagnetic fields in accelerators is called beam
dynamics.
It is necessary to understand the basics of beam dynamics in linacs and
synchrotrons as it is the foundation for the discussions of the operation requirements and the methods of fulfilling the requirements through beam-based
methods. Only single particle dynamics is covered here. The transverse dynamics describes beam motion in its deviation from the design orbit in the
plane perpendicular to the design orbit. The longitudinal dynamics describes
the motion in the direction along the design orbit, which involves oscillations
of beam energy and arrival time.
3
C H A P T E R 1
Basics of beam
dynamics
CONTENTS
1.1
Beam motion in magnet lattices . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1 Hamiltonian and the equations of motion . . . . . . . . .
4
1.1.2 Magnets and magnetic fields . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Transverse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.1 Beam motion in linear components . . . . . . . . . . . . . . . .
9
1.2.2 Transfer matrix and transfer map . . . . . . . . . . . . . . . . . 13
1.2.3 Strong focusing principle and orbit stability . . . . . . 15
1.2.4 Courant-Snyder parametrization . . . . . . . . . . . . . . . . . . 17
1.2.5 Propagation of linear optics functions . . . . . . . . . . . . . 18
1.2.6 Beam distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3
Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A particle beam consists of particles that move roughly with the same speed
and in the same direction within a finite cross-section. In an accelerator, electromagnetic fields of various distributions in space and time are placed along
the path of the beam through the accelerator components to guide the beam
motion, change the beam energy, and provide focusing. Beam motion is also
affected by the electromagnetic fields generated by the beam itself, directly
or through the interactions with the environment. The study of beam motion
under the influence of electromagnetic fields in accelerators is called beam
dynamics.
It is necessary to understand the basics of beam dynamics in linacs and
synchrotrons as it is the foundation for the discussions of the operation requirements and the methods of fulfilling the requirements through beam-based
methods. Only single particle dynamics is covered here. The transverse dynamics describes beam motion in its deviation from the design orbit in the
plane perpendicular to the design orbit. The longitudinal dynamics describes
the motion in the direction along the design orbit, which involves oscillations
of beam energy and arrival time.
3
DOI: 10.1201/9780429434358-1