Chapter 2
Beam Dynamics
E. Wilson and B. J. Holzer
2.1 Linear Transverse Beam Dynamics
Now let us look in detail at the motion of particles in the transverse coordinates of
the coordinate system defined in Fig. 2.1.
2.1.1 Co-ordinate System
The guide field of a synchrotron is usually vertically directed, causing the particle to
follow a curved path in the horizontal plane (Fig. 2.1). The force guiding the particle
in a circle is horizontal and is given by:
F = e· v × B,
(2.1)
where:
v is the velocity of the charged particle in the direction tangential to its path,
B is the magnetic guide field.
The guide field inside a dipole magnet is uniform and the ideal motion of the
particle is simply a circle of (local) radius of curvature, ρ(s). The trajectory of an
ideal particle (ideal in energy and without any amplitude) that is defined by the
arrangement of the dipole magnets is called design orbit. The machine is usually
designed with this orbit at the centre of its vacuum chamber. Now there is no such
E. Wilson · B. J. Holzer ()
CERN (European Organization for Nuclear Research), Meyrin, Geneva, Switzerland
e-mail: Bernhard.holzer@cern.ch
© The Author(s) 2020
S. Myers, H. Schopper (eds.), Particle Physics Reference Library,
https://doi.org/10.1007/978-3-030-34245-6_2
15
Beam Dynamics
E. Wilson and B. J. Holzer
2.1 Linear Transverse Beam Dynamics
Now let us look in detail at the motion of particles in the transverse coordinates of
the coordinate system defined in Fig. 2.1.
2.1.1 Co-ordinate System
The guide field of a synchrotron is usually vertically directed, causing the particle to
follow a curved path in the horizontal plane (Fig. 2.1). The force guiding the particle
in a circle is horizontal and is given by:
F = e· v × B,
(2.1)
where:
v is the velocity of the charged particle in the direction tangential to its path,
B is the magnetic guide field.
The guide field inside a dipole magnet is uniform and the ideal motion of the
particle is simply a circle of (local) radius of curvature, ρ(s). The trajectory of an
ideal particle (ideal in energy and without any amplitude) that is defined by the
arrangement of the dipole magnets is called design orbit. The machine is usually
designed with this orbit at the centre of its vacuum chamber. Now there is no such
E. Wilson · B. J. Holzer ()
CERN (European Organization for Nuclear Research), Meyrin, Geneva, Switzerland
e-mail: Bernhard.holzer@cern.ch
© The Author(s) 2020
S. Myers, H. Schopper (eds.), Particle Physics Reference Library,
https://doi.org/10.1007/978-3-030-34245-6_2
15