Beam dynamics topics 59
2.7 LATTICE MODELING AND PARTICLE TRACKING
While beam dynamics theories are very useful for understanding the nature
of the beam motion, the design and operation of an accelerator often demand
more precise and more detailed description of the beam motion and beam
properties of the particular machine, which can only be provided by a thorough lattice model. A lattice model consists of all the accelerator elements
that affect the beam motion and markers of critical locations at which beam
parameters may need to be evaluated. Beam motion in the machine can be
predicted with the lattice model by sequentially calculating the effects of the
individual elements on the particles. One only needs to improve the accuracy
of the modeling of the individual elements in order to accurately describe the
beam motion and the beam properties in a complex machine.
The beam motion through an element can be described either with a transfer map, or by directly tracking phase space coordinates of particles. For the
transfer map approach to achieve a high accuracy, it is necessary to expand
to the higher orders, which makes the map cumbersome and slow to evaluate. Transfer maps can also be used to track particles. However, the tracking
results are not symplectic as either the map is not symplectic (e.g., Taylor
maps) or the map needs to be truncated during evaluation (Lie maps). Hence
tracking with transfer maps is not suitable for the study of long-term stability.
In practice, it is more common to use element-by-element particle tracking.
In element-by-element particle tracking, the 6-dimensional phase space
coordinates, (x, p x , y, p y , z, δ)
T (or 4-dimensional if the longitudinal motion
is not included), are passed through each element from the entrance face to
the exit face. The coordinate changes in an element depend on the physical
process involved, the element parameters, and the initial coordinates. Using
particle tracking, the lattice features and beam properties can be evaluated.
2.7.1 Tracking different types of accelerator elements
Linear elements:
Tracking through the linear elements, namely, drift spaces, dipole magnets, and quadrupole magnets, is straightforward. The transfer matrices in
Eqs. (1.27), (1.32), and (1.37) (or the corresponding forms with negative gradients) can be directly applied. In these equations, the transfer matrices are
given for x
and y
coordinates, instead of p x and p y . Eq. (1.2) can be used to
convert between the coordinates for each particle before and after the matrices are applied. The gradients and the curvature in Eqs. (1.32) and (1.37) are
scaled with
1
1+δ , hence the effects of chromatic errors on the beam motion are
included in the tracking.
The only effect on the longitudinal coordinates by these linear elements
is a shift in z. This can be derived by integrating Eq. (1.12). The z shift
in drift spaces and quadrupoles are second order functions of the transverse
coordinates at the entrance. For dipole magnets, the transverse motion in the
2.7 LATTICE MODELING AND PARTICLE TRACKING
While beam dynamics theories are very useful for understanding the nature
of the beam motion, the design and operation of an accelerator often demand
more precise and more detailed description of the beam motion and beam
properties of the particular machine, which can only be provided by a thorough lattice model. A lattice model consists of all the accelerator elements
that affect the beam motion and markers of critical locations at which beam
parameters may need to be evaluated. Beam motion in the machine can be
predicted with the lattice model by sequentially calculating the effects of the
individual elements on the particles. One only needs to improve the accuracy
of the modeling of the individual elements in order to accurately describe the
beam motion and the beam properties in a complex machine.
The beam motion through an element can be described either with a transfer map, or by directly tracking phase space coordinates of particles. For the
transfer map approach to achieve a high accuracy, it is necessary to expand
to the higher orders, which makes the map cumbersome and slow to evaluate. Transfer maps can also be used to track particles. However, the tracking
results are not symplectic as either the map is not symplectic (e.g., Taylor
maps) or the map needs to be truncated during evaluation (Lie maps). Hence
tracking with transfer maps is not suitable for the study of long-term stability.
In practice, it is more common to use element-by-element particle tracking.
In element-by-element particle tracking, the 6-dimensional phase space
coordinates, (x, p x , y, p y , z, δ)
T (or 4-dimensional if the longitudinal motion
is not included), are passed through each element from the entrance face to
the exit face. The coordinate changes in an element depend on the physical
process involved, the element parameters, and the initial coordinates. Using
particle tracking, the lattice features and beam properties can be evaluated.
2.7.1 Tracking different types of accelerator elements
Linear elements:
Tracking through the linear elements, namely, drift spaces, dipole magnets, and quadrupole magnets, is straightforward. The transfer matrices in
Eqs. (1.27), (1.32), and (1.37) (or the corresponding forms with negative gradients) can be directly applied. In these equations, the transfer matrices are
given for x
and y
coordinates, instead of p x and p y . Eq. (1.2) can be used to
convert between the coordinates for each particle before and after the matrices are applied. The gradients and the curvature in Eqs. (1.32) and (1.37) are
scaled with
1
1+δ , hence the effects of chromatic errors on the beam motion are
included in the tracking.
The only effect on the longitudinal coordinates by these linear elements
is a shift in z. This can be derived by integrating Eq. (1.12). The z shift
in drift spaces and quadrupoles are second order functions of the transverse
coordinates at the entrance. For dipole magnets, the transverse motion in the