Beam dynamics topics 63
be drawn from a Gaussian distribution with the proper mean and standard
deviation. Since the use of kicks with negative field lengths introduces extra
excitation, the second order symplectic integrator, in which the kick length is
positive, is preferred.
Misalignment:
There are always errors in the positions and orientations of the accelerator
elements in a real machine as compared to the ideal design. These errors
are referred to as misalignment. Misalignment of magnets can significantly
impact the accelerator performance. For example, transverse position shifts
of quadrupole magnets produce dipole kicks to the beam through the feeddown effects. The kicks cause closed orbit offsets, typically much larger than
the alignment errors themselves. The ratio of the induced rms orbit offset
to the rms misalignment of magnets is called the amplification factor, which
typically ranges from 10 to 100. Small misalignment errors, such as 100 µm,
can cause large orbit errors on the order of 1 to 10 mm and in turn optics
errors, coupling, and degradation of nonlinear dynamics performance.
Modeling of small alignment errors can be done by performing coordinate
transformations at the entrance and exit faces of the misaligned elements.
For example, for a horizontal alignment error of ∆x, the x-coordinate of the
particle is first shifted by −∆x at the entrance, and after tracking through
the element, shifted back by ∆x at the exit. When multiple alignment errors
are modeled for one element, the transformations are applied in the opposite
order at the exit and entrance faces.
2.7.2 Calculation of lattice functions and beam parameters
With the ability to track phase space coordinates of particles through the
lattice, various lattice functions can be calculated.
Closed-orbit:
The closed-orbit can be found by solving for a coordinate vector that
satisfies the fixed-point condition M(X c ) = X c , where M represents the oneturn map, here executed by particle tracking simulation.
There are two scenarios. First, the lattice consists of no RF element and
a closed-orbit is found for the on-energy particle or a particle with a given
momentum deviation, δ. The 4-dimensional closed-orbit vector, X c =(x, p x , y,
p y )
T
c , can be found iteratively. At each iteration, the present solution, X n , is
tracked for one-turn. The solution for the next iteration is then given by
X n+1 = X n + [I − R 4 (X n )]
−1 (M(X n ) − X n ),
(2.124)
where R 4 (X n ) is the 4 × 4 transfer matrix on orbit X n (which can be approximated with R 4 (X 0 )), and the longitudinal coordinates of (0, δ) are used in
tracking. The initial solution may be set to X 0 = (0, 0, 0, 0)
T .
In the second scenario, the lattice can have RF elements and other elements
that change the beam energy, for example, radiation damping or impedance
elements. The goal is to find the 6-dimensional closed-orbit. A particle on this
be drawn from a Gaussian distribution with the proper mean and standard
deviation. Since the use of kicks with negative field lengths introduces extra
excitation, the second order symplectic integrator, in which the kick length is
positive, is preferred.
Misalignment:
There are always errors in the positions and orientations of the accelerator
elements in a real machine as compared to the ideal design. These errors
are referred to as misalignment. Misalignment of magnets can significantly
impact the accelerator performance. For example, transverse position shifts
of quadrupole magnets produce dipole kicks to the beam through the feeddown effects. The kicks cause closed orbit offsets, typically much larger than
the alignment errors themselves. The ratio of the induced rms orbit offset
to the rms misalignment of magnets is called the amplification factor, which
typically ranges from 10 to 100. Small misalignment errors, such as 100 µm,
can cause large orbit errors on the order of 1 to 10 mm and in turn optics
errors, coupling, and degradation of nonlinear dynamics performance.
Modeling of small alignment errors can be done by performing coordinate
transformations at the entrance and exit faces of the misaligned elements.
For example, for a horizontal alignment error of ∆x, the x-coordinate of the
particle is first shifted by −∆x at the entrance, and after tracking through
the element, shifted back by ∆x at the exit. When multiple alignment errors
are modeled for one element, the transformations are applied in the opposite
order at the exit and entrance faces.
2.7.2 Calculation of lattice functions and beam parameters
With the ability to track phase space coordinates of particles through the
lattice, various lattice functions can be calculated.
Closed-orbit:
The closed-orbit can be found by solving for a coordinate vector that
satisfies the fixed-point condition M(X c ) = X c , where M represents the oneturn map, here executed by particle tracking simulation.
There are two scenarios. First, the lattice consists of no RF element and
a closed-orbit is found for the on-energy particle or a particle with a given
momentum deviation, δ. The 4-dimensional closed-orbit vector, X c =(x, p x , y,
p y )
T
c , can be found iteratively. At each iteration, the present solution, X n , is
tracked for one-turn. The solution for the next iteration is then given by
X n+1 = X n + [I − R 4 (X n )]
−1 (M(X n ) − X n ),
(2.124)
where R 4 (X n ) is the 4 × 4 transfer matrix on orbit X n (which can be approximated with R 4 (X 0 )), and the longitudinal coordinates of (0, δ) are used in
tracking. The initial solution may be set to X 0 = (0, 0, 0, 0)
T .
In the second scenario, the lattice can have RF elements and other elements
that change the beam energy, for example, radiation damping or impedance
elements. The goal is to find the 6-dimensional closed-orbit. A particle on this