64 Beam-based Correction and Optimization for Accelerators
closed-orbit is synchronous with the RF cavities. The same iterative procedure
as in Eq. (2.124) can be used, except now the one-turn transfer matrix and
the orbit vectors are now 6-dimensional.
Linear optics functions:
The transfer matrix between any two locations in the lattice can be computed with numeric differences of the particle coordinates. For example, the
i’th column of the transfer matrix (i.e., the linear dependence of the exit coordinates on the i’th coordinate at entrance) is calculated by tracking two
particles whose initial i’th coordinate is shifted by ±∆, respectively,
R :i =
M(X 2 ) − M(X 1 )
2∆
,
(2.125)
where X 2 and X 1 are equal to the reference orbit, X 0 , except their i’th components are given by X 2 (i) = X 0 (i) + ∆, X 1 (i) = X 0 (i) − ∆. Numerically
∆ = 1 × 10
−8 may be used for double-precision computers.
The one-turn transfer matrix for a ring lattice can be similarly computed.
The matrix is usually calculated on the closed-orbit. For uncoupled lattices
(or with weak x-y coupling), the 2 × 2 matrices for the horizontal and vertical
planes can be used to calculate the betatron tunes and the Courant-Snyder
parameters, using Eq. (1.58). The C-S parameters at other locations and the
phase advances can be calculated with Eq. (1.70) and (1.71), respectively, using the transfer matrix between the two locations. Parametrization of coupled
motion can be done with the procedure described in Eqs. (2.59-2.63).
The dispersion functions at one location can be calculated from the oneturn transfer matrix using Eq. (2.37). Dispersion functions elsewhere can be
obtained by transporting the dispersion vector with the extended transfer
matrix. The momentum compaction factor can be calculated from the R 56
element of the one-turn transfer matrix with Eq. (2.49).
By calculating the transfer matrix with a small fixed momentum deviation and in turn the corresponding betatron tunes, the chromaticities can be
obtained. Chromatic beta beating can also be calculated with the off-energy
transfer matrix.
Nonlinear beam dynamics performance:
The nonlinear beam dynamics of a circular accelerator can be characterized
with particle tracking simulation. Basic nonlinear dynamics features, such as
betatron tune shifts with oscillation amplitudes and momentum deviation, are
typically computed for fixed-momentum particles (with RF cavities turned off
in the lattice). In these calculations, particles with a series of initial x, y, or δ
coordinates (while all other 5 coordinates are equal) are launched and tracked
for a number of turns (e.g., 1024 turns). For example, for tune shifts with the xamplitude, particles with initial coordinates y = 0.1 mm, p x = p y = z = δ = 0,
and x = −10 mm to 10 mm with a step size of 0.25 mm may be launched for a
ring with a dynamic aperture around 10 mm. A small initial y offset is used for
the purpose of evaluating the vertical tune from the orbit oscillations. Betatron
closed-orbit is synchronous with the RF cavities. The same iterative procedure
as in Eq. (2.124) can be used, except now the one-turn transfer matrix and
the orbit vectors are now 6-dimensional.
Linear optics functions:
The transfer matrix between any two locations in the lattice can be computed with numeric differences of the particle coordinates. For example, the
i’th column of the transfer matrix (i.e., the linear dependence of the exit coordinates on the i’th coordinate at entrance) is calculated by tracking two
particles whose initial i’th coordinate is shifted by ±∆, respectively,
R :i =
M(X 2 ) − M(X 1 )
2∆
,
(2.125)
where X 2 and X 1 are equal to the reference orbit, X 0 , except their i’th components are given by X 2 (i) = X 0 (i) + ∆, X 1 (i) = X 0 (i) − ∆. Numerically
∆ = 1 × 10
−8 may be used for double-precision computers.
The one-turn transfer matrix for a ring lattice can be similarly computed.
The matrix is usually calculated on the closed-orbit. For uncoupled lattices
(or with weak x-y coupling), the 2 × 2 matrices for the horizontal and vertical
planes can be used to calculate the betatron tunes and the Courant-Snyder
parameters, using Eq. (1.58). The C-S parameters at other locations and the
phase advances can be calculated with Eq. (1.70) and (1.71), respectively, using the transfer matrix between the two locations. Parametrization of coupled
motion can be done with the procedure described in Eqs. (2.59-2.63).
The dispersion functions at one location can be calculated from the oneturn transfer matrix using Eq. (2.37). Dispersion functions elsewhere can be
obtained by transporting the dispersion vector with the extended transfer
matrix. The momentum compaction factor can be calculated from the R 56
element of the one-turn transfer matrix with Eq. (2.49).
By calculating the transfer matrix with a small fixed momentum deviation and in turn the corresponding betatron tunes, the chromaticities can be
obtained. Chromatic beta beating can also be calculated with the off-energy
transfer matrix.
Nonlinear beam dynamics performance:
The nonlinear beam dynamics of a circular accelerator can be characterized
with particle tracking simulation. Basic nonlinear dynamics features, such as
betatron tune shifts with oscillation amplitudes and momentum deviation, are
typically computed for fixed-momentum particles (with RF cavities turned off
in the lattice). In these calculations, particles with a series of initial x, y, or δ
coordinates (while all other 5 coordinates are equal) are launched and tracked
for a number of turns (e.g., 1024 turns). For example, for tune shifts with the xamplitude, particles with initial coordinates y = 0.1 mm, p x = p y = z = δ = 0,
and x = −10 mm to 10 mm with a step size of 0.25 mm may be launched for a
ring with a dynamic aperture around 10 mm. A small initial y offset is used for
the purpose of evaluating the vertical tune from the orbit oscillations. Betatron