66 Beam-based Correction and Optimization for Accelerators
Beam distribution parameters:
The 6-dimensional second order moment matrix (also known as the Σmatrix) defined by
Σ = XX
T =
XX
T ρ(X)dX,
(2.126)
completely characterizes a beam in Gaussian distribution, ρ(X) =
1
(2π) 3
√
det Σ
exp(−
1
2 X
T Σ
−1 X) (assumed to be centered on the reference orbit for notation simplicity, i.e., with X = 0). The Σ-matrix is also often
used to characterize beams in other distributions. If M is the transfer matrix between two locations, X 2 = MX 1 , it is straightforward to show that
Σ 2 = MΣ 1 M
T . For a symplectic matrix M (see Eq. (1.44)), it follows that
Σ 2 S = MΣ 1 SM
−1 and hence the eigenvalues of ΣS do not change in symplectic transportations [120]. It can be shown that the 6 eigenvalues are ±ii k ,
k = 1, 2, 3, and det Σ =
2
1
2
2
2
3 , where k are the eigen-emittances for the three
degrees of freedom of particle motion. In the typical case of weak coupling,
the eigen-emittances correspond to the three planes, x, y, and z, respectively.
In electron storage rings, the beam reaches an equilibrium distribution due
to radiation damping and quantum excitation. For an uncoupled lattice, the
emittances can be calculated from radiation integrals, which are determined
by the linear lattice functions [106]. However, it becomes more complicated
when there exists linear coupling between the horizontal and vertical planes.
A general procedure can be applied to track through the lattice elements
to obtain the transfer matrix with radiation damping and the accumulated
quantum diffusion effects, which are then used to solve for the Σ-matrix [90].
The equilibrium distribution can also be found by long-term multi-particle
tracking. Radiation damping, quantum excitation, and RF cavities are turned
on in the lattice. A large number of particles (e.g., ≥ 1000) are launched
with random initial coordinates and are tracked for a few damping times. The
particles will settle down to an equilibrium distribution independent of the
initial conditions. The 6-dimensional Σ-matrix can be evaluated by replacing
the integral in Eq. (2.126) with a summation over all particles, from which
the emittances and beam sizes can be calculated.
Beam distribution parameters:
The 6-dimensional second order moment matrix (also known as the Σmatrix) defined by
Σ = XX
T =
XX
T ρ(X)dX,
(2.126)
completely characterizes a beam in Gaussian distribution, ρ(X) =
1
(2π) 3
√
det Σ
exp(−
1
2 X
T Σ
−1 X) (assumed to be centered on the reference orbit for notation simplicity, i.e., with X = 0). The Σ-matrix is also often
used to characterize beams in other distributions. If M is the transfer matrix between two locations, X 2 = MX 1 , it is straightforward to show that
Σ 2 = MΣ 1 M
T . For a symplectic matrix M (see Eq. (1.44)), it follows that
Σ 2 S = MΣ 1 SM
−1 and hence the eigenvalues of ΣS do not change in symplectic transportations [120]. It can be shown that the 6 eigenvalues are ±ii k ,
k = 1, 2, 3, and det Σ =
2
1
2
2
2
3 , where k are the eigen-emittances for the three
degrees of freedom of particle motion. In the typical case of weak coupling,
the eigen-emittances correspond to the three planes, x, y, and z, respectively.
In electron storage rings, the beam reaches an equilibrium distribution due
to radiation damping and quantum excitation. For an uncoupled lattice, the
emittances can be calculated from radiation integrals, which are determined
by the linear lattice functions [106]. However, it becomes more complicated
when there exists linear coupling between the horizontal and vertical planes.
A general procedure can be applied to track through the lattice elements
to obtain the transfer matrix with radiation damping and the accumulated
quantum diffusion effects, which are then used to solve for the Σ-matrix [90].
The equilibrium distribution can also be found by long-term multi-particle
tracking. Radiation damping, quantum excitation, and RF cavities are turned
on in the lattice. A large number of particles (e.g., ≥ 1000) are launched
with random initial coordinates and are tracked for a few damping times. The
particles will settle down to an equilibrium distribution independent of the
initial conditions. The 6-dimensional Σ-matrix can be evaluated by replacing
the integral in Eq. (2.126) with a summation over all particles, from which
the emittances and beam sizes can be calculated.