Basics of beam dynamics 17
Applying the stability condition to the FODO cell of Eq. (1.50), we find
that the beam motion is stable if and only if
f ≥
L
2
.
(1.57)
1.2.4 Courant-Snyder parametrization
The 2 × 2 transfer matrix of a stable periodic lattice cell can be parametrized
as follows [26]
M =
cos Φ + α sin Φ
β sin Φ
−γ sin Φ
cos Φ − α sin Φ
,
(1.58)
where γ =
1+α
2
β
and Φ is defined as the betatron phase advance over the
period. Parameters α, β, and γ are called Courant-Snyder (C-S) parameters.
The C-S parameters are the main characteristics of the linear optics, which
are often called the linear optics functions. The β parameter is referred to as
the beta function. The cell transfer matrix can be further rewritten as
M = BR(Φ)B
−1 ,
(1.59)
with
B =
√
β
0
−
α
√
β
1
√
β
, and R(ψ) =
cos ψ
sin ψ
− sin ψ cos ψ
.
(1.60)
The transverse coordinate of a particle oscillates in the lattice. This is
called the betatron oscillation. The betatron phase advance for a full revolution of a circular accelerator, Φ, is used to define the betatron tune,
ν =
Φ
2π
.
(1.61)
The tune is the number of oscillations the beam executes in one revolution.
If we observe the motion of a particle traveling through a periodic lattice
in each cell, the coordinates observed at s n = s 0 + nL, will be related to the
initial coordinate, X 0 , via
X n = M
n X 0 = BR(nΦ)B
−1 X 0 ,
(1.62)
where s 0 is the initial location and L is the cell length. In the (y, y
) phase
space the points representing the successive observed coordinates will trace
out an ellipse
y
2 + (αy + βy
)
2 = 2βJ,
(1.63)
where J is an invariant of motion given by the initial coordinates. The shape
and orientation of the ellipse are determined by the C-S parameters.
Applying the stability condition to the FODO cell of Eq. (1.50), we find
that the beam motion is stable if and only if
f ≥
L
2
.
(1.57)
1.2.4 Courant-Snyder parametrization
The 2 × 2 transfer matrix of a stable periodic lattice cell can be parametrized
as follows [26]
M =
cos Φ + α sin Φ
β sin Φ
−γ sin Φ
cos Φ − α sin Φ
,
(1.58)
where γ =
1+α
2
β
and Φ is defined as the betatron phase advance over the
period. Parameters α, β, and γ are called Courant-Snyder (C-S) parameters.
The C-S parameters are the main characteristics of the linear optics, which
are often called the linear optics functions. The β parameter is referred to as
the beta function. The cell transfer matrix can be further rewritten as
M = BR(Φ)B
−1 ,
(1.59)
with
B =
√
β
0
−
α
√
β
1
√
β
, and R(ψ) =
cos ψ
sin ψ
− sin ψ cos ψ
.
(1.60)
The transverse coordinate of a particle oscillates in the lattice. This is
called the betatron oscillation. The betatron phase advance for a full revolution of a circular accelerator, Φ, is used to define the betatron tune,
ν =
Φ
2π
.
(1.61)
The tune is the number of oscillations the beam executes in one revolution.
If we observe the motion of a particle traveling through a periodic lattice
in each cell, the coordinates observed at s n = s 0 + nL, will be related to the
initial coordinate, X 0 , via
X n = M
n X 0 = BR(nΦ)B
−1 X 0 ,
(1.62)
where s 0 is the initial location and L is the cell length. In the (y, y
) phase
space the points representing the successive observed coordinates will trace
out an ellipse
y
2 + (αy + βy
)
2 = 2βJ,
(1.63)
where J is an invariant of motion given by the initial coordinates. The shape
and orientation of the ellipse are determined by the C-S parameters.