The Periodic Transport
203
Denoting the linear normalization transformation (as opposed to the transformation matrix), its inverse and the linear transfer map as
A = ˆ
A
⎛
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎠ ,
A
−1 = ˆ
A
−1
⎛
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎠ ,
M = ˆ
M
⎛
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎠ ,
we obtain
⎛
⎜
⎜
⎜
⎝
x 1
a 1
y 1
b 1
⎞
⎟
⎟
⎟
⎠
=
⎡
⎢
⎢
⎢
⎣
A ◦
⎛
⎜
⎜
⎜
⎝
x
a + k s ds
x
2
− y
2
y
b − 2k s dsxy
⎞
⎟
⎟
⎟
⎠
◦ A
−1
⎤
⎥
⎥
⎥
⎦
◦
A ◦ M ◦ A
−1
=
⎛
⎜
⎜
⎜
⎝
x
a + k s ds
√
β x
β x x
2
− β y y
2
y
b − 2k s ds
√
β x β y xy
⎞
⎟
⎟
⎟
⎠
◦ R(μ x , μ y ),
where
R(μ x , μ y ) =
ˆ
R (μ x )
0
0
ˆ
R (μ y )
⎛
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎠ ,
ˆ
R (μ x ) =
cos μ sin μ
− sin μ cos μ
.
Since the kick of a sextupole is second order in the coordinates, the offmomentum particle has to go through it off the magnetic center in order
to affect the linear motion and hence the tune. In other words, the dispersion
has to be nonzero at the location of the sextupole to correct chromaticity.
With the presence of dispersion, the coordinates are
x → x − D x δ, a → a − D
x δ,
and the normalized coordinates are
x →
x −
D x δ
√
β x
, a → a −
β x D
x δ.