202
An Introduction to Beam Physics
The last step keeps only the leading order effect. The quantity ξ N is called
natural chromaticity.
It will likely lead to beam loss since certain off-energy particles lie on resonant tunes. To remedy this problem, it is common to use sextupoles, because
they do not affect tunes of on-momentum particles but at the same time
provide quadratic nonlinearity that can be used to compensate the natural
chromaticity. The magnetic field of a sextupole is
B y = b 2
x
2
− y
2
, B x = 2b 2 xy,
where b 2 = −3M 3,3 . Defining k s = −b 2 /χ m0 and denoting
ˆ
M =
⎛
⎜
⎜
⎝
cos μ x +α x sin μ x
β x sin μ x
0
0
−γ x sin μ x
cos μ x −α x sin μ x
0
0
0
0
c o s μ y +α y sin μ y
β y sin μ y
0
0
−γ y sin μ y
cos μ y −α y sin μ y
⎞
⎟
⎟
⎠ ,
we obtain
⎛
⎜
⎜
⎝
x 1
a 1
y 1
b 1
⎞
⎟
⎟
⎠ =
⎛
⎜
⎜
⎝
x
a + k s ds
x
2
− y
2
y
b − 2k s dsxy
⎞
⎟
⎟
⎠ ◦
⎡
⎢
⎢
⎣
ˆ
M
⎛
⎜
⎜
⎝
x 0
a 0
y 0
b 0
⎞
⎟
⎟
⎠
⎤
⎥
⎥
⎦
=
⎛
⎜
⎜
⎜
⎝
x
a + k s ds
x
2
− y
2
y
b − 2k s dsxy
⎞
⎟
⎟
⎟
⎠
◦
⎛
⎜
⎜
⎜
⎝
(cos μ x +α x sin μ x ) x 0 + (β x sin μ x ) a 0
− (γ x sin μ x ) x 0 + (cos μ x −α x sin μ x ) a 0
(cos μ y +α y sin μ y ) y 0 + (β y sin μ y ) b 0
− (γ y sin μ y ) y 0 + (cos μ y −α y sin μ y ) b 0
⎞
⎟
⎟
⎟
⎠
.
Again, to make the physical picture clearer, let us apply the normalization
transformation in four-dimensional space, which is
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
= ˆ
A
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
1/
√
β x
0
0
0
α x /
√
β x
√
β x
0
0
0
0 1/
β y
0
0
0 α y /
β y
β y
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
.
The inverse is
ˆ
A
−1 =
⎛
⎜
⎜
⎜
⎝
√
β x
0
0
0
−α x /
√
β x 1/
√
β x
0
0
0
0
β y
0
0
0
−α y /
β y 1/
β y
⎞
⎟
⎟
⎟
⎠
.