228
An Introduction to Beam Physics
and
(a|δ) =
1 + (x|x)
(x|a)
(x|δ).
Note that the two equations from the dispersion and the dispersion prime are
not linearly independent. It is easy to verify that the transverse linear matrix
of a magnetic sector dipole satisfies the above relations. For a linearly stable
cell, i.e., |(x|x)| < 1, mirror symmetry implies that
α x =
(x|x) − (a|a)
2 sin μ x
= 0,
and
D
=
(1 − (x|x))(a|δ) + (a|x)(x|δ)
2 − (x|x) − (a|a)
=
1 − (x|x)
2 + (x|a)(a|x)
[2 − (x|x) − (a|a)] (x|a)
(x|δ) = 0.
Alternatively, we can also express the transfer matrix of a mirror symmetric
cell as functions of the first half of the cell. If the matrix of the first half of a
mirror symmetry cell is
ˆ
M
1
x =
⎛
⎜
⎝
(x|x) 1 (x|a) 1 (x|δ) 1
(a|x) 1 (a|a) 1 (a|δ) 1
0
0
1
⎞
⎟
⎠ ,
the second half is
ˆ
M
2
x =
⎛
⎜
⎝
(a|a) 1 (x|a) 1 −(a|a) 1 (x|δ) 1 + (x|a) 1 (a|δ) 1
(a|x) 1 (x|x) 1 −(a|x) 1 (x|δ) 1 + (x|x) 1 (a|δ) 1
0
0
1
⎞
⎟
⎠ .
The matrix of the whole cell is
ˆ
M
T
x = ˆ
M
I
x
ˆ
M x
=
⎛
⎜
⎝
(a|a) 1 (x|a) 1 −(a|a) 1 (x|δ) 1 + (x|a) 1 (a|δ) 1
(a|x) 1 (x|x) 1 −(a|x) 1 (x|δ) 1 + (x|x) 1 (a|δ) 1
0
0
1
⎞
⎟
⎠
⎛
⎜
⎝
(x|x) 1 (x|a) 1 (x|δ) 1
(a|x) 1 (a|a) 1 (a|δ) 1
0
0
1
⎞
⎟
⎠
=
⎛
⎜
⎝
(x|x) 1 (a|a) 1 + (x|a) 1 (a|x) 1
2(x|a) 1 (a|a) 1
2(x|a) 1 (a|δ) 1
2(x|x) 1 (a|x) 1
(x|x) 1 (a|a) 1 + (x|a) 1 (a|x) 1 2(x|x) 1 (a|δ) 1
0
0
1
⎞
⎟
⎠.
In addition to the relations obtained above, this alternative expression shows
that when (a|δ) 1 = 0, the cell is achromatic. Furthermore it reveals another
relation, which is
(a|δ) =
(x|x) 1
(x|a) 1
(x|δ).