200
An Introduction to Beam Physics
and hence
⎛
⎝
D
D
1
⎞
⎠ =
⎛
⎝
(x|x) − 1 (x|a) (x|δ)
(a|x) (a|a) − 1 (a|δ)
0
0
1
⎞
⎠
−1 ⎛
⎝
0
0
1
⎞
⎠ .
Defining
ˆ
I H =
⎛
⎝
1 0 0
0 1 0
0 0 1
⎞
⎠ ,
we obtain the more compact form
⎛
⎝
D
D
1
⎞
⎠ =
ˆ
M − ˆ
I H
−1
⎛
⎝
0
0
1
⎞
⎠ .
It is straightforward to generalize the above equation to the case of a nonlinear
map. Here the equation to be solved is
⎛
⎝
D (δ)
D
(δ)
δ
⎞
⎠ = M ◦
⎛
⎝
D (δ)
D
(δ)
δ
⎞
⎠ ,
where D (δ) and D
(δ) are polynomials of δ without the constant part. Similar
to the case for the linear map, we obtain
⎛
⎝
D (δ)
D
(δ)
δ
⎞
⎠ = (M − I H )
−1 ◦
⎛
⎝
D (δ)
D
(δ)
δ
⎞
⎠ ,
where
I H =
⎛
⎝
1 0 0
0 1 0
0 0 1
⎞
⎠
⎛
⎝
x
a
δ
⎞
⎠ .
As described in Section 5.4.2, the map (M − I H )
−1 can be obtained using the
Differential Algebraic (DA) technique order-by-order up to any given order.
Thus the periodic solution of dispersion up to arbitrary order can be obtained
without loss of accuracy.
8.2.2 Chromaticity
Now let us turn our attention to the betatron tune of off-momentum particles. We know that higher momentum particles are bent less, hence focal
length of quadrupoles is longer. As a result, we expect that total phase advance decreases as momentum increases. Recall that
k =
q
p
∂B y
∂x
,