198
An Introduction to Beam Physics
x
a
FIGURE 8.7: Illustration of the case where the beam ellipse (dashed) and
the invariant ellipses (solid) are matched. After each revolution, the beam
ellipse is exactly reproduced.
of area.
So it is best to operate a repetitive system in such a way that the beam
ellipse is matched to the accelerator’s invariant ellipse, and to avoid mismatching, the so-called beating.
8.2 Dispersive Effects
8.2.1 The Periodic Solution
Let ˆ
M be the transfer matrix of a periodic cell,
ˆ
M =
⎛
⎝
(x|x) (x|a) (x|δ)
(a|x) (a|a) (a|δ)
0
0
1
⎞
⎠ .
The periodic solution characterized by D, D
satisfies
⎛
⎝
D
D
1
⎞
⎠ =
⎛
⎝
(x|x) (x|a) (x|δ)
(a|x) (a|a) (a|δ)
0
0
1
⎞
⎠
⎛
⎝
D
D
1
⎞
⎠ .
(8.2)
An Introduction to Beam Physics
x
a
FIGURE 8.7: Illustration of the case where the beam ellipse (dashed) and
the invariant ellipses (solid) are matched. After each revolution, the beam
ellipse is exactly reproduced.
of area.
So it is best to operate a repetitive system in such a way that the beam
ellipse is matched to the accelerator’s invariant ellipse, and to avoid mismatching, the so-called beating.
8.2 Dispersive Effects
8.2.1 The Periodic Solution
Let ˆ
M be the transfer matrix of a periodic cell,
ˆ
M =
⎛
⎝
(x|x) (x|a) (x|δ)
(a|x) (a|a) (a|δ)
0
0
1
⎞
⎠ .
The periodic solution characterized by D, D
satisfies
⎛
⎝
D
D
1
⎞
⎠ =
⎛
⎝
(x|x) (x|a) (x|δ)
(a|x) (a|a) (a|δ)
0
0
1
⎞
⎠
⎛
⎝
D
D
1
⎞
⎠ .
(8.2)