270
An Introduction to Beam Physics
Denoting
Δμ =
n
m=1
ΔK m cos (2Ψ m )
2
+
n
m=1
ΔK m sin (2Ψ m )
2
,
we see the unstable interval of the tune is
2ππ +
1
2
n
m=1
ΔK m
<
Δμ
2
,
and Δμ is called the integer stop band. Similar calculation shows that the
same expression also gives the half–integer stop band [16].
In order to obtain the change in the invariant ellipse, we have to go back
to the original space, where
ˆ
M
q =
√
β 0
0
−α 0 /
√
β 0 1/
√
β 0
·
ˆ
M
q
·
1/
√
β 0
0
α 0 /
√
β 0
√
β 0
=
cos μ + α 0 sin μ
β 0 sin μ
−γ 0 sin μ
cos μ − α 0 sin μ
−
1
2
n
m=1
ΔK m
sin μ − α 0 cos μ
−β 0 cos μ
γ 0 cos μ
sin μ + α 0 cos μ
−
1
2
n
m=1
ΔK m
sin
μ + α 0 cos
μ
β 0 cos
μ
1 − α
2
0
cos
μ/β 0 + 2α 0 /β 0 − sin
μ − α 0 cos
μ
.
Immediately we have
cos (μ + Δμ) + (α 0 + Δα) sin (μ + Δμ)
= cos μ + α 0 sin μ −
1
2
n
m=1
ΔK m [sin μ − α 0 cos μ + sin
μ + α 0 cos
μ] ,
(β 0 + Δβ) sin (μ + Δμ)
= β 0 sin μ −
1
2
n
m=1
ΔK m [−β 0 cos μ + β 0 cos
μ] .
To the first order of Δk, we have
Δα = −
1
2 sin μ
n
m=1
ΔK m [sin
μ m + α 0 cos
μ m ] ,
Δβ
β 0
=
1
2 sin μ
n
m=1
ΔK m cos
μ m ,
and we remind ourselves of
μ m = μ − 2φ 0m and ΔK m = (Δkl) m β m . It is
clear that when the tune is close to the half–integer resonance, the size of the
beam becomes larger and eventually goes to infinity as the tune approaches
the half–integer.