Resonances in Repetitive Systems
269
The trace of the matrix is
tr
ˆ
M
q
= 2 cos μ −
n
m=1
ΔK m
sin μ +
n
l,m=1,l ΔK l ΔK m sin φ lm sin(μ − φ lm ).
To the second order, the trace is
tr
ˆ
M
q
= ±2
1 − 2π
2
2
− ππ
n
m=1
ΔK m
+
n
l,m=1,l ΔK l ΔK m sin φ lm sin (2πν − φ lm ) .
The last term can be simplified, which is
n
l,m=1,l ΔK l ΔK m sin φ lm sin (2πν − φ lm )
=
1
4
n
l,m=1
ΔK l ΔK m [cos (2φ lm ) − 1]
=
1
4
n
l,m=1
ΔK l ΔK m cos (2Ψ l ) cos (2Ψ m )
+
1
4
n
l,m=1
ΔK l ΔK m sin (2Ψ l ) sin (2Ψ m ) −
1
4
n
l,m=1
ΔK l ΔK m
=
1
4
n
m=1
ΔK m cos (2Ψ m )
2
+
1
4
n
m=1
ΔK m sin (2Ψ m )
2
−
1
4
n
m=1
ΔK m
2
,
where Ψ m is the phase at the mth quadrupole. For the integer stop band, we
have
tr
ˆ
M
q
= 2
1 − 2π
2
2
− ππ
n
m=1
ΔK m
+
1
4
n
m=1
ΔK m cos (2Ψ m )
2
+
1
4
n
m=1
ΔK m sin (2Ψ m )
2
−
1
4
n
m=1
ΔK m
2
.
The unstable region of the tune is determined by the relation
2π
2
2 + ππ
n
m=1
ΔK m −
1
8
n
m=1
ΔK m cos (2Ψ m )
2
+
1
8
n
m=1
ΔK m sin (2Ψ m )
2
+
1
8
n
m=1
ΔK m
2
< 0.