Resonances in Repetitive Systems
263
When μ → 2πk, sin(nμ/2)/ sin(μ/2) → n. The result is that the motion is
divergent in phase space and eventually the beam will be lost. In other
words, the particle is called in resonance when μ = 2πk for some k. Since
errors in dipole magnets are unavoidable, the only way to avoid this resonance
is to adjust the tune away from any integer.
In theory, the phase space coordinates become arbitrarily large only when
the tune is infinitely close to an integer. But in practice, the finite size of
the beam pipe defines a finite interval around an integer such that the beam
will be lost, which is called the stop band. In order to determine the stop
band of a given machine, the size of the beam pipe and the dipole errors of
the magnets have to be known. The source of dipole errors can be either
the field errors in the dipole magnets and dipole component generated from
misalignment of multipoles (quadrupoles, mainly, and sextupoles to a lesser
extent). Since the dipole errors act on every particle the same way, it is
sufficient to study the motion of the so-called centroid of the beam only. In a
ring, this centroid, which is called the closed orbit, is the periodic solution of
the one turn transfer map. With the presence of dipole errors, the one turn
map is not origin preserving. Because the distorted closed orbit is usually
close to the design orbit, only the linear part of the map is included in the
treatment. The periodic solution of a single error at s 0 is obtained through
the equation
x 0
a 0
=
0
(ΔBl) (s 0 ) /Bρ
+ ˆ
M
x 0
a 0
.
As a result, we have
x 0
a 0
=
ˆ
I − ˆ
M
−1
0
(ΔBl) (s 0 ) /Bρ
=
(ΔBl) (s 0 )
Bρ
1
2 sin(μ/2)
β cos(μ/2)
sin(μ/2) − α cos(μ/2)
.
(11.1)
From eq. (6.10), we obtain the closed orbit at an arbitrary location s
x co (s) =
β s /β s0
cos ¯
φ + α sin ¯
φ
x 0 +
β s β s0 sin ¯
φ a 0
=
(ΔBl) s0
Bρ
1
2 sin(μ/2)
β s /β s0
cos ¯
φ + α s0 sin ¯
φ
β s0 cos
μ
2
+
β s β s0 sin ¯
φ
sin
μ
2
− α s0 cos
μ
2
=
(ΔBl) s0
Bρ
cos
πν − ¯
φ
2 sin(μ/2)
,
where ¯
φ, β s , β s0 , α s0 and (ΔBl) s0 represent φ(s) − φ(s 0 ), β(s), β(s 0 ), α(s 0 )
and (ΔBl) (s 0 ) , respectively. When n dipole errors are present, the closed
orbit is
x co (s) =
1
2 sin (μ/2)
n
m=1
(ΔBl) (s m )
Bρ
cos [μ/2 − (φ(s) − φ(s m ))] .
263
When μ → 2πk, sin(nμ/2)/ sin(μ/2) → n. The result is that the motion is
divergent in phase space and eventually the beam will be lost. In other
words, the particle is called in resonance when μ = 2πk for some k. Since
errors in dipole magnets are unavoidable, the only way to avoid this resonance
is to adjust the tune away from any integer.
In theory, the phase space coordinates become arbitrarily large only when
the tune is infinitely close to an integer. But in practice, the finite size of
the beam pipe defines a finite interval around an integer such that the beam
will be lost, which is called the stop band. In order to determine the stop
band of a given machine, the size of the beam pipe and the dipole errors of
the magnets have to be known. The source of dipole errors can be either
the field errors in the dipole magnets and dipole component generated from
misalignment of multipoles (quadrupoles, mainly, and sextupoles to a lesser
extent). Since the dipole errors act on every particle the same way, it is
sufficient to study the motion of the so-called centroid of the beam only. In a
ring, this centroid, which is called the closed orbit, is the periodic solution of
the one turn transfer map. With the presence of dipole errors, the one turn
map is not origin preserving. Because the distorted closed orbit is usually
close to the design orbit, only the linear part of the map is included in the
treatment. The periodic solution of a single error at s 0 is obtained through
the equation
x 0
a 0
=
0
(ΔBl) (s 0 ) /Bρ
+ ˆ
M
x 0
a 0
.
As a result, we have
x 0
a 0
=
ˆ
I − ˆ
M
−1
0
(ΔBl) (s 0 ) /Bρ
=
(ΔBl) (s 0 )
Bρ
1
2 sin(μ/2)
β cos(μ/2)
sin(μ/2) − α cos(μ/2)
.
(11.1)
From eq. (6.10), we obtain the closed orbit at an arbitrary location s
x co (s) =
β s /β s0
cos ¯
φ + α sin ¯
φ
x 0 +
β s β s0 sin ¯
φ a 0
=
(ΔBl) s0
Bρ
1
2 sin(μ/2)
β s /β s0
cos ¯
φ + α s0 sin ¯
φ
β s0 cos
μ
2
+
β s β s0 sin ¯
φ
sin
μ
2
− α s0 cos
μ
2
=
(ΔBl) s0
Bρ
cos
πν − ¯
φ
2 sin(μ/2)
,
where ¯
φ, β s , β s0 , α s0 and (ΔBl) s0 represent φ(s) − φ(s 0 ), β(s), β(s 0 ), α(s 0 )
and (ΔBl) (s 0 ) , respectively. When n dipole errors are present, the closed
orbit is
x co (s) =
1
2 sin (μ/2)
n
m=1
(ΔBl) (s m )
Bρ
cos [μ/2 − (φ(s) − φ(s m ))] .