4 Impedance and Collective Effects
121
intensity, i.e. considering independently the modes m (which is valid up to a certain
intensity), with the following eigenvalue system
Δω cmq σ mq (l) =
p=+∞
p=−∞
K
m
lp σ mq (p), with Δω cmq = ω cmq − mω s ,
(4.25)
where
K
m
lp = −
2πI b mω s
2
0
ˆ
V T h cos φ s
j
Z l (p)
p
ˆ
τ =+∞
ˆ
τ =0
dg 0
d ˆ
τ
J m
pp 0 ˆ
τ
J m
ll 0 ˆ
τ
d ˆ
τ .
(4.26)
Here, ω cmq is the coherent complex synchrotron frequency shift to be determined, I b = N b eΩ 0 /2π is the bunch current, ω s , ˆ
V T and φ s are the new synchrotron
frequency, total voltage and synchronous phase (taking into account the potentialwell distortion), Z l = Z
0 is the longitudinal impedance, g 0 is the longitudinal
amplitude density function and h the RF harmonic number. The procedure to
obtain first order exact solutions, with realistic modes and a general interaction,
thus consists of finding the eigenvalues and eigenvectors of the infinite complex
matrix whose elements are given by Eq. (4.26). The result is an infinite number
of modes mq of oscillation. To each mode, one can associate a coherent frequency
shift ω cmq = ω cmq − mω s (which is the qth eigenvalue), a coherent spectrum
σ mq (p) (which is the qth eigenvector) and a perturbation distribution g mq
ˆ
τ
. For
numerical reasons, the matrix needs to be truncated, and thus only a finite frequency
domain is explored. For the case of the parabolic amplitude distribution and a
constant inductive impedance (which leads to real tune shifts only and therefore
no instability), the signal at the pick-up electrode shown for several superimposed
turns is depicted on Fig. 4.5. In the case of a complex impedance, the real part will
lead in addition to a growing amplitude with an associated instability rise-time. The
spectrum of mode mq is peaked at f q ≈ (q + 1)/(2τ b ) and extends ∼ ±τ
−1
b , where
τ b is the full bunch length (in second). It can be seen from Fig. 4.5 that there are
q nodes on these “standing-wave” patterns. The longitudinal signal at the pick-up
electrode is given by
S mq (t, ϑ) = S z0 (t, ϑ) + ΔS zmq (t, ϑ) ,
(4.27)
S z0 (t, ϑ) = 2πI b
p=+∞
p=−∞
σ 0 (p)e
jpp 0 t e
−jpϑ , σ 0 (p) =
ˆ
τ =+∞
ˆ
τ =0
J 0
pp 0 ˆ
τ
g 0
ˆ
τ
ˆ
τ d ˆ
τ ,
(4.28)
ΔS zmq (t, ϑ) = 2πI b
p=+∞
p=−∞
σ mq (p)e
j (pp0+mωs+Δωcmq)t e
−jpϑ .
(4.29)
121
intensity, i.e. considering independently the modes m (which is valid up to a certain
intensity), with the following eigenvalue system
Δω cmq σ mq (l) =
p=+∞
p=−∞
K
m
lp σ mq (p), with Δω cmq = ω cmq − mω s ,
(4.25)
where
K
m
lp = −
2πI b mω s
2
0
ˆ
V T h cos φ s
j
Z l (p)
p
ˆ
τ =+∞
ˆ
τ =0
dg 0
d ˆ
τ
J m
pp 0 ˆ
τ
J m
ll 0 ˆ
τ
d ˆ
τ .
(4.26)
Here, ω cmq is the coherent complex synchrotron frequency shift to be determined, I b = N b eΩ 0 /2π is the bunch current, ω s , ˆ
V T and φ s are the new synchrotron
frequency, total voltage and synchronous phase (taking into account the potentialwell distortion), Z l = Z
0 is the longitudinal impedance, g 0 is the longitudinal
amplitude density function and h the RF harmonic number. The procedure to
obtain first order exact solutions, with realistic modes and a general interaction,
thus consists of finding the eigenvalues and eigenvectors of the infinite complex
matrix whose elements are given by Eq. (4.26). The result is an infinite number
of modes mq of oscillation. To each mode, one can associate a coherent frequency
shift ω cmq = ω cmq − mω s (which is the qth eigenvalue), a coherent spectrum
σ mq (p) (which is the qth eigenvector) and a perturbation distribution g mq
ˆ
τ
. For
numerical reasons, the matrix needs to be truncated, and thus only a finite frequency
domain is explored. For the case of the parabolic amplitude distribution and a
constant inductive impedance (which leads to real tune shifts only and therefore
no instability), the signal at the pick-up electrode shown for several superimposed
turns is depicted on Fig. 4.5. In the case of a complex impedance, the real part will
lead in addition to a growing amplitude with an associated instability rise-time. The
spectrum of mode mq is peaked at f q ≈ (q + 1)/(2τ b ) and extends ∼ ±τ
−1
b , where
τ b is the full bunch length (in second). It can be seen from Fig. 4.5 that there are
q nodes on these “standing-wave” patterns. The longitudinal signal at the pick-up
electrode is given by
S mq (t, ϑ) = S z0 (t, ϑ) + ΔS zmq (t, ϑ) ,
(4.27)
S z0 (t, ϑ) = 2πI b
p=+∞
p=−∞
σ 0 (p)e
jpp 0 t e
−jpϑ , σ 0 (p) =
ˆ
τ =+∞
ˆ
τ =0
J 0
pp 0 ˆ
τ
g 0
ˆ
τ
ˆ
τ d ˆ
τ ,
(4.28)
ΔS zmq (t, ϑ) = 2πI b
p=+∞
p=−∞
σ mq (p)e
j (pp0+mωs+Δωcmq)t e
−jpϑ .
(4.29)