4 Impedance and Collective Effects
123
- 3.0
- 2.5
- 2.0
- 1.5
- 1.0
- 0.5
0.0
- 10
- 5
0
5
10
x
Re (
Q ) /
Q
s
- 3.0
- 2.5
- 2.0
- 1.5
- 1.0
- 0.5
0.0
- 10
- 5
0
5
10
x
Re (
Q ) /
Q
s0
- 3.0
- 2.5
- 2.0
- 1.5
- 1.0
- 0.5
0.0
- 1.0
- 0.5
0.0
0.5
1.0
x
Im (
Q ) /
Q
s
- 3.0
- 2.5
- 2.0
- 1.5
- 1.0
- 0.5
0.0
- 1.0
- 0.5
0.0
0.5
1.0
x
Im (
Q ) /
Q
s0
Fig. 4.6 (Left) Comparison between GALACLIC Vlasov Solver [63] (in red) and Laclare’s
approach [53] (in black) of the normalised mode-frequency shifts vs. the normalised parameter
x (proportional to the bunch intensity [63]), in the case of a broad-band resonator impedance (with
a quality factor of 1 and a resonance frequency f r such that f r τ b = 2.8), above transition, without
taking into account the potential-well distortion (this is why the intensity-dependent synchrotron
tune Q s is used) and for a “Parabolic Amplitude Density” (PAD) longitudinal distribution [53].
(Right) Similar plot from GALACLIC only, taking into account the potential-well distortion (this
is why the low-intensity synchrotron tune Q s0 is used)
the chromaticity should be negative below transition and positive above. Sacherer’s
formula is also valid for coupled-bunch instability with M equally-populated
equally-spaced bunches, assuming multi-bunch modes with only one type of internal
motion (i.e. the same head–tail mode number). This analysis was extended in Ref.
[66] to include also the coupling between the modes (and the possibility to have
different head–tail modes in the different bunches). In the case of gaps between
bunch trains, a time-domain approach is usually better suited.
At low intensity (i.e. below a certain intensity threshold), the standing-wave
patterns (head–tail modes) are treated independently. This leads to instabilities
where the head and the tail of the bunch exchange their roles (due to synchrotron
oscillation) several times during the rise-time of the instability. The (approximate)
complex transverse coherent betatron frequency shift of bunched-beam modes is
given by Sacherer’s formula for round pipes [52]. For flat chambers a quadrupolar
effect (see Sect. 4.2) has to be added to obtain the real part of the coherent tune
shift, which explains why the horizontal coherent tune shift is zero in horizontally
flat chambers (of good conductors). As an example, a head–tail instability with
mode q = 10 is shown in Fig. 4.9(left). It is worth mentioning that there is also a
head–tail instability in the longitudinal plane. The longitudinal head–tail instability,
123
- 3.0
- 2.5
- 2.0
- 1.5
- 1.0
- 0.5
0.0
- 10
- 5
0
5
10
x
Re (
Q ) /
Q
s
- 3.0
- 2.5
- 2.0
- 1.5
- 1.0
- 0.5
0.0
- 10
- 5
0
5
10
x
Re (
Q ) /
Q
s0
- 3.0
- 2.5
- 2.0
- 1.5
- 1.0
- 0.5
0.0
- 1.0
- 0.5
0.0
0.5
1.0
x
Im (
Q ) /
Q
s
- 3.0
- 2.5
- 2.0
- 1.5
- 1.0
- 0.5
0.0
- 1.0
- 0.5
0.0
0.5
1.0
x
Im (
Q ) /
Q
s0
Fig. 4.6 (Left) Comparison between GALACLIC Vlasov Solver [63] (in red) and Laclare’s
approach [53] (in black) of the normalised mode-frequency shifts vs. the normalised parameter
x (proportional to the bunch intensity [63]), in the case of a broad-band resonator impedance (with
a quality factor of 1 and a resonance frequency f r such that f r τ b = 2.8), above transition, without
taking into account the potential-well distortion (this is why the intensity-dependent synchrotron
tune Q s is used) and for a “Parabolic Amplitude Density” (PAD) longitudinal distribution [53].
(Right) Similar plot from GALACLIC only, taking into account the potential-well distortion (this
is why the low-intensity synchrotron tune Q s0 is used)
the chromaticity should be negative below transition and positive above. Sacherer’s
formula is also valid for coupled-bunch instability with M equally-populated
equally-spaced bunches, assuming multi-bunch modes with only one type of internal
motion (i.e. the same head–tail mode number). This analysis was extended in Ref.
[66] to include also the coupling between the modes (and the possibility to have
different head–tail modes in the different bunches). In the case of gaps between
bunch trains, a time-domain approach is usually better suited.
At low intensity (i.e. below a certain intensity threshold), the standing-wave
patterns (head–tail modes) are treated independently. This leads to instabilities
where the head and the tail of the bunch exchange their roles (due to synchrotron
oscillation) several times during the rise-time of the instability. The (approximate)
complex transverse coherent betatron frequency shift of bunched-beam modes is
given by Sacherer’s formula for round pipes [52]. For flat chambers a quadrupolar
effect (see Sect. 4.2) has to be added to obtain the real part of the coherent tune
shift, which explains why the horizontal coherent tune shift is zero in horizontally
flat chambers (of good conductors). As an example, a head–tail instability with
mode q = 10 is shown in Fig. 4.9(left). It is worth mentioning that there is also a
head–tail instability in the longitudinal plane. The longitudinal head–tail instability,