Interplay between SPS Impedance and electron cloud

1
Interplay between SPS Impedance and electron
cloud

• K. Cornelis

2
Geometry of Wakefields

• xi (transverse position of inducing particle)
• xs ( spectator
  particle)

xi
xs
3
Transverse picture (horizontal flat chamber)
Inducing
xi
4
spectator
xi-xs
Fsk(xi-xs)
5
Round chamber
xi
xs
Fs k xi
Inducing particle creates a dipole field
6
xi
7
xs
xi
Fsk(xixs)
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In General for an elliptical chamber
B
A
1
a
1
-1
e
-1
9
The classical head-tail
j is phase angle in longitudinal phase space
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The classical two particle head-tail
D
The way matrix D is written implies a Round
chamber (Fsk.xi)!!
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Including geometry
r 1 vertical flat r 0 round r -1
horizontal flat
(Fsk(xir.xs))
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r
-.5
1.25
1
-1
0.5
-1.25
Different kind of coupling. For r0 (round
chamber) head tail coupling gives strongest
instability
0
13
Same calculation but for 8 Modes (8 particles)
Flat vertical
Round
Flat horizontal
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Summary

• For a flat chamber the head tail modes couple
  weakly and at higher currents, giving small
  growth rates (ltQs)
• For a round chamber the coupling is strongest and
  leads to growth rates gtQs for low currents and
  simple modes.

15
E-cloud density during bunch passing (vertical)
16
E-cloud density during bunch passing (horizontal)
17
In the vertical plane Inducing particle creates
e-cloud density that acts as force on spectator
particle. System not linear (sum of causes
sum of results) For small
bunch oscillations impedance with varying
frequency.
4 1010
8 1010
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The Electron Cloud Wake
The interaction between subsequent parts
the bunch via the electron cloud, can be written
as
Fk(xi-xs) This is equivalent to a flat chamber
horizontal impedance. Adding a vertical flat
chamber machine impedance The e-cloud makes
the vertical impedance look more than a round
chamber.
19
Round
Flat vertical
Flat vertical E-cloud
Flat horizontal
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Conclusions

• In flat vacuum chamber the transverse mode
  coupling is weaker than in a round chamber.
• The electron cloud alone, gives very weak
  unstable mode coupling.
• The combination of e-cloud and vertical machine
  impedance enhances growth rates.