Simulation of Transverse Single Bunch Instabilities and Emittance Growth caused by Electron Cloud in LHC and SPS.

1
Simulation of Transverse Single Bunch
Instabilities and Emittance Growth caused by
Electron Cloud in LHC and SPS.

• E. Benedetto, D. Schulte,
• F. Zimmermann, CERN
• G. Rumolo, GSI.

2
Contents

• HEADTAIL code
• Electric boundary conditions
• Sensitivity to numerical parameters
• Instability threshold and emittance growth, in
  the LHC at injection, as a function of
  chromaticity, ec-density and bunch intensity.
• SPS simulations with feedback and in a dipole
  field comparison with observations
• Resonator model for the electron cloud
• Conclusions and Future plans

3
Motivation

• Instabilities, beam loss and beam size blow up
  due to electron cloud observed at CERN PS and
  SPS, KEKB LER, and PEP II a concern for LHC
• Electrons are accumulated around the beam center
  during the bunch passage (pinch)
• If there is a displacement between head and tail,
  the tail feels a net wake force
• Effective short-range wake field -- TMCI type of
  single-bunch instability.
• Causes head-tail motion and emittance growth
• Slow emittance growth

4
The HEADTAIL code

• The interaction between the bunch and the
  cloud is modeled via a finite number of
  Interaction Points (IPs)

• The bunch is divided into slices that enter into
  the e-cloud at successive time step.
• PIC module (2D) to compute the interaction
  between electrons and protons, and vice versa.
• Transfer matrix to transport the protons to the
  successive IPs.
• The electron cloud is regenerated at each bunch
  passage.

5
Boundary Conditions
(Implemented with G. Rumolo D. Schulte)
Beam chamber (perfectly conducting)
Some image charges
2b

• Perfectly conducting rectangular box
  (approximation of the beam chamber)
• The electric potential is zero on the surface
• FFT Poisson Solver (from D. Schultes code
  Guinea-Pig)
• The difference with the solution in open space
  can be significant for small chamber size to beam
  size ratios

beam
electrons
Some image charges
2a
6
Boundary Conditions
ab
a2b
w b.c. w/o b.c.
w b.c. w/o b.c.
Ex (a.u.)
Ex (a.u.)
x/a
x/a

• Ex on the axis y0 the beam size s is 10 times
  smaller than the chamber

• Only small differences w and w/o b.c.
• Ratios at the pipe wall (x?a) can be calculated
  analytically and checked with those given by our
  Poisson solver

7
Boundary Conditions
Ey on the axis y?b/2, for a pipe ten times
larger than the ss of the beam w and w/o b.c.
still do not exhibit large differences for a
square pipe ab (left), but the difference seems
rather significant for a flat pipe with a2b
(right)
ab
a2b
Ey (a.u.)
Ey (a.u.)
x/a
x/a
8
Parameters used in the simulations (LHC at
injection)
? ECLOUD code
9
Interaction Points (IPs) along the ring

• Number of IPs per turn
• Position along the ring and phase advance between
  them

2nd lap
1st lap
3rd lap
Left figure the 3 IPs are equally spaced along
the ring, and their position does not change at
subsequent turns.
Right figure the position and the phase advance
between the IPs change every turn and they are
chosen randomly.
10
of IPs per turn
(Fixed Positions along the ring)
3 IPs
6 IPs
2 IPs
1 IP
7 IPs
5 IPs
4 IPs
5 IPs
3 IPs
Horizontal Emittance m
Vertical Emittance m
2 IPs
7 IPs
1 IP
8 IPs
6 IPs
4 IPs
8 IPs
Time s
Time s

• Horiz. (left) and Vert. (right) Emittance vs Time
  for LHC at inj, for different of IPs (1?9)
  ec-density 6 1011 m-3.
• Evidence of 2 regimes
• of IPs larger than 5 is needed for convergence

11
Evidence of two different regimes
t 0 s t 0.02 s t 0.04 s
t 0 s t 0.02 s t 0.04 s
y m
y m
z m
z m

• Snapshot of the vertical bunch shape (centroid
  and rms beam size) at different time step,
  assuming 1 IP (left) and 5 IPs per turn (right).
• for 1 IP the emittance growth is almost
  incoherent
• for 5 IPs an headtail instability develops

12
Random phase advance between IPs

• The average number of IPs per turn is given, but
  the position along the ring is randomly chosen at
  each turn
• Change more monotonic but poor convergence

1 IP
4 IPs
10 IPs
1 IP
2 IPs
4 IPs
2 IPs
5 IPs
3 IPs
5 IPs
3 IPs
20 IPs
10 IPs
Horizontal Emittance m
50 IPs
20 IPs
Vertical Emittance m
50 IPs
Time s
Time s
13
Number of macroparticles
of macro-protons
of macro-electrons
104
105
3 105
8 104
106
Vertical Emittance m
Vertical Emittance m
1 106
Time s
Time s

• 100000 macro-electrons per IP
• 70 slices

• 300000 macro-protons
• 10 IPs/turn

14
Emittance growth for different Electron Cloud
density
(Chromaticity Q2)
r 3 1012 m-3
r 3 1012 m-3
r 15 1011 m-3
r 9 1011 m-3
r 15 1011 m-3
r 6 1011 m-3
r 9 1011 m-3
Horizontal Emittance m
r 4 1011 m-3
Vertical Emittance m
r 6 1011 m-3
r 4 1011 m-3
r 3.5 1011 m-3
r 3.5 1011 m-3
r 3 1011 m-3
r 3 1011 m-3
Time s
Time s
Horizontal (Left) and Vertical (Right) Emittance
vs Time for different values of ec-density (from
3 1011 to 3 1012 m-3)
15
Rise time Vs EC-density (Chromaticity Q2)

• t is the time during which the emittance
  increases from 7.82 10-9 m (initial value) to 8
  10-9 m (2.3)

16
Extrapolation

• Extrapolated ec-density for 2.3 emittance
  growth during 30min operation in LHC (at
  injection! Q2) is 3 1010 m-3

17
Emittance growth for different Chromaticities
ec-density 6 1011 m-3
T-vert
Q2
Q15
Q10
T-horiz
Q20
Vertical Emittance m
Q25
Q30
Q40
Time s

• The Rise time here is defined as the interval Dt
  in which the emittance passes from 8e-9 to 8.2e-9
  (2.5).
• For high chromaticities we are in another regime
  with a slow emittance growth.

18
Transition between the two regimes

• Chromaticity vs ec-density, at which the
  transition between the two regimes occurs

19
Bunch intensity
Nb 1011
Nb 1011
Nb 8.5 1010
Nb 8.5 1010
Nb 1.15 1011
Horizontal Emittance m
Vertical Emittance m
Nb 1.15 1011
Nb 1.3 1011
Nb 7 1010
Nb 1.3 1011
Nb 7 1010
Nb 5.5 1010
Nb 5.5 1010
Nb 4 1010
Nb 4 1010
Time s
Time s

• Horizontal (left) and Vertical (right) Emittance
  growth vs time for different values of Bunch
  Intensities (0.4 1011 to 1.3 1011)
• ? For half the nominal bunch intensity (green
  curve) the growth is strongly reduced

20
Experimental results in SPS
Courtesy G.Arduini
Total beam intensity
Time

• Poor beam lifetime with LHC beam in the SPS on
  August 13, 2003 (can be explained by electron
  cloud?)

21
HEADTAIL Simulations for SPS
Vertical Emittance
Emittance m
Horizontal Emittance
Time s

• Dipole field
• Feedback system

22
SPS simulations (1)
Q26
Q19.5

• Chromaticity helps only at the very beginning,
  then for larger values of Q does not help any
  more.

Q2
Q13
Q8
Vertical Emittance vs Time, for different
cromaticities, ec-density1012 m-3.
23
SPS simulations (2)
Ec-density1012 m-3 Space Charge
Q3.9
Vertical Emittance m
Q7.5
Q26
Q13
Q19.5
Q3.9
Time s
Q7.5
Q13
Q19.5
Vertical Emittance m
Ec-density6 1011 m-3
Q26
Time s
24
Resonator model

• Broadband impedance model for the ec-interaction
  with the bunch K.Ohmi, F.Zimmermann,
  E.Perevedentsev, Wake field and fast head-tail
  instability caused by an electron cloud, Phys.
  Rev. E 65, 016502 (2002).

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Resonator Model (1)
Reson r 9 1011 m-3
Reson r 6 1011 m-3
Reson r 3 1012 m-3

• Emittance growth for different electron cloud
  density
• comparison between the Resonator Model and
  HEADTAIL PIC module

Reson r 15 1011 m-3
PIC r 9 1011 m-3
PIC r 3 1012 m-3
Vertical emittance m
PIC r 15 1011 m-3
PIC r 6 1011 m-3
PIC r 4 1011 m-3
Reson r 4 1011 m-3
Time s
26
Resonator Model (2)
Rise time of the emittance growth vs ec-density
comparison between the Resonator model and
HEADTAIL PIC module
Rise time s
Ec-density m-3
T1 time during which the emittance increases
from 7.82 10-9 m (initial value) to 8 10-9 m
(2.3) DeltaT interval during which the
emittance passes from 8 10-9 m to 8.2 10-9 m
(2.5).
27
Resonator Model (3)

• At least for the very beginning, the Broadband
  Impedance model seems to agree with HEADTAIL
  simulations, but then
• Non linear effect become important and the
  resonator model is not adequate any more.
• Maybe also the finite size of the grid and the
  cloud play a role.

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Benchmark with QuickPIC code
Collaboration with Ali Ghalam and Tom Katsouleas
QuickPIC
HEADTAIL
Horizontal Beam Size m
Vertical Beam Size m
HEADTAIL
QuickPIC
Time s
Time s

• Horizontal (right) and vertical (left) beam size
  vs. time.
• For purpose of comparison in both HEADTAIL and
  QuickPIC the electron cloud has been modeled
  using 1 IP per turn.

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Conclusions (1)

• Electric boundary conditions were added to
  HEADTAIL
• Sensitivity of HEADTAIL to numerical parameters
  was checked
• of macroparticles for protons electrons
• number of interaction points between cloud and
  bunch and their position around the ring
• Instability thresholds and emittance growth as a
  function of chromaticity, electron density and
  bunch intensity.
• Extrapolation suggests that for electron cloud
  densities of a few 1010m-3 the emittance growth
  over 30 minutes becomes acceptable this density
  is 10 times lower than the expected initial
  density
• At half the nominal bunch intensity emittance
  growth is strongly reduced

30
Conclusions (2)

• Chromaticity is a cure for the strong head-tail
  instability, but it may not be efficient for
  suppressing long-term emittance growth
• Feedback has been implemented into HEADTAIL to
  compare simulations with experimental results in
  SPS.
• The dependence on chromaticity has also been
  simulated for SPS , to be compared with
  observations.
• Resonator model seems to give similar growth
  rates as the electron-cloud simulation. For large
  amplitudes the finite size of the field grid
  and/or the nonlinear force slow down the
  emittance growth induced by the electron cloud.

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Ongoing work future plans

• Collaboration with USC and UCLA and comparison of
  HEADTAIL with QuickPIC code
• Explore need for magnetic boundary conditions
• Check the effect of the lattice of octupoles on
  the emittance growth
• Benchmark against SPS experiments

32
Acknowledgements

• Many thanks to all those who have contributed to
  this work, in particular
• Francesco Ruggiero,
• Gianluigi Arduini, Elias Metral,
• Ali Ghalam, Tom Katsouleas,
• Kazuhito Ohmi