1. A Head-Tail Simulation Code for Electron Cloud
January 30, 2004
Yunhai Cai
Stanford Linear Accelerator Center

2. Outline Motivation and observation Physics and approximation
Threshold of head-tail instability
Spectrum of dipole oscillation
Emittance growth along a bunch train
Effects of chromaticity
Conclusion

3. Bunch-by-Bunch Luminosity
before installation of solenoid and bunches spaced with 4 RF buckets

4. Vacuum Pressure in Straights
4-RF bunch spacing
692 bunch train
30 Gauss solenoid
Courtesy Artem Kulikov, 2001

5. Measurement of Beam Size

6. Longitudinally Sliced Bunch
1. Dynamical slices with static transverse size:
2. Initial longitudinal Gaussian distribution:
where sz is the bunch length and sd is the energy
spread of the positron bunch.

7. Positron Beam Center and Size
3. Centroid motion of the bunch:
4. Rms value of position for all the slices:
5. Effective size of the bunch:
where sy is the size of the slices.

8. Presentation of Electron Cloud
1. Two-dimensional macro particles:
2. Initialized Gaussian distribution every turn:
3. Kick experienced by ith electron from nth slice:
where
is given by the Erskine-Bassetti formula.

9. Kick by a Gaussian Slice
Erskine-Bassetti formula:
where
Relation to the complex error function W:
where Nb is the bunch population.

10. Dynamics of the Positron Slices
1. Kick by the electron cloud:
where
2. Lattice map for transverse coordinates:
3. Lattice map for longitudinal coordinates :
where bz = sz/sd.

11. Beam and Machine Parameters
E (Gev)
energy
3.1
C(m)
circumference
2200
bx/by(m)
beta x/y
16.52/17.83
ex/ey(nm-rad)
emittance x/y
24/1.5
tt(turn)
damping time
9740
sz(cm)
bunch length
1.3 sd
energy spread
7.7E-4
nx/ny/ns
tunes
0.65/0.56/0.025 Nb
bunch population
1E11 Ns
number of slices
1024

12. Parameters of Electron Cloud
ne(m-3)
density
(1-13)E11
sxe(mm)
x size
6.0
sye(mm)
y size
3.0 Nm
# of mp
10240

13. “Emittance Growth” at Different Intensities of Electron Cloud
range of electron density: ne=(1-13)x1011m-3.

14. Threshold of Head-Tail Instability
threshold of electron cloud density: ne=(4-5)x1011m-3.

15. Power Spectrum of Dipole Oscillation below the Threshold
m = -1 mode
betatron tune
threshold
of head-tail instability

16. Observation at PEP-II -1 mode? Courtesy Uli Wienands 2002
synchrotron sideband
-1 mode?
Courtesy Uli Wienands 2002

17. Build up of Electron Cloud Along a Bunch Train
Densities of electron cloud were generated by Furman
and Pivi with 4 RF spacing. t was about 50 ns and the
saturation density was about 2x1011m-3.

18. Beam Blowup Along a Bunch Train
Densities were below the threshold of head-tail instability.

19. Effects of Chromaticity on Beam Centroid Motion
Density of electron cloud was above the threshold.
ne = 8x1011m-3.

20. Effects of Chromaticity on Beam Size

21. Wake Field Due to Electron Cloud
where
and
The wake field extracted from the code fitted well with the
wake from LRC circuit.

22. Bunch Pattern and Luminosity
L = 7.17x1033cm-2s-1, I- = 1310mA, I+ = 1950mA,
Nb = 1319, 78 trains, December 23, 2003

23. Conclusion
The low energy ring was operated (or limited) below the threshold of head-tail instability. The density of electron cloud was about 2x1011m-3. at the maximum beam current with a regular by-4 bunch pattern.
Progress has been made to understand the dynamics between the electron cloud and positron beam. The simulation results can be qualitatively compared to the observations and experimental measurements.
The main mechanism of beam blowup can be explained very well by the spread of centroid for all slices in a single bunch. Positive chromaticity can be used to damp down the centroid motion of a bunch but has little effects on the spread of slice centroid along the bunch.

24. Acknowledgement
Thank F.J. Decker, M. Furman, S. Heifets, R. Holtzapple, A. Kulikov, K. Ohmi, M.Pivi, J. Seeman, and U. Wienands for many helpful discussion and sharing the idea and experimental data.