1. Study of Fast Ions in CESR
M. Billing
&
G. Codner
April 26, 2007
Study of Fast Ions in CESR

2. Motivation: Fast Ion Instability?
- 45 bunch train
- Electrons
- 14ns spacing
- 1.2e10/bunch Qh
0.6 kHz full scale
Instability?
Linear theory predicts
100 turn growth rate for
45th bunch
Vertical beam size Qv
0.5 kHz full scale
April 26, 2007
Study of Fast Ions in CESR

3. Study of Fast Ions in CESR
Basic Plan
Store electron beam
40-45 bunches in 1 train
14 nsec spacing between bunches
Use bunch-by-bunch feedback (H, V & L) for stabilization
Fill to a total current below an instability threshold
Take Grow-Damp measurements
Briefly turn off vertical feedback (automatically turns on when instability amplitude grows to preset value)
Measure the positions of each bunch for 1K turns
Analysis
Reconstruct the multi-bunch spectrum from FFT’s of all of the bunches
Compare with linear model for the Fast Ion Instability
April 26, 2007
Study of Fast Ions in CESR

4. Linear Model of Fast Ion Instability*
Assumptions
Beam-ion force is linear (small amplitude motion < y)
Ignore Landau damping
Model treats ion frequency as a constant
Within bunch-train ions are not over-focused
Number of neutral gas ions is large wrt number of ions
Ions are generated by ionization from beam and synchrotron radiation
Neutralization of ions from electrons from the vacuum chamber walls was ignored
Predicts a Fast Ion Vertical Instability
Growth along train (s): amplitude ~
Amplitude growth turn-to-turn
* Raubenhiemer et al, Phys Rev E, Vol 52, #5, p 5487
April 26, 2007
Study of Fast Ions in CESR

5. Ion Oscillation Frequency
In Drift or Horizontal Bend
Nb - particles/b; Lsep - distance between bunches; x,y - beam sizes; A - atomic mass of ion
f i ~ 1 / (A y)1/2
Model for CESR
Assume H2+ (A = 2) ions
Every 1 cm (in drift or bend) calculate the ion frequency
Histogram to find frequency distribution of ions
April 26, 2007
Study of Fast Ions in CESR

6. Study of Fast Ions in CESR
Plan for Measurements
Two vertical beam sizes
140 m
330 m
Two current distributions
Uniform train of 40 bunches (1.3 mA/b)
Instability threshold ~ 54 mA total current
Case modeled by Raubenhiemer, et al
Expect to see an amplitude growth along the train
15 higher current bunches (3 mA/b) & (1.3 mA/b)
First 15 bunches - initiate beam-ion motion
Subsequent bunches - act like a longer time current probe - expect to get a narrower resolution for the ion frequencies
April 26, 2007
Study of Fast Ions in CESR

7. Ion Frequency Resolution
Calculated resolution for finite number of bunches
(Essentially is an interference pattern in frequency)
For 15 bunches with the same amplitude oscillation:
Width of main spectral peak HWHM = 2 MHz ~ 1/Nbunches
Number of lobes = Nbunches - 1
Measured spectrum is convolved with this resolution
April 26, 2007
Study of Fast Ions in CESR

8. Predicted Ion Spectrum in CESR
Small vertical size m
5.7 MHz
April 26, 2007
Study of Fast Ions in CESR

9. Predicted Ion Spectrum in CESR
Larger vertical size m
3.3 MHz
April 26, 2007
Study of Fast Ions in CESR

10. Vertical Betatron Dipole Motion for Three of the Bunches in the Train
Conditions
40 bunches: mA/b
v = 140 m
1024 turns
Conclusions:
Amplitude of motion grows down the train
Amplitude grows “exponentially” in time
Motion exceeds linear range
Bunch 2
Bunch 36
20 mm
Bunch 41
April 26, 2007
Study of Fast Ions in CESR

11. Construction of Multi-bunch Spectrum from Individual Bunch Spectra
FFT each bunch’s motion
Obtain Vn(') amplitude and (') phase for ' in the range [0,  0] for all Nb bunches where  0 = 2  frev
Construct multi-bunch spectrum
where Vtotal is the complex spectrum Tbb is the time between bunches mod(x, x’) is the modulus function
April 26, 2007
Study of Fast Ions in CESR

12. Multi-bunch Spectrum of Vertical Betatron Dipole Modes
Conditions
40 bunches: 1.36 mA/b
v = 140 m
30 dB
MHz
April 26, 2007
Study of Fast Ions in CESR

13. Multi-bunch Spectrum of Vertical Betatron Dipole Modes
Conditions
40 bunches: 1.36 mA/b
v = 140 m
Trigger at 1/5 the preceding threshold level as a test
Spectral peak shifts upward
April 26, 2007
Study of Fast Ions in CESR

14. Multi-bunch Spectrum of Vertical Betatron Dipole Modes
Conditions
40 bunches: 1.3 mA/b
v = 330 m
April 26, 2007
Study of Fast Ions in CESR

15. Multi-bunch Spectrum of Vertical Betatron Dipole Modes
Conditions
15 bunches: 3 mA/b & 25 bunches: 1.1 mA/b
v = 330 m
April 26, 2007
Study of Fast Ions in CESR

16. Multi-bunch Spectrum of Vertical Betatron Dipole Modes
Conditions
15 bunches: 3 mA/b & 25 bunches: 1.1 mA/b
v = 130 m
April 26, 2007
Study of Fast Ions in CESR

17. Multi-bunch Spectrum of Vertical Betatron Dipole Modes
Conditions (Repeated measurements)
15 bunches: 3 mA/b & 30 bunches: 1.1 mA/b
v = 140 m
Lower ion frequencies due to larger amplitude motion?
April 26, 2007
Study of Fast Ions in CESR

18. Study of Fast Ions in CESR
Conclusions
Observations
See general shape of computed ion frequency distribution
Ion frequencies appear to be higher than those for H2+ -> suggests H+ (H2 consistent with CESR vacuum)
Large amplitude oscillations shift frequencies down
See dependence of 1 / y1/2
Oscillation amplitude grows along the train
Future measurements
Explore smaller amplitude oscillation regime
Compare with positron trains (to separate wakefield effects)
April 26, 2007
Study of Fast Ions in CESR