1. Beam Instability in CEPC
WANG Na, ZHENG Hongjuan, WANG Yiwei, WANG Dou

2. Outline Impedance budget Single-bunch effects Multi-bunch effects
Electron cloud instability
Beam ion instability

3. 1. Impedance budget Broadband impedances Narrowband impedances
Quality factor Q~1
Discontinuities of chamber cross section (steps, tapers, bellows, monitors, …)
Single bunch effects: potential-well distortion, bunch lengthening, microwave instabilities…
Narrowband impedances
Quality factor Q >>1
Cavity like structures
Multi-bunch effects: longitudinal and transverse coupled bunch instabilities
Instability causes beam current limitations, particle losses, generally a degradation of beam quality  Luminosity. 3

4. 1. Impedance budget
Resistive wall wake for a Gaussian bunch in a cylindrical beam pipe is calculated analytically
Aluminum beam pipe will be used in CEPC. The beam pipe has an elliptical cross section with half height of ax/ay=52/28 mm (80% of LEP’s).
A five cell SC RF cavity structure of 650MHz will be used.
Impedance of the RF cavities is calculated with ABCI

5. CEPC ring wake and impedance budget
Object
Contributions
R [k]
L [nH]
kloss [V/pC]
|Z///n|eff []
Resistive wall (Al)
6.6
87.1
210.9
0.0031
RF cavities (N=400)
29.3 --
931.2
---
Total
35.9
1142.1
The longitudinal wake is fitted with the analytical model
The loss is dominated by the RF cavities.
The imaginary part of the RF cavities is capacitive.
Longitudinal wake at nominal bunch length (σz=2.66mm) 5 5

6. 2. Single-bunch effects Parameter Symbol, unit Value Beam energy
E, GeV
120
Circumference
C, km
53.6
Beam current
I0, mA
16.6
Bunch number nb 50
Natural bunch length
l0, mm
2.66
Emittance (horz./vert.)
x/y, nm
6.79/0.02
RF frequency
frf, GHz
0.65
Harmonic number h
116245
Natural energy spread
e0
1.5E3
Momentum compaction factor p
4.15E5
Betatron tune
x/y
179.08/179.22
Synchrotron tune s
0.199
Damping time (H/V/s)
x/y/z, ms
14/14/7
(paramter_lattice )

7. Longitudinal microwave instability
Keil-Schnell criterion:
The threshold of the longitudinal impedance is |Z///n| < .
K. Ohmi: |Z///n|th＝0.013/0.011 (CEPC/KEKB)
LEP: The threshold current for turbulent bunch lengthening was measured. The stability criterion gives effective longitudinal impedance of |Z///n|th＝0.03

8. Bunch lengthening
Steady-state bunch shape is obtained by Haissinski equation
Bunch is shortened due to the capacitive impedance of the RF cavity(only resistive wall and RF cavity considered)
Pseudo-Green function wake (z=0.5mm)
Steady-state bunch shape

9. Bunch lengthening with scaled SuperKEKB’s geometry wake
Scaled LER wake+RW+RF (bunch is lengthened by 9.0%)
Scaled HER wake+RW+RF (bunch is lengthened by 18.5%) 9
The scaling factor is Cir (CEPC)/Cir(SuperKEKB)=53.6e3/

10. Bunch lengthening with beam current (Simulation by Demin Zhou from KEK)
Difference of the impedance models between SuperKEKB LER and HER:
There is ante-chamber in the LER, but not in the HER. Without ante-chamber, the SR masks, pumping ports, and bellows contribute more impedances.
Scale SuperKEKB LER/HER:
(Geometric+RW impedance of SuperKEKB LER/HER) * Cir(CEPC)/Cir(SuperKEKB) 10

11. Bunch lengthening with beam current (Simulation by Demin Zhou from KEK)
RF+RW+Scale SuperKEKB LER/HER:
Impedance data of CEPC RF+RW
Multiply Cir(CEPC)/Cir(SKEKB):bellows, flanges, pumping ports, SR masks, BPMs
No scaling: Feedback kicker, Longitudinal kicker
Scale by number of IP: collimators, IR duct

12. Comments from Demin: CEPC should be safe from MWI
This is still a pessimistic estimation, because CEPC should have not as many discontinuities as those in SuperKEKB.
In the SuperKEKB design procedure, tremendous efforts have been to suppress the impedances. The structure optimizations require advanced techniques for hardware fabrication, and consequently enhance the total costs. There should be some balance between fabrication cost and impedance level.
It is noteworthy that ante-chambers suppress much impedance in SuperKEKB. It should be the same for CEPC. But other considerations (such as SR power, cost, etc.) should be taken into account when we make the choice of using ante-chamber for CEPC or not.

13. Coherent synchrotron radiation
z1/2/h3/2=9.2 (=> CSR shielded)
The threshold of bunch population for CSR is given by
The CSR threshold in BAPS is Nb,Th = 5.01012 >> Nb = 3.71011.
CSR is not a problem in CEPC.
(K. Bane, Y. Cai, G. Stupakov, PRST-AB, 2010)
Space charge tune shift
y = 1.7e4, x = 5.0e6

14. Transverse mode coupling instability (TMCI)
The threshold of transverse impedance is |Z| < 28.3 M/m.
The equivalent longitudinal impedance is 2.66 , which is much larger than that of the longitudinal instability.
Eigen mode analysis
Considering only resistive wall impedance
Beam current threshold:
Ibth=11.6mA (I0th=578mA)

15. 3. Multi-bunch effects Transverse resistive wall instability
with wpn = 2pfrev  (pnb + n + nx,y)
The growth rate for the most dangerous instability mode is 1.1 Hz (=0.9 s) in the vertical plane with mode number of  = 20.
The growth time is much higher than the transverse radiation damping time.
The resistive wall instability is not supposed to happen in the main ring!
Growth rate vs. mode number in the vertical plane

16. The growth rate is not quite sensitive to the chromaticity.
Instability growth rate vs. vertical tune
Instability growth rate vs. chromaticity
Smaller decimal tune are preferred to alleviate the transverse resistive wall instability.
The growth rate is not quite sensitive to the chromaticity. 16

17. Coupled bunch instability induced by the RF HOM’s
Monopole Mode
f (GHz)
R/Q() *
TM011
1.173
84.8
TM020
1.427
54.15
Dipole Mode
R/Q(/m)**
TE111
0.824
832.23
TM110
0.930
681.15
TE122
1.232
544.5
TM112
1.440
101.53
* k∥mode= 2πf·(R/Q)/4 [V/pC]
** k⊥mode = 2πf·(R/Q)/4 [V/(pC·m)] 17 17

18. Instability threshold18 18

19. Monopole Mode f (GHz) R/Q() * Dipole Mode R/Q(/m)** Q
Qlimit
TM011
1.173
84.8
5.1E+5
TM020
1.427
54.15
6.6E+5
Dipole Mode
R/Q(/m)** Q
TE111
0.824
832.23
2.3E+4
TM110
0.930
681.15
2.8E+4
TE122
1.232
544.5
3.5E+4
TM112
1.440
101.53
1.9E+5
* k∥mode= 2πf·(R/Q)/4 [V/pC]
** k⊥mode = 2πf·(R/Q)/4 [V/(pC·m)] 19 19

20. To find the total effects of all the 400 RF cavities, we need to take into account the spread in the resonance frequencies of the cells due to the finite tolerances in the cavity construction.
This will result in an “effective” quality factor Q of the whole RF system which is lower than that of a single cell.
Blue: dfR=0
Red: dfR=3kHz 20 20

22. 4. Electron cloud instability
The threshold value of the volume density of the electron cloud for the head-tail instability
We take Qnl = 7 for analytical estimation, and get the threshold density for the single bunch instability is 1.1×1012 m−3
For the multi-bunch instability, the electron cloud is considered as a rigid Gaussian with the chamber size. The characteristic frequency is
The phase angle between adjacent bunches is ωGLSP/c = 21.2, which means the electrons are not supposed to accumulate and the multipacting effects is low
K = ωeσz/c
Q = min(Qnl, ωeσz/c)
Qnl depends on the nonlinear interaction 22 22

23. Threshold density e,th, 1012m-3
KEKB
SuperKEKB
SuperB
CEPC
Beam energy E, GeV
3.5
4.0
6.7
120
Circumference L, m
3016
1370
53600
Number of e+/bunch, 1010
3.3 9
5.74
37.1
Emittance H/V x/y, nm
18/0.36
3.2/0.01
1.6/0.004
6.79/0.02
Bunch length z, mm 4 6 5
2.66
Bunch space Lsp, ns 2
3575.8
Single bunch effect
Electron freq. e/2, GHz
35.1
150
272
183.9
Phase angle ez/c
2.94
18.8
28.5
10.3
Threshold density e,th, 1012m-3
0.7
0.27
0.4
1.1
Multi-bunch effect
p-e per meter n, p/(m)
5.0E8
1.5E9
3.6E9
1.1E10
Characteristic freq. G, MHz
62.8
87.2
69.6
5.9
Phase angle GLsp/c
0.13
0.35
0.28
21.2
Threshold density for the single bunch effect is considerable high.
The phase angle for the multi-bunch effect is about two orders higher. 23 23

24. 5. Beam ion instability Ion trapping
With uniform filling pattern, the ions with a relative molecular mass larger than Ax,y will be trapped.
The ions will not be trapped by the beam.

25. Fast beam ion instability
With uniform filling, the growth time considering ion oscillation frequency spread is 6.9ms, which is faster than the damping time.
The force between beam and ions is assumed to be linear, this approximation would no longer be valid when the coherent oscillation grow larger than the beam size.
The ions are assumed not overfocused inside a bunch train: ωiLsep/c<2
For CEPC ωiLsep/c=40, the multibunch estimation is invalid.
Nonlinear force slower the growth rate (~1/10)
Smear of ion cloud makes further slower the growth rate.

26. The effect of pretzel on the instability
In the pretzel scheme, there are two beams hosted in the same beam pipe.
When a beam cross a resonator (eg. RF cavity), the wake field excited by the beam will affect the other beam, i.e. the two beams will talk to each other.
For the electron cloud instability, the electron beam will disturb the electron cloud accumulation. For the beam ion instability, the positron beam will affect the distribution of the ions in the beam pipe. In some point of view, both effects will be suppressed by the other counter rotating beam. There might be some new physics. (Discussion with K. Ohmi and G. Stupakov)

27. Thank you!