1. CEPC main ring collective instability study
N. Wang, H. J. Zheng, D. J. Gong, Y. Zhang, Y.S. Sun, G. Xu, Y. W. Wang, D. Wang, Q. Qin, J. Gao, J. He (IHEP, Beijing),
W. Chou (Fermilab)
D. M. Zhou, K. Ohmi (KEK, Ibaraki)
2016/9/2

2. Contents Impedance Collective Effects
Impedance threshold
Impedance budget
Collective Effects
Single bunch effects
Coupled bunch effects
Ion Effects in the electron ring
Electron cloud in the positron ring
Comparison between Higgs and Z-pole
Summary

3. Design parameter 20160325 by D. Wang
Main parameters
Parameter
Symbol, unit
H-High lumi.
H-low power Z
Beam energy
E, GeV
120
45.5
Circumference
C, km 54
Beam current
I0, mA
16.9
10.5
45.4
Bunch number nb 67 44
1100
Bunch Population Ne
2.851011
2.671011
0.461011
Natural bunch length
l0, mm
4.1
4.0
Emittance (horz./vert.)
x/y, nm
2.45/0.0074
2.06/0.0062
0.62/0.002
RF frequency
frf, GHz
0.65
Harmonic number h
117081
Natural energy spread
e0
1.3E3
5.0E4
Momentum compaction factor p
2.5E5
2.2E5
3.5E5
Betatron tune
x/y
319.21/318.42
319.21/318.42?
Synchrotron tune s
0.08
0.04
Synchrotron damping time
x/y/z, ms
14/14/7
Design parameter by D. Wang

4. Impedance threshold Longitudinal broadband impedance
Determined by the microwave instability
Estimated according to the Boussard or Keil-Schnell criteria
Limit on the effective longitudinal impedance at design current is |Z/n|eff<24[mΩ]
Critical due to small momentum compaction along with small energy spread.
Transverse broadband impedance
Determined by the transverse mode coupling instability
Limit on the transverse kick factor at design current is ky, total < 68.9[kV/pC/m]
Less critical since considerably large synchrotron tune and beam pipe aperture are chosen.

5. Impedance threshold Narrowband impedance
Determined by the longitudinal and transverse coupled bunch instability
A conservative assumption is that the bunch spectrum overlaps the resonant frequency and the instability growth rate less than the synchrotron radiation damping.
This limits the shunt impedance at any resonance frequency of the HOMs.
In the longitudinal case:
In the transverse case:
Small beta functions are preferred at large impedance locations
Remedies:
Control the impedance below the threshold
Import feedback systems to cure the instability

6. Impedance budget Main sources:
Resistive wall, RF cavities, BPMs, bellows, pumping ports, flanges (Considered)
Electrostatic separators, transitions, Y-shape chamber, IR chamber, feedback kickers, injection kickers, collimators, …. (To be considered)
Impedance components
Number
Resistive wall -
RF cavities
384
Flanges
~10000
BPMs
2300
Bellows
Pumping ports

7. Resistive wall
Copper beam pipe with NEG coating will be used. The beam pipe has an elliptical cross section with half height of ax/ay=52/28 mm.
Resistive wall impedance in a two layer cylinder is calculated analytically
where In(x) are the modified Bessel functions, κ and M are determined by the parameters of the beam pipe.
Longitudinal and transverse wake potentials are obtained by Fourier transform of the impedances. Cu
NEG
Vacuum
Layers
Thick, mm
Cond, S/m r r Cu 2
5.9107 1
NEG
0.001
1.0106
Vacuum 

8. RF cavities
Two cell SC RF cavity structure with RF frequency of 650MHz will be used.
Given a design accelerating gradient of ~20MV/m, 384 cavities are needed.
The wake of the RF cavities is calculated with ABCI.
The RF cavities contribute both resistive and capacitive impedance, the later will help to compress the bunch length.
Loss factor for one RF cavity is kl=0.8V/pC, corresponding power loss is

9. Flanges Two type of flanges are considered
BEPCII type: 在法兰端面上沿真空盒内截面开槽，内放弹簧圈的方案
Introduce 1.5mm*1.5mm cavity
Possible choice for HEPS: Quick CF components
simulated with ABCI code
r_tube=28mm
L_gap=1mm
d_gap=5mm
sep=3mm
sigz=4mm
Both wake and impedance are much lower with BEPCII type flange.
For the Quick CF type flange, there is a dominant resonance at around 13GHz.

10. Bellows Consider shielding with RF finger
Impedance due to taper of sliding fingers are calculated and compared with the impedance of bellows without shielding.
Without shielding
With shielding

11. BPM Four button BPM, simulated with CST-PS
The impedance is dominated by broadband impedance, and there are two narrow resonances at 10.7 GHz and 12 GHz. Their contribution to the coupled bunch instability will be studied.

12. Ports of vacuum pumps
The pumping port will be shielded by longitudinal slots, as the disign in BEPCII
Gap width=1mm, slot width=5mm
The impedance contribution is considerably small.
Both longitudinal and transverse impedances are dominated by broadband impedance.

13. Impedance budget Suggestions:
Components
Number
R, kΩ
L, nH
Z||/n, mΩ
kloss, V/pC
HOM power, kW
Resistive wall -
6.7
487.7
17.0
138.4
106.3
RF cavities
384
14.9
-132.7
307.5
236.1
Flanges
~10000
0.7
165.5
5.8
15.1
11.6
BPMs
2300
0.6
21.4
8.9
Bellows
5.9
331.5
122.3
93.9
Pumping ports
0.007
3.1
0.1
Total
28.8
876.5
35.2
595.0
456.9
L and R are effective inductance and resistance, determined by fitting the bunch wake potentials.
Suggestions:
Consider lower impedance design for flanges, and better shielding of bellows.
Reduce the large number items.
Optimize the machine parameters.
Longitudinal wake potential at nominal bunch length

14. Collective Effects Microwave instability
The impedance budget gives total effective longitudinal impedance:
Bunch current threshold from Boussard or Keil-Schnell criteria:
The design bunch current is above the threshold, microwave instability and related bunch lengthening maybe a potential issue for CEPC.
For short bunches, the impedance seen by the beam is dominated by resonances at higher frequencies, the threshold current estimated by Boussard or Keil-Schnell criteria maybe too passive.
The high frequency part of the impedance may also lead to turbulent bunch distribution.
Simulation analysis is needed to confirm the threshold current and to estimate the effective bunch lengthening.

15. Tune shift due to transverse impedance
With transverse tune of (319.21, ), beam could become unstable at lower current than that for transverse mode coupling instability.
The tune shift is given by
When the tune approaches to integer, the tune variation is larger than that given by the above equation.
The tune shift due to transverse impedance is −0.015.
Effective impedance:
Z,eff = 917 kΩ/m (CEPC)
Z,eff = 895 kΩ/m (KEKB, bunch shape taken into account)
Z,eff = 409 kΩ/m (LEP, bunch shape taken into account)

16. Transverse mode coupling instability (TMCI)
For Gaussian bunch, the threshold of the instability can be expressed with the transverse loss factor:
The total kick factor is 18.9 kV/pC/m, which gives the threshold bunch current is 0.9 mA (Nbth = 1.0×1012, ~3 times of the design bunch intensity)
Eigen mode analysis
Considering only resistive wall impedance
Beam current threshold:
Ibth=1.9mA

17. Beam tilt due to the transverse wake fields
When a beam passes through a impedance with a transverse offset, the tail particles will receive transverse kicks
This will lead to transverse emittance increase and a transverse displacement of the bunch tail at IP
CEPC case:
Closed orbit of 1mm in vertical

18. Beam tilt due to the RF cavity wake
Transverse Pseudo-Green function wake for one RF cavity with z=0.4mm
Kick angle along the bunch due to single RF cavity
As there are 384 cavities located in 8 positions in the ring, the displacements at IP are
y*=48 *0.17nm=0.023m
(beam size at IP: x*/y*=24.8/0.1m)
The impedance is assumed to be localized at one point in the ring  Distributed impedance will reduce this effect.
Average beta function is used instead of that at the location of impedance  Smaller beta function can reduce this effect.
Detailed simulation studies are ongoing……

19. Coupled-bunch effects
[Jiyuan Zhai, “CEPC SRF system design for partial double ring scheme”, April 2016]
Consider un-uniform filling
Treat the e- and e+ bunch as identical bunch traveling in the same direction
Impedance at one location

20. Consider un-uniform filling with M bunch, bunch spacing = Tb
Vertical beam oscillation with rigid bunch model
bunch 0#
Bunch M-1#
Bunch n#
Solve the above equation, and consider a total of M multi-bunch modes, for Gaussian beam:

21. Transverse resistive wall instability
With bunch space of 159.3ns, the growth rate for the most dangerous instability mode is 2.3 Hz (=0.43s) in the vertical plane with mode number of  = 96.
The growth time is much higher than the transverse radiation damping time.
Uniform filling with 67*2 bunch: =0.45s
Uniform filling with bunch spacing=Tb: =0.05s
Growth rate vs. mode number in the vertical plane

22. HOM of the RF cavities
In resonant condition, the threshold shunt impedances can be determined by

23. Requirement for Qe Higgs: high lumi. & low power, 384 2-cell cavity
monopole
f (MHz)
R/Q (Ω)
(2 cell) Qe
(H-high lumi.)
(H-low power)
(Z)
TM011
63.4
4.94×105
8.03×105
1.26×103
TM020
1.128
2.34×107
3.80×107
5.96×104
TM021
6.9
3.10×106
5.4×106
7.9×103
TM012
16.17
1.23×106
2×106
3.14×103
dipole
R/Q (Ω/m)
TE111
276.62
6.17×104
9.94×104
5.09×102
TM110
414.84
4.12×104
6.63×104
3.39×102
TM&TE (hybrid)
150.41
1.14×105
1.83×105
9.36×102
TE121
12.61
1.35×106
2.18×106
1.12×104
TM120
15.20
1.12×106
1.81×106
9.26×103
20.13
8.5×105
1.37×106
7.0×103
* k∥mode = 2πf ·(R/Q) / 4 [V/pC]
** k⊥mode = 2πf ·(R/Q) / 4 [V/(pC·m)]
Higgs: high lumi. & low power, cell cavity
Z: cell cavity

24. 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 7.5×1011 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 = 5.9, which means the electrons are not supposed to accumulate and the multipacting effects are low.
K = ez/c
Q = min(Qnl, ez/c)
Qnl depends on the nonlinear interaction 24 24

25. Threshold density e,th, 1012m-3
KEKB
SuperKEKB
SuperB
CEPC H-High
Beam energy E, GeV
3.5
4.0
6.7
120
Circumference L, m
3016
1370
54752
Number of e+/bunch, 1010
3.3 9
5.74
37.9
Emittance H/V x/y, nm
18/0.36
3.2/0.01
1.6/0.004
6.12/0.018
Bunch length z, mm 4 6 5
2.88
Bunch space Lsp, ns 2
3653
Single bunch effect
Electron freq. e/2, GHz
35.1
150
272
287
Phase angle ez/c
2.94
18.8
28.5
24.7
Threshold density e,th, 1012m-3
0.7
0.27
0.4
0.75
Multi-bunch effect
p-e per meter n, p/(m)
5.0E8
1.5E9
3.6E9
8.2E9
Characteristic freq. G, MHz
62.8
87.2
69.6
36.7
Phase angle GLsp/c
0.13
0.35
0.28
5.9
Threshold density for the single bunch effect is considerably high.
The phase angle for the multi-bunch effect is about two orders higher. 25 25

26. Ion effects 1) Ion trapping 2) Fast beam-ion instability
With uniform filling pattern, the ions with a relative molecular mass larger than Ax,y will be trapped.
2) Fast beam-ion instability
In facilities with very high beam current and low emittances, the accumulation of ions within a single pass of the beam can excite fast beam-ion instability, which can leads both coherent oscillations of individual bunches and growth of the beam emittance.
The growth of the amplitude of oscillations is 0.5μs in vertical, when consider the frequency spread i , the growth time can increase to 0.5ms.
This can be an critical issue in CEPC. Multi bunch train filling patterns and feedback should be considered.
The critical mass number is quite high, so the ions will not be trapped by the beam.

27. Summary Beam instability H-High lumi. H-low power Z
Microwave instability, |Z||/n|th [mΩ] 24 22 11
CSR threshold Nbth [E11]
170
161 12
Space charge tune shift Δνx,y
-2E-5/-4E-4
-2E-4/-4E-3
Transverse impedance tune shift Δνx,y
-0.015
-6.6E-3
Transverse Mode Coupling, Nbth [E11]
10.3
2.0
Beam tilt Δy*/σy* (Only RF cavities)
23%
20% 8%
Transverse resistive wall instability, t [s]
0.4
0.7
0.06
ECloud, e,th, 1011m-3/Phase angle GLsp/c
0.75/5.9
0.75/7.0
0.14/0.6
Fast beam ion instability, τex/τey, ms
9.0/0.5
12.4/0.7
0.3/0.02

28. Summary
Microwave instability and FBII instability maybe a potential issue for CEPC Higgs, detailed simulations are required.
Estimation on Z-pole shows more critical collective effects than Higgs, both on single bunch and coupled bunch. The beam parameters should be further optimized.
The impedance model need to be further optimized to include the dominant impedance contributions, and to satisfy the requirement from the beam collective effects.

29. Thank you for your attention!