1. Instability issues in CEPC
N. Wang for CEPC AP study group
25 Oct. 2016
ICFA eeFACT October 2016
Cockcroft Institute at Daresbury Laboratory, UK

2. Outline Introduction Impedance model Single bunch effects
Microwave instability, CSR, TMCI, beam tilt, tune shift
Multi bunch effects
Resistive wall, RF HOMs
Summary

3. Introductions
In order to achieve high luminosity, efforts have been made to increase the beam intensity and decrease the bunch length.
High beam induced heating
More easily couple with the high frequency impedances
High beam energy
Beneficial in general from the instability point of view
Instabilities are more critical for low energy operation (Z-pole)
Large circumference
Lower revolution frequency (~kHz) generates dense beam spectrum lines
Enhancement of the machine impedance

4. This instability study is focused on the partial double ring design.
Parameter
Symbol, unit
H-High lumi. Z
Beam energy
E, GeV
120
45.5
Circumference
C, km 54
Beam current
I0, mA
16.9
45.4
Bunch number nb 67
1100
Natural bunch length
l0, mm
4.1
4.0
Energy spread
e0
1.3E3
5.0E4
Momentum compaction p
2.5E5
3.5E5
D. Wang Version
J.Y. Zhai
Time structure of the counter rotating beams

5. Impedance model The impedance is mainly contributed by
Resistive wall impedance
Vacuum components with large quantity (RF cavities, flanges, BPMs, bellows, …)
Vacuum components with large impedance (RF cavities, IP duct, collimators, kickers, …)

6. Impedance model – 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.
The analytical resistive wall formula for multilayer cylindrical pipe is used in the calculation. Cu
NEG
Vacuum
Layers
Thick, mm
Cond, S/m r r Cu 2
5.9107 1
NEG
0.001
1.0106
Vacuum 
where In(x) are the modified Bessel functions, κ and M are determined by the parameters of the beam pipe.

7. Impedance model – Geometrical impedance
Two cell RF cavity
Flanges
Shielded bellows
BPMs
Shielded pumping ports

8. Impedance model – budget
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.
The impedance is dominated by resistive wall, RF cavities, flanges and bellows.
Lower impedance design for large number items should be considered.
More impedance contributions will be included in future studies.
Longitudinal wake potential at the nominal σz

9. Single bunch effects Microwave instability
Coherent synchrotron radiation (CSR)
Transverse mode coupling instability (TMCI)
Tune shift due to transverse impedance
Beam tilt

10. Single bunch effects – Microwave instability
The Boussard or Keil-Schnell criterion gives threshold bunch current
The threshold bunch current is lower than the design value.
The microwave instability can be a potential issue for CEPC.
For short bunches, the impedance seen by the beam is dominated by resonances at higher frequencies, the criterion maybe too passive.
Bunch lengthening by numerical solving the Haissinski equation
Higgs Z
Threshold current [mA]
0.17
0.014
Design current [mA]
0.25
0.04
Bunch lengthening below threshold
Ib=0.2mA
Ib=0.225mA

11. Single bunch effects - CSR
The threshold is calculated with theoretical formula
The beam is assumed to be moving in a circle of radius ρ between two parallel plates at locations y=±h/2.
If we take h=11mm, z1/2/h3/2=24 (=> CSR well shielded)
The threshold bunch population is evaluated by the coasting beam theory
For both cases, the instability threshold is more than one order higher than the designed bunch population.
CSR is not a concern in the present design.
(Y. Cai, IPAC’11)
Higgs Z
Threshold current [mA]
9.4
0.74
Design current [mA]
0.25
0.04

12. Single bunch effects – 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)
(S. Krinsky, PAC’07)
Eigen mode analysis
Higgs Z
Threshold current [mA]
1.9
0.37
Design current [mA]
0.25
0.04
The beam should be safe when consider the transverse mode coupling instability

13. Single bunch effects – Tune shift
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
The tune shift due to transverse impedance is −0.015/−7E-3(Higgs/Z).
When the tune approaches to integer, the tune variation is larger than that given by the above equation.
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)

14. Single bunch effects – Beam tilt
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

15. 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 under going.

16. Coupled bunch instability
Consider uneven filling with M bunches
One bunch train with bunch spacing=Tb
Treat the e- and e+ bunch as identical bunch traveling in the same direction.
Vertical beam oscillation with rigid bunch model
The instability growth rate can be written as

17. Coupled bunch instability – Resistive wall
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.
The growth time is much higher than the transverse radiation damping.
Even fill
Uneven fill
Growth rate vs. mode number in the vertical plane
Growth rate vs. bunch train duration

18. Coupled bunch instability – RF HOMs
In resonant condition, the threshold shunt impedances can be determined by (Considering one beam)

19. Requirement for Qe Both cases with 384 2-cell cavity
Monopole
f (MHz)
R/Q (Ω)
(2 cell)
Qe (H-high lumi.)
Qe (Z)
TM011
63.4
4.94×105
1.26×103
TM020
1.128
2.34×107
5.96×104
TM021
6.9
3.10×106
7.9×103
TM012
16.17
1.23×106
3.14×103
Dipole
R/Q (Ω/m)
TE111
276.62
6.17×104
5.09×102
TM110
414.84
4.12×104
3.39×102
TM&TE (hybrid)
150.41
1.14×105
9.36×102
TE121
12.61
1.35×106
1.12×104
TM120
15.20
1.12×106
9.26×103
20.13
8.5×105
7.0×103
Both cases with cell cavity
Z: cell cavity
* k∥mode = 2πf ·(R/Q) / 4 [V/pC]
** k⊥mode = 2πf ·(R/Q) / 4 [V/(pC·m)]

20. Dependence on manufacture errors
Considering two beams sharing the RF cavities, the threshold value can be further reduced by a factor of two.
Dependence on manufacture errors
Consider the whole RF system with large number of RF cavities, the spread in the resonant frequencies of the HOMs can reduce the effective shunt impedance.
Dependence on filling pattern
fR=1.173GHz
Q=2.54×105
384 RF cavities
σfR=5MHz
σfR=5kHz
Train
Q=1.2e6
Q=1.2e3

21. Summary Strong Medium Weak
Beam instability
H-High lumi. Z
Microwave instability, |Z||/n|th [mΩ] 24 11
CSR threshold Nbth [E11]
170 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]
21.4
4.2
Beam tilt Δy*/σy* (Only RF cavities)
23% 8%
Transverse resistive wall instability, t [s]
0.4
0.06
Strong
Medium
Weak
Design value: Nb=2.85E11/0.46E11 (Higgs/Z), |Z||/n|th~35[mΩ]
Single bunch instability is more critical both for Higgs and Z.
Main difference between the two operation modes:
The coupled bunch instability is more serious for Z.
The threshold impedance of microwave instability for Z is about half of Higgs.
The studies are mainly based on analytical formulae, more detailed simulation studies are required for the critical ones.

22. Thank you!