1. Impedance model and collective effects
for FCC-ee
E.Belli
M. Migliorati, G. Rumolo
Acknowledgments: G. Castorina, G. Iadarola, R. Kersevan, L. Mether, K. Oide,
S. Persichelli, B. Spataro, F. Zimmermann, M. Zobov
FCC Week 2017
May 31, Berlin

2. How much safety margin do we have?
Introduction
E. Belli - Impedance model and collective effects for FCC-ee
Within the FCC studies, FCC-ee would be the first step towards the 100 TeV hadron collider FCC-hh.
Intensity limitation due to collective effects and the impedance budget
Each equipment has an impedance to be characterized and minimized
Good knowledge of the global machine impedance crucial to reach the required performance
Major issues for the operation with nominal parameters
Heat load in specific machine locations
Longitudinal and transverse instabilities due to wake fields
How much safety margin do we have? 1

3. Outline 1. Resistive wall (RW) Single bunch effects
E. Belli - Impedance model and collective effects for FCC-ee
1. Resistive wall (RW)
Single bunch effects
Microwave Instability
Transverse mode coupling instability
Multibunch effects
2. Other impedance sources
3. Electron cloud
4. Conclusions 2

4. FCC-ee parameter list Beam energy [GeV] 45.6 Circumference 𝐶 [km]
E. Belli - Impedance model and collective effects for FCC-ee
Beam energy [GeV]
45.6
Circumference 𝐶 [km]
97.75
RF frequency 𝑓 𝑅𝐹 [MHz]
400
RF voltage 𝑉 𝑅𝐹 [MV]
255
Arc cell
60º/60º
Momentum compaction 𝛼 𝑐 [ 10 −5 ]
1.479
Horizontal tune 𝑄 𝑥
269.14
Vertical tune 𝑄 𝑦
267.22
Synchrotron tune 𝑄 𝑠
0.0413
SR energy loss/turn 𝑈 0 [GeV]
0.0373
Longitudinal damping time 𝜏 𝑙 [ms]
398
Beam current 𝐼 [A]
1340
Bunches/ring
68240
Bunch population 𝑁 [ ]
0.4
Horizontal emittance 𝜀 𝑥 [nm]
0.267
Vertical emittance 𝜀 𝑦 [pm] 1
Energy spread (SR) 𝜎 𝑑𝑝,𝑆𝑅 [%]
0.038
0.073
Bunch length (SR) 𝜎 𝑧,𝑆𝑅 [mm]
2.1
4.1 3

5. + Resistive wall Three layers Non Evaporable Getter (NEG) (No Coating)
E. Belli - Impedance model and collective effects for FCC-ee
Three layers
(No Coating) Cu
2mm
𝜌=1.66∙ 10 −8 Ωm
Dielectric
6mm
𝜌= Ωm
Iron
Infinity
𝜌= 6.89∙10 −7 Ωm
Non Evaporable Getter (NEG)
thk =1𝜇m
𝜌= 10 −6 Ωm
Best choice for vacuum (low SEY, low desorption yield)
MI threshold below nominal intensity
Titanium Nitride (TiN)
thk = 200nm
𝜌=0.5∙ 10 −6 Ωm +
Low SEY, thinner coating
Vacuum pumps for desorption
Amorphous Carbon
(AC)
thk = 200 nm
𝜌= 10 −4 Ωm
Low SEY, thinner coating, low desorption yield
35mm 4

6. Instability threshold
RW – Single bunch effects
E. Belli - Impedance model and collective effects for FCC-ee
Main effects of the RW on the single bunch dynamics
Microwave Instability (MI) in the longitudinal plane
Transverse Mode Coupling Instability (TMCI) in the transverse plane
Bunch intensity
limitation
Frequencies of the intra-bunch modes ( Ω=𝑚 𝜔 𝛽 +𝑙 𝜔 𝑠 ) depend on the bunch intensity
DELPHI
As the bunch intensity increases, the mode frequencies shift and merge, giving rise to the instability
Above the threshold:
Transverse case
Bunch loss
Longitudinal case
Increase of the bunch length and energy spread
Bunch internal oscillations (dangerous in beam-beam collisions)
Instability threshold 5

7. Longitudinal impedance
E. Belli - Impedance model and collective effects for FCC-ee
𝐈𝐦[ 𝒁 𝑵𝑬𝑮 ]
𝐑𝐞[ 𝒁 𝑵𝑬𝑮 ]
Im[ 𝑍 𝑇𝑖𝑁 ]≈Im[ Z AC ]
Re[ 𝑍 𝑁𝐶 ]≈Im[ Z NC ]≈Re[ 𝑍 𝑇𝑖𝑁,𝐴𝐶 ] 6

8. Bunch length and energy spread
E. Belli - Impedance model and collective effects for FCC-ee
No Beamstrahlung ( 𝜎 𝑧 =2.1𝑚𝑚, 𝜎 𝑑𝑝 =0.038% )
MI regime
MI instability threshold
≃1.6 −1.8 ⋅ in the case of no coating
Around the nominal bunch intensity in the case of TiN and AC
Below the nominal bunch intensity in the case of NEG 7

9. Bunch length and energy spread
E. Belli - Impedance model and collective effects for FCC-ee
With Beamstrahlung ( 𝜎 𝑧 =4.1𝑚𝑚, 𝜎 𝑑𝑝 =0.073% )
Stable beams
MI instability threshold far beyond the nominal bunch intensity for all coatings except NEG
NEG still dangerous 8

10. Possible optimizations
E. Belli - Impedance model and collective effects for FCC-ee 1
Does the material thickness affect the MI?
Example: Amorphous Carbon, no Beamstrahlung ( 𝜎 𝑧 =2.1𝑚𝑚, 𝜎 𝑑𝑝 =0.038%)
The MI threshold is 3x higher in case of 25nm thickness 2
Shatilov proposal [1] to mitigate the coherent X-Z instability:
increase in the momentum compaction 𝜶 𝒄
increase in bunch length 𝝈 𝒛 and bunch population 𝑵 𝒑 in the same proportion
Boussard criterion to obtain a scaling with beam parameters
𝑍 ∥ 𝑛 𝑛 ≤ 2 𝜋 𝐸 0 𝑐 𝑒 2 𝛼 𝑐 𝜎 𝑧 𝑁 𝑝 𝜎 𝑑𝑝 2 9
[1] D. Shatilov, “Recent results about the X-Z instability”, 53th FCC-ee Optics Design meeting

11. For all the coatings, TMCI seems to be not dangerous
E. Belli - Impedance model and collective effects for FCC-ee
No Beamstrahlung
Macroparticle simulations with PyHEADTAIL code
Analytic simulations with DELPHI code including the bunch lengthening due to the longitudinal wake
For all the coatings, TMCI seems to be not dangerous 10

12. Robust feedback for instability suppression
Transverse coupled bunch instability
E. Belli - Impedance model and collective effects for FCC-ee
Growth rate for the 𝜇-th mode
𝟏 𝝉 𝟕𝟑𝟓𝟎𝟒 =𝟒𝟗𝟒.𝟖 𝒔 −𝟏 ≃𝟔 𝒕𝒖𝒓𝒏𝒔 =1
with
𝑅𝑒 [ 𝑍 ⊥ 𝜔 ]=𝑠𝑔𝑛(𝜔) 𝐶 2𝜋 𝑏 𝑍 0 𝑐 𝜎 𝑐 |𝜔|
Negative 𝜔  unstable mode with exponential growth
Positive 𝜔  stable mode with damped oscillations
The most dangerous coupled mode when 𝜔 𝑞 ≈0
𝝁=−𝒒𝑴− 𝑸 𝟎 −𝟏=𝟕𝟑𝟓𝟎𝟒
Robust feedback for instability suppression 11

13. Other impedance sources
E. Belli - Impedance model and collective effects for FCC-ee
400 MHz single cell cavities
≈55
SR absorbers
≈ 10000
Beam Position Monitors
≈ 4000
Winglet-to-circular tapers
≈ 4000
Bellows with RF fingers
≈ 8000 12

14. Longitudinal impedance budget
E. Belli - Impedance model and collective effects for FCC-ee
ABCI and CST results ( 𝝈 𝒛 = 4mm)
Contribution of SR absorbers too low to have a reliable estimation
Contributions of tapers and bellows with RF fingers not negligible
𝐤 𝐥𝐨𝐬𝐬 [𝐕/𝐩𝐂]
Resistive Wall
(Cu, circular with r=35mm)
181.6
Bellows with RF fingers
195.2
Tapers
84.4
RF cavities
16.7
BPMs
26.8 13

15. EC build up
E. Belli - Impedance model and collective effects for FCC-ee
Positively charged bunches passing through a section of an accelerator
Primary or Seed Electrons
Residual gas ionization
Molecules of the residual gas in the vacuum chamber can be ionized by the beam
Photoemission due to synchrotron radiation
Emitted photons hitting the wall can have enough energy to extract electrons from the pipe’s wall (photoelectrons)
Heat load on the pipe walls
Transverse instabilities
~300𝑒𝑉
~10𝑒𝑉
Lost t
Bunch spacing
Beam pipe
Bunch 14

16. EC build up studies 𝑵 𝒑𝒉,𝒅 = 𝑵 𝒑𝒉 ∙(𝟏−𝑹) 𝑁 𝑝ℎ = 𝑁 𝛾 ∙𝑌
E. Belli - Impedance model and collective effects for FCC-ee
Photoemission due to SR
𝑵 𝒑𝒉,𝒅 = 𝑵 𝒑𝒉 ∙(𝟏−𝑹)
𝑁 𝑝ℎ = 𝑁 𝛾 ∙𝑌
𝑵 𝒑𝒉,𝒓𝒇 = 𝑵 𝒑𝒉 ∙𝑹
Energy [GeV]
45.6
Bending radius [km] 11
Bunch spacing [ns]
2.5
Bunch population [ ]
0.4
Horizontal emittance [nm]
0.255
Vertical emittance [pm] 1
Bunch length [mm]
2.1
Bunch train patterns
230b + 60e
65b + 50e
Arc elements
L[m]
B[T] or G[T/m]
𝛽 𝑥 [m]
𝛽 𝑦 [m]
Dipole
24.11
0.0135 50 55
QD1
1.4
2.283
88.3
17.6
QF2
3.5
5.639
16.7
94.4
𝑁 𝛾 = 5𝛼 𝛾 𝜌 =𝟎.𝟎𝟖𝟓
Parameter scans
Secondary Emission Yield (SEY)
Photoelectron Yield Y
Reflectivity R
Ionization H2
10-9 mbar pressure in the vacuum chamber
Ionization cross section
𝜎=1.874∙ 10 −24 𝑍 2 𝛽 2 𝑀 2 𝑥+𝐶 𝑚 2 =0.255∙ 10 −18 𝑐 𝑚 2 15

17. Highest contribution from drifts
E. Belli - Impedance model and collective effects for FCC-ee
Effect of photoelectrons on the heat load
Drift
Dipole
Photoemission + Ionization
Trains of 230 bunches + 150ns gap
QD1
QF2
Highest contribution from drifts 16

18. Shorter trains with gaps allow to mitigate the EC
E. Belli - Impedance model and collective effects for FCC-ee
Effect of photoelectrons on the heat load
Drift
Dipole
Photoemission + Ionization
Trains of 65 bunches + 125ns gap
QD1
QF2
Shorter trains with gaps allow to mitigate the EC 17

19. Single bunch head tail instability
E. Belli - Impedance model and collective effects for FCC-ee
Analytic threshold
𝜔 𝑒 = 2𝜆 𝑝 𝑟 𝑒 𝑐 2 𝜎 𝑦 ( 𝜎 𝑥 + 𝜎 𝑦 )
𝜌 e, th = 2𝛾 𝜐 𝑠 𝜔 𝑒 𝜎 𝑧 /𝑐 3 𝐾𝑄 𝑟 0 𝛽𝐶
𝐾= 𝜔 𝑒 𝜎 𝑧 /𝑐,
𝑄=min⁡( 𝜔 𝑒 𝜎 𝑧 /𝑐, 7)
with
and
Beam energy [GeV]
45.6 80
120
175
Circumference [km]
97.75
Bending radius [km]
10.747
RF frequency [MHz]
400
Bunch population [ ]
0.4
0.41
0.71
2.04
Horizontal emittance [nm]
0.267
0.28
0.63
1.34
Vertical emittance [pm] 1
1.2
2.7
Betatron tunes Qx/Qy
269.14/267.22
389.08/389.18
Synchrotron tune Qs
0.0413
0.034
0.0499
0.0684
βy [m]
58.2 40
Bunch length [mm]
2.1
2.0
2.4
Electron frequency 𝜔 𝑒 /2π [GHz]
246.4
295.6
313.3
327.5
Electron oscillation 𝜔 𝑒 𝜎 𝑧 /c
10.85 13
13.13
16.5
Density threshold 𝛒 𝐭𝐡 [𝟏 𝟎 𝟏𝟏 / 𝐦 𝟑 ]
0.38
0.79
1.75
3.5 18

20. Conclusions Resistive wall MI regime in case of no beamstrahlung
E. Belli - Impedance model and collective effects for FCC-ee
Resistive wall
MI regime in case of no beamstrahlung
NEG  MI threshold below the nominal bunch intensity
Stable beams with beamstrahlung
injection with alternating beams would allow a good margin of safety
TMCI threshold far beyond the nominal bunch intensity
Robust feedback system required for the fast (6 turns) transverse instability suppression
Other impedance sources
SR absorbers and RF cavities negligible compared to RW
Good design for BPMs
Contributions of tapers and bellows with RF fingers comparable with the RW
The impedance can be reduced by
Removing the tapers by placing the BPMs on the winglet chamber (at 45º)
Using a different design for the bellows (comb-type RF shield proposed for KEKB?)
Electron cloud build up estimated in the arcs
highest contribution from drifts
Analytic single bunch instability threshold estimated for all the energies
Instability studies with macroparticle simulation codes needed 19

21. Thanks for your attention

22. Backup
E. Belli - Impedance model and collective effects for FCC-ee 1

23. Longitudinal coupled bunch instability
E. Belli - Impedance model and collective effects for FCC-ee
Growth rate for the 𝜇-th mode
1 𝜏 𝜇, ∥ = 𝑒𝜂 𝐼 𝑏 4𝜋𝐸 𝑄 𝑠 𝑞 𝜔 𝑞 𝑅𝑒 𝑍 ∥ 𝜔 𝑞 𝐺 ∥ 𝜎 𝑧 𝑐 𝜔 𝑞 ′ =1
with
𝜔 𝑞 = 𝑞𝑀+𝜇+ 𝑄 𝑠 𝜔 0
𝑅𝑒 𝑍 ∥ 𝜔 𝑞 >0
Stability for negative 𝜔 𝑞
The worst unstable case is 𝜔 𝑟 = 𝜔 𝑞 >0
Coupling with a single frequency line
1 𝜏 𝜇, ∥ = 𝑒𝜂 𝐼 𝑏 4𝜋𝐸 𝑄 𝑠 𝜔 𝑟 𝑅 𝑠
Max 𝑅 𝑠 producing a growth rate compensated by the natural radiation damping
HOM shunt impedance

24. RW transverse and longitudinal impedance
E. Belli - Impedance model and collective effects for FCC-ee
ImpedanceWake2D
ImpedanceWake2D

25. RW longitudinal wake potential
E. Belli - Impedance model and collective effects for FCC-ee
Macroparticle simulations (PyHEADTAIL, SBSC)
Wake potential of a bunch as the convolution between the wake function, i.e. the wake potential of a point charge, and the longitudinal bunch distribution
Variation of the wake function in short distances  Simulations very time consuming due to the large number of slices needed
Wake potential of a very short bunch as Green function for the code
𝑊 ∥ 𝑧 = −∞ 𝑧 𝜔 ∥ 𝑧− 𝑧 ′ 𝜆 𝑧 ′ 𝑑𝑧′

26. RW longitudinal wake potential
E. Belli - Impedance model and collective effects for FCC-ee
𝝈 𝒛 =𝟎.𝟐𝟏𝒎𝒎

27. RW MI – Haissinsky equation
E. Belli - Impedance model and collective effects for FCC-ee
The Haissinski equation gives the bunch length and the distortion of the Gaussian distribution in the presence of wake fields and RF external voltage for intensities below the MI threshold
𝜆 𝑧 = 𝜆 0 𝑒 1 𝐸 0 𝜂 𝜎 𝑑𝑝 2 Ψ(𝑧)
with
Ψ z = 1 𝐶 0 𝑧 𝑒 𝑉 𝑅𝐹 𝑧 ′ − 𝑈 0 𝑑 𝑧 ′ − 𝑒 2 𝑁 𝑝 𝐶 0 𝑧 𝑑𝑧′ −∞ 𝑧′ 𝜆 𝑧 ′′ 𝜔 ∥ 𝑧− 𝑧 ′′ 𝑑𝑧′′
𝝈 𝒛 =𝟐.𝟏𝒎𝒎
𝝈 𝒛 =𝟐.𝟒𝒎𝒎
𝝈 𝒛 =𝟐.𝟔𝒎𝒎