1. FCC-ee: coupling impedances and collective effects
E.Belli
M.Migliorati, G.Rumolo
Acknowledgements: M. Zobov, G.Castorina, B.Spataro, S. Persichelli
Impedance meeting
May 12, 2017

2. Outline Resistive wall (RW) Single bunch effects
Microwave Instability (MI)
Transverse mode coupling instability (TMCI)
Multibunch effects
Other impedance sources
Conclusions

3. FCC-ee parameter list for instabilities studies
K. Oide

4. + Resistive wall Vacuum chamber with 35mm radius
Non Evaporable Getter (NEG)
thk =1𝜇m
𝜌= 10 −6 Ωm
Best choice for vacuum (low SEY, low desorption yield)
MI threshold below nominal intensity
Three layers
(No Coating) Cu
2mm
𝜌=1.66∙ 10 −8 Ωm
Dielectric
6mm
𝜌= Ωm
Stainless Steel
Infinity
𝜌= 6.89∙10 −7 Ωm
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

5. RW - Transverse and longitudinal impedances
ImpedanceWake2D
ImpedanceWake2D

6. RW – Single bunch effects
Main single bunch effects of the RW on the single bunch dynamics
Microwave Instability in the longitudinal plane
Transverse Mode Coupling Instability 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
As the bunch intensity increases, the mode frequencies shift and merge, giving rise to the instability
Instability threshold localized in the point where there is mode coupling. 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)
DELPHI

7. MI - Longitudinal impedance
Higher values of impedance for NEG
Im[ 𝑍 𝑇𝑖𝑁 ]≈Im[ Z AC ]
Re[ 𝑍 𝑁𝐶 ]≈Im[ Z NC ]≈Re[ 𝑍 𝑇𝑖𝑁,𝐴𝐶 ]

8. MI – Longitudinal wake potential
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
𝝈 𝒛 =𝟎.𝟐𝟏𝒎𝒎
𝑊 ∥ 𝑧 = −∞ 𝑧 𝜔 ∥ 𝑧− 𝑧 ′ 𝜆 𝑧 ′ 𝑑𝑧′

9. MI – Haissinski equation
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

10. MI – Bunch length and energy spread
60 ∘ optics (45.6 GeV), No Beamstrahlung ( 𝜎 𝑧 =2.1𝑚𝑚)
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
MI regime

11. MI – Bunch length and energy spread (WBS)
Beamstrahlung, i.e. SR during the collision with the opposite beam
Total energy spread
Total bunch length
𝜎 𝑑𝑝,𝑡𝑜𝑡 = 𝜎 𝑑𝑝,𝑆𝑅 𝜎 𝑑𝑝,𝑆𝑅 4 +𝐴 𝜎 𝑑𝑝,𝑆𝑅 2 𝜎 𝑧,𝑆𝑅 2 =𝟎.𝟏𝟏𝟐%
𝐴=1.4 𝑛 𝐼𝑃 𝜏 𝐸,𝑆𝑅 4 𝑇 0 𝑟 𝑒 5 𝑁 𝑏 3 𝛾 2 𝛼 𝜎 𝑥 3
with
𝜎 𝑧,𝑡𝑜𝑡 = 𝛼 𝑐 𝐶 2𝜋 𝑄 𝑠 𝜎 𝑑𝑝,𝑡𝑜𝑡 =𝟔.𝟐𝟐𝒎𝒎
Stable beams

12. MI – Coating thickness?
60° optics (45.6 GeV), No BS ( 𝜎 𝑧 =2.1𝑚𝑚)
Does the material thickness affect the MI?
Example: AC
The MI instability threshold is 3x higher in the case of 25nm thickness

13. MI – Increasing 𝜎 𝑧 and 𝑁 𝑝
D. Shatilov, “Recent results about the X-Z instability”, 53th FCC-ee Optics Design meeting
in the same proportion
Boussard criterion to obtain a first estimation of the MI threshold
𝑍 ∥ 𝑛 𝑛 ≤ 2 𝜋 𝐸 0 𝑐 𝑒 2 𝛼 𝑐 𝜎 𝑧 𝑁 𝑝 𝜎 𝑑𝑝 2

14. TMCI seems to be not dangerous
TMCI simulations
60 ∘ optics (45.6 GeV),
No BS ( 𝜎 𝑧 =2.1𝑚𝑚)
Macroparticle simulations with PyHEADTAIL
Analytic simulations with DELPHI (N.Mounet), by including the bunch lengthening due to the longitudinal wake
TMCI seems to be not dangerous

15. RW - Transverse coupled bunch instability
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

16. RW - Longitudinal coupled bunch instability
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

17. Other impedance sources
Other vacuum chamber components (absorbers, RF cavities with tapers) already evaluated
Contribution negligible compared to the RW one
4000 Beam Position Monitors
4000 winglet-to-circular double tapers
8000 bellows with RF fingers
CST wakefield simulations (Gaussian bunch of 4mm RMS bunch length)

18. Other impedance sources
RW analytic wake potential (no coating case)
Sirius BPM seems to be a good candidate
Contributions of tapers and bellows with RF fingers not negligible

19. Conclusions
The effects of the resistive wall on the beam dynamics have been analyzed
No Beamstrahlung: MI regime, i.e. the beam would be unstable (for NEG MI threshold is below the nominal bunch intensity)
Beamstrahlung allows to have stable beams
injection with alternating beams would allow a good margin of safety
TMCI not dangerous: threshold far beyond the nominal bunch intensity
Robust feedback system required for the fast (6 turns) transverse instability suppression
Maximum allowed shunt impedance of a HOM as a function of its resonant frequency estimated
Other impedance sources have been evaluated
SR absorbers and RF cavities with tapers negligible w.r.t. RW
The four-button BPM design seems to be a good candidate
The contributions of tapers and bellows with RF fingers are comparable with the RW one
The impedance can be reduced by
Removing the tapers by placing the BPMs on the winglet vacuum chamber
Using a different design for the bellows, for example the comb-type RF shield proposed for KEKB.