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, 2017 - Berlin

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

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

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 [ 10 11 ] 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

+ 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= 10 15 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

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

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

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 ⋅ 10 11 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

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

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 3 2 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

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

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 3 2 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

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

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

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

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 [ 10 11 ] 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 2 3 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

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

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

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 [1 0 11 ] 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

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

Thanks for your attention

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

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

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

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′

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

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