1. Electron cloud simulations for SuperKEKB Y.Susaki,KEK-ACCL 9 Feb, 2010 KEK seminar

2. 1.Positron beam emits synchrotron radiation 2.Electrons are produced at the chamber wall by photoemission 3.The electrons are attracted and interact with the positron beam 4.The electrons are absorbed at the chamber wall after several 10 ns o Secondary electrons are emitted according the circumferences 5.The electrons are supplied continuously for multi-bunch operation with a narrow spacing Electron cloud is built up Electron cloud built-up K.Ohmi, Phys.Rev.Lett,75,1526 (1995) e-e- γ Secondary e - e + beam y x

3. Wake field is left behind in the electron cloud by advanced bunches The wake field induced by the electron cloud affect backward bunches Coherent instability occurs when there is resonance between the wake field and the backward bunches Coupled bunch instability Single bunch instability Coherent instability due to electron cloud e-e-

4. Coupled bunch instability The wake field causes correlations among bunches Threshold is determined by balance with some damping effects Independent of emittance, momentum compaction Depends on electron cloud density, distribution and motion e-e-

5. Single bunch instability The wake filed causes correlations among positrons within a single bunch Threshold is determined by the balance with Landau damping due to the momentum compaction factor Depends on emittance Depends on only local electron cloud density e+e+ e+e+ e-e-

6. The list of parameters UnitSuperKEKBSuperB E+/E-GeV4/7 I+/I-Amp3.6/2.62.7/2.7 Np×10 10 6.254.53/4.53 Nbun25001740 I bunch mA1.4/1.01.6/1.6 β x,y avem12 νsνs Hz0.012 εxεx nm3.2/1.72.8/1.6 εyεy pm12.8/8.27/4 σxσx mm0.20/0.140.18/0.13 σyσy μm12.3/9.99.1/6.9 σzσz mm6/55/5 Lm30161400

7. Number of the produced electrons(1) Number of the photons emitted by one positron per unit meter SuperKEKB-LER γ=8000, L=3016 → Y γ =0.17 m -1 Bunch population SuperKEKB-LER design (3.6A) Np=10 11 The quantum efficiency for photoelectrons (n p.e. /n γ ) Energy distribution 10±5 eV α= 1/137 (fine structure const.)

8. Number of the produced electrons(2) Number of electrons produced by one positron per unit meter SuperKEKB-LER Y p.e = Y ϒ η = 0.017 m -1 Number of electrons produced by one bunch per unit meter Maximum secondary emission yield

9. Analysises for coupled bunch instabilities

10. Increase of electron density  e with multi-bunch (simulation results)  2,max =1.2 Y p,e N p = 1.7×10 9 (  =0.1) Y p,e N p = 1.7×10 8 (  =0.01) Y p,e N p = 1.7×10 7 (  =0.001) Y p,e N p = 1.7×10 7 (  =0.001)  2,max =1.2  2,max =1.1  2,max =1.0

11. Theηshould be reduced to 0.001! Electron density  e as functions of quantum efficiencies (  and  2,max )  2,max =1.2 Y p,e N p = 1.7×10 7 (  =0.001) η=0.003 w/ antechamber (a simulation result) ρ eth =1.1 ✕ 10 11 m -3 The analytical value of the threshold in the case of SBI Together with solenoid it is expected to reduce the actual η to 0.001 (Suetsugu)

12. Effect of antechamber ©Suetsugu

13. Electron distribution and electric potential with  2,max =1.2 Antechamber Cylindrical chamber

14. Electron distribution and electric potential with  2,max =0 Antechamber Cylindrical chamber

15. Reduction factor for the averaged electron density The ratio of the densities at the beam pipe of ante-chamber and cylindrical-chamber The ratio ≈ 0.03 for δ 2,max =0  The antechamber reduces η in 3% effectively!

16. Wake field induced by electron cloud and beam stability(1) Coasting beam model We assume a homogenous stream of the beam We can apply this model even for bunches if ω e σ z /c>>1 The position of the center of mass in the transverse direction : y s

17. Wake field induced by electron cloud and beam stability(2) EOM for the beam and the cloud n b,c : line density of each particle r b,c : classical radius of each particle F becomes linear near the beam y s the betatron oscillation

18. Wake field induced by electron cloud and beam stability(3) The eq. for the cloud can be solved as The eq. for the beam becomes wake force wake field  tune shift Δω β

19. Wake field induced by electron cloud and beam stability(4) Fourier trans. of the eq. for the beam leads Growth rate of instability = Im ω

20. Wake field for bunch correlation Y 1e N p =1.7×10 9 m -1 (  =0.1)1.7×10 8 m -1 (  =0.01)1.7×10 7 m -1 (  =0.001)

21. Unstable modes and growth rate Y 1e N p =1.7×10 9 m -1 (  =0.1)1.7×10 8 m -1 (  =0.01)1.7×10 7 m -1 (  =0.001)

22. Growth rate as a function of η We evaluate the growth rate associated with the unstable modes as a function of η –The growth rate is 0.02 for η=0.001 (Note that η=0.001 corresponds to the threshold of the SBI when η is evaluated as the function of the electron density) –not so severe that the growth could be suppressed by the feedback from the empirical point of view  The CBI could be avoided below the threshold of the SBI

23. Analysis for single bunch instabilities

24. Stability condition for the single bunch instability Landau damping Coherence of the transverse oscillation is weakened by the longitudinal oscillation associated with momentum compaction The stability condition for ω e σ z /c>1 Balance of growth and Landau damping

25. Threshold of the single bunch instability Threshold of the electron cloud density Q nl depends on the nonlinear interaction K characterizes cloud size effect and pinching ω e σ z /c>10 for low emittance rings We use K=ω e σ z /c and Q nl =7 for analytical estimation

26. Threshold for SuperKEKB and SuperB UnitSuperKEKBSuperB Lm30161400 γ8000 I+/I-Amp3.6/2.62.7/2.7 Np×10 10 6.254.53 I bunch mA1.4/1.01.6/1.6 β x,y avem12 νsνs Hz0.012 σxσx mm0.20/0.140.18/0.13 σyσy μm12.3/9.99.1/6.9 σzσz mm6/55/5 Q77 ω e σ z /c10.9 radiation damping time ms(turn)30(3000)20(4600) ρ e threshold×10 11 m -3 1.13

27. Simulation with Particle In Cell Method for the single bunch instability Electron clouds are put at several positions in a ring Beam-cloud interaction is calculated by solving two- dimensional Poisson equation on the transverse plane A bunch is sliced into 20-30 pieces along the longitudinal direction e+e+ e-e- large enough for describing the oscillations

28. Simulations for instability threshold for SuperKEKB Profiles of the beam size η y =0.2η y =0 ρ e,th ≈2.4×10 11 m -3 ρ e,th ≈2.2×10 11 m -3

29. Bunch and e-cloud profiles at 4000 turn Coherent motions for SuperKEKB η y =0.2η y =0

30. FFT spectra below and above the threshold Unstable modes of the instability for SuperKEKB stable unstable η y =0.2η y =0

31. Simulations for instability threshold for SuperB Profiles of the beam size η y =0η y =0.2 ρ e,th ≈4.4×10 11 m -3 ρ e,th ≈2.6×10 11 m -3

32. Bunch and e-cloud profiles at 4000 turn Coherent motions for SuperB η y =0.2η y =0

33. FFT spectra below and above the threshold Unstable modes of the instability for SuperB stable unstable η y =0.2η y =0

34. Summary Multi-bunch numerical simulation The effective quantum efficiency η should be reduced to 0.001 The antechember alone seems not to be sufficient for achieving η=0.001, but together with solenoid it is expected to cure the situation (Suetsugu) The CBI seems not to be severe with η=0.001 Single bunch numerical simulation The threshold of the electron cloud density for the stability has been estimated for SuperKEKB, SuperB