1. Beam observation and Introduction to Collective Beam Instabilities Observation of collective beam instability Collective modes Wake fields and coupling impedances Head-tail instability Microwave instability Beyond T. Toyama KEK

2. Observation of collective beam instability Example: KEK-PS 12 GeV Main Ring At 500 MeV injection plat bottom a beam loss occurs (red curve). Amount and timing of the loss => random. Proton number N B (Feedback CT) Magnetic field

3. Example: KEK-PS 12 GeV Main Ring At phase transition energy~5.4 GeV (in kinetic energy) a large beam loss occurs (red curve). Amount of the loss is at random. Proton number N B (Feedback CT) Magnetic field

4. Observation NBNB Multi-trace of horizontal betatron oscillation NBNB Amplitude of betatron oscillation Magnetic field

5. Observation horizontal betatron tune during acceleration t f f rev 2f rev f rev  - f  f rev  - f  2f rev  - f  2f rev  f  Without external kick, coherent oscillation emerged

6. Measurement by a wall current monitor Real signals may be attenuated by the loss in the cable > 100 m and limited band width of the WCM.

7. Beam loss: collective instabilities --- at random, a kind of positive feedback starting from a random seed direct space charge effects --- regular some mistake in parameterrs --- regular (B, f RF, tune, …)

8. Collective modes Coasting beam / longitudinal n=3 Beam

9. Coasting beam / transverse Collective modes n=3 Beam betatton oscillation x or y

10. Bunched beam / logitudinal Collective modes l=1l=2l=3 dipole quadrupolesextupole z zz charge density Phase space ….. no momopole mode

11. Bunched beam / transverse Collective modes dipole mode density  z  x zz z l=0l=1l=2 monopole dipolequadrupole ….. superimposed

12. Wake fields and coupling impedance Electromagnetic fields is produced by the beam passed by.

14. Wake fields and coupling impedance

17. Wake fields due to a Gaussian beam in a resistive pipe Longitudinal wake potentialTransverse wake potential Acceleration Decceleration Dampen deflection Further deflection

18. Wake fields and coupling impedance Impedance of a resistive pipe

19. Wake fields and coupling impedance Wake fields by cavities Q=1Q=10

20. Wake fields and coupling impedance Impedance of cavities Q=1Q=10

21. Head-Tail Instability Transverse bunched beam instability Time domain picture

22. Head-Tail Instability Chromaticity = 0 Red full line: (z)x(z) Red dushed line: (z)x’(z) Blue: kick due to resistive wall Growth Damp No effect ~Totally no effect (1) (2) (3) (4) head tail

23. Head-Tail Instability Head-tail phase z  p/p   0 phase of betatron oscillation phase space of synchrotron oscillation

24. Head-Tail Instability Damp  ~ 1 Red full line: (z)x(z) Red dushed line: (z)x’(z) Blue: kick due to resistive wall (1) (2) (1) (2) ~Totally damping head tail

25. Head-Tail Instability Growth ~Totally growing (1) (2)  ~  Red full line: (z)x(z) Red dushed line: (z)x’(z) Blue: kick due to resistive wall (1) (2) head tail

26. Head-Tail Instability Summary of Growth rate vs. Chromaticity Head-tail phase Growth rate Chao’s text book mode = 0 mode = 1 mode =2 mode =3 Stable Unstable

27. Head-Tail Instability KEK-PS 12 GeV Main Ring T. Toyama et al., PAC97, APAC98, PAC99 mode=0 mode=1 mode=2 NBNB amplitude of dipole oscillation

28. Head-Tail Instability CERN PS higher order head-tail mode R. Cappi, NIM

29. Head-Tail Instability KEK-PS 12GeV MR Frequency domain analysis growth rate ∝ Re[Z(  )] F(   ) Re[Z T ] Form factor F  (freq. spectrum of the beam)  m=0 m=1 m=2

30. Head-Tail Instability Observation Growth rate mode=0

31. Head-Tail Instability Cure Chromaticity control Landau damping by octupole magnets …

32. Beam response and Landau damping Coasting beam Transverse motion

33. Beam response and Landau damping Driving force Response

34. Driving force Response of the beam Absorbed power by the beam The beam: ensemble of the particles Frequency distribution:  The beam motion approaches steady oscillation. Velocity d /dt: in phase with the force Work is done on the beam Absorbed power by the beam: constant Stored energy in the beam: Macroscopic aspect: a beam driven by a force approaches steady oscillation. Microscopic aspect: Small amount of resonant particles grows infinitely large. Response of particles

35. Longitudinal instability Microwave Instability uniform distribution Wake: V=  Z (z) The seed of density modulation is produced V 1 =  Z (z), slippage, 

36. Landau damping by the spread of  rev =  p/p  phase slippage factor = 1/  t 2  1/  2  t  phase transition energy  p/p Density modulation reduced! Larger  p/p more stable

37. Microwave Instability Observation & simulation K. Takayama et al., Phys. Rev. Lett. 78 (1997) 871

38. Microwave Instability Sources: Narrow-band resonances  res ~ 1GHz

39. Cures Reducing Impedance Landau damping Reducing local beam chaege line density Artificial increasing momentum spread  p/p  >  rev Methods Higher harmonic rf cavity Voltage modulation of foundamental rf cavity …

40. Cures Reducing Impedance Exchange ~ 2/3 BPMs new ESM BPM ~2/3 Pump port new one with slits Growth rate reduction

41. Reducing local beam chaege line density Increasing momentum spread  >  rev Voltage modulation of foundamental rf cavity T. Toyama, NIM A447 (2000) 317

42. Beyond  Impedance calculation  Impedance measurements Beam transfer function  Vlasov equation Coupled bunch instability Mode-coupling instability  Electron-cloud instability  feedback system feedback in RF control system feedback damper = pick-up & kicker

43. “… every increase in machine performance has accompanied by the discovery of new types of instabilities.” - J. Gareyte (CERN)

44. References Schools: CAS, USPAS, and OHO (Japanese) Conferences proceedings: APAC, EPAC, and PAC Textbook etc.: A. W. Chao, PHYSICS OF COLLECTIVE BEAM INSTABILITIES IN HIGH ENERGY ACCELERATORS Editors: A. Chao and M. Tigner, Handbook OF ACCELERATOR PHISICS AND ENGINEERING

45. Good Luck!