wake fields & impedance longitudinal impedance transverse impedance beam-beam effects

1. Impedance (& Beam-Beam) • wake fields & impedance • longitudinal impedance • transverse impedance • beam-beam effects

2. ‘test’ particle traveling a distance z behind driving particle experiences longitudinal and transverse wake forces proportional to the product of the two particles’ charges and depend on the distance z , the transverse force is also proportional to the offset of the drive particle from the pipe center,x

3. transverse wake function longitudinal wake function longitudinal dipole wake function Panofsky-Wenzel theorem

4. longitudinal wake function starts at a finite positive value which represents energy loss by the test particle transverse wake function starts at zero and for small |z| grows with a linear slope; negative sign shows that it is defocusing close to the source both wake functions are zero ahead of the source

5. Fourier transform of the wake function is the impedance usually the real part of the impedance is related to instability growth (or damping) rates; the imaginary part shifts the mode frequencies; energy loss is due to the real part of Z0||

6. longitudinal effects induced voltage parasitic energy loss Gaussian charge distribution

7. transverse effects deflection (tune shift, etc.)

8. common types of impedance • resistive wall W0’~1/a, W1~1/a3 • low frequency impedance • coupled-bunch & single bunch • hhigher-order modes modes in rf cavities • narrow-band resonators • drives coupled-bunch instabilities • discontinuities in beam-pipe cross section • broadband resonator • single bunch

9. how to measure impedance effects • detect variation of observable • (tune, orbit,…) with impedance-related • parameter like • bunch intensity • bunch length • chromaticity • chamber aperture (e.g., collimator or • insertion device)

10. parasitic mode energy loss (real part of Z0) • measurements: • coherent synchrotron tune vs. rf voltage for • different bunch currents (LEP example) • synchronous phase shift with current • dispersive orbit vs. intensity [local?] • orbit change with intensity in transport line [local] • threshold of microwave instability (also ImZ0) • instabilities during debunching [w dependence!] imaginary part of longitudinal impedance • measurements: • bunch lengthening/shortening with current • quadrupole mode frequency (2Qs) vs. current • shift of incoherent synchrotron tune vs. intensity

11. LEP model from localization of rf cavities (computed) determined with 10-3 precision voltage calibration energy loss due to SR and impedance synchrotron tune as a function of total rf voltage in LEP at 60.6 GeV; the two curves are fits to the 640 mA and 10 mA data; the difference due to current-dependent parasitic modes is clearly visible (A.-S. Muller) Qs vs Vrf for Different Beam Intensities

12. Synchronous Phase Shift with Bunch Current the rf component of the bunch signal filtered from an intensity monitor is compared with rf cavity voltage for different bunch intensities by means of a vector voltmeter relative phase and amplitude measured while the current is decreased by a scraper SLC Damping Ring 1985 this gives the energy loss for a given charge; bunch length dependence can be measured as well other possibility: phase distance of 2 bunches with unequal charge (K. Bane in SPEAR)

13. Dispersive Orbit vs. Intensity measure horizontal orbit change with intensity due to energy loss at impedance locations the circumference also changes DC = Dx’ Dx+…!?

14. example: horizontal orbit deviation vs. vertical scraper aperture in ELETTRA E. Karantzoulis et al., PRST-AB 6 (2003) orbit deviation looks like dispersion!!?!

15. example: X orbit change in ANKA when Y collimator is closed example: fitted momentum offset (=energy loss?) vs. collimator position A.-S. Muller, F. Zimmermann, et al., 26.-27. August 2003

16. measurement of energy loss due to longitudinal wake fields in the SLC collider arcs, APAC’98 data and fits for other bunch lengths

17. CERN SPS (E. Shaphoshnikova) detect unstable frequencies during debunching identify different frequencies with ring components (here effect of MKE kicker magnets)

18. SLC Damping Ring 1988 bunch lengthens due to inductive impedance and, at higher current, microwave instability energy spread stays constant up to microwave threshold from where it steeply increases synchronous phase shift was measured as well bunch length was measured by using the RTL bunch compressor as ‘streak camera’

19. quadrupole mode frequency shift vs. intensity CERN SPS (E. Shaphoshnikova)

20. incoherent synchrotron tune can be measured by resonant depolarization on a synchrotron sideband example is from ANKA (A.-S. Muller, EPAC’04) change in ratio corresponds to change in bunch length

21. transverse impedance • measurements of real part: • head-tail growth or damping rate vs. chromaticity, • intensity, bunch length • measurements of imaginary part: • betatron tune shift with intensity Im(Z1) • orbit change with intensity (global or local bumps) • [local] • betatron phase advance with intensity [local] • orbit response matrix at different intensities [local]

22. Head-Tail Growth or Damping Rate measure head-tail growth (or damping) rates for different values of chromaticity, intensity and bunch length growth/damping rate proportional to impedance, chromaticity, intensity and beta function

23. 1/t LEP 45.63 GeV, damping rate 1/t vs. Ibunch for different chromaticities [A.-S. Muller] Q’=14 1/(100 turns) horizontal damping partition number Q’=2.7 1/tSR Ib (A.-S. Muller)

24. Coherent Tune Shift measure coherent vertical tune shift as a function of intensity

25. example: tune shift vs. scraper-blade position in ELETTRA E. Karantzoulis et al., PRST-AB 6 (2003)

26. Orbit Change –Transverse Wake measure orbit change as a function of intensity and beam position (e.g., local bumps), or, e.g., as a function of collimator gap, etc. for large y the dependence becomes nonlinear (Piwinski):

27. example: difference orbit in ANKA with collimator closed and open; red curve is a fit to the optics model which gives the kick A.-S. Muller, F. Zimmermann, et al., 26. August 2003 red curve is a fit, scraper is at s=80.6 m, bx=20.7, by=8.0 m, design Dx=0 m, 0.5 GeV, … longitudinal curve, curve vs. collimator gap, tune shift?

28. example: orbit deflection if either ANKA collimator jaw is closed A.-S. Muller, F. Zimmermann, et al., 26. August 2003

29. example: vertical orbit deviation vs. vertical scraper aperture in ELETTRA E. Karantzoulis et al., PRST-AB 6 (2003)

30. example:nonlinear res.-wall & geometric wake deflection; the collimator position is varied for a constant gap size; SLAC linac s’y=1.35 mrad, sz=1.3 mm, Nb=4x1010, E=33 GeV, sx,y=80 mm K. Bane et al., PAC95 Dallas

31. example: same as previous slide, but deflection plotted vs. distance from lower jaw revealing 1/(a-y0)2 dependence K. Bane et al., PAC95 Dallas

32. Current Dependent Phase Advance Take 1000-turn data for different intensity; analysis of the current-dependent phase advance should yield the impedance location and strength effect of a single localized impedance SPS

33. impedance acts like current-dependent quadrupole response matrix:

34. example: localized SPS impedance from current dependent phase advance for impedance localization take data at variable intensity (without changing quadrupoles) some example data: 26 GeV/c 14 GeV/c

35. tune shift with intensity, average and spread over all BPMs in the following discard all data sets with larger errors

36. typical slope with intensity at a few BPMs from different regions of the SPS ateach BPM fit slope and offset to measured phase vs. intensity (above the offset is already subtracted)

37. result of fitting phase vs. intensity at each BPM offset vs BPM no. 0-tune at 14 GeV/c was quite different from model slope vs BPM no. impedance

38. fit Df/DN (the 3rd fit!) by SVD technique with weight and cut-off to the phase beating response matrix; trick to get defocusing impedance: fit in 10 iterations, at each step increase weight of wrong-sign DK by factor 10

39. the final result: large impedance at 4-7 locations SPS regions with impedance identified at both energies: 119 (MKP), 301-307 (arc?), 417-422 (MKE), 507 (arc?) extraction kickers injection kickers

40. Orbit Response Matrix take LOCO orbit response data for different beam intensities example: LOCO measurements at APS V. Sajaev, PAC2003 Portland (2003)

41. Beam-Beam Effects • head-on and long-range beam-beam • tune spread • drop in beam lifetime, limit on luminosity • beam sizes must be matched at IP • sensitivity to tune modulation • bunch-to-bunch orbit & tune differences • coherent and incoherent effects • each tune splits into two or a larger • number of coupled tunes • various compensation techniques are • under study, e.g., • Tevatron Electron Lens • LHC Wire Compensation (Tests in SPS)

42. head-on beam-beam collision

43. long-range collisions on either side of IP

44. beam-beam deflection vs. offset head-on collision long-range collision (30 of these for each head-on!)

45. and LHC: 4 primary IPs 30 long-range collisions per IP 120 in total partial mitigation by alternating planes of crossing at IP1 & 5 etc.

46. Long-Range Beam-Beam Collisions • perturb motion at large betatron amplitudes, where particles come close to opposing beam • cause ‘diffusive aperture’ (Irwin), high background, poor beam lifetime • increasing problem for SPS, Tevatron, LHC,... that is for operation with larger # of bunches

47. experience from Tevatron Run-II “long-range beam-beam interactions in Run II at the Tevatron are the dominant sources of beam loss and lifetime limitations of anti-protons …” (T. Sen, PAC2003) LR collisions reduce the dynamic aperture by about 3s to a value of 3-4s; little correlation between tune footprint and dynamic aperture drop in ey for first 4 pbar bunches after injection; asymp- totic emittance is measure of their dynamic aperture

48. LHC tune “footprint” due to head-on & long-range collisions in IP1 & IP5 (Courtesy H. Grote) LR vertical crossing head-on LR horizontal crossing

49. total LHC tune “footprint” for regular and “PACMAN” bunch (Courtesy H. Grote) LR collisions ‘fold’ the footprint!

50. Long-Range Beam-Beam Compensation for the LHC • To correct all non-linear effects correction must be local. • Layout: 41 m upstream of D2, both sides of IP1/IP5 current-carrying wires Phase difference between BBLRC & average LR collision is 2.6o (Jean-Pierre Koutchouk)