Understanding Wakefields & Impedance in Particle Accelerators

1. Wakefields & Impedance • wake fields & impedance • longitudinal impedance • transverse impedance

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 for Gaussian charge Distribution:

7. transverse effects deflection (tune shift, focusing change, instability, etc.)

8. practical description – effective impedance longitudinal effective impedance transverse effective impedance

9. practical description – loss factor energy loss per beam particle longitudinal loss factor units: V/pC transverse kick factor units V/pC-m sign convention !?

10. relation between real and imaginary impedance Kramers-Kronig relations due to causality of the wake functions, the impedances cannot have singularities in the upper complex w-plane; then the real and imaginary parts must be related by the above Hilbert transforms

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

12. 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)

13. 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

14. 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

15. Loss factor - 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 early direct measurement at SLAC in 1975 using RF vector voltmeter 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) M. Allen, et al., IEEE Trans. On Nucl. Sci. (1975) L. Rivkin, et. al., IEEE Trans. On Nucl. Sci. (1985) B. Podobedov, R. Siemann, PRST-AB (1998)

16. RF phase measurement using streak camera typically two bunches separated by ~1/2 ring one bunch is weak, used a as a time reference the intensity of the other bunch is varied, and its position w.r.t. weak bunch is measured example measurements from the APS bunch centroid shift bunch profile a loss factor of 10 V/pC was measured according to simulations, cavities are responsible for 70% of the total measured loss factor N. Sereno et al. PAC1997; V. Sajaev, 2014 ICFA impedance workshop

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

18. localized loss factor: DI – measurement at TRISTAN only total non-cavity related loss factor main complications: orbit noise and BPM intensity dependence measurement result: non-cavity loss factor: ~250 V/pC RF section measured loss factor vs. bunch length loss factor around the ring inferred from dispersive orbit shift C. Zhang, Proceedings of EPAC 1996

19. localized loss factor DI – measurement at LEP • by turning off one RF section out of two, loss factor of RF section was measured • measurement results: • loss factor of 1 RF section: 210 V/pC • total loss factor per arc: 90 V/pC D. Brandt, Proceedings of PAC 199 V. Sajaev, 2014 ICFA impedance workshop

20. varying local loss factor– measurement at ELETTRA horizontal orbit deviation vs. vertical scraper aperture g E. Karantzoulis et al., PRST-AB 6 (2003) orbit deviation looks like dispersion!!?!

21. varying local loss factor – measurement at ANKA 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., EPAC’04

22. localized loss factor Dsz – measurement at the SLC measurement of energy loss due to longitudinal wake fields in the SLC collider arcs, APAC’98 note that zFWHM/2.355 becomes less than 50 mm near s = 500 m data and fits for other bunch lengths (different settings of bunch compressor voltage and linac RF phase)

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

24. longitudinal impedance – bunch lengthening bunch length changes are defined mostly by a combination of potential well distortion and microwave instability; bunch lengthening due to microwave instability can be described as streak camera is a straightforward way to measure bunch length; measurements at ALS gave Z/n = 0.2 W J. Byrd, et. al., Proceedings of PAC 1995

25. longitudinal impedance – bunch lengthening often bunch lengthening due to potential-wall distortion is important ; simulations (tracking) can deal with complex cases ; calculated wake function was used to simulate bunch lengthening at APS: Y-C. Chae, Proceedings of PAC 2007

26. 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’

27. longitudinal impedance – quadrupole mode shift quadrupole mode frequency shift vs. intensity CERN SPS (E. Shaphoshnikova)

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

29. longitudinal impedance – beam transfer function longitudinal stability diagram 1/BTF(w) of a dense beam obtained with electron cooling (N=7.7x109 protons circulating in LEAR); the harmonic near f=8 MHz was scanned result: Z/n~300±150 W + i(3500±1000) W imaginary part dominated by space charge J. Bosser et al., 1991

30. 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]

31. 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

32. transverse impedance – head-tail growth rate 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)

33. transverse impedance – head-tail growth rate Concurrent measurement of head-tail growth rate measurement SPS measurements in 1999 gave Re(Zeff,y) → Zt= 8.3±0.6 MW/m (assuming resonator impedance at Q~1) Growth rate of the vertical head-tail mode instability (in units of 10−3 turns−1), as a function of the decrement of chromaticity. H. Burkhardt, F. Zimmermann, M.P. Zorzano

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

35. transverse impedance – tune shift with current tune shift with current is one of the basic measurements ; it was used to monitor impedance changes in the SPS in preparation for the LHC SPS measurements in 1999 gave Im(Zeff,y) → Zt= 28±2 MW/m (resonator impedance at Q~1) horizontal and vertical tune as a function of the bunch population horizontal DQ is small and positive due to contribution from “quadrupole wake” for a flat chamber H. Burkhardt, F. Zimmermann, M.P. Zorzano

36. transverse impedance – tune shift with current at ELETTRA, tune shift was measured after each small-gap chamber installation ; total vertical reactive impedance found: 0.3 MW/m; impedance of one small-gap chamber: 50 kW/m for a dipolar wake and “short” bunches, the tune shift is expected to be negative E. Karantzoulis, et. al., PRST-AB 2003 similar measurements at the SPS by H. Burkhardt et al., around 2000

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

38. local measurements using turn-by-turn BPMs if focusing depends on beam current, phase advance depends on current ; local phase advance can be measured by exciting betatron motion and recording on BPMs ; at LEP phase advance was measured as a function of current; from this the impedance of RF cavities and arcs was extracted ; good agreement with expected values was found D. Brandt, Proceedings of PAC 1995

39. frequency shift of head-tail modes measurements at BESSY Touschek (red) and elastic Coulomb (black) loss rates, and vertical beam size (green) measured as a function of the frequency used for the modulation of the amplitude of the beam exciting 250 MHz voltage. The head-tail modes m=±1 are excited. Head-tail modes as a function of current in a single bunch at Q’=0. Note the strong broadening of the m=+1-mode, the small line emerging from the m=-1-mode, and the hysteresis effects of this mode between up- and down-scans (green and red curves) at higher beam currents. P. Kuske

40. 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):

41. measurement at VEPP-4M (BINP) to avoid beam current dependence of BPMs, two orbits with low and high beam currents are measured first (without the bump) ; then two orbit with the bump are measured for the same currents ; the resulting orbit change is obtained as difference of differences of orbits: V. Kiselev, V. Smaluk, Proceedings of EPAC 1998

42. measurement at APS two types of 4-correctors bumps were tested – parallel and angle bumps ; bump closure is enforced using orbit feedback adjusting 2 correctors; impedance inferred from corrector excitation bump amplitude is scanned as well as the beam current L. Emery, G. Decker, J. Galayda, Proc. PAC 2001

43. measurement at ESRF a BPM vector is generated as a difference between orbits with +1 mm and -1 mm bumps in the insertion device (ID) of interest; two such measurements are performed for the same total current (22 nC in 992 bunches or in a single bunch); the difference orbit around the ring is then fitted with 2 correctors located at ends of the ID straight section L. Farvacque, E. Plouviez, Proc. EPAC 2002

44. 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., EPAC’04 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, …

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

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

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

48. 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

49. 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 - calculated SPS

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