synchronous phase synchrotron tune dispersion momentum compaction chromaticity

1. Longitudinal Optics Measurement and Correction • synchronous phase • synchrotron tune • dispersion • momentum compaction • chromaticity [MCCPB, Chapter 7]

2. 1. synchrotron tune & synchronous phase longitudinal coordinates: equations of motion momentum compaction factor smooth approximation (not correct for rings with strong rf focusing, which require difference equations for localized rf; examples: LEP, DAFNE-II) linearize around synchronous phase synchrotron frequency

3. usually sinusoidal rf voltage: synchrotron tune sidebands of revolution frequency due to modulation of arrival time and for nonzero D also modulation of transverse position harmonic number synchronous phase from synchrotron tune and/or quantum lifetime, or direct measurement (?)

4. d f

5. illustration of synchronous phase

6. measurement of multiple synchrotron sidebands at injection into the SLAC electron damping ring; the synchrotron frequency is given by the difference frequency between the fundamental and the nearest synchrotron sideband

7. 2. dispersion dispersion closed orbit betatron motion in a storage ring in a transport line or linac (1,6) transport matrix element from point where momentum error d is introduced

8. measuring the dispersion a) rf frequency shift momentum deviation detect change in horizontal and vertical orbit

9. ‘static’ dispersion measurement in the PEP-II HER; orbit change induced by a 2-kHz shift in rf frequency; nominal frequency is 476 MHz, harmonic number h=3492, and the momentum compaction factor ac~0.0024

10. ‘static’ dispersion measurement at the KEK/ATF Damping Ringbefore (top) and after applying a correction (bottom) based on exciting steering magnets; the vertical dispersion was measured by a 5-kHz shift in rf frequency; a dispersion of D=10 mm corresponds to Dx~30 mm.

11. measuring the dispersion b) rf modulation rf is modulated at synchrotron frequency; induced orbit variation at this frequency dispersion non-vanishing D in cavities ‘spurious’ dispersion Borer et al., LEP

12. CERN SL/91-38 (AP) ‘Effect of residual dispersion at the RF cavities on the dynamic measurement of dispersion in LEP” by Francesco Ruggiero arc b function dispersion invariant in cavities

13. measuring the dispersion c) rf amplitude or phase jump SLC damping rings (DV) ATF damping ring (Df) also these give spurious results, if there is dispersion in the cavities Dxb=-DDd

14. d) resonant dispersion growth & resonant correction betatron oscillation constant amplitude dispersion dipole feed-down due to trajectory offset in quadrupole leads to resonant excitation D increases linearly (F. Ruggiero, A. Zholents) FFT around ring of normalized Dypeaks near Qy!

15. correct dispersion by special orbit bumps across arcs these produce dispersion with amplification factor Q’cell: chromaticity of single FODO cell Ncell: number of cells covered by bump bumps across various arcs can be combined in a symmetric or anti-symmetric manner to control either D or D’ at the collision points

16. e) higher-order dispersion in transport line or linac several energy steps are made by variation of rf amplitude and phase

17. or, alternatively, write at ith BPM then, for set of M BPMs fit oscillation determine initial conditions of nth order dispersion and correct with linear combination of multipole magnets algorithm developed for SLAC North and South Ring-to-Linac transfer line (P. Emma)

18. evidence of 3rd order dispersion in the SLC ring-to-linac transfer line BPM x large cubic component! <d>

19. 3rd order dispersion for all BPMs in the RTL and in the early linac U1666 BPM data 3rd order dispersion in linac is fitted to compute U1666 and U2666 matrix elements s [m] large 3rd order dispersion led to irrecoverable emittance growth

20. multiknobs for corrrecting 2nd order dispersion, chromaticity, … T211 geometric (insignificant) T216 chromaticity K1=K2 T266 2nd order dispersion K1=-K2

21. at the SLC a pair of octupoles was installed to cancel the U1666 and U2666 terms; emittance was minimized by scanning octupole-pair setting: the octupole strength for which the emittance is minimized agrees with the fit from the BPM data

22. 3. momentum compaction measuring the momentum compaction factor a) synchrotron tune

23. 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; tfe difference due to current-dependent parasitic modes is clearly visible (A.-S. Muller)

24. if the energy is known at one point, i.e., on a spin resonance, the rf voltage can be calibrated from the Qs vs Vc curve fitted beam energy energy known from resonant depolarization voltage calibration factor g (A.-S. Muller)

25. measuring the momentum compaction factor b) bunch length plot bunch length vs. inverse synchrotron tune sd either measured from decoherence due to nonzero chromaticity (see last week) or calculated from optics: can be verified by measuring horizontal emittance ~/(3-Je) or longitudinal damping time ~1/Je

26. rms bunch length measured by streak camera in the PEP-II HER as a function of the inverse synchrotron tune fitted slope determines the momentum compaction factor, if the rms energy spread is known

27. measuring the momentum compaction factor c) quantum lifetime (for electron storage rings) momentum acceptance with note: recipe: measure Qs. sz and tq for different rf voltages and fit for aC! using previous equations (assumes lifetime limited by quantum fluctuations)

28. measuring the momentum compaction factor d) direct measurement using streak camera R56 measurement for the asynchronous arc of the KEKB linac before and after correction; a streak camera was used to measure arrival time as a function of beam energy streak camera trigger was locked to the linac rf frequency upstream of the arc correction was done by changing a few quad strengths

29. measuring the momentum compaction factor e) beam energy via resonant depolarization spin tune if radially oscillation field is in resonance with the fractional part of the spin tune, the effect of the field adds up over many turns and the beam depolarizes; the exact value of the resonant frequency determines the beam energy via the above equation recipe: measure energy change caused by shift in rf frequency slope is momentum compaction factor

30. change of beam energy E as a function of rf frequency frf in LEP only last 4 digits of frequency are displayed (nominal value is 352 254 170 Hz); several strong spin resonances are indicated by the dotted lines; from this measurement the momentum compaction was determined to be 0.000186+/-0.000002, consistent with theoretical value 0.0001859 (R. Assmann)

31. measuring the momentum compaction factor f) change in field strength for unbunched proton beam energy of unbunched proton beam is constant, neglecting SR losses if strength of all magnets (dipoles and quadrupoles) is increased by a factor DB/B, the orbit moves inwards and the revolution time is reduced; this change in revolution time can be detected by a Schottky monitor the momentum compaction factor follows from change in revolution period: this change in revolution period can be detected by a Schottky monitor

32. 4. chromaticity normalized unnormalized relation chromaticity describes the change of focusing with particle energy usually 2 or more families of sextupoles are used to compensate and control the chromatcity small chromaticity is desired to minimize tune spread and amount of synchrobetatron coupling (maximize dynamic aperture) but large positive chromaticity is often employed to damp instabilities (ESRF, Tevatron, SPS,…)

33. measuring the total chromaticity a) tune shift as a function of rf frequency horizontal tune vs change in rf frequency measured at LEP; the dashed line shows the linear chromaticity as determined by measurements at +/- 50 kHz

34. measuring the total chromaticity b) head-tail phase shift deflect bunch transversely and measure the oscillation of head and tail over Ts chromaticity inferred from the measurements of the head-tail phase shift at the CERN SPS top left: head oscillation after kick top right: center oscillation after kick bottom left: phase of head and center and difference bottom right: chromaticity inferred for each turn

35. measuring the total chromaticity c) from de- and recoherence after kick chromaticity measuring the total chromaticity d) from difference in sideband width on Schottky pick up Tevatron, 2004

36. measuring the natural chromaticity (Q’ w/o sextupoles) e) from tune shift vs. dipole field electron ring natural chromaticity measured at PEP-II HER for e-, the orbit is unchanged (determined by rf!) for p, simultaneous change in rf frequency required to keep the same orbit:

37. measuring the local chromaticity f) from tune shift vs. DK at different rf frequencies alternatively and faster, by measuring betatron phase advance around the ring at different rf frequencies

38. g) chromaticity control in s.c. proton rings variation of chromaticity Q’ in time at injection in HERA due to persistent current decay same picture with automatic correction based on continuous sextupole-field measurements in two reference magnets

39. variation of chromaticity during acceleration in HERA • measured w/o correction • variation expected from reference magnets • measured with correction

40. variation of chromaticity at the start of cycle in HERA

41. h) application: measuring the central frequency LEP measure tune vs. rf frequencdy for different sextupole strengths; find intersection -> central frequency; used for monitoring energy changes

42. Summary • synchronous phase • synchrotron tune • dispersion • frequency change, phase modulation, • amplitude or phase jump, transport line, • higher-order dispersion • momentum compaction factor • synchrotron tune, bunch length, lifetime, • path length vs. energy, beam energy vs. frequency, • change in field + Schottky monitor • chromaticity • tune shift due to frequency change, head-tail • phase shift, decoherence, Schottky monitor, • natural chromaticity, chromaticity in s.c. storage • rings, central frequency