Some results on the LHC optics

1. Some results on the LHC optics Yi-Peng Sun AB/ABP Group European Organization for Nuclear Research (CERN) 29 November 2007 *Yipeng.Sun@cern.ch

2. Main conclusions • The paper “Synchrobetatron stop bands due to a single crab cavity” by G. H. Hoffstaetter and A. W. Chao. • Single crab cavity creates larger stop bands than dispersion at RF. • Place the crab cavity at dispersion-free places: • Works if the horizontal tune is close to half integer; • Not if the horizontal tune is close to integer. • Stop bands can be avoided at the favorable side of the integer or half integer tune. • Two crab cavities per ring can weaken the stop bands.

3. Introduction (1) • Crossing angle at IP introduces synchrobetatron resonances. • Use crab cavity to eliminate this coupling source. • However, transverse kicks in the crab cavity also depends the longitudinal bunch position, like the beam beam force, so it will also introduce synchrobetatron resonances by themselves. • Use transport matrix to describe the phase space.

4. Introduction (2) • The transport matrix of the accelerating cavity: • The transport matrix of the crab cavity:

5. Introduction (3) • The crab cavity strength: • The one turn matrix: • When no dispersion at the RF and no crab cavity, that is the decoupled case, the four eigenvalues of the the TM is given by: • That gives:

6. Stability conditions • Need all four eigenvalues to have unit absolute values. • (A) Integer horizontal tune • (B) • Integer and half integer horizontal tune

7. Three cases

8. General results for (A)

9. General results for (B)

10. General results for (C)

11. Horizontal tunes close to integer • The boundary condition is found by solving: • (A) No crab cavity • Expand to second order:

12. (A) No crab cavity

13. (B) No dispersion at both cavities

14. (C) Dispersion at crab cavity

15. (C) Dispersion at crab cavity

16. Horizontal tunes close to half integer • For B factories or e/e+ colliders, horizontal tune is usually close to half integer. The synchrotron tune is small and two eigenvalues will not be brought together by the coupling force. • In that case, one of the eigenvalues will become real and condition (B) is important. • The borders of instability are found by solving:

17. (A) No crab cavity

18. (B) No dispersion at both cavities The stop bands only appear at slightly above half integer. Horizontal tune: 8.51

19. (C) Dispersion at crab cavity

20. (C) Dispersion at crab cavity

21. (C) Dispersion at crab cavity

22. A pair of crab cavities • A pair of crab cavities can strongly reduce the width of the stop bands when the two crab cavities have betatron phases which differ by odd multiples of pi.

23. Comparison (1) Two crab cavities One crab cavity

24. Comparison (2) Two crab cavities One crab cavity

25. Thanks for your attention.