Simulation of the FCC-ee lattice with quadrupole and sextupole misalignments

1. Simulation of theFCC-ee lattice with quadrupole and sextupole misalignments Sergey Sinyatkin 13.10.2016

2. Mission Using ordinary tolerances for magnetic elements preserve small vertical emittance. Transverse shift of elements is Gaussian with RMSx,y = 100 um, truncated at 2*Sigma. Estimation of vertical emittance with misalignmentsof magnets in the area of extremely large betas(IR). Estimation of vertical emittanceafter orbit and betatron coupling correction.

4. Task (Quadrupole misalignment ) • Quadrupolesare shifted randomly (Gauss) in transverse plain with Sigma_x,y = 100 um, truncated at 2*Sigma. • FF quads and those in V and H chromatic sections (extremely large beta) meanwhile are not shifted. • Q shift distorts CO, vertical dispersion and generates betatron coupling (COD at sextupoles) leads to vertical emittance growth. • I installed BPMs (4-5 per betatron wave) and correctors (3-4 per wave). • Skew-quads are installed in the FF quads, in the IR vertical chromatic section and in every hundredth arc quads. • Correctors and skew-quads correct COD, coupling and vertical dispersion. What vertical emittance can we achieve after correction?

5. Correction COD, coupling and Dy is corrected by MADX. The process is time consuming. All sextupoles and bends are perfectly aligned. The results are preliminary

6. Q shift and CO • For the 100 um Q shift the closed orbit w/o correction is hardly can be found (but can) • For the 10 um Q shift w/o correction the probability for CO is ~ 50%. • For the 100 um Q shift after correction the CO exists for ~ 30-40% samples. • CO can not be found when the vertical tune moves to the half integer or integer resonance.

7. CO search by iterations Correction of beam trajectory and betatron phase advancecan be done step by step (BPM by BPM) like in beam line (not closed optics) starting from injection point. (very slowly with MADX) Alternative is closed orbit correction iteratively with misalignments scaledfrom 0 um to 100 um.

8. CO search • vertical tune limits the closed optics • For some seeds MADX failed with CO • E_y/Ex = 0.5-5x10-3 After skew quad correction. Q shift of 100 um (RMS).

9. COD after correction COD around the ring (10 seeds) 100 um quadrupole shift produces average residue COD  70 um (x) and  50 um (y) after orbit correction

10. COD in the IR (after correction) COD in the IR. Max horizontal COD  800 um

11. Optics distortion (9 samples)after correction (βx_err/βx)max = 1.1 (βy_err/βy)max = 1.5 σDx= 1.5 cm σDy= 3.4mm

12. Eigen mode oscillation (1 sample)after orbit correction after Skew quads correction εy/ εx = 0.15 % before Skew quads correction εy/ εx = 1.6 %

13. Vertical emittance excitation CO correction CO correction + SQ in the IR + SQ in the arcs CO correction + SQ in the IR

14. Steerers strength σkickx= 6.6 urad σkicky= 6.5 urad

15. Skew quad strength σK1sL= 79e-6 m-1

16. Q tilt (coupling) Quadrupoles rotate randomly (Gauss) with Sigma_tilt = 100 urad, truncated at 2Sigma. FF quads and those in H/V chromatic sections are not rotated. Tilt excites vertical dispersion and generates betatron coupling  vertical emittance degrades. Skew-Quads are installed in the FF quads, IR vertical chromatic section and in every hundredth arc quad.

17. Optics distortion Before SQ correction σDy= 2.6 mm After SQ correction σDy= 1.7mm

18. Vertical emittance excitation Before correction After correction by SQ (E_y/E_x ~0.5x10-3)

19. Skew quad strength σK1sL= 76e-6 m-1

20. FF quads misalignment FF Quadsare shifted randomly (Gauss) transversely with Sigma_x,y = 50 um, truncated at 2Sigma. Misalignments of other elements are zero. Q shift distorts CO, vertical dispersion  vertical emittance degrades. Skew-quads are installed in the FF quads, IR vertical chromatic section and in every hundredth arc quads. IR Steerers correct COD Coupling and vertical dispersion is corrected by Skew-quads.

21. FF Quads shift and CO There is not the closed orbit w/o correction for the 50 um Q shift. For the 10 um Q shift w/o correction the probability for CO is ~ 50% (in many cases it is MADX problem to find CO). For the 50 (100) um Q shift after correction the CO exists for all samples.

22. FF quads misalignment After orbit correction <|ηy|>= 3 cm εy / εx = 3 % After orbit and vertical dispertion correction <|ηy|>= 0.7 cm εy / εx = 0.3 % FF Quads: Sigma_x,y = 50 um, truncated at 2Sigma

23. Emittances ratio • Unacceptable ratio between ver. and hor. emittances ~ 2 % after dispersion correction by SQ.

24. FF quads misalignmentCOD and dispersion functions

25. Conclusion (Quads) • For the 100 um Q shift w/o correction CO exists for ~few% samples. • To have 100 % CO one has to correct CO and keep the vertical tune far from resonance. • For the 100 um Q shift CO correction by dipole steerers and coupling correction by skew quads provide the emittance ratio less then 0.1%. • Q tilt gives low betatron coupling ~0.2%and after suppressing by skew quadrupoles < 0.05%. • Vertical dispersion is strongly excited by FF quads misalignment. • Acceptable value of FF quads tolerance < 50 um(expected 20 – 30 um).

26. Task (Sextupoles misalignment ) All sextupoles are shifted randomly (Gauss) transversely with Sigma_x,y = 100 um, truncated at 2Sigma The shift distorts CO, generates betatron coupling and vertical dispersion  vertical emittance degrades BPMs (4-5 per betatron wave) and steerers (3-4 per wave) are installed Skew-quads are installed in the FF quads, IR vertical chromatic section and in every fiftieth arc quads Steerers and skew-quads correct COD, coupling and vertical dispersion. What vertical emittance can be achieved after correction?

27. COD COD around the ring (200 seeds) 100 um sextupole shift produces average COD  10 um (x) and  20 um (y)

28. COD in the IR COD in the IR. Max vertical COD  400 um is in FF quad

29. Optics distortion (βx_err/βx)max = 1.7 (βy_err/βy)max = 2.3 σDx= 2.2 cm σDy= 5.2 cm

30. Vertical emittance excitation Too bad samples. Shall not be counted? Average: y  10% x

31. Correction COD, coupling and Dy correction was performed by MADX. The process is time consuming, so not too many samples (10) are available by now. The results are preliminary

32. COD after correction COD around the ring (10 seeds) 100 um sextupole shift produces average COD  0.1 um (x) and  0.2 um (y) m

33. Optics correction A random sample before and after COD and coupling correction (βx_err/βx)max = 1.1 (βy_err/βy)max = 1.9 σDx= 1.2 cm σDy= 0.8 cm 10 samples (βx_err/βx)max = 1.7 (βy_err/βy)max = 2.3 σDx= 2.2 cm σDy= 5.2 cm 200 samples

34. Vertical emittance after correction y/x = 310-3

35. Steering magnets

36. Example of skew quad correction εy / εx = 6% -> 0.5 % N ~ 10 (strong skew quads)

37. Conclusion • 100 um misalignment of sextupoles provides unacceptable vertical emittance y/x  10%. • This is recovered to y/x ≈ 310-3 by COD and coupling correction. • Vertical dispersion is reduced by skew quadrupoles (σDy= 5 cm  σDy= 0.8 cm), the some part of ver. emittance is formed by betatron coupling.