Orbit Distortion calculated by tracking code (PTC)

1. Orbit Distortion calculated by tracking code (PTC) David Kelliher ASTEC/STFC/RAL 4th November, 2007

2. Contents • PTC model of EMMA • Orbit distortion calculated by PTC • Correcting closed orbit distortion • Correcting accelerated orbit distortion • Local orbit correction

3. PTC model of EMMA

4. Polymorphic Tracking Code • PTC is a kick code, allowing symplectic integration through all accelerator elements • Inherently based on a map formalism • New subroutines were written to simulate acceleration in EMMA. • PTC_TWISS parameters calculated in the Ripken style.

5. Select synchronous frequency

6. Calculate phase of rf cavities at synchronous energy

7. Acceleration in longitudinal phase space

8. Orbit distortion calculated by PTC

9. Phase space with vertical misalignments Vertical alignment error standard deviation = 50 microns 120V/cavity assumed

10. Vertical Distortion at rf cavities during acceleration

11. Vertical Distortion during acceleration Cavities rms = 2.4 mm All elements rms = 3.2 mm

12. Vertical orbit distortion and misalignment All elements included

13. Vertical orbit distortion and misalignment rf cavities included

14. Conclusions • PTC has been set up to model EMMA. The characteristic serpentine acceleration curve in longitudinal phase space is produced. • Accelerated orbit distortion is not noticeably influenced by integer tune resonances. • Orbit is distorted by quasi-random kicks instead. • Corrector strengths which reduce closed orbit distortion do not reduce accelerated orbit distortion, i.e. corrector magnets are not useful in reducing orbit distortion in non-scaling FFAGs with acceleration completed in few turns. • Amplification factor (rms of distortion/  misalignment) is of order 100. • To reduce orbit distortion (both vertical and horizontal), local correction of magnets is necessary.

15. Closed orbit distortion correction

16. BPMs and vertical kicker location Neil Bliss 3/4/07

17. Find optimal vertical corrector strengths to reduce closed orbit distortionLeast-Squares Minimisation method For consistency, would like to replace MADX TWISS with PTC_TWISS Input Error (BPM data) Vertical magnet misalignments Apply a set of vertical corrector strengths from within some range 2 > target value Calculate TWISS orbit at select energies Calculate penalty function 2 - Given by the sum of vertical orbit distortion squared over select energies. Output vertical orbit distortion at each energy 2 < target value Use vertical corrector strengths over entire energy range

18. Correction of closed orbit in range 10-11.2 MeV(MADX TWISS)

19. Correction of closed orbit in range 10-11.2 MeV(PTC_TWISS, using corrector settings from MADX)

20. Correction of accelerated orbit in range 10-11.2 MeV(PTC_TWISS, using corrector settings from MADX)

21. Correction of closed orbit over full energy range(MADX TWISS)

22. Correction of closed orbit in full energy range(PTC_TWISS, using corrector settings from MADX)

23. Correction of accelerated orbit over full energy range(PTC_TWISS, using corrector settings from MADX) rms of vertical orbit distortion in both cases is 1.8 mm

24. Failure of Harmonic Correction • Effect of kicker varies since the phase advance between the kicker and any point in the lattice changes as the momentum changes. Therefore, conventional harmonic correction of the error source with kickers is not possible in a nonscaling FFAG.

25. Accelerated orbit distortion correction

26. Introduction • Already showed that corrector strengths which reduce closed orbit distortion do not in general significantly reduce “accelerated orbit” distortion. • This presentation: vary corrector strengths to see if accelerated orbit distortion can be reduced. • Only vertical orbit distortion due to magnet misalignments considered.

27. Method • Vary first corrector strength, run PTC, calculate the orbit distortion rms over the full energy range, find minimum • Improve result by varying corrector strength about this minimum. • Repeat for each corrector and find the best one. • Keeping this optimal corrector, repeat the exercise for a second corrector. • Continue until 16 correctors used.

28. Scan for minimum orbit distortion uncorrected orbit rms = 2.7mm

29. Adding more correctors

30. Correction with 1 kicker Vertical orbit distortion rms reduced from 2.67mm to 0.64mm

31. Dependence of distortion on initial conditions I • In general, the closed orbit co-ordinates at the injection momentum do not give optimal initial conditions. Starting the tracking at a momentum where the integer tune is close to integer results in relatively high orbit distortion.

32. Dependence of distortion on initial conditions II • In general, the closed orbit co-ordinates at the injection momentum do not give optimal initial conditions. A scan over the initial phase space (x,x’,y,y’) allows the distortion to be reduced.

33. Optimise initial (y,y’) Vertical orbit distortion rms reduced from 2.67mm to 0.63mm

34. Conclusions • Important to first establish initial conditions in (x,x’,y,y’) that best reduce the orbit distortion. In general this (x,x’,y,y’) will not be the same as the closed orbit at the injection energy.

35. Optimising the accelerated orbit distortion • For a given set of magnet misalignments, is it possible to calculate the optimal accelerated orbit – i.e. the orbit with the minimum orbit distortion rms over the momentum range? • How many vertical corrector magnets would be needed to kick the beam onto this optimal orbit? Introducing correctors magnets will change the optimal orbit itself.

36. Local vertical orbit correction

37. Infer magnet misalignments from kicker strengths(see my presentation - May 14th) • Misalignment of quadrupole j can be expressed as a kick j • Orbit position measured at ideal BPM i is the sum of these kicks. • MADX CORRECT module finds the set of k that best restores target orbit. • Kickers are placed in the middle of divided magnets. Assume k j.

38. Calculate vertical misalignments at 10 MeV (no source of error other than vertical misalignments) rms error = 1.5 microns

39. Accuracy of vertical misalignment prediction (10-11 MeV) (no source of error other than vertical misalignments)

40. Future Work • Work with Etienne Forest to improve PTC modelling of EMMA (after this workshop) • Investigate if local correction of magnet misalignments is feasible.