Orbit Distortion and Correction

1. Orbit Distortion and Correction David Kelliher ASTeC/STFC/RAL Design Review III, Daresbury December 10th & 11th

2. Contents • PTC model of EMMA • Orbit distortion - Tolerances • A new orbit correction scheme • Vertical corrector magnets Design Review III, Daresbury

3. PTC model of EMMA

4. PTC model of EMMA • PTC is a kick code, allowing symplectic integration through all accelerator elements • Cavities with the appropriate fixed frequency included • Initially, an rbend magnet with a quadrupole component was used to simulate the displaced quadrupoles • Displaced quadrupoles have recently been added. These have straight field lines (unlike the curved field lines in a rbend). • The fringe fields in the displaced quadrupoles have quad symmetry*. • These modifications to PTC carried out by the author, E. Forest, KEK. * “Beam Dynamics”, E Forest, Harwood Academic Publishers, Volume 8, p389 * É Forest and J Milutinović, Nucl. Inst. and Meth. A269 (1988) 474 Design Review III, Daresbury

5. Tune comparison with rbend model Design Review III, Daresbury

6. Orbit Distortion

7. Tracking with acceleration and misalignment misalignment = 50 micron Design Review III, Daresbury

8. Horizontal misalignments Amplification factor => max orbit distortion/ sigma of misalignment =123 Tolerance (1 mm distortion) = 8 microns Design Review III, Daresbury

9. Vertical misalignments Amplification factor = 98 Tolerance (1 mm distortion) = 10 microns Design Review III, Daresbury

10. Longitudinal misalignments Amplification factor = 13 Tolerance (1 mm distortion) = 80 microns Design Review III, Daresbury

11. Rotation misalignments Design Review III, Daresbury

12. Tolerances Assuming 1 mm distortion is limit Design Review III, Daresbury

13. Orbit Correction

14. Orbit Correction • Since the betatron tune varies with momentum, the phase difference between corrector magnets and error sources will change. • As the corrector strength cannot change during the rapid acceleration, it follows that harmonic correction of orbit distortion will not work. • However, local correction of the magnets should be possible Design Review III, Daresbury

15. Local Correction • In horizontal plane there are sliders to enable the misalignments to be corrected • In vertical plane, it is hoped that a once only adjustment to the magnets will be sufficient. • 2 BPMs per cell • To calculate the misalignments based on the BPM measurements, it would be better to run at fixed energy. • A number of turns would allow stochastic BPM errors to be averaged out and the tune to be calculated. • However, the error due to BPM misalignments needs to be considered when attempting to calculate the magnet misalignments Design Review III, Daresbury

16. BPM and Quadrupole misalignments • Assume these quantities are independently misaligned. Not considering other sources we have NQ+NBPM unknowns and NBPM measurements. • However, we can use a characteristic property of the FFAG, namely that the phase shift depends on momentum, to generate another set of NBPM measurements. • If we have 2NBPM=NQ+NBPM we can solve the set of simultaneous equations. Otherwise some sort of least squares fit could be attempted. Design Review III, Daresbury

17. Vertical corrector magnets

18. Motivation • Harmonic correction doesn’t work in a EMMA • Local correction, while possible, may sometimes be impractical in the vertical case • Can we use vertical corrector magnets to reduce the accelerated orbit distortion? Design Review III, Daresbury

19. BPMs and vertical kicker location Neil Bliss 3/4/07 Design Review III, Daresbury

20. Accelerated Orbit Distortion • The accelerated orbit distortion is calculated by tracking with PTC from 10-20 MeV. • Unlike the closed orbit distortion, it is not affected by the integer tune resonances. • In PTC, the initial conditions are given by the closed orbit at the initial energy. Design Review III, Daresbury

21. 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. Design Review III, Daresbury

22. Correction with 1 kicker Vertical orbit distortion rms reduced from 2.67mm to 0.64mm Design Review III, Daresbury

23. Adding more correctors Design Review III, Daresbury

24. Optimise initial (y,y’) Vertical orbit distortion rms reduced from 2.67mm to 0.63mm Design Review III, Daresbury

25. Conclusions • A PTC model of EMMA including displaced quadrupoles has been completed • The simulations predict an amplification factor with respect to quadrupole misalignments ~100. This will place stringent requirements on magnet alignment. • Tolerance levels of various translational and rotational errors were calculated. • Local correction of the misalignments will be necessary. • An new scheme to measure BPM misalignments is presented (only available in FFAG). • Vertical corrector magnets can reduce the accelerated orbit distortion. This may be equivalent to optimising the initial conditions. Design Review III, Daresbury

26. Future Work • The scheme to measure BPM errors and quadrupole misalignments will be tested. • Incorporate field maps of the EMMA magnets into PTC. This work is well underway with the help of Ben Shepherd and Etienne Forest. • This should allow the beam dynamics, and in particular the amplification factor, to be determined with more confidence. Design Review III, Daresbury