1. Suzie Sheehy DPhil Candidate, John Adams Institute 3/9/08 PAMELA lattice studies Dynamics of the Machida lattice

2. Outline  Linear lattice options  Non-linear lattice options  Machida lattice dynamics  Future directions Page 2 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice

3. Linear NS-FFAG lattices  Quadrupole & dipole elements  e.g. KST (Keil, Sessler, Trobjevic) 48 cell lattice  F/D doublet cells  Small orbit excursion  Large tune excursion  Resonance crossing Page 3 E. Keil, A. M. Sessler, and D. Trbojevic. “Three ring FFAG complex for H+ and C6+ therapy” pg. 1681. EPAC Proceedings, 2006. 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice

4. Resonance crossing in linear lattices  Two significant factors:  Acceleration rate – need to cross quickly!  Number of cells  Number of machine resonances crossed  Ability to have long straight sections Page 4 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice Cell tunes Δp/p Horiz. Vert. Total tunes 21.6 2.4 4.8 7.2 9.6 12.0 14.4 19.2 16.8

5. Resonance crossing with alignment errors  Simulated KST Ring 2 using ZGOUBI  Particles on closed orbit accelerated & tracked to extraction  Alignment errors (gaussian distributed) introduced  20 error sizes, 10 lattices simulation for each  Amplification factor defined as:  A x ~ 315  (nb. Without acceleration COD 4.4)  100μm error gives ~31.5mm orbit distortion! Page 5 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice

6. Emittance change through resonances  Real emittance change - crossing through integer resonance  E.g. isolated integer resonance  36 macro particles on 5 π mm mrad emittance ellipse  100 turns from KE=142 MeV 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice Page 6 Linear NS-FFAG(average B n ), T.Yokoi

7. Avoidance strategies…  Cross resonances very quickly ?  Accelerate very quickly  RF voltage must be large  Pushing RF limits as it is…  Reduce number of resonances crossed ?  Reduce number of cells – fewer machine integer resonances  Reduce machine tune excursion to within an integer  i.e. reduce cell tune excursion to 1/N, where N is number of cells Page 7 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice

8. How to reduce tune excursion 1. Take NS-FFAG and add higher order components  Severely reduced dynamic aperture  Small dispersion makes chromatic correction hard OR 2. C.Johnstone – wedge shaped quadrupoles (see T.Yokoi’s talk)  edge focusing & path length of quads a function of momentum  effective focal length of lattice similar & independent of momentum  almost flat tune over the wide momentum range of factor 6 OR 3. S.Machida – different approach  Start from scaling FFAG with large k  Relax scaling law to get “non-scaling” Page 8 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice

9. Machida lattice: aim  Non-linear NS-FFAG (“tune stabilised”)  The aim:  Small orbit excursion  Few cells  Compact magnets  Small tune excursion  Long straight section for inj/extr.  Stability Diagram  Orbit excursion is inv. prop. to k (or dB/dr)  Take the largest possible k-value  For scaling – end of design procedure…? Page 9 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice S. Machida, from PAMELA Design Review, June 08

10. Machida lattice: the idea  8 Cell triplet FDF lattice  Approximate scaling r -k  Remove scaling law – only to decupole  Rectangular magnets (instead of wedge)  Magnets on straight line (easy alignment)  Long straight sections of over 2m  Orbit excursion < 30cm 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice Page 10

11. Machida lattice – ZGOUBI model  Total tune excursion within an integer Page 11 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice Horiz. Vert.

12. Dynamic Aperture  Simulations using ZGOUBI  (no fringe fields at present)  Close to 1/3 at extraction – can easily be shifted to achieve dynamic aperture Page 12 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice D. Kelliher, from PAMELA Design Review, June 08

13. Alignment errors 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice Page 13  ZGOUBI study with alignment errors (as before)  Ax ~ 1 (cf. 315 for linear lattice)  Expected – as no major resonances crossed!  Need to check with fringe fields

14. Future directions  Understand resonance crossing  Include fringe fields in ZGOUBI for Machida lattice  Orbit/optics studies with errors  Optimise lattice Page 14 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice

15. Conclusions  Densely packed, linear lattices very sensitive to alignment errors when crossing resonances  Want sparse lattice, few cells, no resonance crossing  Machida lattice achieves this – though further studies needed! Page 15 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice

16. Questions?  Thanks for listening Page 16 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice

17. Page 17 3/9/08 PAMELA lattice studies – Dynamics of the Machida lattice