A.Poklonskiy (SPbSU, MSU), D.Neuffer (FNAL) C.Johnstone (FNAL), M.Berz (MSU), K.Makino (MSU)

1. Exploring and optimizing Adiabatic Buncher and Phase Rotator for Neutrino Factory in COSY Infinity A.Poklonskiy (SPbSU, MSU), D.Neuffer (FNAL) C.Johnstone (FNAL), M.Berz (MSU), K.Makino (MSU)

2. Goal & Problems Goal: • Build neutrino factory to study different neutrino-related things and/or muon collider… to collide Problems: • Muons are short-living particles  compact lattice, fast beam gymnastics • Muons are produced with large initial momentum spread  cooling • Energy spread is large  energy spread reduction before cooling • Some desired beam manipulations requires new types of field configuration  development of such new elements • All these  small muons production rate (<0.2) and…PRICE!

3. R&D goal: “affordable” e,  -Factory • Improve from baseline: • Collection • Induction Linac “high-frequency” buncher • Cooling • Linear Cooling  Ring Coolers • Acceleration • RLA  “non-scaling FFAG” –  e– + ne +  +  e+ + n + e and/or The Neutrino Factory and Muon Collider Collaboration http://www.cap.bnl.gov/mumu/mu_home_page.html

4. RF Cavity and Solenoid in Pictures

5. Adiabatic buncher + () Rotator (David Neuffer) • Drift (90m) •  decay, beam develops  correlation • Buncher (60m) (~333Mhz200MHz, 04.8MV/m) • Forms beam into string of bunches •  Rotator (~12m)(~200MHz, 10 MV/m) • Lines bunches into equal energies • Cooler (~50m long)(~200 MHz) • Fixed frequency transverse cooling system • Replaces Induction Linacs with • medium-frequency RF (~200MHz)

6. Longitudinal Motion (2D simulations) Buncher Drift   (E) rotator Cooler   System would capture both signs(+, -)

7. Key Parameters • Drift • Length LD • Buncher • Length LB • RF Gradients EB • Final RF frequency RF (LD, LB, RF: (LD + LB) (1/) = RF) • Phase Rotator • Length LR • Vernier offset, spacing NR, V • RF gradients ER

8. Lattice Variations • Shorter bunch trains (for ring cooler, more ’s lost)? • Longer bunch trains (more ’s survived)? • Different final frequencies? (200,88,44Mhz) • Number of different RF frequencies and gradients in buncher and rotator (60…10)? • Different central energies (200MeV, 280MeV, optimal)? • Matching into cooling channel, accelerator • Transverse focusing (150m B=1.25T solenoidal field or…)? • Mixed buncher-rotator? • Cost/perfomance optimum? OPTIMIZATION IS NEEDED

9. COSY Infinity Simulations • COSY Infinitycode (M. Berz,K. Makino, et al.) • Where M – map of the equations of motion (flow), obtained as a set of DA – vectors (Taylor expansions of final coordinates in terms of initial coordinates) • uses DA methods to compute maps toarbitrary order • own programming language allows complicatedoptimization scenarios writing • internal optimization routines and interface to add more • provides DA framework which could significantly simplify use of gradient optimization methods • model is simple now, but much more complicated in future and COSY has large library of standard lattice elements

10. Big Problem Use ofTaylor seriesleads to tricky way of handling beams with large coordinate spread (and that is exactly the case) Relative coordinates should be < 0.5 (empirical fact)

11. Straightforward division

12. Equations of Longitudinal Motion synchronous particle equations in deviations from synchronous particle nonlinear oscillator COSY Infinity uses similar coordinates

13. Consequences of Equations of Motion • Existence of stable regions, where we have oscillatory motion and unstable regions, and, therefore existence of separatrix (depends on frequency, RF gradient, synch. phase, etc… ). Stable area is called the “bucket”.

14. Beam Evolution

15. Clever Division • Use central energies as centres of boxes, use RF period as ranges for the box • Add “jumping” between intervals after each step. We change buckets and particles could be lost in one bucket and re-captured in another

16. Still a Problem • 50 central energies x 60 RF cavities in Buncher = 300 maps… • We are using DA arithmetic, everything is a DA-vector, including elementary functions (sin), so we need relatively high expansion order. Use 2 times more intervals + 5th orderleads to natural advantage in buncher… maybe. • Use COSY’s ability to generate parameter-dependent maps with ease and special law of bunches central energies distribution (smaller energies tends to be closer to each other) 40-50 maps  12-15 maps Potential calculation time reduction. Implementing. • 6000 particles, 50 central energies, 70 RFs • 1st order: 0 hrs 10 min • 7th order: 8 hrs some mins OPTIMIZATION?

17. Sin Taylor expansion

18. Different Order Simualtions

19. Different Order Simualtions

20. Conclusions • Model is implemented in COSY Infinity and checked for consistency with other codes • Some removal of the obstacles is done • Brute-force optimization still seems to be infeasible. Looking for some more sophisticated method.

21. Yet Unanswered Questions • Should longitudinal motion be studied separately, or should it be included on the very early stages? • Are there any map-dependent criterias which could be used for map-based optimization?

22. Second Optimization Approach • From synchronism condition one could derive following relation for kinetic energies of synch particles:

23. Final Kinetic Energy Relation • From the rotator concept one could derive amount of energy gained by n-th synchronous particle in RF • So for final energy n-th particle has after the rotator consists of m RFs we have

24. Evolution of central energies shape T(n,m)

25. Energies shape in buncher and amount of kick they get in rotator

26. Energy Shape Evolution in Rotator

27. Objective Functions • The idea of the whole structure is to reduce overall beam energy spread and to put particles energies around some central energy. So we have general objective function: • First, we can set and get

28. Different optimized paremters (n vs T_fin)

29. Different optimized paremters (T_0 vs T_fin)

30. Evolution of central energies shape (unoptimized)

31. Evolution of central energies shape (optimized)

32. Evolution of central energies shape (optimized)

33. Different optimized paremters (T_0 vs T_fin) + energies distribution

34. Objective Functions • For calculating we can use particle’s energies distribution • n energy particles % • ---------------------------------------------- • -12 963.96 1023 17.050000 • -11 510.85 692 11.533333 • -10 374.64 537 8.950000 • -9 302.98 412 6.866667 • …

35. Different optimized paremters (n vs T_fin)

36. Summary • Model of buncher and phase rotator was written inCOSY Infinity • Simulations of particle dynamics in lattice with different orders and different initial distributions were performed • Comparisons with previous simulations (David Neuffer’s code, ICOOL, others) shows good agreement • Several variations of lattice parameters were studied • Model of lattice optimization using control theory is proposed • Model of central energies distribution is developed. Some results for parameters have been obtained

37. Future Plans • Finish central energies optimizations, try changing more parameters, check optimized parameters for whole distribution (COSY, ICOOL?) • Develop some criteria for simultaneous optimization of central energies and energies of all paritcles in a beam or use control theory approach for the whole longitudinal motion optimization • Study transverse motion and particles loss because of decay and aperture, final emittance cut • Different lattices for different cooling sections/targets/whatever proposed (project is on R&D status)