1. Taylor Models Workhop 20 December 2004 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. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 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 + + n  + e  –  e – + n e +  and/or The Neutrino Factory and Muon Collider Collaboration http://www.cap.bnl.gov/mumu/mu_home_page.html

4. Taylor Models Workshop 20 December 2004 RF Cavity and Solenoid in Pictures

5. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 Longitudinal Motion (2D simulations) Drift Buncher (  E) rotator Cooler System would capture both signs (  +,  - )    

7. Taylor Models Workshop 20 December 2004 Key Parameters Drift –Length L D Buncher –Length L B –RF Gradients E B –Final RF frequency RF (L D, L B, RF : (L D + L B )  (1/  ) = RF ) Phase Rotator –Length L  R –Vernier offset, spacing N  R,  V –RF gradients E  R

8. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 COSY Infinity Simulations COSY Infinity code (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 to arbitrary order  own programming language allows complicated optimization 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. Taylor Models Workshop 20 December 2004 Big Problem Use of Taylor series leads 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. Taylor Models Workshop 20 December 2004 Straightforward division

12. Taylor Models Workshop 20 December 2004 Equations of Longitudinal Motion nonlinear oscillator synchronous particle equations in deviations from synchronous particle COSY Infinity uses similar coordinates

13. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 Beam Evolution

15. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 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 order leads 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 –7 th order: 8 hrs some mins OPTIMIZATION?

17. Taylor Models Workshop 20 December 2004 Sin Taylor expansion

18. Taylor Models Workshop 20 December 2004 Different Order Simualtions

19. Taylor Models Workshop 20 December 2004 Different Order Simualtions

20. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 Second Optimization Approach From synchronism condition one could derive following relation for kinetic energies of synch particles:

23. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 Evolution of central energies shape T(n,m)

25. Taylor Models Workshop 20 December 2004 Energies shape in buncher and amount of kick they get in rotator

26. Taylor Models Workshop 20 December 2004 Energy Shape Evolution in Rotator

27. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 Different optimized paremters (n vs T_fin)

29. Taylor Models Workshop 20 December 2004 Different optimized paremters (T_0 vs T_fin)

30. Taylor Models Workshop 20 December 2004 Evolution of central energies shape (unoptimized)

31. Taylor Models Workshop 20 December 2004 Evolution of central energies shape (optimized)

32. Taylor Models Workshop 20 December 2004 Evolution of central energies shape (optimized)

33. Taylor Models Workshop 20 December 2004 Different optimized paremters (T_0 vs T_fin) + energies distribution

34. Taylor Models Workshop 20 December 2004 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. Taylor Models Workshop 20 December 2004 Different optimized paremters (n vs T_fin)

36. Taylor Models Workshop 20 December 2004 Summary Model of buncher and phase rotator was written in COSY 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. Taylor Models Workshop 20 December 2004 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)