1. Muon Colliders ‘2004 14 December 2004 Optimization of adiabatic buncher and phase rotator for Muon Accelerators A.Poklonskiy (SPbSU, MSU), D.Neuffer (FNAL) C.Johnstone (FNAL), M.Berz (MSU), K.Makino (MSU)

2. Muon Colliders ‘2004 14 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)

3. Muon Colliders ‘2004 14 December 2004 Longitudinal Motion (2D simulations) Drift Buncher (  E) rotator Cooler System would capture both signs (  +,  - )    

4. Muon Colliders ‘2004 14 December 2004 Adiabatic Buncher overview Array of RF cavities Fix RF frequency at the end to 200 Mhz  1.5m, desired central energy and total length of the buncher RF phase is set to be 0 for reference energy through all buncher Find the condition for particles velocities to pass last RF in 0 phase (no energy change) and set frequencies in all RFs in buncher to maintain this condition Example: rf : 0.90  1.5m

5. Muon Colliders ‘2004 14 December 2004 Adiabatic Buncher overview Adiabatically increase RF gradient:

6. Muon Colliders ‘2004 14 December 2004  Rotator overview At end of buncher, change RF to decelerate high-energy bunches, accelerate low energy bunches, i.e. rotation in  phase space With central reference particle at 0 phase, set rf a bit less than bunch spacing (increase RF frequency) Places low/high energy bunches at accelerating/decelerating phases Change frequency along channel to maintain phasing Example: rf : 1.485  1.517m;

7. Muon Colliders ‘2004 14 December 2004  Rotator overview At end of buncher, choose: –Second reference particle TN –Vernier offset  Example: –T0 = 125 MeV –Choose N= 10,  =0.1 – T10 starts at 77.28 MeV Along rotator, keep second ref particle at (N +  ) rf spacing –  10 = 36° at  =0.1 –Bunch centroids change: Use Erf = 10MV/m; L=8.74m –High gradient not needed –Bunches rotate to ~equal energies. Example: rf : 1.485  1.517m ;

8. Muon Colliders ‘2004 14 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

9. Muon Colliders ‘2004 14 December 2004 Central Energies Optimization Approach This is how rotator could look like in reality This is “transit time factor” (percent of the acceleration from maximum which particle could gain in changing E field): g – length of the cavity, w – cyclic frequency, v – particle’s velocity For  1. We can use kick approximation for the particle energy gain 2. We “forget” about the influence of cavity phases and gradients on all beam particles dynamics (could be added lately)  Study dynamics of the central particles of the bunches separately: T_final = T(n,…)

10. Muon Colliders ‘2004 14 December 2004 Centroids Kinetic Energies From buncher synchronism condition one could derive following relation for kinetic energies of central particles: Puts limits on n_min and n_max => n_bunches!

11. Muon Colliders ‘2004 14 December 2004 Final Centroids Kinetic Energy –From the rotator concept one could derive amount of energy gained by n-th synchronous particle in each RF (kept const by changing frequency) or, more generally –So for final energy n-th bunch central particle has after the ROTATOR consists of m RFs we have

12. Muon Colliders ‘2004 14 December 2004 Evolution of central energies shape T(n,m,…)

13. Muon Colliders ‘2004 14 December 2004 Energies shape in buncher and amount of kick they get in rotator

14. Muon Colliders ‘2004 14 December 2004 Energy Shape Evolution in Rotator

15. Muon Colliders ‘2004 14 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:

16. Muon Colliders ‘2004 14 December 2004 Objective function 1 –First, we can set and get

17. Muon Colliders ‘2004 14 December 2004 Different optimized paremters (n vs T_fin), COSY built-in optimizer

18. Muon Colliders ‘2004 14 December 2004 Different optimized paremters (T_0 vs T_fin), COSY built-in optimizer

19. Muon Colliders ‘2004 14 December 2004 Objective Function 2 –As we can use particle’s energies distribution in a beam 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 …

20. Muon Colliders ‘2004 14 December 2004 Objective Function 2

21. Muon Colliders ‘2004 14 December 2004 Modeled Optimization (OBJ1), whole domain search Fixed params: Desired central kinetic energy (T_c) = 125.0000000000000 T_0 in buncher (T_0) = 125.0000000000000 Drift+Buncher length (L_buncher) = 90.00000000000000 Final frequency (final_freq) = 200000000.0000000 Varied params: 1st lever particle (n1) : - 4.000000000000000 ==> 1.000000000000000 2nd lever particle (n2) : - 3.000000000000000 ==> 27.00000000000000 Vernier parameter (vernier) : 0.1000000000000000 ==> 0.3000000000000000 RF gradient (V_RF) : 1.000000000000000 ==> 10.00000000000000 Number of RFs in rotator (m) : 1.000000000000000 ==> 10.00000000000000 Objective functions: !! 188624.7593430086 ==> 7434.672341694457 = 181190.0870013142 21918840.88332907 ==> 1643615.384258914 = 20275225.49907015 215.4599659295154 ==> 238.7961711271633 = - 23.33620519764796 31921138.21770231 ==> 15260635.43728683 = 16660502.78041548

22. Muon Colliders ‘2004 14 December 2004 Modeled Optimization (OBJ2), whole domain search Fixed params: Desired central kinetic energy (T_c) = 125.0000000000000 T_0 in buncher (T_0) = 125.0000000000000 Drift+Buncher length (L_buncher) = 90.00000000000000 Final frequency (final_freq) = 200000000.0000000 Varied params: 1st lever particle (n1) : - 7.000000000000000 ==> 1.000000000000000 2nd lever particle (n2) : - 6.000000000000000 ==> 14.00000000000000 Vernier parameter (vernier) : 0.1000000000000000 ==> 0.4000000000000000 RF gradient (V_RF) : 1.000000000000000 ==> 10.00000000000000 Number of RFs in rotator (m) : 1.000000000000000 ==> 10.00000000000000 Objective functions: 552558.8685890788 ==> 331194.3462835634 = - 221364.5223055154 !! 1120051685.347023 ==> 740898336.4248258 = - 379153348.9221967 790.5115736637570 ==> 714.6396495910632 = - 75.87192407269379 1614049125.098601 ==> 1105871829.498042 = - 508177295.6005592

23. Muon Colliders ‘2004 14 December 2004 Other possible objective functions We may try to incorporate information about buckets widths and lengths We may combine this objective function with any of the first two with any weight coefficients

24. Muon Colliders ‘2004 14 December 2004 Other Optimization (OBJ1) (asked by David), whole domain search Fixed params: Desired central kinetic energy (T_c) = 125.0000000000000 T_0 in buncher (T_0) = 125.0000000000000 Drift+Buncher length (L_buncher) = 75.00000000000000 Final frequency (final_freq) = 100000000.0000000 Varied params: 1st lever particle (n1) : - 3.000000000000000 ==> 1.000000000000000 2nd lever particle (n2) : - 2.000000000000000 ==> 4.000000000000000 Vernier parameter (vernier) : 0.1000000000000000 ==> 0.2000000000000000 RF gradient (V_RF) : 1.000000000000000 ==> 10.00000000000000 Number of RFs in rotator (m) : 1.000000000000000 ==> 10.00000000000000 Objective functions: !! 920802.1870498012 ==> 716144.2078318644 = - 204657.9792179369 2906605812.448390 ==> 2289340374.907225 = - 617265437.5411658 -1154.766872358291 ==> - 1012.150761595708 = 142.6161107625823 2905272325.918894 ==> 2288315925.743026 = - 616956400.1758685

25. Muon Colliders ‘2004 14 December 2004 Other Optimization (OBJ2) (asked by David), whole domain search Fixed params: Desired central kinetic energy (T_c) = 125.0000000000000 T_0 in buncher (T_0) = 125.0000000000000 Drift+Buncher length (L_buncher) = 75.00000000000000 Final frequency (final_freq) = 100000000.0000000 Varied params: 1st lever particle (n1) : - 3.000000000000000 ==> 0.000000000000000 2nd lever particle (n2) : - 2.000000000000000 ==> 5.000000000000000 Vernier parameter (vernier) : 0.1000000000000000 ==> 0.4000000000000000 RF gradient (V_RF) : 1.000000000000000 ==> 10.00000000000000 Number of RFs in rotator (m) : 1.000000000000000 ==> 10.00000000000000 Objective functions: 920802.1870498012 ==> 716325.0523539253 = - 204477.1346958759 !! 2906605812.448390 ==> 2288276641.662843 = - 618329170.7855473 -1154.766872358291 ==> - 1015.263140233114 = 139.5037321251762 2905272325.918894 ==> 2287245882.418927 = - 618026443.4999671

26. Muon Colliders ‘2004 14 December 2004 Summary Model of central energies shape optimization for buncher and phase rotator is proposed. Ready-to-use program is written, It allows to perform optimization on any set of supported parameters (length of the buncher and rotator, final frequency, central energy, E field gradient, phases). We can search for optimal parameters values in any desired range and check some previously chosen params for optimality. (It could take long… ) Some example results are presented

27. Muon Colliders ‘2004 14 December 2004 To do  Check results in (t,E) space as more important (problem: energy is changing  )  Different RF field waveform?  Check optimized parameters for the whole beam distribution (COSY, ICOOL?) Is it really better?  Switch to 3D-motion simulation and optimization