1. 1 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Dejan Trbojevic Work with the Muons Inc. FFAG – Type Multipass Arcs for RLA’s

2. 2 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 FFAG – Type Multipass Arcs for RLA’s: Introduction: Present design of the muon RLA’s A short introduciotn to the non-scaling FFAG Problems: Matching of the circular non-scaling FFAG to the straight linac. Time of flight adjustments for each pass. Goals: Try to make four or five times in muon energy by either a race track or dog- bone acceleration with a single arc (2.5-10 GeV or 10-40 GeV). Match the betatron and dispersion functions from the arc to the linac. Design a chicane to adjust the time of flight for different energy passes.

3. 3 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 ‘ Racetrack’ vs ‘Dogbone’ RLA (both  + and  - species )   E/2                EE better orbit separation at linac’s end ~ energy difference between consecutive passes (2  E) allows both charges to traverse the Linac in the same direction (more uniform focusing profile the droplets can be reduced in size according to the required energy both charge signs can be made to follow a Figure-8 path (suppression of depolarization effects) Chuck Ankenbrandt From Alex Bogacz:

4. 4 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 1-pass, 3-5 GeV phase adv. drops much faster in the horizontal plane Triplet FODO vs Triplet focusing  ‘flat focusing' linac profile* From Alex Bogacz: 256.82 meters

5. 5 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 From CYCLOTRONS to FFAG’s:

6. 6 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Presently runs at 590 MeV with energy on target of 1.2 MW 2 mA in CW mode Upgrade to 1.8 MW at the beam energy of 590 MeV PSI – (Paul Scherrer Institute) CYCLOTRON 4.5 meter outer orbit Compared to TRIUMF 7.6 m

7. 7 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 The first SCALING FFAG  r~60-100 cm           r   sin(  /sin(  r  MURA-KRS-6 Phys.Rev. 103, 1837 (1956) November 12, 1954 K. R. Symon: The FFAG SYNCHROTRON – MARK I This is why FFAG had lost:

8. 8 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Many scaling FFAG’s have been built: latest complex at KURRI-Kyoto Japan

9. 9 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Proton driven reactor: latest complex at KURRI-Kyoto Japan

10. 10 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 First current in KURRI’s FFAG

11. 11 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Non-scaling FFAG concept Orbit offsets are proportional to the dispersion function:  x = D x   p/p To reduce the orbit offsets to ±4 cm range, for momentum range of  p/p ~ ± 50 % the dispersion function D x has to be of the order of: D x ~ 4 cm / 0.5 = 8 cm The size and dependence of the dispersion function is best presented in the normalized space and by the H function:  = D x /  x and  = D’ x  x +  x D x  x with H: H =  2 +  2

12. 12 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 The minimum emittance lattice: The minimum emittance lattice requires reduction of the function H: The normalized dispersion amplitude corresponds to the 1/2 Conditions are for the minimum of the betatron function bx and dispersion function Dx to have small values at the middle of the dipole (combined function dipole makes it even smaller).  min  Ld  15 D xmin =  Ld/24 

13. 13 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 D. Trbojevic E.D. Courant, and A. Garren, September 30, 1999, Montauk, Long Island High Energy Muon Collider Workshop Our first publication: Lee Teng: 1956 Non-scaling FFAG proposal at the conference

14. 14 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Lattice got simplified with smaller number of magnets: just two kinds

15. 15 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Muon acceleration Basic Properties of the non-scaling FFAG - Extremely strong focusing with small dispersion function. - large energy acceptance. - tunes variation - very small orbit offsets -small magnets -linear magnetic field

16. 16 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Basic Properties of the Non-Scaling FFAG A. Particle orbits B. Lattice

17. 17 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Scaling FFAG – Non scaling FFAG Scaling FFAG properties: Zero chromaticity. Orbits parallel for different  p/p Relatively large circumference. Relatively large physical aperture (80 cm – 120 cm). RF - large aperture Tunes are fixed for all energies. Negative momentum compaction. B =B o (r/r o ) k non-linear field Non-Scaling FFAG properties: Chromaticity is changing. Orbits are not parallel. Relatively small circumference. Relatively small physical aperture (0.50 cm – 10 cm). RF - smaller aperture. Tunes move 0.4-0.1 in basic cell. Momentum compaction changes. B = Bo+x G o linear field B = Bo+r G o B =B o (r/r o ) k

18. 18 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 EMMA: first non-scaling FFAG

19. 19 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 EMMA: first non-scaling FFAG Two already built magnets: EMMA’s cavity:

20. 20 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 EMMA: first non-scaling FFAG: six cells on a girder

21. 21 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Design of the arcs – from the densely populated FODO cells for the 2.5 -> 10 GeV muons N=142 cells L=1.9 m L BD =0.9 m L QF =0.6 m For the:  p/p=+-60% B BD =1.64 T B QF =-1.09 T GF=20.8 T/m GD=-10.2 T/m r=42.94 m 85.9 m

22. 22 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 x max =69.7 mm x min = -35.1 mm 1.9 m 0.3 m 1.08 m FODO cell for the  p/p=+-60 % -> 2.5 - 10 GeV Single arc cells 35.1 mm L BD = 1.08 m, L QF =0.6 m

23. 23 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 FODO cell for the  p/p=+-60 % -> 2.5 - 10 GeV

24. 24 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 The matching cell length is: L=3 * 1.9 m = 5.7 m Matching cell – geometrical constraint - arc to linac

25. 25 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Matching cell to the non-scaling FFAG arcs

26. 26 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Non scaling FFA arcs with matching cells without linac Orbits from 2.5 – 10 GeV through the matching cells and arcs:

27. 27 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Non scaling FFAG arcs with matching cells without linac Betatron Functions from 2.5 – 10 GeV through the matching cells and arcs:

28. 28 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Non scaling FFA arcs with matching cells without linac Dispersion from 2.5 – 10 GeV through the matching cells and arcs:

29. 29 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 The linac – Betatron Function dependence on energy

30. 30 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Matching cell with linac – arc to linac Orbits magnified 100 times From 2.5 GeV- 10GeV

31. 31 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Multipass Linac - racetrack FFAG Chicane Non-scaling FFAG arc 20 Cavities Non-scaling FFAG arc

32. 32 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Multipass Linac with combined function triplets Details of the orbits with Chicanes:

33. 33 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Path length difference from the arc cell  s=0.55m for arc)

34. 34 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Details of the chicane calculations: LoLo   L L o /L=cos 

35. 35 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Details of the Chicane CAVIT Y TRIPLET

36. 36 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Phil Meads: IEEE Transactions on Nuclear Science, Vol. NS-30, No. 4, August 1983

37. 37 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Analytical formulae from the combined function magnet: p_max p_min LINAC

38. 38 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 New matching cell ff ff dd lolo x max x min a max u min u max  do  dmin  dmax  fmin  fmax  fo dd dd dd a min Input parameters are: x max and x min from the arc NS-FFAG p max, p o, and p min, D x,  x,  y, Unknowns: B D, B F,  fo,  do, and l o To be matched to the input parameters of the linac:  x,  y,  x,  y

39. 39 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Analytical formulae from the combined function magnet:

40. 40 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Matching Cell - @ zero dispersion end dd  dmax  dmin a max a min  dmin  dmax  do u max u min lolo

41. 41 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Matching Cell @ entrance p min p max popo x max x min a max a min  fmax  fo  fmin  fmax  fmin  fo u max u min  fmin -  fo  fo -  fmax w j lolo u max =a max +l o tan(  fo -  fmax ) u min =a min +l o tan(  fmin -  fo )  fmax  fo -  fmax =  do -  dmax  fmin -  fo =  dmin -  do

42. 42 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 A solution for zero offsets at the end ff ff dd lolo x fp+ x fp- X d+ u min u max  do  dmin  dmax  fmin  fmax  fo dd dd dd X d- low p - high p + popo F/2 D/2 x d+ =0 x d- =0

43. 43 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 p>p cent orbits matched to linac -> zero dispersion for each momentum p=p cent p=p max

44. 44 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Orbits matched to linac -> zero dispersion for each momentum 10 GeV 2.5 GeV FFAG cell Matching cell Linac

45. 45 Muons, Inc. Dejan Trbojevic Muon ColliderDesign Workshop, 9:00, December 11, 2008 Conclusions: A solution of the non-scaling FFAG arcs with RLA’s looks very good. This multiple pass linac is designed by the triplet combined function dipoles cells. Time of flight adjustments is necessary due to 0.6 m delay for the lowest energy pass through the arc. A details of chicane design has to be studied. Initial matching between the two arcs and linac for any energy without orbit offsets in the linac ! The simulation of acceleration with many particles is already set-up by the PTC (Polymorphic Tracking Code). Thanks to Muon Inc. for the support