1. 1 Dejan Trbojevic Muon acceleration by RLA with the non-scaling FFAG

2. 2 RLA for Muons Introduction: Present design of the muon RLA’s Problems: Matching of the circular non-scaling FFAG to the straight linac. Time of flight adjustments for each pass. Goals: Use the permanent magnets for the arcs – Halbach magnets. 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 -60%< δp/p< +60%). 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. Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

3. 3 ‘ 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 presentation at the previous LEMC: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

4. 4 1-pass, 3-5 GeV phase adv. drops much faster in the horizontal plane Triplet FODO vs Triplet focusing  ‘flat focusing' linac profile* Bob Palmer* From Alex Bogacz presentation at the previous LEMC: 256.82 meters Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

5. 5 The linac – Betatron Function dependence on energy Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

6. 6 Multipass Linac - racetrack FFAG Chicane Non-scaling FFAG arc 20 Cavities Non-scaling FFAG arc Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

7. 7 Muon Collider Review Meeting At BNL 2001

8. 8 Design of the arcs with Halbach magnets FODO cells for the 2.5 -> 10 GeV muons N=170 cells L=1.606 m L BD =0.73 m L QF =0.52 m For the:  p/p=+-60% B BD =2.5 T GF=40.0 T/m GD=-50.0 T/m r=43.42 m 86.85 m Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

9. 9 x max =65.8 mm x min = -35.1 mm 1.606 m 0.73/2 m 0.1675 m FODO cell for the  p/p=+-60 % -> 2.5 - 10 GeV Arc cell with Halbach magnets 52 mm Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 0.73/2 m

10. 10 x max =52.9 mm x min = -12.8 mm 1.606 m 0.73 m 0.1675 m FODO cell for the  p/p=+-60 % -> 2.5 - 10 GeV Arc cell with Halbach magnets 52/2 mm Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 0.52/2 m

11. 11 FODO cell for the  p/p=+-60 % -> 2.5 - 10 GeV Arc cells for two different lattices Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

12. 12

13. 13 A total difference in the path length - 10 GeV and the minimum of the parabola is  l=0.2465 m

14. 14 Halbach permanent magnet But in reality it would be limited by: (1)The realistic size (2)The demagnetization effect

15. 15 Permanent Magnet for the BNL additional triplets (K. Halbach) Y (cm) X (cm) Field Quality @ R=6 cm b2 = 10248.0 Gauss (17 T/m) b6 = 44.3 Gauss (4.3E-3) Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

16. 16 Halbach permanent magnets: available material No-Fe-B Type Rare Earth Magnets:

17. 17 Halbach permanent magnets – pictures from the original publication: QL D = 52 cm BL = 16.75 cm QL F = 73 cm G F = 2.7 T/0.068 m = 40 T/m G D = -2.7 T/0.054 m=-50 T/m 14 cm 85 cm B g = B r ln(OD/ID) B r =1.5 T OD=85 cm ID =14 cm ln(OD/ID)=1.8 B g =2.7 T Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

18. 18 Halbach permanent magnets 35 cm Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

19. 19 The matching cell length is: L=3 * 1.605 m = 4.815 m Matching cell – geometrical constraint - arc to linac Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

20. 20 IEEE Transactions on Nuclear Science, Vol. NS-30, No. 4, August 1983 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

21. 21 Yoshiharu Mori – Fermilab FFAG workshop:

22. 22 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 Yoshiharu Mori – Fermilab FFAG workshop:

23. 23 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 Yoshiharu Mori – Fermilab FFAG workshop:

24. 24 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 Yoshiharu Mori – Fermilab FFAG workshop:

25. 25 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 Shinji Machida – Fermilab FFAG workshop

26. 26 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 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

27. 27 Matching Cell - @ zero dispersion end dd  dmax  dmin a max a min  dmin  dmax  do u max u min lolo Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

28. 28 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 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

29. 29 Mike Craddock’s approach: 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 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

30. 30 Matching to linac -> zero dispersion for each momentum Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 Orbit offsets Dispersion yy xx

31. 31 Matching to linac -> zero dispersion for each momentum 10 GeV 2.5 GeV Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

32. 32 p>p cent orbits matched to linac -> zero dispersion for each momentum p=p cent p=p max Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

33. 33 Matching cell to the non-scaling FFAG arcs Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

34. 34 Matching cell to the non-scaling FFAG arcs Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

35. 35 Non scaling FFA arcs with matching cells without linac Orbits from 2.5 – 10 GeV through the matching cells and arcs: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

36. 36 Non scaling FFA arcs with matching cells without linac Betatron Functions from 2.5 – 10 GeV through the matching cells and arcs: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

37. 37 Non scaling FFA arcs with matching cells without linac Dispersion from 2.5 – 10 GeV through the matching cells and arcs: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

38. 38 Matching cell with linac – arc to linac Orbits magnified 100 times From 2.5 GeV- 10GeV Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

39. 39 Multipass Linac with combined function triplets Details of the orbits with Chicanes: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

40. 40 Details of the chicane calculations: LoLo   L L o /L=cos  Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

41. 41 Details of the Chicane CAVIT Y TRIPLET Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

42. 42 Summary: A combination of the non-scaling FFAG with linac is possible. Time of flight adjustments is necessary –maximum of 0.493 m delay. The simulation of acceleration can be set-up by the PTC (Polymorphic Tracking Code). Thanks to Muon Inc. for the support Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

43. 43 Shinji Machida – Fermilab FFAG workshop

44. 44 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 Shinji Machida – Fermilab FFAG workshop

45. 45 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009 Yoshiharu Mori – Fermilab FFAG workshop:

46. 46 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1-3, 2009

47. 47 Yoshiharu Mori – Fermilab FFAG workshop: