1. 1 Muon Acceleration and FFAG II Shinji Machida CCLRC/RAL/ASTeC NuFact06 Summer School August 20-21, 2006

2. 2 Content 1.Acceleration of muons 2.Evolution of FFAG 3.FFAG as a muon accelerator Scaling FFAG Non-scaling FFAG 4.Design example of muon acceleration Japanese scenario US and Europe scenario Appendix:Proton driver Reference (among others): –BNL-72369-2004, FNAL-TM-2259, LBNL-55478 –NuFactJ Design study report

3. 3 3 FFAG as a muon accelerator

4. 4 Betatron oscillation and Tune In transverse plane, a particle in an accelerator oscillates around the centeral orbit. It is called “betatron oscillation” and its oscillation frequency is “tune”. Precisely speaking tune is FFAG as a muon accelerator s x

5. 5 Cardinal conditions Geometrical similarity   : average curvature  : local curvature  : generalized azimuth Constancy of k at corresponding orbit points k : index of the magnetic field The field satisfies the scaling law. Tune is constant independent of momentum: scaling FFAG FFAG as a muon accelerator

6. 6 Beyond scaling FFAG If we can break scaling law, FFAG will be much simpler and magnet will be smaller. Why do we keep scaling (constant tune) during acceleration? Because of resonance in accelerator. Bz(r) r r No gentle slope at low momentum. - Orbit excursion is shorter. Constant gradient. - Linear magnet. FFAG as a muon accelerator

7. 7 “Resonances” in accelerators There are many resonances near operating tune. Once a particle hits one of them, probably it will be lost. In reality, however, operating tune moves due to imperfection of magnet (red zigzag line). Particles can survive after crossing resonances if resonance is weak and crossing is fast. x y Tune diagram of 150 MeV FFAG FFAG as a muon accelerator

8. 8 Non-scaling FFAG Muons circulate only a few (~15) turns in FFAG. Is resonance really harmful to a beam? Maybe not. Forget scaling law ! Let us operate ordinary AG synchrotron without ramping magnet. Orbit moves as momentum increases. –Large  p makes the orbit shift small. Focusing force decreases as momentum increases. FFAG as a muon accelerator

9. 9 Orbit for different momentum Orbit shifts more at larger dispersion section. No similar shape unlike scaling FFAG. non-scaling low p high p FFAG as a muon accelerator

10. 10 Tune variation in a cycle Tune decreases as a beam is accelerated. d (tune)/dT(turn)~1 for muon rings. low phigh p FFAG as a muon accelerator

11. 11 Beam dynamics issues Acceleration out of RF bucket. Crossing of many resonances during acceleration. –Structure resonance has some effects. –With alignment errors, integer resonances have to be considered. Huge acceptance (30,000  mm-mrad) for muons. –Dynamic aperture without acceleration at injection energy. FFAG as a muon accelerator

12. 12 Synchrotron oscillation and RF bucket (1) FFAG as a muon accelerator time voltage time voltage < momentum

13. 13 Synchrotron oscillation and RF bucket (2) FFAG as a muon accelerator time momentum If momentum is too large, it cannot be trapped in sinusoidal RF voltage. -> “RF bucket”.

14. 14 Acceleration in non-scaling FFAG (1) Revolution frequency changes because orbit shifts and path length changes although speed of mouns is already a speed of light. If you look at orbits carefully, path length at the central momentum is shortest. FFAG as a muon accelerator

15. 15 Acceleration in non-scaling FFAG (2) In a first half of a cycle, path length becomes shorter and revolution frequency becomes higher. In a second half of a cycle, path length becomes longer and revolution frequency becomes lower. momentum 1/revolution freq. FFAG as a muon accelerator

16. 16 Acceleration in non-scaling FFAG (3) Suppose we choose RF frequency that is synchronized with revolution frequency at the center. In the first half of a cycle, a particle lags behind the RF. At the center, a particle is synchronized with RF. In the second half, a particle lags again. low center high time voltage FFAG as a muon accelerator

17. 17 Acceleration in non-scaling FFAG (4) In the longitudinal phase space, a particle follows the path with constant color. If there is enough RF voltage, a particle can be accelerated to the top energy. This is called “Gutter acceleration”. momentum time FFAG as a muon accelerator

18. 18 Transverse amplitude effects without transverse amplitude with finite transverse amplitude Longitudinal phase space (phi, momentum) Horizontal is 5,000 pi mm mrad Vertical is zero 5 to 10 GeV ring FFAG as a muon accelerator

19. 19 Source of longitudinal emittance growth A particle with large transverse amplitude has longer path length. This effects become visible because muon emittance (amplitude) is huge. s x FFAG as a muon accelerator

20. 20 Matching between two FFAG rings Second harmonics and 10% increase of RF voltage partially cure the problem. FFAG as a muon accelerator 5GeV 10GeV 20GeV

21. 21 RLA Synchrotron oscillation helps to mix momentum spread. FFAG as a muon accelerator

22. 22 Vertical is 5,000  mm-mrad, normalized, zero horizontal emittance. Shows the coupling due to nx-2ny=0 (structure) resonance. If we start finite horizontal and zero vertical emittance, no exchange of emittance. Resonance crossing without errors 5 GeV 10 GeV vertical emittance horizontal emittance FFAG as a muon accelerator

23. 23 Resonance crossing without errors, amplitude dependence 5,000 pi mm-mrad500 pi mm-mrad 0.5 pi mm-mrad FFAG as a muon accelerator

24. 24 Beam has to face many integer tunes. Resonance crossing with alignment errors tune per cell tune per ring FFAG as a muon accelerator

25. 25 Resonance crossing with alignment errors, envelope Horizontal is 10,000  mm-mrad, normalized, zero vertical emittance. Errors of 0, 0.05, 0.10, 0.20 mm (rms). 0. mm0.05 mm 0.20 mm 0.10 mm Horizontal phase space (x, xp) FFAG as a muon accelerator

26. 26 Scaling vs. non-scaling Scaling machine principle is proven. Large acceptance so that cooling is not needed. Magnet tends to be larger. Cost more. Non-scaling machine can be more compact. Cost less. Need cooling to fit a beam into the acceptance. Principle have to be proven. –Resonance crossing –Gutter acceleration –Demonstration by electron model is scheduled in UK. FFAG as a muon accelerator

27. 27 FFAG as a muon accelerator summary FFAG used to satisfy scaling law, that assures geometrical similarity of orbit and tune independent of momentum. If resonance crossing is not harmful, scaling law is not necessary. Just ordinary synchrotron without ramping magnet makes a new concept of FFAG, namely non-scaling FFAG. Acceleration is so fast that RF frequency cannot be synchronized with revolution frequency. “Gutter acceleration” is one possible way. Transverse amplitude makes longitudinal growth. FFAG as a muon accelerator

28. 28 4 Design example of muon accelerator

29. 29 Japanese scheme Scaling FFAG Acceleration with a bucket of low frequency RF, 5~20 MHz

30. 30 Acceleration No time to modulate RF frequency. 1 MV/m (ave.) RF voltage gives large longitudinal acceptance. From 10 to 20 GeV/c within 12 turns.

31. 31 Accelerator chain Before acceleration –Target and drift –No cooling section Four scaling FFAGs, –0.3 - 1.0 GeV –1.0 - 3.0 GeV –3.0 - 10.0 GeV –10. - 20. Gev If physics demands, another FFAG –20. - 50. GeV

32. 32 Longitudinal emittance vs acceptance (after target and drift) Acceptance of US scheme is 0.167 eV.sec (150 mm). Difference comes from frequency of RF (5 vs. 201 MHz).

33. 33 Transverse emittance ~100 mm (100,000 pi mm-mrad)

34. 34 Hardware R&D (1) Low frequency RF (ferrite loaded) Shunt impedance Ferrite core

35. 35 Hardware R&D (2) Low frequency RF (air core)

36. 36 Hardware R&D (3) Superconducting magnet

37. 37 US scheme (Europe’s similar) Combination of RLA (LA) and Non scaling FFAG High frequency RF, 201 MHz

38. 38 Accelerator chain Before acceleration –Target, drift, buncher, rf rotator, and cooling Linac –0.220 GeV - 1.5 GeV RLA –1.5 - 5. GeV Two non-scaling FFAGs –5. - 10. GeV –10. - 20. GeV If physics demands, another non-scaling FFAG –20. - 50. GeV

39. 39 Before acceleration Three more stages compared to Japanese scheme.

40. 40 A way to make small emittance fit into 201 MHz RF There is some stage to make longitudinal emittance smaller so that 201 MHz RF can be used.

41. 41 Emittance evolution before FFAG injection Cooling is also necessary to fit into the acceptance. transverselongitudinal Path length [m] Emittance [mm]

42. 42 Acceleration system requirements Initial momentum0.3GeV/c Final momentum20GeV/c Normalized transverse acceptance30mm Normalized longitudinal acceptance150mm Bunching frequency201.25MHz Maximum muons per bunch1.1 x 10 11 Muons per bunch train per sign3.0 x 10 12 Bunches in train89 Average repetition rate15Hz Minimum time between pulses20ms From Reference 1.

43. 43 Design example of muon acceleration summary Japanese scheme assumes low frequency (~5 MHz) RF and no cooling is necessary. It uses scaling FFAG. US and Europe scheme assumes high frequency (~200 MHz) RF. It uses non-scaling FFAG. Hardware R&D is going on. Proof of principle model for non-scaling FFAG is scheduled in UK.

44. 44 Appendix FFAG as a proton driver

45. 45 Requirement of proton driver (1) Beam power = energy x current = energy x (particles per bunch) x (repetition rate) Energy –MW using a few GeV or more energetic protons. Particles per bunch and Repetition rate –From accelerator point of view, low ppb is preferable. –Probably rep. rate does not matter as long as the beam power above is obtained.

46. 46 Requirement of proton driver (2) Beam quality –Short bunch is preferable for smaller longitudinal emittance. –Momentum spread of protons is not important because that of muons can not be small. –Beam size (transverse emittance) is not important either.

47. 47 Machine candidate (1) Slow cycling synchrotron (0.1 ~ 1 Hz) J-PARC is one of examples –Maximum energy is 50 GeV. –Particles per bunch is high, 3e14 to obtain 0.75 MW –Should be more to upgrade to a few MW facility –Space charge and beam instability are problems.

48. 48 Machine candidate (2) Rapid cycling synchrotron (10 ~ 50 Hz) ISIS upgrade is one of examples –Maximum energy is 50 GeV. –Particles per bunch can be reduced, –Design of 30 GeV with 50 Hz is feasible.

49. 49 Machine candidate (3) Rapid cycling linac (10 ~ 50 Hz) SPL is one of example –Maximum energy is limited to a few GeV. –More particle per bunch is needed compared with RCS –Space charge and beam instability problem are less because acceleration is quicker.

50. 50 Machine candidate (4) FFAG (100 ~ 1000 Hz) –Maximum energy can be as high as synchrotron. –Particles per bunch can be much less. –Space charge and beam instability problem are less because acceleration is quicker. SCSRCSRCLFFAG energy~50 GeV ~3 GeV~20 GeV rep. rate0.1~110~5050100~1000 ppbhighlow much low Space charge etc. seriousmoderatelessNo problem

51. 51 Subjects to be studied Electron model of non scaling FFAG –New scheme of acceleration –Resonance crossing High intensity operation Optimization of scaling magnet Make the magnet superconducting

52. 52 Exercise (4) Scaling FFAG has magnetic field shape as –Momentum compaction factor  c is defined as –Show momentum compaction factor of scaling FFAG. –RF bucket (half) height is where E is total energy, h is harmonic number,  is slippage factor defined as how much RF voltage is required to accelerate 10 to 20 GeV when h=20 and k=280.

53. 53 Exercise (5) Scaling FFAG has magnetic field shape as where B0 and r0 are magnet strength and average radius at injection. –Ignore the azimuthal dependence and calculate orbit shift between 10 to 20 GeV when k is 280 and r0 is 120m. –The present design assumes the muon emiitance of 30 pi mm (or 30,000 pi mm-mrad) normalized. Calculate beam size at 10 GeV and 20 GeV and compare them with the orbit shift. Use the relation where R is average radius of an accelerator (120 m) and is tune (20).