1. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 1 Dejan Trbojevic Crossing Resonances In a Non-scaling FFAG

2. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 2 Crossing Resonances in a Non-scaling FFAG Scaling – Non-Scaling FFAG

3. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 3 3 1. Introduction: CONCEPT OF SCALING FFAG’s Radial Sector scaling FFAG Spiral Sector Scaling FFAG Okhawa, Kolomensky, and Symon (1953-55)

4. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 4 BASIC RELATIONS IN SCALING FFAG’S  the tunes are given by the same equations as those for imperfectly isochronous cyclotrons: r 2 ≈ 1 + k k=(r /B R ) (dB/dr)=-n y 2 ≈ -k + F 2 (1 + 2 tan 2  ) The constant r requires k=constant implying magnetic field and momentum profiles of the form: The vertical tune also has to stay constant during acceleration this requires the second equation to be F 2 (1 + 2 tan 2  ) = constant. For spiral sectors: choosing constant , so the sector axis is a logarithmic spiral: R=R o e  cot  while for the radial sectors B D = -B F.

5. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 5 MURA-KRS-6 Phys. Rev. 103, 1837 (1956) November 12, 1954 K. R. Symon: The FFAG SYNCHROTRON – MARK I Radial Sector FFAG  r~60-100 cm 11 22 11 11 Wedge angle   qq qq qq qq

6. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 6 Radial Sector scaling FFAG r Then it is isochronous =√(1-n) x

7. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 7 NON-SCALING FFAG - 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 0.9 meters 0.09 meters

8. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 8 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 large Tunes are fixed for all energies no integer resonance crossing. Negative momentum compaction. B =B o (r/r o ) k non-linear field Large acceptance Large magnets Very large range in  p/p= ±90% 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 resonance crossing for protons Momentum compaction changes. B = B o +x G o linear field Smaller acceptance Small magnets Large range in  p/p=±60% B = B o +r G o B =B o (r/r o ) k

9. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 9 Tune variation in the NS-FFAG Carbon Ion acceleration – the third ring in Keil, Seesler, Trbojevic example

10. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 10 Following: Rick Baartman – (Guignard) - May 6, 2005, FFAG workshop in TRIUMF – Vancouver n th -harmonic m th –driving term

11. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 11 Following: Rick Baartman – (Guignard) - May 6, 2005, FFAG workshop in TRIUMF – Vancouver Integer resonance m = 1 (tune is in US) Q = 1 resonance Half-integer, m = 2 n=2

12. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 12 Following: Rick Baartman – (Guignard) - May 6, 2005, FFAG workshop in TRIUMF – Vancouver 1/3-integer, m = 3 This is the intrinsic resonance 3Q = 3 =N p. Even a slight systematic sextupole component in the magnetic field will result in very large b n,3 when n = N cells =N p

13. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 13 Basic definitions: The resonance occurs when m  = n = f N p, where  is the tune of the ring, m is order of the resonance driving term, n corresponds to n th harmonic Fourier component of the (m-1) th derivative of the magnetic field, f is an integer and N p is the number of lattice periods. The intrinsic or systematic resonances of an accelerator depend on the number of lattice periods N p through the resonant condition: m x = N p. The resonance strength parameter b n,m is: average magnetic field:

14. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 14 Eberhard Keil – CYCLOTRONS-2007

15. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 15 Eberhard Keil – CYCLOTRONS-2007

16. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 16 3. Previous Studies 3.1. Suzanne Sheehy (Manchester-2008): (A) NS-FFAG C=43.17 m and later (B) C=26.7 m. 3.2. Shinji Machida – EMMA simulation of the proton acceleration. 3.3. T. Yokoi, J. Cobb, K. Peach, and S. Sheehy – ZGOUBI tracking NS-FFAG acceleration study (PAMELA). S. Sheehy: tracking simulations were made to investigate the effect of misalignment on the resulting orbit distortions first without acceleration. Analysis of the results shows that tolerances in alignment have to be better than 0.1 mm. The result of an error in alignment of  = 0.4 mm causes a maximum orbit distortion during acceleration of 1.7 mm in the larger ring.

17. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 17 (m) 3.1.A. Orbit distortions due to misalignment: NS-FFAG C=43.17 m My experience with the RHIC alignment for the large magnets 3-4 mills = 100  Instruments – laser tracker – 1 mill = 25.4  But for the light sources even 10  is considered easily achievable. (m) 50 

18. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 18 3.1.B. Orbit distortions due to misalignment: NS-FFAG C=27 m Realistic misalignment errors

19. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 19 3.2. Shinji Machida: EMMA study simulating proton acceleration: The average radius of the machine is: R avg =2.5 m and circumference of C EMMA ~16.6 m. The "scaled" B  =0.05 Tm to be compared to the carbon central momentum B  =4.23 Tm or proton central momentum of B  =1.62 Tm The maximum orbit offsets during acceleration in EMMA are x max < 16 mm to be compared with: -20 mm <x max < 80 mm of the C = 43.17 m carbon ring and the -80 mm<x max <180 mm of the C = 26.88 m ring. For correct simulations, beam amplitudes should scale with the size of the orbit offsets and so the beam sizes and emittances in EMMA should be reduced by factors of 6 and 40, respectively. Electron beam emittances of 20  10 -6 rad m used in this study had to be reduced to a typical value of 0.5  10 -6 rad m for a proton therapy ring – why?

20. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 20 3.2. Shinji Machida: EMMA study simulating proton acceleration:

21. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 21 3.2. Shinji Machida: EMMA study simulating proton acceleration: R= 4.33 m B  =4.23 Tm - carbon B  =1.62 Tm - proton = 8.0 - 12.8 = 4.8 T/m  =20  10 -6 rad m Electron normalized beam emittances are 40 times larger than those proposed for the C=43.17 m and C=26.88 m proton therapy rings. The ndings of the EMMA simulations are that even misalignments as low as = 0.015 mm lead to small dynamic apertures and large orbit distortion growth on integer resonance crossing.  =0.5  10 -6 rad m

22. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 22 3.2. Shinji Machida: EMMA study simulating proton acceleration:

23. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 23 3.2. Shinji Machida: EMMA study simulating proton acceleration: The step size is: 20 x 10 - 6  m rad normalized (that is geometrical emittance

24. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 24 3.2. Shinji Machida: EMMA study simulating proton acceleration: V. SUMMARY: With particle tracking simulations, the effects of resonance crossing in a linear non-scaling FFAG were studied when there were alignment errors and when an accelerator was operated with a relatively slow acceleration rate of 100 or 1000 turns. When there were practical errors such as a few times 10  m rms misalignment, the resonance behavior started appearing with 100 turns operation and became clear with 1000 turns operation. In the latter case, dynamic aperture practically disappeared. It was shown that the amplitude growth dependence on acceleration rate and alignment error agreed with the analytical formula by Baartman et al.

25. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 25 3.2. S. Machida and D. Kelliher: EMMA study By tracking simulation, we studied the orbit and optics distortion in a non-scaling FFAG with alignment and gradient errors. Although integer and half-integer resonance crossing was a primary concern because the betatron tune changes a lot during acceleration, the tracking results do not show any excitation by resonances in orbit and optics distortion at the nominal acceleration rate. The orbit and optics distortion has a linear dependence on the magnitude of errors in a ring. Rather, the simulation results indicate that the distortion is excited by random dipole and quadrupole kicks.

26. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 26 3.2. Shinji Machida: EMMA simulation of proton acceleration: The NS-FFAG is a linear machine without tune dependence on amplitude, but the initial emitance size is important as with presence of the magnetic field errors the tune will depend on amplitude. To prevent an increase in transverse space charge levels, the longitudinal extent of the beam should increase by the same factor, which infers that the single EMMA bunch should be replaced by a train of 40 bunches and the need of a much lower frequency acceleration system. T. Yokoi, J. Cobb, K. Peach, and S. Sheehy – NS-FFAG PAMELA study Study using the ZGOUBI code: Emittance blow up was less than two,  < 2, for the input emittances lower than 2 mm mrad if the alignment limits and magnetic field errors were smaller than 0.1 mm and  B/B n < 2  10 -3, respectively.

27. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 27 4. Resonance crossings of the linear NS-FFAG: Introduction: Description of the linear NS-FFAG ring Method of acceleration by the phase jump Acceleration without errors: 10,320 turns through the 1/3 tune in the cell 1300 turns through the 1/3 in the cell Acceleration with ~1000 turns with magnetic field random errors  B/B n < 10 -4  B/B n < 10 -3

28. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 28 Acceleration PHASE JUMP each turn – Mike Blaskiewicz: Acceleration is performed with the phase jump after each turn. The phase jump during acceleration with the fixed frequency by M. Blaskiewicz [28]. -The RF frequency needs to be in a high range, 370 MHz) because of required large number of RF cycles between the passages of bunches in order to achieve higher values of Q and to limit the frequency swing. The total stored energy in the cavity is related to the amplitude V RF of the RF voltage as: where  r is the angular resonant frequency, Q is the quality factor, and R is the resistance. The cavity voltage dependence on the  klystron voltage (driven by the low level drive)

29. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 29 The whole ring with all elements: r=4.278 m 24 doublets 12 cavities Three kickers Circumference = 26.88 m -50 %<  p/p < +50% E k_inj =31 MeV E k_max = 250 MeV D=8.56 m

30. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 30 Requires a loaded quality factor Q=50 Full horizontal aperture 28 cm Full vertical aperture 3 cm, R/Q = 33 Ohm (circuit) for  =0.24 ACCELERATION:

31. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 31 PROTON ACCELERATION 31-250 MeV  The bunch train lls half of the ring at the injection. The changes from injection to the maximum energy of 250 MeV between  inj = 0.251 to  extr. = 0.614. There is 80 ns time to change the cavity frequency when there is no beam. With Q=50 and f=374 MHz the exponential decay time for the eld is 43 ns, where R=Q 33 at =0.25. If the synchronous voltage is 18 kV, a number of turns required for acceleration of protons is: for twelve cavities (ncav=12). It is very clear that higher effective voltage on the cavities could improve the the resonance crossing problem as well as the patient treatment time. A total power for one RF driver is 100 kW, for twelve cavities this make 1.2 MW and this is a price for the fast acceleration.

32. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 32 Acceleration: 26.88 meter circumference 31 MeV < proton kinetic energy < 250 MeV, 0.24 <  < 0.61 Central rf frequency = 374 MHz

33. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 33 Orbit offsets and dimensions in the cell  d =0.1090831  ½  f = ½ 0.15271631 ½ F D ½ QL f =44 cm/2 QL d =22 cm 8 cm 38 cm ½ F  ½  f = ½ 0.15271631 L=1.12 m 14.1 cm 8.21 2.6 -2.5 -10.1 -6.9

34. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 34 Tunes vs. momentum E k =30.96 MeV 250.0 MeV

35. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 35 Betatron Functions Dependence on Momentum

36. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 36 Magnetic Properties: L BD = 22 cm L BF = 30 cm G d = -14.3 T/m G f = 8.73 T/m B do = 0.804 T B fo = 0.563 T Values of the magnetic fields at the maximum orbit offsets: B d max- = 0.804 + (-14.3  0.0484) = 1.496 T B d max+ = 0.804 + (-14.2)  (0.107) = -0.715 T B f max+ = 0.563 + 8.73  0.141 = 1.794 T B f max- = 0.563 + 8.73  0.102) = -0.327 T  p/px 0ff (m) 50 0.140638 40 0.111097 30 0.082114 20 0.053819 10 0.026376 0 0.000000 -10 -0.025024 -20 -0.048317 -30 -0.069370 -40 -0.087506 -50 -0.101838 Offsets at F  p/px 0ff (m) 50 0.107354 40 0.083583 30 0.060737 20 0.039014 10 0.018662 0 0.000000 -10 -0.016560 -20 -0.030484 -30 -0.041077 -40 -0.047447 -50 -0.048481 Offsets at D Minimum horizontal aperture: A min =0.140638+0.101838+6  cm

37. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 37 10,320 turns, very small random magnetic field errors

38. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 38 10,320 turns, very small random magnetic field errors TURN 28 TURN 406 TURN 293 TURN 46 TURN 420 TURN 419 TURN 418 TURN 416 TURN 424 TURN 423 TURN 422 TURN 421

39. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 39 10,320 turns, very small random magnetic field errors TURN 427 TURN 432 TURN 434 TURN 445 TURN 442 TURN 434

40. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 40 x x x x x x x x x’ N turn =1N turn =417N turn =419N turn =420 N turn =425N turn =429N turn =449N turn =6264

41. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 41 x x’ xfxf xf’xf’ Initial conditions in the x, x’ phase space After 1300 turns small field errors going through 1/3 x, x’ phase space ~1300 turns, small random magnetic field errors

42. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 42 y y’ yfyf yf’yf’ Initial conditions in the y, y’ phase space After 1300 turns small field errors going through 1/3 y, y’ phase space ~1300 turns, small random magnetic field errors

43. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 43 x x’ xf’xf’ Initial conditions in the x, x’ phase space After 1300 turns field errors dB/B=10 -3 avoiding 1/3 in cell xo’xo’

44. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 44 y’ y Initial conditions in the y, y’ phase space After 1300 turns field errors dB/B=10 -3 avoiding 1/3 in the cell Random Magnet errors:  B/B n =10 -3

45. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 45 Blow up in x, x’ due to the random errors of 10 -3 third order avoided 941 850 731 77 506 35 1 2 12395 1036 12395/12=1032

46. Dejan Trbojevic, September 21, 2009International Workshop on FFAG09 - Fermilab 46 Summary A conclusion from the six dimensional acceleration tracking study, through the linear non-scaling FFAG, by the PTC – (Polymorphic Tracking Code) of Etienne Forest is that the emittance blow-up due to crossing resonances agrees very well with the Bartmaan – Guignard’s analytical predictions. Slow acceleration with 10300 turns produced large emittance blow-up especially crossing the 1/3 order tune within the cell. Reasonable alignment errors of less than 100  or with random magnetic field errors smaller than  B/B n <10 -3 emittance blow-up was of the order of 1.8. The linear NS-FFAG can cross-resonances without a big problem as long as the alignment tolerances are less < 50-100  and random magnetic field errors are smaller than  B/B n <10 -3.