1. 1 EMMA Tracking Studies Shinji Machida ASTeC/CCLRC/RAL 4 January, 2007 http://hadron.kek.jp/~machida/doc/nufact/ ffag/machida_20070104.ppt & pdf

2. 2 Contents Tracking study as an independent check if beam optics is what we expect. Tracking study to determine machine specifications. Aperture Magnet error tolerance Trim coil of Quadrupole –Orbit distortion due to misalignment –Orbit distortion due to gradient error (H only) –Orbit distortion due to lumped rf cavities (H only) –Optical mismatch due to gradient error Tracking study to predict beam behavior and prepare diagnostics –With gradient error –With misalignment –Strategy of “resonance” crossing study

3. 3 Machine specifications (1) Misalignment –Introduce additional bend for both H and V and create orbit distortion. – What is the alignment tolerance? Gradient error –Introduce additional focusing for both H and V and create optical mismatch. –Introduce additional bend for H, not for V, and create orbit distortion. – What is the gradient tolerance? – What is the specification of trim coil? Lumped (not every cell) rf cavities –Introduce orbit distortion.

4. 4 Machine specifications (2) Previous study results (http://hadron.kek.jp/FFAG/FFAG04_HP/)http://hadron.kek.jp/FFAG/FFAG04_HP/ –Alignment of 30  m (r.m.s Gaussian) by Keil and Sessler. –Alignment of 50  m (100% uniform) and gradient of 0.5% (100% uniform) by Machida. Keil and Sessler showed the longitudinal acceptance limit imposed by misalignment. I showed amplitude growth of single particle.

5. 5 Orbit distortion due to misalignment (1) Distortion pattern changes during acceleration because phase advance is not constant. Magnitude of distortion is a function of acceleration rate. Orbit moves in horizontal plane without misalignment. Simple formula does not work to estimate orbit distortion.

6. 6 Orbit distortion due to misalignment (2) Misalignment:  =0.050 mm, max=0.100 mm (2  Gaussian  –Reduction of aperture of 3 mm most likely and 5 mm in the worst case in horizontal. –Reduction of aperture of 3 mm most likely and 6 mm in the worst case in vertical. horizontal 50 seeds vertical 50 seeds

7. 7 Orbit distortion due to misalignment (3) Misalignment:  =0.050 mm, max=0.100 mm (2 , Gaussian  –Use 50 different seeds to see statistics. –6 examples in vertical are shown.

8. 8 Orbit distortion due to misalignment (4) The slower acceleration gives larger orbit distortion as expected.

9. 9 Orbit distortion due to gradient error (1) Gradient error:  =0.1%, max=0.2% (2 , Gaussian  –Reduction of aperture of 2 mm most likely and 3.5 mm in the worst case. horizontal 50 seeds

10. 10 Orbit distortion due to lumped rf cavities (1) Lumped rf cavities means not every cell has a cavity. More than 14 rf cavities, the maximum distortion is less than 1 mm. Resonance structure with 7 rf cavities.

11. 11 Orbit distortion due to lumped rf cavities (2) Distortion with 7 rf cavities occurs later in a cycle.

12. 12 Orbit distortion due to lumped rf cavities (3) Distortion with 7 rf cavities (every 6 cells) is likely attributed to synchro-beta coupling: (6Qx)-Qs=1, where Qx and Qs are cell tunes and Qs=0. Qx becomes 1/6 later in a cycle. Dispersion at rf also becomes larger later in a cycle.

13. 13 Orbit distortion due to lumped rf cavities (4) Failure of some rf cavities out of 14 rf excites the coupling. #1 fail #1 and 3 fail #1 and 5 fail#1 and 4 fail #1 and 2 fail

14. 14 Orbit distortion due to lumped rf cavities (5) If there are rf cavities every 3 cells and a few cavities are failed, This is the worst case when cavities fail alternatively.

15. 15 Orbit distortion (summary1) For example, sourcemisalignmentgradient errorrf failure magnitude 0.050 mm (  )0.10% (  ) 1 rf cavity horizontal max (average) 5 mm (3 mm) 3.5 mm (2.5 mm) 1 mm (0.5 mm) vertical max (average) 6 mm (3 mm) 0 mm

16. 16 Orbit distortion (summary2) Vertical design aperture is 11mm, where  y=0.8 m, and  y un =0.15  mm. Orbit distortion of 3 mm reduces the acceptance by (8/11) 2 ~0.5 Either correct the orbit distortion or enlarge aperture. –How we can correct the orbit? Phase advance is not constant ! –Beam based alignment? Consider again if 3  mm acceptance is necessary. –What is the rationality behind? Otherwise, simply add margin: 10 mm in H and 6 mm in V.

17. 17 Beam loss due to gradient error (1) Gradient error:  =0.1%, max=0.2% (2 , Gaussian  is necessary to suppress beam loss for nominal acceleration. For slow acceleration, tolerance should be tighter.

18. 18 Beam loss due to gradient error (2) Gradient errors introduce optical mismatch and a beam starts tumbling. When the beam size effectively large, even if emittance does not change, some particles are lost. When single particle emittance (either H or V), becomes more than 1.5 times, the particle is lost.

19. 19 Beam behavior and diagnostics (1) Gradient errors induce optical mismatch and a beam starts tumbling. Within 10 turns, it does not smear out. Although emittance is constant, beam size oscillates. 0 turn 6 turn 3 turn 9 turn

20. 20 Beam behavior and diagnostics (2) Another result of a muon 10 to 20 GeV ring with gradient errors. When emittance is small, there is only tumbling. When emittance is large, nonlinear distortion appears. 0 turn 4 turn 8 turn 12 turn 16 turn 0.003  mm 0.3  mm 30  mm

21. 21 Beam behavior and diagnostics (3) Another result of a muon 10 to 20 GeV ring with gradient errors. Beam beta defines as does not have any clear correlation with total tune.

22. 22 Beam behavior and diagnostics (4) Alignment errors introduce orbit distortion. On the frame of distorted orbit, beam shape does not change. 0 turn 6 turn 3 turn 9 turn

23. 23 Beam behavior and diagnostics (5) When acceleration is fast, there is no resonance behavior in a conventional sense. What we would see is –orbit distortion –beam tumbling –a bit deformation due to nonlinearity Measuring emittance is not a right way to study “resonance” crossing. Because of tumbling and unknown beta function, beam size measurement does not give emittance.

24. 24 Beam behavior and diagnostics (6) Possible alternative is –To survey initial phase space with a pencil beam. –When acceptance is 3  mm, Sqrt[3/0.01]xSqrt[3/0.01]=17x17 grid points in phase space can be surveyed. –Measure beam loss at accurate timing. –Make sure a pencil beam remains as a pencil. Diagnostics –1 pass beam position monitor at every cell for both H and V. –Beam current or beam loss monitor (at every cell). –Beam profile monitor or slits at extraction line to make sure small pencil beam size. –Collimator (at several place) to define or reduce machine aperture.

25. 25 Summary In addition to misalignment, lumped rf cavities introduce orbit distortion in horizontal plane. We need a margin of a few mm in aperture to provide a room for orbit distortion and beam tumbling. Magnitude of margin depends on the alignment and gradient tolerance and rf cavity configuration. For example, 6 mm in vertical if  is 0.05 mm. Trim coil to make 1% focusing error is enough to excite “controlled” error. Identification of beam loss at accurate timing should be more emphasized than emittance or beam size measurement to study “resonance” crossing.

26. 26 Backup slides

27. 27 Beam dynamics parameters (1) lattice function  function at different momentum. 10.5 MeV/c 15.5 MeV/c 20.5 MeV/c

28. 28 Beam dynamics parameters (2) tune and time of flight Nominal operation

29. 29 Beam dynamics parameters (3) difference in tune Relatively large discrepancy around injection momentum.

30. 30 Beam dynamics parameters (4) emittance evolution Initial emittance 3  mm, waterbag Acceleration with constant energy gain 12 turns (nominal)120 turns

31. 31 Beam dynamics parameters (5) end field modeling Tracking code adopts “Multipole symmetry” with Enge fall off. Keep the leading order and truncate the rest. becomes that is same as Berg’s assumption except the form of G 2,0 (z).

32. 32 Beam dynamics parameters (6) magnet (thin lens) and trajectory of 8th December lattice 2 cellLS enlarged

33. 33 Beam dynamics parameters (7) x and x’ at the center of LS, QF, and QD Based on the 13th December lattice with hard edge. P [MeV/c]x [mm]x’ [mrad]tof [ns]QxQy 10.5 LS-3.098-31.19055.47440.356060.34111 QF-15.437+23.330 QD-31.006+16.090 15.5 LS+0.180+0.23055.33530.216010.18997 QF-8.723+0.404 QD-31.253-1.617 20.5 LS+10.322+27.61455.46100.164580.12364 QF+4.410-18.715 QD-24.155-17.785