1. Details of space charge calculations for J-PARC rings

2. J-PARC accelerator complex –Phase 1 + Phase 2 = 1,890 Oku Yen (= $1.89 billions if $1 = 100 Yen). –Phase 1 = 1,527 Oku Yen (= $1.5 billions) for 7 years. –JAERI: 860 Oku Yen (56%), KEK: 667 Oku Yen (44%). JAERI Portion KEK Portion

3. Repetition of 3GeV Synchrotron injection 500μs injection turns ~350 particles per pulse 8.3e13 acceleration 20 ms extraction <1μs injection extraction acceleration

4. Repetition of 50 GeV Synchrotron injection 0.17s particles per pulse 3.3e14 acceleration 1.96 s extraction (slow) 0.7s injection extraction acceleration

5. Two approaches A whole cycle of 3 GeV synchrotron takes 20 ms. –Full simulation with self-consistent model is possible. –Tracking parameters (# of macro particles, grid size, etc) have to be optimized. Only injection period of 50 GeV synchrotron takes 0.6 s (or a bit less). –Not realistic to make self-consistent simulation. –Frozen space charge model might be justified because of well defined particle distribution.

6. Examples of full tracking for 3GeV Syn. Different colors shows results of different number of macro particles. Things are included. Injection painting Multipole errors Misalignment Acceleration Aperture of all elements Image in a circular pipe 3 months (100,000) 5 weeks (50,000) 2 weeks (20,000) Results within 3 months (1,000,000~200,000) Things are not included. Scattering at foil. RF jitter Impedance

7. Other tracking parameters Number of azimuthal mode Number of z grids Max. mode = 4, 8, 16 z grids = 10, 20, 30, 40, 50

8. Detailed study results Correlated and anti-correlated painting COD and beam loss –Coupled with strong chromaticity correction sextupole, COD introduces nonlinearity of all harmonics. Beam intensity dependence

9. Correlated and anti-correlated painting There is particle loss even during injection period. correlated anti-correlated 0.5 s for injection

10. Phase space density right after injection and at 3 ms later horizontal vertical at 0.5 ms at 3 ms at 0.5 ms correlated anti-correlated

11. COD and beam loss rms COD 0 mm 0.2 mm 0.5 mm 1.0 mm Coupled with strong chromaticity correction sextupole, COD introduces nonlinearity of all harmonics.

12. Phase space density for different COD No difference in core density. Tails are developed with COD. rms COD 0 mm 0.2 mm 0.5 mm 1.0 mm rms COD 0 mm 0.2 mm 0.5 mm 1.0 mm Hor. Ver.

13. Beam intensity dependence 30mA 20mA cf. 30mA is design value which deliver 0.6 MW beam from RCS with tune spread of ~0,25.

14. Intensity dependence Core density is reduce with 30 mA. (lower order resonance is involved?) Tails are also developed. 20mA 30mA 20mA 30mA

15. Summary of self-consistent simulation A whole cycle of 3 GeV Syn can be simulated even though it takes a few months. Horizontal and vertical coupling is the source which makes anti- correlated painting worse. Increase of particle loss due to larger COD is attributed to tail development. Higher order effects are involved. Intensity limitation may be explained with lower order resonance. That is a regime where coherence picture is applicable.

16. Example of beam loss during injection with frozen space charge model Model assumes Particle distribution is Gaussian. Emittance is constant. dp/p is finite and there are synchrotron oscillations. Transverse space charge force depends on longitudinal position.

17. Tracking model “Frozen model” of space charge is adopted. –Space charge potential is fixed throughout a tracking. –No self-consistency. –No coherent oscillations. –Gaussian charge distribution in 3D is assumed. Lattice nonlinearities and misalignment errors are included. Aperture of magnets and collimator are included so that we can estimate beam loss. Macro particles (1,000) of 3D Gaussian distribution with 2 sigma cut are tracked for 0.12s (original design value for accumulation) or more.

18. Some numbers Emittance(2sigma)54 pi mm-mrad (36pi, 45pi, 64pi) Acceptance at collimator71 pi mm-mrad for H and V Acceptance at magnets> 81 pi mm-mrad Circulating current10 A (3.3E14 ppp) Incoherent tune shift-0.16 Bare tune(22.42, 20.80)

19. COD Chromaticity sextupoles coupled with COD introduce beta modulation and higher harmonics of nonlinearity. Survival at 0.12s after injection. COD shows a rms value. Maximum is about 3 times. Collimator aperture is adjusted taking a local COD into account. We expect COD(rms) is less than 0.5mm after correction. The loss is not linear as COD. survival at 0.12s (%) 80 85 90 95 100 0 0.5 1.0 1.5 2.0 COD (rms) (mm)

20. Different lattices Although rms COD is almost same, different lattices (seeds) give different results. Previous example is the worst case among three. survival (%) 96 97 98 99 100 0 0.05 0.1 time (s)

21. Beam current The pattern of COD is the same for both. Magnitude is different. The design current is 10A. Blue: COD=0.5mm Red: COD=1.0mm survival at 0.12s (%) 80 85 90 95 100 0 5.0 10 15 beam current (A)

22. Initial emittance Acceptance at collimator is fixed at 71 pi mm-mrad. Space charge force is fixed according to the initial emittance. We expect 54 pi mm-mrad emittance shaped at the 3-50BT collimator. Collimator acceptance should be optimized to have the maximum survival. survival at 0.12s (%) 80 85 90 95 100 30 40 50 60 70 80 initial emittance (pi mm-mrad)

23. Location of loss CODMagnet acceptance Collimator acceptance Initial emittance Loss at collimator Loss at magnet Total number 0mm> 81 pi mm-mrad 71 pi mm-mrad 54 pi mm-mrad 3 (100%)0 (0%)3/1000 0.2> 81 pi71 pi54 pi4 (100%)0 (0%)4/1000 0.5> 81 pi71 pi54 pi11 (92%)1 (0%)12/1000 1> 81 pi71 pi36 pi33 (97%)1 (3%)34/1000 1> 81 pi71 pi45 pi33 (92%)3 (8%)36/1000 1> 81 pi71 pi54 pi21 (88%)3 (12%)24/1000 1> 81 pi71 pi64 pi59 (88%)8 (12%)67/1000

24. Beam loss at collimator (h=18) total 0.72 MW COD0 mm0.2 mm0.5 mm Loss670 W800 W1370W COD (rms) = Red: 0mm Yellow: 0.2mm Green: 0.5mm All the particles hit collimator first.

25. Beam loss at collimator (h=18) total 0.58 MW COD0 mm0.2 mm0.5 mm Loss350 WN/A690 W COD (rms) = Red: 0mm Yellow: 0.2mm Green: 0.5mm All the particles hit collimator first.

26. Frozen model with acceleration phis bunch length dp/p

27. 99% emittance and beam loss Acceleration starts right after injection. Acceleration starts at 0.16 s after injection.

28. Acceleration starts at 0.6 s after injection and h=18.

29. Single particle behavior Tracking without aperture limit to see single particle behavior. Slow growth of amplitude. Not obvious correlation with synchrotron oscillations. Trapping? Timing of hitting collimator 0 0.01 0.02 0.03 0.04 0.05 time (s) horizontal position (m) -0.1 -0.05 0 0.50 0.1

30. Single particle behavior Track a single particle which is lost in 0.6 s. Look at betatron oscillation amplitude and transverse tune as a function of turn until a particle is lost. For example, there are –38 lost particles (out of 1000) when rms COD is 0mm. –40 lost particles (out of 1000) when rms COD is 0.2 mm. –52 lost particles (out of 1000) when rms COD is 0.5 mm.

31. #1 #2 #3 #5#6#4 HVHVHVHV turn number (~ 10,000 turns) amplitude rms COD is 0.5 mm

32. HVHVHVHV #7 #8#9 #11#12#10 turn number (~ 10,000 turns) amplitude rms COD is 0.5 mm

33. HVHV Horizontal amplitude always increases and gets to the aperture limit. Vertical amplitude always decreases. Coupling between H and V is manifest. #16 #15#14#13 HVHV turn number (~ 10,000 turns) amplitude rms COD is 0.5 mm

34. In tune space Blue points are intermediate tune of lost particles. Red points are tune just before particles are lost. Tune before particle loss are same with and without COD. 2 x- y=24 x-2 y=-19 bare tune x-22 y-20

35. Coupling between H and V is manifest, but Tune space plot does not show resonance driving term. –2 x- y=24 is skew and cannot be excited even with finite dp/p and dispersion in a lattice. If there is any way to reduce a driving term. –Since the source is not identified, it is difficult.

36. Summary of frozen space charge simulation Particle loss occurs because horizontal amplitude increases and hits the collimator aperture. The source of the increase is a coupling between H and V. With finite COD, particle loss occurs with less turns. However, transverse tune when a particle loss occurs does not depend on COD magnitude. Loss is very slow process: the order of 10 4 turns. Time scale of horizontal and vertical coupling is also same order.

37. Basic loop of calculation Advance particle coordinates do ip=1,np (200,000) Calculate space charge potential based on particle positions. do imode=1,nmode (16) Apply space charge kicks to all particles do ip=1,np (200,000) Simpsons uses Fourier expansion in azimuthal direction. Make parallel processing of Fourier modes.

38. Distribution of workload (4 CPUs) with MPI imode =3,4,5,6 imode =0,1,2 imode =7~11 imode =12~16 Add up all E-fields 4 CPU works in the same way