1. Main Ring + Space charge effects WHAT and HOW … Alexander Molodozhentsev for AP_MR Group May 10, 2005

2. MR Injection process Basic MR-design: Injection time (W kin =3 GeV) to accumulate the required beam intensity from RCS is about 0.17sec (in the case when just 5% of the total RCS beam intensity is injected into MR). The revolution period at the injection energy for MR is 5.38e-6 sec. Then the injection process for MR is about 31’500 turns. Feb.15,2005

3. Space charge tracking General recommendation for the space-charge simulations 1.Transverse space-charge nodes should be at least 10 (or more) per one betatron oscillation MR…Q x,y ~ 22, C RING = 1567.5 m, then  L TSC ~ 7.5 m (~200 per ring nodes) Our choice: TSC after each element and after L DRIFT < 3 m (~900 nodes per MR) 2.Number of macro-particles should be at least 10 per one mesh- point …for the transverse mesh 32x32 minimum required number of the macro particles should be 10’000. Our choice: time optimization … N mp  10000 Feb.15,2005

4. ORBIT-MPI (TEAPOT tracker) Single Processor.... DELL Latitude D600: Intel Pentium Centrino, 1.4GHz Linux RedHat 9 MR, 3GeV 1turn Macro Particles 1'000mp5'000mp10'000mp20'000mp50'000mp10'000mp+ BeamPipe Pair-wise 23' 592.66' Brute- Force PIC 22.22' 32x32 25.15' FFT-PIC 32x32 40x40 32x32 ~ 7’ 12.3' 30.9' 15.8' 1000 turns 1.9 h 3.4 h 8.6 h 4.4 h 10'000 turns ~ 19 h ~ 34 h ~ 86 h ~ 44 h Estimation of time for simulation Feb.15,2005 Longitudinal N bin = 32

5. Comparison… ORBIT-MPI (J.Holmes,ORNL) 1 turn ~ 7’ (CPU) ACCSIM4 * (F.Jones,TRIUMF) 1turn ~ 30’ (CPU) UAL (N.Malitsky,BNL) 1 turn ~ 22’ (CPU) * Including ACCSIM post-processing Desktop PC-LINUX Single processor Notebook – LINUX Single processor Notebook – LINUX Single processor Feb.15,2005 Conditions: wp2, SpCh – ON … Nppb = 3.3e14/8, RF - ON, CCSX - OFF 2 nd order MAD8 matrixDrift-Kick-Drift-Kick scheme: Q_split = 2 (kicks=8) BM_split =1 (kicks=4) 2 nd order MAD8 matrix Teapot tracker: QM: (kicks=2) BM: (kicks=2)

6. Transverse space charge Pair-wise sum calculates the Coulomb force on one particle by summing the force over all other particles. This method requires ~(n p ) 2 operations to calculate the kicks, where n p is the number of macro-particles. Brute-force PIC a straight forward PIC implementation. The macro particles are binned on a prescribed X-Y grid, the force at each grid point is calculated using the binned particle distribution, the force on each particle is calculated by a bi- linear interpolation from the grid. The operation time is ~(n bin ) 4. FFT-PIC … FFT method is used to calculate the force on the grid using the binned particle distribution. The computation time is ~2n bin (n bin ) 2. Feb.15,2005

7. Longitudinal space charge General way: -bins the longitudinal beam profile; -calculates the longitudinal space charge force; -applies a momentum kick, based on the space charge force to the macro particles. … the longitudinal profile of the beam is used as a weighting factor for the transverse space charge kicks. Couping the longitudinal motion into the transverse one: … the space charge force on a given macro-particles is scaled according to the longitudinal charge density at its position in the bunch.

8. ORBIT-MPI test tracking W kin = 3GeV, N pb = 3.3e14/8 (h 1 =9) RF-ON ChromCorrection SX - ON H1=9, V RF1 = 280kV H2=18, V RF2 = 140kV ‘Bare’ wp: Q x = 22.333 / Q y = 20.77 Gaussian Transverse:  100% = 54  mm.mrad  RMS ~ 6.7  mm.mrad Uniform Longitudinal: (  p/p) MAX =  0.004 (  ) MAX =  60 degree N SYNCH ~ 400 turns

9. ORBIT-MPI: Gaussian T-distribution Uniform L-distribution Gaussian Transverse:  100% = 54   RMS ~ 6.7  Q x =22.333 Q y =20.774 Uniform Longitudinal: (  p/p) MAX =  0.004 (  ) MAX =  60 degree After 200 turns 2Q x -2Q y =3 4Q y =83 -Q x +2Q y =19 4Q x =893Q x =67 Q x +2Q y =64 Q x + Q y =43 2Q x -Q y =24 3Q y =62 2Q x +Q y =65

10. Gaussian Transverse Distribution Kicks number (QM&BM) = 2 (in code=2) SpCh – ON CCSX - ON ORBIT-MPI (Teapot tracker) test tracking  100% = 54   RMS ~ 6.8  Q x =22.333 Q y =20.774

11. UAL test tracking CCSX - ON SpCh - ON Gaussian TD Uniform LD Q x =22.333 Q y =20.774 Kicks number QM = 8 (in code=2) BM = 4 (in code=1)  100% = 54   RMS ~ 6.8 

12. Steps … INJECTION & Acceleration COD RANDOM BM-error  (BL)/(BL) 0 and H/V shift of QM so that to get some realistic COD (  x COD,  y COD ) < 0.7 mm (TDR) … observation NORMAL SEXTUPOLE resonances & NORMAL OCTUPOLE resonances Misalignment TILT of Sextupole Magnets … observation SKEW SEXTUPOLE resonances TILT of Quadrupole Magnets … observation LINEAR COUPLING (skew) resonance Correction of Resonances …