1. Off-momentum DA of RCS 3D_BM & QFF & CC AP meeting / December 8, 2004 Alexander Molodozhentsev Etienne Forest KEK

2. Tune diagram -Q X +2Q y =6 “Sextupole” -Q x +3Q y =12 “Octupole” -Q x +4Q y =18 “Decapole” Structure resonances: “Bare” tunes: Q x = 6.68 Q y = 6.27 -Q x +5Q y =24 RCS_3DBM --- scanning regions Chromatic tune_shift ~  0.08 (dp/p=  0.01)

3. MADX: RCS Twiss parameters (dp=0.0) MADX_Chromaticity: ~ -15 !!! MAX_D x ~ 10 m !!! WHY? Because MADX uses ‘time” as the independent parameter instead of ‘path-length’… W kin = 181 MeV …  = 0.54523

4. Off-momentum particle motion (lattice#1) Observation point : SFX D x ~ ??? -15cm15 cm 5.5 cm  p/p = 0.01 PTC tracking: Use “path” instead of “time”…SAD, MAD8 D x (SFX) ~ 5.5 m PTC: Chromaticity (x/y) ~ -8 RCS_3DBM

5. Chromaticity definition Sextupole field component Path_length in ring If the code uses the time instead of the path, the chromaticity will be different in the case of the low kinetic energy: ds = (c  ) dt W kin = 181 MeV …  = 0.54523 … then  PATH ~ 0.545  TIME !!!

6. Dispersion function definition Periodic dispersion function can be written as [Mario Conte, p.85] … also depends on choice “path” or “time” … MAD dispersion (“path”) is about two time bigger than SAD dispersion for W kin =181 MeV …

7. Off-momentum particle motion (lattice#1)  p/p = 0.01 Observation point : SFX D x ~ 10 m  x  p/p ~ 10 cm -15cm15 cm TDR: Bore radius of SFX is (330/2) mm=165 mm SFX:  x ~ 13m,  y ~ 6m  x center->chamber ~ 16.5-10.0=6.5 cm  x,MAX ~  (  x) 2 /  x ~ (6.5e-2) 2 /13 ~ 325 .mm.mrad … Momentum Acceptance for  p/p = 0.01 PTC tracking: Use “time” instead of “path” …MADX

8. Effects of the quadrupole fringe field (2) De-tuning effect for the off-momentum particles … … the end-field of the quadrupole magnet gives the beam two kicks in the opposite directions, which cancel for the paraxial trajectories, but which gives a net focusing to trajectories passing though at an angle. It means that the fringe field of the quadrupole magnets, which are located in the non- zero dispersion sections, will provide the de-tuning of the off-momentum particles. This effect depends on the particle momentum and the betatron amplitude, the dispersion and the slope of the dispersion at the location of the quadrupole magnet. (1) ‘Octupole-like’ effect … … at the leading order – excitation of the normal octupole resonances (3) Sextupole feeding-down component … … for the off-momentum particles the pseudo-octupole nonlinearity of the quadrupole magnet will be “feeding-down” due to the closed orbit displacement in the quadrupole magnets in the non-zero dispersion regions (in the ARCs). In this case the quadrupole fringe field can contribute additional excitation to the normal sextupole resonances. RCS_3DBM ON-momentum particles… OFF-momentum particles…

9. Tune variation for off-momentum particles Without sextupole magnets for the chromaticity correction… Linear chromaticity ~ (-8) RCS_3DBM

10. Off-momentum DA (3D_BM) 3D_BM Plus: QFF HE NO Sextupole CC Observation point: BM 1 entrance (#47) RCS_3DBM dp/p=0

11. Off-momentum DA (3D_BM) 3D_BM Plus: QFF HE NO Sextupole CC Observation point: BM 1 entrance (#47) Resonance [-1,2] driving terms Resonance [-1,2] driving terms for this case is 3 times smaller than for the corresponding tunes of the on- momentum particles RCS_3DBM

12. Off-momentum DA (3D_BM) 3D_BM Plus: QFF HE NO Sextupole CC Observation point: BM 1 entrance (#47) RCS_3DBM Increasing the resonance 4Q x =27 for the off-momentum particles … by the feeding-down the high-order field components of the 3D_BM …

13. Effect of the quadrupole fringe field for the off-momentum particles 3D_BM Plus: QFF HE NO Sextupole CC RCS_3DBM (1) (1)Contribution to the normal sextupole resonance by the sextupole ‘feeding-down’ of the QFF “pseudo-octupole”… some compensation effect (2) (2) Increasing of the normal octupole resonance…

14. Off-momentum DA (3D_BM) 3D_BM Plus: QFF HE NO Sextupole CC Observation point: BM 1 entrance (#47) RCS_3DBM Main limitation of the off-momentum DA is caused by the normal octupole resonance.

15. Off-momentum DA (3D_BM): dp/p=  0.005

16. Off-momentum DA (3D_BM): dp/p=  0.01

17. … synchrotron oscillation … Some speculation … look at summary012-file …summary012

18. On-momentum particle motion Q X – fix Q Y - vary RCS_3DBM Tune-scanning in the Q y direction Main limitation of the on-momentum DA is caused by the normal sextupole resonance NO CC Sextupoles

19. Off-momentum (3D_BM) plus chromatic sextupole magnets Changing of the tune for different (dp/p) before (1) and after (2) the chromaticity correction RCS_3DBM

20. Off-momentum DA after chromatic correction 3D_BM Plus: QFF HE Sextupole CC Observation point: BM 1 entrance (#47) (X=Y) MIN = 2.75 cm Twiss Parameters: (β x ) 1/2 = 2.7848 m 1/2,  x = 0.2517 (β y ) 1/2 = 3.9030 m 1/2,  y = 0.1990 A x = (X max ) 2  x ~ 190  mm mrad A y = (Y max ) 2  y ~ 150  mm mrad RCS_3DBM

21. On-momentum DA after chromatic correction … result was presented October 27, 2004. RCS_3DBM

22. To DO: … correction of the normal sextupole resonance … … off-momentum particle motion after the correction… RCS_3DBM Presented for the AP-group December 8, 2004