1. Alexander Molodozhentsev KEK for MR-commissioning group September 20, 2005 for RCS-MR commissioning group September 27, 2005 Sextupole effect for MR - beam centroid beating

2. MR Technical Design COD after correction should be less than 1 mm. From the beam point of view it means that the deviation of the beam centroid (center of “mass”) from the machine center (0,0,s) should be less than 1 mm.

3. Quadrupole field A s ~ x·y A x = A y =0  x / s ~ x  y / s ~ y Single particle transverse kick from thin quadrupole Multi-particle transverse kick from thin quadrupole ~ = 0 without COD Vector potential for the quadrupole field Average kick for the beam’s macroparticles is zero around the central orbit…

4. Sextupole field (simplified model) A s = 1/6 B // (x 3 – 3xy 2 ) A x = A y =0  x / s ~ (a x 2 – b y 2 )  y / s ~ xy Single particle transverse kick from thin sextupole Multi-particle transverse kick from thin sextupole ~ (a – b ) ~ ( a  x – b  y ) ~ = 0 without COD Vector potential for the sextupole field Average kick for the beam’s macroparticles is NOT zero around the central orbit … shift of the beam centroid !!!

5. Sextupole effect on the beam centroid E.Forest book “Beam Dynamics: …” (p.285) … sextupoles do move the average position of the beam. The part which does not depend on amplitude is the regular  2 dispersion. It is a non-dynamical effect. It is the change of the fixed point as a function of energy in a coasting beam normalization. The amplitude dependent terms are dynamical. A beam of finite size, on momentum (  =0), appears shifted as if the fixed point moved. This effect of the sextupole field nonlinearity is the ‘leading’ order effect, then it could be observed by using the second-order matrix formalism.

6. Sextupole effect on the beam centroid Estimation of the effect for MR (3GeV_Beam) for a single sextupole MR: ~ 15 m, ~ 1.5 m  =  p/p =  0.004  x 100% =  y 100% = 54  mm.mrad … Uniform:  x,RMS =  y,RMS = 54/4 = 13.5  mm.mrad “Dispersion” term … ~ 18  10 -6 “Amplitude Dependent” term (X) … ~ 196  10 -6 “AD” term is much bigger than “D” term.

7. Study approach … MAD second order transfer matrix between the ring elements for MR lattice for the sextupole magnets (OFF / ON). Teapot-type multi-particle tracker (ORBIT), based on the MAD transfer matrix. NO space charge effects. Observation of the first transverse moments ( & ) around the ring. MR:: Working point :: Q x = 22.428, Q y = 20.82 Transverse particle distribution :: 3GeV … 54  mm.mrad (Gaussian:  100% =8  RMS / Parabolic: 6  RMS / RCS_beam) 40 GeV … 6  mm.mrad

8. Test tracking #1 10’000 mp S [m] [mm] 54  mm.mrad  p/p = 0 NO CCSX

9. Test tracking #2 54  mm.mrad  p/p = 0 NO CCSX 95’254 mp S [m] [mm]

10. Test tracking #3_1 54  mm.mrad  p/p = 0 NO CCSX 250’000 mp S [m] [mm]

11. Test tracking #3_2 54  mm.mrad  p/p = 0 NO CCSX 250’000 mp S [m] [mm]

12. Some conclusion…from Test Tracking Oscillation of and around the ring for the case without CC_Sextupole_Magnets is caused by: (1)Statistical effect (limited number of macro particles) (1)Effect of the fringing field of the bending magnets (‘sextupole’- like effect … will be explained later.

13. Beam centroid motion around MR 3 GeV  100% = 54 .mm.mrad (Gaussian-beam::  RMS = 54/8 .mm.mrad )  p/p = 0 95254 macro_particles S [m] [mm]

14. Beam centroid motion around MR 3 GeV S [m] [mm]  100% = 54 .mm.mrad (Parabolic-beam::  RMS = 54/6 .mm.mrad )  p/p = 0 95254 macro_particles

15. Beam centroid motion around MR 3 GeV RCS-beam (v.050906) ::  RMS x,y =7.190 / 8.325 .mm.mrad  p/p = 0 95254 macro_particles S [m] [mm] Turn_by_turn observation point #OP

16. S [m] [mm] Beam centroid motion around MR  y = 54 .mm.mrad (RCS- beam)  p/p = 0

17. 50 turns S [m] [mm] Beam centroid motion around MR  x = 54 .mm.mrad (Parabolic-beam)  p/p = 0

18. S [m] [mm] Beam centroid motion around MR  y = 54 .mm.mrad (RCS- beam)

19. Beam centroid motion around MR  x = 54 .mm.mrad (Parabolic-beam)

20. Turn_by_Turn changing of the beam centroid location RCS_beam Turn_by_turn observation point #OP Q x = 22.42 2Q x = 45 Turn numbers are presented on Fig.

21. Beam centroid motion around MR 40GeV  x = 6 .mm.mrad (40GeV-beam) [mm] S [m]

22. Beam centroid motion around MR 40GeV  y = 6 .mm.mrad (40GeV- beam) [mm] S [m]

23. Conclusions: (2) At the injection energy of 3 GeV in the case of full linear chromaticity correction the maximum value of the beam centroid beating in the horizontal plane is about 2 mm (for the RCS_beam without any COD). (3) The maximum shift the the beam centroid for MR has been observed at the center of a half of the MR_Arc. (4)The contribution of the dispersion part into the beam centroid beating for MR is negligible in comparison with the amplitude dependent terms. (1) Sextupole magnets, which are used for the chromaticity correction in MR, lead to the beam centroid beating around the ring in the horizontal plane. This is the ‘leading order’ effect of the sextupole field nonlinearity.

24. … to consider … possibility to correct (reduce) the beam centroid shift caused by the sextupole field nonlinearity for MR Possible solutions: … re-arrange the sextupole magnets (SDA) … look at the effect of bump-orbit at the locations where the beam centroid shift is maximum.

25. re-arrange the sextupole magnets (SDA) … Minimum … SDA:  x = 6.5 m,  y = 19.9 m SFA:  x = 16.6 m,  y = 7.78 m Present SDA&SFA location Move SDA to QFX … then …  y (SDA) will be reduced to minimize contribution SDA to the shift of the beam centroid… CHECK !!! Disadvantage … changing of the SDA location at the position with smaller  y will lead to increasing the SDA sextupole strength to keep the chromaticity after correction … CHECK !!! … increasing of the SDA strength could lead to reduction of DA … CHECK !!!

26. MR: Dispersion_SuperPeriod

27. SDA MR: BetaX_ARC SDA SFA  x (SDA) = 6.5m  x (SFA) = 16.6 m

28. MR: BetaY_ARC  y (SDA) = 19.9m  y (SFA) = 7.78 m Drift (SDA_QFX) ~ 2.3 m