Sextupole effect for MR (beam centroid beating)

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

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 As ~ x·y Ax = Ay =0 Vector potential for the quadrupole field Single particle transverse kick from thin quadrupole x/s ~ x y/s ~ y Multi-particle transverse kick from thin quadrupole < x/s > ~ <x> = 0 < y/s > ~ < y > = 0 without COD

4. Sextupole field (simplified model) As = 1/6 B// (x3 – 3xy2) Ax = Ay =0 Vector potential for the sextupole field Single particle transverse kick from thin sextupole x/s ~ (a x2 – b y2) y/s ~ y Multi-particle transverse kick from thin sextupole < x/s > ~ (a <x2> – b <y2>) ~ ( a x – b y ) < y/s > ~ < y > = 0 without COD

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: <x> ~ 15 m, <x> ~ 1.5 m  = p/p =  0.004 x = y = 54  mm.mrad “Dispersion” term … ~ 72  10-6 “Amplitude Dependent” term (X) … ~ 784  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 (<X> & <Y>) around the ring. MR:: • Working point :: Qx = 22.428, Qy = 20.82 • Transverse particle distribution :: 3GeV … 54  mm.mrad (Parabolic, max=6) 40 GeV … 6  mm.mrad

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

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

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

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

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

13. Beam centroid motion around MR3 GeV x = 54 .mm.mrad (RCS-beam) p/p = 0 <x> [mm] S [m] 95254 macro_particles

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

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

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

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

18. Beam centroid motion around MR40GeV x = 6 .mm.mrad (40GeV-beam) <x> [mm] S [m]

19. Beam centroid motion around MR40GeV y = 6 .mm.mrad (40GeV-beam) <y> [mm] S [m]

20. Conclusions: • 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. 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 case without any COD). The maximum shift the the beam centroid for MR has been observed at the center of a half of the MR_Arc. The contribution of the dispersion part into the beam centroid beating for MR is negligible in comparison with the amplitude dependent terms.

21. … 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.

22. MR: Dispersion_SuperPeriod

23. MR: BetaXY_SuperPeriod