1. Study of Slow Extraction Relevant to the FAIR Project Markus Kirk Fair-Synchrotrons Group GSI mbH Blockseminar, Riezlern, 8 th March 2007

2. Overview 1)Theoretical introduction to resonant extraction 2)Technical aspects of the RF „Knock Out“ method 3)Power supply ripple and its effect on the spill 4)Summary and outlook

3. Effect of a sextupole perturbation on the linear motion Equation of motion (horizontal “sheet” beam) Theory – 3rd integer resonance where the B-field of the sextupole is The focussing strength K(s) changes along the lattice. We wish remove this variation by a change of variables… Taken from G. Rees, CERN school, Vol. II

4. …normalized coordinates. Viz., Theory – 3rd integer resonance Considering just the nth harmonic of the driving term Equation of motion can be reduced to SHM and sextupole driving term

5. Change variables again to enable us to find the unstable fixed points, i.e. (u,p)  (r,  ) Theory – 3rd integer resonance Thus eventually obtaining The unstable fixed points (A,B,C in figure) are given by Which yield the conditions on r and  …

6. Theory – 3rd integer resonance … For optimum condition,, one obtains which should be greater than the thickness of the septum wires 0.1mm thick. Furthermore, it may be shown that the area of the separatrix is given by Thereby allowing the jump in x per turn to be calculated Simulation of SIS-300 where

7. Excitation of the resonance Aim: with as little power as possible create a strong enough resonance whereby we do not shift away from the natural chromaticity. A suitable configuration is to alternate the sextupole strengths according to which follows from where subscript variables denote the i th sextupole in the j th arc the change in chromaticity due to a sextupole

8. SIS-300 lattice ~12m SX11 SX12 SV SH SV SH Quadrupole layout modified to accomodate for injection (below)

9. SIS-300 lattice – beta function in x

10. SIS-300 lattice – beta function in y

11. SIS-300 lattice – dispersion in x

12. Magnet parameters of SIS-300 Dipoles 12 short dipoles, 24 long Bending radius 67 m Max. rigidity 300 Tm Physical field length 7.76 m (3.88 m short) Pole gap 1 m Quadrupoles FODO structure 40 focusing and 38 defocusing (2 super periods) Max. B’=45 T/m Physical/effect field length 1.0/0.9 m Pole radius 50 mm Also…4 long F-quads and 4 D-quads (3m each) Fast quadrupole max. 30T/m, 0.5 m effective length, pole radius 55 mm

13. Magnet parameters of SIS-300 Sextupoles Resonance excitation 2 per straight, 12 in total Max. B’’=750 T/m 2 Physical/effect field length 0.75/0.75 m Pole radius 47.5 mm Chomaticity control 4 per arc (V-H-H-V), 24 in total Max. B’’=750 T/m 2 Physical/effect field length 0.75/0.75 m Pole radius 47.5 mm

14. Ripple in power supplies Time variation in tune of a bunched beam subject to ripple from the power supplies to the quadrupoles Therefore area of separatrix will also oscillate (effect of ripple in sextupoles is much smaller) Thus, to minimize sensitivity to ripple in the quadrupoles, extract with as high S as possible without distortion to the separatrix. where

15. RF exciter - parameters Technical details Stripline electrodes in horizontal plane Pseudo random noise from a maximum length sequence register RF carrier digitally modulated with Binary Phase Shift Keying Carrier frequency at Q f f 0 Full width of centre lobe in spectrum 2f bit Spectrum

16. RF K.O./BTF system architecture In operation in SIS-18 P. Moritz, GSI

17. Simulation – Particle tracking Micromap library (Franchetti, GSI) Thin lens sextupoles Thick lens quadrupoles and dipoles Closed orbit bump made with thin dipoles Dispersion and chromaticity treated exactly Kicks from RF exciter treated as thin elements Sextupole strengths for resonance control calculated from the amplitude and phase RF exciter amplitude modulation with commissioned ramps

18. Kick from RF exciter Thin element approximation, viz. where

19. Ripple in power supplies RF-Gap (kV)0 dp/p| HW  1.6E-5 Chromaticities(-1.49,~1.4) Tunes(13.327,9.289) SIS-300: Tune sensitivity to focussing strength K 1

20. Ripple in power supplies SIS-300: Spill under ideal conditions dI/I in quads0 Gap-Voltage (kV)0 dp/p| HW  1.6E-5 xx -1.49

21. Ripple in power supplies SIS-300: Spill degradation due to ripple dI/I in quads5E-5 Gap-Voltage (kV)0 dp/p| HW  1.6E-5 xx -1.49 T ripple

22. Ripple in power supplies RF-Gap (kV)0 dp/p| HW  1.6E-5 Chromaticities(-1.49,~1.4) Tunes(13.327,9.289) SIS-300: Systematic survey on ripple amplitude

23. Summary and outlook Summary Slow extraction model with RF-K.O. developed Systematic survey of ripple in quadrupoles undertaken Outlook Hardt condition Stochastic extraction Introduction of B-clock (all rings) Acknowledgements: P. Spiller, N. Pyka, J. Stadlmann, G. Franchetti from GSI Darmstadt