Questions by Francesco 27.04.2006 to Elias, Giovanni and me

1. coherent tune shift due to collimator impedance - its dependence on gap size, bunch length, chromaticity,beta function, conductivity,beam energy, #bunches-thanks to Elias Metral & Javier Resta Lopez Frank Zimmermann, material for LTC

2. Questions by Francesco 27.04.2006 to Elias, Giovanni and me • “Understand/clarify the scaling of the effective LHC collimator • impedance as a function of the collimator gap. • Does a simple scaling law exist?” • → Impact of Running LHC with non-nominal settings • → Rating of ILC IR upgrade options • Questions by Jean-Pierre 23.05.2006 to Ralph and me • Is the Cu collimator an option for the upgrade? • It was the nominal system, rejected for lack of robustness. • Can it be considered again for the upgrade with much higher • beam power? If yes, impedance problem disappears. • 2) What is dominant and should be used for scaling: • single bunch or coupled bunch? Shall I take the tune shift • to be proportional to the bunch charge or to the product • of the bunch charge by the number of bunches? Frank Zimmermann, material for LTC

3. sC=105W-1m-1 impedance of single flat collimator sCu=5.9x107W-1m-1 b=70 m b=6 s s=192 mm d=3 cm 7 TeV B-L theory C Cu C Cu Frank Zimmermann, material for LTC

4. impedance increase for 2x smaller gap carbon factor 23 factor 22 factor 21 factor 23 factor 22 Frank Zimmermann, material for LTC

5. Gaussian weight functions Q’=0 Q’=10 m=0 Q’=10 m=1 Frank Zimmermann, material for LTC

6. coherent coupled-bunch head-tail tune shifts F. Sacherer, 1974 A. Chao, 1993 Frank Zimmermann, material for LTC

7. coh.tune shift – b=70 m, b=6s, 7TeV Frank Zimmermann, material for LTC

8. coh.tune shift – b=70 m, b=6s, 7 TeV half of the modes unstable all modes unstable all modes damped “Ruggiero graph” Frank Zimmermann, material for LTC

9. coh.tune shift – b=70 m, 7 TeV opening the collimators from 6 to 8s reduces tune shift more than 2 times Frank Zimmermann, material for LTC

10. carbon Q’=0, m=0 b=70 m, 7 TeV critical modes: largest DQ or ImDQ injection always 0 or 3564 changes by a few 100 Frank Zimmermann, material for LTC

11. maximum growth rate ImDQ vs. gap for m=0, Q’=0 variation with conductivity 7 TeV ~1/b! fitted curve ~1/b2! where b is the half gap size nearly inversely linear dependence on gap size! Frank Zimmermann, material for LTC

12. maximum growth rate ImDQ vs. gap for m=0, Q’=0 carbon computed growth rates fit result injection fitted curve ~1/b2! where b is the half gap size nearly inversely quadratic dependence on gap size! Frank Zimmermann, material for LTC

13. maximum tune shift |DQ| vs. gap, m=0, Q’=0, b=70 m variation with conductivity Q’=0, m=0 carbon 7 TeV copper ~1/b2.7~1/b3 fitted curves ~1/b2.3~1/b2 Frank Zimmermann, material for LTC

14. maximum tune shift |DQ| vs. gap for m=0, Q’=0 & 10 variation with chromaticity carbon m=0 Q’=0 7 TeV Q’=10 almost ~1/b3! fitted curves Frank Zimmermann, material for LTC

15. maximum tune shift |DQ| vs. gap for m=0, Q’=0 carbon m=0 computed tune shifts Q’=0 fitted curves injection almost ~1/b3! fitted curves Frank Zimmermann, material for LTC

16. maximum tune shift |DQ| vs. gap for m=0 & 1, Q’=5 variation with head-tail mode carbon Q’=5 m=0 m=1 almost ~1/b3! fitted curves Frank Zimmermann, material for LTC

17. maximum tune shift |DQ| vs. gap, m=0, Q’=0, b varying variation with beta function carbon Q’=0, m=0 computed tune shifts b=70m fitted curves b=700m almost ~1/b3! fitted curves Frank Zimmermann, material for LTC

18. tune shift decrease for 10x larger b ~ about factor2 carbon, Q’=0, m=0 Frank Zimmermann, material for LTC

19. maximum tune shift |DQ| vs. gap, m=0, Q’=0, b varying variation with beta function cont’d carbon Q’=0, m=0 gap=6s fitted curves computed tune shifts Frank Zimmermann, material for LTC

20. maximum tune shift |DQ| vs. gap, m=0, Q’=0, b=70 m variation with bunch length Q’=0, m=0 carbon computed tune shifts fitted curves sz=3.77 cm sz=7.55 cm fitted curves ~1/b3! Frank Zimmermann, material for LTC

21. maximum tune shift |DQ| vs. nb, m=0, Q’=0, b=70 m variation with # bunches ~const. fitted curves ~nb! Frank Zimmermann, material for LTC

22. maximum growth rate vs. nb, m=0, Q’=0, b=70 m variation with # bunches ~nb1.4 fitted curves ~nb1.2 Frank Zimmermann, material for LTC

23. maximum tune shift |DQ| vs. gap, m=0, Q’=0, b=70 m variation with bunch length, #bunches, conductivity, bunch charge Q’=0, m=0 computed tune shifts carbon, 2808 bunches, sz=7.55 cm, Nb=1.15x1011 fitted curves copper, 5616 bunches, sz=3.77 cm, Nb=1.7x1011 ~1/b2.7 fitted curves ~1/b2.2 Frank Zimmermann, material for LTC

24. correction factor from nonlinear wake components derived from A. Piwinski’s wake field, in “Impedance of Elliptical Vacuum Chambers,” DESY 94-068, Eq. (52) Frank Zimmermann, material for LTC

25. nonlinear correction vs. gap NLC round beam b/s Frank Zimmermann, material for LTC

26. can we detect the inductive bypass effect with single bunches in the SPS? • for larger opening nonlinear correction is small, but tune shift is small too • if we go very close to integer resonance, classical formula diverges b=4s ge=1.5 mm 270 GeV 1 bunch Frank Zimmermann, material for LTC

27. conclusions • for carbon jaw DQ ~1/b2.75 , for Cu jaw ~1/b2.5 • value for Cu almost 10 times smaller • weak dependence on b: DQ ~1/b0.25 • halving bunch length increases DQ by ~50% • LHC upgrade with half sz, 1.7x1011 ppb, 5616 bunches, and Cu collimators → 1/3 tune shift of nominal LHC with C jaws • correction from nonlinear wake field a few percent for half gaps of 6s or larger Frank Zimmermann, material for LTC