1. Correction of sextupole resonances

2. w.p. (6.36,6.22) - N=0.6*10^14, Correction of 3Qx=19 at low intensity Total emittance pi mm mrad % outside blue – no errors red – errors, no correction green – correction of islands

3. w.p. (6.36,6.22) – N=2.0*10^14 Blue – no sextupole errors, Grey – sextupole errors, no correction Green – sextupole errors, 3Qx=19 corrected

4. w.p. (6.36,6.22) 3 rd order resonances excited by sextupole errors For intensity >1*10^14 requires simultaneous correction of 3Qx=19 and 2Qy-Qx=6 resonances

5. w.p. (6.36,6.22) – N=2.0*10^14, simultaneous correction of 3Qx=19 and 2Qy-Qx=6 resonances Blue – no errors Grey – errors, no correction Green- errors, correction of 3Qx=19 Pink – errors, simultaneous correction of 3Qx=19 and 2Qy-Qx=6

6. Towards VERY high intensity, w.p. and crossed resonances No that we found that we can have good correction in the presence of space charge, we want to see whether we can push intensity very high by considering the w.p. far away from coherent structure resonance at integer. We then take (6.4,6.3) and perform simultaneous correction of Qx+2Qy=19 sum resonance and 3Qx=19 with only sextupole errors being introduced.

7. w.p. (6.4,6.3) – towards very high intensity, Simultaneous correction of both sum coupling and 1D resonance blueblue Blue – no errors, Red – errors, no correction, N=2*10^14 Green – errors, correction of Qx+2Qy=19 and 3Qx=19 resonances, N=2*10^14 Pink – errors, correction of Qx+2Qy=19 and 3Qx=19 resonances, N=3*10^14 Grey – errors, correction of Qx+2Qy=19 and 3Qx=19 resonances, N=4*10^14

8. Correction of sextupole resonances: w.p. (6.36,6.22) - Correction of 3Qx=19 and 2Qy-Qx=6 Total emittance pi mm mrad % outside N=0.6*10^14 blue – no errors red – errors, no correction green – errors, correction of 3Qx=19 N=2*10^14 blue – no errors grey – errors, no correction green – errors, correction of 3Qx=19 pink – errors, simultaneous correction of 3Qx=19 and 2Qy-Qx=6 resonances 3Qx 2Qy-Qx Qx Qy

9. Correction of sextupole resonances: w.p. (6.4,6.3) - Correction of sum coupling resonance Qx+2Qy=19 and 3Qx=19 resonance N=0.6*10^14 blue- no errors red – errors, no correction pink – errors, correction of Qx+2Qy=19 N=2.0*10^14 blue- no errors red- errors, no correction green- errors, simultaneous corrections of Qx+2Qy=19 and 3Qx=19 resonances N=3.0*10^14 pink – errors, corrections of Qx+2Qy=19 and 3Qx=19 resonances Total emittance pi mm mrad % outside

10. Tentative conclusions (based on correction of sextupole resonances – March 2003) For the test cases of local errors, we found out that detailed correction of resonance without space charge gives good results for the case when such resonance are crossed at high intensity. Strength of correctors used is well within the range. These studies suggest important conclusions: For high intensity operation one can chose working point above the dangerous nonlinear resonance and cross these resonance with relatively low losses at high intensity, provided that one did careful correction of these resonance – it was demonstrated that it is possible to do so.

11. Correction of octupole resonances: w.p. (6.36,6.22), N=1*10^14, Correction of octupole resonance 2Qx+2Qy=25 blue – no errors red – b3=60 units, no correction green – b3=60 units plus correction of 2Qx+2Qy=25 resonance

12. Correction of octupole resonances: w.p. (6.36,6.22)- N=1*10^14 correction of strong octupole resonance 2Qx+2Qy (b3=120 units) blue – no errors gray – b3=120 units, no correction pink – b3=120 units plus correction of 2Qx+2Qy resonance

13. Correction of sextupole and octupole resonance simultaneously: w.p. (6.36,6.22) – N=2*10^14 blue – no errors red – b2=30 units, b3=60 units, no correction green – simultaneous correction of sextupole: 3Qx=19, 2Qy-Qx=6 and octupole: 2Qx+2Qy resonances

14. Correction of skew octupole resonances We do not have skew octupole correctors. However, these resonances are there. For completeness, we thus explore their importance and their correction.

15. Correction of skew octupole resonance: w.p. (6.36, 6.22) – Qx+3Qy=25 –sum resonances red – a3 errors, no correction green – a3 errors plus skew-octupole correction

16. Effective strength of resonances Approximate strength of resonances for the case of lumped errors in units of Ks of Qx+2Qy=19 resonance: Sext. (1,2)=19 –> Ks Skew sext. (2,1)=19 -> Ks Oct. (0,4)=25 -> 0.3*Ks Skew oct. (1,3)=25 -> 0.7*Ks Oct. (2,2)=25 -> 0.9*Ks Sext. (3,0)=19 -> 0.6*Ks Skew Oct. (3,1)=25 -> 0.7*Ks Sext. (1,-2)=6 -> Ks Skew oct. (3,-1)=12 -> 0.7*Ks

17. Error budget In resonance correction studies we assumed lumped b2=30 units and b3, a3 = 60 units which corresponds to 1)5 times stronger random sextupole errors than measured 2)10 times stronger skew octupole errors than measured With the present error budget coming just from arc dipoles and arc quadrupoles the dominant contribution comes from normal sextupole and normal octupole resonances – good correction. If the budget of skew octupole errors becomes factor of 5 stronger than the present - one will need skew octupole correctors.