1. Resonance correction: -Q x +2Q y =6 for the AP-group Etienne Forest Alexander Molodozhentsev KEK January 12, 2005

2. Dynamic Aperture for RCS 3D_BM & QFF & CC {kL} SEXT – SAD simulation 3 independent families Hotchi san’s data 1000 turns Observation: entrance BM 1 (1) (2) Main limitation of DA (1) is caused by the sextupole field nonlinearity used for the chromaticity correction. Additional contribution to the normal octupole resonance (2). modified 16.12.04

3. Resonance correction - simulation approach 1.To provide the differentiation in s-direction… …representation of the TOSCA 3D field data of the RCS bending magnet by the Gaussian wavelet (Daubechies, 1992)… 2. Normal form analysis… 3.Integrated resonance driving term [-1,2] … definition of the required strength of the sextupole correctors to make zero the cosine and sine parts of the resonance driving term.

4. Single particle tracking: before the resonance [-1,2] correction Gaussian wave-let PTC#3: Q x =6.56817 Q y =6.26662 (… min of beam survival)  p/p=0Observation: #38 (rc6H_02) NEGATIVE_BM X-X / Y-Y / Lost -0.100.10 X 0 =Y 0 =0.028m X / 0 =Y / 0 =0

5. Single particle tracking: before & after the resonance [-1,2] correction PTC#3: Q x =6.56817 Q y =6.26662 (… min of beam survival)  p/p=0 Observation: #38 (rc6H_02) NEGATIVE_BM X-X / Y-Y / White … BEFORE correction; Yellow … AFTER correction X 0 =Y 0 =0.028m X / 0 =Y / 0 =0 Lost Stable

6. Sextupole correctors Required integrated strength of the sextupole correctors: k s L (SC1) = 0.112770195 m -2 k s L (SC2) = 0.113920857 m -2 6 sextupole correctors in the dispersion-free straight sections 2 independent families (SC1 & SC2) L eff = 0.15 m Strength of the sextupole magnets for the chromaticity correction: (k s L) SDA:= -0.319012465119 [m -2 ] SFA:= 0.378052080173 [m -2 ] SDB:= -0.304307265885 [m -2 ]] …definition …

7. DA after correction X 0 =Y 0 = 0.028, 0.035, 0.040, 0.0415, 0.0420, 0.0425 (lost) X / 0 = Y / 0 = 0 PTC#3: Q x =6.56817 Q y =6.26662 (… min of beam survival (X=Y) MAX )  p/p=0 Observation: #38 (rc6H_02) NEGATIVE_BM X-X / Y-Y / -0.10 0.10 -0.002 0.002 1000 turns

8. DA and resonance correction 3D_BM QFF Chrom_Sextupoles RC_Sextupoles (2) Dynamic Acceptance_BEFORE (BM_3D&QFF&CC): A X = (X 0,max ) 2  x ~ 216 .mm.mrad A y = (Y 0,max ) 2  y ~ 187 .mm.mrad (1) Dynamic Acceptance_AFTER (BM_3D&QFF&CC&RSC): A X = (X 0,max ) 2  x ~ 529 .mm.mrad A y = (Y 0,max ) 2  y ~ 423 .mm.mrad (1) (2) H-V coupling X, Y – initial particle coordinates, (X / =Y / =0)

9. DA: On- & Off-momentum 3D_BM QFF Chrom_Sextupoles RC_Sextupoles Dynamic Acceptance_AFTER (dp/p=0.01) (BM_3D&QFF&CC&RSC): A X = (X 0,max ) 2  x ~ 410 .mm.mrad A y = (Y 0,max ) 2  y ~ 301 .mm.mrad

10. Conclusion The 3D-field data can be represented by the Gaussian wavelet to provide the resonance analysis. After the [-1,2] resonance correction the DA has been improved about 2 times for the on- and off-momentum particles. The correction scheme requires moderate strength of the sextupole correctors.