1. 1 BROOKHAVEN SCIENCE ASSOCIATES Considerations for Nonlinear Beam Dynamics in NSLS-II lattice design Weiming Guo 05/26/08 Acknowledgement: J. Bengtsson S. Kramer S. Krinsky B. Nash

2. 2 BROOKHAVEN SCIENCE ASSOCIATES Outline Method: Simulation using Elegant 1. Introduction to the linear lattice 2. Nonlinear requirements: injection and Toschek scattering 3. Minimum needed sextupole families 4. Required physical aperture 5. Effects of the multipole field errors Subjects will not be covered: On-going optimization of the nonlinear lattice; Misalignment errors; choice of the working point; and effects of the damping wigglers and IDs

3. 3 BROOKHAVEN SCIENCE ASSOCIATES Main Design Parameters Beam Energy 3 GeV Circumference792 m Circulating current 500 mA Total number of buckets1320 Lifetime3 hours Electron beam stability10% beam size Top-off injection current stability<1% ID straights for undulators>21 Straight length9.3/6.6 m

4. 4 BROOKHAVEN SCIENCE ASSOCIATES Magnets and lattice Quad. Family: 8, 10 magnets per cell Sext. Family: 9, 10 magnets per cell Chromaticity per cell: -3.4/-1.4 Natural emittance with 0, 1, 2, 3, 5 and 8 DWs added Courtesy of S. Kramer

5. 5 BROOKHAVEN SCIENCE ASSOCIATES Linear Lattice design approaches 1. Large bending radius enhances the effect of the damping wigglers 2. Strict double-bend-acromat Two reasons NSLS-II is sensitive to the residual dispersion: 1nm emittance. For β x =2 m, η x =4.5 cm, β x ε x ~(η x σ δ ) 2 Damping wiggler has quantumn excitation effect

6. 6 BROOKHAVEN SCIENCE ASSOCIATES Nonlinear Design Goal 1. Sufficient dynamic aperture (>11mm) for injection On momentum particle Septum Bumped stored beam Injected beam (σ x =0.39 mm, σ´ x =77 μrad) 3+1 mm3.5 mm1+1.2 mm (σ x =0.15 mm) 2. Sufficient dynamic aperture to keep Touschek scattered particles with δ= ±2.5% Off momentum particle Minimum required dynamic aperture δ Courtesy of I. Pinayev

7. 7 BROOKHAVEN SCIENCE ASSOCIATES Source of the nonlinearity Frist order chromatic terms (5) Frist order geometric terms (5) Amplitude tune dependence Second order chromaticity

8. 8 BROOKHAVEN SCIENCE ASSOCIATES Number of chromatic sextupole families Conclusion: Because the betatron phase advance in the dispersive region is small (x: ~ 10 o, y~ 20 o ), the five chromatic terms have only two degree of freedom.  7~8 sextupole families

9. 9 BROOKHAVEN SCIENCE ASSOCIATES Stay-clear aperture 38x12.5 mm

10. 10 BROOKHAVEN SCIENCE ASSOCIATES Dynamic Aperture with Multipole Error Magnet Type Magnet Aperture (mm) Multipole Order Relative Strength (x10 -4 ) Quad6661 Quad66103 Sext6891 Sext68152 DA collapse at negative momentum The Feb.-08 multipole components at 25 mm

11. 11 BROOKHAVEN SCIENCE ASSOCIATES Effects of Multipoles Conclusion: introducing systematic multipole errors doesn’t add resonance lines.

12. 12 BROOKHAVEN SCIENCE ASSOCIATES Amplitude Tune Dependence Conclusion: Multipole terms changes the tunes at large amplitude dramatically, and makes the stable area smaller.

13. 13 BROOKHAVEN SCIENCE ASSOCIATES Why Negative Momentum?

14. 14 BROOKHAVEN SCIENCE ASSOCIATES Minimizing 2 nd Order Dispersion

15. 15 BROOKHAVEN SCIENCE ASSOCIATES Tune Scan Conclusion: moving the tune doesn’t solve this problem. Delta = -3.0% Delta = 3.0% Delta = 0.0% Delta = -3.0% Area of DA

16. 16 BROOKHAVEN SCIENCE ASSOCIATES Squeeze the Tune Excursion Tunes moved, sextupoles reoptimized CDR lattice with multipole errors Red: Jan4th lattice Black: re-tuned lattice The re-tuned lattice

17. 17 BROOKHAVEN SCIENCE ASSOCIATES New Specifications Magnet TypeMagnet Aperture (mm) Multipole OrderRelative Strength (x10 -4 ) Quad9061 Quad90100.5 Quad90140.1 Sext7690.5 Sext76150.5 Sext76210.5 Multipole components of high precision magnets (25 mm)

18. 18 BROOKHAVEN SCIENCE ASSOCIATES Summary Nonlinear design goal to provide sufficient dynamic aperture for injection and Touschek scattered particles up to δ=2.5% There are 8 sextupole families in the lattice, one more might be added. The physical aperture is conservatively retained. The reduction of the dynamic aperture due to multipole errors is show to be the momentum related tune shift with amplitude. Identified the multipole errors in the high dispersion region, and required increasing the pole radius. Further optimization of the dynamic aperture is proceeding.