Analysis of a proposal for the design of the CLIC damping rings wigglers

1. Analysis of a proposal for the design of the CLIC damping rings wigglers Simona Bettoni, Remo Maccaferri

2. Outline • Introduction • The model • 2D (Poisson) • 3D (Opera Vector Fields-Tosca) • The analysis tools • Field uniformity • Multipoles (axis and trajectory) • Tracking studies • The integrals of motion cancellation • Possible options • The final proposal • The prototype analysis • Method to reduce the integrated multipoles • Conclusions

3. Damping ring layout

4. Wigglers/undulators model Large gap & long period Small gap & short period

5. 2D design (R. Maccaferri) • Advantages: • Short period • Small forces on the heads (curved) BEAM

6. The 3D model

7. The 3D model (base plane)

8. The 3D model (extrusions)

9. The 3D model (conductors) Conductors generated using a Matlab script. Grouping of the conductors. • Parameters the script: • wire geometry (l_h, l_v, l_trasv) • winding “shape” (n_layers, crossing positions)

10. The analysis tools • Tracking analysis: • Single passage: ready/done • Multipassage: to be implemented • Field uniformity: ready/done • Multipolar analysis: • Around the axis: ready /done • Around the reference trajectory: ready x and x’ at the exit of the wiggler

11. Prototype analysis (CLIC_Wiggler_7.op3) z x y

12. Field distribution on the conductors BMod (Gauss) • Maximum field and forces (PMAX ~32 MPa) on the straight part • Manufacture: well below the limit of the maximum P for Nb3Sn • Simulation: quick to optimize the margin

13. The 2D/3D comparison 1.9260 T 2D (Poisson) -2.1080 T 1.9448 T 3D (Tosca) -2.1258 T

14. Field uniformity (x range = ±2 cm) z (cm)

15. Multipolar analysis (x range = ±2 cm)

16. Multipolar analysis (x range = ±2 cm)

17. Multipolar analysis (x range = ±2 cm)

18. Tracking studies Trajectory x-shift at the entrance = ± 3 cm z x y

19. Tracking studies: the exit position Subtracting the linear part

20. Tracking studies: the exit angle

21. Integrals of motion = 0 for anti-symmetry 1st integral 2nd integral Offset of the oscillation axis CLIC case (even number of poles anti-symmetric) No offset of the oscillation axis

22. Integrals of motion: the starting point = 0 for anti-symmetry 1st integral 2nd integral (cm)

23. Lowering the 2nd integral: what do we have to do? To save time we can do tracking studies in 2D up to a precision of the order of the difference in the trajectory corresponding to the 2D/3D one (~25 mm) and only after refine in 3D.

24. Lowering the 2nd integral: how can we do? → Highly saturated → → • What we can use: • End of the yoke length/height • Height of the yoke • Terminal pole height (|B| > 5 T) • Effectiveness of the conductors

25. Lowering the 2nd integral: option 1

26. The multipoles of the option 1 CLICWiggler7.op3 CLICWiggler8.op3

27. Lowering the 2nd integral: option 2 (2D)

28. Option 1 vs option 2 • The “advantage” of the option 2: • Perfect cancellation of the 2nd integral • Field well confined in the yoke • Possibility to use only one IN and one OUT (prototype) • The “disadvantage” of the option 2: • Comments? • The “advantage” of the option 1: • Easy to be done • The “disadvantage” of the option 1: • No perfect cancellation of the 2nd integral • Field not completely confined in the yoke • Multipoles get worse → start → → → end → 1st layers (~1/3 A*spire equivalent) All the rest

29. Lowering the 2nd integral: option 2 (3D) If only one IN and one OUT → discrete tuning in the prototype model Fine regulation would be possible in the long model and in the DR (modular)

30. Tracking studies (optimized configuration) Not optimized Optimized

31. Working point: Nb3Sn & NbTi Wire diameter (insulated) = 1 mm Wire diameter (bare) = 0.8 mm Non-Cu fraction = 0.53 Cu/SC ratio = 1 * Nb3Sn NbTi Nb3Sn NbTi *MANUFACTURE AND TEST OF A SMALL CERAMIC-INSULATED Nb3Sn SPLIT SOLENOID, B. Bordini et al., EPAC’08 Proceedings.

32. Possible configurations Possible to increase the peak field of 0.5 T using holmium (Remo), BUT $

33. Working point: comparison

34. Short prototype status & scheduling

35. Reduction of the integrated multipoles S. Bettoni, Reduction of the integrated odd multipoles in periodic magnets, PRST-AB, 10, 042401 (2007), S. Bettoni et al., Reduction of the Non-Linearities in the DAPHNE Main Rings Wigglers, PAC’07 Proceedings.

36. The integrated multipoles in periodic magnets In a displaced system of reference: y y’ xT bAk → defined in the reference centered in OA (wiggler axis) bTk → defined in the reference centered inOT (beam trajectory) O T OA x x’ Even multipoles → Left-right symmetry of the magnet Multipoles change sign from a pole to the next one Sum from a pole to the next one Odd multipoles →

37. The displacement of the magnetic field axis WITHOUT THE POLE MODIFICATION In each semiperiod the particle trajectory is always on one side with respect the magnetic axis Octupole ↑ WITH THE POLE MODIFICATION Opportunely choosing the B axis is in principle possible to make zero the integrated octupole in each semiperiod In each semiperiod the particle travels on both sides with respect to the magnetic axis

38. The application to the DAFNE main rings wigglers Excursion of ±1.3 cm with respect to the axis of the wiggler

39. The results

40. Conclusions • A novel design for the CLIC damping ring has been analyzed (2D & 3D) • Advantages: • Possibility to have a very small period wiggler • Small forces on the heads • Analysis on the prototype: • Maximum force • Multipolar analysis • Tracking studies • Zeroing the integrals of motion • A method to compensate the integrated multipoles has been presented • Even multipoles cancel from a pole to the next one and odd multipoles canceled by the opportune magnetic axis displacement • How to proceed • Optimization of the complete wiggler model (work in progress): • Best working point definition, if not already (margin) • Modeling of the long wiggler • 2nd integral optimization for the long model • Same analysis tools applied to the prototype model (forces, multipoles axis/trajectory, tracking) • Minimization of the integrated multipoles

41. Extra slides

42. Longitudinal field (By = f(y), several x) • Scan varying the entering position in horizontal, variation in vertical: • Dz = 0.1 mm for x-range = ±1 cm • Dz = 2 mm for x-range = ±2 cm

43. Horizontal transverse field (Bx = f(y), several x) • Scan varying the entering position in horizontal, variation in vertical: • Dz = 0.1 mm for x-range = ±1 cm • Dz = 2 mm for x-range = ±2 cm

44. Controlling the y-shift: cancel the residuals W1 W2 W3 W4 W1 W2 W3 W4 2 mm in 10 cm -> 20*2 = 40 mm in 2 m

45. Controlling the x-shift: cancel the residuals (during the operation) Quadrupoles very close to the beginning of the wiggler or at half distance? W1 W2 … Entering at x = 0 cm Entering at x = -DxMAX/2 • Entering at x = +DxMAX/2 (opposite I wiggler … positron used for trick)

46. The fit accuracy: an example

47. Field uniformity (x-range = ±3 cm)

48. Multipolar analysis (x-range = ±3 cm)

49. Tracking at x-range = ±3 cm: exit position Subctracting the linear part

50. Tracking at x-range = ±3 cm: exit angle