1. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 1 Lecture 5  Magnetic Multipoles Magnetic Multipoles, Magnet Design USPAS, Fort Collins, CO, June 10-21, 2013 Alex Bogacz, Geoff Krafft and Timofey Zolkin

2. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 2 Lecture 5  Magnetic Multipoles Maxwell’s Equations for Magnets - Outline Solutions to Maxwell’s equations for magnetostatic fields: in two dimensions (multipole fields) in three dimensions (fringe fields, end effects, insertion devices...) How to construct multipole fields in two dimensions, using electric currents and magnetic materials, considering idealized situations. A. Wolski, University of Liverpool and the Cockcroft Institute, CAS Specialised Course on Magnets, 2009, http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf USPAS, Hampton, VA, Jan. 17-28, 2011

3. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 3 Lecture 5  Magnetic Multipoles Basis Vector calculus in Cartesian and polar coordinate systems; Stokes’ and Gauss’ theorems Maxwell’s equations and their physical significance Types of magnets commonly used in accelerators. following notation used in: A. Chao and M. Tigner, “Handbook of Accelerator Physics and Engineering,” World Scientific (1999). USPAS, Hampton, VA, Jan. 17-28, 2011

4. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 4 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Maxwell’s equations

5. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 5 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Maxwell’s equations

6. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 6 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Physical interpretation of

7. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 7 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Physical interpretation of

8. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 8 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Linearity and superposition

9. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 9 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields

10. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 10 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields

11. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 11 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields

12. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 12 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields

13. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 13 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields

14. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 14 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields

15. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 15 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Generating multipole fields from a current distribution

16. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 16 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

17. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 17 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

18. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 18 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

19. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 19 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

20. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 20 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

21. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 21 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

22. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 22 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

23. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 23 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

24. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 24 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

25. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 25 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields from a current distribution

26. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 26 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Superconducting quadrupole - collider final focus

27. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 27 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

28. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 28 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

29. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 29 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

30. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 30 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

31. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 31 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

32. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 32 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

33. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 33 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

34. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 34 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

35. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 35 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

36. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 36 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

37. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 37 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Multipole fields in an iron-core magnet

38. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 38 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Generating multipole fields in an iron-core magnet

39. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 39 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Generating multipole fields in an iron-core magnet

40. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 40 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Generating multipole fields in an iron-core magnet

41. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 41 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Generating multipole fields in an iron-core magnet

42. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 42 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Maxwell’s Equations for Magnets - Summary

43. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 43 Lecture 5  Magnetic Multipoles Multipoles in Magnets - Outline Deduce that the symmetry of a magnet imposes constraints on the possible multipole field components, even if we relax the constraints on the material properties and other geometrical properties; Consider different techniques for deriving the multipole field components from measurements of the fields within a magnet; Discuss the solutions to Maxwell’s equations that may be used for describing fields in three dimensions. USPAS, Fort Collins, CO, June 10-21, 2013

44. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 44 Lecture 5  Magnetic Multipoles Previous lecture re-cap USPAS, Fort Collins, CO, June 10-21, 2013

45. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 45 Lecture 5  Magnetic Multipoles Previous lecture re-cap USPAS, Fort Collins, CO, June 10-21, 2013

46. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 46 Lecture 5  Magnetic Multipoles Allowed and forbidden harmonics USPAS, Fort Collins, CO, June 10-21, 2013

47. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 47 Lecture 5  Magnetic Multipoles Allowed and forbidden harmonics USPAS, Fort Collins, CO, June 10-21, 2013

48. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 48 Lecture 5  Magnetic Multipoles Allowed and forbidden harmonics USPAS, Fort Collins, CO, June 10-21, 2013

49. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 49 Lecture 5  Magnetic Multipoles Allowed and forbidden harmonics USPAS, Fort Collins, CO, June 10-21, 2013

50. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 50 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Allowed and forbidden harmonics

51. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 51 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Allowed and forbidden harmonics

52. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 52 Lecture 5  Magnetic Multipoles Measuring multipoles USPAS, Fort Collins, CO, June 10-21, 2013

53. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 53 Lecture 5  Magnetic Multipoles Measuring multipoles in Cartesian basis USPAS, Fort Collins, CO, June 10-21, 2013

54. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 54 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Measuring multipoles in Cartesian basis

55. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 55 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Measuring multipoles in Polar basis

56. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 56 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Measuring multipoles in Polar basis

57. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 57 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Measuring multipoles in Polar basis

58. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 58 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Advantages of mode decompositions

59. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 59 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Three-dimensional fields

60. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 60 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Three-dimensional fields

61. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 61 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Three-dimensional fields

62. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 62 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Three-dimensional fields

63. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 63 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Three-dimensional fields

64. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 64 Lecture 5  Magnetic Multipoles Symmetries in multipole magnets restrict the multipole components that can be present in the field. It is useful to be able to find the multipole components in a given field from numerical field data: but this must be done carefully, if the results are to be accurate. Usually, it is advisable to calculate multipole components using field data on a surface enclosing the region of interest: any errors or residuals will decrease exponentially within that region, away from the boundary. Outside the boundary, residuals will increase exponentially. Techniques for finding multipole components in two dimensional fields can be generalized to three dimensions, allowing analysis of fringe fields and insertion devices. In two or three dimensions, it is possible to use a Cartesian basis for the field modes; but a polar basis is sometimes more convenient. USPAS, Fort Collins, CO, June 10-21, 2013 Summary – Part II

65. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 65 Lecture 5  Magnetic Multipoles Appendix A - Field Error Tolerances Focusing ‘point’ error perturbs the betatron motion leading to the Courant-Snyder invariant change: Beam envelope and beta-function oscillate at double the betatron frequency USPAS, Fort Collins, CO, June 10-21, 2013

66. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 66 Lecture 5  Magnetic Multipoles Single point mismatch as measured by the Courant-Snyder invariant change: here, m =1 quadrupole, m =2 sextupole, m=3 octupole, etc Each source of field error (magnet) contributes the following Courant-Snyder variation USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A - Field Error Tolerances

67. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 67 Lecture 5  Magnetic Multipoles multipole expansion coefficients of the azimuthal magnetic field, B  - Fourier series representation in polar coordinates at a given point along the trajectory): multipole gradient and integrated geometric gradient : USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A - Field Error Tolerances

68. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 68 Lecture 5  Magnetic Multipoles Cumulative mismatch along the lattice (N sources): Standard deviation of the Courant-Snyder invariant is given by: Assuming weakly focusing lattice (uniform beta modulation) the following averaging (over the betatron phase) can by applied: USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A - Field Error Tolerances

69. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 69 Lecture 5  Magnetic Multipoles Some useful integrals …. : will reduce the coherent contribution to the C-S variance as follows: Including the first five multipoles yields: USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A - Field Error Tolerances

70. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 70 Lecture 5  Magnetic Multipoles Beam radius at a given magnet is : One can define a ‘good fileld radius’ for a given type of magnet as: Assuming the same multipole content for all magnets in the class one gets: The first factor purely depends on the beamline optics (focusing), while the second one describes field tolerance (nonlinearities) of the magnets: USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A - Field Error Tolerances

71. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 71 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 A scalar potential description of the magnetic field has been very useful to derive the shape for the pole face of a multipole magnet. Appendix B - The vector potential

72. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 72 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

73. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 73 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

74. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 74 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

75. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 75 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

76. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 76 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

77. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 77 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

78. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 78 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

79. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 79 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

80. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 80 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential

81. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 81 Lecture 5  Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B - The vector potential