1. Multipole Magnets from Maxwell’s Equations
Alex Bogacz
USPAS, Hampton, VA, Jan , 2011

2. Maxwell’s Equations for Magnets - Outline
Solutions to Maxwell’s equations for magneto static fields:
in two dimensions (multipole fields)
in three dimensions (fringe fields, insertion devices...)
How to construct multipole fields in two dimensions, using electric currents and magnetic materials, considering idealized situations.
A. Wolski, Academic Lectures, University of Liverpool , 2008
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3. 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).
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4. Maxwell’s equations
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5. Maxwell’s equations
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6. Physical interpretation of
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7. Physical interpretation of
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8. Linearity and superposition
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9. Multipole fields
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10. Multipole fields
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11. Multipole fields
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12. Multipole fields
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13. Multipole fields
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14. Multipole fields
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15. Generating multipole fields from a current distribution
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16. Multipole fields from a current distribution
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17. Multipole fields from a current distribution
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18. Multipole fields from a current distribution
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19. Multipole fields from a current distribution
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20. Multipole fields from a current distribution
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21. Multipole fields from a current distribution
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22. Multipole fields from a current distribution
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23. Multipole fields from a current distribution
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24. Multipole fields from a current distribution
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25. Multipole fields from a current distribution
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26. Superconducting quadrupole - collider final focus
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27. Multipole fields in an iron-core magnet
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28. Multipole fields in an iron-core magnet
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29. Multipole fields in an iron-core magnet
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30. Multipole fields in an iron-core magnet
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31. Multipole fields in an iron-core magnet
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32. Multipole fields in an iron-core magnet
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33. Multipole fields in an iron-core magnet
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34. Multipole fields in an iron-core magnet
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35. Multipole fields in an iron-core magnet
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36. Multipole fields in an iron-core magnet
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37. Multipole fields in an iron-core magnet
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38. Generating multipole fields in an iron-core magnet
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39. Generating multipole fields in an iron-core magnet
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40. Generating multipole fields in an iron-core magnet
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41. Generating multipole fields in an iron-core magnet
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42. Maxwell’s Equations for Magnets - Summary
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43. Maxwell’s Equations for Magnets - Summary
Maxwell’s equations impose strong constraints on magnetic fields that may exist.
The linearity of Maxwell’s equations means that complicated fields may be expressed as a superposition of simpler fields.
In two dimensions, it is convenient to represent fields as a superposition of multipole fields.
Multipole fields may be generated by sinusoidal current distributions on a cylinder bounding the region of interest.
In regions without electric currents, the magnetic field may be derived as the gradient of a scalar potential.
The scalar potential is constant on the surface of a material with infinite permeability. This property is useful for defining the shapes of iron poles in multipole magnets.
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44. 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.
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45. Previous lecture re-cap
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46. Previous lecture re-cap
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47. Allowed and forbidden harmonics
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48. Allowed and forbidden harmonics
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49. Allowed and forbidden harmonics
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50. Allowed and forbidden harmonics
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51. Allowed and forbidden harmonics
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52. Allowed and forbidden harmonics
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53. Measuring multipoles
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54. Measuring multipoles in Cartesian basis
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55. Measuring multipoles in Cartesian basis
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56. Measuring multipoles in Polar basis
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57. Measuring multipoles in Polar basis
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58. Measuring multipoles in Polar basis
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59. Advantages of mode decompositions
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60. Three-dimensional fields
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61. Three-dimensional fields
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62. Three-dimensional fields
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63. Three-dimensional fields
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64. Three-dimensional fields
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65. Measuring multipoles in Polar basis
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66. Multipoles in Polar basis – including x-dependence
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67. Three-dimensional fields
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68. Three-dimensional fields - Cylindrical basis
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69. Three-dimensional fields - Cylindrical basis
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70. Three-dimensional fields - Cylindrical basis
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71. Three-dimensional fields - Cylindrical basis
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72. Three-dimensional fields - Cylindrical basis
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73. Three-dimensional fields - Cylindrical basis
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74. Summary – Part II
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.
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75. Appendix A - The vector potential
A scalar potential description of the magnetic field has been very useful to derive the shape for the pole face of a multipole magnet.
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76. Appendix A - The vector potential
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77. Appendix A - The vector potential
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78. Appendix A - The vector potential
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79. The vector potential
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80. The vector potential
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81. The vector potential
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82. The vector potential
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83. The vector potential
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84. Appendix A - The vector potential
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85. Appendix A - The vector potential
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86. Appendix B - 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
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87. Appendix B - Field Error Tolerances
Single point mismatch as measured by the Courant-Snyder invariant change:
Each source of field error (magnet) contributes the following Courant-Snyder variation
here, m =1 quadrupole, m =2 sextupole, m=3 octupole, etc
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88. Appendix B - Field Error Tolerances
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:
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89. Appendix B - Field Error Tolerances
Some useful integrals …. :
will reduce the coherent contribution to the C-S variance as follows:
Including the first five multipoles yields:
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90. Appendix B - Field Error Tolerances
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:
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91. Appendix B - Field Error Tolerances
Standard deviation of the Courant-Snyder invariant:
where:
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92. Appendix B - Field Error Tolerances
multipole expansion coefficients of the azimuthal magnetic field, Bq - Fourier series representation in polar coordinates at a given point along the trajectory):
multipole gradient and integrated geometric gradient:
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93. Appendix B - Field Error Tolerances
The linear errors, m =1, cause the betatron mismatch – invariant ellipse distortion from the design ellipse without changing its area – no emittance increase.
By design, one can tolerate some level (e.g. 10%) of Arc-to-Arc betatron mismatch due to the focusing errors, df1 (quad gradient errors and dipole body gradient) to be compensated by the dedicated matching quads
The higher, m > 1, multipoles will contribute to the emittance dilution – ‘limited’ by design via a separate allowance per each segment (Arc, linac) (e.g. 1%)
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94. Appendix B - Field Error Tolerances
Here one assumes the following multipole content for the dipoles and quads:
Quads: sextupole (m = 2), octupole (m = 3), duodecapole (m = 5) and icosapole (m = 9)
Dipoles: sextupole (m = 2) and decapole (m = 4)
The values of multipoles are calculated in the extreme case – a given order (m) multipole by itself exhausts the emittance dilution allowance of 1%.
One can use the above analytic formalism to set the magnet error tolerances for specific groups (types) of magnets (dipoles and quads) within each lattice segment (Arc, linac) For each group of magnets within each segment one needs to evaluate:
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95. Cumulative C-S invariant change due to magnet errors
Assuming the same multipole content for all magnets in the class one gets:
C-S mis-match
emittance dilution
‘Beam region’ defined by :
The first factor purely depends on the beamline optics (focusing), while the second one describes field errors of the magnets
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