Magnetic Multipoles, Magnet Design Alex Bogacz, Geoff Krafft and Timofey Zolkin
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Magnetic Multipoles, Magnet Design Alex Bogacz, Geoff Krafft and Timofey Zolkin Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 1 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 Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Hampton, VA, Jan. 17-28, 2011 2 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). Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Hampton, VA, Jan. 17-28, 2011 3 Maxwell’s equations Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 4 Maxwell’s equations Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 5 Physical interpretation of Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 6 Physical interpretation of Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 7 Linearity and superposition Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 8 Multipole fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 9 Multipole fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 10 Multipole fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 11 Multipole fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 12 Multipole fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 13 Multipole fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 14 Generating multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 15 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 16 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 17 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 18 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 19 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 20 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 21 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 22 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 23 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 24 Multipole fields from a current distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 25 Superconducting quadrupole - collider final focus Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 26 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic MultipolesUSPAS, Fort Collins, CO, June 10-21, 2013 27 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 28 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 29 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 30 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 31 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 32 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 33 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 34 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 35 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic MultipolesUSPAS, Fort Collins, CO, June 10-21, 2013 36 Multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 37 Generating multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 38 Generating multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 39 Generating multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 40 Generating multipole fields in an iron-core magnet Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 41 Maxwell’s Equations for Magnets - Summary Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 42 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. Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 43 Previous lecture re-cap Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 44 Previous lecture re-cap Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 45 Allowed and forbidden harmonics Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 46 Allowed and forbidden harmonics Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 47 Allowed and forbidden harmonics Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 48 Allowed and forbidden harmonics Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 49 Allowed and forbidden harmonics Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 50 Allowed and forbidden harmonics Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 51 Measuring multipoles Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 52 Measuring multipoles in Cartesian basis Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 53 Measuring multipoles in Cartesian basis Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 54 Measuring multipoles in Polar basis Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 55 Measuring multipoles in Polar basis Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 56 Measuring multipoles in Polar basis Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 57 Advantages of mode decompositions Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 58 Three-dimensional fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 59 Three-dimensional fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 60 Three-dimensional fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 61 Three-dimensional fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 62 Three-dimensional fields Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 63 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. Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 64 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 Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 65 Appendix A - Field Error Tolerances Single point mismatch as measured by the Courant-Snyder invariant change: ( ) 2 2 ( ) x x 2 2 x 2 x sin , , sin cos - sin Each source of field error (magnet) contributes the following Courant-Snyder variation 2 cos 2 B , grad dl B m x m ,wherem kmdl m 1 here, m =1 quadrupole, m =2 sextupole, m=3 octupole, etc 2 m 1 m m cos sin m1 m m 2 m sin , m Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 66 Appendix A - Field Error Tolerances multipole expansion coefficients of the azimuthal magnetic field, B - Fourier series representation in polar coordinates at a given point along the trajectory): r B r , m 2 r0 m -1 Bm cos m Am sin m multipole gradient and integrated geometric gradient: Gm = 1 r0 m Bm 1 kGauss cm - m kn Gn cm- ( n 1) B n G dl n B cm- n Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 67 Appendix A - Field Error Tolerances Cumulative mismatch along the lattice (N sources): N 1 2 m 1 n 1 N m -1 m cos sin m 2 m1 m -1 m sin m 2 , Standard deviation of the Courant-Snyder invariant is given by: 2 - 2 2 2i i 1 m 1 N i m -1 m cos sin m i m1 2 i m -1 m sin 2 m Assuming weakly focusing lattice (uniform beta modulation) the following averaging (over the betatron phase) can by applied: 2 ... 1 2 d ... 0 Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 68 Appendix A - Field Error Tolerances Some useful integrals …. : cos sin m 0 , m -1 sin m sin m - 2 m 0m odd m - 1!!m even m !! will reduce the coherent contribution to the C-S variance as follows: 2i i 1 m 1 N i m -1 m cos sin m i m1 2 i m -1 m sin 2 m Including the first five multipoles yields: N i 1 2 i 2 12 sin 2 i 2 2 213 sin 4 i 32 215 224 sin 6 ... 1 2 Operated by JSA for the U.S. Department of Energy 1 3 2 4 Thomas Jefferson National Accelerator Facility Lecture 5 - Magnetic Multipoles 1 3 5 2 4 6 USPAS, Fort Collins, CO, June 10-21, 2013 69 Appendix A - Field Error Tolerances Beam radius at a given magnet is : ai 1 i 2 One can define a ‘good fileld radius’ for a given type of magnet as: a Max(ai ) Assuming the same multipole content for all magnets in the class one gets: N 1 2 i 1 2 i 3 5 12 a 2 2 2 213 a 4 32 215 224 ... 2 2 The first factor purely depends on the beamline optics (focusing), while the second one describes field tolerance (nonlinearities) of the magnets: 3 5 12 a 2 2 2 213 a 4 32 215 224 ... 2 2 Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 70 Appendix B - 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. Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 71 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 72 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 73 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 74 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 75 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 76 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 77 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 78 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 79 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 80 Appendix B - The vector potential Thomas Jefferson National Accelerator Facility Operated by JSA for the U.S. Department of Energy Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 81
