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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
1Lecture 5 Magnetic Multipoles
Magnetic Multipoles, Magnet Design
USPAS, Fort Collins, CO, June 10-21, 2013
Alex Bogacz, Geoff Krafft and Timofey Zolkin
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2Lecture 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
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3Lecture 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
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4Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Maxwell’s equations
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5Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Maxwell’s equations
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6Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Physical interpretation of
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7Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Physical interpretation of
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8Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Linearity and superposition
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9Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields
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10Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields
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11Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields
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12Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields
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13Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields
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14Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields
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15Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Generating multipole fields from a current distribution
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16Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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Thomas Jefferson National Accelerator Facility
17Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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Thomas Jefferson National Accelerator Facility
18Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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19Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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20Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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21Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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22Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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23Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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24Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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25Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields from a current distribution
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26Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Superconducting quadrupole - collider final focus
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27Lecture 5 Magnetic MultipolesUSPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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28Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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29Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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30Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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31Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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32Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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33Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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34Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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35Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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36Lecture 5 Magnetic MultipolesUSPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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37Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Multipole fields in an iron-core magnet
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38Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Generating multipole fields in an iron-core magnet
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39Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Generating multipole fields in an iron-core magnet
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Thomas Jefferson National Accelerator Facility
40Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Generating multipole fields in an iron-core magnet
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41Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Generating multipole fields in an iron-core magnet
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42Lecture 5 Magnetic MultipolesUSPAS, Fort Collins, CO, June 10-21, 2013
Maxwell’s Equations for Magnets - Summary
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43Lecture 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
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44Lecture 5 Magnetic Multipoles
Previous lecture re-cap
USPAS, Fort Collins, CO, June 10-21, 2013
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45Lecture 5 Magnetic Multipoles
Previous lecture re-cap
USPAS, Fort Collins, CO, June 10-21, 2013
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46Lecture 5 Magnetic Multipoles
Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 10-21, 2013
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47Lecture 5 Magnetic Multipoles
Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 10-21, 2013
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48Lecture 5 Magnetic Multipoles
Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 10-21, 2013
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49Lecture 5 Magnetic Multipoles
Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 10-21, 2013
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50Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Allowed and forbidden harmonics
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51Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Allowed and forbidden harmonics
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52Lecture 5 Magnetic Multipoles
Measuring multipoles
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53Lecture 5 Magnetic Multipoles
Measuring multipoles in Cartesian basis
USPAS, Fort Collins, CO, June 10-21, 2013
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54Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Measuring multipoles in Cartesian basis
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55Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Measuring multipoles in Polar basis
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56Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Measuring multipoles in Polar basis
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57Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Measuring multipoles in Polar basis
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58Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Advantages of mode decompositions
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59Lecture 5 Magnetic MultipolesUSPAS, Fort Collins, CO, June 10-21, 2013
Three-dimensional fields
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60Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Three-dimensional fields
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61Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Three-dimensional fields
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62Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Three-dimensional fields
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63Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Three-dimensional fields
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64Lecture 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.
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Summary – Part II
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65Lecture 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
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66Lecture 5 Magnetic Multipoles
Single point mismatch as measured by the Courant-Snyder invariant change:
22 cos ,
sin , sin cos - sin
x
2 2
2
( ) 2 ( )
2 ,
x x
x
1
,grad
mm m m
m
B dlx where k dl
B
here, m =1 quadrupole, m =2 sextupole, m=3 octupole, etc
Each source of field error (magnet) contributes the following Courant-Snyder variation
2
1 1
2 cos sin sin ,m m
m mm m
m m
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Appendix A - Field Error Tolerances
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67Lecture 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):
-1
2 0
, cos sin
m
m mm
rB r B m A m
r
multipole gradient and integrated geometric gradient:
1B mm mmG kGauss cm
0
1=r
( 1) nnn
Gk cm
B
n nn
G dlcm
B
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Appendix A - Field Error Tolerances
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68Lecture 5 Magnetic Multipoles
Cumulative mismatch along the lattice (N sources):
Standard deviation of the Courant-Snyder invariant is given by:
2
1 12
1 11
1 2 cos sin sin ,N m m
m mN m m
m mn
22 2 2
1 12
1 1 1
2 cos sin sinN m m
m mi i m i i m
i m m
Assuming weakly focusing lattice (uniform beta modulation) the following averaging (over the betatron phase) can by applied:
2
0
1... ...
2d
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Appendix A - Field Error Tolerances
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69Lecture 5 Magnetic Multipoles
Some useful integrals …. :
will reduce the coherent contribution to the C-S variance as follows:
cos sin 0 ,m
Including the first five multipoles yields:
2
1 12
1 1 1
2 cos sin sinN m m
m mi i m i i m
i m m
22 2 2 2 4 2 61 2 1 3 3 1 5 2 4
1
sin 2 sin 2 2 sin ...N
i i ii
1
2
1 3
2 4
1 3 5
2 4 6
2
0 m odd1
sin sin 1 !!m even
!!
m mmm
mm
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Appendix A - Field Error Tolerances
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70Lecture 5 Magnetic Multipoles
Beam radius at a given magnet is :
1
2i ia
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:
2 2 2 2 4 21 2 1 3 3 1 5 2 4
1
1 3 52 2 2 ...
2 2 2
N
ii
a a
( ) ia Max a
The first factor purely depends on the beamline optics (focusing), while the second one describes field tolerance (nonlinearities) of the magnets:
2 2 2 4 21 2 1 3 3 1 5 2 4
3 52 2 2 ...
2 2a a
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Appendix A - Field Error Tolerances
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71Lecture 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
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Appendix B - The vector potential
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Appendix B - The vector potential
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Appendix B - The vector potential
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Appendix B - The vector potential
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Appendix B - The vector potential
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Appendix B - The vector potential
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Appendix B - The vector potential
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Appendix B - The vector potential
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80Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013
Appendix B - The vector potential