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http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf1
Multipole Magnets from
Maxwell’s Equations
USPAS, Hampton, VA, Jan. 17-28, 2011
Alex Bogacz ,Geoff Krafft and Hisham Sayed
![Page 2: Multipole Magnets from Maxwell’s Equations...Maxwell’s equations and their physical significance Types of magnets commonly used in accelerators. following notation used in: A](https://reader030.vdocuments.us/reader030/viewer/2022040919/5e959dfd11b447194c658600/html5/thumbnails/2.jpg)
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf2
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
Specialized Course on Magnets, 2009,
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf
USPAS, Hampton, VA, Jan. 17-28, 2011
![Page 3: Multipole Magnets from Maxwell’s Equations...Maxwell’s equations and their physical significance Types of magnets commonly used in accelerators. following notation used in: A](https://reader030.vdocuments.us/reader030/viewer/2022040919/5e959dfd11b447194c658600/html5/thumbnails/3.jpg)
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf3
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
![Page 4: Multipole Magnets from Maxwell’s Equations...Maxwell’s equations and their physical significance Types of magnets commonly used in accelerators. following notation used in: A](https://reader030.vdocuments.us/reader030/viewer/2022040919/5e959dfd11b447194c658600/html5/thumbnails/4.jpg)
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf4USPAS, Hampton, VA, Jan. 17-28, 2011
Maxwell’s equations
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf5USPAS, Hampton, VA, Jan. 17-28, 2011
Maxwell’s equations
![Page 6: Multipole Magnets from Maxwell’s Equations...Maxwell’s equations and their physical significance Types of magnets commonly used in accelerators. following notation used in: A](https://reader030.vdocuments.us/reader030/viewer/2022040919/5e959dfd11b447194c658600/html5/thumbnails/6.jpg)
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf6USPAS, Hampton, VA, Jan. 17-28, 2011
Physical interpretation of
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf7USPAS, Hampton, VA, Jan. 17-28, 2011
Physical interpretation of
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf8USPAS, Hampton, VA, Jan. 17-28, 2011
Linearity and superposition
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf9USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf10USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf11USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf12USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf13USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf14USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf15USPAS, Hampton, VA, Jan. 17-28, 2011
Generating multipole fields from a current
distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf16USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf17USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf18USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf19USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf20USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf21USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf22USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf23USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf24USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf25USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields from a current distribution
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf26USPAS, Hampton, VA, Jan. 17-28, 2011
Superconducting quadrupole - collider final focus
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf27USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf28USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf29USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf30USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf31USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf32USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf33USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf34USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf35USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf36USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf37USPAS, Hampton, VA, Jan. 17-28, 2011
Multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf38USPAS, Hampton, VA, Jan. 17-28, 2011
Generating multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf39USPAS, Hampton, VA, Jan. 17-28, 2011
Generating multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf40USPAS, Hampton, VA, Jan. 17-28, 2011
Generating multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf41USPAS, Hampton, VA, Jan. 17-28, 2011
Generating multipole fields in an iron-core magnet
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf42USPAS, Hampton, VA, Jan. 17-28, 2011
Maxwell’s Equations for Magnets - Summary
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Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf43
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, Hampton, VA, Jan. 17-28, 2011
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf44
Previous lecture re-cap
USPAS, Hampton, VA, Jan. 17-28, 2011
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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
http://cas.web.cern.ch/cas/Belgium-2009/Lectures/PDFs/Wolski-1.pdf45
Previous lecture re-cap
USPAS, Hampton, VA, Jan. 17-28, 2011
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Allowed and forbidden harmonics
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Allowed and forbidden harmonics
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Allowed and forbidden harmonics
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Allowed and forbidden harmonics
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Allowed and forbidden harmonics
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Allowed and forbidden harmonics
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Measuring multipoles
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Measuring multipoles in Cartesian basis
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Measuring multipoles in Cartesian basis
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Measuring multipoles in Polar basis
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Measuring multipoles in Polar basis
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Measuring multipoles in Polar basis
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Advantages of mode decompositions
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Three-dimensional fields
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Three-dimensional fields
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Three-dimensional fields
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Three-dimensional fields
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Three-dimensional fields
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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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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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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
m
m 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 m
m m
m m
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Appendix A - Field Error Tolerances
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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 m
m
rB r B m A m
r
multipole gradient and integrated geometric gradient:
1B m
m mmG kGauss cm
0
1=
r( 1) nn
n
Gk cm
B
n n
n
G dlcm
B
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Appendix A - Field Error Tolerances
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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 m
N m m
m mn
2
2 2 21 1
2
1 1 1
2 cos sin sinN m m
m m
i 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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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 m
i i m i i m
i m m
22 2 2 2 4 2 6
1 2 1 3 3 1 5 2 4
1
sin 2 sin 2 2 sin ...N
i i i
i
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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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 2
1 2 1 3 3 1 5 2 4
1
1 3 52 2 2 ...
2 2 2
N
i
i
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 2
1 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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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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Appendix B - The vector potential
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Appendix B - The vector potential