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Thomas Jefferson National Accelerator Facility 1 Lecture 5 - Magnetic Multipoles

Magnetic Multipoles, Magnet Design

USPAS, Fort Collins, CO, June 13-24, 2016

S.A. Bogacz, G.A. Krafft, S. DeSilva and R. Gamage

Jefferson Lab and Old Dominion University



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, 2009

  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



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.

USPAS, Hampton, VA, Jan. 17-28, 2011


Thomas Jefferson National Accelerator Facility 4 Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016

Maxwell’s equations



Maxwell’s equations



Physical interpretation of



Linearity and superposition



Multipole fields



Generating multipole fields from a current distribution



Multipole fields from a current distribution



Fields from a cylindrical current distribution



Superconducting quadrupole - collider final focus



Multipole fields in an iron-core magnet



Generating multipole fields in an iron-core magnet



Maxwell’s Equations for Magnets - Summary



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.




Previous lecture re-cap




Allowed and forbidden harmonics




Measuring multipoles




Measuring multipoles in Cartesian basis




Measuring multipoles in Polar basis



Advantages of mode decompositions



Three-dimensional fields



  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.


Summary – Part II



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




  Single point mismatch as measured by the Courant-Snyder invariant change:

22 cos ,δε εβ µ δθ βδθ= +

( )sin , sin cos - sinεεβ µ θ µ µ α µ

β= =x( )

2 2

2

( ) 2 ( )2 ,

x xx

ε β θ δθ α θ δθ γ

ε βθ α δθ βδθ

ʹ = + + + +

= + + +

δθ δφ δφ δρ =

= = =∫ ∑ ∫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 12 cos sin sin ,

m mm mm m

m mδε εβ εβ δφ µ µ β εβ δφ µ

= =

⎛ ⎞= + ⎜ ⎟

⎝ ⎠∑ ∑





  multipole expansion coefficients of the azimuthal magnetic field, Bθ - Fourier series representation in polar coordinates at a given point along the trajectory):

( ) ( )-1

2 0

, cos sinm

m mm

rB r B m A mrθ θ θ θ

=

⎛ ⎞= +⎜ ⎟

⎝ ⎠∑

  multipole gradient and integrated geometric gradient:

1B mm mmG kGauss cm−

+ ⎡ ⎤⎣ ⎦0

1=r

( 1) nnnGk cmBρ

− +⎡ ⎤= ⎣ ⎦ φρ

−⎡ ⎤= ⎣ ⎦∫ n n

n

G dlcm

B





  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 mm m

N m mm mn

ε ε β εβ δφ µ µ β εβ δφ µ− −

= ==

⎛ ⎞⎛ ⎞= + + ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑ ∑∏

( ) ( )22 2 2

1 12

1 1 12 cos sin sin

N m mm mi i m i i m

i m m

εδε δεσ

β εβ δφ µ µ β εβ δφ µε ε

− −

= = =

⎡ ⎤− ⎛ ⎞⎢ ⎥= = + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑ ∑

  Assuming weakly focusing lattice (uniform beta modulation) the following averaging (over the betatron phase) can by applied:

π

µπ

= ∫2

0

1... ...2

d





  Some useful integrals …. :

will reduce the coherent contribution to the C-S variance as follows:

cos sin 0 ,mµ µ =

  Including the first five multipoles yields:

( ) ( )2

1 12

1 1 12 cos sin sin

N m mm mi i m i i m

i m m

εσ β εβ δφ µ µ β εβ δφ µε

− −

= = =

⎡ ⎤⎛ ⎞⎢ ⎥= + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑ ∑

( ) ( ) ( ){ }22 2 2 2 4 2 61 2 1 3 3 1 5 2 4

1sin 2 sin 2 2 sin ...

N

i i ii

εσ β δφ µ εβ δφ δφδφ µ εβ δφ δφδφ δφ δφ µε =

⎡ ⎤= + + + + + +⎣ ⎦∑

12

1 3 2 4

1 3 5 2 4 6

( )2

0 m odd1sin sin 1 !!

m even!!

µ µ−

−

= = −

m mmmmm





  Beam radius at a given magnet is : 12

εβ=i 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 aεσ β δφ δφ δφδφ δφ δφδφ δφ δφε =

= × + + + + + +∑

( )= 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 aδφ δφ δφδφ δφ δφδφ δφ δφΔΦ = + + + + + +





  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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