1

Magnet BasicsS. Bernal

USPAS 08

U. of Maryland, College Park

IREAPIREAP

2

Magnets: Introduction• Magnets are key components of all accelerators

• Magnet modeling has several stages:1. Simple hardedge models for optics design (energy and type of charged

particle are main considerations)

2. Computer calculations

3. Mechanical/electrical design and construction

4. Magnet measurement:

field/gradient profile and/or multipole measurement

5. Beam testing

• Item 1 requires a magnet strength per amp or volt, and an

effective length. Items 2 and 4 yield actual values.

• Measurement devices: gaussmeters (e.g. Hall-effect), rotating

coils, taut wire techniques, etc.

• Computer codes: OPERA3D/TOSCA, MERMAID, AMPERE, MAG-Li

3

Magnets: Introduction

4

Magnets in UMER: Matching Section

Q2 Solenoid

IC2 IC1

Electron Gun

5

Coordinate Systems and Notation

Bending dipoles define reference trajectory.

6

γγγγmv2/ρρρρ ====qvB → Bρ ρ ρ ρ ====p/q: Magnetic Rigidity

For relativistic e-: 0.3p

B pce

ρ = ≅

Tesla m GeV/c

Bending:

dθ = ds/ρ(s), so

2 2

1 12 1

3.0( )

( )

s s

s s

dsB s ds

s pcθ θ

ρ− = ≅∫ ∫

Focusing: 0x 0yx (z)+ x(z) = 0, y (z)+ y(z) = 0.′′ ′′κ κκ κκ κκ κ

00 0

( )x y

g

Bκ κ

ρ= − =• Quadrupole:

0r (z)+ r(z) = 0′′ κκκκ• Solenoid:( )

2

0 24

ZB

Bκ

ρ=

Defined as: ,p

Bq

ρ = p m cγ β=

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Magnets: Introduction

Separated Function vs. Combined Function

Electrostatic vs. Magnetostatic

Dipoles, Quadrupoles, Sextupoles, Octupoles

Displaced, overlapping (& infinite) solid elliptical cylinders carrying uniform current density generate pure fields:

Pure Dipole

+J -J X

Y

BY

X

Pure Quadrupole

Y +J

+J

-J -J

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UMER PC Dipole and Quadrupole*

Only conductors parallel to z-axiscontribute to integrated B-field.

On a circular cylindrical surface, we want: ∝∫ ZK dz cosnθ

n=order of multipole.Recall…

-J+J

*W.W. Zhang, et al, Phys, Rev. ST Accel. Beams, 3, 122401 (2000).

UMER PC quadrupole

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Multipole Expansion

2D Multipole Expansion:

( )

, (1)

∑n-1

x+iyB(x, y) = B + iB = b +ia ,y x n n rn=1 0

2 2r = x +y < r0

bn = Normal Component,

an = Skew Component

r0 = Aperture Radius

From symmetry, a magnet with quadrupole symmetry has only

multipoles of the form n = 4k+2 (k=0,1,2, …), i.e. quadrupole (n=2),

duodecapole (n=6), 10-pole (n=10), etc.

[ ]

[ ] (2)

∑

∑

n-1

n n0

n-1

n n0

rB (r, θ) = b Sin(nθ)+a Cos(nθ) ,r rn=1

rB (r, θ) = b Cos(nθ)-a Sin(nθ) .θ rn=1

3D Multipole Expansion:

B → BInt

WE WANT SMALL UNDESIRED MULTIPOLES:typically less than 1 part in 104

10

UMER Rotating Coil*

The coil contains ~3000

turns of very fine wire. The whole of the

rotating coil apparatus

is normally enclosed inmu-metal box.

2nd Gen. UMER

Quadrupole

*W.W. Zhang, et al, Phys, Rev. ST Accel. Beams, 3, 122401 (2000).

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-80

-70

-60

-50

-40

-30

-20

-10

0

0 10 20 30 40 50 60FF

T o

f R

ota

tin

g C

oil

Sig

nal

(dB

m)

Frequency (Hz)

PC QUADRUPOLE

-80

-70

-60

-50

-40

-30

-20

-10

0

0 10 20 30 40 50 60FF

T o

f R

ota

tin

g C

oil

Sig

nal

(dB

m)

Frequency (Hz)

PC DIPOLE

-80

-70

-60

-50

-40

-30

-20

-10

0

0 10 20 30 40 50 60FF

T o

f R

ota

tin

g C

oil

Sig

nal

(dB

m)

Frequency (Hz)

NO MAGNET

FFT of Rotating Coil Signal

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UMER Simple Rotating Coil

Rawson-Lush rotating coil gaussmeter

Model 780:

Tip Diameter. A: 6.35 mm

Probe Length B: 50.0 cm

Length to coil center C: 48.9 cm

Tube Diameter E: 6.35 mm

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Short Solenoid: Axial Field Profile Measurement

For axially symmetric B-fields, components Bz, Br can be foundat all (z, r) from knowledge of Bz(r=0,z)=B(z), i.e. the on-axis field profile:

3 3

3( , ) ,

2 16r

r B r BB r z

z z

∂ ∂= − + − ⋅ ⋅ ⋅

∂ ∂

2 2 4 4

2 4( , )

4 64z

r B r BB r z B

z z

∂ ∂= − + − ⋅ ⋅ ⋅

∂ ∂

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Review: Modeling of Lenses

′′δ(z)

x (z)+ x(z) = 0f

“Point” Lens:

Z

X

f

0 0

1x (z)+ x(z) = 0 = l

f′′ →κ κκ κκ κκ κ

Thin Hard-Edge (f>>l):

Zl

Xk0

eff peak

1x (z)+ (z)x(z) = 0 = l

f′′ →κ κκ κκ κκ κ

Smooth-Profile:

1

1

z

eff-z

peak

1l = (z)dzκ

κ ∫

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Effective Length of UMER Quadrupole*

*S. Bernal, et al, Phys, Rev. ST Accel. Beams, 9, 064202 (2006).

Red curve is analytical fit to Mag-Li profile (black curve):

g(s)=g0exp(-s2/d2),

g0=3.61 G/cmA, d = 2.10 cm.

0.0

1.0

2.0

3.0

4.0

-10 -7.5 -5 -2.5 0 2.5 5 7.5 10

HE1HE2gxEquation

g(s

) (G

/cm

pe

r A

mp

)

s (cm)

g0

geff

Standard definition of effective length (which uses hardtop

gradient g0) yields leff = 3.72 cm

However, the same focal length can be obtained with a wider hardedge model with smaller hardtop gradient geff.

For the short UMER quad the correct hardedge model yields

leff = 5.16 cm, geff =0.72×g0

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Effective Length of Short Solenoid*

For UMER short solenoid, new treatment yields

leff(cm)= 6.571 cm – 0.00029××××κpeak(m-2),

κeff(m-2)= 0.6945××××κpeak(m

-2)

*S. Bernal, et al, Phys, Rev. ST Accel. Beams, 9, 064202 (2006).

UMER Solenoid Profile ( ) ( ) ( )2 2 2

0 0(0, ) exp sech sinhZB z B z d z b C z b = − +

Similar issues as with the UMER quad…

Effective length calculated the standard way is leff = 4.50 cm

The effective length has a slight dependence on peak focusing function.

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1. K.G. Steffen, High Energy Beam Optics (Wiley, 1965).

2. H. Wiedemann, Particle Accelerator Physics I-II (Springer-Verlag, 1993).

3. P.J. Bryant and K. Johnsen, The Principles of Circular Accelerators and

Storage Rings (Cambridge U. Press, 1993).

4. H. Wolnik, Optics of Charged Particles (Academic Press, 1987).

5. M. Reiser, Theory and Design of Charged-Particle Beams, (Wiley, 1994).

6. USPAS Proceedings (e.g., USPAS 2004).

7. CERN Accelerator School (CAS) Proceedings (e.g., CERN 98-05, CERN

92-05).

8. On-line Journal: Physical Review ST Accel. Beams.

9. Manuals to computer codes like TRANSPORT, TRACE3D, etc.

10. M. Venturini, Ph.D. Thesis (Dept. of Physics, UMCP, 1998).

11. W.W. Zhang, et al, Phys, Rev. ST Accel. Beams, 3, 122401 (2000).

12. S. Bernal, et al, Phys, Rev. ST Accel. Beams, 9, 064202 (2006).

References