48 macroscopic fields in acceleratorshoff/lectures/06uspas/notes2.pdf · n m n m ψx y z bnm z x y...

[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006

CHESS & LEPPCHESS & LEPP

48

E has a similar effect as v B.For relativistic particles B = 1T has a similar effect asE = cB = 3 108 V/m , such anElectric field is beyond technical limits.

Electric fields are only used for very low energies orFor separating two counter rotating beams withdifferent charge.

)( BvEqpdtd

×+=

+p−p

B

+p−p

B

+ E

Electrostatic separators at CESR

Macroscopic Fields in Accelerators



49

Static magnetic fileds: Charge free space:

(x=0,y=0) is the beam’s design curve

0;0 ==∂ EBt 0=j

0)(0

)(0)(2

00

=∇⇒=⋅∇

∇−=⇒=∂+=×∇

rB

rBEjB tψ

ψεµ

x

y

For finite fields on the design curve,Ψ can be power expanded in x and y: ∑

∞

=

=0,

)(),,(mn

mnnm yxzbzyxψ

Magnetic Fields in Accelerators



50

)()(

0

00

001

0

000

BAsdBsdB

sdBsdBsdB

sdBsdBsdBsdB

X

B

A

X

X

B

B

A

A

X

X

B

B

A

A

X

r

Ψ−Ψ=⋅+⋅≈

⋅+⋅+⋅=

⋅+⋅+⋅=⋅=

∫∫

∫∫∫

∫∫∫∫

µ

)in()out( ⊥⊥ = BB

)i(1)out(

)in()out(

parallelparallel

parallelparallel

nBB

HH

rµ=

=

For large permeability, H(out) is perpendicular to the surface.

For highly permeable materials (like iron) surfaces have a constant potential.

X

BA 0=×∇ B

Surfaces of Equal Potential

)()r( rB Ψ∇−=



51

Knowledge of the field and the scalar magnetic potential on a closed surface inside a magnet determines the magnetic field for the complete volume which is enclosed.

02 =∇ ψ

)(),( 0020 rrrrG −=∇ δ

[ ][ ]

[ ][ ] 02000

02

0000

03

00000

03

020

200

03

00

)()(

)()(

)()(

)()(

)()()(

rdGrBGr

rdrGGr

rdrGGr

rdrGGr

rdrrrr

V

V

V

V

V

∫

∫

∫

∫

∫

∂

∂

⋅+∇=

⋅∇−∇=

∇−∇∇=

∇−∇=

−=

ψ

ψψ

ψψ

ψψ

δψψ

Green function:

Green’s Theorem



52

If field data in a plane (for example the midplane of a cyclotron or of a beam line magnet) is known, the complete filed is determined:

Data of the magnetic field in the plane y=0 is used to determine b0(x,z) and b1(x,z).

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂

∂−=⇒=∑

∞

= ),(),(),(

),0,(),(),,(

0

1

0

0 zxbzxbzxb

zxByzxbzyx

z

x

n

nnψ

[ ]∑

∑∑∞

=+

∞

=

−∞

=

+++∂+∂=

−+∂+∂=∇=

02

22

2

2

0

222

)1)(2()(

)1()(0

n

nnnzx

n

nn

n

nnzx

ybnnb

ybnnybψ

),()()1)(2(

1),( 222 zxbnnzxb nyxn ∂+∂++

−=+

Potential Expansion



53

Iteration equation:

The functions Ψν(z) along a line determine the complete field inside a magnet.

22222222 4)()(

)(,,

zwwzwwwwzyx

wwwwywwx iiiiyxwiyxw

∂+∂∂=∂+∂−∂−∂+∂=∂+∂+∂=∇

∂−∂=∂−∂=∂∂+∂=∂−=+=

( )

0})]()1)(1(4[Im{

})()()(4Im{

})(Im{

,,

0,1

0,0

,,

1,0

12

0,

=++++=

++=∇

⋅=

∑

∑∑

∑

∞

=+

∞

==

∞

==

−

∞

=

νλνλ

λννλ

λν

νλνλ

λν

νλνλ

λν

νλνλ

λνλ

λνλψ

ψ

wwwaa

wwwawwwa

wwwza

)(,)1)(1(4

10

,,1 zaaa νννλνλ λνλ

Ψ=+++

−=+

Complex Potentials



54

Ψν(z) are called the z-dependent multipole coefficients

The index ν describes Cν Symmetryaround the z-axisdue to a sign change after

})(Im{2!)!(

!)1(),,(

)}(4!)!(

!)1(Im{),,(

]2[

0,

2

]2[

0,

νϕλν

λν

νλλ

λν

λν

νλλ

λνλνϕψ

λνλνψ

iezrrzr

zwwwzyx

−∞

=

∞

=

Ψ⎟⎠⎞

⎜⎝⎛

+−

=

Ψ⎟⎠⎞

⎜⎝⎛

+−

=

∑

∑

νπϕ =∆

ze

3=ν

S

S

SN

N

N

Multipole Coefficients



55

Main fields in accelerator physics:

Main field

Only the fringe field region has terms with and 0≠λ 02 ≠∂ ψz

0,0 2 =∂= ψλ z

⎪⎩

⎪⎨⎧

=Ψ≠Ψ

=Ψ0for0for

0 ννν

νϑ

ν

ν

iei

01

)( }Im{),( Ψ+Ψ=∑∞

=

−−

ν

ϑϕνν

ν νϕψ ierr

B

Fringe field

Fringe Fields and Main Fields



56

Main field potential:

Where the rotation of the coordinate system is set to 0

The isolated multipole:

The potentials produced by different multipole components have

a) Different rotation symmetry Cν

b) Different radial dependence rν

∑∞

=

−Ψ−Ψ=1

0 )](sin[ν

ννν ϑϕνψ r

)sin(νϕψ νν Ψ−= r

νϑ

νΨ

Main Field Potential



57

0

)(4

)(

'0

''02

''02

''00

=⋅∇⇒⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Ψ−ΨΨ

=

±Ψ−Ψ=

BB

zwwz

y

x

…ψj

wmBqiw

mqBiw

xy

mzqB

xy

mqB

yx

BBB

zyx

qzyx

m

zz

zz

z

zy

zx

γγ

γγ

γ

2

2

'

'2

'2

−−=

⇓

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

⇓

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

×⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∫==

t

dtgiz eww

mqBg 00,2 γ

02

0

02

0

02

20

0)2(

ygy

xgx

wg

ewgwgiwgiww

t

dtgi

−=

−=

−=

∫−++=

Focusing in a rotatingcoordinate system

Multipoles in Accelerators ν=0: Solenoids



58

Solenoid focusing is weak compared to the deflections created by a transverse magnetic field.

Transverse fields:

Strong focusing

zz

zz rBmqvr

mqBr

ργγ 42

2

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

yyxx eBeBB +=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛⇒

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

x

yzy

x

BB

mqv

yx

BB

zyx

qzyx

mγ

γ0

zz qB

p=ρ

If the solenoids field was perpendicular to the particle’s motion,

its bending radius would be

Weak focusing < Strong focusing by about ρr

Solenoid vs. Strong Focusing



59

The solenoid’s rotation of the beam is often compensated by

a reversed solenoid called compensator.

Solenoid or Weak Focusing:

Weak focusing from natural ring focusing:

Solenoid magnets are used in detectors for particle identification viaqBp

=ρ

Solenoids are also used to focus low γ beams: wm

qBw z2

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

γ

γϕ

mqBz2

−=

rmqBrvrrrr

yxrRyrRxrR

Rrr

∆⎟⎟⎠

⎞⎜⎜⎝

⎛−=∆⎟⎟

⎠

⎞⎜⎜⎝

⎛−=∆−=∆⇒∆−=∆∂

∆+∆=∆∆=∆−∆++∆−∆+

−=∆

2222

00

220

20

)sin(cos:inionLinearizat]sin)[(]cos)[(

γρϕ

ϕϕϕϕ

ϕ

Rrϕ

Solenoid Focusing



60

C1 Symmetry

Dipole magnets are used for steering the beams direction

(+,-) in Ψ

(S,N) in B

yeByiyx 111 }Im{ Ψ=∇−=⇒⋅Ψ−=−Ψ= ψψ

-

+ +

- B⇑

ρ

⊥⊥ ===⇒=⇒×= qB

ppdp

vdtddlqvB

dtdpBvq

dtpd

/ϕρ

ϕdpdp =p

ϕd

qBp

=ρBending radius:

Equipotential.const=y

Multipoles in Accelerators ν=1: Dipoles



61

aHaHlH

aHlHsdHnI

Fe

FeFe

r22

22

0001

0

≈+=

+=⋅= ∫µ

anI

pq 01 µ

ρ=

C-shape magnet: H-shape magnet: Window frame magnet:

)in()out(

)in()out(

⊥⊥

⊥⊥

=

=

HH

BB

rµ

anIB 00 µ= Dipole strength:

Different Dipoles



62

Shims reduce the space that is open to the beam, but they also reduce the fringe field region.

anIB 00 µ=B < 1 T: Region in which

B < 1.5 T: Typically used region

B = 2 T: Typical limit, since the field becomesdominated by the coils, not the iron.Limiting j for Cu is about 100A/mm2

Dipole Fields



63

HERA Tunnel

Where is the vertical Dipole?



64

C2 Symmetry

In a quadrupole particles are focused in one plane and defocused in the other plane. Other modes of strong focusing are not possible.

+

+-

-

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ=∇−=⇒⋅Ψ−=−Ψ=

xy

Bxyiyx 22})Im{( 222

2 ψψ

+

-+

-z

y

x

Multipoles in Accelerators ν=2: Quadrupoles



65

⇒⋅Ψ−= xy22ψ yx const.=Equipotential:

0

2

20 µ

adrHsdHnIa

r Ψ=≈⋅= ∫∫

rerBxy

B 22 2)10(2 Ψ=⇒⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ=

20

012anI

pqB

pqk yx

µ=∂=

Quadrupole strength:

Quadrupole Fields



66

PETRA Tunnel

The coils show that this is an upright quadrupole not a rotated or skew quadrupole.

SLAC

Real Quadrupoles



67

i) Sextupole fields hardly influence the particles close to the center, where one can linearize in x and y.

ii) In linear approximation a by ∆x shifted sextupole has a quadrupole field.

iii) When ∆x depends on the energy, one can build an energy dependent quadrupole.

⎟⎟⎠

⎞⎜⎜⎝

⎛−

Ψ=∇−=⇒−⋅Ψ=−Ψ= 22323

33

3

23)3(})Im{(

yxxy

Byxyiyx ψψ

C3 Symmetry

)(62

3

23

23223

223

xOxy

xyx

xyB

xxxyx

xyB

∆+⎟⎟⎠

⎞⎜⎜⎝

⎛∆Ψ+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

Ψ≈

+∆

⎟⎟⎠

⎞⎜⎜⎝

⎛−

Ψ=∇−= ψ

S

S

SN

N N

Multipoles in Accelerators ν=3: Sextupoles



68

2303yB

xyΨ−=

=

⇒−⋅Ψ= )3( 232 yxyψ2

31const. y

yx +=Equipotential:

0

3

30 µ

adrHsdHnIa

r Ψ=≈⋅= ∫∫ 3002

26anI

pqB

pqk yx

µ=∂=

Quadrupole strength:

Sextupole Fields



69

ESRF

Real Sextupoles



70The CESR Tunnel



71

Higher order multipoles come fromField errors in magnetsMagnetized materialsFrom multipole magnets that compensate such erroneous fieldsTo compensate nonlinear effects of other magnetsTo stabilize the motion of many particle systemsTo stabilize the nonlinear motion of individual particles

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ==⇒−⋅Ψ=−Ψ= −

−1

1 0)0()(})Im{( nnn

nn

n xnyByxniiyx …ψ

Multipole strength: )!1(10, +Ψ=∂= += npqB

pqk nyxy

nxn units: 1m

1+n

p/q is also called Bρ and used to describe the energy of multiply charge ions

Names: dipole, quadrupole, sextupole, octupole, decapole, duodecapole, …

Higher order Multipoles



72

The discussed multipolesproduce midplane symmetric motion. When the field is rotated by π/2,

i.e , one speaks of a skew multipole.

),,(),,(),,(),,(),,(),,(

),,(),,(),,(),,(),,(),,(

zyxBzyxBzyxBzyxBzyxBzyxB

zyxBvzyxBvBvBvzyxBvzyxBvBvBvzyxBvzyxBvBvBv

zz

yy

xx

xyyxxyyx

zxxzzxxz

yzzyyzzy

−=−=−−=−

⇒−+−=−−−−=−−−−−=−

y

y−

yp

yp−

nn 2πϑ =

),(),(

),,(),,(

⊕⊕⊕

⊕

⊕

=⇒=

−=

−=

prFpprFp

ppppzyxr

dtd

dtd

zyx

),,(),,( zyxzyx ψψ −=−

{ } { }{ }[ ] 00)(Re2Im

)(Im)(Im

=⇒=+Ψ⇒

+Ψ−=+Ψ

nnin

n

ninn

ninn

iyxe

iyxeiyxen

nn

ϑϑ

ϑϑ

Midplane Symmetric Motion



73

Above 2T the field from the bare coils dominate over the magnetization of the iron.But Cu wires cannot create much filed without iron poles:5T at 5cm distance from a 3cm wire would require a current density of

Cu can only support about 100A/mm2.

Superconducting cables routinely allow current densities of 1500A/mm2 at 4.6 K and 6T. Materials used are usually Nb aloys, e.g. NbTi, Nb3Ti or Nb3Sn.Superconducting magnets are not only used for strong fields but also when there is no space for iron poles, like inside a particle physics detector.

20

22 mmA138921 ===

µπ Br

ddIj

Superconducting 0.1T magnets for inside the HERA detectors.

Superconducting Magnets



74

Problems:Superconductivity brakes down for too large fieldsDue to the Meissner-Ochsenfeldeffect superconductivity current only flows on a thin surface layer.

Remedy:Superconducting cable consists of many very thin filaments (about 10µm).

Superconducting Magnets



75

Straight wire at the origin: ⎟⎟⎠

⎞⎜⎜⎝

⎛−==⇒=×∇

xy

rIe

rIrBjB

πµ

πµµ ϕ 22

)( 000Wire at :

{ }{ } ( ) aaa

aa

a

a

ia

IawI

awwI

awwI

waww

wI

aww

aw

I

aa

aIyx

ew

i

iwwww

wwiiBB

ϕννπ

µν

ν

νν

νπµ

πµ

πµ

πµ

πµ

πµ

νψ 1121

122

22

22

02

02

0

20

20

2

200

Im)1ln(Im

)1ln(Im2)1ln(

1))(()(

=Ψ⇒⎭⎬⎫

⎩⎨⎧

−=−=

−∂−=−∂=

−

−=

−−−−

=+

∑∞

=

a

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

−=

x

y

axay

arIyxB

][)(2

),( 20

πµ

This can be represented by complex multipole coefficients νΨ

( ) ψψ wyxyx iiBByxB ∂−=∂+∂−=+⇒Ψ∇−= 2),(

Complex Potential of a Wire



76

Creating a multipole be created by an arrangement of wires:

addIi

ade

a

a ϕϕπ

ϕννπ

µν ν∫=Ψ

2

0

112

0

Inannˆ11

20µ

νν δ=Ψ aa nII ϕϕ cosˆ)( =if

aϕa

adϕ

x

y

y y

Ideal multipole Approximate multipole

Air-coil Multipoles



77

LHC dipole

Quadrupole corrector

RHIC Tunnel

Real Air-coil Multipoles



78

LHC double quadrupole

RHIC SiberianSnake dipole

Accuracy

Special SC Air-coil Magnets

48 macroscopic fields in acceleratorshoff/lectures/06uspas/notes2.pdf · n m n m ψx y z bnm z x y...

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