!*(9:7*=/

2. Charged particles in magnetic fields 2.2 Magnets (synchrotron magnet) 2.3 Multipole expansion 2.4 Superconducting magents

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 1

,_,=+&,3*98

(42'.3*)=+:3(9.43a=8>3(-749743=2&,3*9

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 2 942'.3*)=+:3(9.43=2&,3*9a=8>3(-749743=2&,3*9

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 3

,_-=+:19.541* *=5&38.43=

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 4 >*3*7&1=2:19.541* *=5&38.43

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 5

>*3*7&1=2:19.541* *=5&38.43a=9>1.3)7.(&1= (447).3&9*=7*57*8*39&9.43=@

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 6 >*3*7&1=2:19.541* *=5&38.43a=9>1.3)7.(&1= (447).3&9*=7*57*8*39&9.43=@@

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 7

>*3*7&1=2:19.541* *=5&38.43a=9>1.3)7.(&1= (447).3&9*=7*57*8*39&9.43=@@@

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 8 >*3*7&1=2:19.541* *=5&38.43a=9>1.3)7.(&1= (447).3&9*=7*57*8*39&9.43=@B

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 9 >*3*7&1=2:19.541* *=5&38.43a=9&79*8.&3=(447).3&9*= 7*57*8*39&9.43=@

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 12 >*3*7&1=2:19.541* *=5&38.43a=9&79*8.&3=(447).3&9*= 7*57*8*39&9.43=@@

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 13

C.*1)=).897.':9.43=4+=9-*=+.789=+*<=2:19.541*8

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 14 C.*1)=).897.':9.43=4+=9-*=+.789=+*<=2:19.541*8

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 15

E.,-*7=47)*7=+:19.541*8

q n−−−1 1 Multipole strength: kn === ∂∂∂ x By units: p x, y===0 mn+1

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

Higher order multipoles come from  Field errors in magnets  Magnetized materials  From multipole magnets that compensate such erroneous fields  To compensate nonlinear effects of other magnets  To stabilize the motion of many particle systems  To stabilize the nonlinear motion of individual particles ,_.=G:5*7(43):(9.3,=2&,3*98

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 17

G:5*7(43):(9.3,=+&,3*98

Conventional iron core magnets: Above 1.5 T 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 I 1 2π r B A j = 2 = 2 =1389 2 d d µ0 mm Normal-conducting Cu can only support about 100A/mm2. Solution: Superconducting cables! Superconducting cables routinely allow current densities of 1500A/mm2 at 4.6 K and 6T. Materials used are usually Nb alloys, e.g. NbTi (Tc ≈≈≈ 10K), Nb3Ti or Nb3Sn. Superconducting: dipole fields >6T, quadrupole gradients >100 T/m 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.

Superconducting 0.1T magnets for inside the HERA detectors. G:5*7(43):(9.3,=+&,3*98

Problems:  Superconductivity brakes down for too large fields (quenching) Critical field: F 2 V C T S B (T ) = B )0( G1− D T W c c G D T T W H E c U X  Due to the Meissner-Ochsenfeld effect magnetic field is pushed out of superconductor. -> current can only flow in a thin surface layer (~ 50 nm deep).

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

H.78(4.1=+:19.541*8 @

! ! ! ! µ I ! µ I C− yS Straight wire at the origin: ; 0 0 D T ∇× B = µ0 j B(r) = eϕ = D T 2π r 2π r E x U ! ! µ I C−[y − a ]S Wire at a : 0 D y T B(x, y) = ! ! 2 D T 2π (r − a) E x − ax U ! ! Potential: B(x, y) = −∇Ψ → potential of single wire Use principle of superposition to get potential of arrangement of wires (see Wille) y For cylindrical current shell with given current distribution I(ϕa): dϕ 2π a dI ! Ψ(r,ϕ) = Ψwire dϕa a O dϕa 0 ϕa x

y H.78(4.1=+:19.541*8 @@

Can create a given multipole by an arrangement of wires: ˆ µ0I n if ˆ Ψn ()r,ϕ = − r sin()nϕ I(ϕa ) = I cosnϕa an

y

Ideal multipole (diople) Approximate multipole

9:77*39=).897.':9.43=+47=&3=.)*&1=&.78(47*=2&,3*9

Schematic of superconducting dipole

Corresponding yoke profile of Current distribution conventional magnet

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 22 9:77*39=).897.':9.43=+47=&3=.)*&1=&.78(47*=2&,3*9

Corresponding yoke profile of conventional magnet Current distribution

Corresponding yoke profile of conventional magnet Current distribution

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 23

J*&1=H.78(4.1=+:19.541*8

RHIC Tunnel Quadrupole corrector

LHC dipole •Large magnetic fields >2T required for protons/ ions at high energies >100 GeV (RHIC, HERA, LHC) J*&1=H.78(4.1=+:19.541*8

•Forces of >106 N/m between conductors! -> copper sleeve around cords; aluminum clamps hold current shells together •If field too high: Magnet quenches (transitions to normal- conducting) -> Important to distribute stored energy via heating evenly over entire coil! LHC dipole HERA dipole

G5*(.&1=G9=H.78(4.1=+&,3*98

LHC double quadrupole

1e-5 Accuracy

RHIC Siberian Snake dipole (2 pi rotation)