Carmine SENATORE

Superconductivity and its applications

Lecture 10

http://supra.unige.ch

Group of Applied SuperconductivityDepartment of Quantum Matter Physics

University of Geneva, Switzerland

Previously, in lecture 9HTS materials for applications

Copper oxides with highly anisotropic structures, texturing (grain orientation) is required in technical superconductors

Produced by Powder-In-Tube methodAg matrix materialTape geometry to achieve c-axis texturingIc depends on the magnetic field orientation

Produced by Powder-In-Tube methodAg matrix materialRound wire, spontaneous radial texturing of the c-axisIsotropic properties of Ic

Tc [K] texture

Bi2223 110 c-axis

Bi2212 91c-axis

(radial)

Y123 92 biaxial

Coated conductorsY123 layer deposited on a metallic substrateTexturing along c and in the ab planeIc depends on the magnetic field orientation

Previously, in lecture 9

Superconducting magnets, field shapes and winding configurations

Solenoids (NMR, MRI and laboratory magnets)

0 ,B JaF

2 2

02

, ln1 1

F

2 ( )

NIJ

l b a

where

l

a

b

a

z

Bz Bw

and have an influence on the field uniformity (Bz and Bw)

Previously, in lecture 9

Subdivide the winding into a number of concentric sections to improve the efficiency of superconductor utilization

Each section operates at the maximum current density allowed by the local field level

All sections take the same current, but each section has its own J, and

General guidelines for magnet design

Notched solenoids and homogeneity along z

2 4 0E E sixth order

2 4 6

0 2 4 61 ...z

z z zB B E E E

a a a

Transverse fields: accelerator magnets

flat 'race track' coils

Sketch of coils to generate transverse fields

curved saddle coils

Uniform field (dipole)

x

y z

(a)

x

y z

(b) Gradient field (quadrupole)

Bending the beam Focusing the beam

Transverse fields: accelerator magnets

x

y

r

B +J

Infinite cylinders carrying uniform J

02

JrB

Within a cylinder carrying uniform J, the field is

Two cylinders with their centres spaced apart by d and carrying uniform J but oppositely directed

01 1 2 2 0cos cos

2 2y

J JdB r r

01 1 2 2sin sin 0

2x

JB r r

d x

y

+J -J

r1

1

r22

By is uniform over the whole aperture

Perfect dipole field

Multipole magnets

x

y

+J-JA cylindrical current sheet, infinite along z, carrying a cos current distribution generates a perfect dipole field

It can be demonstrated that:

x

y

+J+J

-J

-J

A cylindrical current sheet, infinite along z, carrying a cos2 current distribution generates a perfect quadrupole field

Intersecting ellipses carrying uniform J

y

xd

b

c +J-J

+J+J

-J

-J

x

y

b

c

0( )

y

JdcB

b c

0xB

Perfect dipole

0 ( )x

J b cB x

b c

0 ( )y

J b cB y

b c

Perfect quadrupole

0 ( )J b cg

b c

Uniform gradient

Real coil configurations for multipole magnets

+J -J

x

y

d

P

Approximation for the ideal dipole current distribution

x

y+J

-JP

ab

Approximation for the ideal quadrupole current distribution

Harmonic generation in real coils

x

y

+J -J

y

x

+J

-J =

x

y

+J

-J

-J

-J

+J

+J

+ +…

Uniform current shell =

Magnetic field =

cos current

dipole field

+ cos3 current + …

+ sextupole term + …

LHC dipoles

LHC quadrupoles

Toroidal fields

In a 'perfect' torus the distribution of current density is perfectly uniform in the direction

No field in the r- or z-directions

By applying Ampere's law

02B rd B r NI

0

2

NIB

r

A practical torus is constructed from a number of discrete circular coils

Toroidal fields: fusion magnets

Toroidal windings produce fields in which themagnetic lines of force close up on themselves

It is difficult for charged particles to diffuse indirections perpendicular to the magnetic field lines

Closed-line configurations provide good confinementfields for the hot ionized plasmas which are needed toproduce controlled thermonuclear fusion

ITER Reactor

Introduction to Forces and Stresses in a Magnet

Infinite solenoidthin wall

B=B0 B=0

0 0B JdThe magnetic field is

The expression of the magnetic force density (per unit of volume) is

f J B

The average field in the winding is 0

2

B

F

dr

0 2

BJ r z d

20

0

2

Br z

20

02m

Bs p s

The (radial) force on a winding volume element is

0

2

BF J v

0( 1 ) 4 mp B T bar 0( 10 ) 400 mp B T bar

Hoop stress in a ring

r

loop

F dvf dvJ B

A ring carrying a current I in a field B

2 R I BB

I = J·Swire

The total radial magnetic force on the ring is

R

x x x x x x x x

x x x x x x x x

x x x x x x x x

x x x x x x x x

x x x x x x x x

x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

x x x x x x x x x x x x x

Hoop stress in a ring

r

loop

F dvf dvJ B

A ring carrying a current I in a field B

2 R I B

radial force2

rF F

A tension is developed within the ringTT /2/2

2 sin 22 2

T T T

R I B2

rFT

The so-called “hoop stress” on the wire is wire

TR J B

S

The total radial magnetic force on the ring is

Hoop stress levels above 100 MPa are common, the NHMFL 32 T magnet operates at 400 MPa

Electromagnetic stresses in a finite solenoidIn a winding adjacent turns will press on each other and develop a radial stress r which modifies the hoop stress

The condition for equilibrium between radial stress r , hoop stress and body force BJr is given by the equation

2

2

1 1d du ur BJ

r dr dr Er

Considering the solenoid as a continuous uniform medium, from the Hooke’s law

21r

E du u

dr r

21

E u du

r dr

and

where u is the local displacement in the radial direction, E is the Young’s modulus and is the Poisson’s ratio

F

Electromagnetic stresses in a finite solenoid

magnetic lines of force vectors of electromagnetic force per unit volume

20

02

Bp

r

bore radius

' BJr

Example: for B0 of the order of 10 T, on the winding of > 200 MPa (> 2000 bar)

Electromagnetic stresses in an accelerator dipole

magnetic lines of force vectors of electromagnetic force per unit volume

In straight-sided coils such as dipolesand quadrupoles the conductor is unableto support the magnetic forces in tension

20

02

Bp

For B0=6T and (a+b)/2=100 mm

Fx=200 tons/m

How to make the dipole able to sustain the stress

Rutherford cable

SC wireSC filaments

How to make the dipole able to sustain the stress

• Clamping the winding in a solid collar

175 tons/m

85 tons/mF

Magnetic forces in the ITER TF coils

Central Solenoid

D-coil

in-plane forces

out-of-plane forces

Wire cabling for the ITER coils

CICC = Cable-In-Conduit Conductor

Thermal precompression of the superconductor

Mismatch in thermal contraction within the composite induces a precompression in the SC

650°C 4.2 K

m

When the SC is cooled at the operating temperature,its crystal structure deforms and this induces changeson Tc , Bc2 (and thus on Jc)

MgB2

NbTi

Nb3Sn

All technical superconductors are composite

Bi2223

Bi2212

YBCO

Strain-induced changes in the critical current

m irr applied strain

crit

ica

l cu

rren

t

The applied force reduces the effects of the thermal precompression and the critical current increases up to a maximum

The applied force induces breakages within the filaments and the critical current is irreversibly degraded

0

Effects of the longitudinal strain

Strain-induced changes in the critical current

m irr applied strain

crit

ica

l cu

rren

t

0

Effects of the longitudinal strain

REVERSIBLE REGIONIc recovers after force unload

IRREVERSIBLE REGIONIc does not recover after force unload

Technology = m-irr [%]

Bronze Route 0.4 – 0.6

Internal Sn 0.05 – 0.2

Powder-In-Tube

0.15 – 0.3

Wire cabling for the ITER coilsReversible variation of Ic under strain matters

CICC = Cable-In-Conduit Conductor

Because of the thermal precompression, wires in a CICC experience an effective axial compressive strain ranging between -0.6% and -0.8%

-0.8 -0.6 -0.4 -0.2 0.00.0

0.2

0.4

0.6

0.8

1.0 01UL0299A01

no

rmal

ize

d I c

intrinsic strain [%]

@ 4.2 K, 12 T

The performance of the superconductor is limited to ~40% of the maximum achievable current

Why superconducting properties depend on strain

Reversible strain effects

Under strain the crystal structure deforms and this induces change both in the phonon spectrum and the electronic bands and thus on Tc and Bc2

Cubic-shaped stress-free cell for Nb3Sn

Precompression induced by cooling to low temperature has two components

HYDROSTATICChange of the cell volume

DEVIATORICChange of the cell shape

Reversible strain effects in Nb3Sn wires

Reversible strain effects in Nb3Sn wires: lattice parameters

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.85.260

5.265

5.270

5.275

5.280

5.285

5.290

5.295

5.300

latt

ice

par

ame

ter

[Å]

applied strain [%]

C. Scheuerlein et al., IEEE TAS 19 (2009) 2653

XRD experiments @ ESRF Grenoble

L. Muzzi et al., SUST 25 (2012) 054006

Bronze route wire: Nb3Sn lattice parameters vs uniaxial strain @ 4.2 K

ZERO APPLIED STRAINNb3Sn is precompressedHydrostatic+Deviatoric

CROSSING POINTCubic cell recoveredHydrostatic component still present

HIGH APPLIED STRAIN Again Hydrostatic+Deviatoric

Lattice parameters and Ic under axial strain

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.85.260

5.265

5.270

5.275

5.280

5.285

5.290

5.295

5.300

#26

latt

ice

par

ame

ter

[Å]

applied strain [%]

@ 19T, 4.2K

50

100

150

200

250

300

crit

ical

cu

rren

t [

A]

Bronze Route wire

MAXIMUM OF Ic

The maximum of the critical current occurs when the cubic cell is restored

Bibliography

IwasaCase Studies in Superconducting MagnetsChapter 3 Section 3

Papers cited in the slides

WilsonSuperconducting MagnetsChapter 3Chapter 4

EkinExperimental Techniques for Low-Temperature MeasurementsChapter 10 Section 5