superconductivity and its applications · the magnetic field is the expression of the magnetic...
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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