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Flux line motion
In principlea Type II superconductor should be “better” than a Type I for most
applications - it remains superconducting to much higher fields ……..but is it?
J
Flux lines
FL
FL
FL
FL
FL
Flux lines tend to move easily so as
to reach equilibrium.
But if a current flows in a superconductor
above Hc1 there will be a Lorentz force
acting between the current (ie charge
carriers) and the flux.
FL  J  o
So the flux lines move perpendicular to
the current and induce an electric field
E  B v
v = velocity
of flux lines
Now, E is parallel to J, so acts like a resistive voltage, and power is dissipated
1 watt of dissipated power at 4.2K requires 300MW of refrigerator power!
The solution is to introduce “pins” by creating defects within the superconductor
Lecture 8
Superconductivity and Superfluidity
The Bean Model
A Type II superconductor without any pinning is said to be reversible
- flux enters abruptly at Hc1 and produces a uniform flux density throughout.
If there are pinning centres within the Type II superconductor they hold the flux
lines back near the surface creating a gradient of flux lines - such
superconductors are said to be irreversible
surface
Lecture 8
Superconductivity and Superfluidity
The Bean Model
The pins can be thought of as
introducing “friction” inhibiting the
movement of flux lines into the
supercoductor
In this respect the superconductor is a
little like a sandpile with the flux lines
behaving like grains of sand
However big we make the sandpile the
sloping sides always have the same
gradient
In an analogous fashion, however large
the magnetic field the gradient of flux
lines remains constant:
This is the basis of the Bean Model
Lecture 8
superconductor
B2
B2
B*
B*
B1
B1
D
Superconductivity and Superfluidity
The Bean Model
Remember that curl B=oJ So a field gradient implies
that a current is flowing perpendicular to B
If B is in the z-direction, and the gradient is in the xdirection, then
B
in the y-direction
o J 
z
y
x
The Bean Model assumes that the effect of pinning is to:
(a) produce a maximum gradient
B and therefore to
x
(b) produce a maximum current density J
x
B2
B2
B*
B*
B1
B1
From another viewpoint the Bean model assumes
(a) there exists a maximum current density Jc
D
(b) any emf, however small induces this current to flow
The Bean model is therefore also known as the critical state model :
only three current states are allowed - zero current for regions that have not felt B
and ±Jc (ie the critical current density) depending upon the sense of the emf
generated by the last field change
Lecture 8
Superconductivity and Superfluidity
Critical state model - increasing field
+Jc
2B*
2B*
B*
B*
B*/2
B*/2
current density
superconductor
2B*
and
B*
0
B*/2
-Jc
D
Lecture 8
D
Superconductivity and Superfluidity
Critical state model - decreasing field
First increase B to a value of, say Bo
then reduce B to zero again
Because the flux density gradient must remain constant, flux is
trapped inside the superconducting sample, even at B=0
+Jc
a
b
c
Bo
e
Bo
B=0
B=0
D
Lecture 8
d
current density
superconductor
a
B
bo
c
d
e
0
-Jc
D
Superconductivity and Superfluidity
Predictions of the Bean model
The magnetisation of a sample can be calculated using the Bean model from
diagrams such as the previous ones, with B* as the only free parameter
Lecture 8
Superconductivity and Superfluidity
Calculating the critical current density
2
Magnetisation .
From the Bean model the critical
current density is easily calculated
from the hysteretic magnetisation
loop
2M
Jc 
D
in SI units, where Jc is measured
in A.m-2 , in A.m-1 and D, the
diameter of the sample in m
The strength of the pinning force, F,
can also be determined:
F  JB 
2M
Bapplied
D
F usually shows a peak, at a field
corresponding to that at which the pins
“break”
Lecture 8
1
0
2M
F
-1
B
-2
-1.0
-0.5
0.0
0.5
1.0
Applied field (T)
0
Superconductivity and Superfluidity
Magnetic superconductors
Notice that in the 2nd quadrant of the
hysteresis loop the magnetisation of the
sample is positive, ie strongly (or even very
strongly) paramagnetic.
2
Magnetisation .
3
2
b
1
a
c
So - depending upon the magnetic history
of the sample - the superconductor can be
attracted to a magnet!
...can become this
So this…..
0
-1
4
1
-2
-1.0
-0.5
0.0
0.5
1.0
Applied field (T)
0
gnetisation (emu)
As the magnetic falls away the field decreases from a-b, so the magnetisation
increases.
The magnet therefore moves closer to the superconductor (b-c) and the
-1
field increases, but the magnetisation decreases and the magnet falls away….etc
Lecture 8
Superconductivity and Superfluidity
Critical current densities
Lecture 8
Superconductivity and Superfluidity
Critical current densities
Lecture 8
Superconductivity and Superfluidity
Small Angle Neutron Scattering
multi-detector
64x64cm2
R
Bragg’s Law: =2dsinq
neutron wavelength  = 10Å
flux lattice spacing, d = 1000Å
sinq ~R/L = 1/100
L
B
scattering
angle
2q
Superconducting
sample
incident
neutron
beam
Lecture 9
Flux distribution determined from SANS
Superconductivity and Superfluidity
Flux lattice melting
multi-detector
64x64cm2
R
L
B
scattering
angle
2q
Superconducting
sample
incident
neutron
beam
Lecture 9
Superconductivity and Superfluidity