Indeed, in a magnetic field a force should act upon the vortices and leads them to move.
But moving vortices should produce nonsteady magnetic fields and, consequently, energy
loss.
To make the estimates assume that there is one vortex with the current
j
and the exter
nal current density is equal to
j
ex
, the total current being
j
+
j
ex
.
Assuming that the corre
sponding contribution to the free energy is
n
s
mv
2
s
/
2 = (
n
s
m/
2)(
j/n
s
e
)
2
= (4
π/c
2
δ
2
L
)
j
2
/
2
we get for the interaction energy
U
=
4
π
c
2
δ
2
L
Z
j
ex
·
j
d
V
.
Then we remember that
j
depends only on the difference
r

r
L
(where
r
is the 2D co
ordinate in the plane perpendicular to the line). The force is
F
k
=

∂
U
∂r
Lk
∝ 
X
i
Z
d
V
j
ex
i
∂j
i
∂r
Lk
=
X
i
Z
d
V
j
ex
i
∂j
i
∂r
k
=
348
CHAPTER 17.
MAGNETIC PROPERTIES TYPE II
=
X
i
Z
d
V
j
ex
i
∂j
i
∂r
k

∂j
k
∂r
i
+
X
i
Z
d
V
j
ex
i
∂j
k
∂r
i
.
The last item vanishes because by integration by parts we get div
j
ex
= 0
.
Thus
F
=
4
π
c
2
δ
2
L
Z
[
j
ex
×
curl
j
]
d
V
.
Substituting the expression for curl
j
we get
f
L
=
Φ
0
c
[
j
ex
×
z
]
for a separate vortex. The total force acting upon the vortex structure is just the Lorentz
force
F
L
=
1
c
[
j
ex
×
B
]
.
Now let us assume that a vortex moves with a given velocity
v
L
and there is a viscous
braking force
f
v
=

η
v
L
.
As a result of force balance,
v
L
=
Φ
0
ηc
[
j
ex
×
z
]
.
We see that
v
L
⊥
j
ex
⊥
B
.
According to the laws of electrodynamics, such a motion produces
the electric field
E
=
1
c
[
B
×
v
L
] =
Φ
0
B
ηc
2
j
ex
.
We observe the Ohm’s law with the resistivity
ρ
=
Φ
0
B
ηc
2
.
If we assume that at
B
=
H
c
2
ρ
=
ρ
n
(i.e. to the resistivity of the normal phase) we get
η
(
H
c
2
) =
Φ
0
H
c
2
ρ
n
c
2
→
ρ
=
ρ
n
B
H
c
2
.
This expression is only a very rough orderofmagnitude estimate. In fact the viscosity
η
is a very complicated and interesting function of both the temperature and magnetic field.
From this point of view we get the conclusion that the superconductivity is destroyed at
H
=
H
c
1
.
Fortunately, this statement is wrong. In real materials there is
pinning
, i.e. the vortices
become pinned by the defects. One kind of pinning is the surface barrier we have discussed
earlier. It is clear that largescale defects with size greater than
ζ
should be very effective.
To get a simple estimate let us consider a cavity in the SC with
d.
ζ.
Suppose that
the core is in the normal state that leads to the extra energy
∼
H
2
c
ζ
2
per unit length. If the
17.4.
NONEQUILIBRIUM PROPERTIES. PINNING.
349
vortex passes through the cavity, this energy is absent. Consequently, there is attraction
between the line and the cavity the force being of the order
f
p
∼
H
2
c
ζ
(we have taken into account that at the distance
∼
ζ
the vortex collides with its image).
Combining this expression with the expression for the Lorentz force we find the critical
current density able to start the motion
j
c
∼
H
c
c
δ
L
(we have used the relations
f
L
=
j
ex
Φ
0
/c
and
H
c
∼
Φ
0
/δ
L
ζ
).
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 Physics, Cubic crystal system, periodic structures, Reciprocal lattice, Lattice Vibrations