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5 Flux Pinning Phenomena
When the size of the superconductor is comparable to or only slightly larger than the
characteristic length, the reversible phenomena become prominent.
It is assumed that a superconductor in the normal state is cooled down in a magnetic
field. In this condition, the magnetic flux density in the superconductor is expected
to be constant on a macroscopic scale and is denoted by B. There is no macroscopic
current inside the superconductor. In this case, a group of flux lines, which will move
collectively if necessary, are considered to stay at the bottom of the pinning potential.
It is assumed that these flux lines are displaced by distance u by an increase in the
external magnetic field or by an applied current. Then, the force that the flux lines
receive from the pinning potential will be given by
F = −α L u
(5.45)
in a unit volume, where α L is the Labusch parameter that represents the strength of the
flux pinning. The variation in the magnetic flux density caused by the displacement
of flux lines is denoted by b. Integrating the continuity equation for flux lines, (5.34),
with time, we obtain
∇ × (B × u) = −b.
(5.46)
For simplicity, we assume that the magnetic field is applied along the z-axis and the
displacement is along the x-axis. Then, the spatial variation occurs only along the
x-axis, and (5.46) is reduced to
B
∂u
∂x
= −b,
(5.47)
where b is the z-component of b. The restoring force, i.e., the Lorentz force, acts to
reduce the strain that appears in the flux line system. This acts along the x-axis as
F L = −
(B + b)
μ 0
∂
∂x
(B + b) ∼ = −
B
μ 0
∂b
∂x
.
(5.48)
Since this is balanced with the pinning force given by (5.45), we have
B
μ 0
∂b
∂x
= −α L u.
(5.49)
Elimination of u using (5.47) leads to
B
2
μ 0
∂
2 b
∂x 2 = α L b.
(5.50)
Under the boundary condition that b = 0 on the surface x = 0 and the condition that
b is finite at infinity (x → ∞), (5.50) is solved as
5 Flux Pinning Phenomena
When the size of the superconductor is comparable to or only slightly larger than the
characteristic length, the reversible phenomena become prominent.
It is assumed that a superconductor in the normal state is cooled down in a magnetic
field. In this condition, the magnetic flux density in the superconductor is expected
to be constant on a macroscopic scale and is denoted by B. There is no macroscopic
current inside the superconductor. In this case, a group of flux lines, which will move
collectively if necessary, are considered to stay at the bottom of the pinning potential.
It is assumed that these flux lines are displaced by distance u by an increase in the
external magnetic field or by an applied current. Then, the force that the flux lines
receive from the pinning potential will be given by
F = −α L u
(5.45)
in a unit volume, where α L is the Labusch parameter that represents the strength of the
flux pinning. The variation in the magnetic flux density caused by the displacement
of flux lines is denoted by b. Integrating the continuity equation for flux lines, (5.34),
with time, we obtain
∇ × (B × u) = −b.
(5.46)
For simplicity, we assume that the magnetic field is applied along the z-axis and the
displacement is along the x-axis. Then, the spatial variation occurs only along the
x-axis, and (5.46) is reduced to
B
∂u
∂x
= −b,
(5.47)
where b is the z-component of b. The restoring force, i.e., the Lorentz force, acts to
reduce the strain that appears in the flux line system. This acts along the x-axis as
F L = −
(B + b)
μ 0
∂
∂x
(B + b) ∼ = −
B
μ 0
∂b
∂x
.
(5.48)
Since this is balanced with the pinning force given by (5.45), we have
B
μ 0
∂b
∂x
= −α L u.
(5.49)
Elimination of u using (5.47) leads to
B
2
μ 0
∂
2 b
∂x 2 = α L b.
(5.50)
Under the boundary condition that b = 0 on the surface x = 0 and the condition that
b is finite at infinity (x → ∞), (5.50) is solved as