102
5 Flux Pinning Phenomena
interactions must be virtually switched off. In fact, if all pinning forces are zero, the
pinning correlation length diverges, and perfect long-range order is achieved. Another
problem is that Labusch’s theory is not a mean field theory, although the Labusch
parameter α L is used to represent the mean strength of the pinning interactions. Here,
a simple discrepancy is shown. Assume that the flux pinning is not effective because
the pinning force is lower than the threshold. Thus, flux lines can move to any place,
even with an infinitesimal driving force. Hence, they can move to places effective
for pinning, resulting in a nonzero pinning force. In practice, flux lines cannot freely
move under the interaction field of the surrounding pinning centers. It is expected,
however, that there is a compatible situation between the statistically summed pinning
force density and the interaction field. This is the essential standpoint of the mean
field theory. In Labusch’s theory, the spring constant in (5.60) is determined under the
fixed condition of the flux line lattice at infinity. This may be based on the idea that
the elastic interaction does not extend to infinity when only one pinning interaction
is switched on. The elastic constant takes on a large value due to this condition,
resulting in the large threshold value. The boundary condition at infinity must be
free, however, if the pinning is not effective. In order to realize long-range order in
the flux line lattice, all other pinning forces must also be virtually switched off. It is
important to clarify how this virtual step contributes to deriving the spring constant.
Here, the coherent potential approximation theory [17] is introduced. This theory
is a kind of mean field theory used for random systems. The mean interaction field of
surrounding pinning centers is assumed first, and then, the pinning force density is
formally determined under the influence of the mean field strength using a statistical
method. Finally, the pinning force density and mean field strength are determined
self-consistently. In the case of flux pinning phenomena in superconductors, the
Labusch parameter α L is used as a parameter representing the mean field strength of
flux pinning. The mean spacing of pinning centers is denoted by d p . Then, we have
N p = d
−3
p . The easiest way to perform the statistical summation is to employ a model
in which each elementary region of flux lines includes a single pinning center. In this
case, all elementary regions are equivalent to each other, and it is not necessary to
consider the interaction among pinning centers in each elementary region. Hence,
the volume of each elementary region is V = d
3
p . The details of the shape of the
elementary region will be shown later.
If all the pinning interactions are virtually switched off at the same time to realize
the long-range order in the flux line lattice, no information can be obtained. Hence,
we first switch off the pinning force in a representative elementary region, as done
by Labusch. We assume that the flux line interacting with the pinning center at x
moves to x 0 . The force balance in this case is
k f (x 0 − x) + f (x) = 0,
(5.80)
where k f is the spring constant. The first term is the restoring force against the
displacement by x 0 − x. The result of Labusch’s theory can be used for k f . Next, we
assume that all remaining pinning interactions are switched off. In this case the flux
line lattice is perfect, and the observed flux line is assumed to move to . This means
5 Flux Pinning Phenomena
interactions must be virtually switched off. In fact, if all pinning forces are zero, the
pinning correlation length diverges, and perfect long-range order is achieved. Another
problem is that Labusch’s theory is not a mean field theory, although the Labusch
parameter α L is used to represent the mean strength of the pinning interactions. Here,
a simple discrepancy is shown. Assume that the flux pinning is not effective because
the pinning force is lower than the threshold. Thus, flux lines can move to any place,
even with an infinitesimal driving force. Hence, they can move to places effective
for pinning, resulting in a nonzero pinning force. In practice, flux lines cannot freely
move under the interaction field of the surrounding pinning centers. It is expected,
however, that there is a compatible situation between the statistically summed pinning
force density and the interaction field. This is the essential standpoint of the mean
field theory. In Labusch’s theory, the spring constant in (5.60) is determined under the
fixed condition of the flux line lattice at infinity. This may be based on the idea that
the elastic interaction does not extend to infinity when only one pinning interaction
is switched on. The elastic constant takes on a large value due to this condition,
resulting in the large threshold value. The boundary condition at infinity must be
free, however, if the pinning is not effective. In order to realize long-range order in
the flux line lattice, all other pinning forces must also be virtually switched off. It is
important to clarify how this virtual step contributes to deriving the spring constant.
Here, the coherent potential approximation theory [17] is introduced. This theory
is a kind of mean field theory used for random systems. The mean interaction field of
surrounding pinning centers is assumed first, and then, the pinning force density is
formally determined under the influence of the mean field strength using a statistical
method. Finally, the pinning force density and mean field strength are determined
self-consistently. In the case of flux pinning phenomena in superconductors, the
Labusch parameter α L is used as a parameter representing the mean field strength of
flux pinning. The mean spacing of pinning centers is denoted by d p . Then, we have
N p = d
−3
p . The easiest way to perform the statistical summation is to employ a model
in which each elementary region of flux lines includes a single pinning center. In this
case, all elementary regions are equivalent to each other, and it is not necessary to
consider the interaction among pinning centers in each elementary region. Hence,
the volume of each elementary region is V = d
3
p . The details of the shape of the
elementary region will be shown later.
If all the pinning interactions are virtually switched off at the same time to realize
the long-range order in the flux line lattice, no information can be obtained. Hence,
we first switch off the pinning force in a representative elementary region, as done
by Labusch. We assume that the flux line interacting with the pinning center at x
moves to x 0 . The force balance in this case is
k f (x 0 − x) + f (x) = 0,
(5.80)
where k f is the spring constant. The first term is the restoring force against the
displacement by x 0 − x. The result of Labusch’s theory can be used for k f . Next, we
assume that all remaining pinning interactions are switched off. In this case the flux
line lattice is perfect, and the observed flux line is assumed to move to . This means