5.5 Pinning Loss Energy Density
101
i.e., in the continuous region of −3a f /4 < x < x 0 , and this leads to an appreciable
loss, where x 0 is given by (5.66).
As shown above, the description of the summation theory for flux pinning and
that of the loss energy caused by unstable flux motion are formally valid, and the
essential problem is summarized as the determination of the threshold value of the
elementary pinning force. In the next section, the summation theory is treated again
to explore the threshold value problem.
5.6 Coherent Potential Approximation Theory
Larkin and Onchinnikov [16] proposed a new summation theory after the threshold
value problem in the statistical theory of Labusch [11] was pointed out. The main
point of this theory was that, since the pinning correlation lengths are finite, individual
pinning forces in each correlated region are not perfectly cancelled out, resulting in
a nonzero macroscopic pinning force density. One of the correlated lengths is Campbell’s AC penetration depth, given by (5.52). This length gives the distance over which
a perturbation such as displacement of flux lines extends and becomes shorter for
stronger pinning. It was assumed in Labusch’s theory that a perfect flux line lattice
with long-range order, which gives the lattice point , can be achieved by virtually switching off the interaction force of the representative pinning center alone.
Since strains remain in the flux line lattice due to the many surrounding pinning
centers, as pointed out by Larkin and Onchinnikov, the assumption in Labusch’s
theory is not correct. There are also points, however, in the Larkin-Onchinnikov
theory. Although the pinning forces remain without being perfectly cancelled out in
each correlated region, there is no theoretical proof of the assumption that these forces
must be directed opposite to the Lorentz force. That is, it is impossible to explain
the assumption in terms of the randomness alone. If individual pinning forces obey a
Gaussian distribution with standard deviation σ , the pinning force in each correlated
region can be approximated by a sum of pinning forces sampled randomly from the
Gaussian distribution. If the number of samples is N, the pinning forces in each correlated region are known to obey another Gaussian distribution with standard deviation
√
N σ . The expected pinning force density, which is proportional to the average of
this distribution, is zero. Since there is no interaction among correlated regions, the
speculation by Larkin and Ovchinnikov might be allowed. This is, however, only a
hope, and it is not possible to prove it. In addition, the correlation extends over the
whole superconductor in the critical state. The unstable flux motion is inevitable for
a finite pinning force density, which is also effective in the critical state. In addition,
the distinction between the reversible and irreversible phenomena is not clear. In
particular, the mechanism causing the irreversibility is not clearly given.
For this reason, we have to employ the statistical summation theory that can
describe the irreversibility. Here, we will examine the problems in Labusch’s theory.
One of them is the point made by Larkin and Ovchinnikov on the pinning correlation
lengths. In order to achieve long- range order in the flux line lattice, all pinning
101
i.e., in the continuous region of −3a f /4 < x < x 0 , and this leads to an appreciable
loss, where x 0 is given by (5.66).
As shown above, the description of the summation theory for flux pinning and
that of the loss energy caused by unstable flux motion are formally valid, and the
essential problem is summarized as the determination of the threshold value of the
elementary pinning force. In the next section, the summation theory is treated again
to explore the threshold value problem.
5.6 Coherent Potential Approximation Theory
Larkin and Onchinnikov [16] proposed a new summation theory after the threshold
value problem in the statistical theory of Labusch [11] was pointed out. The main
point of this theory was that, since the pinning correlation lengths are finite, individual
pinning forces in each correlated region are not perfectly cancelled out, resulting in
a nonzero macroscopic pinning force density. One of the correlated lengths is Campbell’s AC penetration depth, given by (5.52). This length gives the distance over which
a perturbation such as displacement of flux lines extends and becomes shorter for
stronger pinning. It was assumed in Labusch’s theory that a perfect flux line lattice
with long-range order, which gives the lattice point , can be achieved by virtually switching off the interaction force of the representative pinning center alone.
Since strains remain in the flux line lattice due to the many surrounding pinning
centers, as pointed out by Larkin and Onchinnikov, the assumption in Labusch’s
theory is not correct. There are also points, however, in the Larkin-Onchinnikov
theory. Although the pinning forces remain without being perfectly cancelled out in
each correlated region, there is no theoretical proof of the assumption that these forces
must be directed opposite to the Lorentz force. That is, it is impossible to explain
the assumption in terms of the randomness alone. If individual pinning forces obey a
Gaussian distribution with standard deviation σ , the pinning force in each correlated
region can be approximated by a sum of pinning forces sampled randomly from the
Gaussian distribution. If the number of samples is N, the pinning forces in each correlated region are known to obey another Gaussian distribution with standard deviation
√
N σ . The expected pinning force density, which is proportional to the average of
this distribution, is zero. Since there is no interaction among correlated regions, the
speculation by Larkin and Ovchinnikov might be allowed. This is, however, only a
hope, and it is not possible to prove it. In addition, the correlation extends over the
whole superconductor in the critical state. The unstable flux motion is inevitable for
a finite pinning force density, which is also effective in the critical state. In addition,
the distinction between the reversible and irreversible phenomena is not clear. In
particular, the mechanism causing the irreversibility is not clearly given.
For this reason, we have to employ the statistical summation theory that can
describe the irreversibility. Here, we will examine the problems in Labusch’s theory.
One of them is the point made by Larkin and Ovchinnikov on the pinning correlation
lengths. In order to achieve long- range order in the flux line lattice, all pinning