5.4 Summation Problem and Irreversibility
97
Here, we have used the relationship d/dx =
f pt + f p
/f pt from (5.64), and x 0
represents the position of the flux line at the left edge given by
x 0 =
a f
4
·
f p − 3f pt
f p + f pt
.
(5.66)
To obtain a nonzero pinning force density the elementary pinning force f p should be
larger than f pt , which is called the threshold value of elementary pinning force.
The statistical calculation method was employed by Labusch [11] and Campbell [5]. Later, Kramer [1] found, however, that there was a significant discrepancy between the results of theoretical calculations and those of experiments on the
threshold value. This raised doubt about the reliability of the statistical calculation
method. As shown in Fig. 5.4, even defects with an elementary pinning force smaller
by a factor of 10
4 than the threshold value can effectively work as pinning centers.
This result may indicate that there is no threshold value.
The problem with the threshold value will be discussed later, but we will now
discuss the situation in which the flux lines are displaced back in the direction of
the negative x-axis from the condition shown in the lower panel of Fig. 5.25. The
distribution changes through that in the middle panel of Fig. 5.25 to the symmetric
distribution with respect to x = 0 in the lower panel. In the final state, the pinning
force density is −F p . For a further displacement, the critical state is kept. This shows
that the macroscopic irreversibility originates from the biased distribution of flux
lines inside the pinning potential, which is associated with the instability of flux
lines. That is, the unstable region appears on the opposite side to the direction of the
Lorentz force, and the resultant pinning force works opposite to the Lorentz force.
Such an instability is an assumption at this moment, but it will be proved theoretically
in Sect. 5.6. The difference between the general irreversibility and the irreversibility
inherent to the flux pinning will be discussed in Sect. 5.7. Zero pinning force density,
i.e., reversibility, for flux pinning strength below f pt is a consequence of the fact that
the local pinning interaction is reversible. This indicates also that the irreversibility
comes from the above point, which is another mechanism. Thus, both reversibility
and irreversibility are explained by Campbell [5]. Note that the displacement of flux
lines measured by experiments is estimated from that of .
5.5 Pinning Loss Energy Density
In this section the pinning loss energy density is treated. All the loss energy is
ohmic and caused by motion of normal electrons in the normal core, as mentioned
in Sect. 4.3. Hence, it has been speculated that the loss energy depends on the flow
resistivity ρ f or viscous coefficient η. As shown in (5.39), however, the AC loss
energy density does not depend on ρ f . The material parameter that affects the loss
energy is the critical current density J c or the pinning force density F p . How can we
understand this result?
97
Here, we have used the relationship d/dx =
f pt + f p
/f pt from (5.64), and x 0
represents the position of the flux line at the left edge given by
x 0 =
a f
4
·
f p − 3f pt
f p + f pt
.
(5.66)
To obtain a nonzero pinning force density the elementary pinning force f p should be
larger than f pt , which is called the threshold value of elementary pinning force.
The statistical calculation method was employed by Labusch [11] and Campbell [5]. Later, Kramer [1] found, however, that there was a significant discrepancy between the results of theoretical calculations and those of experiments on the
threshold value. This raised doubt about the reliability of the statistical calculation
method. As shown in Fig. 5.4, even defects with an elementary pinning force smaller
by a factor of 10
4 than the threshold value can effectively work as pinning centers.
This result may indicate that there is no threshold value.
The problem with the threshold value will be discussed later, but we will now
discuss the situation in which the flux lines are displaced back in the direction of
the negative x-axis from the condition shown in the lower panel of Fig. 5.25. The
distribution changes through that in the middle panel of Fig. 5.25 to the symmetric
distribution with respect to x = 0 in the lower panel. In the final state, the pinning
force density is −F p . For a further displacement, the critical state is kept. This shows
that the macroscopic irreversibility originates from the biased distribution of flux
lines inside the pinning potential, which is associated with the instability of flux
lines. That is, the unstable region appears on the opposite side to the direction of the
Lorentz force, and the resultant pinning force works opposite to the Lorentz force.
Such an instability is an assumption at this moment, but it will be proved theoretically
in Sect. 5.6. The difference between the general irreversibility and the irreversibility
inherent to the flux pinning will be discussed in Sect. 5.7. Zero pinning force density,
i.e., reversibility, for flux pinning strength below f pt is a consequence of the fact that
the local pinning interaction is reversible. This indicates also that the irreversibility
comes from the above point, which is another mechanism. Thus, both reversibility
and irreversibility are explained by Campbell [5]. Note that the displacement of flux
lines measured by experiments is estimated from that of .
5.5 Pinning Loss Energy Density
In this section the pinning loss energy density is treated. All the loss energy is
ohmic and caused by motion of normal electrons in the normal core, as mentioned
in Sect. 4.3. Hence, it has been speculated that the loss energy depends on the flow
resistivity ρ f or viscous coefficient η. As shown in (5.39), however, the AC loss
energy density does not depend on ρ f . The material parameter that affects the loss
energy is the critical current density J c or the pinning force density F p . How can we
understand this result?