196
Appendix
mechanism for the work
E · Jdt done by the general restoring force has not been
clarified in association with energy conservation. In reality this is the work done by
the torque, which exceeds the pinning interaction during the rotational motion of
flux lines, similarly to the usual cases in the transverse magnetic field. We simply
discuss the process of the flux cutting here, however, neglecting such an argument
on the fundamental aspects. To realize the flux cutting, two flux lines with different
angles must come close to each other against a strong repulsive force, but there is no
force to make the flux lines come close to each other. Even if it were assumed that the
flux cutting could happen, the theoretically estimated threshold value for the needed
current density is too high for both the inter-cutting and the intra-cutting processes,
and the observed critical current densities cannot be explained. It may be pointed
out that flux cutting maybe easily take place at normal precipitates, etc. If this is so,
however, the flux cutting maybe more easily occur in a superconductor with more
pinning centers, resulting in a lower critical current density. The practical situation is
the opposite, and such an insistence is meaningless. In reality, the flux cutting does
not take place, since the rotational flux motion with no threshold occurs first.
Here, we introduce another experimental result that clearly shows that the flux
cutting cannot take place. The experimental result shown in Fig. 6.15 clarifies that
the magnetic flux distribution is exactly in the force-free state. The penetration depth
of the AC magnetic flux approaches the value of about 10 μm in the limit of zero AC
field amplitude, indicating that the motion of flux lines is in the regime of reversible
flux pinning. Since the flux line spacing is about 90 nm under a DC magnetic field
of 0.290 T, the variation in the external magnetic field reaches the distance of about
100 rows of flux lines from the surface in this situation. On the other hand, since the
flux cutting is irreversible, the flux cutting process must be completed by the row
behind to start the interaction with flux lines on the next row. That is, the penetration
depth of the flux cutting is on the order of the flux line spacing, and the extension
of such a reversible phenomenon to 10 μm cannot be explained [5]. In reality the
rotational motion of flux lines occurs, and it can extend to the shielding distance of
flux pinning in the reversible regime. Thus, the observed reversible phenomenon can
be explained by the rotational motion of flux lines. Since energy dissipation does not
occur by the mechanism of flux cutting in this regime, the energy is not conserved
in the case of flux cutting, as discussed in Sect. 6.3, resulting in a contradiction.
In addition, relativity is not satisfied for the flux cutting between the case of
rotating the external magnetic field and the case of rotating the superconductor in the
opposite direction in a stationary external magnetic field (see p. 151 in [1]), although
the details are not shown here.
(3) No possibility of explanation by the flux cutting
As described in Chap. 6, the following phenomena cannot be explained by the
mechanism of flux cutting:
(a) the paramagnetic effect in the usual process of application of current after
applying an external magnetic field,
(b) a critical current density that depends on the flux pinning strength, especially,
very small critical current densities for weak pinning,
Appendix
mechanism for the work
E · Jdt done by the general restoring force has not been
clarified in association with energy conservation. In reality this is the work done by
the torque, which exceeds the pinning interaction during the rotational motion of
flux lines, similarly to the usual cases in the transverse magnetic field. We simply
discuss the process of the flux cutting here, however, neglecting such an argument
on the fundamental aspects. To realize the flux cutting, two flux lines with different
angles must come close to each other against a strong repulsive force, but there is no
force to make the flux lines come close to each other. Even if it were assumed that the
flux cutting could happen, the theoretically estimated threshold value for the needed
current density is too high for both the inter-cutting and the intra-cutting processes,
and the observed critical current densities cannot be explained. It may be pointed
out that flux cutting maybe easily take place at normal precipitates, etc. If this is so,
however, the flux cutting maybe more easily occur in a superconductor with more
pinning centers, resulting in a lower critical current density. The practical situation is
the opposite, and such an insistence is meaningless. In reality, the flux cutting does
not take place, since the rotational flux motion with no threshold occurs first.
Here, we introduce another experimental result that clearly shows that the flux
cutting cannot take place. The experimental result shown in Fig. 6.15 clarifies that
the magnetic flux distribution is exactly in the force-free state. The penetration depth
of the AC magnetic flux approaches the value of about 10 μm in the limit of zero AC
field amplitude, indicating that the motion of flux lines is in the regime of reversible
flux pinning. Since the flux line spacing is about 90 nm under a DC magnetic field
of 0.290 T, the variation in the external magnetic field reaches the distance of about
100 rows of flux lines from the surface in this situation. On the other hand, since the
flux cutting is irreversible, the flux cutting process must be completed by the row
behind to start the interaction with flux lines on the next row. That is, the penetration
depth of the flux cutting is on the order of the flux line spacing, and the extension
of such a reversible phenomenon to 10 μm cannot be explained [5]. In reality the
rotational motion of flux lines occurs, and it can extend to the shielding distance of
flux pinning in the reversible regime. Thus, the observed reversible phenomenon can
be explained by the rotational motion of flux lines. Since energy dissipation does not
occur by the mechanism of flux cutting in this regime, the energy is not conserved
in the case of flux cutting, as discussed in Sect. 6.3, resulting in a contradiction.
In addition, relativity is not satisfied for the flux cutting between the case of
rotating the external magnetic field and the case of rotating the superconductor in the
opposite direction in a stationary external magnetic field (see p. 151 in [1]), although
the details are not shown here.
(3) No possibility of explanation by the flux cutting
As described in Chap. 6, the following phenomena cannot be explained by the
mechanism of flux cutting:
(a) the paramagnetic effect in the usual process of application of current after
applying an external magnetic field,
(b) a critical current density that depends on the flux pinning strength, especially,
very small critical current densities for weak pinning,