110
5 Flux Pinning Phenomena
δu(z) =
μ 0 z
2
2B 0
δJ ,
(5.116)
where δB x (z) = μ 0 δJz is the increment of the x component of the magnetic flux
density when the current is slightly increased. We have used the boundary condition
δu(0) = 0. If we denote the elastic restoring force density by f , we have
f =
1
δz
lim
δJ →0
δW
δu
= JB 0 .
(5.117)
Thus, the driving force density f is the Lorentz force, i.e., the line tension. On the
other hand, the pinning force density is derived from the pinning energy density U p
and is formally the same as that for the isolated flux line system. Hence, the force
balance (5.104) is obtained also for non-isolated systems [19].
As shown above, the energy penetrating the superconductor during the virtual
displacement, which is obtained using Poynting’s vector, is larger than the increase
in the magnetic energy, and the additional energy density δW is independent of the
magnetic flux distribution and cannot be seen in it. This will be discussed in more
detail in Appendix A.7. In the above case the additional energy is absorbed as an
increase in the pinning energy to realize the assumed magnetic flux distribution,
i.e., the increase from the initial isolated state to the new equilibrium state after the
displacement driven by the restoring force. Since the pinning energy is a kind of
thermodynamic energy, it cannot be seen from the viewpoint of electromagnetism.
The same result is obtained for the case of magnetic pressure.
As shown above, the driving force is proved to be the Lorentz force, and the Gibbs
free energy density is generally described as
(5.118)
The force balance equation in the critical state model is derived from the condition
of the minimum Gibbs free energy,
(see Appendix A.8).
The pinning force discussed up to now is reversible in nature and derived from the
pinning energy. Finally, it is necessary to extend such a reversible pinning force to
the irreversible one under practical conditions. This is done by the summation theory
for flux pinning in Sect. 5.4. In the final state, the pinning force density F changes
to the irreversible pinning force density with the maximum value F p , and the force
balance (5.104) changes to (5.6) in the critical state model.
As shown here, the critical state model that can explain various experimental
results is no longer a phenomenological model but a rigorous theory. In particular, the
summation theory used in the proof is regarded as a useful theory that can statistically
derive a macroscopic irreversibility from the random interactions of many bodies
among pinning potentials. The resultant irreversibility is caused by the biased flux
distribution due to unstable flux motion inside pinning potentials. Hence, the origin
of the irreversibility in this case is independent of the usual breaking of time reversal
symmetry. Namely, the irreversibility in the flux pinning results from the difference
5 Flux Pinning Phenomena
δu(z) =
μ 0 z
2
2B 0
δJ ,
(5.116)
where δB x (z) = μ 0 δJz is the increment of the x component of the magnetic flux
density when the current is slightly increased. We have used the boundary condition
δu(0) = 0. If we denote the elastic restoring force density by f , we have
f =
1
δz
lim
δJ →0
δW
δu
= JB 0 .
(5.117)
Thus, the driving force density f is the Lorentz force, i.e., the line tension. On the
other hand, the pinning force density is derived from the pinning energy density U p
and is formally the same as that for the isolated flux line system. Hence, the force
balance (5.104) is obtained also for non-isolated systems [19].
As shown above, the energy penetrating the superconductor during the virtual
displacement, which is obtained using Poynting’s vector, is larger than the increase
in the magnetic energy, and the additional energy density δW is independent of the
magnetic flux distribution and cannot be seen in it. This will be discussed in more
detail in Appendix A.7. In the above case the additional energy is absorbed as an
increase in the pinning energy to realize the assumed magnetic flux distribution,
i.e., the increase from the initial isolated state to the new equilibrium state after the
displacement driven by the restoring force. Since the pinning energy is a kind of
thermodynamic energy, it cannot be seen from the viewpoint of electromagnetism.
The same result is obtained for the case of magnetic pressure.
As shown above, the driving force is proved to be the Lorentz force, and the Gibbs
free energy density is generally described as
(5.118)
The force balance equation in the critical state model is derived from the condition
of the minimum Gibbs free energy,
(see Appendix A.8).
The pinning force discussed up to now is reversible in nature and derived from the
pinning energy. Finally, it is necessary to extend such a reversible pinning force to
the irreversible one under practical conditions. This is done by the summation theory
for flux pinning in Sect. 5.4. In the final state, the pinning force density F changes
to the irreversible pinning force density with the maximum value F p , and the force
balance (5.104) changes to (5.6) in the critical state model.
As shown here, the critical state model that can explain various experimental
results is no longer a phenomenological model but a rigorous theory. In particular, the
summation theory used in the proof is regarded as a useful theory that can statistically
derive a macroscopic irreversibility from the random interactions of many bodies
among pinning potentials. The resultant irreversibility is caused by the biased flux
distribution due to unstable flux motion inside pinning potentials. Hence, the origin
of the irreversibility in this case is independent of the usual breaking of time reversal
symmetry. Namely, the irreversibility in the flux pinning results from the difference