190
Appendix
where H 0 is the external magnetic field. The condition of the equilibrium state is
obtained from
, and it satisfies
∂F
∂B
= H 0 .
(A.8.2)
The variation in the volume integral of the free energy density from (A.1.9) when
the vector potential changes by δA is given by
−
V
(H 0 · ∇ × δA)dV .
(A.8.3)
This is partially integrated, and neglecting the surface integral, we have
−
V
(δA · ∇ × H 0 )dV = 0.
(A.8.4)
Thus, the Ginzburg-Landau equation (4.4) does not change. Equation (4.3) does not
change either.
The Gibbs free energy density should also be used for the electromagnetic
phenomena in a non-isolated flux line system in a transverse magnetic field. For
example, the Gibbs free energy density is given by
(A.8.5)
when a force of density F is added to the flux line system, where u is the displacement
of flux lines. This is minimized with respect to u, and from the condition
,
we have
F + F p = 0,
(A.8.6)
where we used F p = −∂F/∂u. The variation in the free energy density as a function
of the displacement of the flux line system in the regime of reversible flux motion is
illustrated in Fig. A.2. The spatial variation with period a comes from the first term
of (A.8.5), including the pinning energy density, and the decrease with increasing
displacement comes from the second term. Point A at which the energy density is
locally a minimum gives the stable equilibrium position. The Legendre term for the
second term in (A.8.5) is derived using Poynting’s vector when we extend the flux
line system from the isolated condition to the non-isolated one in the critical state
theory in Sect. 5.7. Such a washboard pinning potential is usually assumed when
discussing the flux creep phenomena.
Appendix
where H 0 is the external magnetic field. The condition of the equilibrium state is
obtained from
, and it satisfies
∂F
∂B
= H 0 .
(A.8.2)
The variation in the volume integral of the free energy density from (A.1.9) when
the vector potential changes by δA is given by
−
V
(H 0 · ∇ × δA)dV .
(A.8.3)
This is partially integrated, and neglecting the surface integral, we have
−
V
(δA · ∇ × H 0 )dV = 0.
(A.8.4)
Thus, the Ginzburg-Landau equation (4.4) does not change. Equation (4.3) does not
change either.
The Gibbs free energy density should also be used for the electromagnetic
phenomena in a non-isolated flux line system in a transverse magnetic field. For
example, the Gibbs free energy density is given by
(A.8.5)
when a force of density F is added to the flux line system, where u is the displacement
of flux lines. This is minimized with respect to u, and from the condition
,
we have
F + F p = 0,
(A.8.6)
where we used F p = −∂F/∂u. The variation in the free energy density as a function
of the displacement of the flux line system in the regime of reversible flux motion is
illustrated in Fig. A.2. The spatial variation with period a comes from the first term
of (A.8.5), including the pinning energy density, and the decrease with increasing
displacement comes from the second term. Point A at which the energy density is
locally a minimum gives the stable equilibrium position. The Legendre term for the
second term in (A.8.5) is derived using Poynting’s vector when we extend the flux
line system from the isolated condition to the non-isolated one in the critical state
theory in Sect. 5.7. Such a washboard pinning potential is usually assumed when
discussing the flux creep phenomena.