Appendix
185
K = α
1/2
L d
2
p
C
4
44
C 66
1/6
.
(A.5.9)
Using this equation, the right-hand side of (5.84) is written as
K
2 a f
ζ d 4
p
C 66
C
4
44
1/3
.
(A.5.10)
Using (5.62), (5.82), and (5.85), the following equation is obtained:
K =
k f t
k f a f /4f p
− t
.
(A.5.11)
Thus, (5.86) is obtained with
β =
ζ d p
4k f
C
4
44
C 66
1/3
.
(A.5.12)
The spring constant k f is given by the inverse compliance of a representative flux line
for the lattice fixed at infinity as [3]
k
−1
f
= G
(0) =
1
4
2
√
3π
1/2 1
a f
(C 44 C 66 )
−1/2
.
(A.5.13)
Thus, (5.87) is obtained from (A.5.12) and (A.5.13).
A.6 Derivation of (5.98)
Here we derive the condition of the minimum free energy given by (5.94) in volume
V of the superconductor. The displacement of flux lines during minimization of the
free energy is denoted by u, and an additional small displacement is denoted by δu.
In this case the following condition should be satisfied:
V
[F(u + δu) − F(u)]dV = 0.
(A.6.1)
The first term is expanded as
F(u + δu) = F(u) +
∂F
∂u
· δu +
∂F
∂u
· δu
,
(A.6.2)
where u
is defined as