Page 403 - Electrical Properties of Materials

The Landau–Ginzburg theory 385
2
1 dA

4
2
G s (B)= G n + a 1 ψ + a 2 ψ + B a –
μ 0 dx
2
1 2 ∂ψ 2 2 2
+ +4e A ψ . (14.52)
2m ∂x
The solution of the variational problem is now considerably easier. As
shown in Appendix IV, ψ(x) and A(x) will minimize the integral if they satisfy
the following differential equations:
∂G s (B) d ∂G s (B)
– = 0 (14.53)
∂ψ dx ∂(∂ψ/∂x)
and

∂G s (B) d ∂G s (B)
– = 0. (14.54)
∂A dx ∂(∂A/∂x)

Substituting eqn (14.52) into eqn (14.53) and performing the differenti-
ations, we get

1 2 2 d 1 2 ∂ψ
3
2a 1 ψ +4a 2 ψ + 8e A ψ – 2 = 0, (14.55)
2m dx 2m ∂x
which after rearrangement yields

2
d ψ m 2e 2 2 4m 3
= 2a 1 1+ A ψ + a 2 ψ . (14.56)
dx 2 2 a 1 m 2
Similarly, substituting eqn (14.52) into eqn (14.54) we get

2
2
2
d A 4e ψ μ 0
= A, (14.57)
dx 2 m
which must be solved subject to the boundary conditions,
B = B a = μ 0 H a , dψ/dx =0 at x = 0 (14.58)
2
2
B =0, ψ = ψ , dψ/dx =0 at x = ∞. (14.59)
0
The boundary conditions for the flux density simply mean that at the bound-
ary with the vacuum the flux density is the same as the applied flux density,
and it declines to zero far away inside the superconductor. The condition for
dψ/dx comes from the more stringent general requirement that the normal
component of the momentum should vanish at the boundary. But since in the
one-dimensional case A is parallel to the surface, A · i x is identically zero, and
the boundary condition reduces to the simpler dψ/dx = 0. Since A is determ-
ined except for a constant factor, we can prescribe its value at any point. We
shall choose A(∞)=0.