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lattice, while the two-mode PFC model describes the FCC lattice and 2D square
lattice. Detailed derivation of the one-mode PFC will be derived below, and the
two-mode PFC model can be derived following the same process.
Transforming the ˆ
C 2 (k) back into real space, it should be noted that multiplying
by a quadratic term of k in Fourier space is equivalent to a Laplacian in real space.
2
Therefore, in real space, the expression in Eq. (5.14) becomes:
dr 2 n 1 C 2 (r 1 , r 2 )n 2 ≈ n 1 (−ˆ c 0 − ˆ
c 2 ∇
2
− ˆ
c 4 ∇
4
)n 1
(5.15)
Substituting Eqs. (5.15) into (5.13):
F =
dr
n
2
2
−
n
3
6
+
n
4
12
+
¯
ρ
2
n(ˆ c 0 + ˆ
c 2 ∇
2
+ ˆ
c 4 ∇
4
)n
=
dr
1 + ¯
ρ ˆ
c 0
2
n
2
+ n
¯
ρ ˆ
c 2
2
∇
2 n + n
¯
ρ ˆ
c 4
2
∇
4 n −
n
3
6
+
n
4
12
=
dr
B
l
2
n
2
+
B
x
2
n(2R
2
∇
2
+ R
4
∇
4
)n −
n
3
6
+
n
4
12
(5.16)
where B
l
≡ 1 + ¯
ρ ˆ
c 0 , B
x
≡ ¯
ρ(ˆ c 2 )
2
/4ˆ c 4 , and R ≡
2
ˆ
c 4
/ˆ c 2 . This is the general form
of free energy functional in the one-mode PFC model [7, 8].
As mentioned earlier, the periodic crystal structure described by one-mode PFC
model is BCC lattice, which can be written as:
n = n 0 + φ
cos(qx) cos(qy) + cos(qx) cos(qz) + cos(qy) cos(qz)
(5.17)
3D visualizations of n are provided in Fig. 5.2. Substituting Eq. (5.17) expression
into Eq. (5.16) and integrate over a unit cell, i.e., −π/q x, y, z π/q, a function
F (q, n 0 , φ) with respect to q, n 0 and φ is given. To find the stable structures is to
determine the minima of function F . The corresponding expression with respect to
q is:
dF
dq
= 3B
x
(−q + 2q
3
) = 0
(5.18)
which gives q = 1/
√
2 for the BCC lattice. The results for n 0 and φ are more complicated:
2 Write the inverse Fourier transform of C(k) and take second order differentials:
c(x) = F
−1 C(k) =
C(k)e
i2π kx dk, F
∂ 2 c(x)
∂x 2
= −(2π k)
2 C(k)