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is transformed by using Parseval's theorem to give
∞
∞
2
2
3
(
1
2
(
2
π
iuG
(
u
,
v
)
−
F
a
(
u
,
v
)+
F
b
(
u
,
v
)
−
F
c
(
u
,
v
)))
−
∞
−
∞
2
dudv
1
√
3
(
+
2
π
ivG
(
u
,
v
)
−
F
b
(
u
,
v
)+
F
c
(
u
,
v
))
.
(64)
Inconsistency on hexagonal lattices is therefore defined as
2
dudv
)
+
)
2
J
hex
G
x
(
G
y
(
≡
u
,
v
u
,
v
,
(65)
D
where
2
3
(
1
2
(
G
x
(
u
,
v
)
≡
2
π
iuG
(
u
,
v
)
−
F
a
(
u
,
v
)+
F
b
(
u
,
v
)
−
F
c
(
u
,
v
)))
(66)
1
√
3
(
G
y
(
u
,
v
)
≡
2
π
ivG
(
u
,
v
)
−
F
b
(
u
,
v
)+
F
c
(
u
,
v
))
.
(67)
And, gradient intensity
J
hex
1
is defined as follows.
J
hex
1
2
2
2
u
2
v
2
2
≡
D
(
|
G
x
(
u
,
v
)
|
+
|
G
y
(
u
,
v
)
|
)
dudv
=
D
(
4
π
(
+
)
|
G
(
u
,
v
)
|
)
dudv
.
The following expressions are defined to simplify the above expressions.
1
√
3
u
2
3
v
1
2
(
η
b
mn
c
mn
a
mn
b
mn
c
mn
ψ
mn
≡
(
η
+
η
)
−
(
η
+
−
η
))
(68)
2
3
u
1
2
(
η
1
√
3
v
mn
mn
mn
mn
mn
φ
mn
≡
(
η
+
−
η
))+
(
η
+
η
)
(69)
)
≡
∂
∂
2
3
(
1
2
(
g
x
(
x
,
y
x
g
(
x
,
y
)
−
f
a
(
x
,
y
)+
f
b
(
x
,
y
)
−
f
c
(
x
,
y
)))
(70)
)
≡
∂
∂
1
√
3
(
g
y
(
x
,
y
y
g
(
x
,
y
)
−
f
b
(
x
,
y
)+
f
c
(
x
,
y
))
(71)
G
x
(
u
,
v
)
|
2
G
y
(
u
,
v
)
|
2
Ψ
0
(
u
,
v
)
≡
|
+
|
|
F
(
u
,
v
)
|
2
1
1
√
3
u
(
H
b
(
u
,
v
)+
H
c
(
u
,
v
))
−
2
3
v
(
H
a
(
u
,
v
)+
1
2
(
H
b
(
u
,
v
)
−
H
c
(
u
,
v
)))
|
2
=
+
v
2
|
u
2
16
2
+
v
2
(
m
,
n
a
mn
ψ
mn
)
=
(72)
u
2
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