Environmental Engineering Reference
In-Depth Information
corresponding to
g
i
(
ω
), may be written
hω
j
A
i
J
⊥
(
Q
)
G
(
ij, ω
)=2
(
ij
)
G
(
j
j, ω
)
,
(6
.
1
.
27)
−A
i
J
A
i
δ
ij
−
obtained from the RPA equation (3.5.7) by multiplying with the energy
denominator in
g
i
(
ω
). We introduce the Fourier transforms
N
ij
G
n
(
q
,ω
)=
1
G
(
ij, ω
)
e
−i
q
·
(
R
i
−
R
j
)
e
−in
Q
·
R
i
,
(6
.
1
.
28)
where
n
is an integer, and the coupling parameter
−
2
A
J
(
Q
)
−J
⊥
(
q
+
n
Q
)
,
γ
n
(
q
)=
(6
.
1
.
29)
which is always negative (
A
>
0), as the stability of the structure re-
quires
J
(
Q
)
−J
⊥
(
Q
)
>
0. From (6.1.27), we then obtain the infinite
set of equations
hωG
n
(
q
,ω
)+
γ
n
+1
(
q
)
G
n
+1
(
q
,ω
)+
γ
n−
1
(
q
)
G
n−
1
(
q
,ω
)=
A
(
δ
n,
1
+
δ
n,−
1
)
(6
.
1
.
30)
whenever
Q
is incommensurable.
In a commensurable structure, for
which
m
Q
=
p
, we determine
G
n
(
q
,ω
)=
G
n
+
m
(
q
,ω
) by the corre-
sponding finite set of
m
equations. Of the infinite number of Green
functions, we wish to calculate the one with
n
= 0, as the transverse
τ
scattering function is proportional to Im
G
0
(
q
,ω
)
.
It is possible to rewrite eqn (6.1.30) so that
G
0
(
q
,ω
) is expressed
in terms of
infinite continued fractions
. In order to derive such an ex-
pression, we shall introduce the semi-infinite determinant
D
n
,with
n
positive,
hω
γ
n
+1
0
0
0
0
···
γ
n
hω
γ
n
+2
0
0
0
···
0
γ
n
+1
hω
γ
n
+3
0
0
···
D
n
=
(6
.
1
.
31)
0
0
γ
n
+2
hω
γ
n
+4
0
···
.
.
.
.
.
.
.
.
.
.
.
leaving out the variables
q
and
ω
. Expanding the determinant in terms
of the first column, we have
D
n
=
hωD
n
+1
−
γ
n
γ
n
+1
D
n
+2
.
(6
.
1
.
32)
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