Environmental Engineering Reference
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where the approximate result follows from the usual RPA decoupling
introduced by eqn (3.5.16). According to eqn (5.4.25),
i
=
k
(
ikF
k
/
2)(
β
k
+
β
+
−
k
)exp(
i
k
·
R
i
)
,
where we assume, for simplicity, only one phonon mode. From the equa-
tions of motion determining the two Green functions
β
q
;
a
rs
(
i
)
and
β
+
−
q
;
a
rs
(
i
)
,weobtain
β
q
+
β
+
−
q
;
a
rs
(
i
)
=
B
γ
2
i
(7
.
3
.
5)
(
iqF
q
/
2)
D
(
q
,ω
)
e
−i
q
·
R
i
N
νµ
a
νµ
(
i
);
a
rs
(
i
)
,
νµ
where
D
(
q
,ω
) is the phonon Green function for the mode considered:
2
ω
ν
q
h
ω
2
ω
ν
q
.
D
ν
(
q
,ω
)=
(7
.
3
.
6)
−
If this is introduced into (7.3.4), and the resulting expression is added
to the l.h.s. of (3.5.18), the procedure leading to eqn (3.5.21) yields the
equivalent result
χ
o
(
ω
)
(
q
,ω
)
χ
(
q
,ω
)=
χ
o
(
ω
)
.
χ
(
q
,ω
)
−
J
(7
.
3
.
7)
However, these quantities are now four-dimensional matrices in the vec-
tor space defined by the operators
J
ix
,
J
iy
,
J
iz
,and
O
−
2
(
J
i
), or more
accurately by these operators minus their expectation values. The only
extra element in
J
(
q
,ω
), in addition to the normal Cartesian compo-
nents
J
αβ
(
q
), is
J
44
(
q
,ω
)=
N
i
2
qF
q
B
γ
2
2
D
(
q
,ω
)
.
(7
.
3
.
8)
The excitation energies are determined by the condition
1
(
q
,ω
)
=0
.
χ
o
(
ω
)
−
J
When
q
is along an
a
-or
b
-direction, and the external fields are applied
in the basal plane, parallel or perpendicular to
q
,then
J
(
q
)andthe
3 Cartesian components of
χ
o
(
ω
), at low frequencies, are diagonal
with respect to the (
ξηζ
)-axes. In this case, the most phonon-like pole
is found at a frequency determined by
1
×
3
(
q
,ω
)
α
1
J
αα
(
q
)
=1
χ
o
(
ω
)
χ
αα
(
ω
)
−
J
−
−
Ξ(
q
,ω
)
J
44
(
q
,ω
)=0
,
(7
.
3
.
9
a
)
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