Environmental Engineering Reference
In-Depth Information
leaving out the common factor
c
i
. Although this means that
K
(
i, ω
)
and
(
ij, ω
) may be non-zero even when
c
i
is zero, this has no conse-
quences in eqn (5.6.10). In order to derive the configurational average of
this equation, we make the assumption that each site is surrounded by
thesameeffectivemedium. Hence
K
(
i, ω
)
T
K
(
ω
)isconsideredtobe
independent of the site considered, and therefore we have, from (5.6.9),
χ
(
ii, ω
)=
c
i
χ
(
ω
);
χ
(
ω
)=
1
χ
o
(
ω
)
K
(
ω
)
−
1
χ
o
(
ω
)
.
−
(5
.
6
.
12)
With this replacement, the configurational average of eqn (5.6.11) may
be derived straightforwardly, as we can take advantage of the condition
that, for instance,
c
j
only occurs once in the sum over
j
.Itisimportant
here that the common factor
c
i
was cancelled, because
(
j
j, ω
)involves
T
(
j
j, ω
) more complicated. Intro-
the site
i
, making the averaging of
c
i
T
ducing the notation
T
E
(
ij, ω
)=
T
(
ij, ω
)
cf
, we get from (5.6.11) the
CPA equation
(
ij
)
χ
(
ω
)+
j
c
2
(
ij
)
χ
(
ω
)
T
E
(
j
j, ω
)
χ
(
ω
)
δ
ij
+
c
J
J
=
{
1+
c K
(
ω
)
χ
(
ω
)
}{
δ
ij
+
c T
E
(
ij, ω
)
χ
(
ω
)
}
(5
.
6
.
13)
for the effective medium, which may be diagonalized by a Fourier trans-
formation. Introducing the effective coupling parameter
J
E
(
q
)=
J
(
q
)
−
K
(
ω
)
,
(5
.
6
.
14)
where the scalar appearing in a matrix equation is, as usual, multiplied
by the unit matrix, we get
J
E
(
q
)
D
E
(
q
,ω
)
−
1
T
E
(
q
,ω
)=
;
D
E
(
q
,ω
)=1
−
c χ
(
ω
)
J
E
(
q
)
(5
.
6
.
15)
and, from (5.6.10),
χ
(
q
,ω
)=
c χ
(
ω
)+
c
2
χ
(
ω
)
T
E
(
q
,ω
)
χ
(
ω
)=
D
E
(
q
,ω
)
−
1
c χ
(
ω
)
.
(5
.
6
.
16)
Hence the result is similar to that obtained in the VCA, except that the
parameters are replaced by the effective quantities introduced by eqns
(5.6.12) and (5.6.14). These effective values are determined from the
'bare' parameters in terms of
K
(
ω
). It is easily seen that we retain the
VCA result, i.e.
K
(
ω
) cancels out of (5.6.15), if (5.6.12) is replaced by
the corresponding VCA equation
χ
(
ω
)
1
c χ
o
(
ω
)
K
(
ω
)
−
1
χ
o
(
ω
).
−
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