Environmental Engineering Reference
In-Depth Information
In the case
c
= 1, both the VCA and the CPA results coincide with the
usual RPA result.
K
(
ω
) is itself determined by the effective parameters,
and (5.6.13), with
i
=
j
, leads to the following self-consistent equation
N
q
K
(
ω
)=
1
c
J
(
q
)
χ
(
ω
)
T
E
(
q
,ω
)
.
(5
.
6
.
17
a
)
This result may be written
K
(
ω
)=
1
N
q
(
q
)
D
E
(
q
,ω
)
−
1
=
q
(
q
)
χ
(
q
,ω
)
q
J
J
χ
(
q
,ω
)
,
(5
.
6
.
17
b
)
corresponding to the condition that the
effective
T-matrix vanishes when
summed over
q
,
q
T
E
(
q
,ω
) = 0, in accordance with our starting as-
sumption, (5.6.10).
In order to derive the effective medium result (5.6.13),
χ
(
j
j
,ω
)
in (5.6.11) was replaced by
c
j
χ
(
ω
), which is an approximation, as this
response depends on the actual surroundings, including the sites
i
and
j
. The CPA incorporates the same type of mistake as in the VCA, but
it is clear that the frequency of such errors is substantially reduced. The
dependence of
χ
(
j
j
,ω
)on
c
i
and
c
j
, corresponding to a site dependence
of
K
(
j
,ω
), becomes relatively unimportant if the configuration number
Z
is large, since
i
or
j
may only be one of the
Z
neighbours of the site
j
.
The effective medium procedure is straightforwardly generalized to
the case where
χ
2
(
i, ω
) is non-zero (Jensen 1984). Again the CPA result
may be expressed in the same way as the VCA result, (5.6.5-6), except
that all the quantities are replaced by their effective CPA counterparts;
J
J
E
(
q
), given by (5.6.14), and
χ
o
(
q
) becomes
r
(
ω
) in (5.6.6) is replaced
by
χ
r
(
ω
)=
1
γ
rr
χ
r
(
ω
)
K
(
ω
)
−
1
χ
r
(
ω
)
,
−
(5
.
6
.
18)
where the effective-medium parameter
K
(
ω
) is determined by the same
self-consistent equation (5.6.17) as above. To a first approximation,
D
E
(
q
,ω
)
−
1
in this equation may be replaced by the simpler virtual-
crystal result. Because of the poles in
D
(
q
,ω
)
−
1
, both the real and
imaginary parts of
K
(
ω
) are usually non-zero, and the imaginary con-
tribution then predicts a finite lifetime for the excitations, due to the
static disorder. This leading-order result may serve as the starting point
in an iterative calculation of
K
(
ω
), and thus of a more accurate CPA
result.
It is much more complicated to include the effects of off-diagonal
disorder. They have been considered in the papers referred to above, but
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