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 ) .
(5 . 6 . 17 a )
This result may be written
K ( ω )= 1
N
q
( q ) D E ( q ) 1 =
q
( q ) χ ( q )
q
J
J
χ ( q ) ,
(5 . 6 . 17 b )
corresponding to the condition that the effective T-matrix vanishes when
summed over q , q T E ( q ) = 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 )
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 )on c i and c j , corresponding to a site dependence
of K ( j ), 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 ) 1 in this equation may be replaced by the simpler virtual-
crystal result. Because of the poles in D ( q ) 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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