Environmental Engineering Reference
In-Depth Information
approximation. It is known (Yonezawa 1968) that the cumulant ex-
pansion, in solving the dilute RPA equation (5.6.8), includes all terms
proportional to
P
2
(
ij
)=
c
2
, but that this occurs at the ex-
pense of 'self-containedness', leading to unphysical features in the final
results. Compared to this, the CPA neglects some of the products of
P
2
(
ii
)
P
2
(
jj
) for neighbouring sites, which are of the order (1
/Z
)
2
(see
the discussion following (5.6.17)), but it is self-contained and the results
are well-behaved and accurate if
Z
is not small, as discussed by Elliott
et al.
(1974). Hence, referring to the analyses of the dilute systems, we
expect the effective-medium approximation to be more adequate than
the unrestricted cumulant expansion in the first order of 1
/Z
.Moreim-
portantly, the Hartree-Fock decoupling of the higher-order cumulants,
i.e.
c
i
c
j
−
(6)
=(
(2)
)
3
+3
(2)
(4)
(2)
)
2
to first order in 1
/Z
,which
is one of the basic ideas behind the cumulant-expansion method consid-
ered here, does not appear to be a good approximation. The effective-
medium model is not solved 'exactly', as this would require a determi-
nation of the whole series for
G
(
iω
n
)in(7
.
2
.
5
a
), but a consideration
of the second- and higher-order diagrams in this series indicates that
the Dyson-equation generalization is much more reasonable. The sum
rules, like (3.3.18) or the 'monotopic restriction' discussed by Haley and
Erdos (1972), are satisfied to the considered order in 1
/Z
. This is ob-
viously true for the unrestricted cumulant expansion, but it also holds
for the effective-medium approximation, as this is derived directly from
the behaviour of the single sites. One may ask (Galili and Zevin 1987)
whether there exists any other 'conservation law' which permits a more
stringent distinction between the various possibilities. For this purpose,
we propose to use the condition that the resultant Green function should
be independent of adding the following constant to the Hamiltonian:
S
S
S
{S
−
(
S
}
λ
i
∆
H
=
−
J
i
·
J
i
=
−
NλJ
(
J
+1)
,
(7
.
2
.
9)
corresponding to a replacement of
(
q
)+
λ
. This change does
not affect the effective-medium equation (5.6.9), other than by adding
the constant to
J
(
q
)by
J
(
q
), so
K
(
iω
n
) is still determined by (7
.
2
.
7
c
), with
λ
added on the r.h.s. A replacement of
K
(
iω
n
)by
K
(
iω
n
)+
λ
in (7
.
2
.
5
a
)
J
does not make any difference, as (1
/β
)
n
g
(
iω
n
)
u
(
n, n
)=
g
(
iω
n
)
when
n
0
+
n
1
=1,sothat
J
iα
J
iα
is a constant. The additions of
λ
to
both
J
(
q
)and
K
(
iω
n
) cancel out in the
q
-dependent Green function
expressed in terms of the single-site Green function, as may be seen
from (7
.
2
.
7
a
), so that the final result is independent of
λ
. Thisisnot
the case when the unrestricted cumulant expansion is used. Formally,
the occurrence of
λ
is a (1
/Z
)
2
-effect, but this is an unphysical feature
which is a serious defect, since
λ
may assume an arbitrary value. This
−
Search WWH ::
Custom Search