Environmental Engineering Reference
In-Depth Information
approach of Jensen (1984). We first consider the case where
χ
2
(
ω
)
vanishes identically, corresponding to the presence of non-magnetic im-
purities with a concentration 1
c
.TheRPAequation(5
.
6
.
4
a
)may
then be solved formally by iteration:
χ
(
ij, ω
)=
c
i
χ
o
(
ω
)
δ
ij
+
c
i
χ
o
(
ω
)
−
(
ij
)
c
j
χ
o
(
ω
)
J
+
j
c
i
χ
o
(
ω
)
(
ij
)
c
j
χ
o
(
ω
)
(
j
j
)
c
j
χ
o
(
ω
)+
J
J
···
.
(5
.
6
.
8)
c
j
cf
=
c
n
, which is incorrect
since
c
j
cf
=
c
j
cf
=
c
. Consequently, the VCA leads to errors already
in the fourth term in this expansion, or in the third term if
i
=
j
,even
though
The VCA result is obtained by assuming
(
ii
) is zero. In order to ameliorate these deficiencies, we first
consider the series for
χ
(
ii, ω
), where
i
=
j
. The different terms in this
series may be collected in groups according to how many times the
i
th
site appears, which allows us to write
χ
(
ii, ω
)=
c
i
χ
o
(
ω
)+
χ
o
(
ω
)
K
(
i, ω
)
χ
o
(
ω
)
+
χ
o
(
ω
)
K
(
i, ω
)
χ
o
(
ω
)
K
(
i, ω
)
χ
o
(
ω
)+
J
···
=
c
i
1
− χ
o
(
ω
)
K
(
i, ω
)
−
1
χ
o
(
ω
)
,
(5
.
6
.
9)
where
K
(
i, ω
) is the infinite sum of all the 'interaction chains' involv-
ing the
i
th site only at the ends, but nowhere in between. A similar
rearrangement of the terms in the general RPA series leads to
χ
(
ij, ω
)=
χ
(
ii, ω
)
δ
ij
+
χ
(
ii, ω
)
T
(
ij, ω
)
χ
(
jj,ω
)
,
(5
.
6
.
10)
where
=
j
and, by exclusion, is the sum
of all the interaction chains in which the
i
th site appears only at the
beginning, and the
j
th site only at the end of the chains. Introducing
this expression in the RPA equation (5.6.4), we may write it
T
(
ij, ω
) is only non-zero if
i
χ
(
ij, ω
)=
c
i
χ
o
(
ω
)
δ
ij
+
J
(
ij
)
χ
(
jj,ω
)+
j
J
(
ij
)
χ
(
j
j
,ω
)
T
(
j
j, ω
)
χ
(
jj,ω
)
.
From (5.6.9), we have
χ
o
(
ω
)
−
1
χ
(
ii, ω
)=
c
i
{
,anda
comparison of this equation for
χ
(
ij, ω
) with (5.6.10), leads to the result:
1+
K
(
i, ω
)
χ
(
ii, ω
)
}
(
ij
)
χ
(
jj,ω
)+
(
ij
)
χ
(
j
j
,ω
)
(
j
j, ω
)
χ
(
jj,ω
)
δ
ij
+
J
j
J
T
=
{
1+
K
(
i, ω
)
χ
(
ii, ω
)
}{
δ
ij
+
T
(
ij, ω
)
χ
(
jj,ω
)
}
,
(5
.
6
.
11)
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