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so close to the average that individual variations can be neglected.
Thus we replace the actual MF Hamiltonian of the
i
th ion with the
configurationally-averaged MF Hamiltonian and, considering a type 1
ion (
c
i
= 1), obtain
H
MF
(
i
)
H
MF
(
i
)
cf
=
(5
.
6
.
3)
(
J
1
i
−
2
J
1
i
H
1
(
J
1
i
)
−
)
·
J
(
ij
)
{
c
J
1
j
+(1
−
c
)
γ
J
2
j
}
.
j
From this equation, and the similar one for
c
i
=0,wemaydetermine
the MF values of the two moments
, and the corresponding
susceptibilities
χ
1
(
ω
)and
χ
2
(
ω
). For a paramagnetic or ferromagnetic
system these quantities are all site-independent, in the present approx-
imation. We note that (5.6.3) is correct in the case of a paramagnet,
as possible environmental variations on the individual ions are already
neglected in the starting Hamiltonian. The next step is the introduction
of a 2
J
1
and
J
2
2 matrix of susceptibility tensors
χ
rs
(
ij, ω
), where the elements
with
r
= 1 or 2 are defined in terms of
c
i
J
1
i
or (1
×
c
i
)
J
2
i
respectively,
and
s
= 1 or 2 similarly specifies the other component. We may then
write the RPA equation (3.5.7):
χ
rs
(
ij, ω
)=
χ
r
(
i, ω
)
δ
rs
δ
ij
+
j
−
(
ij
)
χ
s
s
(
j
j, ω
)
,
γ
rs
J
(5
.
6
.
4
a
)
s
where
χ
1
(
i, ω
)=
c
i
χ
1
(
ω
)
c
i
)
χ
2
(
ω
)
,
;
χ
2
(
i, ω
)=(1
−
(5
.
6
.
4
b
)
recalling that
c
i
=
c
i
(= 0 or 1), and defining
J
rs
(
ij
)=
γ
rs
J
(
ij
), with
γ
22
=
γ
2
.
γ
11
=1
;
γ
12
=
γ
21
=
γ
;
(5
.
6
.
4
c
)
In spite of the great simplification introduced through the random-phase
approximation, the RPA equation for the alloy is still very complicated,
because
χ
r
(
i, ω
) depends on the randomness, and it cannot be solved
without making quite drastic approximations. The simplest result is
obtained by neglecting completely the site-dependence of
χ
r
(
i, ω
), and
consequently replacing
c
i
in (5
.
6
.
4
b
) by its average value
c
.Thispro-
cedure corresponds to the replacement of each individual angular mo-
mentum
J
ri
c
)
J
2
i
, and it is known as the
virtual crystal approximation
(VCA). In this approximation, (5.6.4) may
be solved straightforwardly after a Fourier transformation, and defining
the
T-matrices
according to
by the average
c
J
1
i
+(1
−
χ
rs
(
q
,ω
)=
χ
r
(
ω
)
δ
rs
+
χ
r
(
ω
)
T
rs
(
q
,ω
)
χ
s
(
ω
)
,
(5
.
6
.
5
a
)
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