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where
χ
1
(
ω
)=
cχ
o
c
)
χ
o
1
(
ω
)and
χ
2
(
ω
)=(1
−
2
(
ω
)
,
(5
.
6
.
5
b
)
we find that these T-matrices are given by
(
q
)
D
(
q
,ω
)
−
1
,
T
rs
(
q
,ω
)=
γ
rs
J
(5
.
6
.
6
a
)
with
cχ
1
(
ω
)+(1
c
)
γ
2
χ
2
(
ω
)
D
(
q
,ω
)=1
−
−
J
(
q
)
.
(5
.
6
.
6
b
)
This result is simplified by the assumption, (5.6.2) or (5
.
6
.
4
c
), that
J
12
(
q
) is the geometric mean of
J
22
(
q
). In this and in
more complex cases, the introduction of the T-matrices in (5.6.5) makes
it somewhat easier to handle the RPA equations. The configurationally-
averaged susceptibility is
χ
(
q
,ω
)=
rs
χ
rs
(
q
,ω
), but this does not di-
rectly determine the inelastic neutron-scattering cross-section. We must
take into account the difference in the form factor
J
11
(
q
)and
1
for the two
kinds of ions, in the differential cross-section (4.2.1). At small scattering
vectors,
F
(
{
2
gF
(
κ
)
}
) is generally close to one and the most important variation
is due to the
g
-factor. In this case, the inelastic scattering is proportional
to the susceptibility:
κ
g
2
χ
(
q
,ω
)
≡
g
r
g
s
χ
rs
(
q
,ω
)
rs
=
g
1
c χ
o
c
)
χ
o
1
(
ω
)+
g
2
(1
(
q
)
D
(
q
,ω
)
−
1
χ
3
(
ω
)
,
(5
.
6
.
7
a
)
−
2
(
ω
)+
χ
3
(
ω
)
J
with
χ
3
(
ω
)=
g
1
c χ
o
c
)
γ χ
o
1
(
ω
)+
g
2
(1
−
2
(
ω
)
.
(5
.
6
.
7
b
)
If
χ
r
(
i, ω
) only depends on
c
i
, as assumed in (5
.
6
.
4
b
), the RPA
equation (5
.
6
.
4
a
) is equivalent to that describing the phonons in a crys-
tal with
diagonal disorder
, in the harmonic approximation. The possible
variation of the molecular field (or other external fields) from site to site,
which is neglected in (5.6.3), introduces
off-diagonal disorder
.Ifsuch
off-diagonal disorder is neglected, the main effects of the randomness, in
3-dimensional systems, are very well described in the
coherent potential
approximation
(CPA) (Taylor 1967; Soven 1967; Elliott
et al.
1974; Lage
and Stinchcombe 1977; Whitelaw 1981). In the CPA, the different types
of ion are treated separately, but they are assumed to interact with a
common surrounding medium. This configurationally-averaged medium,
i.e. the
effective medium
, is established in a self-consistent fashion. The
method may be described in a relatively simple manner, following the
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