Environmental Engineering Reference
In-Depth Information
RPA equation (3.5.7):
χ
11
(
q
,ω
)=
χ
o
(
ω
)
1+
J
12
(
q
)
χ
21
(
q
,ω
)
J
11
(
q
)
χ
11
(
q
,ω
)+
(5
.
1
.
2)
χ
21
(
q
,ω
)=
χ
o
(
ω
)
J
21
(
q
)
χ
11
(
q
,ω
)+
J
22
(
q
)
χ
21
(
q
,ω
)
,
assuming that the MF susceptibility
χ
o
(
ω
) is the same for all the sites,
as in a paramagnet or a ferromagnet. These matrix equations may be
solved straightforwardly, and using (5
.
1
.
1
b
) we find
χ
o
(
ω
)
χ
o
(
ω
)
χ
11
(
q
,ω
)=
D
(
q
,ω
)
−
1
{
−
J
1
(
q
)
}
1
(5
.
1
.
3
a
)
χ
21
(
q
,ω
)=
D
(
q
,ω
)
−
1
χ
o
(
ω
)
2
J
2
(
−
q
)
,
where
D
(
q
,ω
)=
1
|
2
=
1
− χ
o
(
ω
)
{J
1
(
q
)+
|J
2
(
q
)
|}
1
− χ
o
(
ω
)
{J
1
(
q
)
−|J
2
(
q
)
|}
,
(5
.
1
.
3
b
)
J
1
(
q
)
2
−
χ
o
(
ω
)
χ
o
(
ω
)
−
|J
2
(
q
)
and, by symmetry,
−
q
,ω
)
.
(5
.
1
.
3
c
)
If
χ
o
(
ω
) contains only one pole, as in the case of the Heisenberg ferro-
magnet, then
D
(
q
,ω
)
−
1
in (5
.
1
.
3
a
) generates two poles, corresponding
to the existence of both an
acoustic
and an
optical
mode at each
q
-vector.
J
2
(
0
) must be real and, since it is also positive in a ferromagnet, the
acoustic mode arises from the zero of the first factor in (5
.
1
.
3
b
), its
energy therefore being determined by the effective coupling parameter
J
1
(
q
)+
χ
22
(
q
,ω
)=
χ
11
(
q
,ω
)
d
χ
12
(
q
,ω
)=
χ
21
(
J
2
(
0
) is negative, as it is in
paramagnetic Pr, it is the second factor which gives the acoustic mode.
The nomenclature results from the circumstance that the deviations of
the moments from their equilibrium values are in phase in the acoustic
mode in the limit of
q
→
0
, and it therefore dominates the neutron
cross-section. The inelastic neutron scattering is determined by (4.2.2)
and (4.2.3), i.e. by
|J
2
(
q
)
|
.
On the other hand, if
N
ij
2
ss
,ω
)=
1
χ
(
ij, ω
)
e
−i
κ
·
(
R
i
−
R
j
)
=
1
χ
(
κ
χ
ss
(
κ
,ω
)
,ω
)
−
1
1
)]
χ
o
(
ω
)
,
(5
.
1
.
4)
χ
o
(
ω
)
J
1
(
−
2
=
D
(
κ
−
κ
)
[
J
2
(
κ
)+
J
2
(
−
κ
where
N
=2
N
0
,with
q
lying in the primitive zone, we may write this result as a sum of the
acoustic and optical response functions:
χ
Ac
(
q
,ω
)=
1
is the number of atoms. Introducing
κ
=
q
+
τ
)
−
1
χ
o
(
ω
)
χ
o
(
ω
)(
−
J
1
(
q
)+
ν
|J
2
(
q
)
|
(5
.
1
.
5)
χ
Op
(
q
,ω
)=
1
)
−
1
χ
o
(
ω
)
,
χ
o
(
ω
)(
−
J
1
(
q
)
−
ν
|J
2
(
q
)
|
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