Environmental Engineering Reference
In-Depth Information
S
i
must be nearly independent of time, i.e.
S
i
S
z
.Inthis
random-
phase approximation
(RPA) the commutator reduces to
S
j
−
S
i
,
[
S
i
S
z
,
H
]
−
J
(
ij
)
j
and the equations of motion lead to the following linear set of equations:
hωG
±
(
ii
,ω
)+
j
G
±
(
ji
,ω
)
G
±
(
ii
,ω
)
S
z
J
(
ij
)
−
(3
.
4
.
7)
[
S
i
,S
i
S
z
=
]
=2
δ
ii
.
The infinite set of RPA equations is diagonal in reciprocal space. Intro-
ducing the Fourier transform
G
±
(
q
,ω
)=
i
G
±
(
ii
,ω
)
e
−i
q
·
(
R
i
−
R
i
)
,
(3
.
4
.
8)
we obtain
J
(
0
)
G
±
(
q
,ω
)
=2
hωG
±
(
q
,ω
)+
S
z
(
q
)
G
±
(
q
,ω
)
S
z
−J
,
or
S
z
2
G
±
(
q
,ω
) = lim
→
0
+
,
(3
.
4
.
9)
hω
+
ih
−
E
q
where the
dispersion relation
is
S
z
E
q
=
{J
−J
}
.
(3
.
4
.
10)
(
0
)
(
q
)
G
±
(
q
,ω
), we obtain
Introducing the susceptibility
χ
+
−
(
q
,ω
)=
−
S
z
2
S
z
χ
+
−
(
q
,ω
)=
hω
+
iπ
2
δ
(
hω
−
E
q
)
.
(3
.
4
.
11
a
)
E
q
−
Defining
χ
−
+
(
q
,ω
) analogously to
χ
+
−
(
q
,ω
), but with
S
+
and
S
−
in-
terchanged, we obtain similarly, or by the use of the symmetry relation
(3.2.15),
S
z
E
q
+
hω
−
2
S
z
χ
−
+
(
q
,ω
)=
iπ
2
δ
(
hω
+
E
q
)
,
(3
.
4
.
11
b
)
so that the absorptive susceptibility is
χ
+
−
(
q
,ω
)=
χ
−
+
(
q
,
S
z
−
−
ω
)=2
π
δ
(
hω
−
E
q
)
.
(3
.
4
.
11
c
)
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