Environmental Engineering Reference
In-Depth Information
S
z
value
0
, reflecting the lack of self-consistency in this analysis. As a
supplement to the previous results, we find that
χ
zz
(
ω
)
β
(
δS
z
)
2
χ
zz
(
q
,ω
)=
=
(
q
)
δ
ω
0
,
(3
.
5
.
27
a
)
1
−
χ
zz
(
ω
)
J
(
q
)
1
−
β
(
δS
z
)
2
J
and the corresponding correlation function is
(
δS
z
)
2
S
zz
(
q
,ω
)=2
πh
(
q
)
δ
(
hω
)
.
(3
.
5
.
27
b
)
1
−
β
(
δS
z
)
2
J
The
zz
-response vanishes in the zero-temperature limit and, in this ap-
proximation, it is completely elastic, since (
δS
z
)
2
is assumed indepen-
dent of time. However, this assumption is violated by the dynamic
correlation-effects due to the spin waves. For instance, the (
n
=1)-sum-
rule (3
.
3
.
18
b
) indicates that the second moment
(
hω
)
2
zz
is non-zero,
when
q
=
0
and
T>
0, which is not consistent with a spectral function
proportional to
δ
(
hω
).
Although this procedure leads to a less accurate analysis of the
Heisenberg ferromagnet than that applied previously, it has the advan-
tage that it is easily generalized, particularly by numerical methods, to
models with single-ion anisotropy, i.e. where
H
J
(
J
i
) in (3.5.1) is non-
zero. The simplicity of the RPA result (3.5.8), or of the more general
expression (3.5.7), furthermore makes it suitable for application to com-
plex systems. As argued above, its validity is limited to low tempera-
tures in systems with relatively large coordination numbers. However,
these limitations are frequently of less importance than the possibility of
making quantitative predictions of reasonable accuracy under realistic
circumstances. Its utility and effectiveness will be amply demonstrated
in subsequent chapters.
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