Environmental Engineering Reference
In-Depth Information
The imaginary term linear in
ω
implies that the correlation function
(4.2.3), which is proportional to
χ
αβ
(
q
,ω
)
/βhω
for
ω
0, is non-zero
in this limit. Hence the inelastic-scattering spectrum of the incommen-
surable system contains a tail down to zero frequency, with a magnitude
at
ω
= 0 proportional to
T
. At non-zero frequencies, eqn (6.1.35) can
only be solved in special cases, such as if
γ
n
is independent of
n
, corre-
sponding to
→
J
⊥
(
q
) = 0 (Liu 1980). In general, numerical methods must
be applied. We may, for example, replace
ω
by
ω
+
i
and, instead of
considering the limit
0
+
, allow
to stay small but non-zero (e.g.
→
=0
.
01
). Then
G
0
(
q
,ω
) becomes insensitive to the value of
z
±n
,
if
n
is suciently large (
n>
50).
acts as a coarse-graining param-
eter, of the type mentioned at the beginning of this section, and any
energy gaps in the spectrum, smaller than
h
, are smeared out. A more
careful treatment of this problem has been given by Lantwin (1990).
Solutions of eqn (6.1.35) have been presented by Ziman and Lindg ard
(1986), Lovesey (1988), and Lantwin (1990), for various values of
Q
and the axial anisotropy parameter
| ω |
(
Q
). The most impor-
tant result is that the imaginary part of
G
0
(
q
,ω
)
/ω
contains a number
J
(
Q
)
−J
⊥
Fig. 6.3.
The imaginary part of the response function
χ
ξξ
(
q
,ω
)
/ω
for an incommensurable longitudinal structure, as a function of
ω
and
q
,
from Ziman and Lindg ard (1986). The sharp peaks indicate the presence
of well-defined excitations in this structure.
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