Environmental Engineering Reference
In-Depth Information
Fig. 7.4.
The temperature dependence of the neutron-scattering in-
tensities for the
q
= 0 magnetic optical-mode on the hexagonal sites of
Pr. The instrumental resolution and the overall scaling of the intensity
are extracted from the experimental results at 6 K, and thereafter used
unchanged in the calculations, which are based on a 1
/Z
-expansion, as
describedinthetext.
accurately described by the effective-medium theory developed above,
than by his unrestricted cumulant expansion.
An analysis of the 1
/Z
-corrections to the MF-RPA theory for the
singlet-triplet model does not exist in the literature, to our knowledge.
We shall not attempt such an analysis here, but we will discuss one
aspect, that the elastic response due to the triplet states is predicted
to be a diffusive peak of non-zero width, to first order in 1
/Z
,within
the effective-medium approximation. In order to consider this matter,
we can omit the singlet and use instead the (
J
= 1) Heisenberg model,
corresponding to the above model with ∆ = 0. In this case, the diagonal
components of the single-site Green function are
3
β
1
βK
(0)
δ
n
0
+
3(
ihω
n
)
2
K
(0)
K
(
iω
n
)
+
4
2
−
6
···
(7
.
2
.
10)
to first order in 1
/Z
. In zero order, the response is purely elastic and
K
(
iω
n
)
G
(
iω
n
)=
−
−
δ
n
0
. If this is introduced into (7.2.10), the second term pre-
dicts an inelastic contribution to
G
(
ω
), which further diverges propor-
tionally to
ω
−
2
in the zero-frequency limit. This divergence indicates
that the elastic peak must broaden out to a Lorentzian, with a non-
zero half-width Γ, as in (3.3.10-11), corresponding to the replacement
of (
hω
)
2
in the denominator by (
hω
)
2
+Γ
2
, when the higher-order terms
in the series are included. The classification of
K
(
iω
n
∝
=0)asahigher-
order term in the series (7.2.9) is not consistent with a simple Lorentzian,
Search WWH ::
Custom Search