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gap also vanishes in the other limit when
σ
0, as does the amplitude-
mode energy gap, reflecting the soft-mode nature of the transition in
this approximation. In the above discussion, we have assumed that the
system is incommensurable. In a commensurable structure, the free en-
ergy is no longer invariant to an overall phase shift of the structure, and
the longitudinal susceptibility does not diverge at
Q
. Because of the
small value of
σ
8
, however, it is close to divergence. The phason-mode
energy gap stays non-zero at
T
= 0 in the commensurable case, but it
is estimated to be only about 0.03 meV at 1 kbar.
Even at the lowest temperatures reached in the inelastic neutron-
scattering experiments, quite strong line-broadening of the low-lying lon-
gitudinal excitations was observed in the ordered phase. There are sev-
eral mechanisms which may lead to non-zero linewidths. One possibil-
ity, if the ordering is incommensurable, is a broadening of the excitation
peaks analogous to that illustrated in Fig. 6.3 in Section 6.1.2. How-
ever, the off-diagonal coupling terms, corresponding to
γ
n
in (6.1.30), are
here multiplied by
σ
2
, which means that the continued-fraction solution,
although infinite, converges very rapidly without producing linewidth
effects of any importance. The 1
/Z
-expansion, discussed in Section
7.2, accounts very well in first order in 1
/Z
for the lifetime effects ob-
served in paramagnetic Pr, as shown in Fig. 7.4. In this order, the
intrinsic-linewidth effects vanish exponentially at low temperature, and
they should be negligible in the temperature range of the ordered phase,
with the important exception that the elastic RPA response acquires a
non-zero width. To first order in 1
/Z
,
δ
ω
0
in eqn (7
.
4
.
8
a
) is replaced by
aLorentzian
i
Γ
/
(
hω
+
i
Γ), with 1
/
Γ
→
n
01
(
π/
2)
N
η
(
E
)is
the density of states of the
η
-polarized part of the excitation spectrum.
Due to the large value of Γ, estimated to be about 1 meV, the RPA
predictions for the behaviour of the phason modes near the ordering
wave-vector are strongly modified. Instead of an elastic diffusive and an
inelastic, adiabatic phason mode, the theory to this order predicts only
one mode at zero energy, but with non-zero width, when
q
is close to
Q
. An inelastic low-energy peak develops only at a distance of about
0
.
03
N
η
(∆), where
from
Q
. The exchange-enhancement factor in the scattering
function causes the width of the Lorentzian near
Q
to be much less than
2Γ. Formally the width tends to zero when
q
→
Q
, but it is more pre-
cisely the intensity which diverges, while strong inelastic tails remain at
q
=
Q
, in accordance with the experimental results.
As was mentioned in Section 7.2, the 1
/Z
-expansion of the effective
medium theory was extended to second order in 1
/Z
by Jensen
et al.
(1987). The second-order modifications are important here, but not in
the zero-stress case considered in Fig. 7.4, because the low-temperature
energy gap of about 1 meV in the excitation spectrum is suppressed by
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