Environmental Engineering Reference
In-Depth Information
where Σ(
ω
) is the previous function with
K
(
ω
) replaced by
K
(
ω
). The
most interesting effects of the scattering of the magnetic excitations
against the electron-hole pair excitations of the conduction electrons de-
rive from the first term in the self-energy, which already appears in the
'RPA' in (7.3.16). The lifetime of the excitations becomes
q
-dependent
and remains finite in the zero-temperature limit, whereas the imagi-
nary part of Σ(
ω
), and therefore also of Σ(
ω
), vanishes exponentially at
low temperatures, in the order 1
/Z
. The importance of the higher-order
contributions associated with this scattering mechanism, as compared to
those of the intrinsic processes, i.e. the relative magnitudes of
hω
and
K
(
ω
), may depend on the system considered, but in Pr, for exam-
ζ
(
ω
)
ple, Im
K
(
ω
)
is much the dominant term at frequencies lying within
the excitonic band. Hence,
may be neglected in
K
(
ω
)attemper-
atures where the linewidths are still somewhat smaller than the overall
bandwidth.
In Pr, the effect of the conduction electrons on the linewidths at
low temperatures only becomes visible due to the strong increase in the
value of
ζ
(
q
) in the limit of small
q
, where it is approximately propor-
tional to 1
/q
.Houmann
et al
. (1979) were thus able to observe the
remarkable broadening of the acoustic modes illustrated in Fig. 7.6, as
q
was reduced at 6 K. The width at
q
=0
.
2 A is only slightly greater than
the experimental resolution, but the peak has become very broad by
0.05 A, and it has almost vanished into the background at
q
=0,even
though the integrated intensity is expected to increase as the energy
decreases. This behaviour is in sharp contrast to that observed in Tb
where, as shown in Fig. 5.13 on page 269, the width at small
q
is greatly
reduced by the spin-splitting of the Fermi surface, in accordance with
eqn (5.7.37). Since the spin-splitting of the Pr Fermi surface becomes
very substantial in a large field, as illustrated in Fig. 1.10, the scattering
of the long-wavelength magnetic excitations by the conduction electrons
should be quenched by the application of a field. A careful study of
this phenomenon would allow a detailed investigation of the interaction
between the conduction electrons and the 4
f
moments.
The modification of
K
(
ω
) also contributes to the broadening of the
diffusive peak and, instead of (7.2.11), the result for
J
=1isnow
ζ
(
ω
)
Γ
2
(
hω
+
i
Γ)
2
,
G
(
ω
)=
G
(0)
i
Γ
1
hω
−
(7
.
3
.
19
a
)
with
Γ = Γ
1
+
2
K
(0)
/β.
ζ
(0)
/β
(7
.
3
.
19
b
)
Γ
1
=2
and
The term linear in
, introduced in (7.2.10), predicts Lorentzian
broadening, if
K
(0) is neglected. The intrinsic contribution may also
ζ
(0)
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