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used. However, it is both more convenient and, in general, more correct
to use the true Brillouin zone for the dhcp structure, as in Fig. 7.2.
The excitations in this figure are polarized in the plane, and may also
be described by (7.1.4), with parameters appropriate to the cubic sites.
The
z
-modes were not observed in these experiments, on account of the
neutron scans employed. The dispersion is much smaller than that on
the hexagonal sites and, in particular, it is negligible in the
c
-direction,
indicating very weak coupling between planes of cubic ions normal to
this axis. Again in contrast to the hexagonal ions, the splitting between
modes of different polarization is not resolved, demonstrating that the
anisotropy in the two-ion coupling is smaller.
7.2 Beyond the MF-RPA theory
When the temperature is raised, the available magnetic scattering inten-
sity, from eqn (4.2.7) proportional to
J
(
J
+ 1), is divided more and more
equally among the (2
J
)! different dipolar transitions, and in the high-
temperature limit half the intensity is transferred to the emissive part of
the spectrum. This means that the different crystal-field excitations be-
come weaker and less dispersive, and correspondingly correlation effects
become less important as the temperature is raised. An additional mech-
anism diminishing the correlation effects at elevated temperatures is the
scattering of the excitations against random fluctuations, neglected in
the MF-RPA theory. In this theory, all the ions are assumed to be in
the same MF state, thus allowing an entirely coherent propagation of
the excitations. However, at non-zero temperatures, the occupations of
the different crystal-field levels differ from site to site, and these
single-
site fluctuations
lead to a non-zero linewidth for the excitations. In
fact, if two-ion interactions are important, such fluctuations already ex-
ist at zero temperature, as the MF ground state
i
|
0
i
>
cannot be
the true ground state, because
i
|
does not commute with
the two-ion part of the Hamiltonian. Hence, the occupation
n
0
of the
'ground-state' is reduced somewhat below 1 even at
T
=0. There-
sponse functions derived above already predict such a reduction of
n
0
but, as discussed earlier in connection with eqn (3.5.23), the MF-RPA
theory is not reliable in this order. A more satisfactory account of the
influence of fluctuations, both at zero and non-zero temperatures, can
only be obtained by calculations which go beyond the MF-RPA.
One way to proceed to higher order is to postpone the use of the
RPA decoupling to a later stage in the Green-function hierarchy gener-
ated by the equations of motion. Returning to our derivation of the MF-
RPA results in Section 3.5; instead of performing the RPA decoupling on
the Green function
0
i
><
0
i
|
a
νξ
(
i
)
a
ν
µ
(
j
);
a
rs
(
i
)
, as in eqn (3.5.16), we first
apply this decoupling to the higher-order Green functions appearing in
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