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only in relatively simple models like the dilute Heisenberg ferromagnet
with nearest-neighbour interactions. This model may be considered as
the extreme example of off-diagonal disorder, and the CPA concept of
an effective medium loses its meaning completely below the
percolation
concentration, where all clusters of interacting spins are of finite size,
precluding any long-range order. If the molecular field is independent
of the site considered, i.e.
H
MF
(
i
)=
H
MF
(
i
)
cf
in (5.6.3), as happens
in the paramagnetic case or if
, then the CPA result above
should apply. However, except in a pure boson or fermion system, the
'dynamical' disorder due to thermal fluctuations introduces corrections
to the RPA equation (5.6.4), with consequences of the same order of
magnitude as
K
(
ω
) in (5.6.16), at least at elevated temperatures. In
most magnetic systems, the two kinds of disorder may lead to damping
effects of the same magnitude, and furthermore the use of the CPA result
(5.6.16), without taking into account the dynamic renormalization of the
RPA, occasionally leads to misleading results, as discussed for instance
by Jensen (1984).
The excitations of binary heavy-rare-earth alloys have been studied
much less extensively than their magnetic structures. However, the effect
of 10% of Y, Dy, Ho, and Tm on the spin-wave spectrum of Tb has
been examined, and the characteristic influence of the different solutes
observed. The results of Larsen
et al.
(1986) for the Y and Dy alloys
could be interpreted in terms of a simple average-crystal model, in which
all sites are considered as equivalent, and the effect of the solute atoms
is to modify the average exchange and the effective single-ion anisotropy.
Thus Dy reduces the effective hexagonal anisotropy, and the spin-wave
energy gap therefore decreases. On the other hand, Y dilutes the two-
ion coupling, and therefore decreases
T
N
and the spin-wave energies,
although the relative magnitude of the peak in
J
1
=
γ
J
2
(
q
) increases, extending
the temperature range over which the helical structure is stable. The first
excited state of the Ho ion in the Tb host lies in the spin-wave energy
band, and the dispersion relation is consequently strongly perturbed
(Mackintosh and Bjerrum Møller 1972).
However, the most pronounced effects were observed by Larsen
et
al.
(1988) in Tb
90
Tm
10
, where the Tm ions, with a spin
S
=1,are
relatively weakly coupled to the surrounding Tb moments, with
S
=3.
Furthermore, the axial anisotropy of the Tm ions is large and of opposite
sign to that of Tb. As a result, well-defined quasi-localized states may
be excited on the Tm sites, as shown in Fig. 5.11. These rather complex
results were interpreted by means of a VCA calculation, in which the
crystal-field parameters for the Tm ions were deduced from the dilute-
alloy experiments of Touborg (1977), while the single-ion anisotropy and
the two-ion coupling between the Tb ions were taken from the analysis
J
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