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perature. The effects of these interactions on the ground state and the
spin waves are therefore most pronounced in the low-temperature limit,
whereas the behaviour of the system at high temperatures which, in the
heavy rare earths, includes the critical region around the phase transi-
tion between the ordered and paramagnetic phases, is dominated by the
coupling between the dipolar moments, and the single-ion quadrupole
interaction, i.e. by the terms in eqn (5.5.14) with
l
+
l
=2.
5.6 Binary rare earth alloys
The great similarity in the chemical properties of the different rare earth
metals allows almost complete mutual solubility. It is therefore possible
to fabricate rare earth alloys with nearly uniform electronic properties,
but containing ions with disparate magnetic properties, distributed ran-
domly on a single lattice. By a judicious choice of the constituents,
the macroscopic magnetic properties, such as the ordering temperatures
and the anisotropy parameters, may be continuously adjusted as desired.
From a macroscopic viewpoint, such an alloy resembles a uniform and
homogeneous crystal, with magnetic properties reflecting the character-
istics and concentrations of the constituents. The spectrum of magnetic
excitations also displays such average behaviour (Larsen
et al.
1986),
but in addition, there are effects which depend explicitly on the dispar-
ity between the different sites.
We restrict ourselves to binary alloys, which are described by the
Hamiltonian,
=
i
H
{
c
i
H
1
(
J
1
i
)+(1
−
c
i
)
H
2
(
J
2
i
)
}
2
1
−
i
=
j
J
(
ij
)
{
c
i
J
1
i
+
γ
(
ij
)(1
−
c
i
)
J
2
i
}·{
(
c
j
J
1
j
+
γ
(
ij
)(1
−
c
j
)
J
2
j
}
,
(5
.
6
.
1)
where
c
i
is a variable which is 1 if the ion on site
i
is of type 1, and
0ifthe
i
th ion is of type 2. The configurational average of
c
i
is the
atomic concentration of the type-1 ions,
c
i
cf
=
c
. In addition to the
simplifications made earlier in the Hamiltonian, we shall assume that
γ
(
ij
)isaconstant
γ
, independent of
i
and
j
. This approximation is
consistent with a model in which the indirect exchange is assumed to
dominate the two-ion coupling, in which case
γ
(
ij
)=
γ
=(
g
2
−
1)
/
(
g
1
−
1)
,
(5
.
6
.
2)
where the indices 1 and 2 refer to the two types of ions with angular
momenta
J
1
and
J
2
.
In order to derive the excitation spectrum of the alloy system,
we first make the assumption that the surroundings of each ion are
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