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the magnetoelastic parameter (
B
γ
2
)
2
/c
γ
. This agrees with the value of
B
γ
2
for the hexagonal ions which accounts for the coupling between the
magnetic excitations and the phonons, and for the field dependences of
c
66
(Fig. 7.5) and of the
γ
-strains (Hendy
et al.
1979). It furthermore
allowed the accurate prediction of the strain-induced antiferromagnetic
transition in Pr, shown in Fig. 7.10.
The low-temperature magnetic properties of Pr are dominated by
the hexagonal ions. One consequence of this is that it is more dicult
to construct a reliable model for the cubic ions, based on experimentally
derivable parameters. Although the model proposed by Houmann
et al.
(1979) accounts accurately for the bulk of the low-temperature experi-
mental results, it is not uniquely specified and some deficiencies appear
in comparison with experiments at elevated temperatures. The model
is based on the following crystal-field Hamiltonian for the cubic ions:
H
J
=
i∈
cub
.
ions
B
4
(
c
)
20
√
2
O
−
3
4
O
4
(
J
i
)
{
−
(
J
i
)
}
,
(7
.
4
.
6)
which neglects the departure of the local symmetry of these sites from
cubic. We shall not present an extensive discussion of the model here
(more details may be found in Houmann
et al.
(1979) and Jensen (1979b,
1982)), but it is clear that this MF model, which is the simplest descrip-
tion of Pr consistent with its magnetic behaviour in the low-temperature
limit, must be extended in order to describe, for instance, the magne-
tostriction measurements of Hendy
et al.
(1979) and Ott (1975). In
addition to introducing a non-zero value of
B
α
2
for the hexagonal ions,
of the magnitude used to obtain agreement with experiment in Fig. 7.13,
it is probably also necessary to include
B
α
1
. Moreover, the magnetoe-
lastic parameters for the cubic ions are presumably of the same order of
magnitude as those on the hexagonal ions. This probably also applies
to
B
2
(
c
), neglected in eqn (7.4.6). The separation of the contributions
from the hexagonal and the cubic ions to the
c
-axis bulk susceptibility as
a function of temperature, accomplished through neutron-diffraction ex-
periments by Rainford
et al.
(1981), indicates that not only is
B
2
(
c
)non-
zero, but the exchange between the
c
-axis components of the moments is
also different from the corresponding coupling between the basal-plane
components. The development of a MF model for Pr which describes
its properties more accurately at elevated temperatures would benefit
greatly from a more detailed examination of the excitations on the cu-
bic sites, i.e. a determination of the energies of the excitations polarized
in the
c
-direction, and the field-induced changes of these excitations, and
of those polarized in the basal-plane and shown in Fig. 7.2.
4
√
2
O
−
3
O
6
(
J
i
)+
35
77
8
O
6
(
J
i
)
+
B
6
(
c
)
{
(
J
i
)
−
}
6
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