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condition for
(
Q
), but the qualitative behaviour is unchanged. It is
therefore possible to realize a system in which the moments are rela-
tively strongly coupled to each other, but which remains paramagnetic
at low temperatures, i.e. a crystal-field system in which cooperative ef-
fects are important. Perhaps the best example is elemental Pr, which is
only slightly undercritical, with
R
0
J
0
.
92, and therefore exhibits a rich
variety of unusual magnetic phenomena.
Pr crystallizes in the double hexagonal-close-packed (dhcp) struc-
ture, illustrated in Fig. 1.3, with the stacking sequence ABAC along the
c
-axis. This implies that there are two non-equivalent types of site of
different symmetry in the crystal. The ions in A layers are in an ap-
proximately cubic environment, with nearest neighbours close to the fcc
configuration, while those in the B and C layers experience a crystal
field of hexagonal symmetry and form together an hcp structure. The
tripositive Pr ion, with two 4
f
electrons, is a
non-Kramers
ion (
S
=1,
L
=5,and
J
= 4 for the ground-state multiplet) allowing the occurrence
of singlet crystal-field states. Experimental observations, particularly of
neutron scattering, have revealed that both kinds of site in fact have
a singlet as the ground state. The lowest states of the
hexagonal
ions
are the singlet
|
J
ζ
=0
>
followed by the doublet
|
J
ζ
=
±
1
>
,with
an energy difference of ∆
h
3
.
5 meV, as illustrated in Fig. 1.16. If
the distortion of the point symmetry of the
cubic
ions, due to the non-
ideal
c/a
ratio, is neglected, their ground state is the Γ
1
-singlet, with
the Γ
4
-triplet lying ∆
c
8
.
4 meV above it. A complete survey of the
classification and energies of crystal-field states in cubic surroundings
has been given by Lea, Leask, and Wolf (1962). The possibility that
the Γ
4
state is split into a singlet and a doublet, due to the deviation
from cubic symmetry, has not yet been investigated experimentally. At
temperatures well below 40 K (
∼
3.5 meV), only the two ground states
are populated significantly, and Pr may be considered to be a coupled
singlet-doublet and singlet-triplet system. Furthermore, the difference
between ∆
h
and ∆
c
is so large, compared to the two-ion interactions,
that the excitation spectrum can be divided into two parts, related re-
spectively to the crystal-field transitions on each kinds of ion. The weak
coupling of the two components may be accounted for by second-order
perturbation theory (Jensen 1976a), leading to an effective decoupling,
with the two-ion parameters replaced by slightly different, effective val-
ues. Hence, at low temperature, Pr may be treated as a combination of a
singlet-doublet system on an hcp lattice and a singlet-triplet system on
a simple hexagonal lattice. Of these, the singlet-doublet system is much
the more important because of the smaller value of the crystal-field split-
ting. The singlet-doublet scheme corresponds to an effective
J
= 1 and,
if the two doublet states are defined to be
1
s
>
=
|
1
>
/
√
2
|
+1
>
+
|−
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