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transition, at a temperature determined by
∆
2
− E
Q
E
Q
R
0
=
3
=
3
I
(
I
+1)
A
2
χ
J
(0)
I
(
I
+1)
A
2
χ
J
(0)
k
B
T
N
,
1
−
R
0
(7
.
3
.
28)
to a modulated phase described by the wave-vector
Q
at which
(
q
)
has its maximum value, where
R
0
is the critical parameter defined by
eqn (7.1.6). With ∆ = 3
.
52 meV and
E
Q
=1
.
0 meV for the hexagonal
excitations in Pr, the electronic system is just undercritical, with a crit-
ical ratio
R
0
J
0
.
92. This means that the importance of the hyperfine
interaction is much enhanced, and eqn (7.3.28) predicts
T
N
=45mKfor
the cooperative ordering of the nuclear and electronic moments in Pr.
The transition is no longer accompanied by a soft mode, but there is
rather an elastic peak, with a scattering intensity given by
(2
M
ξ
/E
q
)
2
ξξ
d
0) =
3
I
(
I
+1)
A
2
(
q
)
δ
(
hω
)
,
(7
.
3
.
29)
in the paramagnetic phase, which diverges at
q
=
Q
when
T
approaches
T
N
, analogously to the behaviour of the singlet-triplet case described
by (7.1.13).
S
(
q
,ω
≈
1
−
χ
J
(0)
{
1+
A
2
χ
J
(0)
χ
I
(0)
}J
7.4 Magnetic properties of Praseodymium
The magnetic behaviour of Pr has already been extensively discussed in
this chapter, in order to illustrate a number of the phenomena which
occur in crystal-field systems. In this section, we will collect together
these threads into a coherent description of the magnetic ordering which
may be induced by various perturbations, and of the excitations in the
paramagnetic and ordered phases.
7.4.1 Induced magnetic ordering
As discussed at the end of the preceding section, the coupling of the
nuclear spins to the electronic moments in Pr gives rise to a magnetic
system whose ground state is degenerate. According to the third law
of thermodynamics, this degeneracymustbeliftedatsucientlylow
temperatures and, within the MF approximation, this is accomplished
by magnetic ordering at a temperature determined by eqn (7.3.28). The
enhancement factor
R
0
/
(1
R
0
) is about 12 for the hexagonal sites, so
that the calculated collective-ordering temperature for the nuclear spins
and the electronic moments is raised into the more readily accessible
range of about 45 mK. The strong neutron-diffraction peak illustrated
in Fig. 7.8 was observed at 40 mK by Bjerrum Møller
et al.
(1982), at a
value of
Q
close to the minimum in the dispersion relations of the mag-
netic excitons. This mode of excitation comprises magnetic fluctuations
−
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