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7.4.2 The magnetic excitations
The magnetic-excitation spectrum in Pr has been investigated experi-
mentally in great detail as a function of various external constraints,
such as the temperature, a magnetic field applied in the basal plane, and
uniaxial pressure. Most of the knowledge about the (low-temperature)
coupling parameters in the model Hamiltonian for Pr, which we have al-
ready utilized several times in the preceding sections, has been derived
from these experiments. The first inelastic neutron-scattering exper-
iments on Pr (Rainford and Houmann 1971; Houmann
et al.
1975b)
showed that the excitations behave as expected in a singlet ground-
state system, and that the two-ion coupling is just below the threshold
value for inducing magnetic ordering. A MF analysis of the tempera-
ture dependence of the excitations, shown by the dashed lines in Fig.
7.3, indicated that the crystal-field splitting ∆ between the
|
0
>
ground-
state and the first excited
1
>
-doublet state of the hexagonal ions is
about 3.2 meV. An important discovery (Houmann
et al.
1975b) was the
observation, illustrated in Fig. 7.1, of a strong splitting of the doublet
excitations, whenever such a splitting is allowed by symmetry, i.e. when
q
is not along the
c
-axis. This effect demonstrates that the anisotropic
contribution to the two-ion Hamiltonian of Pr,
|±
2
1
H
JJ
=
−
ij
J
(
ij
)
J
i
·
J
j
2
(
ij
)
(
J
iξ
J
jξ
−
J
iη
J
jη
)cos2
φ
ij
+(
J
iξ
J
jη
+
J
iη
J
jξ
)sin2
φ
ij
,
+
1
ij
K
(7
.
4
.
5)
is important. Here
φ
ij
is the angle between the
ξ
-axis and the projection
of
R
i
−
R
j
on the basal plane. Real-space coupling parameters
J
(
ij
)
(
ij
) derived from the excitation energies shown in Fig. 7.1, using
the MF-RPA expression for the energies with ∆ = 3
.
52 meV, are shown
in Fig. 1.18. This somewhat larger value of ∆ was obtained from a
study of the field dependence of the excitations (Houmann
et al.
1979),
but it is still consistent with their temperature dependence, as shown
by the results of the self-consistent RPA, the solid lines in Fig. 7.3.
Besides leading to the more accurate value of ∆, the field experiments
revealed the presence of a rather strong magnetoelastic
γ
-strain coupling
in Pr, which creates energy gaps proportional to the field at the crossing
points of the magnetic-exciton and transverse-phonon branches in the
basal-plane directions, as illustrated in Fig. 7.14.
The model Hamiltonian, with the two-ion and magnetoelastic terms
given respectively by (7.4.5) and (7.4.3), together with the usual single-
ion crystal-field Hamiltonian for a hexagonal system, describes very well
the excitation-energy changes observed by Houmann
et al.
(1979) when
K
and
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