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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 J
J J )cos2 φ ij +( J J + J J )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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