CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 338

external magnetic field, the coupling between the dipolar crystal-field
excitations and the long-wavelength phonons must therefore vanish by
symmetry, within the present approximation, and Ξ( q , 0) = χ 44 (0) in
eqn (7.3.10). In the presence of an external magnetic field, the mixed
dipolar-quadrupolar susceptibility-components may become non-zero,
and hence produce a direct coupling of the elastic waves and the dipo-
lar excitations. In this case, the magnetic dipole coupling, which gives
rise to a directional dependence of

J αα ( q ), as discussed in Section 5.5,
leads to different values of c 66 (as determined from the transverse sound
velocity in the b ( η )-direction), depending on whether the field is parallel
to the ξ -orthe η -axis or, if the field is fixed along one of these two axes,
whether q is along the ξ -orthe η -direction. As mentioned earlier, this
anisotropy is similar to that introduced by rotational invariance, and
has a comparable magnitude in paramagnetic systems (Jensen 1988b).
The dynamic coupling between the magnetic and elastic excitations
in Pr has been studied in the long-wavelength limit by Palmer and Jensen
(1978), who measured the elastic constant c 66 by ultrasonic means, as a
function of temperature and magnetic field. At 4 K, it was found to be
very sensitive to a field applied in the basal plane, but insensitive to a
field along the c -axis, reflecting the anisotropy of the susceptibility. At
non-zero fields in the basal plane, there is furthermore a considerable
anisotropy, due to B 6 . Using the crystal-field level scheme illustrated in
Fig. 1.16, and a value of B γ 2 consistent with that deduced from the field
dependence of the magnetic excitations (Houmann et al. 1979), they
were able to obtain a very good fit to the observed dependence of c 66 on
field, shown in Fig. 7.5, and on temperature.
The above theory is also valid at non-zero frequencies. However, if
q is no longer small, we must take account of the discreteness of the lat-
tice and replace q in (7.3.8) by a sinusoidal function of q and the lattice
parameters, as in (5.4.43) in Section 5.4. Except for the change in the
q -dependence of J 44 ( q ,ω ), eqn (7.3.7) still applies, and it predicts hy-

bridization effects between the phonons and the crystal-field excitations,
equivalent to those derived from the linear magnon-phonon coupling in
Section 5.4. The time-reversal symmetry of the paramagnetic system in
zero magnetic field does not exclude the possibility that the phonons at
non-zero frequencies are coupled to the crystal-field dipolar excitations
and, in the case of Pr, the doublet excitations are allowed to interact
with the transverse phonons, when q is in the c -direction. Neverthe-
less, the application of a magnetic field will generally introduce new
interactions via χ 4 α ( ω ), leading to hybridization effects proportional to
the field, as observed in Pr by Houmann et al. (1979) and interpreted
by Jensen (1976a). Interactions between crystal-field excitations and