CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 319

is frozen out exponentially in the low-temperature limit.
As shown in Fig. 7.1, the dispersion relations for the magnetic exci-
tations on the hexagonal sites in Pr, measured by Houmann et al. (1979),
illustrate many of the characteristic features of the ( J =1) XY -model.
As mentioned above, when q is along ΓM, the excitations are pure x -
or y -modes. The hexagonal ions constitute an hcp structure, so there
are an optical and an acoustic mode for each polarization. The excita-
tion energies (7.1.4) are then generalized analogously to eqn (5.1.9), and
since J 2 ( 0 ) is negative in this case, the lower two branches are the opti-

cal modes. From intensity measurements of the type illustrated in Fig.
4.2, it may readily be deduced that the lowest branch is the longitudinal
optical y -mode. The experimental dispersion relations show clearly that
J xx ( q )and J yy ( q ) have very different dependences on wave-vector, and

that the anisotropic component is a substantial fraction of the two-ion
coupling.

Fig. 7.1. Dispersion relations for the magnetic excitations propagat-
ing on the hexagonal sites of Pr at 6 K. In the basal plane, the squares
and circles denote the experimental results for the acoustic and optical
modes respectively. The double degeneracy of these excitations is lifted
by anisotropic exchange, and the lower and upper branches correspond
respectively to polarizations predominantly parallel and transverse to the
wave-vector. The double-zone representation is used for the ΓA direction,
along which the two transverse excitations are degenerate by symmetry,
and therefore form a single branch.