SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 238

form of the resolution function. With a spherical resolution, the scatter-
ing will be quite sharply peaked at the intermediate energy of the spin
waves propagating in the c -direction, as illustrated in the case of Tb
in Fig. 5.8. The calculated sharp peak at about 1.8 meV and the high-
energy shoulder are clearly apparent in the experimental measurements
of Houmann et al. (1975a). In the measurements of Bjerrum Møller
and Mackintosh (1979), on the other hand, the resolution function was
such that the modes propagating in the basal plane perpendicular to
the magnetization were most heavily weighted, so that the predominant
peak occurs at about 2.0 meV. In Ho, the effect of the dipole interaction
on the long-wavelength spin waves is even more pronounced. This is an
example of the opposite extreme, where A 0 ( T )+ B 0 ( T ) is much smaller
than A 0 ( T )

− B 0 ( T ), so that the maximum value of E T ( q ≈ 0 ) occurs
when q lies along the c -direction. As illustrated in Fig. 5.9, the dipolar
splitting in this case is suciently great that the neutron scattering at
q ≈ 0 can be resolved into two peaks.
Another consequence of the strong directional dependence of the
dipolar contributions to the spin-wave energies is found in the behaviour
of the coupled magneto-acoustic sound waves, discussed in the previ-
ous section. The region in q -space sampled in ultrasonic measurements
(with frequencies in the MHz regime) is just that in which eqn (5.5.12)

Fig. 5.8. The state density of the long-wavelength spin-wave mode
E T ( q ≈ 0 ) in Tb at 4 K, calculated from eqn (5.5.12), taking into account
the splitting of the dispersion relations by the dipole-dipole interaction.
The sharp peak is due to the branch in the c -direction.