SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 231

of 1 /J . Equation (5.4.52) shows that the 'zero-point' motion of the ions,
at T = 0, has a slight effect on the magnons. A similar effect occurs due
to the magnon-magnon interactions, but only in an anisotropic ferro-
magnet where B is non-zero, as we discussed in the previous section. In
most cases, the contributions due to the magnon-magnon interactions
are expected to predominate, because the magnon-phonon coupling pa-
rameters are usually quite small, in comparison with the spin-wave inter-
actions. Although the interactions in (5.4.50) may not be important for
the magnons, they may have observable effects on the phonons at finite
temperatures. For instance, they affect the velocity of the transverse
sound waves propagating in the c -direction and polarized perpendicular
to the magnetization, but not those polarized parallel to the magneti-
zation, which are modified by the linear couplings as discussed above.
Deriving the perturbed phonon Green functions in the same way as the
magnon Green function, and taking the long-wavelength limit, we find
(Jensen 1971a,b)
c 44
c 44 NJ

q

1 n q

E q ( T )

f k ⊥ J . =1 − Λ ε when (5 . 4 . 53)

We note that this result is of higher order in 1 /J than the effect due
to the linear coupling, given in (5.4.42). However, the extra factor 1 /J
may be compensated by the magnon population-factor n q in the sum
over q , at elevated temperatures.
Modifications of the results obtained above may occur, due to an-
harmonic terms of third order in the strains, or magnetoelastic terms
quadratic in the strains. These higher-order contributions may possi-
bly be of some importance for the temperature dependence of the elas-
tic constants and the spin-wave parameters. However, they should be
of minor significance under the nearly constant-strain conditions which
obtain, for instance, when the magnetic-field dependence of the elastic
constants is considered.

5.5 Two-ion anisotropy
In this section, we discuss the components of the two-ion coupling which
cannot be included in the isotropic Heisenberg Hamiltonian considered
hitherto, i.e. the two-ion term in eqn (5.2.1). We first consider the clas-
sical magnetic dipole-dipole interaction in some detail, and show how
it may affect the spin-wave energies and ultrasonic velocities. There-
after we discuss some of the complexities resulting from the presence of
general two-ion couplings , which are consistent with the symmetry prop-
erties of the magnetic phase. The experimental manifestations of such
interactions, which either have been observed in the excitation spectrum
of Tb, or could in principle be observed, are finally summarized.