SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 211

symmetry axis, but the first-order parameters are replaced by effective
values. These effective parameters are determined self-consistently in
terms of the spin-wave parameters A q ( T ) B q ( T ), which depend on T ,

and on an eventual applied magnetic field. One advantage of the use of
1 /J as the expansion parameter is that the second-order modifications
are smallest for the low-rank couplings, which are quite generally also
the largest terms. If the magnetization is not along a symmetry axis,
the elementary excitations may no longer be purely transverse. This
additional second-order phenomenon may, however, be very dicult to
detect experimentally within the regime of validity of the second-order
spin-wave theory.

±

5.4 Magnetoelastic effects
The magnetoelastic coupling between the magnetic moments and the
lattice modifies the spin waves in two different ways. The static de-
formations of the crystal, induced by the ordered moments, introduce
new anisotropy terms in the spin-wave Hamiltonian. The dynamic time-
dependent modulations of the magnetic moments furthermore interfere
with the lattice vibrations. We shall start with a discussion of the
static effects, and then consider the magnon-phonon interaction. The
magnetoelastic crystal-field Hamiltonian was introduced in Section 1.4,
where the different contributions were classified according to the symme-
try of the strain parameters. The two-ion coupling may also change with
the strain, as exemplified by eqn (2.2.32). We shall continue consider-
ing the basal-plane ferromagnet and, in order to simplify the discussion,
we shall only treat the low-rank magnetoelastic couplings of single-ion
origin. In the ferromagnetic case, the magnetoelastic two-ion couplings
do not introduce any effects which differ qualitatively from those due
to the crystal-field interactions. Consequently, those interactions which
are not included in the following discussion only influence the detailed
dependence of the effective coupling parameters on the magnetization
and, in the case of the dynamics, on the wave-vector.

5.4.1 Magnetoelastic effects on the energy gap
The static effects of the α -strains on the spin-wave energies may be
included in a straightforward manner, by replacing the crystal-field pa-
rameters in (5.2.1) with effective strain-dependent values, i.e. B 2 →

B 2 + B (2)

α 1 α 1 + B (2)

Q 2

.Equiva-
lent contributions appear in the magnetic anisotropy, discussed in Sec-
tion 2.2.2. This simplification is not possible with the γ -orthe -strain
contributions, because these, in contrast to the α -strains, change the
symmetry of the lattice. When θ = π/ 2, the -strains vanish, and the

α 2 α 2 ,with α -strains proportional to