SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 299

in the presence of arbitrarily large anisotropy, by a numerical calcula-
tion of the MF susceptibility χ o ( ω ), as determined by the crystal-field
Hamiltonian and the exchange field, given by (6 . 1 . 15 b ). In the general
case, it is necessary to include the total interaction-matrix

( q ), and
not only the ( xy )-part as in (6.1.14), when deriving the final suscepti-
bility matrix (6.1.7). A numerical calculation of the excitation energies,
for a model which also accounts fairly accurately for the anisotropy of
Er, leads to energies which are very well described by the linear spin-
wave theory (Jensen 1976c), the discrepancies being only of the order
of a few per cent. The spin waves are not purely transverse, as the
individual moments are calculated to precess in a plane whose normal
makes an angle of about 33 ◦ with the c -axis. The relation between the
difference and the sum of E q and E q is still found to be obeyed, when
the two-ion anisotropy is neglected. The experimental results therefore
attest to the importance of such anisotropy effects. Except for the tilt-
ing of the plane in which the moments precess, which is not easy to
detect experimentally, the linear spin-wave theory is found to give an
accurate account of the excitations at low temperatures in Er. In spite
of this, it is not a good approximation to consider only the ground state
and the first excited state of the MF Hamiltonian, when calculating the
excitation spectrum, because 10-15% of the dispersive effects are due
to the coupling between the spin waves and the higher-lying MF levels.
These effects are included implicitly, to a first approximation, in the
linear spin-wave theory, which gives an indication of the ecacy of the
Holstein-Primakoff transformation (when J is large).
We have so far neglected the hexagonal anisotropy. In Section 2.1.3,
we found that B 6 causes a bunching of the moments about the easy
axes in the plane, leading to (equal) 5th and 7th harmonics in the
static modulation of the moments. The cone is distorted in an anal-
ogous way, but the hexagonal anisotropy is effectively multiplied by the
factor sin 6 θ 0 ≈

J

0 . 01 in Er. The effects of B 6 on the spin waves are
therefore small, and may be treated by second-orderperturbationthe-
ory, which predicts energy gaps in the spectrum whenever E q = E q ± 6 Q
(for a further discussion, see Arai and Felcher, 1975). In the experi-
mental spin-wave spectrum of Er, shown in Fig. 6.2, energy gaps are
visible, but not at the positions expected from the coupling due to the
hexagonal anisotropy. It seems very likely that the two gaps observed
close to q =0 . 4(2 π/c ) are due to an interaction with the transverse
phonons. Although the normal magnetoelastic ε -coupling, which leads
to energy gaps when E q = hω q ± Q , might be significant for the lower
branch, the positions of both gaps agree very well with those expected
from an acoustic-optical coupling, occurring when E q = hω q ± 2 Q + b 3 (in
the double-zone representation), as indicated in Fig. 6.2. Although the