SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 186

prediction and the observed behaviour at low temperatures may be due
to changes of ( q ). At higher temperatures, the RPA renormalization

breaks down completely. The spin-wave energy at the zone boundary
has only fallen by about a factor two at 292 K, very close to T C .Fur-
thermore, strongly-broadened neutron peaks are observed even at 320 K,
well above the transition, close to the zone boundary in the basal plane,
with energies of about k B T C . On the other hand, the low-energy spin
waves progressively broaden out into diffusive peaks as T C is approached
from below.

J

5.2 Spin waves in the anisotropic ferromagnet
In the heavy rare earth metals, the two-ion interactions are large and
of long range. They induce magnetically-ordered states at relatively
high temperatures, and the ionic moments approach closely their sat-
uration values at low temperatures. These circumstances allow us to
adopt a somewhat different method, linear spin-wave theory ,fromthose
discussed previously in connection with the derivation of the correlation
functions. We shall consider the specific case of a hexagonal close-packed
crystal ordered ferromagnetically, with the moments lying in the basal
plane, corresponding to the low-temperature phases of both Tb and Dy.
For simplicity, we shall initially treat only the anisotropic effects intro-
duced by the single-ion crystal-field Hamiltonian so that, in the case of
hexagonal symmetry, we have

gµ B J i · H

=
i

2

1

B l Q l ( J i )+ B 6 Q 6 ( J i ) H − −

i = j J

( ij ) J i · J j .

l =2 , 4 , 6

(5 . 2 . 1)
The system is assumed to order ferromagnetically at low temperatures,
a sucient condition for which is that the maximum of

( q ) occurs at
q = 0 . Q l ( J i ) denotes the Stevens operator of the i th ion, but defined
in terms of ( J ξ ,J η ,J ζ ) instead of ( J x ,J y ,J z ), where the ( ξ, η, ζ )-axes
are fixed to be along the symmetry a -, b -and c -directions, respectively,
of the hexagonal lattice. The ( x, y, z )-coordinate system is chosen such
that the z -axis is along the magnetization axis, specified by the polar
angles ( θ, φ )inthe( ξ, η, ζ )-coordinate system. Choosing the y -axis to
lie in the basal plane, we obtain the following relations:

J

J ξ = J z sin θ cos φ − J x cos θ cos φ + J y sin φ

J η = J z sin θ sin φ − J x cos θ sin φ − J y cos φ (5 . 2 . 2)

J ζ = J z cos θ + J x sin θ,

from which

Q 2 =3 J z cos 2 θ + J x sin 2 θ +( J z J x + J x J z )cos θ sin θ { }− J ( J +1) . (5 . 2 . 3)