SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 246

parameters decrease with increasing temperature or decreasing mag-
netization. The three anisotropy component all decrease very rapidly,
roughly proportionally to σ 15 (like ∆ M ), which means that they are only
important at low temperatures and may be neglected above about 150 K.
The strong temperature dependence of the anisotropic components indi-
cates, according to the Callen-Callen theory, that higher-rank couplings
make the dominant contribution. The lowest-order term in eqn (5.5.14)
which contributes to ( q )involves K 33

44 , and it should renormalize ap-
proximately as σ 19 . The renormalization is observed to be q -dependent
for all the parameters, being slower at larger wave-vectors, and it lies in
the range σ 2 - σ 0 . 1 for

C

( q ) may include higher-rank con-
tributions besides the RKKY-exchange term, but the way in which it
renormalizes resembles quite closely the behaviour observed in Gd and
shown in Fig. 5.1. A q -dependent renormalization may partially be ac-
counted for, in the self-consistent RPA theory, by the k -sum terms in
(5.2.38).
When q is not along the c -axis, there are other ways in which the
presence of two-ion anisotropy may be detected in the ferromagnetic
excitation spectrum:
(iv) Spin-wave energy gaps may appear at the boundaries of
the Brillouin zone.
The isotropic two-ion coupling alone does not lead to energy gaps at
these boundaries, whereas anisotropic two-ion couplings may lift those
degeneracies which are not dictated by symmetry. In fact, the first in-
dication of the presence of two-ion anisotropy in the rare earth metals,
other than the classical magnetic-dipolar interaction, was the splitting
shown in Fig. 5.2 along the KH edge of the Brillouin zone in the ferro-
magnetic phase of Tb (Lindg ard and Houmann 1971). Finally, we have
the related effect:
(v) The spin-wave energies, at a particular q , in domains with
different angles between the q -vector and the magnetiza-
tion vectors, need not be equal if two-ion anisotropy is
important.
In a single domain, the two-ion anisotropy forces may lift the 'accidental'
degeneracies between spin waves at q -vectors which are equivalent in the
paramagnetic phase, but which are no longer equivalent in the Brillouin
zone of the ferromagnet. This manifestation of the two-ion anisotropy
has not yet been subjected to experimental investigation, but it may
provide a useful supplement to studies of the q -dependence of B q ( T ).
As we have seen, the expectation values

I ( q ). In Tb, I

O lm are approximately

proportional to σ l ( l +1) / 2 , if the extra modification due to the elliptical
polarization of the spin-waves is neglected. This means that the impor-
tance of the higher-rank couplings declines relatively rapidly with tem-