SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 205

of (5.2.37), in eqns (5.2.36), (5.2.38), and (5.2.40). If the out-of-plane
anisotropy is stronger than the in-plane anisotropy, as in Tb and Dy, B
is positive and b is negative. This means that η + and η − are respectively
smaller and greater than 1 (for small b ), with the result that the axial
contributions to A 0 ( T )+ B 0 ( T ) are increased, whereas the planar con-
tribution to A 0 ( T ) B 0 ( T ) is diminished, due to b . This is consistent

with the fact that the out-of-plane fluctuations are suppressed in com-
parison with the in-plane fluctuations by the anisotropy. Hence we find,
as a general result, that the elliptical polarization of the spin waves en-
hances, in a self-consistent fashion, the effects of the anisotropy. We note
that Q 6 , which depends on both θ and φ , contributes to both anisotropy
parameters, but that the anisotropy of the fluctuations affects the two
contributions differently.
If b and the k -sums in (5.2.38) are neglected, the above result for
the spin-wave energies E q ( T ) reduces to that derived by Cooper (1968b).
The modifications due to the non-spherical precession of the moments,
b

−

= 0, were considered first by Brooks et al. (1968) and Brooks (1970),
followed by the more systematic and comprehensive analysis of Brooks
and Egami (1973). They utilized directly the equations of motion of
the angular-momentum operators, without introducing a Bose repre-
sentation. Their results are consistent with those above, except that
they did not include all the second-order modifications considered here.
We also refer to Tsuru (1986), who has more recently obtained a re-
sult corresponding to eqn (5.2.31), when B 6 is neglected, using a varia-
tional approach. The procedure outlined above essentially follows that
of Lindg ard and Danielsen (1974, 1975), which was further developed
by Jensen (1975). This account only differs from that given by Jensen
in the use of η instead of b as the basis for the 'power-law' general-

ization (and by the alternative choice of sign for B and b ) and, more
importantly, by the explicit use of 1 /J as the expansion parameter.
As illustrated in Fig. 5.1 for Gd, and in Fig. 5.3 for Tb, the observed
temperature dependence of the spin-wave spectrum is indeed substan-
tial, both in the isotropic and the anisotropic ferromagnet. In the case
of Tb, the variation of the exchange contribution is augmented by the
temperature dependence of the anisotropy terms, which is reflected pre-
dominantly in the rapid variation of the energy gap at q = 0 .Acom-
parison of Figs. 5.1 and 5.3 shows that the change in the form of

±

( q )
appears to be more pronounced in Tb than in Gd. In Tb, the variation
of

J

( q )with q at a particular temperature is also modified if the mag-
netization vector is rotated from the b -axis to a hard a -axis (Jensen et
al. 1975). Most of these changes with magnetization can be explained
as the result of two-ion anisotropy, which we will consider in Section 5.5.
Anisotropic two-ion terms may also affect the energy gap. In addition,

J