SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 304

of sharp peaks, as a function of ω at a constant value of q ;onesuch
example is shown in Fig. 6.3. These peaks indicate the presence of well-
defined excitations. The variation of the energy with the component of
q parallel to Q is very small, but the spectral weights of the different
peaks change. This pattern indicates that the excitations propagating
parallel to the ordering wave-vector are quasi-localized modes of com-
posite angular momenta. This behaviour may be explained by a closer
examination of the single-site response function (6.1.26). g i ( ω ) becomes
nearly zero, at non-zero frequencies, whenever is small, which

generally occurs twice in every period. This explains the low-frequency
diffusive response, and implies that the excitations become essentially
trapped between the sites with small moments.
This theory may, with some modifications, be applicable to a de-
scription of Er in its high-temperature, longitudinally polarized phase
( T N <T <T N ). The excitations in this temperature interval have been
studied by Nicklow and Wakabayashi (1982). They found no sharp peaks
in the transverse spectrum, but saw indications of relatively strong dis-
persive effects at small values of q ± Q . The absence of sharp peaks in
the spectrum may be explained by intrinsic linewidth effects, neglected
in the RPA theory utilized above, which may be quite substantial at the
relatively high temperatures of the experiments. However, the strong
dispersive effects detected close to the magnetic Bragg peaks are not
consistent with the results discussed above. One modification of the sim-
ple model which may be important is the squaring-up of the moments,
which has been considered by Lantwin (1990). The higher harmonics
lead to additional coupling terms in (6.1.30), and the analysis becomes
correspondingly more complex. However, a simple argument shows that
the higher harmonics result in less localized modes, and thus lead to a
stronger dispersion, as also concluded by Lantwin. It is because the in-
tervals along the c -axis in which the moments are small become narrower
when the moments square up, so that the excitations may tunnel more
easily through these regions. Another limitation of the theory, which
may be important for Er, is that the single-site crystal-field anisotropy,
neglected in the model, is probably more important than the two-ion ax-
ial anisotropy. The single-ion anisotropy splits the levels, even when the
exchange field vanishes, and excited dipolar states may occur at energies
suitable for allowing the excitations to propagate across sites with small
moments, more freely than in the simple model. In the limit where the
exchange field is small compared to the crystal-field splittings, which we
shall discuss in the next chapter, the corresponding continued fractions
in G 0 ( q ,ω ) converge rapidly (Jensen et al. 1987), and the results become
largely independent of whether the ordering is commensurable or not.

A i = J iζ