SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 181

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SPIN WAVES IN THE FERROMAGNETIC
HEAVY RARE EARTHS

As discussed in Section 1.5, the exchange interaction dominates the mag-
netic behaviour of the heavy rare earth metals, and the ordered moments
at low temperatures are consequently close to the saturation values. The
excitations of such a system are spin waves , which may be viewed semi-
classically as coupled precessions of the moments about their equilib-
rium directions, with well-defined frequencies which are determined by
the phase relations between the precessing moments on different sites.
From the viewpoint of quantum mechanics, these modes are magnons ,
which are linear combinations of single-ion excitations from the ground
state to the first excited molecular-field state, which is to a good approx-
imation

1 > , with phase factors between the coecients for
different ions which determine the dispersion relation E q for the magnon
energy. A useful review of the excitations of magnetic systems has been
given by Stirling and McEwen (1987).
These spin waves have been very extensively studied in the heavy
rare earths, both experimentally and theoretically. In this chapter, we
consider the simplest case of the ferromagnet , in which all the sites
are equivalent. Since the magnetic heavy rare earths are all hcp, we
begin by extending the earlier treatment of the linear response of the
isotropic Heisenberg ferromagnet to this structure. These results are
immediately applicable to Gd, where the anisotropy is indeed negligible,
with the consequence that the excitation spectrum is the simplest to
be found among the magnetic rare earths. Crystal-field and magneto-
elastic anisotropies modify the excitation spectrum significantly, induc-
ing an elliptical polarization of the precessing moments, and a spin-wave
energy gap at long wavelengths. To treat such systems, we employ linear
spin-wave theory , determining the magnon energies via the Holstein-
Primakoff transformation . We consider in particular the basal-plane
ferromagnet, comparing the calculated excitation spectrum throughout
with experimental measurements on Tb, which has been very compre-
hensively studied. The magnon energies and their temperature depen-
dence are discussed, and the energy gap associated with the uniform
spin-wave mode is treated in some detail and related to the macro-
scopic magnetic anisotropy. The contribution to this energy gap of the
magnetoelastic coupling ,viathe static deformation of the crystal, is then

| J z = J −