SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 229

to the magnetization, as is assumed in eqn (5.4.43), whereas in a b -axis
magnet, the acoustic-optical coupling involves the transverse phonons
polarized perpendicular to the magnetization (i.e. still along an a -axis).
The symmetry arguments show that this coupling must be quadratic in
k in the long wavelength limit, instead of linear as is W k ( ε ). It there-
fore has no influence on the uniform strains or the elastic constants,
and accordingly no counterpart in the magnetoelastic Hamiltonian. Liu
(1972a) has discussed the possible origin of such an acoustic-optical in-
teraction, and he concludes that it cannot be a crystalline-field effect,
but must be mediated indirectly via the conduction electrons and be
proportional to the spin-orbit coupling forces. As is illustrated in Fig.
5.6, the acoustic-optical magnon-phonon interaction is clearly observed
in Tb, where it leads to the energy gap ∆ 2 , the strongest hybridization
effect seen in the metal. However, a closer examination (Jensen and
Houmann 1975) shows that the transverse phonon modes involved are
those polarized parallel to the magnetization, in spite of the fact that Tb
has its magnetization vector in the b -direction. Hence this interaction
violates the selection rules deduced from the general symmetry argu-
ments, leading to the conclusion that the ground-state of Tb cannot be
asimple b -axis ferromagnet as assumed. The 4 f moments are undoubt-
edly along an easy b -axis, but the spins of the conduction electrons are
not necessarily polarized collinearly with the angular momenta of the
core electrons, because of their spin-orbit coupling. If the ground-state
spin-density wave of the conduction electrons in Tb has a polarization
which varies in space within a single unit cell, a coupling mediated by this
spin-density wave may violate the selection rules based on the symmetry
properties of the simple ferromagnet. The presence of the 'symmetry-
breaking' acoustic-optical interaction in Tb demonstrates that the con-
duction electrons play a more active role than passively transmitting the
indirect-exchange interaction. This magnon-phonon coupling is directly
dependent on spin-orbit effects in the band electrons, in accordance with
Liu's explanation, and its appearance demonstrates that the polarization
of the conduction-electron spins must have a component perpendicular
to the angular momenta.
To complete this section, we shall briefly discuss the additional
magnon-phonon interaction terms which are linear in the phonon oper-
ators, but quadratic in the magnon operators:

mp =
qk ν

U ν ( k , q ) α q + k α q + 2

(2)

V ν ( k , q ) α q + k α + H

− q

− q − k ( β ν k + β ν− k ) .

+ 2

V ν (

− k , − q ) α q α (5 . 4 . 50)

Referring back to the magnetoelastic Hamiltonian, we find that such an