SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 219

of ∆ M , compared to the contributions of C and A to the energy gap,
effectively precludes this possibly. We must therefore ascribe it to two-
ion anisotropy.
In the analysis of the field dependence of the magnon energy gap,
the possible dependences of the renormalization parameters σ and η ±
on magnetic field and the orientation of the moments were neglected
at zero temperature, but included at non-zero temperatures, assuming
the different parameters effectively to be functions of σ only. In the
case of Dy, the zero-temperature change of the renormalization as a
function of φ is of some importance (Egami 1972; Jensen 1975; Egami
and Flanders 1976), whereas in Tb we have estimated by various means
that both approximations are justified. There are some indications that
there might be a systematic error involved in the determination of the
φ -dependent energy-gap parameters P 6 ( ), possibly arising from the

influence of the classical dipole forces on the inelastic neutron-scattering
at long wavelength, discussed in Section 5.5.1. An extrapolation of the
results found at non-zero wave-vectors to q = 0 suggests that both
P 6 (+) and P 6 (

±

) may be about a factor of two smaller than shown
in Fig. 5.5. If this were the case, ∆ M would still be too large to be
explained by the γ -strain couplings, but δ 6 (

−

) would be reduced almost
to the level of the experimental uncertainties.

−

Otherwise a non-zero

value of δ 6 ( ) can only be explained by theories beyond the RPA, e.g.

by effects, proportional to the frequency, due to the interaction between
the spin-waves and the electron-hole pair-excitations of the conduction
electrons.

−

5.4.2 The magnon-phonon interaction
The displacement of the i th ion from its equilibrium position, δ R i =
u ( R i ), can be expanded in normal phonon coordinates in the usual way:
u ( R i )=
ν k

F k ( β ν k + β ν− k ) e i k · R i , (5 . 4 . 24 a )

with

F k ,α = h
2 NMω ν k

2 f k ,α .

(5 . 4 . 24 b )

M is the mass of the ions and f k ,α is the α -component of the phonon-
polarization vector. β ν k is the phonon-annihilation operator and ω ν k
the corresponding phonon frequency, where ν denotes one of the three
(acoustic) branches. The polarization vectors are normalized and are
mutually orthogonal:

( f k ,α ) ∗ f ν

k ,α = δ νν .

(5 . 4 . 24 c )

α