SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 242

considered by Kaplan and Lyons (1963) and Kasuya and Lyons (1966).
In order to obtain an estimate of the importance of the corrections, they
assumed plane-wave states for the conduction electrons, expanded in a
series of spherical Bessel functions centred at the ion. These calculations
indicated anisotropic two-ion couplings with a magnitude of the order of
10% of the isotropic coupling, or greater (Specht 1967). As discussed in
Section 1.3, the free-electron model does not provide a very satisfactory
description of the conduction electrons in the rare earths. It is partic-
ularly inadequate when orbital effects are involved, since the expansion
of the plane-wave states clearly underestimates the ( l = 2)-character of
the d -like band-electrons, which dominates the exchange interaction in
the ( L = 0)-case of Gd (Lindg ard et al. 1975). When L is non-zero, the
Kaplan-Lyons terms may be of comparable importance to the RKKY
interaction in the rare earth metals. The relativistic modification of the
band states, due to the spin-orbit coupling, may enhance the orbital
effects and also lead to anisotropic interactions in Gd. In addition to
the exchange, the direct Coulomb interaction between the 4 f and the
band electrons may contribute to eqn (5.5.14), with terms in which l
and l are both even. This coupling mechanism, via the conduction elec-
trons, is probably more important for this kind of term than the direct
electrostatic contribution mentioned above.
The RKKY interaction is derived on the assumption that the 4 f
electrons are localized in the core, and that their mixing with the conduc-
tion electrons is exclusively due to the exchange. However, the Coulomb
interaction may lead to a slight hybridization of the localized 4 f states
with the band states. In recent years, Cooper and his co-workers (Cooper
et al. 1985; Wills and Cooper 1987) have analysed the consequences of
a weak hybridization between an ion with one or two f electrons and
the band electrons, with special reference to the magnetic behaviour of
Ce compounds and the actinides. They find that the magnetic two-ion
coupling becomes highly anisotropic in the Ce compounds. Although
Ce is the rare earth element in which the strongest hybridization effects
would be expected to occur, these results and the analysis of Kaplan and
Lyons (1963) suggest that the presence of anisotropic two-ion couplings
should be a common feature in rare earth metals with orbital angular
momentum on the ion.
As is clear from the above discussion, an analysis from first prin-
ciples cannot at present give a reliable estimate of the relative magni-
tude of the Heisenberg exchange interaction and the various possible
anisotropic two-ion couplings in the rare earth metals. We cannot a
priori exclude any terms of the form given by eqn (5.5.14). In order to
arrive at such an estimate, it is necessary to calculate the consequences
of the anisotropic two-ion terms and compare the predictions with exper-