ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 32

theory gives a realistic description of the conduction electron states, and
may therefore be used as a basis for the calculation of properties which
depend on the electronic structure.
This conclusion could already be drawn, though with slightly less
confidence, from the pioneering measurements of Mattocks and Young
(1977) of the dHvA effect in ferromagnetic Gd. Because of the ferro-
magnetic moment, the exchange interaction (1.3.22) separates the energy
bands of different spin even in zero field, and the exchange splitting is
essentially independent of field. The results were interpreted in terms
of the paramagnetic energy bands, originally calculated by Dimmock
and Freeman (1964) and, with relativistic effects, by Keeton and Loucks
(1968), taking account of the ferromagnetic structure by a rigid split-
ting of the bands. The resulting two majority-spin hole surfaces and the
minority-spin electron surface could account for all of the observed large
orbits, with a value of I close to that later deduced for Pr, and with
a comparable variation through the zone. However, many small orbits
were observed which could not be explained with this model, nor have
subsequent band calculations, culminating in those of Temmerman and
Sterne (1990), in which the exchange splitting of the conduction bands
was included a priori, fully accounted for the small pieces of the Fermi
surface. Although the general features of the electronic structure of Gd
may therefore be considered as well understood, a further theoretical
effort, taking into account the effect on the band structure of the spin-
orbit coupling in the presence of both an exchange field and an external
field, would be necessary to explain the finer details.
The positron-annihilation experiments of Williams and Mackintosh
(1968), although at a much lower level of resolution, were also in gen-
eral accord with the calculations of Keeton and Loucks (1968). They
studied a number of heavy rare earths in their paramagnetic phases,
showing that their Fermi surfaces are highly anisotropic and rather sim-
ilar to each other. A calculation based upon energy-band theory gave
a good account of the experimental results for Y. The distributions of
the annihilation photons displayed a feature which is sensitive to the
form of the hole surface shown in Fig. 1.11, namely the shape of the
'webbing' which may join the 'toes' on the surface near L. This charac-
teristic is very dependent on the relative positions of the s and d bands,
and the calculations indicated that the webbing is absent in Gd, very
narrow in Tb, and fully developed, forming a kind of plateau, in the
other heavy rare earths. These conclusions were in accordance with the
positron-annihilation results, which further indicated that the webbing
is destroyed in the magnetically ordered phase of Ho. The relation of
these observations to the occurrence of periodic magnetic structures will
be discussed in the following section.