ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 39

which increases through these structural sequences, is the essential de-
terminant of the structure, and made an approximate calculation of the
energy differences using canonical-band theory. The results of Skriver
(1983) in Fig. 1.15 show how well the d occupancy indeed correlates
with the structure. To complete the picture, Min et al. (1986b) demon-
strated that increasing the pressure on Lu should produce a series of
phase transitions following the above sequence, the first of which has
been observed experimentally.

1.4 Magnetic interactions
In the metallic state, the 4 f electrons on a rare earth ion are subjected
to a variety of interactions with their surroundings. These forces may
be broadly classified into two categories. The single-ion interactions act
independently at each ionic site, so that their influence on the state of
the 4 f electrons at a particular site is unaffected by the magnetic state
of its neighbours. The corresponding contribution to the Hamiltonian
therefore contains sums over terms located at the ionic sites i of the
crystal, but without any coupling between different ions. On the other
hand, the two-ion interactions couple the 4 f -electron clouds at pairs of
ions, giving terms which involve two sites i and j .
The charge distribution around an ion produces an electric field,
with the local point-symmetry, which acts on the 4 f electrons and gives
rise to the large magnetic anisotropies which are characteristic of the rare
earth metals. This crystal field makes a contribution to the potential
energy

v cf ( r )= eρ ( R ) d R , (1 . 4 . 1)

| r − R |

where ρ ( R ) is the charge density of the surrounding electrons and nuclei.
If these do not penetrate the 4 f charge cloud, v cf ( r ) is a solution of
Laplace's equation, and may be expanded in spherical harmonics as

v cf ( r )=
lm

A l r l Y lm ( r ) , (1 . 4 . 2)

where

eρ ( R )
R l +1

4 π
2 l +1

Y l−m ( R ) d R ,

A l 1) m =( − (1 . 4 . 3)

which is a special case of the multipole expansion (1.3.7). We can
thus look upon (1.4.2) as arising from the interaction of the multipoles
r l Y lm ( r )ofthe4 f electrons with the appropriate components of the
electric field. If part of the charge which is responsible for the crystal
field lies within the 4 f cloud, v cf ( r ) can still be expanded in spherical