ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 25

are then known respectively as the top and bottom of the nl band, even
though this designation is not precisely accurate.
Over the energy range A nl - B nl , the potential function may be
parametrized with reasonable accuracy as

1
γ nl

ε − C nl

P l ( ε ) . (1 . 3 . 18)

ε − V nl

It is convenient to define the related mass parameter µ nl by

C nl

dP l ( ε )
dε

1
γ nl ( C nl −

µ nl S 2 =

V nl ) .

(1 . 3 . 19)

It is determined by the probability that an electron described by the par-
tial wave R l ( C nl ,r ) reaches the atomic sphere and, if the wavefunction
is normalized within the sphere, it may be shown that
µ nl S 2 = 2 SR l ( C nl ,S ) − 1 . (1 . 3 . 20)

For free electrons, µ nl ≡ 1 for all values of n and l .

With this parametrization, we may write down an explicit expres-
sion for the unhybridized band energies. From eqns (1.3.12), (1.3.18)
and (1.3.19) these are given by

k
lj

S

1
µ nl S 2

ε lj ( k )= C nl + .

(1 . 3 . 21)

lj

1 − γ nl S

The pure l bands are thus obtained from the corresponding canonical
bands by fixing the position with C nl , scaling the bandwidth by µ nl S 2 ,
and distorting them with γ nl .
Hybridization between bands of different l is taken into account
by including the structure constants with l

= l in (1.3.11), causing
a repulsion between energy levels with the same k and symmetry, as
specified by the labels in Fig. 1.6. Bands of the same symmetry are
thus not allowed to cross, and strong hybridization instead creates an
energy gap. In addition, weak hybridization gives rise to a mixing and
repulsion between bands which do not cross in the absence of hybridiza-
tion. In order to complete the calculation of the band structure, the
inaccuracies due to approximating the atomic polyhedron by a sphere,
and to neglecting higher partial waves, may be conveniently treated to-
gether by perturbation theory. In practice, the energy bands are not
of course calculated step-wise as described above, but all the steps are
performed simultaneously on a computer. Nevertheless, the conceptual
description of the procedure as a placing, scaling and distortion of the