Environmental Engineering Reference
In-Depth Information
canonical bands, according to the values of the potential parameters,
with a final hybridization between bands of the same symmetry, allows
a clear visualization of the way in which the relatively complex band
structure is built up from simpler elements, and of the relation between
the eigenstates of the atom and those in the solid.
Tab l e 1 . 3 .
Electronic parameters for
α
-Ce.
6
s
6
p
5
d
4
f
A
l
(Ry)
2.234
2.698
1.198
0.648
C
l
(Ry)
0.620
1.452
0.792
0.628
B
l
(Ry)
0.330
0.909
0.409
0.587
µ
l
0.61
0.70
2.18
45.36
n
l
0.509
0.245
2.091
1.154
N
l
(
ε
F
)(Ry
−
1
)
1.81
1.50
6.48
21.11
P
l
Ω (Ry)
0.195
0.152
−
0.219
−
0.163
This procedure may be illustrated by considering the construction
of the band structure of
α
-Ce from its component parts. Partial waves in
the atomic sphere at the energies of the band-centres, where
P
l
(
ε
)=0,
are shown in Fig. 1.5, and the corresponding potential parameters are
given in Table 1.3. In this section, we express the energies in Rydbergs,
following our general principle of using throughout the topic those units
which are favoured by practitioners of research in the subject currently
under discussion. The
s
and
p
effective masses are somewhat below 1,
and the relative positions of the band centres correspond quite closely to
those of the free-electron gas. Through the influence of the
l
-dependent
centrifugal-potential term in (1.2.12), the
d
and
f
states are in contrast
constrained to the inner regions of the atomic sphere, with the conse-
quence that the
d
mass is relatively large (though not as large as in a
typical transition metal) and the
f
mass is extremely large.
The energy bands of Fig. 1.8 were calculated by an iterative pro-
cedure, by Skriver (private communication). The electron density
n
(
r
)
is first estimated by, for example, overlapping
atomic
charge densities
situated on the lattice sites, and from it the periodic potential
v
eff
(
r
)
is constructed, using the local-density approximation (1.2.9). The band
structure is then determined for this potential and
n
(
r
) recalculated,
in analogy with (1.2.6), by summing over occupied states, those be-
neath the Fermi level. This procedure is repeated until the potential
self-consistently reproduces itself, and the energy bands have converged
to the desired accuracy. The band structure can be considered as being
Search WWH ::
Custom Search