Environmental Engineering Reference
In-Depth Information
This effect is relatively modest for the
d
bands, but much greater for
the
s
and
p
bands. The relative shift of the
s
and
d
bands is reduced by
the adjustment of the lattice to its equilibrium configuration, but only
by a small amount. As may be seen from Fig. 1.12, (
B
d
−
B
s
)increases
from 101 mRy for La to 373 mRy for Lu at constant
S
, whereas the cor-
responding values for the equilibrium atomic volumes are 136 mRy and
380 mRy. The band masses also change across the series;
µ
d
at constant
volume increases from 2.1 in La to 3.0 in Lu, so that the
d
bands narrow
as they fall, while
µ
s
increases slightly with atomic number, but remains
below 1 throughout (Skriver 1983).
The canonical-band theory may be used to calculate the electronic
pressure and its partitioning between the different angular momentum
components. According to the
force theorem
(see Mackintosh and Ander-
sen 1980) the change in the total energy, due to an infinitesimal change
in the lattice constant, may be determined as the difference in the band
energies, calculated while maintaining the potential unchanged. We may
thus write
dU
=
δ
ε
F
εN
(
ε
)
dε,
(1
.
3
.
24)
where
N
(
ε
) is the total electronic density of states, and
δ
indicates the
restricted variation with a
frozen potential
. The electronic pressure is
then given by
dU
d
Ω
,
P
=
−
(1
.
3
.
25)
where Ω is the volume of the atomic polyhedron. The expression (1.3.21)
for the canonical-band energies then leads to the approximate result for
the
l
partial pressure:
C
l
)
δ
ln
µ
l
S
2
δ
ln
S
δC
l
δ
ln
S
+
n
l
(
ε
l
−
3
P
l
Ω=
−
n
l
,
(1
.
3
.
26)
where
n
l
is the occupation number of the
l
states and
ε
F
εN
l
(
ε
)
dε
1
n
l
ε
l
=
(1
.
3
.
27)
is their mean energy. Equation (1.3.26) is useful for purposes of inter-
pretation, but the results which we shall present are based upon a more
accurate procedure, involving the fully hybridized self-consistent band
structure (Skriver 1983).
The partial occupation numbers, state-densities and electronic pres-
sures for
α
-Ce, at the equilibrium lattice constant, are given in Table
1.3. The
s
and
p
electrons make a positive, repulsive contribution to
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