Environmental Engineering Reference
In-Depth Information
Fig. 1.4.
The Brillouin zones for the hexagonal and fcc lattices.
effective procedure for the rare earths is to adopt one of the linear meth-
ods of Andersen (1975). In the following, we will use the
Atomic Sphere
Approximation
(ASA) which will allow us to illustrate the construction
and characteristics of the energy bands in a transparent way. This ap-
proximation, and the closely-related
Linear Mun-Tin Orbitals Method
(LMTO), which allows computationally very ecient calculations of ar-
bitrarily precise energy bands, for a given potential, have been concisely
described by Mackintosh and Andersen (1980) and, in much more detail,
by Skriver (1984).
In a close-packed solid, the electrons may to a very good approxima-
tion be assumed to move in a
mun-tin potential
, which is spherically
symmetric in a sphere surrounding each atomic site, and constant in the
interstitial regions. We recall that the
atomic polyhedron
,or
Wigner-
Seitz cell
, is bounded by the planes which perpendicularly bisect the
vectors joining an atom at the origin with its neighbours, and has the
same volume as the
atomic sphere
, whose radius
S
is chosen accordingly.
If we surround each site in the crystal with an atomic sphere, the po-
tential within each of these overlapping regions will, to a high degree of
accuracy, be spherically symmetric. Neglecting the spin, we may there-
fore write the solutions of the Schrodinger equation for a single atomic
sphere situated at the origin in the form
ψ
lm
(
ε,
r
)=
i
l
R
l
(
ε, r
)
Y
lm
(
r
)
,
(1
.
3
.
2)
where the radial function
R
l
(
ε, r
) satisfies eqn (1.2.12) and is a function
of the continuous energy variable
ε
. Examples of such radial functions
are shown in Fig. 1.5.
Search WWH ::
Custom Search