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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.
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