Environmental Engineering Reference
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Fig. 1.7. The n th period of the logarithmic derivative function D l ( ε ),
and the corresponding potential function P l ( ε ). The bottom, centre, and
top of the nl band are defined respectively by P l ( B nl )= 2(2 l +1)( l +1) /l
( D l ( B nl )=0), P l ( C nl ) = 0, and P l ( A nl )= l ( D l ( A nl )= −∞ ).
lj = 0 and, since the average over the Brillouin zone may be
shown to vanish, i.e.
which
S
2 l +1
k
BZ S
lj d k =0 ,
(1 . 3 . 15)
j =1
the designation of C nl as the centre of the band is appropriate. Equa-
tion (1.3.12) may be satisfied, and energy bands thereby formed, over
an energy range around C nl which, to a good approximation, is defined
by the Wigner-Seitz rule, which states that, by analogy with molecular
binding, the top and bottom of the band occur where the radial wave-
function and its derivative respectively are zero on the atomic sphere.
The corresponding energies, defined by
D l ( A nl )=
−∞
(1 . 3 . 16)
and
D l ( B nl )=0 ,
(1 . 3 . 17)
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