Environmental Engineering Reference
In-Depth Information
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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