Chemistry Reference
In-Depth Information
Table 3.2 Roots of the closed chains for N
¼
5 and N
¼
15
N
¼
5
N
¼
15
2 cos
2
p
5
2 cos
2
p
k
15
k
x
k
¼
k
x
k
¼
0
2
0
2
1
0.618
1
1.827
2
1.618
2
1.338
3
0.618
4
0.219
5
1
6
1.618
7
1.956
, we have the plots sketched in Figure 3.5, that on the left
giving x(
k
) versus
k
, that on the right the energy levels
«ðkÞ
For N
!1
versus the
density of states N(
«
), and where the Fermi level
«
F
is apparent.
In the crystal, the periodic potential of the nuclei will perturb the energy
levels, removing the double degeneracy of the two states corresponding to
p=
a, the splittingmanifesting itself as a band gap in the energy spectrum.
For the values of
2a this originates discontinuities in the
spectrum, giving gaps that divide the
for which
l ¼
k
j
k
j
space into zones called Brillouin
zones. The region from
0 is the first break, called the first Brillouin
zone, from there up to the second break is the second Brillouin zone, and
so on. These zones have the dimensions of reciprocal length, and are
schematically plotted in Figure 3.6.
j
k
j¼
Figure 3.4 Plots of the roots for the closed chain with
N
¼
5 (left) and N
¼
15
(right)
