Consider a
one-dimensional (1D) solid with lattice constant
*
a
*
and
length
*
L
*
with nearly free electrons of effective mass
*
m
*
subject
to a weak symmetric lattice potential
*
V
*
(
*
x
*
):

*
V
*
(
*
x
*
) = -
*
V
_{
0
}
*

(a) Show that the energy gap at the
*
n
*
^{
th
}
Brillouin
zone (BZ) boundary can be expressed as
*
E
_{
gn
}
*
= 2|<

(b) Plot
*
E
*
versus
*
k
*
from -π/a <
*
k
*
<
π/a
(i.e., in the reduced-zone scheme), showing the
magnitude of all energy gaps and the energy at
*
k
*
= 0.