Environmental Engineering Reference
In-Depth Information
Figure 3.5 Sketch of standing wave probability P¼ (cos kx)
2
expected at k just below the Bragg
point p/a; the cos(kx) combination of solutions concentrates charge near the positive ions,
stabilizing the state. (Courtesy of M. Medikonda).
located at
x¼
0,
x¼na
. The probability density
P¼
(cos
kx
)
2
peaks at ion locations.
For the sin
kx
combination, for
k
larger than
p
/
a
, the peaks of
P
(
x
) lie between
the ions.
The Kronig
-
Penney potential
V
(
x
) model for perfect conduction along a row of
atoms is sketched in Figure 3.6.
This model assumes a linear array of
N
atoms spaced by
a
along the
x
-axis,
0
<
x
<
Na¼L
. The 1D potential
U
(
x
) is a square wave with period
a
:
UðxÞ¼
0
< ðawÞ;
V
o
; ðawÞ <
;
0
<
x
ð
3
:
13
Þ
x
<
a
:
The model potential
U
(
x
)
¼
0 except for periodic barriers of height
V
o
and width
w
.
The solutions
Y¼u
k
(
x
)e
ikx
are compatible with the 1D Schrodinger equation
introduced in Chapter 2
2
2
mÞd
2
d
x
2
Y ¼ u
k
ðxÞ
e
ikx
ð
h
=
Y=
þ½UðxÞEY ¼
0
;
with
ð
2
:
9
Þ
with periodic
U
(
x
) of Equation 3.13,
only
if the following condition is satis
ed:
cos
ka ¼ bð
sin
qa
=
qaÞþ
cos
qa ¼ RðEÞ;
ð
3
:
14
Þ
Figure 3.6 Sketch of square wave potential
U(x) assumed by Kronig and Penney. The band-
determining condition is obtained by matching
solutions of types I and II at the boundary,
x ¼aw. A useful simplification is to
approximate the periodic square wave potential
by repulsive delta function potentials at x ¼ a
w/2, preserving the barrier potential area wV
o
as
w ! 0. (Courtesy of M. Medikonda).