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 0, x¼na . The probability density (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).
Search WWH ::




Custom Search