Page 125 - Science at the nanoscale

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June 9, 2009
6.1. From 3D to 0D Nanostructures
n
z
=
+
n n i
j + n k
n y
x
z
n
y
n
x
Figure 6.4.
Spherical volume of radius R encompassing a number of
possible states in positive 3D n-space.
The Rayleigh scheme for counting modes treats the “quantum
numbers” n as if they exist in a 3D coordinate space such that a
given coordinate of n values (n x , n y , n z ) determines a point (hence
state) in that space. The number of possible states is proportional
to the “volume” in “n-space”.
It is convenient to define a radius R in n-space as shown in
Fig. 6.4, where:
q
2
2
2
(6.6)
n + n + n
R =
y
z
x
The energy can be expressed in terms of R and vice versa. From 115 ch06
Eqs. (6.5) and (6.6):
√
2 2mEL
R = (6.7)
h
Note that the n-space associated with the particle-in-a-box
involves only positive values of n, so the volume must be divided
by 8. It must also be multiplied by 2 to account for the two possi-
ble spin values of the electron. Hence the total number of available
states N is:
1 4 3 8π 3/2 L
3
N = 2 πR = (2mE) (6.8)
8 3 3 h 3