Environmental Engineering Reference
In-Depth Information
that hydrogen atoms (see Section 3.1) when packed together at a density [37]
3
N
Mott
ð
0
:
3
=
a
H
Þ
ð
3
:
31
Þ
revert to an ionized, free electron state. While Mott did not discuss the physics in
much detail, the lower kinetic energy of the delocalized electrons certainly is part of
this transition (the second aspect is electron screening).
In Motts formula
a
H
is the expected Bohr radius in the situation (we will see later
that this value is affected by the principal quantum number
n
, a possible effective
mass parameter and by the permittivity). For hydrogen atoms in vacuum
N
Mott
¼
(0.3/0.0529 nm)
3
¼
18.2
10
28
m
3
. This value is much less than the proton density
in the core of the sun,
N
p
¼
7.14
10
31
m
3
, so that hydrogen there is certainly
ionized. However, before comparing with the density of atoms in gold, 5.9
10
28
m
3
, we note that the gold valence electron has quantum number
n¼
6, and
therefore (see Equation 3.3) the Bohr radius is
n
2
a
o
¼
6
2
a
o
¼
1.9 nm. So the relevant
Mott concentration is reduced to
N
Mott
(0.3/(36
a
o
))
3
¼
3.91
10
24
m
3
. This
allows us to proceed with our assumption that the electrons enclosed in a box of side
L
, imagined to be built of gold atoms, are correctly assumed to be in free particle states
such as those in Equation 2.16, but more accurately, those of Equation 3.13.
We assume that the positive charge arising from the gold ions neutralizes the
negative charge of the electrons, and proceed assuming the potential seen by the
Figure 3.13 Schematic representation of (a) density distributions at ametal surface and (b) various
energies relevant to a study of the workfunction [38].
