Biomedical Engineering Reference
In-Depth Information
conductors in Chapter 12. The Pauli model is the simplest one to take account of the
quantum mechanical nature of the electrons: the electrons exist in a three-dimensional
infinite potential well, the wavefunction obeys the Pauli principle and at 0 K the pairs of
electrons occupy all orbitals having energy less than or equal to energy ε
F
(which of course
defines the Fermi energy). The number
N
of conduction electrons can be related to ε
F
and
we find
3
N
π
L
3
2/3
h
2
8
m
e
ε
F
=
(20.5)
Now
N
/
L
3
is the number density of conduction electrons and so Pauli's model gives a
simple relationship between the Fermi energy and the number density of electrons. Physi-
cists very often use the symbol
n
for the number density; it can vary at points in space and
so we write
n
(
r
) but in this simple model, the number density is constant throughout the
dimensions of the metallic box:
N
L
3
n
=
8π
3
h
2
=
(
2
m
e
)
3/2
(
ε
F
)
3/2
(20.6)
We now switch on an
external potential U
ext
(
r
) that is slowly varying over the dimensions
of the metallic conductor, making the number density inhomogeneous. This could be (for
example) due to the set of metallic cations in such a conductor, or due to the nuclei in a
molecule. A little analysis suggests that the number density at position
r
should be written
8π
3
h
2
(2
m
e
)
3/2
(ε
F
−
U
ext
(
r
))
3/2
n
(
r
)
=
(20.7)
This
Thomas-Fermi relation
therefore relates the number density at points in space to
the potential at points in space. The number density of electrons at a point in space is just
the charge density
P
1
(
r
) discussed in several previous chapters and so we can write in more
familiar chemical language
8π
3
h
2
(2
m
e
)
3/2
(ε
F
−
U
ext
(
r
))
3/2
P
1
(
r
)
=
(20.8)
Thomas and Fermi suggested that such a statistical treatment would be appropriate for
molecular systems where the number of electrons is 'large' and in the case of a molecule
we formally identify the external potential as the electrostatic potential generated by the
nuclei. The Thomas-Fermi approach replaces the problem of calculating an
N
-electron
wavefunction by that of calculating the electron density in three-dimensional position space.
Dirac (1930) studied the effects of exchange interactions on the Thomas-Fermi model,
and discovered that these could be modelled by an extra term
C
(
P
1
(
r
))
1/3
U
X
(
r
)
=
(20.9)
where
C
is a constant. R. Gáspár is also credited with this result, which Slater (1951)
rediscovered, but with a slightly different numerical coefficient of
2
3
C
. The disagreement
