Environmental Engineering Reference
In-Depth Information
E
n
¼ E
o
r
2
;
ð
3
:
15
Þ
where
E
o
¼
(
h
2
/8
mL
2
), as an aid to counting the number of states
lled up to an energy
E
, in connection with Figure 3.9. In coordinates labeled by integers
n
x
,
n
y
, and
n
z
,
constant energy surfaces are spherical and two electron states occupy a unit volume.
Since the states are labeled by positive integers, only one octant of a sphere is
involved.
The number
N
of states out to radius
r
(energy
E
n
)is
8
Þð
4
pr
3
3
Þ¼pr
3
3
=
2
N ¼ð
2
Þð
1
=
=
=
3
¼ðp=
3
ÞðE
=
E
o
Þ
;
ð
3
:
16
Þ
for a box of side
L
. This is equivalent to
2
=
3
2
=
3
E
F
¼ðh
2
=p L
3
¼ðh
2
=
8
mÞð
3
N
Þ
=
8
mÞð
3
N
=pVÞ
:
ð
3
:
17
Þ
Setting
N
/V
¼n
, the number of states/m
3
, some algebra gives
2
ÞE
1
=
2
E
3
=
2
F
¼ c
0
E
1
=
2
d
n
=
d
E ¼ gðEÞ¼ð
3
n
=
ð
3
:
18
Þ
as the density of electron states per unit energy and per unit volume at energy
E
with
c
0
a constant. This is the preferred form of the formula, and shows the characteristic
dependence on
E
1/2
. We can use this to
find the average kinetic energy at
T¼
0as
E
av
¼n
Ð
E
0
EgðEÞ
d
E ¼
3
5
E
F
. This is quite different from the classical value, (3/2)
k
B
T
, and a changed form of the
P
(
V
) gas law relation, from an ideal gas to a
degenerate gas is a consequence we will return to.
The Fermi velocity
v
F
is defined by
mv
F
2
/2
¼E
F
, and
T
F
is defined by
kT
F
¼E
F
. The
metallic Fermi velocity and Fermi temperature exceed their thermal counterparts.
The reason for this is that the boundary conditions prescribe allowed states, limited to
two electrons per state, raising the energy and velocity, as particles are added. A test is
to compare the de Broglie wavelength of the electron to the interatomic spacing
n
1/3
.
Taking the highest energy state
E¼E
F
(for
n¼
5.9
10
28
m
3
,
E
F
¼
(
h
2
/8
m
)
(3
n
/
p
)
2/3
=
l ¼h
/
p ¼h
/(2
m
e
E
)
1/2
¼
6.6
10
34
/(2
9.1
10
31
¼
5.53 eV), we
nd
5.53
1.6
10
19
)
1/2
¼
0.52 nm. The spacing between atoms is
n
1/3
¼
(5.9
10
28
)
1/3
¼
0.257 nm. So, on this criterion, electrons at the Fermi energy in gold
behave predominantly as waves rather than classical particles.
We will return to amore realistic discussion of electrical conduction in an empty box
metal after we have presented a more accurate description of electronic shells in atoms.
As an application of this simple development, let us estimate the pressure of the
dense electron gas in the core of the sun, following the discussion inChapter 2. While
we found that the protons in the sun act as classical particles, we must discover if the
free electrons in the sun act as quantum particles or whether they are also classical in
their behavior. If the electrons are following the quantum description, their equation
of state is modi
ed from the ideal gas form
P¼RT
/
V
(where
R
, the gas constant, is
N
A
k
B
, the product of Avogadros number and the Boltzmann constant), to
P ¼
2
NE
F
=
5
V ¼
0
:
4
nE
F
:
ð
3
:
19
Þ
