Environmental Engineering Reference
In-Depth Information
(This formula [31] derives from two statements,
PV¼
2/3
NE
av
, which is a general
ideal gas result, and, for the Fermi gas,
E
av
¼
3/5
E
F
).
In the new formula, as expected, the Fermi energy,
E
F
¼
(
h
2
/8
m
e
)(3
n
/
p
)
2/3
(3.17)
replaces the thermal energy
k
B
T
.
To return to the condition of electrons in the suns core, we recall that
n¼
N
p
þ
2
N
He
¼
6.026
10
31
m
3
(see Equation 2.3) and we
nd,
rst,
E
F
¼
(
h
2
/8
m
e
)
(3
n
/
p
)
2/3
¼
8.90
10
17
J
¼
556 eV.
Since this number is less than the thermal energy
k
B
T¼
1293 eV, we can assume
that the electrons at the center of the sun actually behave in a classical fashion, and
formula (3.19) is not called for.
So to estimate the pressure in the suns core, we can use the classical relation
PV
¼
2/3
NE
av
.
N
is the total particle density, protons, helium, and electrons, which
sums to (6.026
þ
3.106
þ
1.46)
10
31
m
3
¼
((6.6
10
34
)
2
/8
m
)(3
6.026
10
31
/
p
)
2/3
¼
10.59
10
31
m
3
. The pressure then
is 10.59
10
31
1293 eV
1.6
10
19
Pa
¼
2.19
10
16
Pa
¼
217 Gbar, where 1 bar
¼
101 kPa. This is close to the value 232
10
9
bar [32] for the total hydrostatic
pressure at the core of the sun. The total hydrostatic pressure is the sum of pressures
fromelectrons, protons, He ions, and radiation. The radiation pressure is smaller, see
Chapter 1 following Equation 1.2, where 0.126Gbar was found for 15million K. So
these numbers are in good agreement, and the electrons in the sun behave classically.
To return to the properties of a system of electrons at zero temperature, states
below
E
F
are
filled and states above
E
F
are empty. At nonzero temperatures, the
occupation (probability that the state contains one electron) is given by the Fermi
function
k
B
TÞþ
1
1
f
FD
¼½
exp
ðfEE
F
g=
:
ð
3
:
20
Þ
The energy width of transition of
f
FD
from 1 to 0 is about 2
k
B
T
.
Some of these features are sketched in Figure 3.10, for the three-dimensional
case, using notation
n
(
E
)
¼ f
FD
(
E
) and
N
(
E
)
¼ g
(
E
), note the characteristic
E
1/2
of the
upper two curves.
3.4
Atoms, Molecules, and the Covalent Bond
The Schrodinger equation introduced in Chapter 2, together with the laws of
electricity and magnetism, are capable of describing details of atoms, molecules,
and solids, and their interaction with photons. We need to better understand these
essential methods, to allow us to extend the approach to semiconductors, PN
junctions, and solar cells. We have seen that the Bohr model of the atom gives the
correct energies, and allows an initial understanding of the magnetic and optical
properties of one-electron atoms. But a more thorough approach, available through
Schrodingers equation in spherical polar coordinates, is necessary to incorporate the
wave aspects of the electron, and to understand the nature of covalent bonding.
The atom is basically spherical, since the potential energy
U
of the electron in the
electric
field of the nucleus depends only on the radius
r
. The Schrodinger
