Environmental Engineering Reference
In-Depth Information
Multiplying by
A
4/3
and setting d
U
/d
A¼
0, we
nd
A¼
62. This is not far from the
observed value, usually quoted as Fe at
A¼
56.
The total BE/
A
function is then approximately
08
A
1
=
3
104
A
2
=
3
40
Þ
The first term is the attractive short-range force with a surface correction that
roughly describes the increase in BE per nucleon as the number of nearest neighbors
increases. The second term is the Coulomb repulsion that is long range and is seen to
increase on a per nucleon basis as
A
2/3
. This function describes approximately the
data curve shown in Figure 1.5.
Using this we can estimate the energy release in a hypothetical
=
A ¼
12
½
1
1
:
0
:
ð
in MeV
Þ:
ð
2
:
BE
fission reaction
236
U
!
2
118
Pd using the formula (2.40). The BE for
236
U is 1400MeV, while the
energy for two
118
Pd nuclei is 1617.9MeV. The energy release is 217.9MeV, which is
close to a typical
gure quoted as 200MeV per
fission. It is clear that the primary
change is in the Coulomb energy. This is not a practical reaction, although
236
Uisa
starting point reached by capture of a slow neutron by
235
U. The
fission products are
typically two nuclei of different
A
, for example, Kr and Ba, plus an average of
2.5 neutrons.
On the other hand, for the fusion reaction 2D
!
4
He we
find from our approx-
imate formula an energy release of 8.89MeV. This is seriously wrong, since the
quoted reaction 2D
!
4
He
þc
lists
Q¼
23.85MeV. It is clear also from the plot in
Figure 1.5 that the
4
He or alpha particle is exceptionally stable. Shell structures in
nuclei, such as this strongly bound unit of four nucleons, play a role beyond the
general picture of a liquid of closely interacting nucleons.
While our understanding makes clear that the size of atomic nuclei is limited to
A
values less than about 240, with radius around 7.5 f, extended neutron matter is
believed to exist in neutron stars. This neutron matter is apparently stabilized, in the
absence of protons, by pressure.
We can nd the
mass density
of nuclear matter from our working radius formula,
since the mass is
Am
p
¼A
1.67
10
27
kg. Setting
R¼
1
m
, volume
¼
(4/3)
pR
3
¼
(4/3)
p
(1.2
10
15
)
3
A
m
3
with mass
A
1.67
10
27
kg. The density is 2.307
10
17
kg/m
3
. The densities of neutron stars are estimated as two or three times this
value. These values are seen to bemuch greater than density of the solar core, given as
1.62
10
5
kg/m
3
.
Large nuclei like
238
U have several isotopes, corresponding to different neutron
numbers,
N
, for the same
Z
. The chemical identity is controlled by
Z
, which sets the
number of electrons that will collect around that nucleus.
This
238
U
!
236
U
!
is
shown by
the uranium alpha decay
series
234
U
!
228
U, in each case by emitting an alpha particle. The decay
lifetimes in this series range from about 6
10
9
years to about 300 s, all accurately
predicted by the Gamow tunneling model. The plot of lifetime versus
E
1/2
gives a
straight line, consistent with our earlier discussion of Equation 2.24. In the simpli
ed
expression for
c
h
1
(2
m
r
E
)
1/2
(
pr
2
/2
2(
r
1
r
2
)
1/2
), note that
r
2
¼ k
c
90
2
e
2
/
E
(for
uranium,
Z¼
92) and also that the
pr
2
/2 term in the square bracket dominates. Thus,
c
is nearly proportional to
E
1/2
. These considerations make clear that a plot of log of
232
U
!
230
U
!
