Environmental Engineering Reference
In-Depth Information
It is easy to use this idea to understand the rising portion of the BE curve in
Figure 1.5. If we imagine a large spherical nucleus of radius
R
, with
A
nucleons, the
binding energy is
U¼AU
o
and the volume
V¼
4/3
pR
3
¼
(4/3)
pR
o
3
A
, since
R¼
R
o
A
1/3
. The binding energy per unit volume is then
U
V
¼U
o
/((4/3)
pR
o
). However,
the nucleons within a distance
d
of the surface, (
Rd
)
r R
, where
d
is a measure
of the range of the nuclear force, will see only
half
their binding energy, since there are
no nucleons present beyond
R
. The surface energy is related to the volume
D
V¼
4
pR
2
d¼
4
pR
o
A
2/3
d
, and the energy loss is
1
/
2
U
V
D
V
.
The formula for the binding energy, BE, with surface correction, is
2
U
o
=½ð
4
3
ÞpR
o
4
pR
o
A
2
=
3
d
BE
¼ AU
o
1
2
U
V
D
V ¼ AU
o
1
=
ð
2
:
34
Þ
¼ AU
o
½
1
3
d=ð
2
R
o
A
1
=
3
Þ
and the corrected BE per particle is
BE
=A ¼ U
o
½
1
3
d=ð
2
R
o
A
1
=
3
Þ:
ð
2
:
35
Þ
With the empirical choices 3
d
/2
R
o
¼
1.08 (
d¼
0.72
R
o
) and
U
o
¼
12MeV, the
working formula
BE
A
1
=
3
=
A ¼
12 MeV
½
1
1
:
08
=
ð
2
:
36
Þ
fits reasonably well the trend of BE values shown in Figure 1.5 for
A¼
2, 3, 4, and 56.
The second basic aspect is the Coulomb repulsion between the
Z
positive charges.
The electrostatic repulsive energy of a uniform spherical distribution of charge
Q¼Ze
with radius
R
is
U
Coul
¼
3/5
k
C
(
Ze
)
2
/
R
.
A more subtle aspect in
uencing the stability of a nucleus tends to promote the
nearly equal number of neutrons and protons. It is believed that this is similar to the
filling of states in atoms as guided by a rule that will allow an electron of spinup and an
electron of spindown in the same state, but will not allow two electrons of the same
spin. In the context of nuclei, the charge, neutron versus proton, is similar to spinup
or -down, and the lowest energy is achieved with similar numbers of each.
The cost of the Coulomb energy is important for large
Z
, which also is a reasonwhy
the proton number
Z
is less than the neutron number
N
. For large nuclei, looking at
Figure 2.5, the ratio
Z
/(
Z þ N
)
¼Z
/
A
is about 0.38.
We can incorporate these two ideas into an estimated empirical Coulomb repul-
sion energy per particle
U
Coul
/
A
(in MeV), taking
Z¼
0.38
A
, to get
2
U
Coul
=A ¼ð
3
=
5
k
C
e
2
=R
o
A
4
=
3
¼
0
:
104 MeV
A
2
=
3
Þð
0
:
38
AÞ
:
ð
2
:
37
Þ
If we evaluate this for
A¼
238 (uranium) we get 3.99MeV per nucleon as the
Coulomb repulsion.
We can estimate the value of
A
at maximum in the BE curve, from the derivative of
the sum of two energy terms.
104 MeV
A
2
=
3
08
A
1
=
3
d
U
=
d
A ¼ d
=
d
Af
0
:
þ
12 MeV
½
1
1
:
g
ð
2
:
38
Þ
d
U=
d
A ¼ð
2
=
3
Þ
0
:
104 MeV
A
1
=
3
1
=
3
ð
12 MeV
Þ
1
:
08
A
4
=
3
:
ð
2
:
39
Þ
