22
1 Energy Release in Nuclear Reactions, Neutrons, Fission, and Characteristics …
that this nucleus splits into two spherical product nuclei of radii R 1 and R 2 as shown
in part (b) of the figure; after this, they will repel each other due to the Coulomb
force and fly away at high speeds as shown in part (c) of the figure.
Presuming that the density of nuclear matter is constant, conservation of nucleon
number demands that volume be conserved:
R
3
O = R
3
1 + R
3
2 .
(1.44)
Define the mass ratio of the fission products as
f =
R
3
1
R
3
2
.
(1.45)
This ratio could be defined as the inverse of that adopted here, a point to which
we shall return below. In terms of this ratio, the radii of the product nuclei are
R 1 = R O
f
1 + f
1/3
(1.46)
and
R 2 = R O
1
1 + f
1/3
.
(1.47)
Following Bohr and Wheeler, we take the energy of the system at any moment to
comprise two contributions: (i) A surface energy proportional to the surface area of
the system, and (ii) The Coulombic self-energy of the system. The surface area of
the original nucleus is proportional to R
2
O , so its surface energy can be written as
U
orig
S
= a S R
2
O ,
(1.48)
where the “surface energy coefficient” a S is to be determined. The surface energy
for the fissioned nucleus will be
U
f iss
S
= a S
R
2
1 + R
2
2
= a S R
2
O α,
(1.49)
where
α =
f
2/3
+ 1
(1 + f )
2/3
.
(1.50)
For the Coulomb self-energy, begin with the result that the electrostatic self-energy
of a charged sphere of radius r is given by