Nuclear Reactions - Simple Theories
We consider some simple theories regarding nuclear reactions here.
Reaction Cross Sections
In a nuclear reaction, the area of the target nucleus seen by
the incidant particle is proportional to the rate of reaction, and
hence the term cross section is used to mean a constant related
to the rate of a reaction.
The unit for the cross section is a barn (b, or 10-28).
However, the cross section is really an experimental value obtained
from experiment. The rate of reaction (number per unit time)
in an experiment is equal to the product of the cross section,
the number of target atoms per unit area N, and the intensity
of the flux (number of particles per unit area per unit time s-1
cm-2) I. That is,
rate = s N I.
A sample irradiated in the core of a nuclear reactor differs from
irradiated by a unidirectional beam. Neutrons in reactor bombard
the sample from all directions. For neutron irradiation in reactor
core, the cross section is calculated by dividing the rate of
reaction by the total number of nuclei, and the intensity of the flux,
Note that the unit of the cross section so calculated is cm-2
or m-2, depending on the unit used for I.
The unit barn (=10-28 m2 or 10-24
cm2) has been used for the tabulation of cross sections
of nuclides. Cross sections have a very large range, 106
to 10-6. The range indicates a very complicated
Thus, the cross section is really a measure of the probability of a
given reaction, and the total cross section of absorption of a particular
accelerated particle is the sum of all partial cross sections.
In an experiment, 1.0 g of 59Co is placed in a neutron flux
with an intensity of 1015 neutrons s-1 cm-2.
A handbook gives the cross section for 59Co as 17 b for the
reaction 59Co (n, g) 60Co.
What is the rate of producing 60Co.
The rate of production is estimated below:
Rate = 17x10-24 cm2 x
(6.022x1023 / 59) x 1015 cm-2 s-1
= 1.7x10-14 60Co nuclei s-1
1.0 g of colbalt is 1/59 of a mole.
Energy Dependence of Cross Sections
Cross sections of reactions depend on both the bombarding particle
and the nuclide. They not only have a very large range, they also depend
on the (kinetic) energies of the incident particles.
For example, a sketch of the cross section for neutron absorption is
shown here in a figure on the right. In general, the cross section decreases
as the energy of the neutron increases. However, the cross section
increases suddenly at some specific energies of the neutron, but
the cross section rapidly decreases from the high points.
The sudden increase has been attributed to the energy states of nuclei.
Neutrons moving with these particular energies can be accommodated
easily by the target nuclide. The rise in their capture cross section
is known as resonance absorption.
Multiple mode of reactions
There are cases when the energies of the bombarding particles dictate
the mode of reactions. For example, bombardment of 209Bi nuclei
by a particles produces various isotopes of astatine. These reactions
result in the release of neutrons. The number of neutrons released
depends on the kinetic energy of the incident a
particles. Low energy (15 - 30 MeV) a particle bombardment
favours the reaction 209Bi (a, n)
212At, but some 209Bi (a, 2 n)
211At also take place. The latter is dominant if the
a particles have energy between
25 to 35 Mev. Alpha particles with yet higher energy
(greater than 35 MeV) tends to eject 3 or more neutrons
209Bi (a, 3n) 210At.
Still higher energy results in the fragmentation of the Bi into nuclei
of light elements. The variations of these cross sections
are sketched in the diagram shown here.