# 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, s, 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,
 s = rate------N I
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 particle-nucleus interaction.

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.

Example 1

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.

Solution

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.

E-mail: cchieh@uwaterloo.ca