Microscopic cross-sections. The microscopic cross section measures
the probability of occurrence of a particular nuclear reaction.
In only very simple cases it is possible to get theoretical expression
for the cross section. Thus as usual microscopic cross sections are
experimentally measured. We will consider the following experiment:
a parallel monoenergetic beam of neutrons of intensity of I is striking
a thin foil with area A [cm squared] containing N nuclei per unit volume.
It is obvious that the neutron interaction rate is proportional to
the intensity of the neutron beam and the total number of atoms in the foil.
The proportionality coefficient in this dependence is called
the effective microscopic section of interaction of neutrons with nuclei,
and it is denoted by Greek lower case sigma (σ) and expressed
in units of barns, where 1 barn is equal to 1×10^–24 centimeters squared.
Another interpretation of microscopic cross sections, in terms of
probability — sigma is a collision probability of a neutron with a nucleus.
As the probability is dimensionless, it should be concluded that sigma
is the effective area given by the nucleus to an incident neutron
for a nuclear reaction. Every nuclear reaction has the probability
corresponding to their microscopic cross section. So the microscopic cross
section of scattering is the sum of elastic and inelastic scattering cross sections.
Absorption is the sum of radioactive capture, fission, spallation and other similar processes.
The total cross section is the sum of cross sections for absorption and scattering.
It is necessary to mention that a process of interaction of neutrons with nuclei depends
on the energy of a neutron. Dependence of the microscopic cross section
(a quantitative characteristic of the probability of nuclear reactions)
on the energy carries a complicated characterr. For a large number of isotopes,
especially for those whose mass number exceeds 100, the
examination of the absorption cross section variations, depending
on the neutron energy, reveals the existence of three main regions.
As scattering cross sections are generally small, the total cross section,
which measures absorption plus scattering, shows the same behavior as the
absorption one. In the first place there is the region of low energy,
in which the cross section decreases with increasing neutrons’ energy.
As depicted for 238U, the total cross section varies
proportionally to the inverse square root of the neutron energy and, as this energy
is of the kinetic nature, the cross section is inversely proportional
to the speed of the neutron. After the 1 divided to V region for slow neutrons,
there is a resonance region, which corresponds generally to neutrons
energies between 0.1 to 1000 eV. This region is characterized by the
presence of several peaks corresponding to certain values of the neutron
energy. These energies are related to the excitation energy levels of the
compound nucleus. Immediately after the resonance region there may be
some secondary peaks, which are difficult to notice with the resolution
of available instrumentation. The figure demonstrates the cross section
of absorption and total cross section for u235. Elastic scattering with the exception
of hydrogen, whose cross section value is 20 barn when it is in
a free state, the scattering cross section for low energy neutrons of nearly
all elements are between 2 and 10 barn. In most cases,
the scattering cross sections do not vary significantly with the neutron energy,
although there may be a general tendency to a decrease of the value
in the region of high energy. The typical behaviour of the elastic
scattering cross section is presented there. There are three
distinct regions. The first — the elastic scattering is almost constant.
In this region scattering does not occur with formation of a
compound nucleus, this type of scattering is called potential elastic
scattering. The second — it is called the resonance region, a formation
of the compound nucleus takes place. The third — it is called the high energy region,
the resonances come together in such a way that cross section
shows a continuous behaviour. The second and third regions are called
resonant elastic scattering. You can exanimate the behaviour
of all cross sections for all isotopes by using a Java based Software — JANIS.
You can type in google the word JANIS and feel free to download the data base.
Macroscopic cross-section. It is defined as the probability of incident
neutron interacting with the target nucleus per unit length of travel
of the incident neutron. The macroscopic cross section denoted
by the Greek upper case sigma (Σ) and is expressed in units per centimeter
travel of the neutron in a medium. To calculate the
macroscopic cross section you need the nuclei density multiplied by
the microscopic cross section. Summary. The mmicroscopic cross section
determines properties of a single nucleus with respect
to the interaction with a neutron. The macroscopic cross section determines
properties of the whole medium with respect to the interaction with a neutron.
Yury VolkovSenior lecturer, PhD in Nuclear Engineering
