In some locations within the reactor, the flux level may be significantly lower than in other areas due to a phenomenon referred to as neutron shadowing or self-shielding. For example, the interior of a fuel pin or pellet will "see" a lower average flux level than the outer surfaces since an appreciable fraction of the neutrons will have been absorbed and therefore cannot reach the interior of the fuel pin. This is especially important at resonance energies, where the absorption cross sections are large.

Summarv

The important information in this chapter is summarized below.

Nuclear Cross Section and Neutron Flux Summary

Atom density (N) is the number of atoms of a given type per unit volume of material.

Microscopic cross section () is the probability of a given reaction occurring between a neutron and a nucleus.

Microscopic cross sections are measured in units of barns, where 1 barn = 10" cm^Z.

Macroscopic cross section () is the probability of a given reaction occurring per unit length of travel of the neutron. The units for macroscopic cross section are cm ¹.

The mean free path () is the average distance that a neutron travels in a material between interactions.

Neutron flux () is the total path length traveled by all neutrons in one cubic centimeter of material during one second.

The macroscopic cross section for a material can be calculated using the equation below.

The absorption cross section for a material usually has three distinct regions. At low neutron energies (<1 eV) the cross section is inversely proportional to the neutron velocity.

Resonance absorption occurs when the sum of the kinetic energy of the neutron and its binding energy is equal to an allowed nuclear energy level of the nucleus.

Resonance peaks exist at intermediate energy levels. For higher neutron energies, the absorption cross section steadily decreases as the neutron energy increases.

The mean free path equals .

The macroscopic cross section for a mixture of materials can be calculated using the equation below.

Self-shielding is where the local neutron flux is depressed within a material due to neutron absorption near the surface of the material.

REACTION RATES

It is possible to determine the rate at which a nuclear reaction will take place based on the neutron f ux, cross section for the interaction, and atom density of the target. This relationship illustrates how a change in one of these items affects the reaction rate.

EO 2.10 Given the neutron flux and macroscopic cross section, CALCULATE the reaction rate.

EO 2.11 DESCRIBE the relationship between neutron flux and reactor power.

Reaction Rates

If the total path length of all the neutrons in a cubic centimeter in a second is known, (neutron flux (), and if the probability of having an interaction per centimeter path length is also known (macroscopic cross section (), multiply them together to get the number of interactions taking place in that cubic centimeter in one second. This value is known as the reaction rate and is denoted by the symbol R. The reaction rate can be calculated by the equation shown below.

where:

Substituting the fact that into Equation (2-6) yields the equation below.

where:

The reaction rate calculated will depend on which macroscopic cross section is used in the calculation. Normally, the reaction rate of greatest interest is the fission reaction rate.

Example:

If a one cubic centimeter section of a reactor has a macroscopic fission cross section of 0.1 cm 1, and if the thermal neutron flux is 10¹³ neutrons/cm²-sec, what is the fission rate in that cubic centimeter?

Solution:

In addition to using Equation (2-6) to determine the reaction rate based on the physical properties of material, it is also possible to algebraically manipulate the equation to determine physical properties if the reaction rate is known.

Example:

A reactor operating at a flux level of 3 x 10¹³ neutrons/cm²-sec contains 10²⁰ atoms of uranium-235 per cm³. The reaction rate is 1.29 x 10¹² fission/cm³. Calculate .

Solution:

Step 1:The macroscopic cross section can be determined by solving Equation (2-6) for and substituting the appropriate values.

Step 2:To find the microscopic cross section, replace and solve for .