Most of the neutrons absorbed in the fuel cause fission, but some do not. The reproduction factor () is defined as the ratio of the number of fast neurtons produces by thermal fission to the number of themal neutrons absorbed in the fuel. The reproduction factor is shown below.

number of fast neutrons produced by thermal fission

number of thermal neutrons absorbed in the fuel

The reproduction factor can also be stated as a ratio of rates as shown below.

The rate of production of fast neutrons by thermal fission can be determined by the product of the fission reaction rate () and the average number of neutrons produced per fission (v). The average number of neutrons released in thermal fission of uranium-235 is . The rate of absorption of thermal neutrons by the fuel is Substituting these terms into the equation above results in the following equation.

Table 1 lists values of v and rl for fission of several different materials by thermal neutrons and

fast neutrons.

In the case where the fuel contains several fissionable materials, it is necessary to account for each material. In the case of a reactor core containing both uranium-235 and uranium-238, the reproduction factor would be calculated as shown below.

Example:

Calculate the reproduction factor for a reactor that uses 10% enriched uranium fuel. The microscopic absorption cross section for uranium-235 is 694 barns. The cross section for uranium-238 is 2.71 barns. The microscopic fission cross section for uranium-235 is 582 barns. The atom density of uranium-235 is 4.83 x 10²¹ atoms/cm³. The atom density of uranium-238 is 4.35 x 10²² atoms/cm³. v is 2.42.

Solution:

Use Equation (3-2) to calculate the reproduction factor.

As temperature varies, each absorption and fission microscopic cross section varies according to the 1/v relationship (see Module 2). Since both the numerator and the denominator change equally, the net change in rl is zero. Therefore, rl changes only as uranium-235 enrichment changes. rl increases with enrichment because there is less uranium-238 in the reactor making it more likely that a neutron absorbed in the fuel will be absorbed by uranium-235 and cause fission.

To determine the reproduction factor for a single nuclide rather than for a mixture, the calculation may be further simplified to the one shown below.