## Reproduction Factor

The thermal utilization factor gives the fraction of the thermal neutrons that are absorbed in the nuclear fuel, in **all isotopes** of the nuclear fuel. But the nuclear fuel is isotopically rich material even in this case, in which we consider only the fissionable nuclei of in the fuel. In the** fresh uranium fuel**, there are only three fissionable isotopes that have to be included in the calculations – ^{235}U, ^{238}U, ^{234}U. In the power reactors, the fuel significantly **changes its isotopical content** as the **fuel burnup** increases. The isotope of ^{236}U and also trace amounts of ^{232}U appears. The major consequence of increasing fuel burnup is that the content of the plutonium increases (especially ^{239}Pu, ^{240}Pu and ^{241}Pu). All these isotopes have to be included in the calculations of **the reproduction factor**.

Another fact is that **not all** the absorption reactions that occur in the fuel results in fission. If we consider the thermal neutron and the nucleus of ^{235}U, then about **15%** of all absorption reactions result in radiative capture of neutron. About** 85%** of all absorption reactions result in fission. Each of fissionable nuclei have different fission probability and these probabilities are determined by microscopic cross-sections.

It is obvious at this point the neutrons finish one generation and new generation of neutrons may be created. The number of neutrons created in the new generation is determined by **the neutron reproduction factor**. **The reproduction factor, η**, is defined as the ratio of the number of fast neutrons produced by thermal fission to the number

of thermal neutrons absorbed in the fuel. The reproduction factor is shown below.

**495**

**↓**

**η** ~ 2.02

**↓**

**1000**

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

This factor is determined by the **probability** that fission reaction will occur times the average **number of neutrons produced** per one fission reaction. In the case of fresh uranium fuel we consider only one fissile isotope ** ^{235}U** and the numerical value of

**η**is given by following equation:

in which **ν** is the average neutrons production of ** ^{235}U**, N

_{5}and N

_{8}are the atomic number densities of the isotopes

**and**

^{235}U**(when using other uranium isotopes or plutonium the equation is modified in a trivial way). This equation can be also written in terms of**

^{238}U**uranium enrichment**:

where **e** is the atomic degree of enrichment **e = N _{5}/(N_{5}+N_{8})**. The reproduction factor is determined by the composition of the nuclear fuel and strongly depends on the neutron flux spectrum in the core. For

**natural uranium**in the thermal reactor

**η = 1.34**. As a result of the ratios of the microscopic cross sections,

**η increases**strongly in the region of

**low enrichment fuels**. This dependency is shown on the picture. It can be seen there is the limit value about

**η = 2.08**.

**η**does not change with core temperature over the range considered for most thermal reactors. There is essentially

**small change in η**over the lifetime of the reactor core (decreases).This is due to the fact there is a continuous decrease in

**Σ**, but on the other hand this decrease is partially offset by the increase in

_{f}^{U}**Σ**. As the fuel burnup increases, the

_{f}^{Pu}^{239}Pu begins to contribute to the neutron economy of the core.

See also: Nuclear Breeding

There are significant differences in **reproduction factors** between fast reactors and thermal reactors. The differences are in both the **number of neutrons** produced per one fission and, of course, in **neutron cross-sections**, that exhibit significant energy dependency. The differences in cross-sections can be characterized by capture-to-fission ratio, which is **lower in fast reactors**. Furthermore, the number of neutrons produced per one fission is also higher in fast reactors than in thermal reactors. These two features are of importance in the **neutron economy** and contributes to the fact the** fast reactors have a large excess of neutrons** in the core.