**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 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**.

The numerical value of **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 _{f}^{U}**, but on the other hand this decrease is partially offset by the increase in

**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.