## Reproduction Factor

**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**.

Table of key prompt and delayed neutrons characteristics.

^{235}U,

^{238}U or even

^{232}Th). What is crucial the fission of such nuclei produces

**2, 3 or more**free neutrons.

But not all neutrons are released **at the same time following fission**. Even the nature of creation of these neutrons is different. From this point of view we usually divide the fission neutrons into two following groups:

**Prompt Neutrons.**Prompt neutrons are emitted**directly from fission**and they are emitted within**very short time of about 10**.^{-14}second**Delayed Neutrons.**Delayed neutrons are emitted by**neutron rich fission fragments**that are called**the delayed neutron precursors**. These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo**neutron emission**. The fact the neutron is produced via this type of decay and this happens**orders of magnitude later**compared to the emission of the prompt neutrons, plays an extremely important role in the control of the reactor.

See also: Prompt Neutrons

See also: Delayed Neutrons

See also: Reactor control with and without delayed neutrons – Interactive chart

**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.

**Nuclear and Reactor Physics:**

- J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
- J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
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**Advanced Reactor Physics:**

- K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
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- D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
- E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

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