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 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 – 235U, 238U, 234U. In the power reactors, the fuel significantly changes its isotopical content as the fuel burnup increases. The isotope of 236U and also trace amounts of 232U appears. The major consequence of increasing fuel burnup is that the content of the plutonium increases (especially 239Pu, 240Pu and 241Pu). 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 235U, 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.


η ~ 2.02


The capture-to-fission ratioThe capture-to-fission ratio may be used as an indicator of “quality” of fissile isotopes. The ratio depends strongly on the incident neutron energy.

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

reproduction factor

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 235U and the numerical value of η is given by following equation:

reproduction factor - equation

in which ν is the average neutrons production of 235U, N5 and N8 are the atomic number densities of the isotopes 235U and 238U (when using other uranium isotopes or plutonium the equation is modified in a trivial way). This equation can be also written in terms of uranium enrichment:

reproduction factor - enrichment

Reproduction factor as a function of the uranium enrichment
Reproduction factor as a function of the uranium enrichment

where e is the atomic degree of enrichment e = N5/(N5+N8). 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.

Neutron Life Cycle
Nuclear Fission Chain Reaction
Number of Neutrons per Fission
Table of key prompt and delayed neutrons characteristics

Table of key prompt and delayed neutrons characteristics.

Neutron Production - Prompt Neutrons
Most of the neutrons produced in fission are prompt neutrons. Usually more than 99 percent of the fission neutrons are the prompt neutrons, but the exact fraction is dependent on certain nuclide to be fissioned and is also dependent on an incident neutron energy (usually increases with energy).
Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library
Prompt and Delayed Neutrons
It is known the fission neutrons are of importance in any chain-reacting system. Neutrons trigger the nuclear fission of some nuclei (235U, 238U or even 232Th). 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

Table of capture-to-fission ratios for various nuclides
capture-to-fission ratio

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 ΣfU, but on the other hand this decrease is partially offset by the increase in ΣfPu. As the fuel burnup increases, the 239Pu 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.

Thermal vs. Fast Reactor - reproduction factors

Nuclear and Reactor Physics:
  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

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See above:

Chain Reaction

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