{"id":13752,"date":"2017-02-06T09:51:15","date_gmt":"2017-02-06T09:51:15","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=13752"},"modified":"2022-10-21T06:14:18","modified_gmt":"2022-10-21T06:14:18","slug":"reproduction-factor","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reproduction-factor\/","title":{"rendered":"Reproduction Factor"},"content":{"rendered":"
<\/div>\n
The reproduction factor, \u03b7<\/strong>, 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.<\/div><\/div>\n

The thermal utilization factor<\/a> gives the fraction of the thermal neutrons<\/a> absorbed in the nuclear fuel<\/a>\u00a0in all isotopes<\/strong> of the nuclear fuel. But the nuclear fuel is an isotopically rich material even in this case, in which we consider only the fissionable nuclei<\/a> in the fuel. In the fresh uranium fuel<\/strong>, only three fissionable isotopes must be included in the calculations – 235<\/sup>U<\/a>, 238<\/sup>U<\/a>, 234<\/sup>U<\/a>. In power reactors, the fuel significantly changes its isotopic content<\/strong> as the fuel burnup<\/strong> increases. The isotope of 236<\/sup>U<\/a> and also trace\u00a0amounts of 232<\/sup>U<\/a> appears. The major consequence of increasing fuel burnup is that the content of the plutonium<\/a> increases (especially 239<\/sup>Pu<\/a>, 240<\/sup>Pu<\/a>, and 241<\/sup>Pu<\/a>). All these isotopes have to be included in the calculations of the reproduction factor<\/strong>.<\/p>\n

Another fact is that not all<\/strong> the absorption reactions<\/a> that occur in the fuel result in fission. If we consider the thermal neutron and the nucleus of 235<\/sup>U<\/a>, then about 15%<\/strong> of all absorption reactions result in radiative capture<\/a> of a neutron. About 85%<\/strong> of all absorption reactions result in fission<\/a>. Each fissionable nuclei have a different fission probability, and microscopic cross-sections<\/a> determine these probabilities.<\/p>\n

The neutrons<\/a> finish one generation<\/a>, and a new generation of neutrons may be created. The neutron reproduction factor<\/strong> determines the number of neutrons created in the new generation. The reproduction factor, \u03b7<\/strong>, is the ratio of the number of fast neutrons produced by thermal fission to the number
\nof thermal neutrons absorbed in the fuel. The reproduction factor is shown below.<\/p><\/div><\/div>

<\/div>\n

495<\/span><\/strong><\/p>\n

\u2193<\/span><\/strong><\/p>\n

\u03b7<\/strong>\u00a0~ 2.02<\/span><\/strong><\/p>\n

\u2193<\/span><\/strong><\/p>\n

1000<\/span><\/strong><\/p>\n

<\/div>\"The<\/a>The capture-to-fission ratio<\/a> may be used as an indicator of \u201cquality\u201d of fissile isotopes<\/a>. The ratio depends strongly on the incident neutron energy.\n

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library<\/p><\/div><\/div>\n

\"reproduction<\/a><\/p>\n

This factor is determined by the probability<\/strong> that fission reaction will occur times the average number of neutrons produced<\/strong> per one fission reaction. In the case of fresh uranium fuel, we consider only one fissile isotope, 235<\/sup>U,<\/strong> and the numerical value of \u03b7<\/strong> is given by the following equation:<\/p>\n

\"reproduction<\/a><\/p>\n

in which \u03bd<\/strong> is the average neutrons production of 235<\/sup>U<\/strong>, N5<\/sub> and N8<\/sub> are the atomic number densities<\/a> of the isotopes 235<\/sup>U<\/strong>\u00a0and\u00a0238<\/sup>U<\/strong> (when using other uranium isotopes or plutonium, the equation is modified trivially). This equation can also be written in terms of uranium enrichment<\/strong>:<\/p>\n

\"reproduction<\/a><\/p>\n

\"Reproduction<\/a>
Reproduction factor as a function of the uranium enrichment<\/figcaption><\/figure>\n

where e<\/strong> is the atomic degree of enrichment e = N5<\/sub>\/(N5<\/sub>+N8<\/sub>)<\/strong>, the reproduction factor is determined by the nuclear fuel<\/a> composition and strongly depends on the neutron flux spectrum<\/a> in the core<\/a>, for\u00a0natural uranium<\/strong> in the thermal reactor, \u03b7 = 1.34<\/strong>. As a result of the ratios of the microscopic cross-sections, \u03b7 increases <\/strong>strongly in the region of low enrichment fuels<\/strong>. This dependency is shown in the picture. It can be seen there is a limit value about \u03b7 = 2.08<\/strong>.<\/p>\n

<\/span>Neutron Life Cycle<\/div>
\"Nuclear<\/a><\/div><\/div>
<\/span>Number of Neutrons per Fission<\/div>
\"Table<\/a><\/p>\n

Table of key prompt<\/a> and delayed neutrons<\/a> characteristics.<\/p>\n

\"Neutron<\/a>
Most of the neutrons produced in fission are prompt neutrons. Usually, more than 99 percent of the fission neutrons are prompt neutrons. Still, the exact fraction is dependent on certain nuclides 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<\/figcaption><\/figure>\n<\/div><\/div>\n
<\/span>Prompt and Delayed Neutrons<\/div>
It is known the fission neutrons are of importance in any chain-reacting system. Neutrons<\/a> trigger the nuclear fission<\/a> of some nuclei (235<\/sup>U<\/a>, 238<\/sup>U<\/a>, or even 232<\/sup>Th<\/a>). What is crucial the fission of such nuclei produces 2, 3, or more<\/strong> free neutrons<\/a>.<\/p>\n

But not all neutrons are released at the same time following fission<\/strong>. Even the nature of the creation of these neutrons is different. From this point of view, we usually divide the fission neutrons into two following groups:<\/p>\n