Editing Nuclear chain reaction (section)

==== Subcritical multiplication ====
When a nuclear system is subcritical, an introduction of neutrons to the system will result in that population decaying away; however, if neutrons are introduced at a constant rate (i.e. from a neutron source), a nuclear system can appear critical while not actually maintaining true criticality. This is called '''source criticality''' and due to a phenomenon called subcritical multiplication.

The neutron population equation can be modified to be written as follows:

<math>{d \over dt}\left(n(t)\right) = \left({{k_\mathrm{eff}-1} \over l}\right)n(t)+S(t)</math>

This is a much more difficult differential equation to solve. In this case, we assume that all neutrons are from the source, and that each generation of neutrons is of equal magnitude. In this case, we can approximate using a geometric series:

<math>n(t) = n_0 + n_0 k_\mathrm{eff} + n_0 \left(k_\mathrm{eff}\right)^2 + n_0 \left(k_\mathrm{eff}\right)^3 + \cdots = n_0 \sum_{i=0}^\infty (k_\mathrm{eff})^i = \left({1 \over {1-k_\mathrm{eff}}}\right) n_0</math>
[[File:Graphical Representation of Subcritical Multiplication.jpg|alt=A bar graph showing a series of colored bars, with one color each representing an exponentially decreasing generation of neutrons. The bars stack on each other, such that as each generation decreases, the total population in the next time frame is larger, resulting in the total behavior is increasing, approaching a stable value.|thumb|350x350px|Graphical representation of subcritical multiplication]]
We take the above equation and define a new factor <math>M</math>, called the subcritical multiplication factor:

<math>M = {1 \over {1-k_{\mathrm{eff}}}}</math>

Multiplying this factor by the source strength (in neutrons/sec) will give the stable neutron population, as long as <math>k_{\mathrm{eff}}</math> is known:

<math>n_\infty = S_0 \times M</math>

Much more commonly, this equation is used to estimate <math>k_{\mathrm{eff}}</math>, as the stable neutron population is easy to measure, but it is difficult to know the strength of a neutron source. To get around this, as a system approaches criticality, <math>M</math> approaches infinity; therefore, it is much more practical to measure <math>1/M</math>, which approaches zero as a system approaches criticality. <math>1/M</math> can be approximated by the ratio of count rates before and after a reactivity addition. 

<math>{CR_0 \over CR} \approx {1 \over M} \quad \therefore \quad \lim_{ CR_0/CR \rightarrow 0 } \left({k_\mathrm{eff}}\right) = 1</math>

Most neutron sources are a combination of an alpha particle emitter and beryllium. [[Beryllium-9]], the only naturally-occurring stable isotope of beryllium, is capable of emitting a neutron when an alpha particle is absorbed. This (<math>\alpha, n</math>) binary reaction is what generates neutrons. The most common of these are americium-beryllium (AmBe), plutonium-beryllium (PuBe), and polonium-beryllium (PoBe) sources.

<chem>{^9_4 Be} + {^4_2 \alpha} \Rightarrow {^1_0 n} + {^{12}_{6} C}</chem>

[[Antimony-124]] is also used in conjunction with beryllium to generate neutrons, as the gamma ray emitted by antimony-124 is at a unique energy that can be absorbed by beryllium and cause it to emit a neutron. This is called a (<math>\gamma, n</math>) reaction. Antimony-124 sources are commonly used to locate beryllium ore by mining companies. 

<chem>{^9_4 Be} + {\gamma} \Rightarrow {^1_0 n} + {^{8}_{4} Be}</chem>

Other sources of neutrons are from accelerators that use fusion to generate neutrons using deuterium and tritium fusion via this reaction

<chem>{^2_1 D} + {^2_1 D} \Rightarrow {^1_0 n} + {^3_2 He}</chem>

<chem>{^2_1 D} + {^3_1 T} \Rightarrow {^1_0 n} + {^4_2 He}</chem>