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| In [[nuclear physics]], '''secular equilibrium''' is a situation in which the quantity of a [[radioactive]] [[isotope]] remains constant because its production rate (e.g., due to decay of a parent isotope) is equal to its decay rate.
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| ==Secular equilibrium in radioactive decay==
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| <!-- Image with unknown copyright status removed: [[Image:Secular Equilibrium.GIF]] -->
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| Secular equilibrium can only occur in a radioactive decay chain if the [[half-life]] of the daughter radionuclide B is much shorter than the half-life of the parent radionuclide A. In such a situation, the decay rate of A, and hence the production rate of B, is approximately constant, because the half-life of A is very long compared to the timescales being considered. The quantity of radionuclide B builds up until the number of B atoms decaying per unit time becomes equal to the number being produced per unit time; the quantity of radionuclide B then reaches a constant, ''equilibrium'' value. Assuming the initial concentration of radionuclide B is zero, full equilibrium usually takes several half-lives of radionuclide B to establish.
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| The quantity of radionuclide B when secular equilibrium is reached is determined by the quantity of its parent A and the half-lives of the two radionuclide. This can be seen from the time rate of change of the number of atoms of radionuclide B:
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| :<math>\frac{dN_B}{dt} = \lambda_A N_A - \lambda_B N_B</math> ,
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| where λ<sub>A</sub> and λ<sub>B</sub> are the [[exponential decay|decay constants]] of radionuclide A and B, related to their half-lives t<sub>1/2</sub> by <math>\lambda = ln(2)/t_{1/2}</math>, and N<sub>A</sub> and N<sub>B</sub> are the number of atoms of A and B at a given time.
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| Secular equilibrium occurs when <math>dN_B/dt = 0</math>, or
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| :<math>N_B = \frac{\lambda_A}{\lambda_B}N_A</math> .
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| Over long enough times, comparable to the half-life of radionuclide A, the secular equilibrium is only approximate; N<sub>A</sub> decays away according to
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| :<math>N_A(t) = N_A(0) e^{-\lambda_A t}</math> ,
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| and the "equilibrium" quantity of radionuclide B declines in turn. For times short compared to the half-life of A, <math>\lambda_A t \ll 1</math> and the exponential can be approximated as 1.
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| ==See also==
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| * [[Bateman Equation]]
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| *[[Transient equilibrium]]
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| ==References==
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| *[http://www.iupac.org/goldbook/S05532.pdf IUPAC definition]
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| *[http://www.epa.gov/rpdweb00/understand/equilibrium.html EPA definition]
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| {{physics-stub}}
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| [[Category:Radioactivity]]
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