Secular equilibrium
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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 (due, e.g., to decay of a parent isotope) is equal to its decay rate.
[edit] Secular equilibrium in radioactive decay
Secular equilibrium can only occur in a radioactive decay chain if the half-life of the daughter isotope B is much shorter than the half-life of the parent isotope 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 isotope 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 isotope B then reaches a constant, equilibrium value.
The quantity of isotope B when secular equilibrium is reached is determined by the quantity of its parent A and the half-lives of the two isotopes. This can be seen from the time rate of change of the number of atoms of isotope B:
where λA and λB are the decay constants of isotopes A and B, related to their half-lives t1/2 by λ = ln(2) / t1 / 2, and NA and NB are the number of atoms of A and B at a given time.
Secular equilibrium occurs when dNB / dt = 0, or
Over long enough times, comparable to the half-life of isotope A, the secular equilibrium is only approximate; NA decays away according to
,
and the "equilibrium" quantity of isotope B declines in turn. For times short compared to the half-life of A, 
and the exponential can be approximated as 1.
[edit] References
EPA definition [1]
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