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Secular equilibrium

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.

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Secular equilibrium in radioactive decay

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.

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:

$\frac{dN_B}{dt} = \lambda_A N_A - \lambda_B N_B$

where λ_A and λ_B are the decay constants of radionuclide A and B, related to their half-lives t_1/2 by $λ = l n (2) / t 1 / 2$ , and N_A and N_B are the number of atoms of A and B at a given time.

Secular equilibrium occurs when $d N B / d t = 0$ , or

$N_B = \frac{\lambda_A}{\lambda_B}N_A$

Over long enough times, comparable to the half-life of radionuclide A, the secular equilibrium is only approximate; N_A decays away according to

$N_A(t) = N_A(0) e^{-\lambda_A t}$ ,

and the "equilibrium" quantity of radionuclide B declines in turn. For times short compared to the half-life of A, $\lambda_A t \ll 1$ and the exponential can be approximated as 1.