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.

ecular 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. Assuming the initial concentration of radionuclide B is zero, full equilibrium usually takes several half-lives of radionuclide B to establish.

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 $lambda\; =\; ln(2)/t\_\{1/2\}$, and N_A and N_B are the number of atoms of A and B at a given time.

Secular equilibrium occurs when $dN\_B/dt\; =\; 0$, 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.

ee also

*Transient equilibrium

References

[http://www.iupac.org/goldbook/S05532.pdf IUPAC definition]

EPA definition [http://www.epa.gov/radiation/understand/equilibrium.htm]

Radioacktiva