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.

## 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. 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:

where *λ*_{A} and *λ*_{B} are the decay constants of radionuclide *A* and *B*, related to their half-lives *t*_{1/2} by , and *N*_{A} and *N*_{B} are the number of atoms of *A* and *B* at a given time.

Secular equilibrium occurs when , or

Over long enough times, comparable to the half-life of radionuclide *A*, the secular equilibrium is only approximate; *N*_{A} decays away according to

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