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 can occur in a radioactive decay chain only if the half-life of the daughter radionuclide B is much shorter than the half-life of the parent radionuclide A. In such a case, 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 time scales 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. That can be seen from the time rate of change of the number of atoms of radionuclide B:
| dNB | |
| dt |
=λANA-λBNB,
where λA and λB are the decay constants of radionuclide A and B, related to their half-lives t1/2 by
λ=ln(2)/t1/2
Secular equilibrium occurs when
dNB/dt=0
NB=
| λA | |
| λB |
NA.
Over long enough times, comparable to the half-life of radionuclide A, the secular equilibrium is only approximate; NA decays away according to
NA(t)=NA(0)
| -λAt | |
| e |
,
and the "equilibrium" quantity of radionuclide B declines in turn. For times short compared to the half-life of A,
λAt\ll1