Secular equilibrium
# Secular equilibrium

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.

## In radioactive decay

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:

    ${\frac {dN_{B}}{dt}}=\lambda _{A}N_{A}-\lambda _{B}N_{B},$

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.

