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. An example of this is the germanium-68/gallium-68 generator commonly used for the preparation of gallium-68 radiopharmaceuticals for PET imaging.

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:

d N B d t = λ A N A − λ B N B , {\displaystyle {\frac {dN_{B}}{dt}}=\lambda _{A}N_{A}-\lambda _{B}N_{B},}

where λA and λB are the decay constants of radionuclide A and B, related to their half-lives t1/2 by λ = ln ⁡ ( 2 ) / t 1 / 2 {\displaystyle \lambda =\ln(2)/t_{1/2}}, and NA and NB are the number of atoms of A and B at a given time.

Secular equilibrium occurs when d N B / d t = 0 {\displaystyle dN_{B}/dt=0}, or

N B = λ A λ B N A . {\displaystyle 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; NA decays away according to

N A ( t ) = N A ( 0 ) e − λ A t , {\displaystyle 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, λ A t ≪ 1 {\displaystyle \lambda _{A}t\ll 1} and the exponential can be approximated as1.

Secular equilibrium

In radioactive decay

See also