nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies t ...

, secular equilibrium is a situation in which the quantity of a

radioactive Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consi ...

isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbers ...

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 Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable at ...

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 = \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_

, 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 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^,

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.

References

IUPAC definition
*
"secular equilibrium", IUPAC definition
(IUPAC Compendium of Chemical Terminology 2nd Edition, 1997) {{in lang, en

radioactivity.eu.com, IN2P3, EDP Science Radioactivity

In radioactive decay

See also

References