KAHIBARO
Table of Contents
When one radioactive decay feeds another
Secular equilibrium happens in a decay chain when a long lived parent nuclide produces a short lived daughter nuclide, and the daughter decays away almost as soon as it is created. After some time, the amount of daughter present settles into a nearly constant value, even though both parent and daughter are still radioactive.
This idea belongs specifically to chains such as
$$
P \rightarrow D \rightarrow \text{further products}
$$
where the parent $P$ has a much longer half life than the daughter $D$.
The basic physical picture
Imagine a tank with water flowing in and draining out. If water enters slowly but the drain removes water quickly, the water level can reach a steady value where inflow equals outflow. Secular equilibrium is the radioactive version of this balance.
The parent decays and creates daughter nuclei at a rate
$$
\text{production rate of daughter} = \lambda_P N_P
$$
The daughter itself decays at a rate
$$
\text{decay rate of daughter} = \lambda_D N_D
$$
Here, $N_P$ and $N_D$ are the numbers of parent and daughter nuclei, and $\lambda_P$ and $\lambda_D$ are their decay constants.
When the daughter has adjusted to the parent source, these two rates become nearly equal.
In secular equilibrium, the key balance is
$$
\lambda_P N_P \approx \lambda_D N_D
$$
This means, daughter nuclei are produced as fast as they decay.
Condition for secular equilibrium
The essential condition is that the parent must live much longer than the daughter. In terms of decay constants,
$$
\lambda_D \gg \lambda_P
$$
Since half life and decay constant are related by
$$
T_{1/2} = \frac{\ln 2}{\lambda},
$$
this is equivalent to saying
$$
T_{1/2,D} \ll T_{1/2,P}.
$$
Because the parent changes only very slowly over the timescale of the daughter decay, the daughter sees an almost constant source.
Secular equilibrium requires a very long lived parent and a much shorter lived daughter:
$$
T_{1/2,P} \gg T_{1/2,D}
$$
How the daughter activity behaves
At first, if no daughter is present, the parent starts creating it. The daughter activity rises because more daughter nuclei are accumulating. After several daughter half lives, the daughter decay rate catches up to its production rate.
The activity of a nuclide is
$$
A = \lambda N
$$
So in secular equilibrium,
$$
A_D \approx A_P
$$
This is one of the most important results. Even though the number of daughter nuclei is usually much smaller than the number of parent nuclei, their activities become nearly equal because the daughter has a much larger decay constant.
In secular equilibrium, the activities are approximately equal:
$$
A_D \approx A_P
$$
Do not confuse equal activity with equal number of nuclei. Usually,
$$
N_D \neq N_P
$$
Number of daughter nuclei in equilibrium
Using the balance condition,
$$
\lambda_P N_P \approx \lambda_D N_D
$$
we get
$$
N_D \approx \frac{\lambda_P}{\lambda_D} N_P
$$
Since $\lambda_D \gg \lambda_P$, this ratio is small, so
$$
N_D \ll N_P
$$
That means only a small number of daughter nuclei is needed to produce an activity comparable to the parent activity.
Why the equilibrium is called "secular"
The word "secular" means that the parent changes on a very long timescale. Over the short timescale in which the daughter adjusts, the parent appears almost constant. The equilibrium is therefore not perfect forever, but it changes only very slowly because the parent itself is slowly decreasing.
So the daughter activity tracks the parent activity. If the parent activity gradually decreases, the daughter activity also gradually decreases with it, staying nearly equal.
Time needed to reach equilibrium
Secular equilibrium is not established instantly. The daughter needs time to build up. A good rule is that after several daughter half lives, the daughter activity becomes very close to the parent activity.
The exact buildup depends on the decay constants, but for beginners the main idea is simple. The daughter reaches equilibrium on the timescale of its own half life, not the parent half life.
A simple example pattern
A classic kind of situation is a parent with a half life of years, centuries, or longer, producing a daughter with a half life of hours or days. Then the daughter rises quickly and settles into equilibrium while the parent remains almost unchanged.
The table below summarizes the comparison.
| Quantity | Parent | Daughter in secular equilibrium |
|---|---|---|
| Half life | Very long | Much shorter |
| Decay constant | Small | Large |
| Number of nuclei | Large | Smaller |
| Activity | Slowly varying | Nearly equal to parent activity |
Mathematical form of the balance
If the daughter is fed only by the parent and loses nuclei only by its own decay, then
$$
\frac{dN_D}{dt} = \lambda_P N_P - \lambda_D N_D
$$
In secular equilibrium, the daughter number changes very little, so approximately
$$
\frac{dN_D}{dt} \approx 0
$$
Therefore,
$$
\lambda_P N_P - \lambda_D N_D \approx 0
$$
which again gives
$$
\lambda_P N_P \approx \lambda_D N_D
$$
This is the mathematical statement of the equilibrium.
Visual picture
The blue line shows the parent activity, which changes only slowly. The red curve shows the daughter activity rising from zero and approaching the same value.
Common misunderstanding
A common mistake is to think that equilibrium means the decay has stopped. It has not. Parent nuclei are still decaying, daughter nuclei are still being produced, and daughter nuclei are still decaying. Equilibrium means only that the daughter population has reached a nearly steady balance.
Secular equilibrium does not mean no decay occurs.
It means
$$
\text{rate of daughter production} \approx \text{rate of daughter decay}
$$
so the daughter amount stays nearly constant.
Practical importance
Secular equilibrium is important in radioactive sources, natural decay series, and radiation measurements. If a daughter is in secular equilibrium with its parent, measuring the daughter activity can tell us the parent activity, and vice versa. This is especially useful when one member of the chain is easier to detect than the other.
Final idea
Secular equilibrium is the special situation in a decay chain where a very long lived parent feeds a much shorter lived daughter. The daughter quickly builds up until its activity nearly matches the parent activity. After that, both activities decrease together very slowly, controlled by the long lived parent.