A decay chain appears when one radioactive nuclide decays into a daughter that is itself radioactive. The simplest
nontrivial case is a two-member sequential chain:
Parent \(\to\) Daughter \(\to\) later products
This already captures the most important qualitative behavior: the parent activity decreases monotonically, but the
daughter activity can first build up, reach a maximum, and only later decay. That is why
daughter-activity curves in radioactive chains often show a rise-and-fall shape rather than a simple monotonic drop.
Parent decay law
If the initial parent amount is \(N_{1,0}\), then the parent population after time \(t\) is the standard exponential law
Parent population.
\[
\begin{aligned}
N_1(t) &= N_{1,0} e^{-\lambda_1 t}
\end{aligned}
\]
where the parent decay constant is related to its half-life by
\[
\begin{aligned}
\lambda_1 &= \frac{\ln 2}{T_{1/2,1}}.
\end{aligned}
\]
Daughter buildup and decay
The daughter receives new nuclei from the parent while simultaneously losing nuclei by its own radioactive decay. That is
why the daughter does not usually follow a simple single exponential at first. For a branching fraction \(b\) from the
parent into the daughter, the two-member Bateman solution is
Daughter population (Bateman form).
\[
\begin{aligned}
N_2(t)
&= N_{1,0}\,b\,\frac{\lambda_1}{\lambda_2-\lambda_1}
\left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right),
\qquad \lambda_1 \ne \lambda_2.
\end{aligned}
\]
Here \(\lambda_2\) is the daughter decay constant:
\[
\begin{aligned}
\lambda_2 &= \frac{\ln 2}{T_{1/2,2}}.
\end{aligned}
\]
The branching fraction \(b\) is written as a fraction rather than a percentage, so \(b=1\) means \(100\%\) of parent
decays feed the daughter, while \(b=0.5\) means only half do.
Activities of the parent and daughter
Activity is the decay rate, so for each member of the chain:
Activity relations.
\[
\begin{aligned}
A_1(t) &= \lambda_1 N_1(t), \\
A_2(t) &= \lambda_2 N_2(t).
\end{aligned}
\]
This is the quantity most often plotted in practical chain calculations because activity directly reflects how strongly
each nuclide is contributing to the total radioactivity of the system.
Why the daughter can peak
At early times, the daughter is continuously fed by the parent, so its activity increases. Later, once parent feeding is
no longer strong enough to offset daughter decay, the daughter activity reaches a maximum and starts to fall. Setting the
derivative to zero leads to the peak time
Daughter peak time.
\[
\begin{aligned}
t_{\mathrm{peak}}
&= \frac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1},
\qquad \lambda_1 \ne \lambda_2.
\end{aligned}
\]
This formula works whenever the daughter actually has time to build up and then decay. It explains why a short-lived
daughter fed by a much longer-lived parent can reach its maximum relatively quickly and then settle into an equilibrium-like
relationship with the parent.
Secular and transient equilibrium
Two classic regimes appear often in decay chains:
| Regime |
Half-life relation |
Typical behavior |
| Secular equilibrium |
\(T_{1/2,1} \gg T_{1/2,2}\) |
After a transient, daughter activity can approach the parent activity |
| Transient equilibrium |
\(T_{1/2,1} > T_{1/2,2}\) but not enormously larger |
Daughter builds up and then tracks the parent approximately for a while |
The long-lived parent and short-lived daughter example is exactly the kind of system where secular-equilibrium behavior
can emerge after the daughter has had time to build up.
Sample interpretation
Suppose the parent half-life is \(4.5\) billion years while the daughter half-life is \(24\) days. Then the parent changes
extremely slowly on the daughter timescale. The daughter rises from zero, reaches a peak, and then closely follows the
parent’s activity level for a long interval. This is why the activity curve of the daughter often shows a pronounced early
peak when plotted with a logarithmic time axis.
Why logarithmic time can help
When the two half-lives differ by many orders of magnitude, a linear time axis can hide the interesting short-time
daughter behavior. A logarithmic time axis spreads out the early-time region and makes the build-up phase visible.
That is why this simulator allows a log-time display option.
Worked structure of the calculation
Step 1. Convert each half-life into a decay constant.
\[
\begin{aligned}
\lambda_1 &= \frac{\ln 2}{T_{1/2,1}}, \\
\lambda_2 &= \frac{\ln 2}{T_{1/2,2}}.
\end{aligned}
\]
Step 2. Compute the parent population.
\[
\begin{aligned}
N_1(t) &= N_{1,0} e^{-\lambda_1 t}.
\end{aligned}
\]
Step 3. Compute the daughter population.
\[
\begin{aligned}
N_2(t)
&= N_{1,0}\,b\,\frac{\lambda_1}{\lambda_2-\lambda_1}
\left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right).
\end{aligned}
\]
Step 4. Convert both populations into activities.
\[
\begin{aligned}
A_1(t) &= \lambda_1 N_1(t), \\
A_2(t) &= \lambda_2 N_2(t).
\end{aligned}
\]
Step 5. Locate the daughter peak if desired.
\[
\begin{aligned}
t_{\mathrm{peak}}
&= \frac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}.
\end{aligned}
\]
Advanced note
At university level, one extends this picture to longer chains and general branching networks using the full
Bateman equations or matrix methods. Those tools can describe uranium-series chains, thorium chains,
daughter buildup with several branches, and secular-equilibrium limits in greater detail. This calculator intentionally
focuses on the two-member sequential case because it already captures the essential rise-and-fall behavior of daughter
activity and provides the clearest introduction to decay-chain dynamics.
| Concept |
Main relation |
Meaning |
| Parent population |
\(N_1 = N_{1,0} e^{-\lambda_1 t}\) |
Standard exponential decay of the parent |
| Daughter population |
\(N_2 = N_{1,0} b \lambda_1(\mathrm{e}^{-\lambda_1 t}-\mathrm{e}^{-\lambda_2 t})/(\lambda_2-\lambda_1)\) |
Bateman buildup-and-decay form for a two-member chain |
| Activities |
\(A_1=\lambda_1 N_1,\ A_2=\lambda_2 N_2\) |
Decay rates of the parent and daughter |
| Daughter peak time |
\(t_{\mathrm{peak}}=\ln(\lambda_2/\lambda_1)/(\lambda_2-\lambda_1)\) |
Time of maximum daughter activity |
| Secular equilibrium |
\(T_{1/2,1}\gg T_{1/2,2}\) |
Daughter activity can approach the parent activity |
| Branching fraction |
\(b\) |
Fraction of parent decays that actually feed the daughter |