Serial Decay, the Bateman Equations, and Radioactive Equilibrium

The exponential law $N(t)=N_0{e}^{-\lambda t}$ describes a single unstable species in isolation. Most radioactive nuclei are not isolated: the daughter of one decay is frequently unstable itself, so activity flows through a chain, and a nucleus under a neutron beam is produced and destroyed at the same time. Each of these situations is a linear rate equation with a source term, and the family of solutions organizes the observable activities of every decay series.

Activity and its units

The activity of a sample is the number of disintegrations per unit time,

not the number of nuclei present. A detector counts decays, so activity, not $N$, is the measured quantity. For a single species $A(t)=A_0{e}^{-\lambda t}$ with the same decay constant as the population.

A short half-life packs a large activity into few atoms: $1\mu \mathrm{g}$ of ${}^{131}\mathrm{I}$ ($t_{1/2}=8.02\mathrm{d}$) carries about $4.6\mathrm{GBq}$, while $1\mathrm{g}$ of ${}^{238}\mathrm{U}$ ($t_{1/2}=4.47\times 10^9\mathrm{y}$) carries only $12\mathrm{kBq}$.

The two-member chain

Let a parent (species 1) decay to a daughter (species 2) that itself decays to a stable end product. Writing $N_1,N_2$ for the populations and ${\lambda }_1,{\lambda }_2$ for the decay constants,

The daughter is fed at the rate the parent decays, ${\lambda }_1{N}_1$, and removed at its own rate ${\lambda }_2{N}_2$. The first equation gives $N_1(t)=N_1(0)e^{-{\lambda }_{1}t}$. Substituting into the second yields a linear first-order equation with an exponential source,

Multiplying by the integrating factor $e^{{\lambda }_{2}t}$ makes the left side an exact derivative, $\frac{\mathbf{d}}{\mathbf{d}t}\left(N_2{e}^{{\lambda }_{2}t}\right)={\lambda }_1{N}_1(0)e^{({\lambda }_2-{\lambda }_1)t}$. Integrating from $0$ to $t$ with the initial condition $N_2(0)=0$ gives the two-member result,

The daughter activity follows as $A_2={\lambda }_2{N}_2$,

Starting from zero, the daughter activity rises as its stock builds, reaches a maximum, and then falls once the parent supply thins. The maximum sits where $\mathbf{d}{N}_2/\mathbf{d}t=0$, that is where production balances loss, ${\lambda }_1{N}_1={\lambda }_2{N}_2$:

At $t_{\max }$ the two activities are momentarily equal, $A_1(t_{\max })=A_2(t_{\max })$.

The Bateman equations

For a straight chain $1\to 2\to \cdots \to n$ with only the head populated at $t=0$, the same integrating-factor method applied down the chain produces the Bateman solution,¹

Each member contributes a decaying exponential in every downstream population, weighted so that the initial conditions $N_k(0)=0$ for $k\ge 2$ hold. The formula assumes distinct decay constants; equal constants are handled by a limiting form. The two-member result above is the $n=2$ case.

Radioactive equilibrium

The long-time behavior of a two-member chain depends entirely on which member lives longer. For $t\gg 1/{\lambda }_2$ the term $e^{-{\lambda }_{2}t}$ vanishes and the activity ratio approaches a limit set by the two decay constants:

Three regimes follow from the sign and size of ${\lambda }_2-{\lambda }_1$.

• Secular equilibrium (${\lambda }_1\ll {\lambda }_2$, parent effectively stable on the daughter's timescale). The ratio tends to ${\lambda }_2/({\lambda }_2-{\lambda }_1)\approx 1$, so $A_2\to A_1$: the daughter decays exactly as fast as it is produced, ${\lambda }_1{N}_1={\lambda }_2{N}_2$. Every member of a long chain headed by a very long-lived parent reaches the same activity.
• Transient equilibrium (${\lambda }_1<{\lambda }_2$, comparable). The ratio tends to the constant ${\lambda }_2/({\lambda }_2-{\lambda }_1)>1$, so the daughter activity settles above the parent's and both then fall with the parent's slow constant ${\lambda }_1$.
• No equilibrium (${\lambda }_1>{\lambda }_2$, parent shorter-lived). No constant ratio exists. The parent disappears first, and the accumulated daughter afterward decays with its own constant ${\lambda }_2$.

Secular equilibrium is the working principle of the natural series: in an ancient uranium ore every daughter down to the stable lead end has an activity equal to that of ${}^{238}\mathrm{U}$, so counting one member measures the whole chain. A transient-equilibrium pair is exploited in the ${}^{99}\mathrm{Mo}{/}^{99\mathrm{m}}\mathrm{Tc}$ generator: the $66\mathrm{h}$ molybdenum parent feeds the $6.0\mathrm{h}$ technetium daughter, which is eluted for imaging while its activity tracks the parent.

Production and decay under irradiation

A target held in a reactor or accelerator beam is a source with a constant production rate. Let $R$ be the number of daughter nuclei formed per unit time (for a thin target, $R=\varphi \sigma N_{\text{target}}$ with flux $\varphi$ and cross section $\sigma$), and let the product decay with constant $\lambda$. The population obeys

whose solution is $N(t)=\frac{R}{\lambda }\left(1-e^{-\lambda t}\right)$. The activity is

Activity climbs toward the saturation value $A_{\infty }=R$, the point where formation and decay balance. Saturation is independent of the decay constant: a short-lived product saturates quickly at the same ceiling a long-lived one approaches slowly. After $n$ half-lives the activity has reached $1-2^{-n}$ of saturation, so $50\%$ after one, $75\%$ after two, and $87.5\%$ after three; irradiating far beyond a few half-lives yields diminishing returns.

Branching and partial decay constants

A nucleus that can decay by more than one route has a partial decay constant${\lambda }_i$ for each mode. The modes are independent competing processes, so the total decay probability per unit time is their sum,

and the population still falls as $e^{-\lambda t}$ with the total constant. The fraction of decays through mode $i$ is the branching ratio

Each partial constant defines a partial half-life $t_{1/2,i}=\ln 2/{\lambda }_i=t_{1/2}/b_i$, the half-life the nucleus would have if that mode acted alone. Partial half-lives are always longer than the observed half-life. For ${}^{40}\mathrm{K}$, electron capture and ${\beta }^{+}$ to ${}^{40}\mathrm{Ar}$ take about $b_{\mathrm{EC}}=0.11$ of decays and ${\beta }^{-}$ to ${}^{40}\mathrm{Ca}$ the remaining $0.89$; the argon branch is what makes potassium-argon dating possible.

Natural decay series and radiometric dating

Alpha decay lowers the mass number by four and beta decay leaves it unchanged, so every heavy chain conserves $A\mathrm{mod}4$. Three of the four series survive in nature because their heads outlive the age of the Earth.

The neptunium series is extinct: its head decays in a few million years, short against the $4.5$-billion-year age of the solar system, so it long ago reached its stable bismuth end. In an undisturbed mineral the surviving series sit in secular equilibrium, every intermediate at the activity of its long-lived head.

Radiometric dating reads the clock set by a long-lived parent decaying to a stable daughter. If a mineral begins with $N_0$ parent atoms and no daughter, then at time $t$ the parent count is $N=N_0{e}^{-\lambda t}$ and the accumulated stable daughter is $D=N_0-N=N(e^{\lambda t}-1)$. Solving for the age,

The ratio $D/N$ of daughter to remaining parent, both measured by mass spectrometry, fixes the age without knowing the original amount. Uranium-lead (${}^{238}\mathrm{U}\to {}^{206}\mathrm{Pb}$), rubidium-strontium, and potassium-argon clocks date rocks over billions of years.