Chapter 12 — Radioactivity Fundamentals: Decay Law, Activity, Half-Life, and Decay Chains

"The atoms of radioactive substances are unstable, and each atom has a certain probability of breaking up in a definite way during a given interval of time." — Ernest Rutherford, Radioactive Substances and their Radiations (1913)

12.1 Introduction: The Statistical Nature of Nuclear Decay

Radioactivity is, at its core, a quantum mechanical phenomenon governed entirely by probability. We cannot predict when any single nucleus will decay — not because we lack sufficient information, but because the outcome is genuinely indeterminate until the moment of decay. Yet when we gather $10^{23}$ such nuclei into a macroscopic sample, the statistical regularity is breathtaking: the fraction that decays per unit time is constant, yielding the beautifully simple exponential decay law that governs everything from smoke detectors to geological dating.

This chapter develops the mathematical framework of radioactive decay from first principles. We begin with the probabilistic foundation and derive the exponential law, then introduce the concepts of activity, half-life, and mean life. From there we move to decay chains — sequences of radioactive transformations — and the Bateman equations that describe them. We explore the physically important regimes of secular and transient equilibrium. Finally, we apply these ideas to two spectacular domains: radiometric dating (allowing us to determine ages from thousands to billions of years) and the four natural radioactive series that connect the heaviest elements to stable lead and bismuth isotopes.

Spaced Review — Chapter 1 Connection: Recall from Chapter 1 that the binding energy per nucleon $B/A$ reaches its maximum near $A \approx 56$ (iron-nickel region). Nuclei far from this peak — especially heavy nuclei with $A > 200$ — are energetically "eager" to shed mass through $\alpha$ decay or fission. The $Q$-value of a decay (the energy released) is simply the difference in total binding energies between products and parent. Every decay we study in this chapter is driven by the nucleus seeking a state of greater binding energy per nucleon.

Spaced Review — Chapter 4 Connection: In Chapter 4 we introduced the valley of stability and the liquid drop model. Nuclei that lie off the valley floor — too neutron-rich or too proton-rich — undergo $\beta$ decay to move toward stability. The competition between $\alpha$ decay (reducing $A$ and $Z$ simultaneously) and $\beta$ decay (changing $Z$ at constant $A$) produces the characteristic zigzag patterns of radioactive decay chains that we will trace in detail in Section 12.7.

12.2 The Decay Constant and the Exponential Decay Law

12.2.1 The Fundamental Postulate

The starting point is a single, experimentally verified postulate:

Each radioactive nucleus has a constant probability $\lambda \, dt$ of decaying in an infinitesimal time interval $dt$, independent of its age, environment, temperature, pressure, or chemical state.

The constant $\lambda$ is called the decay constant and has units of inverse time (s$^{-1}$, yr$^{-1}$, etc.). Its value is characteristic of the specific nuclear transition — it encodes all the quantum mechanics of barrier penetration ($\alpha$ decay), weak interaction matrix elements ($\beta$ decay), or electromagnetic transition rates ($\gamma$ decay).

The independence from external conditions is remarkable and was one of the earliest confirmations that radioactivity is a nuclear, not atomic, phenomenon. Chemical bonds, temperature, and pressure affect the electron cloud but leave the nucleus essentially untouched. (There are tiny exceptions — electron capture rates can be modified by extreme ionization, and bound-state $\beta$ decay can occur for fully stripped ions — but these effects are typically $< 1\%$ and require extraordinary conditions.)

12.2.2 Derivation of the Decay Law

We present two complementary derivations — one from the macroscopic rate equation and one from the microscopic probability argument — to emphasize that the exponential law is an inevitable consequence of constant decay probability.

Macroscopic approach (rate equation). Consider a sample containing $N(t)$ identical radioactive nuclei at time $t$. In the interval $[t, t + dt]$, the number that decay is:

$$dN = -\lambda \, N(t) \, dt$$

The negative sign indicates that $N$ decreases. Separating variables:

$$\frac{dN}{N} = -\lambda \, dt$$

Integrating both sides from $t = 0$ (where $N = N_0$) to time $t$:

$$\int_{N_0}^{N(t)} \frac{dN'}{N'} = -\lambda \int_0^t dt'$$

$$\ln N(t) - \ln N_0 = -\lambda t$$

$$\ln\frac{N(t)}{N_0} = -\lambda t$$

Exponentiating:

$$\boxed{N(t) = N_0 \, e^{-\lambda t}}$$

where $N_0 = N(t=0)$ is the initial number of nuclei. This is the exponential decay law, also called the radioactive decay law.

Microscopic approach (probability argument). We can also derive the same result by considering a single nucleus and asking: what is the probability that it has not decayed by time $t$?

Divide the interval $[0, t]$ into $n$ tiny subintervals, each of duration $\Delta t = t/n$. The probability of surviving one subinterval is $(1 - \lambda \Delta t)$. Since the decay probability in each subinterval is independent of the nucleus's history (the fundamental postulate), the probability of surviving all $n$ subintervals is:

$$P_{\text{survive}}(t) = (1 - \lambda \Delta t)^n = \left(1 - \frac{\lambda t}{n}\right)^n$$

Taking the limit $n \to \infty$ and using the definition of the exponential:

$$P_{\text{survive}}(t) = \lim_{n \to \infty} \left(1 - \frac{\lambda t}{n}\right)^n = e^{-\lambda t}$$

For a sample of $N_0$ nuclei, the expected number surviving at time $t$ is $N(t) = N_0 \cdot P_{\text{survive}}(t) = N_0 e^{-\lambda t}$, recovering the macroscopic result.

This derivation makes the statistical nature explicit: the exponential law is a consequence of the memorylessness of the decay process. The probability of decaying in the next $dt$ depends only on $\lambda$, not on how long the nucleus has already existed. In probability theory, the exponential distribution is the unique continuous distribution with this memoryless property.

Statistical fluctuations. For a finite sample, the actual number of decays in a time interval fluctuates around the mean. The number of decays in time $\Delta t$ follows a Poisson distribution with mean $\mu = \lambda N \Delta t$. The standard deviation is $\sigma = \sqrt{\mu}$. For a measurement counting $C$ total counts, the statistical uncertainty is $\sqrt{C}$ and the fractional uncertainty is $1/\sqrt{C}$. This is why nuclear counting experiments require large numbers of counts for precision — a theme that pervades experimental nuclear physics.

Let us pause to appreciate the logic. We assumed nothing about the mechanism of decay — only that each nucleus has the same constant probability per unit time. The exponential follows purely from this statistical assumption. The same mathematics describes the discharge of a capacitor, the cooling of an object (Newton's law), and the depletion of a first-order chemical reactant.

12.2.3 A Probabilistic Interpretation

For a single nucleus, the probability of surviving to time $t$ without decaying is:

$$P_{\text{survive}}(t) = e^{-\lambda t}$$

The probability of decaying between $t$ and $t + dt$ is:

$$P_{\text{decay}}(t) \, dt = \lambda \, e^{-\lambda t} \, dt$$

This is a normalized probability density (integrating from $0$ to $\infty$ gives 1). The expectation value of the decay time — the mean life — is:

$$\tau = \langle t \rangle = \int_0^{\infty} t \, \lambda \, e^{-\lambda t} \, dt = \frac{1}{\lambda}$$

We will return to $\tau$ in Section 12.4.

💡 Threshold Concept: Radioactive decay is fundamentally probabilistic — we can predict the behavior of $10^{23}$ atoms but not one. This is not a limitation of our knowledge; it is a feature of quantum mechanics. The decay constant $\lambda$ is not a hidden clock inside the nucleus counting down to some predetermined moment. It is a genuine probability rate, and the exponential law emerges only from the statistics of large numbers.

12.3 Activity and Its Units

12.3.1 Defining Activity

The activity $A$ of a radioactive sample is the number of decays per unit time:

$$A(t) = -\frac{dN}{dt} = \lambda \, N(t) = \lambda \, N_0 \, e^{-\lambda t} = A_0 \, e^{-\lambda t}$$

Activity is what a detector measures — it is the observable quantity, whereas $N(t)$ (the number of undecayed nuclei) is typically not directly accessible. Note that activity also decays exponentially with the same time constant.

12.3.2 The Becquerel and the Curie

The SI unit of activity is the becquerel (Bq):

$$1 \text{ Bq} = 1 \text{ decay/s}$$

Named after Henri Becquerel, who discovered radioactivity in 1896 through the fogging of photographic plates by uranium salts.

The older (but still widely used) unit is the curie (Ci), originally defined as the activity of 1 gram of $^{226}$Ra:

$$1 \text{ Ci} = 3.7 \times 10^{10} \text{ Bq} = 37 \text{ GBq}$$

The curie is a large unit. Practical subunits include:

A typical household smoke detector contains about 1 $\mu$Ci (37 kBq) of $^{241}$Am. A PET scan involves injection of approximately 10 mCi (370 MBq) of $^{18}$F-FDG.

12.3.3 Specific Activity

The specific activity is the activity per unit mass of a radioactive substance:

$$a = \frac{A}{m} = \frac{\lambda N_A}{M}$$

where $M$ is the molar mass (g/mol) and $N_A = 6.022 \times 10^{23}$ mol$^{-1}$ is Avogadro's number.

Example: The specific activity of $^{226}$Ra ($t_{1/2} = 1600$ yr, $M = 226$ g/mol):

$$\lambda = \frac{\ln 2}{1600 \times 3.156 \times 10^7 \text{ s}} = 1.373 \times 10^{-11} \text{ s}^{-1}$$

$$a = \frac{(1.373 \times 10^{-11})(6.022 \times 10^{23})}{226} = 3.66 \times 10^{10} \text{ Bq/g} \approx 1 \text{ Ci/g}$$

This confirms the historical definition of the curie — it was indeed the activity of 1 gram of radium, to within experimental precision of Marie Curie's era.

Example: The specific activity of $^{238}$U ($t_{1/2} = 4.468 \times 10^9$ yr, $M = 238$ g/mol):

$$a = \frac{\ln 2 \cdot N_A}{t_{1/2} \cdot M} = \frac{0.6931 \times 6.022 \times 10^{23}}{(4.468 \times 10^9 \times 3.156 \times 10^7)(238)}$$

$$a = 1.244 \times 10^4 \text{ Bq/g} = 12.44 \text{ kBq/g}$$

This is about 3 million times less than $^{226}$Ra. Long-lived isotopes have low specific activity — they decay slowly precisely because each nucleus is unlikely to decay in any given second.

📊 Real Data: Specific activities of selected isotopes:

Isotope	$t_{1/2}$	Specific Activity
$^{3}$H (tritium)	12.32 yr	356 TBq/g (9,650 Ci/g)
$^{14}$C	5,730 yr	165 GBq/g (4.46 Ci/g)
$^{60}$Co	5.271 yr	41.8 TBq/g (1,130 Ci/g)
$^{137}$Cs	30.08 yr	3.22 TBq/g (87 Ci/g)
$^{226}$Ra	1,600 yr	36.6 GBq/g (0.989 Ci/g)
$^{238}$U	$4.468 \times 10^9$ yr	12.4 kBq/g (0.336 $\mu$Ci/g)
$^{232}$Th	$1.405 \times 10^{10}$ yr	4.06 kBq/g (0.110 $\mu$Ci/g)

12.4 Half-Life and Mean Life

12.4.1 Half-Life

The half-life $t_{1/2}$ is the time required for half the nuclei (or half the activity) to decay:

$$N(t_{1/2}) = \frac{N_0}{2} \implies e^{-\lambda t_{1/2}} = \frac{1}{2}$$

$$\boxed{t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.6931}{\lambda}}$$

After $n$ half-lives, the fraction remaining is $(1/2)^n$:

A useful rule of thumb: after 10 half-lives, less than 0.1% remains; after 20 half-lives, less than one part per million.

Known half-lives span an extraordinary range — more than 55 orders of magnitude:

The ratio between the shortest and longest is $\sim 10^{55}$. Understanding this enormous range requires the quantum mechanical theory of each decay mode, which we develop in Chapters 13-15.

12.4.2 Mean Life

The mean life (or average lifetime) is:

$$\boxed{\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{\ln 2} \approx 1.443 \, t_{1/2}}$$

At time $t = \tau$, the fraction remaining is $e^{-1} \approx 0.368$ (36.8%), not 50%. The mean life is always longer than the half-life by the factor $1/\ln 2 \approx 1.443$.

The mean life has a clean physical interpretation: if all $N_0$ nuclei decayed at a constant rate equal to the initial activity $A_0 = \lambda N_0$, the sample would be completely exhausted in time $\tau$:

$$A_0 \cdot \tau = \lambda N_0 \cdot \frac{1}{\lambda} = N_0$$

Particle physicists prefer $\tau$ because it appears directly in the exponential ($e^{-t/\tau}$) without the factor of $\ln 2$. Nuclear physicists and radiochemists prefer $t_{1/2}$ because it is more intuitive and easier to apply in counting arguments.

The variance of the decay time. The variance of the decay time distribution is:

$$\text{Var}(t) = \langle t^2 \rangle - \langle t \rangle^2 = \int_0^\infty t^2 \lambda e^{-\lambda t} \, dt - \tau^2 = \frac{2}{\lambda^2} - \frac{1}{\lambda^2} = \frac{1}{\lambda^2} = \tau^2$$

The standard deviation equals the mean life: $\sigma_t = \tau$. This large spread (the coefficient of variation is 100%) reflects the enormous variability of individual nuclear decay times — some nuclei decay almost immediately, while a long exponential tail means others persist for many mean lives. Despite this large spread per nucleus, the sample average over $N$ nuclei has uncertainty $\tau/\sqrt{N}$, which is negligible for macroscopic samples.

The median lifetime. The time at which exactly half the nuclei have decayed is, by definition, the half-life $t_{1/2}$. The relationships between the three characteristic times are:

$$t_{1/2} = \tau \ln 2 \approx 0.693\tau \quad (\text{median})$$ $$\tau = 1/\lambda \quad (\text{mean})$$ $$t_{\text{mode}} = 0 \quad (\text{most probable decay time})$$

The most probable decay time is zero — the exponential distribution peaks at $t = 0$, meaning a nucleus is most likely to decay immediately. This may seem counterintuitive, but it follows directly from the constant hazard rate: the probability density $\lambda e^{-\lambda t}$ is largest at $t = 0$ and decreases monotonically.

12.4.3 Decay Rate in Practical Terms

Consider a medical $^{99\text{m}}$Tc source (the most widely used medical radioisotope, with $t_{1/2} = 6.006$ h). A nuclear medicine department receives a 20 mCi (740 MBq) vial at 8:00 AM. What is the activity at 5:00 PM (9 hours later)?

$$A(t) = A_0 \, e^{-\lambda t} = A_0 \, 2^{-t/t_{1/2}}$$

$$A(9\text{ h}) = 20 \text{ mCi} \times 2^{-9/6.006} = 20 \times 2^{-1.499} = 20 \times 0.354 = 7.07 \text{ mCi}$$

The activity has dropped to about a third of its morning value. This is why $^{99\text{m}}$Tc must be produced daily — typically in a $^{99}$Mo/$^{99\text{m}}$Tc generator ("moly cow") that exploits the secular equilibrium we will discuss in Section 12.6.

12.5 Decay Chains and the Bateman Equations

12.5.1 Sequential Decays

Many radioactive nuclei do not decay directly to a stable isotope. Instead, they produce radioactive daughters, which themselves decay, forming a decay chain:

$$N_1 \xrightarrow{\lambda_1} N_2 \xrightarrow{\lambda_2} N_3 \xrightarrow{\lambda_3} \cdots \xrightarrow{\lambda_{n-1}} N_n \text{ (stable)}$$

The coupled differential equations describing such a chain are:

$$\frac{dN_1}{dt} = -\lambda_1 N_1$$

$$\frac{dN_2}{dt} = \lambda_1 N_1 - \lambda_2 N_2$$

$$\frac{dN_3}{dt} = \lambda_2 N_2 - \lambda_3 N_3$$

$$\vdots$$

$$\frac{dN_n}{dt} = \lambda_{n-1} N_{n-1}$$

The first equation is just our familiar exponential decay. Each subsequent equation has a "production" term (from the parent) and a "loss" term (from its own decay). The last member is stable ($\lambda_n = 0$), so it only accumulates.

12.5.2 Two-Member Chain: Parent → Daughter (→ Stable)

Consider $N_1 \xrightarrow{\lambda_1} N_2 \xrightarrow{\lambda_2} N_3$ (stable), with initial conditions $N_1(0) = N_{1,0}$, $N_2(0) = 0$, $N_3(0) = 0$.

The solutions are:

$$N_1(t) = N_{1,0} \, e^{-\lambda_1 t}$$

$$N_2(t) = \frac{\lambda_1}{\lambda_2 - \lambda_1} N_{1,0} \left( e^{-\lambda_1 t} - e^{-\lambda_2 t} \right)$$

$$N_3(t) = N_{1,0} \left[ 1 + \frac{1}{\lambda_1 - \lambda_2}\left( \lambda_2 e^{-\lambda_1 t} - \lambda_1 e^{-\lambda_2 t} \right) \right]$$

The daughter activity is:

$$A_2(t) = \lambda_2 N_2(t) = \frac{\lambda_1 \lambda_2}{\lambda_2 - \lambda_1} N_{1,0} \left( e^{-\lambda_1 t} - e^{-\lambda_2 t} \right)$$

The daughter activity starts at zero, rises to a maximum at:

$$t_{\max} = \frac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

and then decreases. The behavior depends critically on the ratio of the two decay constants, leading to the three equilibrium regimes discussed in Section 12.6.

12.5.3 Three-Member Chain: Explicit Solution

For longer chains, the algebra becomes more involved but the method is the same. Consider a three-member chain:

$$N_1 \xrightarrow{\lambda_1} N_2 \xrightarrow{\lambda_2} N_3 \xrightarrow{\lambda_3} N_4 \text{ (stable)}$$

with $N_1(0) = N_{1,0}$ and $N_2(0) = N_3(0) = N_4(0) = 0$. The coupled equations are:

$$\frac{dN_1}{dt} = -\lambda_1 N_1, \quad \frac{dN_2}{dt} = \lambda_1 N_1 - \lambda_2 N_2, \quad \frac{dN_3}{dt} = \lambda_2 N_2 - \lambda_3 N_3$$

The solution for $N_3(t)$ requires solving for $N_2(t)$ first (given above), then using the integrating factor method on the $N_3$ equation. The result is:

$$N_3(t) = \lambda_1 \lambda_2 \, N_{1,0} \left[ \frac{e^{-\lambda_1 t}}{(\lambda_2 - \lambda_1)(\lambda_3 - \lambda_1)} + \frac{e^{-\lambda_2 t}}{(\lambda_1 - \lambda_2)(\lambda_3 - \lambda_2)} + \frac{e^{-\lambda_3 t}}{(\lambda_1 - \lambda_3)(\lambda_2 - \lambda_3)} \right]$$

This pattern generalizes: the solution for the $n$-th member involves a sum of $n$ exponentials, each weighted by products of decay constants divided by differences of decay constants.

Example: ${}^{222}\text{Rn} \to {}^{218}\text{Po} \to {}^{214}\text{Pb} \to {}^{214}\text{Bi}$

Consider the portion of the uranium series: ${}^{222}\text{Rn}$ ($\lambda_1 = 0.1814\,\text{d}^{-1}$, $t_{1/2} = 3.823\,\text{d}$) $\to$ ${}^{218}\text{Po}$ ($\lambda_2 = 224\,\text{d}^{-1}$, $t_{1/2} = 3.10\,\text{min}$) $\to$ ${}^{214}\text{Pb}$ ($\lambda_3 = 37.3\,\text{d}^{-1}$, $t_{1/2} = 26.8\,\text{min}$). Since $\lambda_2 \gg \lambda_3 \gg \lambda_1$, the short-lived intermediates rapidly come into secular equilibrium with the radon parent. After a few hours (many ${}^{218}\text{Po}$ and ${}^{214}\text{Pb}$ half-lives), all three activities are approximately equal:

$$A_{\text{Rn}} \approx A_{\text{Po}} \approx A_{\text{Pb}}$$

This is the practical basis of radon detection: a measurement of the total alpha activity in a sealed chamber after equilibrium is established gives $A_{\text{Rn}}$ directly, because the short-lived daughters are in secular equilibrium.

12.5.4 The General Bateman Equations

For a chain of $n$ species with initial conditions $N_1(0) = N_{1,0}$ and $N_i(0) = 0$ for $i > 1$, Harry Bateman (1910) derived the general solution:

$$N_n(t) = \frac{N_{1,0}}{\lambda_n} \sum_{i=1}^{n} \lambda_i \, c_i \, e^{-\lambda_i t}$$

where the coefficients are:

$$c_i = \prod_{\substack{j=1 \\ j \neq i}}^{n} \frac{\lambda_j}{\lambda_j - \lambda_i}$$

For the last (stable) member where $\lambda_n = 0$, a limiting procedure is needed.

The Bateman equations assume: 1. Only the parent is present at $t = 0$. 2. No branching (each species has exactly one decay mode producing the next member). 3. All decay constants are distinct ($\lambda_i \neq \lambda_j$ for $i \neq j$).

When branching occurs, the equations are modified by replacing $\lambda_i$ in the production term with $\lambda_i \cdot BR_i$, where $BR_i$ is the branching ratio for the specific channel (see Section 12.6.4).

For chains with many members (e.g., the 14-step $^{238}$U series), the Bateman equations become unwieldy. Numerical integration using standard ODE solvers is the practical approach — which is precisely what we implement in this chapter's progressive project (see decay_chains.py).

⚠️ Numerical Note: The Bateman equations can suffer from severe numerical cancellation when two decay constants are nearly equal (e.g., $\lambda_i \approx \lambda_j$). The products of large numbers and small differences can exceed floating-point precision. For production calculations, numerical ODE integration is more robust.

12.6 Equilibrium in Decay Chains

The relative magnitudes of parent and daughter decay constants determine the long-term behavior of the chain. Three regimes are physically important.

12.6.1 Secular Equilibrium

Condition: $\lambda_1 \ll \lambda_2$, equivalently $t_{1/2,1} \gg t_{1/2,2}$.

When the parent half-life is enormously longer than the daughter's, the parent activity is essentially constant over timescales relevant to the daughter. In this limit ($\lambda_1 \ll \lambda_2$), the daughter activity approaches:

$$A_2(t) \approx A_1(t) \left(1 - e^{-\lambda_2 t}\right)$$

After several daughter half-lives ($t \gg t_{1/2,2}$), the exponential vanishes and:

$$\boxed{A_2 \approx A_1 \quad \text{(secular equilibrium)}}$$

The daughter activity equals the parent activity. Moreover, in a long chain, all activities become equal:

$$A_1 = A_2 = A_3 = \cdots$$

This is the condition of secular equilibrium. The word "secular" comes from the Latin saeculum (an age or generation), reflecting the very long parent half-life.

Physical example: $^{226}$Ra ($t_{1/2} = 1600$ yr) $\to$ $^{222}$Rn ($t_{1/2} = 3.823$ d). In any sealed radium source older than a few weeks, the radon activity equals the radium activity. More broadly, in undisturbed uranium ore that has existed for billions of years, every member of the $^{238}$U chain is in secular equilibrium — from $^{238}$U down to the short-lived daughters like $^{214}$Po ($t_{1/2} = 164$ $\mu$s).

The moly cow: The $^{99}$Mo/$^{99\text{m}}$Tc generator is a practical medical application. $^{99}$Mo ($t_{1/2} = 65.94$ h) decays to $^{99\text{m}}$Tc ($t_{1/2} = 6.006$ h). Since $t_{1/2,\text{Mo}} \approx 11 \times t_{1/2,\text{Tc}}$, this is not quite secular equilibrium but approaches it. The $^{99\text{m}}$Tc is "milked" by eluting with saline, and it regrows toward equilibrium. The maximum $^{99\text{m}}$Tc activity reaches about 95% of the $^{99}$Mo activity after about 23 hours.

12.6.2 Transient Equilibrium

Condition: $\lambda_1 < \lambda_2$ but not by an extreme ratio; roughly $t_{1/2,1}$ is a few to $\sim$10 times $t_{1/2,2}$.

After the initial transient dies out ($t \gg 1/(\lambda_2 - \lambda_1)$), the daughter activity becomes:

$$A_2(t) \approx \frac{\lambda_2}{\lambda_2 - \lambda_1} A_1(t)$$

The daughter activity exceeds the parent activity by the factor $\lambda_2/(\lambda_2 - \lambda_1) > 1$, and both decay at the parent's rate. The daughter is in lockstep with the parent, but at a higher level.

Physical example: $^{99}$Mo ($t_{1/2} = 65.94$ h) $\to$ $^{99\text{m}}$Tc ($t_{1/2} = 6.006$ h). After transient equilibrium is established:

$$\frac{A_{\text{Tc}}}{A_{\text{Mo}}} = \frac{\lambda_{\text{Tc}}}{\lambda_{\text{Tc}} - \lambda_{\text{Mo}}} = \frac{t_{1/2,\text{Mo}}}{t_{1/2,\text{Mo}} - t_{1/2,\text{Tc}}} = \frac{65.94}{65.94 - 6.006} = 1.10$$

The $^{99\text{m}}$Tc activity is about 10% higher than the $^{99}$Mo activity at transient equilibrium.

12.6.3 No Equilibrium

Condition: $\lambda_1 > \lambda_2$, equivalently $t_{1/2,1} < t_{1/2,2}$.

When the parent decays faster than the daughter, no equilibrium is possible. The parent vanishes, and the daughter is left to decay independently at its own rate. The daughter activity rises to a peak at $t_{\max} = \ln(\lambda_2/\lambda_1)/(\lambda_2 - \lambda_1)$ and then falls with its own characteristic decay constant $\lambda_2$.

The peak daughter activity can be found by substituting $t_{\max}$ into the daughter activity expression. The result is always less than $A_{1,0}$ (the initial parent activity), and the ratio $A_{2,\max}/A_{1,0}$ decreases as the parent-to-daughter half-life ratio decreases.

Physical example: $^{218}$Po ($t_{1/2} = 3.10$ min) $\to$ $^{214}$Pb ($t_{1/2} = 26.8$ min). The polonium decays away in about 15 minutes (5 half-lives), and the lead isotope is left to decay on its own 26.8-minute timescale. The ${}^{214}\text{Pb}$ activity peaks at about 7 minutes after the start, then decays exponentially.

Practical importance. The no-equilibrium regime is common in nuclear medicine when the parent is a short-lived activation product. For instance, ${}^{11}\text{C}$ ($t_{1/2} = 20.4$ min) produced by a cyclotron for PET imaging decays to stable ${}^{11}\text{B}$ — there is no daughter activity to manage. But in reactor-produced isotopes where the daughter is longer-lived than the parent, the "daughter accumulation" problem must be carefully managed for waste handling and radiation safety.

12.6.4 Branching Ratios

Many nuclides can decay by more than one mode. For example, $^{212}$Bi can undergo either $\alpha$ decay or $\beta^-$ decay:

$$^{212}\text{Bi} \xrightarrow[\alpha]{36\%} {}^{208}\text{Tl} \quad \text{and} \quad ^{212}\text{Bi} \xrightarrow[\beta^-]{64\%} {}^{212}\text{Po}$$

The total decay constant is the sum of the partial decay constants:

$$\lambda_{\text{total}} = \lambda_\alpha + \lambda_\beta$$

The branching ratio for each mode is:

$$BR_\alpha = \frac{\lambda_\alpha}{\lambda_{\text{total}}} = 0.36, \quad BR_\beta = \frac{\lambda_\beta}{\lambda_{\text{total}}} = 0.64$$

The total half-life is determined by $\lambda_{\text{total}}$:

$$t_{1/2} = \frac{\ln 2}{\lambda_{\text{total}}}$$

The partial half-life for each mode is longer than the total:

$$t_{1/2,\alpha} = \frac{\ln 2}{\lambda_\alpha} = \frac{t_{1/2}}{BR_\alpha}$$

For $^{212}$Bi ($t_{1/2} = 60.55$ min):

$$t_{1/2,\alpha} = \frac{60.55}{0.36} = 168 \text{ min}, \quad t_{1/2,\beta} = \frac{60.55}{0.64} = 94.6 \text{ min}$$

Another important example is $^{40}$K, which has three decay modes:

• $\beta^-$ to $^{40}$Ca: 89.28%
• Electron capture to $^{40}$Ar: 10.72% (includes a small $\beta^+$ contribution of 0.001%)

The total half-life of $^{40}$K is $1.248 \times 10^9$ yr, but the partial half-life for the EC branch (which produces the $^{40}$Ar used in K-Ar dating) is $1.248 \times 10^9 / 0.1072 = 1.164 \times 10^{10}$ yr.

12.7 Radioactive Dating Methods

The predictable mathematics of radioactive decay provides nature's most reliable clocks. By measuring the ratio of parent to daughter isotopes in a sample, we can determine when the "clock started" — typically the moment the sample became a closed system (no exchange of parent or daughter with the environment).

12.7.1 The General Principle

For a parent isotope $P$ decaying to a daughter $D$:

$$N_P(t) = N_{P,0} \, e^{-\lambda t}$$

$$N_D(t) = N_{D,0} + N_{P,0}\left(1 - e^{-\lambda t}\right) = N_{D,0} + N_P(t)\left(e^{\lambda t} - 1\right)$$

If we measure $N_P(t)$ and $N_D(t)$ today, and know or can determine $N_{D,0}$ (the initial daughter content), we can solve for $t$:

$$t = \frac{1}{\lambda} \ln\left(1 + \frac{N_D(t) - N_{D,0}}{N_P(t)}\right)$$

Different dating methods handle the $N_{D,0}$ problem in different ways.

12.7.2 Carbon-14 Dating

Willard Libby developed $^{14}$C dating in 1949, for which he received the Nobel Prize in Chemistry in 1960. It remains the most widely known radiometric method.

Production: $^{14}$C is continuously produced in the upper atmosphere by cosmic-ray neutrons:

$$^{14}\text{N} + n \to {}^{14}\text{C} + p$$

Equilibrium: Living organisms continuously exchange carbon with the atmosphere (through respiration, photosynthesis, and the food chain), maintaining a constant $^{14}$C/$^{12}$C ratio of approximately $1.2 \times 10^{-12}$ in their tissues.

Clock starts: When the organism dies, exchange ceases. The $^{14}$C decays with $t_{1/2} = 5,730 \pm 40$ yr, while the stable $^{12}$C remains constant. The ratio $^{14}$C/$^{12}$C decreases exponentially.

Age determination:

$$t = \frac{t_{1/2}}{\ln 2} \ln\left(\frac{A_0}{A}\right) = 8267 \text{ yr} \times \ln\left(\frac{A_0}{A}\right)$$

where $A_0$ is the activity of a modern reference standard (corrected to 1950) and $A$ is the measured activity of the sample.

Range: Practical range is about 300 to 50,000 years. Beyond $\sim$10 half-lives ($\sim$57,000 yr), too little $^{14}$C remains for reliable measurement. At the lower end, samples younger than about 300 years have experienced the Suess effect and the bomb pulse (see below), complicating interpretation.

Accelerator Mass Spectrometry (AMS). Modern ${}^{14}\text{C}$ dating uses AMS rather than the original decay-counting method. Instead of waiting for ${}^{14}\text{C}$ atoms to decay and counting the beta particles (which requires large samples and long measurement times), AMS directly counts the ${}^{14}\text{C}$ atoms by accelerating them to MeV energies and separating them from ${}^{12}\text{C}$ and ${}^{13}\text{C}$ by mass in a tandem Van de Graaff accelerator and magnetic spectrometer. AMS can measure ${}^{14}\text{C}/{}^{12}\text{C}$ ratios as small as $10^{-15}$, requires only milligram-sized samples (compared to grams for decay counting), and extends the datable range to about 60,000 years. AMS has revolutionized radiocarbon dating in archaeology, allowing dates from individual seeds, parchment scraps, and even the glue on Egyptian mummy wrappings.

Calibration: The atmospheric $^{14}$C/$^{12}$C ratio has not been constant. Variations in cosmic ray flux, Earth's magnetic field strength, and (since 1950) nuclear weapons testing have altered the production rate. Calibration curves, built from tree-ring chronologies (dendrochronology) extending back $\sim$14,000 years and from coral and cave deposits extending further, correct for these variations. The internationally agreed calibration curve is IntCal20 (Reimer et al., 2020, Radiocarbon 62:725-757).

The Suess effect (dilution of atmospheric $^{14}$C by fossil fuel CO$_2$, which is $^{14}$C-free) and the bomb pulse (doubling of atmospheric $^{14}$C from nuclear testing peaking in 1963) are major perturbations that must be accounted for.

Worked example: ${}^{14}$C dating with real data. The Shroud of Turin, long claimed to be the burial cloth of Jesus of Nazareth, was radiocarbon dated in 1988 by three independent laboratories (Arizona, Oxford, Zurich). The measured ${}^{14}\text{C}/{}^{12}\text{C}$ ratio, expressed as a fraction of the modern standard, was $0.9207 \pm 0.0058$. What age does this imply?

The fraction modern $F = N(t)/N_{\text{modern}} = e^{-\lambda t}$, so:

$$t = -\frac{1}{\lambda} \ln F = -\frac{t_{1/2}}{\ln 2} \ln(0.9207) = -8267 \text{ yr} \times (-0.0826) = 683 \text{ yr}$$

This places the manufacture of the linen at approximately 1260–1390 CE (after calibration), consistent with the medieval period and inconsistent with a first-century origin. The result, published in Nature (Damon et al., 1989, Nature 337:611-615), remains one of the most famous applications of radiocarbon dating.

Limitations. Radiocarbon dating assumes that (1) the sample has remained a closed system (no exchange of carbon with the environment after death), (2) the initial ${}^{14}\text{C}/{}^{12}\text{C}$ ratio is known (from calibration curves), and (3) the sample is not contaminated by older or younger carbon. Violation of any of these assumptions introduces systematic errors. Contamination by modern carbon is particularly insidious for old samples — adding just 1% modern carbon to a 40,000-year-old sample makes it appear 7,000 years younger.

12.7.3 Potassium-Argon Dating

$^{40}$K decays to $^{40}$Ar (by electron capture, 10.72%) and to $^{40}$Ca (by $\beta^-$, 89.28%) with a total half-life of $1.248 \times 10^9$ yr.

For K-Ar dating, we use only the argon branch:

$$^{40}\text{Ar}^* = \frac{\lambda_{\text{EC}}}{\lambda_{\text{total}}} \, ^{40}\text{K} \left(e^{\lambda_{\text{total}} t} - 1\right)$$

where $^{40}$Ar$^*$ denotes radiogenic argon (excess over atmospheric).

Clock starts: When a volcanic rock solidifies, argon (a noble gas) escapes from the molten material. The solidified rock starts with essentially zero $^{40}$Ar. As $^{40}$K decays, $^{40}$Ar accumulates and is trapped in the crystal lattice.

Range: $\sim$100,000 yr to the age of the Earth (4.5 Gyr). The method was refined into $^{40}$Ar/$^{39}$Ar dating, which irradiates the sample with neutrons to convert $^{39}$K to $^{39}$Ar, allowing the potassium content to be measured simultaneously with the argon content from the same grain. This technique was used to date the Chicxulub impact layer at $66.038 \pm 0.025$ Ma (Renne et al., 2013, Science 339:684-687).

12.7.4 Uranium-Lead Dating and the Concordia Diagram

The twin decay chains provide two independent clocks:

$$^{238}\text{U} \to {}^{206}\text{Pb} + 8\alpha + 6\beta^- \quad (t_{1/2} = 4.468 \times 10^9 \text{ yr})$$

$$^{235}\text{U} \to {}^{207}\text{Pb} + 7\alpha + 4\beta^- \quad (t_{1/2} = 7.04 \times 10^8 \text{ yr})$$

For a mineral that incorporated uranium but no initial lead:

$$\frac{^{206}\text{Pb}}{^{238}\text{U}} = e^{\lambda_{238} t} - 1$$

$$\frac{^{207}\text{Pb}}{^{235}\text{U}} = e^{\lambda_{235} t} - 1$$

The concordia diagram plots these two ratios against each other. For a system that has remained closed since time $t$, the point lies on the concordia curve — the locus of concordant ages. If lead has been partially lost (a common problem with zircon crystals), the data points fall below concordia along a straight line called a discordia line. The upper intercept of this line with concordia gives the crystallization age; the lower intercept gives the time of lead loss.

The concordia method is extraordinarily powerful because it provides a self-consistency check: two independent chronometers must agree. The mineral zircon (ZrSiO$_4$) is the workhorse of U-Pb geochronology because it readily substitutes U$^{4+}$ for Zr$^{4+}$ but excludes Pb$^{2+}$, providing a near-zero initial lead content. U-Pb zircon dating has achieved precisions of $\pm 0.1\%$, dating the oldest terrestrial minerals (Jack Hills zircons, Western Australia) at $4.404 \pm 0.008$ Ga (Wilde et al., 2001, Nature 409:175-178).

12.7.5 Rubidium-Strontium Dating and the Isochron Method

$^{87}$Rb undergoes $\beta^-$ decay to $^{87}$Sr with $t_{1/2} = 4.92 \times 10^{10}$ yr (the longest half-life used in geochronology).

The isochron method elegantly eliminates the need to know the initial daughter content. The isotope ratio equation is:

$$\frac{^{87}\text{Sr}}{^{86}\text{Sr}} = \left(\frac{^{87}\text{Sr}}{^{86}\text{Sr}}\right)_0 + \frac{^{87}\text{Rb}}{^{86}\text{Sr}} \left(e^{\lambda t} - 1\right)$$

where $^{86}$Sr is a stable, non-radiogenic reference isotope. This has the form $y = b + mx$, where: - $y = {}^{87}\text{Sr}/{}^{86}\text{Sr}$ (measured) - $x = {}^{87}\text{Rb}/{}^{86}\text{Sr}$ (measured) - $b = ({}^{87}\text{Sr}/{}^{86}\text{Sr})_0$ (initial ratio, the $y$-intercept) - $m = e^{\lambda t} - 1$ (the slope, which gives the age)

By measuring several cogenetic minerals with different Rb/Sr ratios, one fits a straight line — the isochron — whose slope gives the age and whose intercept gives the initial $^{87}$Sr/$^{86}$Sr ratio. No assumption about initial conditions is needed beyond the requirement that all samples started with the same $^{87}$Sr/$^{86}$Sr ratio.

Worked example: Rb-Sr isochron. Four mineral separates from a granitic gneiss yield the following isotope ratios:

A linear regression through these four points gives slope $m = 0.005546$ and intercept $b = 0.7274$. The age is:

$$m = e^{\lambda t} - 1 \implies t = \frac{\ln(1 + m)}{\lambda} = \frac{\ln(1.005546)}{1.420 \times 10^{-11} \text{ yr}^{-1}} = \frac{0.005531}{1.420 \times 10^{-11}} = 3.89 \times 10^8 \text{ yr}$$

The rock is approximately 389 million years old (Late Devonian). The initial ${}^{87}\text{Sr}/{}^{86}\text{Sr}$ ratio of 0.7274 is close to the bulk Earth value of about 0.699 at Earth's formation, consistent with crustal evolution over time.

12.7.6 Summary of Dating Methods

Each method exploits the same fundamental principle — the predictable exponential decay of a radioactive parent — but each has its own systematics, assumptions, and potential pitfalls. The most reliable ages come from concordant results across multiple independent methods.

12.8 The Four Natural Radioactive Series

All naturally occurring heavy radioactive isotopes belong to one of four decay series, classified by the residue of the mass number modulo 4:

The mass number classification arises because $\alpha$ decay reduces $A$ by 4, and $\beta$ decay does not change $A$. Therefore, the value of $A \pmod{4}$ is preserved throughout the chain.

12.8.1 The Uranium Series ($4n+2$)

The $^{238}$U series contains 14 steps and passes through 8 $\alpha$ decays and 6 $\beta^-$ decays:

$$^{238}\text{U} \xrightarrow{\alpha} {}^{234}\text{Th} \xrightarrow{\beta^-} {}^{234}\text{Pa} \xrightarrow{\beta^-} {}^{234}\text{U} \xrightarrow{\alpha} {}^{230}\text{Th} \xrightarrow{\alpha} {}^{226}\text{Ra} \xrightarrow{\alpha} {}^{222}\text{Rn}$$ $$\xrightarrow{\alpha} {}^{218}\text{Po} \xrightarrow{\alpha} {}^{214}\text{Pb} \xrightarrow{\beta^-} {}^{214}\text{Bi} \xrightarrow{\beta^-} {}^{214}\text{Po} \xrightarrow{\alpha} {}^{210}\text{Pb} \xrightarrow{\beta^-} {}^{210}\text{Bi} \xrightarrow{\beta^-} {}^{210}\text{Po} \xrightarrow{\alpha} {}^{206}\text{Pb}$$

Selected half-lives along this chain:

The $^{222}$Rn (radon) in this chain is the leading cause of lung cancer after smoking. Being a noble gas, it seeps out of soil and rock and accumulates in enclosed spaces. The EPA action level of 4 pCi/L corresponds to about 150 Bq/m$^3$. The health risk comes not from radon itself (which is inhaled and exhaled without significant dose) but from its short-lived solid daughters — ${}^{218}\text{Po}$ and ${}^{214}\text{Po}$ — which deposit on lung tissue and deliver alpha radiation directly to the epithelial cells. The estimated 21,000 lung cancer deaths per year in the United States attributed to indoor radon make it a significant public health concern and one of the most practically important applications of the decay chain physics developed in this chapter.

The chain also has archaeological significance: the ${}^{230}\text{Th}/{}^{234}\text{U}$ disequilibrium method (often called uranium-thorium dating) exploits the fact that uranium is soluble in water but thorium is not. When calcium carbonate precipitates from groundwater (in corals, speleothems, or travertine), it incorporates uranium but not thorium. The ${}^{230}\text{Th}$ then grows in from the decay of ${}^{234}\text{U}$ with a 75,380-year half-life, providing a chronometer valid from a few hundred years to approximately 500,000 years — bridging the gap between ${}^{14}\text{C}$ dating and K-Ar dating.

12.8.2 The Thorium Series ($4n$)

$$^{232}\text{Th} \xrightarrow{\alpha} {}^{228}\text{Ra} \xrightarrow{\beta^-} {}^{228}\text{Ac} \xrightarrow{\beta^-} {}^{228}\text{Th} \xrightarrow{\alpha} {}^{224}\text{Ra} \xrightarrow{\alpha} {}^{220}\text{Rn}$$ $$\xrightarrow{\alpha} {}^{216}\text{Po} \xrightarrow{\alpha} {}^{212}\text{Pb} \xrightarrow{\beta^-} {}^{212}\text{Bi} \xrightarrow{\alpha(36\%)/\beta^-(64\%)} \cdots \to {}^{208}\text{Pb}$$

The branch at $^{212}$Bi is noteworthy: 64% proceeds by $\beta^-$ to $^{212}$Po (which $\alpha$-decays with $t_{1/2} = 0.299$ $\mu$s to $^{208}$Pb), and 36% by $\alpha$ to $^{208}$Tl (which $\beta^-$-decays with $t_{1/2} = 3.053$ min to $^{208}$Pb). Both branches converge on stable $^{208}$Pb.

12.8.3 The Actinium Series ($4n+3$)

Beginning with $^{235}$U ($t_{1/2} = 7.04 \times 10^8$ yr), this chain passes through 11 steps (7$\alpha$ + 4$\beta^-$) to reach $^{207}$Pb. Notable members include $^{231}$Pa ($t_{1/2} = 3.276 \times 10^4$ yr), $^{227}$Ac ($t_{1/2} = 21.77$ yr, the namesake of the series), and $^{219}$Rn ($t_{1/2} = 3.96$ s, called "actinon" historically).

12.8.4 The Neptunium Series ($4n+1$) — Extinct

The $^{237}$Np series is unique: its longest-lived member, $^{237}$Np itself, has $t_{1/2} = 2.144 \times 10^6$ yr. Since the Solar System is $\sim 4.6 \times 10^9$ yr old, this is more than 2,000 half-lives. The primordial $^{237}$Np has long since vanished — the fraction remaining would be $(1/2)^{2000} \approx 10^{-600}$, which is effectively zero.

The series terminates at $^{209}$Bi, which was long considered the heaviest stable nuclide. However, in 2003, a French team at the Institut d'Astrophysique Spatiale measured the half-life of $^{209}$Bi as $(1.9 \pm 0.2) \times 10^{19}$ yr — about a billion times the age of the universe (de Marcillac et al., 2003, Nature 422:876-878). For practical purposes, it is stable.

Today, trace amounts of $^{237}$Np exist from neutron capture on $^{236}$U in nuclear reactors and from nuclear weapons fallout. The entire $4n+1$ chain can be studied in the laboratory. The principal members of the neptunium series are:

$${}^{237}\text{Np} \xrightarrow{\alpha} {}^{233}\text{Pa} \xrightarrow{\beta^-} {}^{233}\text{U} \xrightarrow{\alpha} {}^{229}\text{Th} \xrightarrow{\alpha} {}^{225}\text{Ra} \xrightarrow{\beta^-} {}^{225}\text{Ac} \xrightarrow{\alpha} {}^{221}\text{Fr} \xrightarrow{\alpha} {}^{217}\text{At} \xrightarrow{\alpha} {}^{213}\text{Bi}$$ $$\xrightarrow{\beta^-(97.8\%)} {}^{213}\text{Po} \xrightarrow{\alpha} {}^{209}\text{Pb} \xrightarrow{\beta^-} {}^{209}\text{Bi}$$

The isotope ${}^{225}\text{Ac}$ in this chain has drawn intense interest for targeted alpha therapy (TAT) in cancer treatment. Its 10-day half-life and cascade of four alpha emissions make it an extraordinarily potent radiotherapy agent when attached to tumor-targeting antibodies or peptides. Clinical trials using ${}^{225}\text{Ac}$-PSMA-617 for metastatic prostate cancer have shown remarkable responses in patients refractory to all other treatments (Kratochwil et al., 2016, Journal of Nuclear Medicine 57:1941-1944). Producing sufficient quantities of ${}^{225}\text{Ac}$ is a major challenge — it is currently obtained from milking ${}^{229}\text{Th}$ sources or from proton irradiation of ${}^{226}\text{Ra}$.

12.8.5 Secular Equilibrium in Natural Ores

In any undisturbed uranium or thorium ore that has existed for much longer than the half-life of every intermediate member, the entire chain is in secular equilibrium. This has a remarkable practical consequence: the activity of every member of the chain is equal to the activity of the long-lived parent.

For a rock containing 1 ppm (by mass) of ${}^{238}\text{U}$, the equilibrium activity of every member of the 14-step chain is:

$$A = a_{\text{U}} \times m_{\text{U}} = 12.44 \text{ kBq/g} \times 10^{-6} \text{ g/g}_{\text{rock}} = 12.44 \text{ mBq/g}_{\text{rock}}$$

This applies to every member: ${}^{234}\text{Th}$, ${}^{234}\text{Pa}$, ${}^{234}\text{U}$, ${}^{230}\text{Th}$, ${}^{226}\text{Ra}$, ${}^{222}\text{Rn}$, all the way down to ${}^{210}\text{Po}$. The total activity is therefore $14 \times 12.44 = 174\,\text{mBq/g}$ from the uranium chain alone. Adding the thorium chain (approximately 40 mBq/g for typical crustal rock) and ${}^{40}\text{K}$ (approximately 600 mBq/g), the total natural radioactivity in typical granite is about 1 Bq/g — detectable but not hazardous at ordinary exposure levels.

Disruption of equilibrium occurs when any member of the chain is physically or chemically separated from its parent. The classic example is radon (${}^{222}\text{Rn}$): being a noble gas, it diffuses out of rock and soil and enters the atmosphere (or indoor air). Once removed from its ${}^{226}\text{Ra}$ parent, the radon decays with its own 3.82-day half-life. Its short-lived daughters (${}^{218}\text{Po}$, ${}^{214}\text{Pb}$, ${}^{214}\text{Bi}$, ${}^{214}\text{Po}$) rapidly come into equilibrium with the radon — not with the original radium. This is the basis of indoor radon monitoring: the alpha activity of the short-lived daughters provides a measure of the radon concentration.

12.9 Worked Examples

Example 12.1: Activity Calculation

A research laboratory has a 5.00 $\mu$g sample of $^{60}$Co ($t_{1/2} = 5.271$ yr, $M = 59.93$ g/mol). Find its activity in Bq and Ci.

Solution:

Number of atoms: $$N = \frac{m N_A}{M} = \frac{5.00 \times 10^{-6} \times 6.022 \times 10^{23}}{59.93} = 5.024 \times 10^{16}$$

Decay constant: $$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.6931}{5.271 \times 3.156 \times 10^7 \text{ s}} = 4.167 \times 10^{-9} \text{ s}^{-1}$$

Activity: $$A = \lambda N = 4.167 \times 10^{-9} \times 5.024 \times 10^{16} = 2.093 \times 10^{8} \text{ Bq} = 209 \text{ MBq}$$

$$A = \frac{2.093 \times 10^8}{3.7 \times 10^{10}} = 5.66 \text{ mCi}$$

Example 12.2: Carbon-14 Dating

A wooden beam from an archaeological site has a $^{14}$C activity of 8.2 disintegrations per minute per gram of carbon (dpm/g). Modern wood has 15.3 dpm/g (corrected to 1950). Estimate the age.

Solution:

$$t = \frac{t_{1/2}}{\ln 2} \ln\left(\frac{A_0}{A}\right) = \frac{5730}{0.6931} \ln\left(\frac{15.3}{8.2}\right)$$

$$t = 8267 \times \ln(1.866) = 8267 \times 0.6245 = 5163 \text{ yr}$$

The beam dates to approximately 3200 BCE (uncalibrated). Calibration against the IntCal20 curve would refine this estimate.

Example 12.3: Specific Activity of Tritium

Tritium (${}^{3}\text{H}$) is used in self-luminous devices and as a tracer in hydrology. Calculate the specific activity of pure tritium ($t_{1/2} = 12.32$ yr, $M = 3.016$ g/mol).

Solution:

$$\lambda = \frac{\ln 2}{12.32 \times 3.156 \times 10^7 \text{ s}} = \frac{0.6931}{3.888 \times 10^8 \text{ s}} = 1.783 \times 10^{-9} \text{ s}^{-1}$$

$$a = \frac{\lambda N_A}{M} = \frac{(1.783 \times 10^{-9})(6.022 \times 10^{23})}{3.016} = 3.560 \times 10^{14} \text{ Bq/g} = 356 \text{ TBq/g}$$

In curies: $a = 356 \times 10^{12} / (3.7 \times 10^{10}) = 9620 \text{ Ci/g}$. This extraordinarily high specific activity is why tritium requires careful handling — a tiny mass represents enormous radioactivity.

Example 12.4: Secular Equilibrium

A sealed sample of $^{226}$Ra ($t_{1/2} = 1600$ yr) has an activity of 1.00 MBq. What is the activity of $^{222}$Rn ($t_{1/2} = 3.823$ d) in the sample after it has been sealed for 30 days?

Solution:

Since $t_{1/2,\text{Ra}} \gg t_{1/2,\text{Rn}}$, secular equilibrium applies. The approach to equilibrium is:

$$A_{\text{Rn}}(t) = A_{\text{Ra}} \left(1 - e^{-\lambda_{\text{Rn}} t}\right)$$

$$\lambda_{\text{Rn}} = \frac{\ln 2}{3.823 \text{ d}} = 0.1814 \text{ d}^{-1}$$

$$A_{\text{Rn}}(30 \text{ d}) = 1.00 \text{ MBq} \times \left(1 - e^{-0.1814 \times 30}\right) = 1.00 \times (1 - e^{-5.44})$$

$$A_{\text{Rn}}(30 \text{ d}) = 1.00 \times (1 - 0.00435) = 0.996 \text{ MBq}$$

After 30 days ($\approx 7.8$ Rn half-lives), the radon is at 99.6% of secular equilibrium.

Example 12.5: Uranium-Lead Age of a Zircon Crystal

A zircon crystal extracted from a granite contains $1.200 \times 10^{-4}$ mol of ${}^{238}\text{U}$ and $4.73 \times 10^{-6}$ mol of radiogenic ${}^{206}\text{Pb}$ (corrected for common lead). Assuming the system has remained closed, calculate the crystallization age.

Solution:

The age equation for ${}^{238}\text{U} \to {}^{206}\text{Pb}$ is:

$$\frac{{}^{206}\text{Pb}}{{}^{238}\text{U}} = e^{\lambda_{238} t} - 1$$

The measured ratio is:

$$\frac{{}^{206}\text{Pb}}{{}^{238}\text{U}} = \frac{4.73 \times 10^{-6}}{1.200 \times 10^{-4}} = 0.03942$$

Solving for $t$:

$$e^{\lambda_{238} t} = 1.03942$$

$$\lambda_{238} t = \ln(1.03942) = 0.03866$$

$$t = \frac{0.03866}{\lambda_{238}} = \frac{0.03866}{1.551 \times 10^{-10} \text{ yr}^{-1}} = 2.49 \times 10^8 \text{ yr}$$

The zircon crystallized approximately 249 million years ago (Triassic period). The concordia method would require the corresponding ${}^{235}\text{U}/{}^{207}\text{Pb}$ measurement to verify concordance.

Example 12.6: Potassium-Argon Dating of a Basalt

A basalt sample from a volcanic eruption contains $0.500\,\text{wt\%}$ K (potassium). Mass spectrometry measures $1.78 \times 10^{-10}$ mol of radiogenic ${}^{40}\text{Ar}$ per gram of rock. The natural abundance of ${}^{40}\text{K}$ is 0.01167% of total K. Calculate the age.

Solution:

First, the amount of ${}^{40}\text{K}$ per gram of rock:

$$n_{{}^{40}\text{K}} = \frac{0.00500 \times 0.0001167}{39.10 \text{ g/mol}} = 1.492 \times 10^{-8} \text{ mol/g}$$

Using the K-Ar age equation with the electron capture branching ratio:

$${}^{40}\text{Ar}^* = \frac{\lambda_{\text{EC}}}{\lambda_{\text{total}}} \cdot {}^{40}\text{K} \cdot (e^{\lambda_{\text{total}} t} - 1)$$

where $\lambda_{\text{EC}}/\lambda_{\text{total}} = 0.1072$ and $\lambda_{\text{total}} = 5.543 \times 10^{-10}\,\text{yr}^{-1}$:

$$\frac{{}^{40}\text{Ar}^*}{{}^{40}\text{K}} = \frac{1.78 \times 10^{-10}}{1.492 \times 10^{-8}} = 0.01193$$

$$e^{\lambda_{\text{total}} t} - 1 = \frac{0.01193}{0.1072} = 0.1113$$

$$\lambda_{\text{total}} t = \ln(1.1113) = 0.1054$$

$$t = \frac{0.1054}{5.543 \times 10^{-10}} = 1.90 \times 10^8 \text{ yr}$$

The basalt erupted approximately 190 million years ago (Early Jurassic).

Example 12.7: How Many Atoms in a "Dead" Source?

A laboratory has an old ${}^{60}\text{Co}$ source ($t_{1/2} = 5.271$ yr) that was originally 1.00 mCi when manufactured 30 years ago. What is its current activity?

Solution:

$$t/t_{1/2} = 30/5.271 = 5.69 \text{ half-lives}$$

$$A(t) = A_0 \cdot 2^{-t/t_{1/2}} = 1.00 \text{ mCi} \times 2^{-5.69} = 1.00 \times 0.0194 = 19.4 \text{ }\mu\text{Ci} = 0.718 \text{ MBq}$$

The source retains about 2% of its original activity. It is no longer useful for most applications (industrial radiography typically requires $> 10$ Ci sources), but at 0.718 MBq it still requires proper radiation safety handling.

12.10 Summary

This chapter established the mathematical foundations of radioactive decay:

1. The decay law $N(t) = N_0 e^{-\lambda t}$ follows from the single assumption that each nucleus has a constant probability $\lambda \, dt$ of decaying per unit time. This is a fundamentally quantum mechanical statement — no classical deterministic clock exists inside the nucleus.

2. Activity $A = \lambda N$ measures the observable decay rate. The SI unit is the becquerel (1 Bq = 1 decay/s); the historical unit is the curie (1 Ci = $3.7 \times 10^{10}$ Bq).

3. Half-life $t_{1/2} = \ln 2 / \lambda$ and mean life $\tau = 1/\lambda$ are related by $\tau = t_{1/2}/\ln 2 \approx 1.443 \, t_{1/2}$.

4. Decay chains are described by the Bateman equations. The long-term behavior depends on the ratio of parent-to-daughter decay constants: - Secular equilibrium ($\lambda_1 \ll \lambda_2$): $A_{\text{daughter}} = A_{\text{parent}}$ - Transient equilibrium ($\lambda_1 < \lambda_2$): $A_{\text{daughter}} > A_{\text{parent}}$, both decay at the parent's rate - No equilibrium ($\lambda_1 > \lambda_2$): parent vanishes, daughter decays independently

5. Radiometric dating exploits the predictability of nuclear decay to measure time. Different parent-daughter pairs cover different time ranges: $^{14}$C for $10^2$-$10^4$ yr, K-Ar for $10^5$-$10^9$ yr, U-Pb for $10^6$-$10^{10}$ yr.

6. Four natural radioactive series (classified by $A \bmod 4$) connect heavy unstable nuclei to stable lead (or bismuth) through chains of $\alpha$ and $\beta^-$ decays. The neptunium series ($4n+1$) is extinct due to its relatively short parent half-life.

In the next chapter, we will examine the quantum mechanics of $\alpha$ decay, where the Gamow theory of barrier tunneling explains the extraordinary range of $\alpha$-decay half-lives through the Geiger-Nuttall relation.

Key Equations Summary