Radioactivity, Decay Kinetics & Radiometric Dating

1. The Segrè Chart of Nuclides ($N$ vs. $Z$)

Nuclear stability is dictated by the delicate quantum balance between attractive short-range nucleon-nucleon strong forces and long-range disruptive Coulomb electrostatic repulsion between protons. Plotting neutron number $N$ against atomic number $Z$ for all known nuclides yields the Segrè Chart:

• Light Stable Nuclei ($Z \le 20$): Follow the $N = Z$ line of exact symmetry ($N/Z = 1.0$), as in $^4_2\text{He}, {^{12}_6\text{C}}, {^{16}_8\text{O}}, {^{40}_{20}\text{Ca}}$.
• Heavy Stable Nuclei ($Z > 20$): Curve progressively toward the neutron axis to provide additional strong-force binding without adding disruptive Coulomb charges, reaching $N/Z \approx 1.54$ at $^{208}_{82}\text{Pb}_{126}$.
• Proton Drip Line & Neutron Drip Line: Boundaries beyond which proton or neutron separation energies vanish ($S_p \le 0$ or $S_n \le 0$); nucleons drip spontaneously from the nucleus on strong-interaction timescales ($\sim 10^{-21}\text{ s}$).
• Upper Limit of Natural Stability: Terminated at Bismuth-209 ($Z = 83$). All heavier elements ($Z \ge 84$) are naturally radioactive.

2. The Fundamental Radioactive Decay Law

Radioactive decay is an intrinsically stochastic, memoryless quantum process governed by the uncertainty principle. For a population of $N(t)$ identical, unstable radioactive parent nuclei at time $t$, the probability of any given nucleus disintegrating per unit time is a fundamental constant, the decay constant $\lambda$ ($\text{s}^{-1}$):

Integrating with the initial condition $N(0) = N_0$ at $t = 0$ yields the exponential Rutherford-Soddy Radioactive Decay Law:

3. Half-Life $T_{1/2}$ and Mean Lifetime $\tau$

• Half-Life ($T_{1/2}$): The elapsed time required for exactly one-half of the initial radioactive nuclei to decay ($N(T_{1/2}) = N_0 / 2$):
  $$\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}} \implies T_{1/2} = \frac{\ln(2)}{\lambda} = \frac{0.693147}{\lambda}$$
• Mean Lifetime ($\tau$): The statistical average lifetime of a radioactive nucleus:
  $$\tau = \langle t \rangle = \frac{\int_0^\infty t \left( \lambda N_0 e^{-\lambda t} \right) dt}{N_0} = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln(2)} \approx 1.4427 T_{1/2}$$

1. Radioactivity and Decay Rate

The activity $A(t)$ of a radioactive sample is defined as the number of disintegrations occurring per unit time:

2. Units of Radioactivity

• Becquerel ($\text{Bq}$): The SI unit of activity, defined as precisely one nuclear disintegration per second:
  $$1\text{ Bq} \equiv 1\text{ disintegration/s}$$
• Curie ($\text{Ci}$): The historical unit, historically based on the activity of 1 gram of pure Radium-226:
  $$1\text{ Ci} \equiv 3.700 \times 10^{10}\text{ Bq} = 37\text{ GBq}$$
• Specific Activity ($a$): Activity per unit mass of the pure radioisotope:
  $$a_{\text{spec}} = \frac{A}{m} = \frac{\lambda N_A}{M} = \frac{\ln(2) N_A}{T_{1/2} M}$$

3. Counting Statistics: Poisson and Gaussian Distributions

Because nuclear disintegrations are independent random events occurring with small probability $p \ll 1$ in a large population $N \gg 1$, the probability of recording exactly $n$ counts in a time interval $\Delta t$ follows the Poisson distribution:

where $\mu = \langle n \rangle$ is the mean count. The variance equals the mean: $\sigma^2 = \mu$. The standard deviation in any single nuclear counting measurement of $N_{\text{counts}}$ is:

The fractional statistical counting uncertainty is $\frac{\sigma}{N} = \frac{1}{\sqrt{N_{\text{counts}}}}$. To achieve a precision of $1\%$, one must accumulate at least $N_{\text{counts}} = (1/0.01)^2 = 10,000$ counts.

1. Multi-Step Radioactive Decay Series

In nature, heavy radioactive isotopes decay through successive chains (e.g., the Uranium-238, Thorium-232, and Actinium-235 series). Consider a general radioactive cascade:

The system of coupled linear differential equations governing the populations is:

2. Derivation of the Daughter Population $N_2(t)$

Assuming initially pure parent at $t = 0$ ($N_1(0) = N_0$ and $N_2(0) = 0$):

Multiplying by the integrating factor $e^{\lambda_2 t}$:

Applying initial condition $N_2(0) = 0 \implies C = - \frac{\lambda_1 N_0}{\lambda_2 - \lambda_1}$ yields:

The daughter activity is $A_2(t) = \lambda_2 N_2(t) = \frac{\lambda_1 \lambda_2 N_0}{\lambda_2 - \lambda_1} (e^{-\lambda_1 t} - e^{-\lambda_2 t})$.

3. General Bateman Solution for the $n$-th Isotope

Harry Bateman (1910) solved the general $n$-step system in closed analytical form:

1. Classification of Radioactive Equilibria

The physical behavior of a parent-daughter system depends on the ratio of parent half-life $T_{1/2, 1}$ to daughter half-life $T_{1/2, 2}$ (or decay constants $\lambda_1$ vs. $\lambda_2$):

2. Secular Equilibrium ($\lambda_1 \ll \lambda_2$ or $T_{1/2, 1} \gg T_{1/2, 2}$)

When the parent is extremely long-lived compared to the daughter (e.g., $^{226}\text{Ra}$ with $T_{1/2} = 1600\text{ y}$ decaying to $^{222}\text{Rn}$ with $T_{1/2} = 3.82\text{ d}$, or $^{238}\text{U}$ with $4.47 \times 10^9\text{ y}$):

• $\lambda_2 - \lambda_1 \approx \lambda_2$ and $e^{-\lambda_1 t} \approx 1$.
• For times $t \gg \tau_2 = 1/\lambda_2$, the transient exponential vanishes ($e^{-\lambda_2 t} \to 0$):

In secular equilibrium, the activity of the daughter equals the activity of the parent. Across an entire undisturbed natural decay series:

3. Transient Equilibrium ($\lambda_1 < \lambda_2$, comparable orders of magnitude)

When the parent is longer-lived than the daughter, but parent decay is non-negligible (e.g., $^{99}\text{Mo}$ with $T_{1/2} = 66\text{ h}$ decaying to $^{99m}\text{Tc}$ with $T_{1/2} = 6.0\text{ h}$, widely used in nuclear medicine generators):

For $t \gg 1/\lambda_2$, $e^{-\lambda_2 t} \ll e^{-\lambda_1 t}$, yielding:

In transient equilibrium, the daughter activity decays with the half-life of the parent, but exceeds the parent activity by the constant factor $\frac{\lambda_2}{\lambda_2 - \lambda_1}$.

4. Non-Equilibrium ($\lambda_1 > \lambda_2$)

If the parent is shorter-lived than the daughter, no equilibrium can ever be established. The parent decays rapidly, leaving the daughter to decay independently with its own characteristic decay constant $\lambda_2$.

1. Radiocarbon ($^{14}\text{C}$) Dating

Willard Libby (1949, Nobel Prize 1960) developed radiocarbon dating based on cosmic-ray production of $^{14}\text{C}$ in the upper atmosphere via neutron capture on nitrogen:

The radioactive $^{14}\text{C}$ ($T_{1/2} = 5730\pm40\text{ y}$, $\beta^-$ emitter) oxidizes to $^{14}\text{CO}_2$ and mixes into the biosphere via photosynthesis and the food chain, establishing an equilibrium specific activity in living organic tissue:

Upon death, biological carbon exchange terminates, and $^{14}\text{C}$ decays exponentially without replenishment: $A(t) = A_0 e^{-\lambda t}$. The age of the archaeological specimen is calculated as:

2. Potassium-Argon ($^{40}\text{K}$-$^{40}\text{Ar}$) Dating

Used for geological samples ($10^5$ to $4.5 \times 10^9$ years). Natural Potassium contains $0.0117\%$ $^{40}\text{K}$ ($T_{1/2} = 1.248 \times 10^9\text{ y}$), which exhibits branched decay:

• $89.3\%$ decays via $\beta^-$ to $^{40}\text{Ca}$.
• $10.7\%$ decays via electron capture to noble gas $^{40}\text{Ar}$.

When volcanic rock crystallizes from magma, molten lava outgasses all volatile Argon ($^{40}\text{Ar}_{\text{initial}} = 0$). Trapped radiogenic $^{40}\text{Ar}$ subsequently accumulates in the solid mineral lattice:

3. Uranium-Lead ($^{238}\text{U}$-$^{206}\text{Pb}$) Isochron Dating

In zircon crystals ($\text{ZrSiO}_4$), initial lead contamination is eliminated during crystallization. For mineral samples containing non-radiogenic common lead ($^{204}\text{Pb}$), measuring the isotopic ratios yields the isochron equation:

Plotting $^{206}\text{Pb}/^{204}\text{Pb}$ vs. $^{238}\text{U}/^{204}\text{Pb}$ for different co-genetic mineral grains yields a straight line whose slope $m = e^{\lambda_{238} t} - 1$ determines the geological age $t$ independently of the initial lead content.