Nuclear and radiochemistry

Radiometric Dating

How radioactive decay constants and parent-daughter isotope ratios provide geochronological clocks, from radiocarbon in archaeology to isochrons and the age of the Earth.

IntuitionIntuition: radioactive hourglasses frozen in stone

An hourglass measures elapsed time because sand grains trickle from the upper chamber to the lower chamber at an invariant speed. In the natural world, radioactive nuclei act as atomic hourglasses: unstable parent atoms transform into radiogenic daughter atoms governed strictly by quantum nuclear probabilities, completely indifferent to ambient temperature, burial pressure, or chemical bonding. When a magma cools into solid mineral crystals or an organism ceases metabolic respiration, the hourglass is turned: parent atoms are locked into crystal lattices, and the steady accumulation of daughter products records deep time.

SchoolSchool level: the fundamental age equation and radiocarbon dating

Radiocarbon dating exploits cosmic-ray spallation in the upper atmosphere, where thermal neutrons react with nitrogen: $\ce{^{14}_7N + ^1_0n -> ^{14}_6C + ^1_1p}$. The produced carbon-14 is oxidized to $\ce{^{14}CO2}$ and assimilated via photosynthesis into the biosphere, maintaining a steady-state ratio $\ce{^{14}C/^{12}C} \approx 1.2 \times 10^{-12}$ in living organisms. Upon death, metabolic assimilation ceases, and $\ce{^{14}C}$ decays back to $\ce{^{14}N}$ by $\beta^-$ emission with a half-life of $t_{1/2} = 5730\pm 40\text{ years}$ (Cambridge half-life). By convention, raw ages are reported in 'radiocarbon years BP' (relative to 1950 CE) using the Libby half-life ($5568\text{ years}$). Natural fluctuations in solar activity and geomagnetic shielding require calibration curves like IntCal20 based on tree rings and varves. Modern Accelerator Mass Spectrometry (AMS) counts individual atoms directly from sub-milligram samples, extending the limit to $\sim 50,000\text{ years}$.

UndergraduateUndergraduate: branched K-Ar decay, 40Ar/39Ar, and Rb-Sr isochrons

Potassium-40 undergoes branched decay: electron capture to noble gas $\ce{^{40}Ar}$ ($10.72\%$, $\lambda_e = 0.580 \times 10^{-10}\ \text{y}^{-1}$) and $\beta^-$ emission to $\ce{^{40}Ca}$ ($89.28\%$, $\lambda_\beta = 4.963 \times 10^{-10}\ \text{y}^{-1}$), yielding total decay constant $\lambda = 5.543 \times 10^{-10}\ \text{y}^{-1}$ ($t_{1/2} = 1.248\text{ Gyr}$). Molten magma completely outgasses volatile argon, resetting initial $\ce{^{40}Ar}_0 \approx 0$. Upon crystallization, radiogenic $\ce{^{40}Ar}^*$ is retained inside tight potassium lattices like sanidine or biotite. The modern $\ce{^{40}Ar/^{39}Ar}$ technique irradiates rock samples in a nuclear reactor to convert $\ce{^{39}K}(n,p)\ce{^{39}Ar}$. Stepwise laser heating in a noble gas mass spectrometer releases argon incrementally, generating an age spectrum plateau that detects argon loss or trapped excess argon with sub-per-mil precision.

When non-zero initial daughter atoms $D_0$ are present at crystallization, single-sample measurements cannot determine the age. The isochron method overcomes this by normalizing to a non-radiogenic stable isotope of the daughter element, such as $\ce{^{86}Sr}$ for rubidium-strontium dating ($\ce{^{87}Rb -> ^{87}Sr}$, $t_{1/2} = 48.8\text{ Gyr}$, $\lambda = 1.42 \times 10^{-11}\ \text{y}^{-1}$). Different minerals co-crystallizing from a single magma chamber incorporate different elemental $\ce{Rb/Sr}$ ratios due to crystal-chemical fractionation, but share identical initial isotopic ratios $(\ce{^{87}Sr/^{86}Sr})_0$. Plotting $\ce{^{87}Sr/^{86}Sr}$ against $\ce{^{87}Rb/^{86}Sr}$ yields a straight line whose slope is $m = e^{\lambda t} - 1 \approx \lambda t$, determining crystallization age $t$ without requiring prior knowledge of initial daughter concentrations.

AdvancedAdvanced: U-Pb concordia, Jack Hills zircons, and the age of the Earth

Zircon ($\ce{ZrSiO4}$) is the gold standard of geochronology. During magmatic crystallization, the zirconium site ($r = 84\text{ pm}$) admits $\ce{U^4+}$ ($r = 100\text{ pm}$) into the tetragonal orthosilicate lattice but categorically rejects $\ce{Pb^2+}$ ($r = 119\text{ pm}$, charge $+2$) by an exclusion factor exceeding $10^5$, establishing initial radiogenic daughter $\ce{Pb}_0 \approx 0$. Because zircon possesses two coupled radioactive decay systems with vastly different half-lives—$\ce{^{238}U -> ^{206}Pb}$ ($t_{1/2} = 4.468\text{ Gyr}$, $\lambda_{238} = 1.55125 \times 10^{-10}\ \text{y}^{-1}$) and $\ce{^{235}U -> ^{207}Pb}$ ($t_{1/2} = 0.7038\text{ Gyr}$, $\lambda_{235} = 9.8485 \times 10^{-10}\ \text{y}^{-1}$)—plotting $\ce{^{206}Pb}^*/\ce{^{238}U}$ versus $\ce{^{207}Pb}^*/\ce{^{235}U}$ generates the Wetherill concordia curve. Undisturbed crystals plot concordantly on the curve; samples that suffered episodic lead loss during metamorphism define a straight discordia line whose upper intercept gives the primary crystallization age and lower intercept dates the metamorphic event. Detrital zircons from the Jack Hills in Western Australia plot concordantly at $4.404\text{ Ga}$, preserving relics of the Hadean Earth.

Taking the ratio of the two radiogenic lead equations eliminates the need to measure absolute parent uranium concentrations: $\frac{\ce{^{207}Pb}^*}{\ce{^{206}Pb}^*} = \frac{\ce{^{235}U}}{\ce{^{238}U}}\left(\frac{e^{\lambda_{235} t} - 1}{e^{\lambda_{238} t} - 1}\right) = \frac{1}{137.88}\left(\frac{e^{\lambda_{235} t} - 1}{e^{\lambda_{238} t} - 1}\right)$, where $137.88$ is the modern terrestrial $\ce{^{238}U/^{235}U}$ ratio. In 1956, Clair Patterson determined the primordial lead composition of the solar system by analyzing the uranium-free troilite phase ($\ce{FeS}$) of the Canyon Diablo iron meteorite. By plotting meteorite lead alongside modern oceanic sediments on a common Pb-Pb isochron, Patterson calculated the age of the Earth to be $4.55 \pm 0.07\text{ Ga}$ ($4.54\text{ Ga}$). Modern high-precision U-Pb analyses of calcium-aluminium-rich inclusions (CAIs) in chondritic meteorites date the condensation of the solar protoplanetary disk to $4.567\text{ Ga}$.