Grade 12

Half-life and the radioactive decay law

The mathematics of nuclear survival: exponential decay, activity in becquerels, parent-daughter Bateman chains, and the transient equilibrium of technetium generators.

IntuitionIntuition: a clock built from pure probability

Imagine tossing ten thousand coins simultaneously every minute and discarding every coin landing on heads. In the first minute approximately half disappear, leaving five thousand. In the second minute, half of what remains vanishes, leaving twenty-five hundred. No individual coin can predict when its turn will come, and a coin that has survived ten minutes is no more likely to land on heads than a newly minted one. Radioactive nuclei behave with this identical memoryless randomness: each unstable nucleus possesses a constant, unalterable probability of decaying in any given second, entirely independent of its past age.

Because macroscopic matter contains astronomical ensembles of nuclei—a single microgram of an isotope contains trillions of atoms—this microscopic randomness averages out into an extraordinarily smooth, predictable exponential law. The number of surviving nuclei halves across every fixed time window called the half-life. Whether tracking an imaging tracer decaying over hours or uranium charting terrestrial geology across billions of years, the exponential decay law furnishes science with an absolute, unyielding atomic chronometer.

SchoolCore concepts: half-life, decay constant and activity

UndergraduateUniversity tier: Bateman equations and decay equilibria

When a radioactive parent 1 disintegrates into a radioactive daughter 2, the daughter inventory obeys Harry Bateman’s coupled differential equations. Depending on the ratio of the half-lives, two primary equilibria govern radioactive families: 1. Secular equilibrium arises when the parent is enormously longer-lived than the daughter ($\lambda_1 \ll \lambda_2$, such as $\ce{^{226}Ra}$ with $t_{1/2} = 1600\text{ y}$ decaying into $\ce{^{222}Rn}$ with $t_{1/2} = 3.82\text{ d}$). After several daughter half-lives, $\mathrm{e}^{-\lambda_2 t} \to 0$ and the daughter activity reaches exact parity with the parent: $A_2(t) \to A_1(t)$. 2. Transient equilibrium develops when the parent is longer-lived than the daughter but has a comparable lifespan ($\lambda_1 < \lambda_2$), exemplified by the $\ce{^{99}Mo} \to \ce{^{99m}Tc}$ medical generator ($t_{1/2} = 65.94\text{ h}$ vs $6.01\text{ h}$). Daughter activity surpasses parent activity and attains a distinct peak at $t_{\max} = \frac{\ln(\lambda_2 / \lambda_1)}{\lambda_2 - \lambda_1} \approx 22.8\text{ h}$ (approx. 23 hours), after which both isotopes decay synchronously governed by the parent half-life: $A_2(t) \to \frac{\lambda_2}{\lambda_2 - \lambda_1} A_1(t) \approx 1.10 \, A_1(t)$.

In radiopharmacology and environmental biophysics, administered radionuclides undergo simultaneous physical decay and biological clearance. The effective decay constant is the direct sum of physical and biological rates ($\lambda_{\text{eff}} = \lambda_{\text{phys}} + \lambda_{\text{bio}}$), yielding the reciprocal harmonic relation for effective half-life: $\frac{1}{T_{\text{eff}}} = \frac{1}{T_{\text{phys}}} + \frac{1}{T_{\text{bio}}}$. If a nuclide disintegrates via competing paths (branching decay with partial constants $\lambda_a$ and $\lambda_b$), the total decay constant is $\lambda = \lambda_a + \lambda_b$, and the branching ratio for mode $a$ is $f_a = \lambda_a / \lambda$.

AdvancedAdvanced physical depth: Poisson statistics and quantum tunneling

Because individual radioactive disintegrations are independent Poisson events occurring in enormous populations, counting $N$ disintegrations carries an irreducible statistical uncertainty with standard deviation $\sigma = \sqrt{N}$ and relative uncertainty $\sigma / N = 1/\sqrt{N}$. In low-background metrology or paleolithic radiocarbon dating, accumulating substantial total counts is the sole physical method to suppress counting noise. Moreover, nuclear decay constants span more than twenty orders of magnitude across the isotope chart. In 1928, George Gamow (along with Gurney and Condon) demonstrated that alpha half-lives originate from quantum mechanical tunneling: an alpha cluster strikes the nuclear Coulomb barrier roughly $10^{21}\text{ times per second}$, where the exponential barrier transmission probability $P \propto \exp[-2\pi\eta]$ explains the empirical Geiger-Nuttall law, linking tiny increases in alpha particle kinetic energy to enormous multi-order drops in half-life.