Half Life Explained: How Radioactive Decay Is Measured

Understanding Half-Life

Radioactive decay is a statistical process. You cannot predict exactly when any individual atom will decay, but you can predict with extraordinary precision what fraction of a large sample will decay in a given time. If you start with 1,000 atoms of a radioactive isotope with a half-life of one hour, after one hour approximately 500 will remain unchanged (the other 500 having decayed into daughter atoms). After two hours, about 250 remain. After three hours, about 125. Each successive half-life reduces the remaining quantity by half, regardless of how many atoms you started with.

Mathematically, the number of remaining atoms N at time t follows an exponential decay curve: N(t) = N0 x (1/2)^(t/t_half), where N0 is the initial number of atoms and t_half is the half-life. This can also be expressed using the decay constant (lambda): N(t) = N0 x e^(-lambda x t), where lambda = ln(2) / t_half = 0.693 / t_half. The decay constant represents the probability per unit time that any given atom will decay. For carbon-14 with its 5,730-year half-life, lambda equals about 1.21 x 10^-4 per year, meaning each C-14 atom has about a 0.012% chance of decaying in any given year.

Activity, measured in becquerels (Bq, one decay per second) or curies (Ci, 3.7 x 10^10 decays per second), describes the rate at which a sample is decaying. Activity is proportional to the number of remaining radioactive atoms: A = lambda x N. As atoms decay and N decreases, the activity decreases at the same exponential rate. A freshly produced medical isotope is highly active (decaying rapidly), while aged nuclear waste has much lower activity (most of the short-lived isotopes have already decayed away).

The Remarkable Range of Half-Lives

Half-lives span at least 50 orders of magnitude across known isotopes. At the shortest extreme, highly unstable isotopes created in particle accelerators exist for mere fractions of a second. Lithium-4 has a half-life of about 9.1 x 10^-23 seconds, barely enough time for the nucleus to vibrate once. Beryllium-8 lasts only 8.2 x 10^-17 seconds, a fact that creates a critical bottleneck in stellar nucleosynthesis (the triple-alpha process must combine three helium nuclei almost simultaneously to bypass unstable Be-8 and produce carbon-12).

Common isotopes span a more moderate but still impressive range. Iodine-131 (half-life 8 days) is used in thyroid treatment because it delivers radiation and then decays away within weeks. Cobalt-60 (5.27 years) is used in cancer radiotherapy and industrial radiography. Strontium-90 (28.8 years) and cesium-137 (30.2 years) are concerning fission products because their half-lives are long enough to persist for decades but short enough to remain highly active. Carbon-14 (5,730 years) enables radiocarbon dating of organic materials up to about 50,000 years old. Uranium-238 (4.47 billion years) has a half-life comparable to the age of the Earth, which is why it still exists in significant quantities in Earth's crust despite being radioactive.

At the extreme long end, tellurium-128 holds the record for the longest directly measured half-life: 2.2 x 10^24 years, roughly 160 trillion times the age of the universe. Despite this inconceivably slow decay rate, scientists confirmed it by carefully measuring the tiny number of decays occurring in a large sample over an extended observation period. Theoretical predictions suggest some nuclei might have half-lives exceeding 10^40 years, far beyond any practical measurement capability.

Half-Life Cannot Be Changed

A remarkable property of radioactive decay is its insensitivity to external conditions. Unlike chemical reaction rates, which can be sped up by heating or adding catalysts, nuclear decay rates are determined entirely by the internal quantum mechanical properties of the nucleus. No amount of heating, freezing, compressing, dissolving in acid, or bombarding with electromagnetic radiation will significantly alter a nuclear half-life. This immutability is what makes radioactive decay such a reliable clock for geological and archaeological dating.

There are a few exotic exceptions that prove the rule. Electron capture decay (where the nucleus absorbs an inner-shell electron) can be affected by extreme ionization: if all electrons are stripped from the atom, electron capture becomes impossible. The isotope beryllium-7, which normally decays by electron capture with a half-life of 53 days, becomes stable if fully ionized. Similarly, bound-state beta decay (where the emitted electron goes directly into an atomic orbital rather than escaping) can occur in highly ionized atoms that would be stable as neutral atoms. These effects are relevant in stellar interiors and heavy-ion physics but have no practical impact under normal conditions on Earth.

Applications of Half-Life

Radiocarbon dating exploits the known half-life of carbon-14 to determine the age of organic materials. Living organisms continuously exchange carbon with the atmosphere, maintaining a constant C-14/C-12 ratio. When an organism dies, this exchange stops and C-14 begins to decrease through decay. By measuring the remaining C-14 fraction and applying the known half-life (5,730 years), scientists can calculate when the organism died. This technique reliably dates materials up to about 50,000 years old (roughly 9 half-lives, by which point less than 0.2% of the original C-14 remains).

Geological dating uses longer-lived isotopes to date rocks and minerals spanning millions to billions of years. The uranium-lead method measures both U-238 to Pb-206 (half-life 4.47 billion years) and U-235 to Pb-207 (half-life 704 million years), providing two independent clocks in the same sample. Potassium-argon dating (K-40, half-life 1.25 billion years) is used for volcanic rocks. Rubidium-strontium dating (Rb-87, half-life 48.8 billion years) works for ancient igneous and metamorphic rocks. These methods collectively provide the absolute timescale for Earth history, confirming our planet's age at approximately 4.54 billion years.

Nuclear medicine relies on isotopes whose half-lives match clinical needs. Technetium-99m (half-life 6 hours) is ideal for diagnostic imaging: active long enough to complete a scan, but decaying rapidly enough that patient radiation exposure is minimal. Fluorine-18 (half-life 110 minutes) is used in PET scans because its positron emission and short half-life enable high-resolution imaging with limited dose. Therapeutic isotopes like iodine-131 (8 days) deliver sustained radiation to tumors over a treatment-relevant timescale before decaying to negligible levels.

Nuclear waste management must account for half-lives spanning many orders of magnitude. Short-lived fission products (cesium-137, strontium-90) dominate waste radioactivity for the first few hundred years. Long-lived actinides (plutonium-239 with a 24,100-year half-life, neptunium-237 with 2.14 million years) determine the timescale for geological isolation. The goal of advanced fuel cycles and transmutation research is to convert long-lived waste into shorter-lived or stable isotopes, reducing the required isolation period from hundreds of thousands of years to a few hundred years.

Decay Chains and Secular Equilibrium

Many radioactive isotopes do not decay directly to a stable product but instead produce another radioactive isotope, which decays again, forming a decay chain. The uranium-238 decay chain passes through 14 intermediate radioactive isotopes (including radium-226 and radon-222) before reaching stable lead-206. Each isotope in the chain has its own characteristic half-life, ranging from microseconds (polonium-214) to billions of years (uranium-238). When a long-lived parent isotope feeds a shorter-lived daughter, secular equilibrium develops over time: the daughter's activity becomes equal to the parent's activity, and both appear to decay at the parent's half-life rate. This principle is exploited in technetium-99m generators (where 66-hour molybdenum-99 feeds 6-hour technetium-99m) and in geological dating techniques where ratios of parent and daughter isotopes reveal the time elapsed since a rock crystallized.

The concept of activity (measured in becquerels, where 1 Bq equals one decay per second, or curies, where 1 Ci equals 37 billion decays per second) quantifies how rapidly a radioactive sample is decaying. Activity depends on both the number of atoms present and the decay constant: A = lambda x N = (ln2 / half-life) x N. A short-lived isotope has high specific activity (activity per gram) while a long-lived isotope has low specific activity. One gram of radium-226 (half-life 1,600 years) has an activity of 37 billion becquerels (1 curie, which is how the unit was historically defined). One gram of uranium-238 (half-life 4.5 billion years) has only 12,400 becquerels, making it nearly three million times less radioactive per gram despite being the same mass of material.