The radiocarbon clock starts in the upper atmosphere. Cosmic-ray neutrons collide with atmospheric nitrogen-14 and convert it to carbon-14 by the reaction N-14 + n → C-14 + p. The resulting carbon-14 oxidises to CO₂, mixes through the atmosphere on a timescale of years, and enters the biosphere through photosynthesis. The steady-state ratio of carbon-14 to ordinary carbon-12 in the atmosphere is approximately 1.2 × 10^-12, which is the same ratio that ends up in every living plant and in every animal that eats the plant. When the organism dies, carbon uptake stops, the carbon-14 in its tissue decays with a half-life of 5,730 years, and the ratio drops as a clock.
The dating equation is straightforward. The age t since death is t = (t1/2 / ln 2) × ln(N0/N), where N0 is the carbon-14 abundance at death (taken as the modern reference value) and N is the carbon-14 abundance now. For a 5,000-year-old sample, the ratio has dropped to about 55 percent of modern; for a 25,000-year-old sample, to about 5 percent; for a 50,000-year-old sample, to about 0.2 percent, which is where the radiocarbon clock runs out against detector background.
Three measurement methods historically competed. Conventional liquid scintillation counting needs a gram or more of carbon, converts the sample to benzene through a wet-chemistry chain, and counts the carbon-14 beta decays in a low-background scintillation counter for several days to several weeks. Gas proportional counting needs similar sample sizes and converts the carbon to methane or CO₂ for counting. Both are slow, both are insensitive, both struggle below 30,000 years.
Accelerator mass spectrometry (AMS) is the modern method and the only one that matters for forensic casework. The sample is graphitised (typically combusted to CO₂, then reduced to elemental graphite over an iron catalyst) and pressed into an aluminium target holder. A caesium sputter ion source generates negative carbon ions from the graphite; a low-energy mass spectrometer selects mass 14; a tandem accelerator stripping foil destroys molecular interferences (especially the troublesome N-14 hydride and C-13 hydride); a high-energy mass spectrometer separates the surviving atomic carbon-14 ions; and a gas-ionisation or silicon-strip detector counts them individually. The carbon-12 and carbon-13 ions are measured in Faraday cups in the same run for the ratio. AMS needs only about a milligram of carbon, completes a measurement in tens of minutes (plus sample preparation), and extends the practical dating range to about 50,000 years.