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Accelerator mass spectrometry uses ion acceleration and particle identification to count rare isotopes relative to abundant ones. For radiocarbon dating, it measures carbon-14 directly rather than waiting to count its beta decays.

Carbon-14 has six protons and eight neutrons. Atmospheric production is dominated by ${}^{14}\mathrm{N}(n,p){}^{14}\mathrm{C}$ using secondary cosmic ray neutrons. It undergoes beta-minus decay to nitrogen-14, with a physical half-life of approximately $5730$ years.

Dating principle. Atmospheric cosmic rays produce secondary neutrons, which can generate carbon-14 through ${}^{14}\mathrm{N}(n,p){}^{14}\mathrm{C}$. Carbon then enters atmospheric carbon dioxide and living organisms. While an organism exchanges carbon with its environment, its carbon-14 to carbon-12 ratio approximately tracks that reservoir. After exchange stops, surviving carbon-14 undergoes beta-minus decay, whereas the stable carbon remains. On the closed-sample, constant-initial-ratio model, radioactive decay givesThus radiocarbon dating infers elapsed time from the loss of an isotope ratio or of radioactive activity per fixed carbon mass. It dates cessation of carbon exchange, which need not equal the date an object was manufactured or buried. Actual calendar dating also requires radiocarbon calibration and control of contamination and reservoir differences.

Mean lifetime. For exponential decay, the half-life obeys $e^{-\lambda T_{1/2}}=1/2$, while the mean lifetime is $\tau =1/\lambda$. HenceUsing $1\mathrm{yr}=365.25\times 86400\mathrm{s}$ gives $\lambda =3.83325\times 10^{-12}{\mathrm{s}}^{-1}$ and $\tau =2.60875\times 10^{11}\mathrm{s}$.

The mass minimum and beta stability. Treat the supplied masses as nuclear rest energies in $\mathrm{MeV}$, and omit nuclear pairing as instructed. At fixed mass number $A$, the volume and surface terms of the semi-empirical mass formula are constant. Expanding the remaining terms gives a convex nuclear mass parabola,The continuous minimum in atomic number is thereforeAt $A=14$, the quadratic coefficient is $7.01302\mathrm{MeV}$ and $Z_{\ast }=6.79450$. The nearest integer is $Z=7$; the nuclear isobar of minimum mass is nitrogen-14. Maximizing nuclear binding energy alone would omit the neutron-proton rest-mass difference; its continuous optimum is slightly different, but also selects integer $Z=7$ here.

The values most useful for checking neighboring decay thresholds areConsequently carbon-14 can lower its nuclear mass by changing a neutron into a proton:For bare nuclear masses, the Q value must include the emitted electron mass. Taking $m_e{c}^2=0.511\mathrm{MeV}$,The opposite proton-to-neutron conversion would lead to $Z=5$, whose nuclear mass is $18.1567\mathrm{MeV}$ higher than that of carbon-14. Neither beta-plus decay nor electron capture can bridge that gap by supplying an electron-mass correction. For the nitrogen-14 daughter, both neighboring nuclear masses are higher: $Z=6$ by $4.13070\mathrm{MeV}$ and $Z=8$ by $9.89535\mathrm{MeV}$. Even electron capture to $Z=6$ has $Q=0.511-4.13070<0$. Thus the daughter has no further energetically allowed single beta decay in this model; physical nitrogen-14 is stable.

These threshold numbers belong to the supplied smooth semi-empirical mass formula, not to an accurate prediction of the carbon-14 electron spectrum or its long half-life. Semi-empirical mass errors can change beta-decay thresholds. Also, the instruction to omit nuclear pairing is a model simplification: even $A=14$ does not make physical pairing vanish, and even-even carbon and odd-odd nitrogen need not have the same pairing correction. No omitted pairing coefficient is guessed here.

Activity of modern carbon. Neglect the small contribution of other stable carbon isotopes when counting atoms in a gram. The Avogadro constant givesThe expected radioactive activity of that one gram isor approximately $15.0$ decays per minute per gram of carbon. A becquerel is one decay per second; the normalization is per gram of carbon, rather than per gram of the entire organic sample.

Artefact age. Its remaining activity fraction is $0.13/0.25008=0.51983$. Inverting the exponential decay law,This is the radiocarbon age on the supplied constant-initial-activity, $5730$-year half-life model. It is not a calibrated calendar date; the sample's historical initial ratio and any later carbon exchange are not supplied.

Radiocarbon dating measures how long carbon-bearing material has been isolated from a carbon reservoir. Atmospheric cosmic rays produce secondary neutrons, which create carbon-14 chiefly throughThe carbon-14 enters atmospheric carbon dioxide and then living organisms through carbon exchange, including photosynthesis and food consumption. While exchange continues, the organism's ${}^{14}\mathrm{C}/{}^{12}\mathrm{C}$ ratio tracks the relevant environmental ratio approximately.

After exchange ceases, the remaining carbon-14 undergoes beta-minus decay,while stable carbon-12 remains. With half-life $T_{1/2}\simeq 5730\mathrm{yr}$, the radioactive decay law givesThus measuring the isotope ratio $R$, or the radioactive activity per fixed amount of carbon, givesHere $a=\lambda N_{14}$ is compared at equal carbon content; $R_0,a_0$ describe the appropriate initial reservoir. Accelerator mass spectrometry counts the rare carbon-14 relative to stable carbon directly; a decay-counting method instead measures its radioactive activity.

The initial ratio is not perfectly constant. Atmospheric production changes and reservoir differences therefore require radiocarbon calibration, often using independently dated tree rings. Contamination and subsequent carbon exchange must also be controlled. The date measures closure of the carbon system, not automatically the date of every later event involving the object. The physical half-life is about $5730$ years; conventional radiocarbon ages historically use $5568$ years and must be calibrated before interpretation as calendar dates.^[ref]

A radiocarbon age estimates elapsed time from a measured carbon-14 ratio $R$, an assumed initial ratio $R_0$ and a specified nuclear decay constant $\lambda$. The displayed exponential decay inversion gives the age on that model's timescale. A conventional radiocarbon age uses a prescribed reference and decay convention; it is not generally a calendar age. Radiocarbon calibration accounts for changes in the atmospheric isotope ratio and maps the measurement to a calendar-age estimate.

Radiocarbon calibration converts conventional radiocarbon ages into calendar-age information using independently dated material and models of changes in reservoir isotope ratios. Conventional ages use the historical $5568$-year half-life, whereas the physical carbon-14 half-life is about $5730$ years.^[ref]

Radiocarbon dating uses the loss of carbon-14 relative to stable carbon after carbon exchange stops. It can measure either an isotope ratio by accelerator mass spectrometry or radioactive activity per fixed carbon content. Interpreting an age requires radiocarbon calibration and control of contamination and reservoir differences.

A radionuclide is a species of atomic nucleus with an unstable nuclear state that undergoes radioactive decay. A sample's expected surviving population decreases with its nuclear decay constant. Carbon-14 is the radioisotope used in radiocarbon dating.