One of the most frequent uses of radiocarbon dating is to estimate the age of organic remains from archaeological sites. When plants fix atmospheric carbon dioxide into organic material during photosynthesis they incorporate a quantity of 14C that approximately matches the level of this isotope in the atmosphere (a small difference occurs because of isotope fractionation, but this is corrected after laboratory analysis). After plants die or they are consumed by other organisms (for example, by humans or other animals) the 14C fraction of this organic material declines at a fixed exponential rate due to the radioactive decay of 14C. Comparing the remaining 14C fraction of a sample to that expected from atmospheric 14C allows the age of the sample to be estimated.
The technique of radiocarbon dating was developed by Willard Libby and his colleagues at the University of Chicago in 1949. Libby estimated that the steady state radioactivity concentration of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram. In 1960, he was awarded the Nobel Prize in chemistry for this work. He first demonstrated the accuracy of radiocarbon dating by accurately measuring the age of wood from an ancient Egyptian royal barge whose age was known from historical documents.
Carbon has two stable, nonradioactive isotopes: carbon-12 (12C), and carbon-13 (13C). In addition, there are trace amounts of the unstable isotope carbon-14 (14C) on Earth. Carbon-14 has a half-life of 5730 years and would have long ago vanished from Earth were it not for the unremitting cosmic ray impacts on nitrogen in the Earth's atmosphere, which create more of the isotope. The neutrons resulting from the cosmic ray interactions participate in the following nuclear reaction on the atoms of nitrogen molecules (N2) in the atmospheric air:
The highest rate of carbon-14 production takes place at altitudes of 9 to 15 km (30,000 to 50,000 ft), and at high geomagnetic latitudes, but the carbon-14 spreads evenly throughout the atmosphere and reacts with oxygen to form carbon dioxide. Carbon dioxide also permeates the oceans, dissolving in the water. For approximate analysis it is assumed that the cosmic ray flux is constant over long periods of time; thus carbon-14 is produced at a constant rate and the proportion of radioactive to non-radioactive carbon is constant: ca. 1 part per trillion (600 billion atoms/mole). In 1958 Hessel de Vries showed that the concentration of carbon-14 in the atmosphere varies with time and locality. For the most accurate work, these variations are compensated by means of calibration curves. When these curves are used, their accuracy and shape are the factors that determine the accuracy and age obtained for a given sample.
Plants take up atmospheric carbon dioxide by photosynthesis, and are ingested by animals, so every living thing is constantly exchanging carbon-14 with its environment as long as it lives. Once it dies, however, this exchange stops, and the amount of carbon-14 gradually decreases through radioactive beta decay.
The solution to this equation is:
Two related times can be defined:
It can be shown that:
Notice that dates are customarily given in years BP which implies t(BP) = -t because the time arrow for dates runs in reverse direction from the time arrow for the corresponding ages. From these considerations and the above equation, it results:
For a raw radiocarbon date:
After replacing values, the raw radiocarbon age becomes any of the following equivalent formulae:
using logs base e and the average life:
using logs base 2 and the half-life:
The sensitivity of the method has been greatly increased by the use of Accelerator Mass Spectrometry (AMS). With this technique 14C atoms can be detected and counted directly vs only detecting those atoms that decay during the time interval allotted for an analysis. AMS allows dating samples containing only a few milligrams of carbon.
Raw radiocarbon ages (i.e., those not calibrated) are usually reported in "years Before Present" (BP). This is the number of radiocarbon years before 1950, based on a nominal (and assumed constant - see "calibration" below) level of carbon-14 in the atmosphere equal to the 1950 level. These raw dates are also based on a slightly-off historic value for the radiocarbon half-life. Such value is used for consistency with earlier published dates (see "Radiocarbon half-life" below). See the section on computation for the basis of the calculations.
Radiocarbon dating laboratories generally report an uncertainty for each date. For example, 3000±30BP indicates a standard deviation of 30 radiocarbon years. Traditionally this included only the statistical counting uncertainty. However, some laboratories supplied an "error multiplier" that could be multiplied by the uncertainty to account for other sources of error in the measuring process. More recently, the laboratories try to quote the overall uncertainty, which is determined from control samples of known age and verified by international intercomparison exercises . In 2008, a typical uncertainty better than ±40 radiocarbon years can be expected for samples younger than 10,000 years. This, however, is only a small part of the uncertainty of the final age determination (see section Calibration below).
As of 2007, the limiting age for a 1 milligram sample of graphite is about ten half-lives, approximately 60,000 years. This age is derived from that of the calibration blanks used in an analysis, whose 14C content is assumed to be the result of contamination during processing (as a result of this, some facilities will not report an age greater than 60,000 years for any sample).
A variety of sample processing and instrument-based constraints have been postulated to explain the upper age-limit. To examine instrument-based background activities in the AMS instrument of the W. M. Keck Carbon Cycle Accelerator Mass Spectrometry Laboratory of the University of California, a set of natural diamonds were dated. Natural diamond samples from different sources within rock formations with standard geological ages in excess of 100 my yielded 14C apparent ages 64,920±430 BP to 80,000±1100 BP as reported in 2007.
A raw BP date cannot be used directly as a calendar date, because the level of atmospheric 14C has not been strictly constant during the span of time that can be radiocarbon dated. The level is affected by variations in the cosmic ray intensity which is in turn affected by variations in the earth's magnetosphere. In addition, there are substantial reservoirs of carbon in organic matter, the ocean, ocean sediments (see methane hydrate), and sedimentary rocks. Changes in the Earth's climate can affect the carbon flows between these reservoirs and the atmosphere, leading to changes in the atmosphere's 14C fraction.
Aside from these changes due to natural processes, the level has also been affected by human activities. From the beginning of the industrial revolution in the 18th century to the 1950s, the fractional level of 14C decreased because of the admixture of large quantities of CO2 into the atmosphere, the combustion production of fossil fuel. This decline is known as the Suess effect, and also affects the 13C isotope. However, atmospheric 14C was almost doubled for a short period during the 1950s and 1960s due to atomic bomb tests.
The calibration curves can vary significantly from a straight line, so comparison of uncalibrated radiocarbon dates (e.g., plotting them on a graph or subtracting dates to give elapsed time) is likely to give misleading results. There are also significant plateaus in the curves, such as the one from 11,000 to 10,000 radiocarbon years BP, which is believed to be associated with changing ocean circulation during the Younger Dryas period. Over the historical period from 0 to 10,000 years BP, the average width of the uncertainty of calibrated dates was found to be 335 years, although in well-behaved regions of the calibration curve the width decreased to about 113 years while in ill-behaved regions it increased to a maximum of 801 years. Significantly, in the ill-behaved regions of the calibration curve, increasing the precision of the measurements does not have a significant effect on increasing the accuracy of the dates.
The 2004 version of the calibration curve extends back quite accurately to 26,000 years BP. Any errors in the calibration curve do not contribute more than ±16 years to the measurement error during the historic and late prehistoric periods (0 - 6,000 yrs BP) and no more than ±163 years over the entire 26,000 years of the curve, although its shape can reduce the accuracy as mentioned above.
These effects were first confirmed when samples of wood from around the world, which all had the same age (based on tree ring analysis), showed deviations from the dendrochronological age. Calibration techniques based on tree-ring samples have contributed to increase the accuracy since 1962, when they were accurate to 700 years at worst.