Radiocarbon dating
Radiocarbon dating is a radiometric dating method that uses the naturally occurring radioisotope carbon-14 (14C) to determine the age of carbonaceous materials up to about 60,000 years. Raw, i.e. uncalibrated, radiocarbon ages are usually reported in radiocarbon years "Before Present" (BP), "Present" being defined as AD 1950. Such raw ages can be calibrated to give calendar dates.
The technique of radiocarbon dating was discovered by Willard Libby and his colleagues in 1949 during his tenure as a professor at the University of Chicago. 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dating by accurately measuring the age of wood from an ancient Egyptian royal barge whose age was known from historical documents.
One of the frequent uses of the technique is to date organic remains from archaeological sites. Plants fix atmospheric carbon during photosynthesis, so the level of 14C in living plants and animals equals the level of 14C in the atmosphere.

# Basic physics
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.

# Computation of ages and dates
The radioactive decay of carbon-14 follows an exponential decay. 
A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value.  Symbolically, this can be expressed as the following differential equation, where N is the quantity and  is a positive number called the decay constant:
The solution to this equation is:
where, for a given sample of carbonaceous matter:
- mean- or average-life: mean or average time each radiocarbon atom spends in a given sample until it decays.
- half-life: time lapsed for half the number of radiocarbon atoms in a given sample, to decay,
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:
and for a raw radiocarbon age:
After replacing values, the raw radiocarbon age becomes any of the following equivalent formulae:
using logs base e and the average life:
and
using logs base 2 and the half-life:

# Measurements and scales
Measurements are traditionally made by counting the radioactive decay of individual carbon atoms by gas proportional counting or by liquid scintillation counting. For samples of sufficient size (several g carbon) this method is still widely used in the 2000s. Among others, all the tree ring,samples used for the calibration curves (see below), were determined by these counting techniques. Such decay counting, however, is relatively insensitive and subject to large statistical uncertainties for small samples. When there is little carbon-14 to begin with, the long radiocarbon half-life means that very few of the carbon-14 atoms will decay during the time allotted for their detection, resulting in few disintegrations per minute.
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-tfe" below). See the section on computation for the basis of the calculations.
Radiocarbon dating laboratories generally report an uncertainty for
each date. For example, 300030BP 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 uncertaintainty of the final age determination (see section Calibration below).
At present (2007), the limiting age is about ten half-lives, i.e. 60,000 years .
A variety of sample processing and instrument-based constraints have been postulated for this limit. To examine instrument-based backgrounds in the University of California Keck Carbon Cycle AMS spectrometer, measurements were performed on a set of natural diamonds. Natural diamond samples from different sources within rock formations with standard geological ages in excess of 100 my yielded raw 14C ages 64,920430 BP to 80,0001100 BP as reported in 2007. In contrast to the sample processing and instrument-based background theories, the authors of an AMS instrument background study conclude: "14C from the actual sample is probably the dominant component of the 'routine' background." .

## Calibration

### The need for calibration
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 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. Changing climate can sometimes disrupt the carbon flow between these reservoirs and the atmosphere. The level has also been affected by human activitiesit was almost doubled for a short period due to atomic bomb tests in the 1950s and 1960s and has been lowered by the admixture of large amounts of CO2 from ancient organic sources relatively depleted in 14C the combustion products of fossil fuels used in industry and transportation, known as the Suess effect.
The raw radiocarbon dates, in BP years, are calibrated to give calendar dates. Standard calibration curves are available, based on comparison of radiocarbon dates of samples that can be dated independently by other methods such as examination of tree growth rings (dendrochronology), deep ocean sediment cores, lake sediment varves, coral samples, and speleothems (cave deposits).
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.

# Radiocarbon half-life

## Libby vs Cambridge values
Carbon dating was developed by a team led by Willard Libby.  Originally a carbon-14 half-life of 556830 years was used, which is now known as the Libby half-life.  Later a more accurate figure of 573040 years was determined, which is known as the Cambridge half-life. This is, however, not relevant for radiocarbon dating. If calibration is applied, the half-life cancels out, as long as the same value is used throughout the calculations. Laboratories continue to use the Libby figure to avoid inconsistencies with previous publications.

# Carbon exchange reservoir
Libby's original exchange reservoir hypothesis assumes that the exchange reservoir is constant all over the world. The calibration method also assumes that the temporal variation in 14C level is global, such that a small number of samples from a specific year are sufficient for calibration. However, since Libby's early work was published (1950 to 1958),  latitudinal and continental variations in the carbon exchange reservoir have been observed by Hessel de Vries (1958; as reviewed by Lerman et al., 1959, 1960). Subsequently, methods have been developed that allow the correction of these so-called reservoir effects, including:
- When CO2 is transferred from the atmosphere to the oceans, it initially shares the 14C concentration of the atmosphere. However, turnaround times of CO2 in the ocean are similar to the half-life of 14C (making 14C also a dating tool for ocean water, ). Marine organisms feed on this "old" carbon, and thus their radiocarbon age reflects the time of CO2 uptake by the ocean rather than the dead of the organism. This marine reservoir effect is partly handled by a special marine calibration curve , but local deviation of several 100 years exist.
- Erosion and immersion of carbonate rocks (which are generally assumed to be older than 80,000 years and so wouldn't contain measurable 14C) causes an increase in 12C and 13C in the exchange reservoir, which depends on local weather conditions and can vary the ratio of carbon that living organisms incorporate. This is believed negligible for the atmosphere and atmosphere-derived carbon since most erosion will flow into the sea.. The atmospheric 14C concentration may differ substantially from the concentration in local water reservoirs. Eroded from CaCO3 or organic deposits, old carbon may be assimilated easily and provide diluted 14C carbon into trophic chains. So the method is less reliable for such materials as well as for samples derived from animals with such plants in their food chain.
- Volcanic eruptions eject large amount of carbonate into the air, causing an increase in 12C and 13C in the exchange reservoir and can vary the exchange ratio locally. This explains the often irregular dating achieved in volcanic areas.
- The earth is not affected evenly by cosmic radiation, the magnitude of the radiation depends on land altitude and earth's magnetic field strength at any given location, causing minor variation in the local 14C production. This is accounted for by having calibration curves for different locations of the globe. However this could not always be performed, as tree rings for calibration were only recoverable from certain locations in 1958.
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.
Fomenko's recent analyses and criticism of the radiocarbon dating method are, thus, invalid because they refer to the earlier, now obsolete, radiocarbon dates.

# Speleothem studies extend 14C calibration
Relatively recent (2001) evidence has allowed scientists to refine the knowledge of  one of the underlying assumptions. A peak in the amount of carbon-14 was discovered by scientists studying speleothems in caves in the Bahamas.  Stalagmites are calcium carbonate deposits left behind when seepage water, containing dissolved carbon dioxide,  evaporates. Carbon-14 levels were found to be twice as high as modern levels.  These discoveries improved the calibration for the radiocarbon technique and extended its usefulness to 45,000 years into the past.

# Examples
- Ancient footprints of Acahualinca
- Chauvet Cave
- Dolaucothi
- Haraldskr Woman
- Kennewick Man
- Skeleton Lake
- Shroud of Turin
- Thera eruption
- Vinland map