### Radiocarbon date

Radiocarbon dating (or simply carbon dating) is a radiometric dating technique that uses the decay of carbon-14 () to estimate the age of organic materials, such as wood and leather, up to about 58,000 to 62,000 years Before Present (BP, present defined as 1950).[1] Carbon dating was presented to the world by Willard Libby in 1949, for which he was awarded the Nobel Prize in Chemistry. Since its introduction it has been used to date many items, including samples of the Dead Sea Scrolls, the Shroud of Turin, enough Egyptian artifacts to supply a chronology of Dynastic Egypt,[2] and Ötzi the Iceman.[3]

The Earth's atmosphere contains various isotopes of carbon, roughly in constant proportions. These include the main stable isotope () and an unstable isotope (). Through photosynthesis, plants absorb both forms from carbon dioxide in the atmosphere. When an organism dies, it contains the standard ratio of to , but as the decays with no possibility of replenishment, the proportion of carbon 14 decreases at a known constant rate. The time taken for it to reduce by half is known as the half-life of . The measurement of the remaining proportion of in organic matter thus gives an estimate of its age (a raw radiocarbon age).[4] However, over time there are small fluctuations in the ratio of to in the atmosphere, fluctuations that have been noted in natural records of the past, such as sequences of tree rings and cave deposits. These records allow fine-tuning, or "calibration", of the raw radiocarbon age, to give a more accurate estimate of the calendar date of the material.

One of the most frequent uses of radiocarbon dating is to estimate the age of organic remains from archaeological sites.

## Physical and chemical background

Carbon has two stable, nonradioactive isotopes: carbon-12 (), and carbon-13 (), and a radioactive isotope, carbon-14 (), also known as radiocarbon. The half-life of (the time it takes for half of a given amount of to decay) is about 5,730 years, so its concentration in the atmosphere might be expected to reduce over thousands of years. However is constantly being produced in the lower stratosphere and upper troposphere by cosmic rays, which generate neutrons that in turn create when they strike nitrogen-14 () atoms.[5] The process is described by the following nuclear reaction, where n represents a neutron and p represents a proton:[6]

$n + \mathrm\left\{^\left\{14\right\}_\left\{7\right\}N^+\right\} \rightarrow \mathrm\left\{^\left\{14\right\}_\left\{6\right\}C\right\} + p$

Once produced, the quickly combines with the oxygen in the atmosphere to form carbon dioxide (). Carbon dioxide produced in this way diffuses in the atmosphere, is dissolved in the ocean, and is taken up by plants via photosynthesis. Animals eat the plants, and ultimately the radiocarbon is distributed throughout the biosphere. The combination of the ocean, the atmosphere and the biosphere is referred to as the carbon exchange reservoir.[7]

If it is assumed that the cosmic ray flux has been constant over the last ~100,000 years, then carbon-14 has been produced at a constant rate, and since it is also lost through radioactivity at a constant rate, the proportion of radioactive to non-radioactive carbon is constant. The ratio of to in the carbon exchange reservoir is 1.5 parts of to 1012 parts of .[7] In addition, about 1% of the reservoir is made up of the stable isotope .[5]

## Invention of the method

In the mid-1940s, Willard Libby, then at the University of Chicago, realized that the decay of carbon-14 might lead to a method of dating organic matter. Libby published a paper in 1946 in which he proposed that the carbon in living matter might include carbon-14 as well as non-radioactive carbon.[8][9] Libby and several collaborators proceeded to experiment with methane collected from sewage works in Baltimore, and after isotopically enriching their samples they were able to demonstrate that they contained radioactive carbon-14. By contrast, methane created from petroleum had no radiocarbon activity. The results were summarized in a paper in Science in 1947, and the authors commented that their results implied it would be possible to date materials containing carbon of organic origin.[8][10] Libby and James Arnold proceeded to experiment with samples of wood of known age. For example, two wood samples taken from the tombs of two Egyptian kings, Zoser and Sneferu, independently dated to 2625 BC plus or minus 75 years, were dated by radiocarbon measurement to an average of 2800 BC plus or minus 250 years.[11][12] These measurements, published in Science in 1949, launched the "radiocarbon revolution" in archaeology, and soon led to dramatic changes in scholarly chronologies.[12] In 1960, Libby was awarded the Nobel Prize in chemistry for this work.[13]

## Calculating ages

While a plant or animal is alive, it is exchanging carbon with its surroundings, so the carbon it contains will have the same proportion of as the biosphere. Once it dies, it ceases to acquire , but the it contains will continue to decay, and so the proportion of radiocarbon in its remains will gradually reduce. Because decays at a known rate, the proportion of radiocarbon can be used to determine how long it is since a given sample stopped exchanging carbon—the older the sample, the less will be left.[7]

The equation governing the decay of a radioactive isotope is[5]

$N = N_0e^\left\{-\lambda t\right\}\,$

where N0 is the number of atoms of the isotope in the original sample (at time t = 0), and N is the number of atoms left after time t.[5] λ is a constant that depends on the particular isotope; for a given isotope it is equal to the reciprocal of the mean-life—i.e. the average or expected time a given atom will survive before undergoing radioactive decay.[5] The mean-life, denoted by τ, of is 8,267 years, so the equation above can be rewritten as:[14]

$t = 8267 \cdot \ln\left(N_0/N\right)$

The ratio of atoms in the original sample, N0, is taken to be the same as the ratio in the biosphere, so measuring N, the number of atoms currently in the sample, allows the calculation of t, the age of the sample.[7]

The half-life of a radioactive isotope (the time it takes for half of the sample to decay, usually denoted by T1/2) is a more familiar concept than the mean-life, so although the equations above are expressed in terms of the mean-life, it is more usual to quote the value of 's half-life than its mean-life. The currently accepted value for the half-life of radiocarbon is 5,730 years.[5] The mean-life and half-life are related by the following equation:[5]

$T_\frac\left\{1\right\}\left\{2\right\} = \tau \cdot \ln 2$

The above calculations make several assumptions: for example, that the level of in the biosphere has remained constant over time.[15] In fact the level of in the biosphere has varied significantly, and as a result the values provided by the equation above have to be corrected by using data from other sources, using a calibration curve, which is described in more detail below.[16] For over a decade after Libby's initial work, the accepted value of the half-life for was 5,568 years; this was improved in the early 1960s to 5,730 years, which meant that many calculated dates in published papers were now incorrect (the error is about 3%). However, it is possible to incorporate a correction for the half-life value into the calibration curve, and so it has become standard practice to quote measured radiocarbon dates in "radiocarbon years", meaning that the dates are calculated using Libby's half-life value and have not been calibrated.[17] This approach has the advantage of maintaining consistency with the early papers, and also avoids the risk of a double correction for the Libby half-life value.[18]

## Carbon exchange reservoir

The different elements of the carbon exchange reservoir vary in how much carbon they store, and in how long it takes for the generated by cosmic rays to fully mix with them.[19] The atmosphere, which is where is generated, contains about 1.9% of the total carbon in the reservoirs, and the it contains mixes in less than 7 years.[20][21] The ratio of to in the atmosphere is taken as the baseline for the other reservoirs: if another reservoir has a lower ratio of to , it indicates that the carbon is older, and hence some of the has decayed.[16] The ocean surface is an example: it contains 2.4% of the carbon in the exchange reservoir,[20] but there is only about 95% as much as would be expected if the ratio were the same as in the atmosphere.[19] The time it takes for carbon from the atmosphere to mix with the surface ocean is only a few years,[22] but the surface waters also receive water from the deep ocean, which has over 90% of the carbon in the reservoir.[16] Water in the deep ocean takes about 1,000 years to circulate back through surface waters, and so the surface waters contain a combination of older water, with depleted , and water recently at the surface, with in equilibrium with the atmosphere.[16]

Creatures living at the ocean surface have the same ratios as the water they live in. Using the calculation method given above to calculate the age of marine life typically gives an age of about 400 years.[23][note 2] Organisms on land, however, are in closer equilibrium with the atmosphere and have the same / ratio as the atmosphere.[19] These organisms contain about 1.3% of the carbon in the reservoir; sea organisms have a mass of less than 1% of those on land and are not shown on the diagram.[20] Accumulated dead organic matter, of both plants and animals, exceeds the mass of the biosphere by a factor of nearly 3, and since this matter is no longer exchanging carbon with its environment it has a / ratio lower than that of the biosphere.[19]

## Dating considerations

The variation in the / ratio in different parts of the carbon exchange reservoir means that a straightforward calculation of the age of a sample based on the amount of it contains will often give an incorrect result. There are several other possible sources of error that need to be considered; the errors are of four general types:

• Variations in the / ratio in the atmosphere, both geographically and over time
• Isotopic fractionation
• Variations in / ratio in different parts of the reservoir
• Contamination

### Atmospheric variation

In the early years of using the technique, it was not assumed that the atmospheric / ratio had been the same over the preceding few thousand years. To verify the accuracy of the method, several artefacts that were datable by other techniques were tested; the results of the testing were in reasonable agreement with the true ages of the objects. However, over the next few years significant discrepancies were found, in particular with the chronology of the early Egyptian dynasties: artefact ages derived from radiocarbon testing were several centuries younger than what were thought to be the true ages. The discrepancy was resolved by the study of tree-rings. Comparison of overlapping series of tree-rings allowed the construction of a continuous sequence of tree-ring data that spanned 8,000 years. Carbon-dating the wood from the tree-rings themselves provided the check needed on the atmospheric / ratio: with a sample of known date, and a measurement of the value of N (the number of atoms of remaining in the sample), the carbon-dating equation allows the calculation of N0 (the number of atoms of in the original sample), and hence the original ratio.[24] Armed with the results of carbon-dating the tree rings it became possible to construct calibration curves designed to correct the errors caused by the variation over time in the / ratio.[25] These curves are described in more detail below. There are three main reasons for these variations in the historical / ratio: fluctuations in the rate at which is created; changes caused by glaciation; and changes caused by human activity.[24]

#### Variations in production

Two different trends can be seen in the tree ring series. First, there is a long term oscillation with a period of about 9,000 years, which causes radiocarbon dates to be older than true dates for the last 2,000 years, and too young before that. The known fluctuations in the earth's magnetic field strength match up quite well with this oscillation: cosmic rays are deflected by magnetic fields, so when there is a lower magnetic field, more is produced, leading to a younger apparent age for samples from those periods. Conversely, a higher magnetic field leads to lower production and an older apparent age. A secondary oscillation is thought to be caused by variations in sunspot activity, which has two separate periods: a longer-term, 200-year oscillation, combined with a shorter 11-year cycle. Sunspots cause changes in the solar system's magnetic field and corresponding changes to the cosmic ray flux, and hence to the production of .[24]

Over geological timescales the earth's magnetic field can reverse, both locally and globally. These global geomagnetic reversals, and shorter, often localized polar excursions, would have had a significant impact on global production, since the geomagnetic field falls to a low value for thousands of years. However, there are no well-established occurrences of either of these events in the recent enough past for there to have been an appreciable effect on present-day measurements. There is some evidence for polarity excursions, but they may not have been global; if they were local they would not have had any noticeable impact on production.[26]

Since the earth's magnetic field varies with latitude, the rate of production changes with latitude too, but atmospheric mixing is rapid enough that these variations amount to less than 0.5% of the global concentration.[24] This is close to the limit of detectability in most years,[27] but the effect can seen clearly in tree rings from years such as 1963, when from nuclear testing rose sharply through the year.[28] The latitudinal variation in was much larger than normal that year, and tree rings from different latitudes show corresponding variations in their content.[28]

can also be produced at ground level, primarily by cosmic rays that penetrate the atmosphere as far as the earth's surface, and by spontaneous fission of naturally occurring uranium. These sources of neutrons only produce at a rate of 1 x 10−4 atoms per gram per second, which is not enough to have a significant impact on dating.[28][29] At higher altitudes, the neutron flux can be substantially higher,[30][note 3] and in addition, trees at higher altitude are more likely to be struck by lightning, which produces neutrons. However, experiments in which wood samples have been irradiated with neutrons indicate that the effect on content is minor, though for very old trees (such as some bristlecone pines) that grow at altitude some effect can be seen.[30]

#### Impact of climatic cycles

Because the solubility of in water increases with lower temperatures, glacial periods would have led to faster absorption of atmospheric by the oceans. In addition, any carbon stored in the glaciers would be depleted in over the life of the glacier; when the glacier melted, as the climate warmed, the depleted carbon would be released, reducing the global / ratio. The changes in climate would also cause changes in the biosphere, with warmer periods leading to more plant and animal life. The effect of these factors on radiocarbon dating is not known.[24]

#### The effects of human activity

Coal and oil began to be burned in large quantities during the 1800s. Both coal and oil are sufficiently old that they contain little detectable , and as a result the released substantially diluted the atmospheric / ratio. Dating an object from the early 20th century hence gives an apparent date older than the true date; and for the same reason, concentrations in the neighbourhood of large cities are lower than the atmospheric average. This fossil fuel effect (also known as the Suess effect, after Hans Suess, who first reported it in 1955) would only amount to a reduction of 0.2% in activity if the additional carbon from fossil fuels were distributed throughout the carbon exchange reservoir, but because of the long delay in mixing with the deep ocean, the actual effect is a 3% reduction.[24][34]

A much larger effect comes from above-ground nuclear testing, which released large numbers of neutrons and created . From about 1950 until 1963, when atmospheric nuclear testing was banned, it is estimated that several tonnes of were created. If all this extra had immediately been spread across the entire carbon exchange reservoir, it would have led to an increase in the / ratio of only a few percent, but the immediate effect was to almost double the amount of in the atmosphere, with the peak level occurring in about 1965. The level has since dropped again, as the "bomb carbon" (as it is sometimes called) percolates into the rest of the reservoir.[24][34][35]

### Fractionation

Photosynthesis is the primary process by which carbon moves from the atmosphere into living things. Two different photosynthetic processes exist: the C3 pathway, and the C4 pathway. About 90% of all plant life uses the C3 process; the remaining plants either use C4 or are CAM plants, which can use either C3 or C4 depending on the environmental conditions. Both the C3 and C4 photosynthesis pathways show a preference for lighter carbon, with being absorbed slightly more easily than , which in turn is more easily absorbed than . The differential uptake of the three carbon isotopes leads to different ratios of the isotopes in plants and in the atmosphere. This effect is known as isotopic fractionation.[30][36]

To determine the degree of fractionation that takes place in a given plant, the amounts of both and are measured, and the resulting / ratio is then compared to a standard ratio known as PDB. The resulting value, known as δ13C, is calculated as follows:[30]

$\mathrm\left\{\delta ^\left\{13\right\}C\right\} = \Biggl\left( \mathrm\left\{\frac\left\{\bigl\left( \frac\left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \bigr\right)_\left\{sample\right\}\right\}\left\{\bigl\left( \frac\left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \bigr\right)_\left\{PDB\right\}\right\}\right\} -1 \Biggr\right) \times 1000\ ^\left\{o\right\}\!/\!_\left\{oo\right\}$

where the ‰ (permil) sign indicates parts per thousand.[30] This can be rewritten as:[37]

$\mathrm\left\{\delta ^\left\{13\right\}C\right\} = \frac\left\{\mathrm\left\{\Bigl\left( \frac\left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \Bigr\right)_\left\{sample\right\}\right\} - \left\{\Bigl\left( \frac\left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \Bigr\right)_\left\{PDB\right\}\right\} \right\}\left\{\bigl\left( \frac\left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \bigr\right)_\left\{PDB\right\}\right\} \times 1000\ ^\left\{o\right\}\!/\!_\left\{oo\right\}$

which makes it apparent that δ13C is proportional to the difference between the / ratios in the PDB standard and in the sample.[37] Because the PDB standard contains an unusually high proportion of ,[note 4] most measured δ13C values are negative. Values for C3 plants typically range from −30‰ to −22‰, with an average of −27‰; for C4 plants the range is −15‰ to −9‰, and the average is −13‰.[36] Atmospheric has a δ13C of −8‰.[30]

For marine organisms, the details of the photosynthesis reactions are less well understood. Measured δ13C values for marine plankton range from −31‰ to −10‰; most lie between −22‰ and −17‰. The δ13C values for marine photosynthetic organisms also depend on temperature. At higher temperatures, has poor solubility in water, which means there is less available for the photosynthetic reactions. Under these conditions, fractionation is reduced, and at temperatures above 14°C the δ13C values are correspondingly higher, reaching −13‰. At lower temperatures becomes more soluble and hence more available to the marine organisms; fractionation increases and δ13C values can reach −32‰.[36] The δ13C value for animals depends on their diet. An animal that eats food with high δ13C values will have a higher δ13C than one that eats food with lower δ13C values.[30] The animal's own biochemical processes can also impact the results: for example, both bone minerals and bone collagen typically have a higher concentration of than is found in the animal's diet, though for different biochemical reasons. When additional fractionation takes place between an animal's diet and its organic matter, it also implies that excreted material is depleted in relative to the diet.[38]

Since makes up about 1% of the carbon in a sample, the / ratio can be accurately measured by mass spectrometry.[16] Typical values of δ13C have been found by experiment for many plants, as well as for different parts of animals such as bone collagen, but when dating a given sample it is better to determine the δ13C value for that sample directly than to rely on the published values.[30] The depletion of relative to is proportional to the difference in the atomic masses of the two isotopes, so once the δ13C value is known, the depletion for can be calculated: it will be twice the depletion of .[16]

The carbon exchange between atmospheric and carbonate at the ocean surface is also subject to fractionation, with in the atmosphere more likely than to dissolve in the ocean. The result is an overall increase in the / ratio in the ocean of 1.5%, relative to the / ratio in the atmosphere. This increase in concentration almost exactly cancels out the decrease caused by the upwelling of water (containing old, and hence depleted, carbon) from the deep ocean, so that direct measurements of radiation are similar to measurements for the rest of the biosphere. Correcting for isotopic fractionation, as is done for all radiocarbon dates to allow comparison between results from different parts of the biosphere, gives an apparent age of about 400 years for ocean surface water.[16]

### Reservoir effects

Libby's original exchange reservoir hypothesis assumed that the exchange reservoir is constant all over the world,[39] but it has since been discovered that there are several causes of variation in the / ratio across the reservoir.[23]

#### Marine effect

The in the atmosphere transfers to the ocean by dissolving in the surface water as carbonate and bicarbonate ions; at the same time the carbonate ions in the water are returning to the air as .[39] This exchange process brings from the atmosphere into the surface waters of the ocean, but the thus introduced takes a long time to percolate through the entire volume of the ocean. The deepest parts of the ocean mix very slowly with the surface waters, and the mixing is known to be uneven. The main mechanism that brings deep water to the surface is upwelling. Upwelling is more common in regions closer to the equator; it is also influenced by other factors such as the topography of the local ocean bottom and coastlines, the climate, and wind patterns. Overall, the mixing of deep and surface waters takes far longer than the mixing of atmospheric with the surface waters, and as a result water from some deep ocean areas has an apparent radiocarbon age of several thousand years. Upwelling mixes this "old" water with the surface water, giving the surface water an apparent age of about several hundred years (after correcting for fractionation).[23] This effect is not uniform—the average effect is about 440 years, but there are local deviations of several hundred years for areas that are geographically close to each other.[23][40] The effect also applies to marine organisms such as shells, and marine mammals such as whales and seals, which have radiocarbon ages that appear to be hundreds of years old.[23] These marine reservoir effects vary over time as well as geographically; for example, there is evidence that during the Younger Dryas, a period of cold climatic conditions about 12,000 years ago, the apparent difference between the age of surface water and the contemporary atmosphere increased from between 400 and 600 years to about 900 years until the climate warmed again.[40]

#### Hard water effect

If the carbon in freshwater is partly acquired from aged carbon, such as rocks, then the result will be a reduction in the / ratio in the water. For example, rivers that pass over limestone, which is mostly composed of calcium carbonate, will acquire carbonate ions. Similarly, groundwater can contain carbon derived from the rocks through which it has passed. These rocks are usually so old that they no longer contain any measurable , so this carbon lowers the / ratio of the water it enters, which can lead to apparent ages of thousands of years for both the affected water and the plants and freshwater organisms that live in it.[16] This is known as the hard water effect, because it is often associated with calcium ions, which are characteristic of hard water; however, there can be other sources of carbon that have the same effect, such as humus. The effect is not necessarily confined to freshwater species—at a river mouth, the outflow may affect marine organisms. It can also affect terrestrial snails that feed in areas where there is a high chalk content, though no measurable effect has been found for land plants in soil with a high carbonate content—it appears that almost all the carbon for these plants is derived from photosynthesis and not from the soil.[23]

It is not possible to deduce the impact of the effect by determining the hardness of the water: the aged carbon is not necessarily immediately incorporated into the plants and animals that are affected, and the delay has an impact on their apparent age. The effect is very variable and there is no general offset that can be applied; the usual way to determine the size of the effect is to measure the apparent age offset of a modern sample.[23]

#### Volcanoes

Volcanic eruptions eject large amounts of carbon into the air. The carbon is of geological origin and has no detectable , so the / ratio in the vicinity of the volcano is depressed relative to surrounding areas. Dormant volcanoes can also emit aged carbon. Plants that photosynthesize this carbon also have lower / ratios: for example, plants on the Greek island of Santorini, near the volcano, have apparent ages of up to a thousand years. These effects are hard to predict—the town of Akrotiri, on Santorini, was destroyed in a volcanic eruption thousands of years ago, but radiocarbon dates for objects recovered from the ruins of the town show surprisingly close agreement with dates derived from other means. If the dates for Akrotiri are confirmed, it would indicate that the volcanic effect in this case was minimal.[23]

#### Hemisphere effect

The northern and southern hemispheres have atmospheric circulation systems that are sufficiently independent of each other that there is a noticeable time lag in mixing between the two. The atmospheric / ratio is lower in the southern hemisphere, with an apparent additional age of 30 years for radiocarbon results from the south as compared to the north. This is probably because the greater surface area of ocean in the southern hemisphere means that there is more carbon exchanged between the ocean and the atmosphere than in the north. Since the surface ocean is depleted in because of the marine effect, is removed from the southern atmosphere more quickly than in the north.[23]

#### Island effect

It has been suggested that an "island effect" might exist, by analogy with the same mechanism thought to explain the hemisphere effect—since islands are surrounded by water, the carbon exchange between the water and atmosphere might reduce the / ratio on an island. Within a hemisphere, however, atmospheric mixing is apparently rapid enough that no such effect exists: two calibration curves assembled in Seattle and Belfast laboratories, with results from North American trees and Irish trees, respectively, are in close agreement, instead of the Irish samples appearing to be older, as would be the case if there were an island effect.[23]

### Contamination

Any addition of carbon to a sample of a different age will cause the measured date to be inaccurate. Contamination with modern carbon causes a sample to appear to be younger than it really is: the effect is greater for older samples. If a sample that is in fact 17,000 years old is contaminated so that 1% of the sample is actually modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old the same amount of contamination would cause an error for 4,000 years. Contamination with old carbon, with no remaining , causes an error in the other direction, which does not depend on age—a sample that has been contaminated with 1% old carbon will appear to be about 80 years older than it really is, regardless of the date of the sample.[41] The equation for the radioactivity of a sample that has been contaminated with other carbon is

$A_m = fA_x + \left(1-f\right)A_s$

where Am is the measured radioactivity of the sample, Ax is the radioactivity of the contaminating material, As is the radioactivity of the original sample prior to contamination, and f is the fraction of the carbon in the sample that is from the contaminant.[42]

Contamination can occur if the sample is brought into contact with or packed in materials that contain carbon. Cotton wool, cigarette ash, paper labels, cloth bags, and some conservation chemicals such as polyvinyl acetate can all be sources of modern carbon.[42] Labels should be added to the outside of the container, not placed inside the bag or vial with the sample. Glass wool is acceptable as packing material instead of cotton wool.[43] Samples should be packed in glass vials or aluminium foil if possible;[42][44] polyethylene bags are also acceptable but some plastics such as PVC can contaminate the sample.[43] Contamination can also occur before the sample is collected: humic acids or carbonate from soil can leach into a sample, and for some sample types, such as shells, there is the possibility of carbon exchange between the sample and the environment, depleting the sample's content.[42]

## Samples

Samples for dating need to be converted into a form suitable for measuring the content; this can mean conversion to gaseous, liquid, or solid form, depending on the measurement technique to be used. Before this can be done, however, the sample must be treated to remove any contamination and any unwanted constituents.[42] This includes removing visible contaminants, such as rootlets that may have penetrated the sample since its burial.[45]

### Pretreatment

Two common contaminants are humic acid, which can be removed with an alkali wash, and carbonates, which can be removed with acid. These treatments can damage the structural integrity of the sample and remove significant volumes of material, so the exact treatment decided on will depend on the sample size and the amount of carbon needed for the chosen measurement technique.[46]

#### Wood and charcoal

Wood contains cellulose, lignin, and other compounds; of these, cellulose is the least likely to have exchanged carbon with the sample's environment, so it is common to reduce a wood sample to just the cellulose component before testing. However, this can reduce to the volume of the sample down to 20% of the original size, so testing of whole wood is often performed as well. Charcoal is less likely than wood to have exchanged carbon with its environment, but a charcoal sample is likely to have absorbed humic acid and/or carbonates, which must be removed with alkali and acid washes.[45][46]

#### Bone

Unburnt bone was once thought to be a poor candidate for radiocarbon dating,[47] but is now possible to test it accurately. The constituents of bone include proteins, which contain carbon; bone's structural strength comes from calcium hydroxyapatite, which is easily contaminated with carbonates from ground water. Removing the carbonates also destroys the calcium hydroxyapatite, and so it is usual to date bone using the remaining protein fraction after washing away the calcium hydroxyapatite and contaminating carbonates. This protein component is called collagen. Collagen is sometimes degraded, in which case it may be necessary to separate the proteins into individual amino acids and measure their respective ratios and activity. It is possible to detect if there has been any degradation of the sample by comparing the relative volume of each amino acid with the known profile for bone. If so, separating the amino acids may be necessary to allow independent testing of each one—agreement between the results of several different amino acids indicates that the dating is reliable. Hydroxyproline, one of the constituent amino acids in bone, was once thought to be a reliable indicator as it was not known to occur except in bone, but it has since been detected in groundwater.[45]

For burnt bone, testability depends on the conditions under which the bone was burnt. The proteins in burnt bone are usually destroyed, which means that after acid treatment, nothing testable will be left of the bone. Degradation of the protein fraction can also occur in hot, arid conditions, without actual burning; then the degraded components can be washed away by groundwater. However, if the bone was heated under reducing conditions, it (and associated organic matter) may have been carbonized. In this case the sample is often usable.[45]

#### Shell

Shells from both marine and land organisms consist almost entirely of calcium carbonate, either as aragonite or as calcite, or some mixture of the two. Calcium carbonate is very susceptible to dissolving and recrystallizing; the recrystallized material will contain carbon from the sample's environment, which may be of geological origin. The recrystallized calcium carbonate is generally in the form of calcite, and often has a powdery appearance; samples of a shiny appearance are preferable, and if in doubt, examination by light or electron microscope, or by X-ray diffraction and infrared spectroscopy, can determine whether recrystallization has occurred.[48]

In cases where it is not possible to find samples that are free of recrystallization, acid washes of increasing strength, followed by dating part of the sample after each wash, can be used: the dates obtained from each sample will vary with the degree of contamination, but when the contaminated layers are removed, consecutive measurements will be consistent with each other. It is also possible to test conchiolin, which is an organic protein found in shell, but this only constitutes 1-2% of shell material.[46]

#### Other materials

• Peat. The three major components of peat are humic acid, humins, and fulvic acid. Of these, humins give the most reliable date as they are insoluble in alkali and less likely to contain contaminants from the sample's environment.[46] A particular difficulty with dried peat is the removal of rootlets, which are likely to be hard to distinguish from the sample material.[45]
• Soil and sediments. Soil contains organic material but because of contamination by humic acid of more recent origin it is very difficult to get satisfactory radiocarbon dates. It is preferable to sieve the soil for fragments of organic origin, and date the fragments with methods that are tolerant of small sample sizes.[46]
• Other types of sample that have been successfully dated include ivory, paper, textiles, individual seeds and grains, straw from within mud bricks, and charred food remains found in pottery.[46]

### Isotopic enrichment

Particularly for older samples, it may be useful to enrich the amount of in the sample before testing. This can be done with a thermal diffusion column. The process takes about a month, and requires a sample about ten times as large as would be needed otherwise, but it allows more precise measurement of the / ratio in old material, and extends the maximum age that can be reliably reported.[49]

### Preparation

Once contamination has been removed, samples must be converted to a form suitable for the measuring technology to be used.[50] A common approach is to produce a gas, for gas counting devices: is widely used, but it is also possible to use other gases, including methane, ethane, ethylene and acetylene.[50][51] For samples in liquid form, for liquid scintillation counters, benzene is used, though other liquids were tried during the early decades of the technique. Libby's first measurements were made with lamp black,[50] but this technique is no longer in use; these methods were susceptible to problems caused by the created by nuclear testing in the 1950s and 1960s.[50] Solid targets can be used for accelerator mass spectrometry, however; usually these are graphite, though and iron carbide can also be used.[52][53]

The steps to convert the sample to the appropriate form for testing can be long and complex. To create lamp black, Libby began with acid washes if necessary to remove carbonate, and then converted the carbon in the sample to by either combustion (for organic samples) or the addition of hydrochloric acid (for shell material). The resulting gas was passed through hot copper oxide to convert any carbon monoxide to , and then dried to remove any water vapour. The gas was then condensed, and converted to calcium carbonate in order to allow the removal of any radon gas and any other combustion products such as oxides of nitrogen and sulphur. The calcium carbonate was then converted back to again, dried, and converted to carbon by passing it over heated magnesium. Hydrochloric acid was added to the resulting mixture of magnesium, magnesium oxide and carbon, and after repeated boiling, filtering, and washing with distilled water, the carbon was ground with a mortar and pestle and a half gram sample taken, weighed, and combusted. This allowed Libby to determine how much of the sample was ash, and hence to determine the purity of the carbon sample to be tested.[54]

To create benzene for liquid scintillation counters, the sequence begins with combustion to convert the carbon in the sample to . This is then converted to lithium carbide, and then to acetylene, and finally to benzene.[50] Targets for accelerator mass spectrometry are created from by catalysing the reduction of the gas in the presence of hydrogen. This results in a coating of filamentous carbon (usually referred to as graphite) on the powdered catalyst—typically cobalt or iron.[53]

### Sample sizes

How much sample material is needed to perform testing depends on what is being tested, and also which of the two testing technologies is being used: detectors that record radioactivity, known as beta counters, or atomic mass spectrometers (AMS). A rough guide follows; the weights given, in grams, are for dry samples, and assume that a visual inspection has been done to remove foreign objects.[50]

 Sample material Mass (g) For betacounters For AMS Whole wood 10–25 0.05–0.1 Wood (for cellulose testing) 50–100 0.2–0.5 Charcoal 10–20 0.01–0.1 Peat 50–100 0.1–0.2 Textiles 20–50 0.02–0.05 Bone 100–400 0.5–1.0 Shell 50–100 0.05–0.1 Sediment/soils 100–500 5.0–25.0

## Measurement

For decades after Libby performed the first radiocarbon dating experiments, the only way to measure the in a sample was to detect the radioactive decay of individual carbon atoms.[52] In this approach, what is measured is the activity, in number of decay events per unit mass per time period, of the sample.[51] This method is also known as "beta counting", because it is the beta particles emitted by the decaying atoms that are detected.[55] In the late 1970s an alternative approach became available: directly counting the number of and atoms in a given sample, via accelerator mass spectrometry, usually referred to as AMS.[52] AMS counts the / ratio directly, instead of the activity of the sample, but measurements of activity and / ratio can be converted into each other exactly.[51]

### Beta counting

Libby's first detector was a Geiger counter of his own design. He coated the inner surface of a cylinder with carbon in the form of lamp black (soot), and inserted it into the counter in such a way that the counting wire was inside the sample cylinder, in order that there should be no material between the sample and the wire.[50] Any interposing material would have interfered with the detection of radioactivity; the beta particles emitted by decaying are so weak that half are stopped by a 0.01 mm thickness of aluminium.[51]

Libby's method was soon superseded by gas proportional counters, which were less affected by bomb carbon. These counters record bursts of ionization caused by the beta particles emitted by the decaying atoms; the bursts are proportional to the energy of the particle, so other sources of ionization, such as background radiation, can be identified and ignored. The counters are surrounded by lead or steel shielding, to eliminate background radiation and to reduce the incidence of cosmic rays. In addition, anticoincidence detectors are used; these record events outside the counter, and any event recorded simultaneously both inside and outside the counter is regarded as an extraneous event and ignored.[51]

The other common technology used for measuring activity is liquid scintillation counting, which was invented in 1950, but which had to wait until the early 1960s, when efficient methods of benzene synthesis were developed, to become competitive with gas counting; after 1970 liquid counters became the more common technology choice for newly constructed dating laboratories. The counters work by detecting flashes of light caused by the beta particles emitted by as they interact with a fluorescing agent added to the benzene. Like gas counters, liquid scintillation counters require shielding and anticoincidence counters.[56][57]

For both types of counter, what is measured is a number of beta particles detected in a given time period. Since the mass of the sample is known, this can be converted to a standard measure of activity in units of either counts per minute per gram of carbon (cpm/g C), or becquerels per kg (Bq/kg C, in SI units). Each measuring device will also be used to measure the activity of a blank sample—a sample prepared from carbon old enough to have no activity. This provides a value for the background radiation, which must be subtracted from the original sample's measured activity to get the activity due to the sample's . In addition, a sample with a standard activity will be measured, in order to provide a baseline for comparison.[58]

### Accelerator mass spectrometry

AMS counts the atoms of and atoms in a given sample, determining the / ratio directly. The sample, often in the form of graphite, is made to emit negatively charged C- ions, which are injected into an accelerator. The ions are accelerated, and passed through a stripper, which removes several electrons, so that the ions emerge with a positive charge. The C3+ ions are then passed through a magnet that curves their path; the heavier ions are curved less than the lighter ones, so the different isotopes emerge as separate streams of ions. A particle detector then records the number of ions detected in the stream, but counts (and counts, needed for calibration) are determined by measuring the electric current created in a Faraday cup, since the volume of these is too great for individual ion detection. This method allows dating samples containing only a few milligrams of carbon, such as individual seeds.[59] The use of AMS, as opposed to simpler forms of mass spectrometer, is necessary because of the need to distinguish the carbon isotopes from other atoms or molecules that are very close in mass to them, such as and .[52] As with beta counting, both a blank sample and a standard sample are also measured, in order to determine the level of background radiation, and to check the accuracy of the setup.[59] Two different kinds of blank may be measured: a sample of dead carbon that has undergone no chemical processing, in order to detect any machine background, and a sample known as a process blank made from dead carbon that is processed into target material in exactly the same way as the sample itself. Any signal from the machine background blank is likely to be caused either by beams of ions that have not followed the expected path inside the detector, or by carbon hydrides such as or . A signal from the process blank measures the amount of contamination introduced during the preparation of the sample. These measurements are used in the subsequent calculation of the age of the sample.[60]

### Calculations

The calculations to be performed on the measurements taken depend on the technology used, since beta counters measure the sample's radioactivity, whereas AMS determines the ratio of the three different carbon isotopes in the sample.[61]

#### Standards

The calculations to convert measured data to an estimate of the age of the sample require the use of several standards. One of these, the standard for normalizing δ13C values, has been discussed above: Pee Dee Belemnite, which had a / ratio of 1.12372%. A related standard is the use of wood, which has a δ13C of -25‰, as the material for which radiocarbon ages are calibrated. Since different materials have different δ13C values, it is possible for two samples of different materials, of the same age, to have different levels of radioactivity and different / ratios. To compensate for this, the measurements are converted to the activity, or isotope ratio, that would have been measured if the sample had been made of wood. This is possible because the δ13C of wood is known, and the δ13C of the sample material can be measured, or taken from a table of typical values. The details of the calculations for beta counting and AMS are given below.[17]

Another standard is the use of 1950 as "present", in the sense that a calculation that shows that a sample's likely age is 500 years "before present" means that it is likely to have come from about the year 1450. This convention is necessary in order to keep published radiocarbon results comparable to each other; without this convention a given radiocarbon result would be of no use unless the year it was measured was also known—an age of 500 years published in 2010 would indicate a likely sample date of 1510, for example. In order to allow measurements to be converted to the 1950 baseline, a standard activity level is defined for the radioactivity of wood in 1950. Because of the fossil fuel effect, this is not actually the activity level of wood from 1950; the activity would have been somewhat lower.[62] The fossil fuel effect was eliminated from the standard value by measuring wood from 1890, and using the radioactive decay equations to determine what the activity would have been at the year of growth. The resulting standard value, Aabs, is 226 becquerels per kilogram of carbon.[63]

Both beta counting and AMS measure standard samples as part of their methodology. These samples contain carbon of a known activity.[59] The first standard, Oxalic Acid SRM 4990C, also referred to as HOxI, was a 1,000 lb batch of oxalic acid created in 1955 by the National Institute of Standards and Technology (NIST). Since it was created after the start of atomic testing, it incorporates bomb carbon, so measured activity is higher than the desired standard. This is addressed by defining the standard to be 0.95 times the activity of HOxI.[63]

All of this first standard has long since been consumed, and later standards have been created, each of which has a given ratio to the desired standard activity. A secondary oxalic acid standard, HOxII, 1,000 lb of which was prepared by NIST in 1977 from French beet harvests, is now in wide use.[64]

#### Calculations for beta counting devices

To determine the age of a sample whose activity has been measured by beta counting, the ratio of its activity to the activity of the standard must be found. The equation:[58]

$A_s = A_\left\{std\right\} \left \left( \frac\left\{M_s - M_b\right\}\left\{M_\left\{std\right\} - M_b\right\} \right \right)$

gives the required ratio, where As is the true activity of the sample, Astd is the true activity of the standard, Ms is the measured activity of the sample, Mstd is the measured activity of the standard, and Mb is the measured activity of the blank.[58]

A correction must also be made for fractionation. The fractionation correction converts the / ratio for the sample to the ratio it would have had if the material were wood, which has a δ13C value of -25‰. This is necessary because determining the age of the sample requires a comparison the amount of in the sample with what it would have had if it newly formed from the biosphere. The standard used for modern carbon is wood, with a baseline date of 1950.[17]

Correcting for fractionation changes the activity measured in the sample to the activity it would have if it were wood of the same age as the sample. The calculation requires the definition of a fractionation factor, which is defined for any sample material as[62]

$Frac_\left\{13/12 \left(sample\right)\right\} = \frac \left\{\left(^\left\{13\right\}C/^\left\{12\right\}C\right)_\left\{wood\right\}\right\}\right\}$

The fractionation factor, Frac14/12, is approximately the square of this, to an accuracy of 1‰:[62]

$Frac_\left\{14/12 \left(sample\right)\right\} = \left(Frac_\left\{13/12 \left(sample\right)\right\}\right)^2$

Multiplying the measured activity for the sample by the fractionation factor converts it to the activity that it would have had had the sample been wood:[62]

$A_\left\{sn\right\} = A_sFrac_\left\{14/12\left(s\right)\right\}$

where Asn is the normalized activity for the sample, and Frac14/12 (s) is the fractionation factor for the sample.[62]

The equation for δ13C given earlier can be rearranged to[62]

$\left \left( \frac \left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \right \right)_\left\{sample\right\} = \left \left( 1 + \frac \left\{\delta^\left\{13\right\}C\right\}\left\{1000\right\} \right \right) \left \left( \frac \left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \right \right)_\left\{PDB\right\}$

Substituting this in the fractionation factor, and also substituting the value for δ13C for wood of -25‰, gives the following expression:[62]

$A_\left\{sn\right\} = A_s \left \left( \frac \left\{\left \left(1 - \frac \left\{25\right\}\left\{1000\right\} \right \right)\left \left( \frac \left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \right \right)_\left\{PDB\right\}\right\}\left\{\left \left(1 + \frac \left\{\delta^\left\{13\right\}C\right\}\left\{1000\right\} \right \right)\left \left( \frac \left\{^\left\{13\right\}C\right\}\left\{^\left\{12\right\}C\right\} \right \right)_\left\{PDB\right\}\right\} \right \right)^2$

where the δ13C value remaining in the equation is the value for the sample itself. This can be measured directly, or simply looked up in a table of characteristic values for the type of sample material—this latter approach leads to increased uncertainty in the result, as there is a range of possible δ13C values for each possible sample material. Cancelling the PDB / ratio reduces this to:[62]

$A_\left\{sn\right\} = A_s \left \left( \frac \left\{\left \left(1 - \frac \left\{25\right\}\left\{1000\right\} \right \right)\right\}\left\{\left \left(1 + \frac \left\{\delta^\left\{13\right\}C\right\}\left\{1000\right\} \right \right)\right\} \right \right)^2$

#### AMS calculations

The results from AMS testing are in the form of ratios of , , and . These ratios are used to calculate Fm, the "fraction modern", defined as

$F_m = \frac\left\{R_\left\{norm\right\}\right\}\left\{R_\left\{modern\right\}\right\}$

where Rnorm is the / ratio for the sample, after correcting for fractionation, and Rmodern is the standard / ratio for modern carbon.[60]

The calculation begins by subtracting the ratio measured for the machine blank from the other sample measurements. That is:

$R\text{'}_s = R_s - R_\left\{mb\right\}$
$R\text{'}_\left\{std\right\} = R_\left\{std\right\} - R_\left\{mb\right\}$
$R\text{'}_\left\{pb\right\} = R_\left\{pb\right\} - R_\left\{mb\right\}$

where Rs is the measured sample / ratio; Rstd is the measured ratio for the standard; Rpb is the measured ratio for the process blank, and Rmb is the measured ratio for the machine blank. The next step, to correct for fractionation, can be done using either the / ratio or the / ratio, and also depends on which of the two possible standards was measured: HOxI or HoxII. R'std is then R'HOxI or R'HOxII, depending on which standard was used. The four possible equations are as follows. First, if the / ratio is used to perform the fractionation correction, the following two equations apply, one for each standard.[60]

$R_\left\{HOxI,-19\right\} = R\text{'}_\left\{HoxI\right\}\left \left( \frac \left\{1 + \frac \left\{-19\right\} \left\{1000\right\}\right\} \left\{1 + \frac \left\{\delta^\left\{13\right\}C_\left\{HoXI\right\}\right\} \left\{1000\right\}\right\}\right \right)^2$
$R_\left\{HOxII,-25\right\} = R\text{'}_\left\{HoxII\right\}\left \left( \frac \left\{1 + \frac \left\{-25\right\} \left\{1000\right\}\right\} \left\{1 + \frac \left\{\delta^\left\{13\right\}C_\left\{HoXII\right\}\right\} \left\{1000\right\}\right\} \right \right)^2$

If the / ratio is used instead, then the equations for each standard are:[60]

$R_\left\{HOxI,-19\right\} = R\text{'}_\left\{HoxI\right\}\left \left( \frac \left\{1 + \frac \left\{-19\right\} \left\{1000\right\}\right\} \left\{1 + \frac \left\{\delta^\left\{13\right\}C_\left\{HoXI\right\}\right\} \left\{1000\right\}\right\} \right \right)$
$R_\left\{HOxII,-25\right\} = R\text{'}_\left\{HoxII\right\}\left \left( \frac \left\{1 + \frac \left\{-25\right\} \left\{1000\right\}\right\} \left\{1 + \frac \left\{\delta^\left\{13\right\}C_\left\{HoXII\right\}\right\} \left\{1000\right\}\right\} \right \right)$

The δ13C values in the equations measure the fractionation in the standards as , prior to their conversion to graphite to use as a target in the spectrometer. This assumes that the conversion to graphite does not introduce significant additional fractionation.[60]

Once the appropriate value above has been calculated, Rmodern can be determined; it is[60]

$R_\left\{modern\right\} = 0.95R_\left\{HOxI,-19\right\} = .7459R_\left\{HOx2,-25\right\}$

The values 0.95 and 0.7459 are part of the definition of the two standards; they convert the / ratio in the standards to the ratio that modern carbon would have had in 1950 if there had been no fossil fuel effect.[60]

Since it is common practice to measure the standards repeatedly during an AMS run, alternating the standard target with the sample being measured, there are multiple measurements available for the standard, and these measurements provide a couple of options in the calculation of Rmodern. Different labs use this data in different ways; some simply average the values, while others consider the measurements made on the standard target as a series, and interpolate the readings that would have been measured during the sample run, if the standard had been measured at that time instead.[60]

Next, the uncorrected fraction modern is calculated; "uncorrected" means that this intermediate value does not include the fractionation correction.[60]

$Fm_\left\{uc\right\} = \frac \left\{R\text{'}_s\right\} \left\{R_\left\{modern\right\}\right\}$

Now the measured fraction modern can be determined, by correcting for fractionation. As above there are two equations, depending on whether the / or / ratio is being used. If the / ratio is being used:[60]

$Fm_\left\{ms\right\} = Fm_\left\{uc\right\}\left \left( \frac \left\{1 + \frac \left\{-25\right\} \left\{1000\right\}\right\} \left\{1 + \frac \left\{\delta 13C_s\right\} \left\{1000\right\}\right\} \right \right)^2$

If the / ratio is being used:[60]

$Fm_\left\{ms\right\} = Fm_\left\{uc\right\}\left \left( \frac \left\{1 + \frac \left\{-25\right\} \left\{1000\right\}\right\} \left\{1 + \frac \left\{\delta13C_s\right\} \left\{1000\right\}\right\} \right \right)$

The δ13Cs value is from the sample itself, measured on prepared while converting the sample to graphite.[60]

The final step is to adjust Fmms for the measured fraction modern of the process blank, Fmpb, which is calculated as above for the sample. One approach[note 5] is to determine the mass of the measured carbon, Cms, along with Cpb, the mass of the process blank, and Cs, the mass of the sample. The final fraction modern,Fms is then[60]

$Fm_s = \frac \left\{Fm_\left\{ms\right\}C_\left\{ms\right\} - Fm_\left\{pb\right\}C_\left\{pb\right\}\right\} \left\{C_s\right\}$

The fraction modern is then converted to an age in "radiocarbon years", meaning that the calculation uses Libby's half-life of 5,568 years, not the more accurate modern value of 5,730 years, and that no calibration has been done:[65]

$Age = -8033 ln \left(Fm\right)$

### Errors and reliability

There are several possible sources of error in both the beta counting and AMS methods.

For samples of sufficient size (several grams of carbon), these methods are still widely used. For instance, 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.

## Radiocarbon ages

Raw radiocarbon ages (i.e. before calibration) 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 incorrect 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 ± 30 BP 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, laboratories try to quote the overall uncertainty, which is determined from control samples of known age and verified by international intercomparison exercises.[66] 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).

Samples older than the upper age limit cannot be dated because the small number of remaining intrinsic atoms will be obscured by the background atoms introduced into the samples while they still resided in the environment, during sample preparation, or in the detection instrument. As of 2007, the limiting age for a 1 mg sample of graphite is about ten half-lives, approximately 60,000 years.[67] This age is derived from that of the calibration blanks used in an analysis, whose content is assumed to be the result of contamination during processing (as a result of this, some facilities[67] 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 Mya yielded apparent ages 64,920 ± 430 BP to 80,000 ± 1100 BP, as reported in 2007.[68]

## Calibration

### The need for calibration

Dates may be expressed as either uncalibrated or calibrated years (the latter abbreviated as cal or cal.). A raw BP date cannot be used directly as a calendar date, because the level of atmospheric has not been strictly constant during the span of time that can be radiocarbon dated, producing radiocarbon plateaus. The level is affected by variations in the cosmic ray intensity, which is, in turn, affected by variations in the Earth's magnetosphere.[70] 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 fraction.

As the graph to the right shows, the uncalibrated, raw BP date underestimates the actual age by 3,000 years at 15000 BP. The underestimation generally runs about 10% to 20%, with 3% of that underestimation attributable to the use of 5,568 years as the half-life of instead of the more accurate 5,730 years. To maintain consistency with a large body of published research, the out-of-date half-life figure is still used in all radiocarbon measurements.[71]

An uncalibrated radiocarbon date is abbreviated as yr BP or C14 yr BP or simply BP, although the last is ambiguously also sometimes used with dating methods other than radiocarbon, such as stratigraphy. A calibrated, or calendar date, is abbreviated as cal yr BP or cal BP, interpretable as "calibrated years before present" or "calendar years before present". In academic practice calibrated dates are generally presented along with their source uncalibrated dates, as the accuracy of the presently established calibration curve varies by time period.

The standard radiocarbon calibration curve is continuously being refined on the basis of new data gathered from tree rings, coral, and other studies. In addition to the natural variation of the curve throughout time, the carbon-14 level has also been affected by human activities in recent centuries. From the beginning of the industrial revolution in the 18th century to the 1950s, the fractional level of decreased because of the admixture of into the atmosphere from the combustion of fossil fuels. This decline, which is known as the Suess effect, also affects the isotope. However, atmospheric was almost doubled during the 1950s and 1960s, due to atmospheric atomic bomb tests.[72]

The calibration method also assumes that the temporal variation in level is global, such that a small number of samples from a specific year are sufficient for calibration.

Marine reservoir variations can be partly handled by a special marine calibration curve.[73]

### Calibration methods

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 - 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.[74]

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.[75]

In late 2009, the journal Radiocarbon announced agreement on the INTCAL09 standard, which extends a more accurate calibration curve to 50,000 years.[76][77] The results of research on varves in Lake Suigetsu, Japan, which was announced in 2012, realised this aim. "In most cases, the radiocarbon levels deduced from marine and other records have not been too far wrong. However, having a truly terrestrial record gives us better resolution and confidence in radiocarbon dating," said Bronk Ramsey. "It also allows us to look at the differences between the atmosphere and oceans and study the implications for our understanding of the marine environment as part of the global carbon cycle."[78]

In 2012, it was argued that there is an error in the way that the most-commonly-used calibration programs calculate calibrated radiocarbon ages.[79] As yet, no correction for this error has been implemented. The inaccuracy in the calibrated ages is typically small, but sometimes can be large (particularly in Bayesian analyses).

### Speleothem studies extend calibration

Speleothems (such as stalagmites) are calcium carbonate deposits that form from drips in limestone caves. Individual speleothems can be tens of thousands of years old.[80] Scientists are attempting to extend the record of atmospheric carbon-14 by measuring radiocarbon in speleothems that have been independently dated using uranium-thorium dating.[81][82] These results are improving the calibration for the radiocarbon technique and extending its usefulness to 45,000 years into the past.[83] Initial results from a cave in the Bahamas suggested a peak in the amount of carbon-14 that was twice as high as modern levels.[84] A recent study does not reproduce this extreme shift and suggests that analytical problems may have produced the anomalous result.[82]

## References

• Boutton, Thomas W. & Yamasaki, Shin-ichi (eds.) (1996). Mass Spectrometry of Soils. New York: Marcel Dekker, Inc. ISBN 0-8247-9699-3
• Eriksson Stenström, Kristina; Skog, Göran; Georgiadou, Elisavet; Genberg, Johan; & Johansson, Anette. "A guide to radiocarbon units and calculations". 2011. Lund:Lund University.
• Goudie, Andrew & Cuff, David J. (eds.) (2001). Encyclopedia of Global Change: Environmental Change and Human Society, Volume 1. Oxford: Oxford University Press. ISBN 0-19-514518-6
• Gove, H. E. (1999) From Hiroshima to the Iceman. The Development and Applications of Accelerator Mass Spectrometry. Bristol: Institute of Physics Publishing.
• Grootes (1992). "Subtle Signals: The Influence of Atmospheric Mixing, Growing Season and In-Situ Production". Radiocarbon 34 (2): 219–225.
• L'Annunziata, Michael F. (ed.) (2012). Handbook of Radioactivity Analysis. Oxford: Academic Press. ISBN 978-0-12-384873-4
• L'Annunziata, Michael F. (2007). Radioactivity: Introduction and History. Oxford: Elsevier. ISBN 978-0-444-52715-8
• Larsen, Clark Spencer (ed.) (2010). A Companion to Biological Anthropology. Oxford: Blackwell. ISBN 978-1-4051-8900-2
• Leng, Melanie J. (ed.) (2006). Isotopes in Palaeoenvironmental Research. Dordrecht: Springer. ISBN 978-1-4020-2503-7
• Mahaney, W.C. (ed.) (1984). Quaternary Dating Methods. Amsterdam: Elsevier. ISBN 0-444-42392-3
• Noller, Jay Stratton; Sowers, Janet M.; & Lettis, William R. (eds.) (2000). Quaternary Geochronology: Methods and Applications. Washington: American Geophysical Union. ISBN 0-87590-950-7
• Tykva, Richard and Berg, Dieter (eds.) (2004). Man-made and Natural Radioactivity in Environmental Pollution and Radiochronology. Dordrecht: Kluwer Academic Publishers. ISBN 1-4020-1860-6
• Weart, S. (2004) The Discovery of Global Warming - Uses of Radiocarbon Dating.
• Willis, E.H. (1996) Radiocarbon dating in Cambridge: some personal recollections. A Worm's Eye View of the Early Days.