Radiocarbon dating

"ports takes 50 ds at once' (or simply carbon dating) is a radiometric dating technique that uses the decay of carbon-14 (¹⁴
C) 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 CE 1950). Carbon dating was presented to the world by Willard Libby in 1949, for which he was awarded the Nobel Prize in Chemistry.

Since the introduction of carbon dating, the method has been used to date many items, including samples of the Dead Sea Scrolls, the Shroud of Turin, enough Egyptian artefacts to supply a chronology of Dynastic Egypt, and Ötzi the Iceman.

The Earth's atmosphere contains various isotopes of carbon, roughly in constant proportions. These include the main stable isotope (¹²
C) and an unstable isotope (¹⁴
C). Through photosynthesis, plants absorb both forms from carbon dioxide in the atmosphere. When an organism dies, it contains the standard ratio of ¹⁴
C to ¹²
C, but as the ¹⁴
C 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 ¹⁴
C. The measurement of the remaining proportion of ¹⁴
C in organic matter thus gives an estimate of its age (a raw radiocarbon age). However, over time there are small fluctuations in the ratio of ¹⁴
C to ¹²
C 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 (¹²
C), and carbon-13 (¹³
C), and a radioactive isotope, carbon-14 (¹⁴
C), also known as radiocarbon. The half-life of ¹⁴
C (the time it takes for half of a given amount of ¹⁴
C to decay) is about 5,730 years, so its concentration in the atmosphere might be expected to reduce over thousands of years. However, ¹⁴
C is constantly being produced in the lower stratosphere and upper troposphere by cosmic rays, which generate neutrons that in turn create ¹⁴
C when they strike nitrogen-14 (¹⁴
N) atoms.

1: Formation of carbon-14
2: Decay of carbon-14
3: The equation is for living organisms, and the inequality is for dead organisms, in which the ¹⁴
C then decays (See 2).

The carbon-14 creation process is described by the following nuclear reaction, where n represents a neutron and p represents a proton:

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

Once produced, the ¹⁴
C quickly combines with the oxygen in the atmosphere to form carbon dioxide (CO
₂). 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.

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. Since it is also lost through radioactivity at a constant rate and the proportion of radioactive to non-radioactive carbon is constant, the rate of production of carbon-14 must be equal to the rate of depletion. The ratio of ¹⁴
C to ¹²
C in the carbon exchange reservoir is 1.5 parts of ¹⁴
C to 10¹² parts of ¹²
C. In addition, about 1% of the reservoir is made up of the stable isotope ¹³
C.

Invention of the methods

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. 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. 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. These measurements, published in Science in 1949, launched the "radiocarbon revolution" in archaeology, and soon led to dramatic changes in scholarly chronologies. In 1960, Libby was awarded the Nobel Prize in chemistry for this work.

Calculating ages

While a plant or animal is alive, it is exchanging carbon with its surroundings, so that the carbon it contains will have the same proportion of ¹⁴
C as the biosphere. Once it dies, it ceases to acquire ¹⁴
C, but the ¹⁴
C that it contains will continue to decay, and so the proportion of radiocarbon in its remains will gradually reduce. Because ¹⁴
C decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon—the older the sample, the less ¹⁴
C will be left.

The equation governing the decay of a radioactive isotope is

$N=N_{0}e^{-\lambda t}\,$

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. 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. The mean-life, denoted by τ, of ¹⁴
C is 8,267 years, so the equation above can be rewritten as:

$t=8267\cdot \ln(N_{0}/N)$

The ratio of ¹⁴
C atoms in the original sample, N₀, is taken to be the same as the ratio in the biosphere, so measuring N, the number of ¹⁴
C atoms currently in the sample, allows the calculation of t, the age of the sample.

The half-life of a radioactive isotope (the time it takes for half of the sample to decay, usually denoted by T_1/2) 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 ¹⁴
C's half-life than its mean-life. The currently accepted value for the half-life of radiocarbon is 5,730 years. The mean-life and half-life are related by the following equation:

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

The above calculations make several assumptions: for example, that the level of ¹⁴
C in the biosphere has remained constant over time. In fact, the level of ¹⁴
C 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. For over a decade after Libby's initial work, the accepted value of the half-life for ¹⁴
C 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. 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.

Carbon exchange reservoir

Simplified version of the carbon exchange reservoir, showing proportions of carbon and relative activity of the ¹⁴
C in each reservoir

The different elements of the carbon exchange reservoir vary in how much carbon they store, and in how long it takes for the ¹⁴
C generated by cosmic rays to fully mix with them. The atmosphere, which is where ¹⁴
C is generated, contains about 1.9% of the total carbon in the reservoirs, and the ¹⁴
C it contains mixes in less than 7 years. The ratio of ¹⁴
C to ¹²
C in the atmosphere is taken as the baseline for the other reservoirs: if another reservoir has a lower ratio of ¹⁴
C to ¹²
C, it indicates that the carbon is older, and hence some of the ¹⁴
C has decayed. The ocean surface is an example: it contains 2.4% of the carbon in the exchange reservoir, but there is only about 95% as much ¹⁴
C as would be expected if the ratio were the same as in the atmosphere. The time it takes for carbon from the atmosphere to mix with the surface ocean is only a few years, but the surface waters also receive water from the deep ocean, which has over 90% of the carbon in the reservoir. 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 ¹⁴
C, and water recently at the surface, with ¹⁴
C in equilibrium with the atmosphere.

Creatures living at the ocean surface have the same ¹⁴
C 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. Organisms on land, however, are in closer equilibrium with the atmosphere and have the same ¹⁴
C/¹²
C ratio as the atmosphere. 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. 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 ¹⁴
C/¹²
C ratio lower than that of the biosphere.

Dating considerations

The variation in the ¹⁴
C/¹²
C 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 ¹⁴
C 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 ¹⁴
  C/¹²
  C ratio in the atmosphere, both geographically and over time
• Isotopic fractionation
• Variations in the ¹⁴
  C/¹²
  C ratio in different parts of the reservoir
• Contamination

Atmospheric variation

In the early years of using the technique, it was understood that it depended on the atmospheric ¹⁴
C/¹²
C ratio having remained 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, in 1958, Hessel de Vries pointed out that this was not the case, by testing wood samples of known ages and showing there was a significant deviation from the expected ¹⁴
C/¹²
C ratio. This discrepancy, often called the de Vries effect, was resolved by the study of tree-rings. The 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 ¹⁴
C/¹²
C ratio: with a sample of known date, and a measurement of the value of N (the number of atoms of ¹⁴
C remaining in the sample), the carbon-dating equation allows the calculation of N₀ (the number of atoms of ¹⁴
C in the original sample), and hence the original ratio. 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 ¹⁴
C/¹²
C ratio. These curves are described in more detail below. There are three main reasons for these variations in the historical ¹⁴
C/¹²
C ratio: fluctuations in the rate at which ¹⁴
C is created; changes caused by glaciation; and changes caused by human activity.

Variations in ¹⁴
C 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 ¹⁴
C is produced, leading to a younger apparent age for samples from those periods. Conversely, a higher magnetic field leads to lower ¹⁴
C 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 ¹⁴
C.

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 ¹⁴
C 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 ¹⁴
C 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 ¹⁴
C production.

Since the earth's magnetic field varies with latitude, the rate of ¹⁴
C production changes with latitude too, but atmospheric mixing is rapid enough that these variations amount to less than 0.5% of the global concentration. This is close to the limit of detectability in most years, but the effect can be seen clearly in tree rings from years such as 1963, when ¹⁴
C from nuclear testing rose sharply through the year. The latitudinal variation in ¹⁴
C was much larger than normal that year, and tree rings from different latitudes show corresponding variations in their ¹⁴
C content.

¹⁴
C 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 ¹⁴
C at a rate of 1 x 10⁻⁴ atoms per gram per second, which is not enough to have a significant impact on dating. At higher altitudes, the neutron flux can be substantially higher, 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 ¹⁴
C content is minor; though for very old trees (such as some bristlecone pines) that grow at altitude, some effect can be seen.

Impact of climatic cycles

Because the solubility of CO
₂ in water increases with lower temperatures, glacial periods would have led to the faster absorption of atmospheric CO
₂ by the oceans. In addition, any carbon stored in the glaciers would be depleted in ¹⁴
C over the life of the glacier; when the glacier melted, as the climate warmed, the depleted carbon would be released, reducing the global ¹⁴
C/¹²
C 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.

The effects of human activity

Atmospheric ¹⁴
C, New Zealand and Austria. The New Zealand curve is representative of the Southern Hemisphere; the Austrian curve is representative of the Northern Hemisphere. Atmospheric nuclear weapon tests almost doubled the concentration of ¹⁴
C in the Northern Hemisphere. The date that the Partial Test Ban Treaty (PTBT) went into effect is marked on the graph.

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 ¹⁴
C and, as a result, the CO
₂ released substantially diluted the atmospheric ¹⁴
C/¹²
C ratio. Dating an object from the early 20th century hence gives an apparent date older than the true date; and for the same reason, ¹⁴
C 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 ¹⁴
C 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.

A much larger effect comes from above-ground nuclear testing, which released large numbers of neutrons and created ¹⁴
C. From about 1950 until 1963, when atmospheric nuclear testing was banned, it is estimated that several tonnes of ¹⁴
C were created. If all this extra ¹⁴
C had immediately been spread across the entire carbon exchange reservoir, it would have led to an increase in the ¹⁴
C/¹²
C ratio of only a few per cent, but the immediate effect was to almost double the amount of ¹⁴
C in the atmosphere, with the peak level occurring in about 1965. The level has since dropped, as the "bomb carbon" (as it is sometimes called) percolates into the rest of the reservoir.

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 ¹²
C being absorbed slightly more easily than ¹³
C, which in turn is more easily absorbed than ¹⁴
C. The differential uptake of the three carbon isotopes leads to ¹³
C/¹²
C and ¹⁴
C/¹²
C ratios in plants that differ from the ratios in the atmosphere. This effect is known as isotopic fractionation.

To determine the degree of fractionation that takes place in a given plant, the amounts of both ¹²
C and ¹³
C are measured, and the resulting ¹³
C/¹²
C ratio is then compared to a standard ratio known as PDB. The resulting value, known as δ¹³C, is calculated as follows:

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

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

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

which makes it apparent that δ¹³C is proportional to the difference between the ¹³
C/¹²
C ratios in the PDB standard and in the sample. Because the PDB standard contains an unusually high proportion of ¹³
C, 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‰. Atmospheric CO
₂ has a δ¹³C of −8‰.

Sheep on the beach in North Ronaldsay. In the winter, these sheep eat seaweed, which has a higher δ¹³C content than grass; samples from these sheep have a δ¹³C value of about −13‰, which is much higher than for sheep that feed on grasses.

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, CO
₂ has poor solubility in water, which means there is less CO
₂ 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, CO
₂ becomes more soluble and hence more available to the marine organisms; fractionation increases and δ¹³C values can be as low as −32‰.

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. 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 ¹³
C than is found in the animal's diet, though for different biochemical reasons. The enrichment of bone ¹³
C also implies that excreted material is depleted in ¹³
C relative to the diet.

Since ¹³
C makes up about 1% of the carbon in a sample, the ¹³
C/¹²
C ratio can be accurately measured by mass spectrometry. 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. The depletion of ¹³
C relative to ¹²
C is proportional to the difference in the atomic masses of the two isotopes, so once the δ¹³C value is known, the depletion for ¹⁴
C can be calculated: it will be twice the depletion of ¹³
C.

The carbon exchange between atmospheric CO
₂ and carbonate at the ocean surface is also subject to fractionation, with ¹⁴
C in the atmosphere more likely than ¹²
C to dissolve in the ocean. The result is an overall increase in the ¹⁴
C/¹²
C ratio in the ocean of 1.5%, relative to the ¹⁴
C/¹²
C ratio in the atmosphere. This increase in ¹⁴
C concentration almost exactly cancels out the decrease caused by the upwelling of water (containing old, and hence ¹⁴
C depleted, carbon) from the deep ocean, so that direct measurements of ¹⁴
C 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.

Reservoir effects

Libby's original exchange reservoir hypothesis assumed that the exchange reservoir is constant all over the world, but it has since been discovered that there are several causes of variation in the ¹⁴
C/¹²
C ratio across the reservoir.

Marine effect

The CO
₂ 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 CO
₂. This exchange process brings¹⁴
C from the atmosphere into the surface waters of the ocean, but the ¹⁴
C 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 CO
₂ 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). 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. 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. 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.

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 ¹⁴
C/¹²
C 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 ¹⁴
C, so this carbon lowers the ¹⁴
C/¹²
C 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. 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.

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.

Volcanoes

Volcanic eruptions eject large amounts of carbon into the air. The carbon is of geological origin and has no detectable ¹⁴
C, so the ¹⁴
C/¹²
C 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 ¹⁴
C/¹²
C 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.

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 ¹⁴
C/¹²
C 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 ¹⁴
C because of the marine effect, ¹⁴
C is removed from the southern atmosphere more quickly than in the north.

Island effect

It has been suggested that an "island effect" might exist, by analogy with the mechanism thought to explain the hemisphere effect—since islands are surrounded by water, the carbon exchange between the water and atmosphere might reduce the ¹⁴
C/¹²
C 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.

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 of 4,000 years. Contamination with old carbon, with no remaining ¹⁴
C, 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. The equation for the radioactivity of a sample that has been contaminated with other carbon is

$A_{m}=fA_{x}+(1-f)A_{s}$

where A_m is the measured radioactivity of the sample, A_x is the radioactivity of the contaminating material, A_s 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.

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. 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. Samples should be packed in glass vials or aluminium foil if possible; polyethylene bags are also acceptable but some plastics, such as PVC, can contaminate the sample. Contamination can also occur before the sample is collected: humic acids or carbonate from the 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 ¹⁴
C content.

Samples

Samples for dating need to be converted into a form suitable for measuring the ¹⁴
C 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. This includes removing visible contaminants, such as rootlets that may have penetrated the sample since its burial.

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.

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 the volume of the sample down to 20% of the original size, so testing of the 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.

Bone

Unburnt bone was once thought to be a poor candidate for radiocarbon dating, 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 ¹⁴
C 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.

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.

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.

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.

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. A particular difficulty with dried peat is the removal of rootlets, which are likely to be hard to distinguish from the sample material.
• 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.
• 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.

Isotopic enrichment

Particularly for older samples, it may be useful to enrich the amount of ¹⁴
C 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 ¹⁴
C/¹²
C ratio in old material, and extends the maximum age that can be reliably reported.

Preparation

Once contamination has been removed, samples must be converted to a form suitable for the measuring technology to be used. A common approach is to produce a gas, for gas counting devices: CO
₂ is widely used, but it is also possible to use other gases, including methane, ethane, ethylene and acetylene. 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, but this technique is no longer in use; these methods were susceptible to problems caused by the ¹⁴
C created by nuclear testing in the 1950s and 1960s. Solid targets can be used for accelerator mass spectrometry, however; usually these are graphite, though CO
₂ and iron carbide can also be used.

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 CO
₂ 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 CO
₂, 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 CO
₂ 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.

To create benzene for liquid scintillation counters, the sequence begins with combustion to convert the carbon in the sample to CO
₂. This is then converted to lithium carbide, and then to acetylene, and finally to benzene. Targets for accelerator mass spectrometry are created from CO
₂ 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.

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.

Sample material	Mass (g)
	For beta counters	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 ¹⁴
C in a sample was to detect the radioactive decay of individual carbon atoms. In this approach, what is measured is the activity, in number of decay events per unit mass per time period, of the sample. This method is also known as "beta counting", because it is the beta particles emitted by the decaying ¹⁴
C atoms that are detected. In the late 1970s an alternative approach became available: directly counting the number of ¹⁴
C and ¹²
C atoms in a given sample, via accelerator mass spectrometry, usually referred to as AMS. AMS counts the ¹⁴
C/¹²
C ratio directly, instead of the activity of the sample, but measurements of activity and ¹⁴
C/¹²
C ratio can be converted into each other exactly. For some time, beta counting methods were more accurate than AMS, but there is now little to choose between them, though AMS still cannot compete with the very highest-precision beta counting laboratories, which can provide results with a standard error of ± 20 years.

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. Any interposing material would have interfered with the detection of radioactivity; the beta particles emitted by decaying ¹⁴
C are so weak that half are stopped by a 0.01 mm thickness of aluminium.

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 ¹⁴
C 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.

The other common technology used for measuring ¹⁴
C 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 ¹⁴
C as they interact with a fluorescing agent added to the benzene. Like gas counters, liquid scintillation counters require shielding and anticoincidence counters.

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 ¹⁴
C. In addition, a sample with a standard activity will be measured, in order to provide a baseline for comparison.

Accelerator mass spectrometry

Simplified schematic layout of an accelerator mass spectrometer used for counting carbon isotopes for carbon dating

AMS counts the atoms of ¹⁴
C and ¹²
C atoms in a given sample, determining the ¹⁴
C/¹²
C 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 ¹⁴
C stream, but ¹²
C counts (and ¹³
C 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.

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 ¹⁴
N and ¹³
CH. 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. 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 ¹⁴
C 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 ¹²
CH
₂ or ¹³
CH. A ¹⁴
C 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.

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.

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 ¹³
C/¹²
C 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 ¹⁴
C/¹²
C 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.

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. 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, A_abs, is 226 becquerels per kilogram of carbon.

Both beta counting and AMS measure standard samples as part of their methodology. These samples contain carbon of a known activity. 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.

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.

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:

$A_{s}=A_{std}\left({\frac {M_{s}-M_{b}}{M_{std}-M_{b}}}\right)$

gives the required ratio, where A_s is the true activity of the sample, A_std is the true activity of the standard, M_s is the measured activity of the sample, M_std is the measured activity of the standard, and M_b is the measured activity of the blank.

A correction must also be made for fractionation. The fractionation correction converts the ¹⁴
C/¹²
C ratio for the sample to the ratio it would have had if the material was wood, which has a δ¹³C value of -25‰. This is necessary because determining the age of the sample requires a comparison of the amount of ¹⁴
C 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.

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 ¹³
C fractionation factor, which is defined for any sample material as

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

The ¹⁴
C fractionation factor, Frac_14/12, is approximately the square of this, to an accuracy of 1‰:

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

Multiplying the measured activity for the sample by the ¹⁴
C fractionation factor converts it to the activity that it would have had had the sample been wood:

$A_{sn}=A_{s}Frac_{14/12(s)}$

where A_sn is the normalized activity for the sample, and Frac_{14/12 (s)} is the ¹⁴
C fractionation factor for the sample.

The equation for δ13C given earlier can be rearranged to

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

Substituting this in the ¹⁴
C fractionation factor, and also substituting the value for δ13C for wood of -25‰, gives the following expression:

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

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 ¹³
C/¹²
C ratio reduces this to:

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

AMS calculations

The results from AMS testing are in the form of ratios of ¹²
C, ¹³
C, and ¹⁴
C. These ratios are used to calculate F_m, the "fraction modern", defined as

$F_{m}={\frac {R_{norm}}{R_{modern}}}$

where R_norm is the ¹⁴
C/¹²
C ratio for the sample, after correcting for fractionation, and R_modern is the standard ¹⁴
C/¹²
C ratio for modern carbon.

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

$R'_{s}=R_{s}-R_{mb}$

$R'_{std}=R_{std}-R_{mb}$

$R'_{pb}=R_{pb}-R_{mb}$

where R_s is the measured sample ¹⁴
C/¹²
C ratio; R_std is the measured ratio for the standard; R_pb is the measured ratio for the process blank, and R_mb is the measured ratio for the machine blank. The next step, to correct for fractionation, can be done using either the ¹⁴
C/¹²
C ratio or the ¹⁴
C/¹³
C 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 ¹⁴
C/¹²
C ratio is used to perform the fractionation correction, the following two equations apply, one for each standard.

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

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

If the ¹⁴
C/¹³
C ratio is used instead, then the equations for each standard are:

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

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

The δ13C values in the equations measure the fractionation in the standards as CO
₂, 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.

Once the appropriate value above has been calculated, R_modern can be determined; it is

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

The values 0.95 and 0.7459 are part of the definition of the two standards; they convert the ¹⁴
C/¹²
C ratio in the standards to the ratio that modern carbon would have had in 1950 if there had been no fossil fuel effect.

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 R_modern. 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.

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

$Fm_{uc}={\frac {R'_{s}}{R_{modern}}}$

Now the measured fraction modern can be determined, by correcting for fractionation. As above there are two equations, depending on whether the ¹⁴
C/¹²
C or ¹⁴
C/¹³
C ratio is being used. If the ¹⁴
C/¹²
C ratio is being used:

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

If the ¹⁴
C/¹³
C ratio is being used:

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

The δ13C_s value is from the sample itself, measured on CO
₂ prepared while converting the sample to graphite.

The final step is to adjust Fm_ms for the measured fraction modern of the process blank, Fm_pb, which is calculated as above for the sample. One approach is to determine the mass of the measured carbon, C_ms, along with C_pb, the mass of the process blank, and C_s, the mass of the sample. The final fraction modern,Fm_s is then

$Fm_{s}={\frac {Fm_{ms}C_{ms}-Fm_{pb}C_{pb}}{C_{s}}}$

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:

$Age=-8033ln(Fm)$

Errors and reliability

There are several possible sources of error in both the beta counting and AMS methods, although laboratories vary in how they report errors. All laboratories report counting statistics—that is, statistics showing possible errors in counting the decay events or number of atoms—with an error term of 1σ (i.e. 68% confidence that the true value is within the given range). These errors can be reduced by extending the counting duration: for example, testing a modern benzene sample will find about eight decay events per minute per gram of benzene, and 250 minutes of counting will suffice to give an error of ± 80 years, with 68% confidence. If the benzene sample contains carbon that is about 5,730 years old (the half-life of ¹⁴
C), then there will only be half as many decay events per minute, but the same error term of 80 years could be obtained by doubling the counting time to 500 minutes. Note that the error term is not symmetric, though the effect is negligible for recent samples; for a sample with an estimated age of 30,600 years, the error term might be +1600 to -1300.

To be completely accurate, the error term quoted for the reported radiocarbon age should incorporate counting errors not only from the sample, but also from counting decay events for the reference sample, and for blanks. It should also incorporate errors on every measurement taken as part of the dating method, including, for example, the δ13C term for the sample, or any laboratory conditions being corrected for such as temperature or voltage. These errors should then be mathematically combined to give an overall term for the error in the reported age, but in practice laboratories differ, not only in the terms they choose to include in their error calculations, but also in the way they combine errors. The resulting 1σ estimates have been shown to typically underestimate the true error, and it has even been suggested that doubling the given 1σ error term results in a more accurate value.

The usual presentation of a radiocarbon date, as a specific date plus or minus an error term, obscures the fact that the true age of the object being measured may lie outside the range of dates quoted. In 1970, the British Museum radiocarbon laboratory ran weekly measurements on the same sample for six months. The results varied widely (though consistently with a normal distribution of errors in the measurements), and included multiple date ranges (of 1σ confidence) that did not overlap with each other. The extreme measurements included one with a maximum age of under 4,400 years, and another with a minimum age of over 4,500 years.

It is also possible for laboratories to have systematic errors, caused by weaknesses in their methodologies. For example, if 1% of the benzene in a modern reference sample is allowed to evaporate, the resulting radiocarbon age will be too young by about 80 years. Laboratories work to detect these errors both by testing their own procedures, and by periodic inter-laboratory comparisons of a variety of different samples; any laboratories whose results differ from the consensus radiocarbon age by too great an amount may be suffering from systematic errors. Even if the systematic errors are not corrected, the laboratory can estimate the magnitude of the effect and include this in the published error estimates for their results.

The limit of measurability is approximately eight half-lives, or about 45,000 years. Samples older than this will typically be reported as having an infinite age. Some techniques have been developed to extend the range of dating further into the past, including isotopic enrichment, or large samples and very high precision counters. These methods have in some cases increased the maximum age that can be reported for a sample to 60,000 and even 75,000 years.

Calibration

The calculations given above produce dates in radiocarbon years: that is, dates which represent the age the sample would be if the ¹⁴
C/¹²
C ratio had been constant historically. Although Libby had pointed out as early as 1955 the possibility that this assumption was incorrect, it was not until discrepancies began to accumulate between measured ages and known historical dates for artefacts that it became clear that a correction would need to be applied to radiocarbon ages to obtain calendar dates.

To produce a curve that can be used to relate calendar years to radiocarbon years, a sequence of securely dated samples is needed which can be tested to determine their radiocarbon age. The study of tree-rings (dendrochronology) led to the first such sequence: tree-rings from individual pieces of wood show characteristic sequences of rings that vary in thickness because of environmental factors such as the amount of rainfall in a given year. These factors affect all trees in an area, so examining tree-ring sequences from old wood allows the identification of overlapping sequences. In this way, an uninterrupted sequence of tree-rings can be extended far into the past. The first such published sequence, based on bristlecone pine tree-rings, was created in the 1960s by Wesley Ferguson. Hans Suess used this data to publish the first calibration curve for radiocarbon dating in 1967. The curve showed two types of variation from the straight line: a long term fluctuation with a period of about 9,000 years, and a shorter term variation, often referred to as "wiggles", with a period of decades. Suess said he drew the line showing the wiggles by "cosmic schwung" – freehand, in other words. It was unclear for some time whether the wiggles were real or not, but they are now well-established.

The calibration method also assumes that the temporal variation in ¹⁴
C level is global, such that a small number of samples from a specific year are sufficient for calibration. This was experimentally verified in the 1980s.

Over the next thirty years many calibration curves were published,using a variety of methods and statistical approaches. These were superseded by the INTCAL series of curves, beginning with INTCAL98, published in 1998, and updated in 2004, 2009, and, most recently, 2013. The improvements to these curves are based on new data gathered from tree rings, varves, coral, and other studies. Significant additions to the datasets used for INTCAL13 include non-varved marine foraminifera data, and U-Th dated speleothems. Marine reservoir variations are partly handled by a special marine calibration curve.

Calibration curve for the radiocarbon dating scale. Data sources: Reimer, P. J., et al. (1998). Samples with a real date more recent than AD 1950 are dated and/or tracked using the N- & S-Hemisphere graphs.

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 ¹⁴
C instead of the more accurate 5,730 years.

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.

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.

Speleothem studies extend ¹⁴
C 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. 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. These results are improving the calibration for the radiocarbon technique and extending its usefulness to 45,000 years into the past. 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. A recent study does not reproduce this extreme shift and suggests that analytical problems may have produced the anomalous result.

Reporting dates

Several different formats for citing radiocarbon results have been used since the first samples were dated. As of 2014, the standard format required by the journal Radiocarbon is as follows.

Uncalibrated dates should be reported as "<laboratory>: <¹⁴
C year> ± <range> BP", where:

• <laboratory> identifies the laboratory that tested the sample, and the sample ID
• <¹⁴
  C year> is the laboratory's determination of the age of the sample, in radiocarbon years
• <range> is the laboratory's estimate of the error in the age, at 1 σ confidence.
• BP stands for "before present", referring to the reference date of 1950, so that 500 BP means the year 1450 AD.

For example, the uncalibrated date "UtC-2020: 3510 ± 60 BP" indicates that the sample was tested by the Utrecht van der Graaf Laboratorium, where it has a sample number of 2020, and that the uncalibrated age is 3510 years before present, ± 60 years. Related forms are sometimes used: for example, "10 ka BP" means 10,000 radiocarbon years before present, and ¹⁴
C yr BP might be used to distinguish the uncalibrated date from a date derived from another dating method such as thermoluminescence.

Radiocarbon gives two options for reporting calibrated dates. A common format is "cal <date-range> <confidence>", where:

• <date-range> is the range of dates corresponding to the given confidence level
• <confidence> indicates the confidence level for the given date range.

For example, "cal 1220–1281 AD (1σ)" means a calibrated date for which the true date lies between 1220 AD and 1281 AD, with the confidence level given as 1σ, or one standard deviation. Calibrated dates can also be expressed as BP instead of using BC and AD. The curve used to calibrate the results should be the latest available INTCAL curve. Calibrated dates should also identify any programs, such as OxCal, used to perform the calibration.

Examples

• Ancient footprints of Acahualinca
• Chauvet Cave
• The Dead Sea Scrolls
• Dolaucothi
• Eve of Naharon
• Haraldskær Woman
• Kennewick Man
• Shroud of Turin
• Skeleton Lake
• Minoan eruption
• Vinland map

See also

• Cosmogenic isotopes
• Old wood

References

• Aitken, M. J. (1990). Science-based Dating in Archaeology. London: Longman. ISBN 0-582-49309-9.
• Boutton, Thomas W. & Yamasaki, Shin-ichi (eds.) (1996). Mass Spectrometry of Soils. New York: Marcel Dekker, Inc. ISBN 0-8247-9699-3
• Bowman, Sheridan (1995) [1990]. Radiocarbon Dating. London: British Museum Press. ISBN 0-7141-2047-2.
• Brown, Larry; Brown, Lawrence Steven; Holme, Tom (2010) [2006]. Chemistry for Engineering Students (2nd ed.). Belmont, CA: Brooks/Cole. ISBN 978-1-4390-4791-0.
• Burke, Heather; Smith, Claire; Zimmerman, Larry J. (2009). The Archaeologist's Field Handbook (North American ed.). Lanham, MD: AltaMira Press. ISBN 978-0-7591-0882-0.
• Cronin, Thomas M. (2010). Paleoclimates: Understanding Climate Change Past and Present. New York: Columbia University Press. ISBN 978-0-231-14494-0.
• Currie, L. (2004). "The Remarkable Metrological History of Radiocarbon Dating II" (PDF). J. Res. Natl. Inst. Stand. Technol. 109: 185–217. doi:10.6028/jres.109.013.
• Eriksson Stenström, Kristina; Skog, Göran; Georgiadou, Elisavet; Genberg, Johan; & Johansson, Anette. "A guide to radiocarbon units and calculations". 2011. Lund:Lund University.
• Fairchild, Ian J.; Baker, Andy (2012). Speleothem Science: From Process to Past Environments. Oxford: John Wiley & Sons. ISBN 978-1-4051-9620-8.
• Friedrich, M. (2004). "The 12,460-Year Hohenheim Oak and Pine Tree-Ring Chronology from Central Europe—a Unique Annual Record for Radiocarbon Calibration and Paleoenvironment Reconstructions". Radiocarbon. 46: 1111–1122. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• 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 ¹⁴
  C Signals: The Influence of Atmospheric Mixing, Growing Season and In-Situ Production". Radiocarbon 34 (2): 219–225.
• Kovar, Anton J. (1966). "Problems in Radiocarbon Dating at Teotihuacan". American Antiquity. 31 (3). Society for American Archaeology: 427–430. doi:10.2307/2694748. JSTOR 2694748.
• 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
• McNichol, A. P.; Jull, A. T. S.; Burr, G. S. (2001). "Converting AMS Data To Radiocarbon Values: Considerations And Conventions". Radiocarbon. 43: 313–320.
• Miller, Charles B.; Miller, Patricia A. (2012) [2003]. Biological Oceanography (2nd ed.). Oxford: John Wiley & Sons. ISBN 978-1-4443-3301-5.
• Lorenz, R. D.; Jull, A. J. T.; Lunine, J. I.; Swindle, T. (2002). "Radiocarbon on Titan". Meteoritics and Planetary Science. 37 (6): 867–874. Bibcode:2002M&PS...37..867L. doi:10.1111/j.1945-5100.2002.tb00861.x.
• Mook, W. G.; van der Plicht, J. (1999). "Reporting ¹⁴
  C activities and concentrations" (PDF). Radiocarbon. 41: 227–239.
• 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
• Olsson, Ingrid U. (ed.) (1970). Radiocarbon Variations and Absolute Chronology. New York: John Wiley & Sons, Inc.
• "Radiocarbon: Information for Authors" (PDF). Radiocarbon. University of Arizona. May 25, 2011. Retrieved January 1, 2014.
• Rasskazov, Sergei V.; Brandt, Sergei Borisovich; Brandt, Ivan S. (2009). Radiogenic Isotopes in Geologic Processes. Dordrecht: Springer. ISBN 978-90-481-2998-0.
• Taylor, R.E. (1987). Radiocarbon Dating. London: Academic Press. ISBN 0-12-433663-9.
• Theodórsson, Páll (1996). Measurement of Weak Radioactivity. Singapore: World Scientific Publishing. ISBN 9810223153.
• 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
• Walker, Mike (2005). Quaternary Dating Methods. Chichester: John Wiley & Sons. ISBN 978-0-470-86927-7.
• Warneck, Peter (2000). Chemistry of the Natural Atmosphere. London: Academic Press. ISBN 0-12-735632-0.
• 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.

External links

• Radiocarbon - The main international journal of record for research articles and date lists relevant to ¹⁴
  C
• C14dating.com - General information on Radiocarbon dating
• calib.org - Calibration program, Marine Reservoir database, and bomb calibration
• NOSAMS: National Ocean Sciences Accelerator Mass Spectrometry Facility at the Woods Hole Oceanographic Institution
• Discussion of calibration (from University of Oxford)
• Several calibration programs can be found at www.radiocarbon.org
• CalPal Online (Cologne Radiocarbon Calibration & Paleoclimate Research Package)
• OxCal program (Oxford Calibration)
• Fairbanks' Radiocarbon Calibration program (for prior to 12400 BP)
• Notes on radiocarbon dating, including movies illustrating the atomic physics (from UC Santa Barbara)
• Carbon Dating-How it works?
• How is physics used in archaeology? (from physics.org)
• RADON - database for European ¹⁴C dates