(Redirected from Carbon-14 dating)

Radiocarbon dating is a method of determining the age of an object by using the properties of radiocarbon, a radioactive isotope of carbon. The method was invented by Willard Libby in the late 1940s and soon became a standard tool for archaeologists. It depends on the fact that radiocarbon, often abbreviated as 14
C
, is constantly being created in the atmosphere by cosmic rays. The resulting radioactive carbon dioxide is incorporated into plants by photosynthesis, and animals acquire 14
C
by eating the plants. When the animal or plant dies, it stops exchanging carbon with its environment, and from that point the amount of 14
C
it contains begins to reduce as the 14
C
undergoes radioactive decay. Measuring the amount of 14
C
in a sample from a dead plant or animal such as piece of old wood or a fragment of bone provides information that can be used to calculate when the animal or plant died. The oldest dates that can be reliably measured by radiocarbon dating are around 50,000 years ago, though special preparation methods occasionally permit dating of older samples.

While the idea behind radiocarbon dating is straightforward, years of additional work were required to develop the technique to the point where accurate dates could be obtained. Research has been going on since the 1960s to determine what the proportion of 14
C
in the atmosphere has been over the past fifty thousand years. The resulting data, in the form of a calibration curve, is now used to convert a given measurement of radiocarbon in a sample into an estimate of the sample's actual calendar age. In addition to this curve, other corrections must be made to account for different proportions of 14
C
in different types of organism (fractionation) and different 14
C
levels in different parts of the biosphere (reservoir effects).

Measurement of radiocarbon was originally done by beta-counting devices, so called because they counted the amount of beta radiation emitted by decaying 14
C
atoms in a sample. More recently, accelerator mass spectrometry has become the method of choice; it can be used with much smaller samples (as small as individual plant seeds), and gives results much more quickly.

The development of radiocarbon dating has had a profound impact on archaeology. In addition to permitting more accurate dating within archaeological sites than did methods previously in use, it also allows comparison of dates of events across great distances. Histories of archaeology often refer to the early impact of the new method as the “radiocarbon revolution”. Occasionally, the method is used for items of popular interest such as the Shroud of Turin, which is claimed to show an image of the body of Jesus Christ. A sample of linen from the shroud was tested in 1988 and found to date from the 1200s or 1300s, casting doubt on its authenticity.

## Background

### History

In the early 1930s Willard Libby was a chemistry student at the University of Berkeley, receiving his Ph.D. in 1933. He remained there as an instructor until the end of the decade. In 1939 the Radiation Laboratory at Berkeley began experiments to determine if any of the elements common in organic matter had isotopes with half-lives long enough to be of value in biomedical research. It was soon discovered that 14
C
's half-life was far longer than had been previously thought, and in 1940 this was followed by proof that the interaction of slow neutrons with 14
N
was the main pathway by which 14
C
was created. It had previously been thought 14
C
would be more likely to be created by deuterons interacting with 13
C
. At about this time Libby read a paper by W. E. Danforth and S. A. Korff, published in 1939, which predicted the creation of 14
C
in the atmosphere by neutrons from cosmic rays which had been slowed down by collisions with molecules of atmospheric gas. It was this paper that first gave Libby the idea that radiocarbon dating might be possible.[1]

In 1945, Libby moved to the University of Chicago. He published a paper in 1946 in which he proposed that the carbon in living matter might include 14
C
as well as non-radioactive carbon.[2][3] 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 14
C
. 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.[2][4]

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 results were published in Science in 1949.[5][6] In 1960, Libby was awarded the Nobel Prize in Chemistry for this work.[2]

### Physical and chemical details

Main article: Carbon-14

In nature, carbon exists as 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.[2] The 14
C
creation process is described by the following nuclear reaction:

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

where n represents a neutron and p represents a proton.[7]

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.[8]

If it is assumed that the cosmic ray flux has been constant over the last ~100,000 years,[2] then 14
C
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 14
C
is equal to the rate of its depletion. The ratio of 14
C
to 12
C
in the carbon exchange reservoir is approximately 1.5 parts of 14
C
to 1012 parts of 12
C
C
.[2]

The equation for the radioactive decay of 14
C
is:[9]

$\mathrm{~^{14}_{6}C}\rightarrow\mathrm{~^{14}_{7}N}+ e^- + \bar{\nu}_e$

By emitting a beta particle (an electron) and an electron antineutrino, one of the neutrons in the 14
C
nucleus changes to a proton and the 14
C
nucleus changes back into the stable (non-radioactive) isotope nitrogen-14 14
N
.[10]

### Principles

During its life, a plant or animal is exchanging carbon with its surroundings, so the carbon it contains will have the same proportion of 14
C
as the biosphere and the carbon exchange reservoir. Once it dies, it ceases to acquire 14
C
, but the 14
C
within its biological material at that time will continue to decay, and so the ratio of 14
C
to 12
C
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.[8]

The equation governing the decay of a radioactive isotope is:[2]

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

where N0 is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t.[2] 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.[2] The mean-life, denoted by τ, of 14
C
is 8,267 years, so the equation above can be rewritten as:[11]

$t = 8267 \cdot \ln(N_0/N) years = 19035 \cdot \log(N_0/N) years$

The sample is assumed to have originally had the same 14
C
/12
C
ratio as the ratio in the biosphere, and since the size of the sample is known, the total number of atoms in the sample can be calculated, yielding N0, the number of 14
C
atoms in the original sample. Measurement of N, the number of 14
C
atoms currently in the sample, allows the calculation of t, the age of the sample, using the equation above.[8]

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 14
C
's half-life than its mean-life.[note 1] The currently accepted value for the half-life of 14
C
is 5,730 years.[2] This means that after 5,730 years, only half of the initial 14
C
will have remained; a quarter will have remained after 11,460 years; an eighth after 17,190 years; and so on.

The above calculations make several assumptions, such as that the level of 14
C
in the biosphere has remained constant over time.[2] 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 in the form of a calibration curve, which is described in more detail below.[12] 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.[13][note 2] 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.[15]

### Carbon exchange reservoir

Simplified version of the carbon exchange reservoir, showing proportions of carbon and relative activity of the 14
C
in each reservoir[2][note 3]

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.[2] 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 seven years.[16][17] 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 that some of the 14
C
has decayed.[12] The ocean surface is an example: it contains 2.4% of the carbon in the exchange reservoir,[16] but there is only about 95% as much 14
C
as would be expected if the ratio were the same as in the atmosphere.[2] The time it takes for carbon from the atmosphere to mix with the surface ocean is only a few years,[18] but the surface waters also receive water from the deep ocean, which has over 90% of the carbon in the reservoir.[12] 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.[12]

Creatures living at the ocean surface have the same 14
C
ratios as the water they live in, and as a result of the reduced 14
C
/12
C
ratio, the radiocarbon age of marine life is typically about 400 years.[19][note 4] Organisms on land, however, are in closer equilibrium with the atmosphere and have the same 14
C
/12
C
ratio as the atmosphere.[2] 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.[16] 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.[2]

## 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 was able to demonstrate that the 14
C
/12
C
ratio had changed over time by testing wood samples of known ages and showing there was a significant deviation from the expected ratio. This discrepancy, often called the de Vries effect, was resolved by the study of tree rings.[20][21] Comparison of overlapping series of tree rings allowed the construction of a continuous sequence of tree-ring data that spanned 8,000 years.[20] (Since that time the tree-ring data series has been extended to 13,900 years.)[22] 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 sample at the time the tree ring was formed – and hence the 14
C
/12
C
ratio in the atmosphere at that time.[20] 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.[23] These curves are described in more detail below.

Atmospheric 14
C
, New Zealand[24] and Austria.[25] 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.[9] 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. 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.[20][26]

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.[20][26][27]

### Isotopic fractionation

Photosynthesis is the primary process by which carbon moves from the atmosphere into living things. In both photosynthetic pathways (C3 and C4) 12
C
is 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.[28][29]

To determine the degree of fractionation that takes place in a given plant, the amounts of both 12
C
and 13
C
isotopes are measured, and the resulting 13
C
/12
C
ratio is then compared to a standard ratio known as PDB.[note 5] The 13
C
/12
C
ratio is used instead of 14
C
/12
C
because the former is much easier to measure, and the latter can be easily derived: the depletion of 13
C
relative to 12
C
is proportional to the difference in the atomic masses of the two isotopes, so the depletion for 14
C
is twice the depletion of 13
C
.[12] The fractionation of 13
C
, known as δ13C, is calculated as follows:[28]

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

where the ‰ sign indicates parts per thousand.[28] Because the PDB standard contains an unusually high proportion of 13
C
,[note 6] most measured δ13C values are negative.

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.[28]
Material Typical δ13C range
PDB 0‰
Marine plankton −22‰ to −17‰[29]
C3 plants −30‰ to −22‰[29]
C4 plants −15‰ to −9‰[29]
Atmospheric CO
2
−8‰[28]
Marine CO
2
−32‰ to −13‰[29]

For marine organisms, the details of the photosynthesis reactions are less well understood and 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, while at lower temperatures, CO
2
becomes more soluble and hence more available to the marine organisms.[29] 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.[28] 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.[32]

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.[12] 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.[28]

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.[12]

### Reservoir effects

Libby's original exchange reservoir hypothesis assumed that the 14
C
/12
C
ratio in the exchange reservoir is constant all over the world,[33] but it has since been discovered that there are several causes of variation in the ratio across the reservoir.[19]

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
.[33] 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).[19] 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.[19][34] 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.[19]

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.[19]

Other effects

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.[12] 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.[19] 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.[19]

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.[19]

### 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.[35]

## 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.[36] This includes removing visible contaminants, such as rootlets that may have penetrated the sample since its burial.[36] Alkali and acid washes can be used to remove humic acid and carbonate contamination, but care has to be taken to avoid destroying or damaging the sample.[37]

### Material considerations

• It is common to reduce a wood sample to just the cellulose component before testing, but since this can reduce the volume of the sample to 20% of the original size, testing of the whole wood is often performed as well. Charcoal is often tested but is likely to need treatment to remove contaminants.[36][37]
• Unburnt bone can be tested; it is usual to date it using collagen, the protein fraction that remains after washing away the bone's structural material. 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.[36]
• For burnt bone, testability depends on the conditions under which the bone was burnt. 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.[36]
• 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. If testing recrystallized shell is unavoidable, it is sometimes possible to identify the original shell material from a sequence of tests.[38] It is also possible to test conchiolin, which is an organic protein found in shell, but this only constitutes 1-2% of shell material.[37]
• 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.[37] A particular difficulty with dried peat is the removal of rootlets, which are likely to be hard to distinguish from the sample material.[36]
• Soil contains organic material, but because of the likelihood 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.[37]
• Other materials 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.[37]

### Preparation and size

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.[39]

Once contamination has been removed, samples must be converted to a form suitable for the measuring technology to be used.[40] Where gas is required, CO
2
is widely used.[40][41] For samples to be used in liquid scintillation counters, the carbon must be in liquid form; the sample is typically converted to benzene. For accelerator mass spectrometry, solid targets are used; while these are usually graphite, CO
2
and iron carbide can also be used.[40][42]

The quantity of sample material needed to perform testing depends on what is being tested and on the testing technology that is being used. There are two types of testing technology: detectors that record radioactivity, known as beta counters, or accelerator mass spectrometers. For beta counters, a sample weighing at least 10 grams is typically required. Accelerator mass spectrometry (AMS) labs are much more sensitive and can deal with samples as small as 10 milligrams (for charcoal) and generally require less than a gram of most sample materials.[40]

## Measurement and results

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.[40] In this approach, what is measured is the activity, in number of decay events per unit mass per time period, of the sample.[41] This method is also known as "beta counting", because it is the beta particles emitted by the decaying 14
C
atoms that are detected.[43] 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.[40] 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.[41] For some time, beta counting methods were more accurate than AMS, but there is now little to choose between them.[44] AMS has two significant advantages over beta counting: it can perform accurate testing on samples much too small for beta counting; and it is much faster—an accuracy of 1% can be achieved in minutes with AMS, which is far quicker than would be achievable with the older technology.[45]

### Beta counting

Libby's first detector was a Geiger counter of his own design. He converted the carbon in his sample to lamp black (soot) and coated the inner surface of a cylinder with it. This cylinder was inserted 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.[40] 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.[41]

Libby's method was soon superseded by gas proportional counters, which were less affected by bomb carbon (the additional 14
C
created by nuclear weapons testing). 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.[41]

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.[46][47]

For both the gas proportional counter and liquid scintillation counter, what is measured is the 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 is also 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 measured activity of the sample which is being dated to get the activity due solely to that sample's 14
C
. In addition, a sample with a standard activity is measured, in order to provide a baseline for comparison.[48]

### 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
in a given sample, determining the 14
C
/12
C
ratio directly. The sample, often in the form of graphite, is made to emit C- ions (carbon atoms with a single negative charge), 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 since the volume of 12
C
(and 13
C
, needed for calibration) is too great for individual ion detection, counts are determined by measuring the electric current created in a Faraday cup.[49] Some AMS facilities are also able to evaluate a sample's fractionation, another piece of data necessary for calculating the sample's radiocarbon age.[50]

The use of AMS, as opposed to simpler forms of mass spectrometry, 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
.[40] As with beta counting, both blank samples and standard samples are used.[49] 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 which is being dated. 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.[51]

### 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.[51]

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. To determine this, a blank sample (of old, or dead, carbon) is measured, and a sample of known activity is measured. The additional samples allow errors such as background radiation and systematic errors in the laboratory setup to be detected and corrected for.[48] A common standard sample is HOxII, 1,000 lb of which was prepared by NIST in 1977 from French beet harvests.[52][53]

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 Fm, the "fraction modern". This is defined as the ratio between the 14
C
/12
C
ratio in the sample and the 14
C
/12
C
ratio in modern carbon, which is in turn defined as the 14
C
/12
C
ratio that would have been measured in 1950 had there been no fossil fuel effect.[51]

Both beta counting and AMS results have to be corrected for fractionation. This is necessary because different materials of the same age, which because of fractionation have naturally different 14
C
/12
C
ratios, will appear to be of different ages because the 14
C
/12
C
ratio is taken as the indicator of age. To avoid this, all radiocarbon measurements are converted to the measurement that would have been seen had the sample been made of wood, which has a known δ13
C
value of −25‰.[13]

Once the corrected 14
C
/12
C
ratio is known, a "radiocarbon age" is calculated using:[54]

$Age = -8033 \cdot ln (Fm)$

The calculation uses Libby's half-life of 5,568 years, not the more accurate modern value of 5,730 years. Libby’s value for the half-life is used in order to maintain consistency with early radiocarbon testing results; calibration curves include a correction for this, so the accuracy of final reported calendar ages is assured.[54]

### Errors and reliability

The reliability of the results can be improved by lengthening the testing time. For example, if counting beta decays for 250 minutes is enough to give an error of ± 80 years, with 68% confidence, then doubling the counting time to 500 minutes will allow a sample with only half as much 14
C
to be measured with the same error term of 80 years.[55][56]

Radiocarbon dating is generally limited to dating samples no more than 50,000 years old, as samples older than that have insufficient 14
C
to be measurable. Older dates have been obtained by using special sample preparation techniques, large samples, and very long measurement times. These techniques can allow dates up to 60,000 and in some cases up to 75,000 years before the present to be measured.[44]

Radiocarbon dates are generally presented with a range of one standard deviation (usually represented by the Greek letter sigma: σ) on either side of the mean. This 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.[57]

Errors in procedure can also lead to errors in the results. If 1% of the benzene in a modern reference sample accidentally evaporates, scintillation counting will give a radiocarbon age that is too young by about 80 years.[58]

### Calibration

The Northern hemisphere curve from INTCAL13. As of 2014 this is the most recent version of the standard calibration curve. There are separate graphs for the southern hemisphere and for calibration of marine data. The diagonal line shows where the curve would lie if radiocarbon ages and calendar ages were the same.[22]

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.[59] 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.[60]

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 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.[61] Hans Suess used this data to publish the first calibration curve for radiocarbon dating in 1967.[20][21][60] 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.[20][21]

A calibration curve is used by taking the radiocarbon date reported by a laboratory, and reading across from that date on the vertical axis of the graph. The point where this horizontal line intersects the curve will give the calendar age of the sample on the horizontal axis. This is the reverse of the way the curve is constructed: a point on the graph is derived from a sample of known age, such as a tree ring; when it is tested, the resulting radiocarbon age gives a data point for the graph.[23]

Over the next thirty years many calibration curves were published using a variety of methods and statistical approaches.[23] 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. The INTCAL13 data includes separate curves for the northern and southern hemispheres, as they differ systematically because of the hemisphere effect; there is also a separate marine calibration curve.[62] For a set of samples with a known sequence and separation in time such as a sequence of tree rings, the samples' radiocarbon ages form a small subset of the calibration curve. The resulting curve can then be matched to the actual calibration curve by identifying where, in the range suggested by the radiocarbon dates, the wiggles in the calibration curve best match the wiggles in the curve of sample dates. This "wiggle-matching" technique can lead to more precise dating than is possible with individual radiocarbon dates.[63] Wiggle-matching can be used in places where there is a plateau on the calibration curve, and hence can provide a much more accurate date than the intercept or probability methods are able to produce.[64] The technique is not restricted to tree rings; for example, a stratified tephra sequence in New Zealand, known to predate human colonization of the islands, has been dated to 1314 AD ± 12 years by wiggle-matching.[65]

Bayesian statistical techniques can be applied when there are several radiocarbon dates to be calibrated. For example, if a series of radiocarbon dates is taken from different levels in a given stratigraphic sequence, Bayesian analysis can help determine if some of the dates should be discarded as anomalies, and can use the information to improve the output probability distributions.[63]

### Reporting dates

Several 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.[66]

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 1σ 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.[66]

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.[66] In addition, an article in Radiocarbon in 2014 about radiocarbon date reporting conventions recommends that information should be provided about sample treatment, including the sample material, pretreatment methods, and quality control measurements; that the citation to the software used for calibration should specify the version number and any options or models used; and that the calibrated date should be given with the associated probabilities for each range.[67]

## Use in archaeology

### Interpretation

A key concept in interpreting radiocarbon dates is archaeological association: what is the true relationship between two or more objects at an archaeological site? It frequently happens that a sample for radiocarbon dating can be taken directly from the object of interest, but there are also many cases where this is not possible. Metal grave goods, for example, cannot be radiocarbon dated, but they may be found in a grave with a coffin, charcoal, or other material which can be assumed to have been deposited at the same time. In these cases a date for the coffin or charcoal is indicative of the date of deposition of the grave goods, because of the direct functional relationship between the two. There are also cases where there is no functional relationship, but the association is reasonably strong: for example, a layer of charcoal in a rubbish pit provides a date which has a relationship to the rubbish pit.[68]

As a tree grows, only the outermost tree ring exchanges carbon with its environment, so the age measured for a wood sample depends on where the sample is taken from. This means that radiocarbon dates on wood samples can be older than the date at which the tree was felled. In addition, if a piece of wood is used for multiple purposes, there may be a significant delay between the felling of the tree and the final use in the context in which it is found.[69] This is often referred to as the "old wood" problem.[2] One example is the Bronze Age trackway at Withy Bed Copse, in England; the trackway was built from wood which had clearly been worked for other purposes before being re-used in the trackway. Another example is driftwood, which may be used as construction material. It is not always possible to recognize re-use. Other materials can present the same problem: for example, bitumen is known to have been used by some neolithic communities to waterproof baskets; the bitumen's radiocarbon age will be greater than is measurable by the laboratory, regardless of the actual age of the context. A separate issue, related to re-use, is that of lengthy use, or delayed deposition. For example, a wooden object that remains in use for a lengthy period will have an apparent age greater than the actual age of the context in which it is deposited.[69]

### Impact

Soon after the publication of Libby's 1949 paper in Science, radiocarbon dating laboratories were being established at universities around the world, and by the end of the 1950s there were over 20 active 14
C
research laboratories. It was quickly apparent that the principles of radiocarbon dating were correct, despite discrepancies whose cause was not then known.[70]

The development of radiocarbon dating has had a profound impact on archaeology; it is often described as the "radiocarbon revolution".[71] In the words of anthropologist R. E. Taylor, "14
C
data made a world prehistory possible by contributing a time scale that transcends local, regional and continental boundaries". It provides more accurate dating within sites than the previous methods, which were usually derived from either stratigraphy or typologies (e.g. of stone tools or pottery); it also allows comparison and synchronization of events across great distances. The advent of radiocarbon dating may even have led to better field methods in archaeology, since better data recording leads to firmer association of objects with the samples to be tested. These improved field methods were sometimes motivated by attempts to prove that a 14
C
date was incorrect. Taylor also suggests that the availability of definite date information freed archaeologists from the need to focus so much of their energy on determining the dates of their finds, and led to an expansion of the questions archaeologists were willing to research. For example, questions about the evolution of human behaviour were much more frequently seen in archaeology, beginning in the 1970s.[72]

The dating framework provided by radiocarbon led to a change in the prevailing view of how innovations spread through prehistoric Europe. It had previously been thought that many ideas spread by diffusion through the continent, or by invasions of peoples bringing new cultural ideas with them. As radiocarbon dates began to prove these ideas wrong in many instances, it became apparent that these innovations must sometimes have arisen locally. This has been described as a "second radiocarbon revolution", and with regard to British prehistory one archaeologist has characterized the impact of radiocarbon dating as "radical...therapy" for the "progressive disease of invasionism". More broadly, the success of radiocarbon dating stimulated interest in analytical and statistical approaches to archaeological data.[72]

Occasionally, radiocarbon dating techniques are used to date an object of popular interest. An example is the Shroud of Turin, a piece of linen cloth thought by some to bear an image of Jesus Christ after his crucifixion. The Shroud of Turin was tested in 1988; the results, from three separate laboratories, dated the sample of linen tested to the 1300s, raising doubts about the authenticity of the shroud.[9]

Other radioactive isotopes created by cosmic rays have been studied to determine if they could be used to assist in dating objects of archaeological interest. These include 3
He
, 10
Be
, 21
Ne
, 26
Al
, and 36
Cl
. With the development of AMS in the 1980s it became possible to measure these isotopes precisely enough for them to be the basis of useful dating techniques. They have been primarily applied to dating rocks.[73] Naturally occurring radioactive isotopes can also form the basis of dating methods: this is the case with potassium-argon dating, argon-argon dating, and uranium series dating.[74] Other dating techniques of interest to archaeologists include thermoluminescence, optically stimulated luminescence, electron spin resonance dating, and fission track dating, as well as techniques that depend on annual bands or layers, such as dendrochronology, tephrochronology, and varve chronology.[75]

## Notes

1. ^ The mean-life and half-life are related by the following equation:[2]
$T_\frac{1}{2} = 0.693 \cdot \tau$
2. ^ The terms "conventional radiocarbon age", or "radiocarbon age", is also used: these terms refer to an age calculated by using the following standards: a) using the Libby half-life of 5568 years, rather than the currently accepted actual half-life of 5730 years; (b) the use of an NIST standard known as HOxII to define the activity of radiocarbon in 1950; (c) the use of 1950 as the date from which years "before present" are counted; (d) a correction for fractionation, based on a standard isotope ratio, and (e) the assumption that the 14
C
/12
C
ratio has not changed over time.[14]
3. ^ The data on carbon percentages in each part of the reservoir is drawn from an estimate of reservoir carbon for the mid-1990s; estimates of carbon distribution during pre-industrial times are significantly different.[16]
4. ^ The age only appears to be 400 years once a correction for fractionation is made.
5. ^ "PDB" stands for "Pee Dee Belemnite", a fossil from the Pee Dee formation in South Carolina.[30]
6. ^ The PDB value is 11.2372‰.[31]

## Footnotes

1. ^ Taylor (1987), pp. 149−151.
2. Bowman (1995), pp. 9–15.
3. ^ Libby, W.F. (1946). "Atmospheric helium three and radiocarbon from cosmic radiation". Physics Review 69: 671–672. Bibcode:1946PhRv...69..671L. doi:10.1103/PhysRev.69.671.2.
4. ^ Anderson, E.C.; Libby, W.F.; Weinhouse, S.; Reid, A.F.; Kirshenbaum, A.D.; Grosse, A.V. (1947). "Radiocarbon from cosmic radiation". Science 105 (2765): 576–577. Bibcode:1947Sci...105..576A. doi:10.1126/science.105.2735.576.
5. ^ Arnold, J.R.; Libby, W.F. (1949). "Age determinations by radiocarbon content: checks with samples of known age". Science 110 (2869): 678–680. Bibcode:1949Sci...110..678A. doi:10.1126/science.110.2869.678. JSTOR 1677049. PMID 15407879.
6. ^ Aitken (1990), pp. 60–61.
7. ^ Bianchi & Canuel (2011), p. 35.
8. ^ a b c d Aitken (1990), pp. 56–58.
9. ^ a b c Currie, Lloyd A. (2004). "The remarkable metrological history of radiocarbon dating II". Journal of Research of the National Institute of Standards and Technology 109: 185–217. doi:10.6028/jres.109.013.
10. ^ Taylor (1987), p. 72.
11. ^ Aitken (1990), p. 59.
12. Aitken (1990), pp. 61–66.
13. ^ a b Aitken (1990), pp. 92–95.
14. ^ Taylor (1987), p. 4.
15. ^ Bowman (1995), p. 42.
16. ^ a b c d Post (2010) pp. 128–129.
17. ^ Warneck (2000), p. 690.
18. ^ Ferronsky & Polyakov (2012), p. 372.
19. Bowman (1995), pp. 24–27.
20. Bowman (1995), pp. 16–20.
21. ^ a b c Suess (1970), p. 303.
22. ^ a b Reimer, Paula J. et al. (2013). "IntCal13 and Marine13 radiocarbon age calibration curves 0–50,000 years cal BP". Radiocarbon 55: 1869–1887. doi:10.2458/azu_js_rc.55.16947.
23. ^ a b c Bowman (1995), pp. 43–49.
24. ^ "Atmospheric δ14
C
record from Wellington"
. Carbon Dioxide Information Analysis Center. Archived from the original on 1 February 2014. Retrieved 1 May 2008.
25. ^ 14
CO
2
record from Vermunt"
. Carbon Dioxide Information Analysis Center. Retrieved 1 May 2008.
26. ^ a b Aitken (1990), pp. 71–72.
27. ^ "Limited Test Ban Treaty". Science Magazine. Retrieved July 26, 2013.
28. Bowman (1995), pp. 20–23.
29. Maslin & Swann (2006), p. 246.
30. ^ Taylor (1987), p. 100
31. ^ Dass (2007), p. 276.
32. ^ Schoeninger (2010), p. 446.
33. ^ a b Libby (1965), p. 6.
34. ^ Cronin (2010), p. 35.
35. ^ Aitken (1990), pp. 85–86.
36. Bowman (1995), pp. 27–30.
37. Aitken (1990), pp. 86-89.
38. ^ Šilar (2004), p. 166.
39. ^ Bowman (1995), pp. 37-42.
40. Bowman (1995), pp. 31-37.
41. Aitken (1990), pp. 76-78.
42. ^ Trumbore (1996), p. 318.
43. ^ Walker (2005), p. 20.
44. ^ a b Walker (2005), p. 23.
45. ^ Malainey (2010), p. 96.
46. ^ Theodórsson (1996), p. 24.
47. ^ L'Annunziata & Kessler (2012), p. 424.
48. ^ a b Eriksson Stenström et al. (2011), p. 3.
49. ^ a b Aitken (1990), pp. 82-85.
50. ^ Tuniz, Zoppi & Barbetti (2004), p. 395.
51. ^ a b c McNichol, A.P.; Jull, A.T.S.; Burr, G.S. (2001). "Converting AMS data to radiocarbon values: considerations and conventions". Radiocarbon 43: 313–320.
52. ^ Terasmae (1984), p. 5.
53. ^ L'Annunziata (2007), p. 528.
54. ^ a b "Radiocarbon Data Calculations: NOSAMS". Woods Hole Oceanographic Institution. 2007. Retrieved August 27, 2013.
55. ^ Bowman (1995), pp. 38–39.
56. ^ Taylor (1987), p. 124.
57. ^ Taylor (1987), pp. 125−126.
58. ^ Bowman (1995), pp. 40−41.
59. ^ Taylor (1987), p. 133.
60. ^ a b Aitken (1990), p. 66–67.
61. ^ Taylor (1987), pp. 19–21.
62. ^ Stuiver, M.; Braziunas, T.F. (1993). "Modelling atmospheric 14
C
influences and 14
C
ages of marine samples to 10,000 BC"
63. ^ a b Walker (2005), pp. 35−37.
64. ^ Aitken (1990), pp. 103−105.
65. ^ Walker (2005), pp. 207−209.
66. ^ a b c "Radiocarbon: Information for Authors". Radiocarbon. University of Arizona. May 25, 2011. pp. 5–7. Retrieved January 1, 2014.
67. ^ Millard, Andrew R. (2014). "Conventions for Reporting Radiocarbon Determinations". Radiocarbon 56 (2): 555–559. doi:10.2458/56.17455. Retrieved 5 July 2014.
68. ^ Mook & Waterbolk (1985), pp. 48–49.
69. ^ a b Bowman (1995), pp. 53−54.
70. ^ Taylor (1987), pp. 167−169.
71. ^ Taylor (1997), p. 70.
72. ^ a b Taylor (1987), pp. 143−146.
73. ^ Walker (2005), pp. 77−79.
74. ^ Walker (2005), pp. 57−77.
75. ^ Walker (2005), pp. 93−162.

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