Calculation of radiocarbon dates

Last updated November 09, 2024

The calculation of radiocarbon dates determines the age of an object containing organic material by using the properties of radiocarbon (also known as carbon-14), a radioactive isotope of carbon.

Radiocarbon dating methods produce data based on the ratios of different carbon isotopes in a sample that must then be further manipulated in order to calculate a resulting "radiocarbon age". Radiocarbon dating is also referred to as carbon dating or carbon-14 dating. Calculations of radiocarbon dates are typically made based on measurements from beta counting devices or from accelerator mass spectrometers (AMS). There are several possible sources of error in both the beta counting and AMS methods.

Calculations

The calculations to be performed depend on the measurements taken based on the technology used, since beta counters measure the sample's radioactivity, whereas accelerator mass spectrometers (AMS) determine the ratio of the three different carbon isotopes in the sample.^[1]

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 $δ$ ¹³C_{values, is Pee Dee Belemnite (PDB), a fossil which has a ¹³
C/¹²
C ratio of 1.12372%.^[2] A related standard is the use of wood, which has a $δ$ ¹³C_{of -25 ‰, as the material for which radiocarbon ages are calibrated. Since different materials have different $δ$ ¹³C_{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 $δ$ ¹³C_{of wood is known, and the $δ$ ¹³C_{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.^[3]}}}}}

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.^[4] 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.^[5]

Both beta counting and AMS measure standard samples as part of their methodology. These samples contain carbon of a known activity.^[6] The first standard, Oxalic Acid SRM 4990B, also referred to as HOxI, was a 1,000 lb batch of oxalic acid prepared in 1955 by the National Institute of Standards and Technology (NIST). Since it was produced after the start of atomic testing, it incorporates bomb carbon, so the measured activity is higher than the desired standard. This is addressed by defining the standard to be 0.95 times the activity of HOxI.^[5]

All of this first standard has long since been consumed, and later standards have been prepared, each of which has a given ratio to the desired standard activity. A secondary standard, Oxalic Acid SRM 4990C, also referred to as HOxII, 1,000 lb of which was prepared by NIST in 1977 from French beet harvests, is now in wide use.^[7]

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:^[8]

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.^[8]

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 $δ$ ¹³C_{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.^[3]}

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^[4]

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

The ¹⁴
C fractionation factor, Frac_14/12, is approximately the square of this, to an accuracy of 1‰:^[4]

{\text{Frac}}_{\text{14/12 (sample)}}=({\text{Frac}}_{\text{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:^[4]

A_{sn}=A_{s}{\text{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.^[4]

The equation for δ13C given earlier can be rearranged to^[4]

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

Substituting this in the ¹⁴
C fractionation factor, and also substituting the value for δ13C for wood of -25‰, gives the following expression:^[4]

A_{sn}=A_{s}\left({\frac {\left(1-{\frac {25}{1000}}\right)\left({\frac {{\ce {^{13}C}}}{{\ce {^{12}C}}}}\right)_{{\ce {PDB}}}}{\left(1+{\frac {{\ce {\delta^{13}C}}}{1000}}\right)\left({\frac {{\ce {^{13}C}}}{{\ce {^{12}C}}}}\right)_{{\ce {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:^[4]

A_{sn}=A_{s}\left({\frac {\left(1-{\frac {25}{1000}}\right)}{\left(1+{\frac {{\ce {\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.^[9]

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.^[9]

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

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

If the ¹⁴
C/¹³
C ratio is used instead, then the equations for each standard are:^[9]

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

R_{{\ce {HOxII,-25}}}=R'_{{\ce {HoxII}}}\left({\frac {1+{\frac {-25}{1000}}}{1+{\frac {{\ce {\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.^[9]

Once the appropriate value above has been calculated, $R modern$ can be determined; it is^[9]

R_{modern}=0.95R_{\text{HOxI,-19}}=.7459R_{\text{HOxII,-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.^[9]

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.^[9]

Next, the uncorrected fraction modern is calculated; "uncorrected" means that this intermediate value does not include the fractionation correction.^[9]

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:^[9]

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

If the ¹⁴
C/¹³
C ratio is being used:^[9]

Fm_{ms}=Fm_{uc}\left({\frac {1+{\frac {-25}{1000}}}{1+{\frac {{\ce {\delta^{13}C}}_{s}}{1000}}}}\right)

The δ13C_s value is from the sample itself, measured on CO^₂ prepared while converting the sample to graphite.^[9]

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^{[note 1]} 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^[9]

Fm_{s}={\frac {Fm_{ms}{\ce {C}}_{ms}-Fm_{pb}{\ce {C}}_{pb}}{{\ce {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:^[10]

{\text{Age}}=-8033\ln(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).^[11] 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.^[12]^[13] 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.^[11]

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.^[11]^[12]

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.^[14]

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, scintillation counting will give a radiocarbon age that is 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.^[15]

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.^[16]^[17]

Notes

↑ McNichol & Burr give two other calculations, one of which can be shown to be equivalent to the one given here. The other depends on the process blank being the same mass as the sample.^[9]

Footnotes

↑ McNichol, Jull & Burr, "Converting AMS Data to Radiocarbon Values: Considerations and Conventions", pp. 313.
↑ Dass (2007), p.276.
1 2 Aitken, Science-based Dating in Archaeology, p. 92–95.
1 2 3 4 5 6 7 8 Eriksson Stenstrom et al. A Guide to Radiocarbon Units and Calculations, p. 6.
1 2 L'Annunziata, Radioactivity, p. 528.
↑ Aitken, Science-based Dating in Archaeology, pp. 82-85.
↑ J. Terasmae, "Radiocarbon Dating: Some Problems and Potential Developments", in Mahaney, Quaternary Dating Methods, p. 5.
1 2 Eriksson Stenström et al., "A guide to radiocarbon units and calculations", p. 3.
1 2 3 4 5 6 7 8 9 10 11 12 13 McNichol, Jull & Burr, "Converting AMS Data to Radiocarbon Values: Considerations and Conventions", pp. 315-318.
↑ "Radiocarbon Data Calculations: NOSAMS". Woods Hole Oceanographic Institution. 2007. Retrieved August 27, 2013.
1 2 3 Taylor, Radiocarbon Dating, p. 102−104.
1 2 Bowman, Radiocarbon Dating, pp. 38–39.
↑ Taylor, Radiocarbon Dating, p. 124.
↑ Taylor, Radiocarbon Dating, pp. 125−126.
↑ Bowman, Radiocarbon Dating, pp. 40−41.
↑ Mike Walker, Department of Archaeology and Anthropology, University of Wales, Lampeter, UK Quaternary Dating Methods, John Wiley & Sons, Ltd. 2005, page 23. Online, pdf 9 MB Archived 2014-07-14 at the Wayback Machine
↑ Shennan, Ian; Long, Antony; Horton, Benjamin (2014). Handbook of Sea-Level Research. John Wiley & Sons. p. 350. ISBN 978-1118452561 . Retrieved 31 October 2018.

Related Research Articles

Conversion of units is the conversion of the unit of measurement in which a quantity is expressed, typically through a multiplicative conversion factor that changes the unit without changing the quantity. This is also often loosely taken to include replacement of a quantity with a corresponding quantity that describes the same physical property.

In chemistry, pH, also referred to as acidity or basicity, historically denotes "potential of hydrogen". It is a logarithmic scale used to specify the acidity or basicity of aqueous solutions. Acidic solutions are measured to have lower pH values than basic or alkaline solutions.

Radiocarbon dating is a method for determining the age of an object containing organic material by using the properties of radiocarbon, a radioactive isotope of carbon.

Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in decibels. A ratio higher than 1:1 indicates more signal than noise.

In chemistry, the molar mass of a chemical compound is defined as the ratio between the mass and the amount of substance of any sample of the compound. The molar mass is a bulk, not molecular, property of a substance. The molar mass is an average of many instances of the compound, which often vary in mass due to the presence of isotopes. Most commonly, the molar mass is computed from the standard atomic weights and is thus a terrestrial average and a function of the relative abundance of the isotopes of the constituent atoms on Earth. The molar mass is appropriate for converting between the mass of a substance and the amount of a substance for bulk quantities.

In physical chemistry, Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulated by the English chemist William Henry, who studied the topic in the early 19th century. In simple words, we can say that the partial pressure of a gas in vapour phase is directly proportional to the mole fraction of a gas in solution.

In physical organic chemistry, a kinetic isotope effect (KIE) is the change in the reaction rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes. Formally, it is the ratio of rate constants for the reactions involving the light (k_L) and the heavy (k_H) isotopically substituted reactants (isotopologues): KIE = k_L/k_H.

The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

Isotope geochemistry is an aspect of geology based upon the study of natural variations in the relative abundances of isotopes of various elements. Variations in isotopic abundance are measured by isotope-ratio mass spectrometry, and can reveal information about the ages and origins of rock, air or water bodies, or processes of mixing between them.

Air–fuel ratio (AFR) is the mass ratio of air to a solid, liquid, or gaseous fuel present in a combustion process. The combustion may take place in a controlled manner such as in an internal combustion engine or industrial furnace, or may result in an explosion ,The air–fuel ratio determines whether a mixture is combustible at all, how much energy is being released, and how much unwanted pollutants are produced in the reaction. Typically a range of fuel to air ratios exists, outside of which ignition will not occur. These are known as the lower and upper explosive limits.

Carbon (₆C) has 14 known isotopes, from ⁸
C to ²⁰
C as well as ²²
C, of which ¹²
C and ¹³
C are stable. The longest-lived radioisotope is ¹⁴
C, with a half-life of 5.70(3)×10³ years. This is also the only carbon radioisotope found in nature, as trace quantities are formed cosmogenically by the reaction ¹⁴
N + n → ¹⁴
C + ¹
H. The most stable artificial radioisotope is ¹¹
C, which has a half-life of 20.3402(53) min. All other radioisotopes have half-lives under 20 seconds, most less than 200 milliseconds. The least stable isotope is ⁸
C, with a half-life of 3.5(1.4)×10⁻²¹ s. Light isotopes tend to decay into isotopes of boron and heavy ones tend to decay into isotopes of nitrogen.

An isotopic signature is a ratio of non-radiogenic 'stable isotopes', stable radiogenic isotopes, or unstable radioactive isotopes of particular elements in an investigated material. The ratios of isotopes in a sample material are measured by isotope-ratio mass spectrometry against an isotopic reference material. This process is called isotope analysis.

The environmental isotopes are a subset of isotopes, both stable and radioactive, which are the object of isotope geochemistry. They are primarily used as tracers to see how things move around within the ocean-atmosphere system, within terrestrial biomes, within the Earth's surface, and between these broad domains.

In geochemistry, paleoclimatology and paleoceanography δ¹⁸O or delta-O-18 is a measure of the deviation in ratio of stable isotopes oxygen-18 (¹⁸O) and oxygen-16 (¹⁶O). It is commonly used as a measure of the temperature of precipitation, as a measure of groundwater/mineral interactions, and as an indicator of processes that show isotopic fractionation, like methanogenesis. In paleosciences, ¹⁸O:¹⁶O data from corals, foraminifera and ice cores are used as a proxy for temperature.

Equilibrium constants are determined in order to quantify chemical equilibria. When an equilibrium constant $K$ is expressed as a concentration quotient,

In geochemistry, paleoclimatology, and paleoceanography δ¹³C is an isotopic signature, a measure of the ratio of the two stable isotopes of carbon—¹³C and ¹²C—reported in parts per thousand. The measure is also widely used in archaeology for the reconstruction of past diets, particularly to see if marine foods or certain types of plants were consumed.

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:

The δ³⁴S value is a standardized method for reporting measurements of the ratio of two stable isotopes of sulfur, ³⁴S:³²S, in a sample against the equivalent ratio in a known reference standard. The most commonly used standard is Vienna-Canyon Diablo Troilite (VCDT). Results are reported as variations from the standard ratio in parts per thousand, per mil or per mille, using the ‰ symbol. Heavy and light sulfur isotopes fractionate at different rates and the resulting δ³⁴S values, recorded in marine sulfate or sedimentary sulfides, have been studied and interpreted as records of the changing sulfur cycle throughout the earth's history.

Methane clumped isotopes are methane molecules that contain two or more rare isotopes. Methane (CH₄) contains two elements, carbon and hydrogen, each of which has two stable isotopes. For carbon, 98.9% are in the form of carbon-12 (¹²C) and 1.1% are carbon-13 (¹³C); while for hydrogen, 99.99% are in the form of protium (¹H) and 0.01% are deuterium (²H or D). Carbon-13 (¹³C) and deuterium (²H or D) are rare isotopes in methane molecules. The abundance of the clumped isotopes provides information independent from the traditional carbon or hydrogen isotope composition of methane molecules.

Photosynthesis converts carbon dioxide to carbohydrates via several metabolic pathways that provide energy to an organism and preferentially react with certain stable isotopes of carbon. The selective enrichment of one stable isotope over another creates distinct isotopic fractionations that can be measured and correlated among oxygenic phototrophs. The degree of carbon isotope fractionation is influenced by several factors, including the metabolism, anatomy, growth rate, and environmental conditions of the organism. Understanding these variations in carbon fractionation across species is useful for biogeochemical studies, including the reconstruction of paleoecology, plant evolution, and the characterization of food chains.

References

Aitken, M. J. (1990). Science-based Dating in Archaeology. London: Longman. ISBN 0-582-49309-9.
Bowman, Sheridan (1995) [1990]. Radiocarbon Dating . London: British Museum Press. ISBN 0-7141-2047-2.
Dass, Chhabil (2007). Fundamentals of Contemporary Mass Spectrometry. Hoboken NJ: John Wiley & Sons. ISBN 978-0-471-68229-5.
Eriksson Stenström, Kristina; Skog, Göran; Georgiadou, Elisavet; Genberg, Johan; & Johansson, Anette. "A guide to radiocarbon units and calculations". 2011. Lund:Lund University.
L'Annunziata, Michael F. (2007). Radioactivity: Introduction and History. Oxford: Elsevier. ISBN 978-0-444-52715-8
McNichol, A. P.; Jull, A. T. S.; Burr, G. S. (2001). "Converting AMS Data To Radiocarbon Values: Considerations And Conventions". Radiocarbon. 43 (2A): 313–320. doi: 10.1017/S0033822200038169 . S2CID 55072620.
Taylor, R.E. (1987). Radiocarbon Dating . London: Academic Press. ISBN 0-12-433663-9.
Walker, Mike (2005). Quaternary Dating Methods. Chichester: John Wiley & Sons. ISBN 978-0-470-86927-7.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[10] McNichol & Burr give two other calculations, one of which can be shown to be equivalent to the one given here. The other depends on the process blank being the same mass as the sample.^[9]

[1] McNichol, Jull & Burr, "Converting AMS Data to Radiocarbon Values: Considerations and Conventions", pp. 313.

[2] Dass (2007), p.276.

[:12-3] 1 2 Aitken, Science-based Dating in Archaeology, p. 92–95.

[:18-4] 1 2 3 4 5 6 7 8 Eriksson Stenstrom et al. A Guide to Radiocarbon Units and Calculations, p. 6.

[:13-5] 1 2 L'Annunziata, Radioactivity, p. 528.

[:9-6] Aitken, Science-based Dating in Archaeology, pp. 82-85.

[7] J. Terasmae, "Radiocarbon Dating: Some Problems and Potential Developments", in Mahaney, Quaternary Dating Methods, p. 5.

[:10-8] 1 2 Eriksson Stenström et al., "A guide to radiocarbon units and calculations", p. 3.

[:11-9] 1 2 3 4 5 6 7 8 9 10 11 12 13 McNichol, Jull & Burr, "Converting AMS Data to Radiocarbon Values: Considerations and Conventions", pp. 315-318.

[11] "Radiocarbon Data Calculations: NOSAMS". Woods Hole Oceanographic Institution. 2007. Retrieved August 27, 2013.

[Taylor_102-104-12] 1 2 3 Taylor, Radiocarbon Dating, p. 102−104.

[Bowman_38-39-13] 1 2 Bowman, Radiocarbon Dating, pp. 38–39.

[Taylor_124-14] Taylor, Radiocarbon Dating, p. 124.

[15] Taylor, Radiocarbon Dating, pp. 125−126.

[16] Bowman, Radiocarbon Dating, pp. 40−41.

[:14-17] Mike Walker, Department of Archaeology and Anthropology, University of Wales, Lampeter, UK Quaternary Dating Methods, John Wiley & Sons, Ltd. 2005, page 23. Online, pdf 9 MB Archived 2014-07-14 at the Wayback Machine

[18] Shennan, Ian; Long, Antony; Horton, Benjamin (2014). Handbook of Sea-Level Research. John Wiley & Sons. p. 350. ISBN 978-1118452561 . Retrieved 31 October 2018.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[note 1]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]