Editing Radiocarbon dating (section)

===Principles===
During its life, a plant or animal is in equilibrium with its surroundings by exchanging carbon either with the atmosphere or through its diet. It will, therefore, have the same proportion of {{chem|14|C}} as the atmosphere, or in the case of marine animals or plants, with the ocean. Once it dies, it ceases to acquire {{chem|14|C}}, but the {{chem|14|C}} within its biological material at that time will continue to decay, and so the ratio of {{chem|14|C}} to {{chem|12|C}} in its remains will gradually decrease. Because {{chem|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&nbsp;– the older the sample, the less {{chem|14|C}} will be left.<ref name=Aitken_56/>

The equation governing the decay of a radioactive isotope is:<ref name=Bowman_9/>
: <math> N = N_0 \, e^{-\lambda t}\, </math>
where ''N''<sub>0</sub> 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''.<ref name=Bowman_9/> ''λ'' is a constant that depends on the particular isotope; for a given isotope it is equal to the [[Multiplicative inverse|reciprocal]] of the [[Radioactive decay#Time constant and mean-life|mean-life]]&nbsp;– i.e. the average or expected time a given atom will survive before undergoing radioactive decay.<ref name=Bowman_9/> The mean-life, denoted by ''τ'', of {{chem|14|C}} is 8,267 years,{{#tag:ref|The half-life of {{chem|14|C}} (which determines the mean-life) was thought to be 5568 ± 30 years in 1952.<ref>Libby (1965), p. 42.</ref> The mean-life and half-life are related by the following equation:<ref name=Bowman_9/>
: <math> T_\frac{1}{2} = 0.693 \cdot \tau </math>|group=note}} so the equation above can be rewritten as:<ref>Aitken (1990), p. 59.</ref>
: <math> t = \ln(N_0/N) \cdot \text{8267 years} </math>
The sample is assumed to have originally had the same {{chem|14|C}}/{{chem|12|C}} ratio as the ratio in the atmosphere, and since the size of the sample is known, the total number of atoms in the sample can be calculated, yielding ''N''<sub>0</sub>, the number of {{chem|14|C}} atoms in the original sample. Measurement of ''N'', the number of {{chem|14|C}} atoms currently in the sample, allows the calculation of ''t'', the age of the sample, using the equation above.<ref name=Aitken_56/>

The half-life of a radioactive isotope (usually denoted by t<sub>1/2</sub>) 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 {{chem|14|C}}'s half-life than its mean-life. The currently accepted value for the half-life of {{chem|14|C}} is 5,700 ± 30 years.<ref name=Nubase2020>{{NUBASE2020|page=030001-22}}</ref> This means that after 5,700 years, only half of the initial {{chem|14|C}} will remain; a quarter will remain after 11,400 years; an eighth after 17,100 years; and so on.

The above calculations make several assumptions, such as that the level of {{chem|14|C}} in the atmosphere has remained constant over time.<ref name=Bowman_9/> In fact, the level of {{chem|14|C}} in the atmosphere has varied significantly and as a result, the values provided by the equation above have to be corrected by using data from other sources.<ref name=Aitken1990>Aitken (1990), pp. 61–66.</ref> This is done by calibration curves (discussed below), which convert a measurement of {{chem|14|C}} in a sample into an estimated calendar age. The calculations involve several steps and include an intermediate value called the "radiocarbon age", which is the age in "radiocarbon years" of the sample: an age quoted in radiocarbon years means that no calibration curve has been used − the calculations for radiocarbon years assume that the atmospheric {{chem|14|C}}/{{chem|12|C}} ratio has not changed over time.<ref name=renamed_from_12_on_20200701175743/><ref name=renamed_from_0_on_20200701175743/>

Calculating radiocarbon ages also requires the value of the half-life for {{chem|14|C}}. In Libby's 1949 paper he used a value of 5720 ± 47 years, based on research by Engelkemeir et al.<ref>{{Cite journal|last1=Engelkemeir|first1=Antoinette G.|last2=Hamill|first2=W.H.|last3=Inghram|first3=Mark G.|last4=Libby|first4=W.F.|date=1949|title=The Half-Life of Radiocarbon (C<sup>14</sup>)|journal=Physical Review|volume=75|issue=12|pages=1825|doi=10.1103/PhysRev.75.1825|bibcode=1949PhRv...75.1825E}}</ref> This was remarkably close to the modern value, but shortly afterwards the accepted value was revised to 5568 ± 30 years,<ref name=Johnson>{{cite journal|author=Frederick Johnson|year=1951|title=Introduction|jstor=25146610|journal=Memoirs of the Society for American Archaeology|issue=8|pages=1–19}}</ref> and this value was in use for more than a decade. It was revised again in the early 1960s to 5,730 ± 40 years,<ref name=Godwin>{{ cite journal
|author=H. Godwin
|title=Half-life of Radiocarbon
|journal=Nature
|year=1962
|volume=195
|issue=4845
|pages=984
|bibcode=1962Natur.195..984G
|doi=10.1038/195984a0
|s2cid=27534222
}}</ref><ref name=Plicht>{{ cite journal
|author=J.van der Plicht and A.Hogg
|title=A note on reporting radiocarbon
|journal=Quaternary Geochronology
|year=2006
|volume=1
|issue=4
|pages=237–240
|url=http://www.ees.nmt.edu/outside/courses/hyd558/downloads/Set_9-10_Carbon-14/van_der_Plicht2006.pdf
|doi=10.1016/j.quageo.2006.07.001
|bibcode=2006QuGeo...1..237V
|access-date=9 December 2017}}</ref> which meant that many calculated dates in papers published prior to this were incorrect (the error in the half-life is about 3%).{{#tag:ref|Two experimentally determined values from the early 1950s were not included in the value Libby used: ~6,090 years, and 5900 ± 250 years.<ref name=Taylor_287>Taylor & Bar-Yosef (2014), p. 287.</ref>|group=note}} For consistency with these early papers, it was agreed at the 1962 Radiocarbon Conference in Cambridge (UK) to use the "Libby half-life" of 5568 years. Radiocarbon ages are still calculated using this half-life, and are known as "Conventional Radiocarbon Age". Since the calibration curve (IntCal) also reports past atmospheric {{chem|14|C}} concentration using this conventional age, any conventional ages calibrated against the IntCal curve will produce a correct calibrated age. When a date is quoted, the reader should be aware that if it is an uncalibrated date (a term used for dates given in radiocarbon years) it may differ substantially from the best estimate of the actual calendar date, both because it uses the wrong value for the half-life of {{chem|14|C}}, and because no correction (calibration) has been applied for the historical variation of {{chem|14|C}} in the atmosphere over time.<ref name=renamed_from_12_on_20200701175743>Aitken (1990), pp. 92–95.</ref><ref name=renamed_from_0_on_20200701175743>Bowman (1995), p. 42.</ref><ref name=INTCAL13/>{{#tag:ref|The term "conventional radiocarbon age" is also used. The definition of radiocarbon years is as follows: the age is calculated by using the following [[w:Calculation of radiocarbon dates#Standards|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 [[Radiocarbon dating#Isotopic fractionation|fractionation]], based on a standard isotope ratio, and (e) the assumption that the {{chem|14|C}}/{{chem|12|C}} ratio has not changed over time.<ref name=Taylor_4>Taylor & Bar-Yosef (2014), pp. 26–27.</ref>|group=note}}