Radiocarbon dating

Radiocarbon dating (sometimes simply known as carbon dating) is a radiometric dating method that uses the naturally occurring radioisotope carbon-14 (¹⁴C) to estimate the age of carbonaceous materials up to about 58,000 to 62,000 years.^[1] Raw, i.e. uncalibrated, radiocarbon ages are usually reported in radiocarbon years "Before Present" (BP), "Present" being defined as 1950 CE. Such raw ages can be calibrated to give calendar dates. One of the most frequent uses of radiocarbon dating is to estimate the age of organic remains from archaeological sites. When plants fix atmospheric carbon dioxide () into organic material during photosynthesis they incorporate a quantity of ¹⁴C that approximately matches the level of this isotope in the atmosphere (a small difference occurs because of isotope fractionation, but this is corrected after laboratory analysis). After plants die or they are consumed by other organisms (for example, by humans or other animals) the ¹⁴C fraction of this organic material declines at a fixed exponential rate due to the radioactive decay of ¹⁴C. Comparing the remaining ¹⁴C fraction of a sample to that expected from atmospheric ¹⁴C allows the age of the sample to be estimated.

The technique of radiocarbon dating was developed by Willard Libby and his colleagues at the University of Chicago in 1949. Emilio Segrè asserted in his autobiography that Enrico Fermi suggested the concept to Libby in a seminar at Chicago that year. Libby estimated that the steady state radioactivity concentration of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram. In 1960, he was awarded the Nobel Prize in chemistry for this work. He first demonstrated the accuracy of radiocarbon dating by accurately estimating the age of wood from an ancient Egyptian royal barge for which the age was known from historical documents.^[2]^[3]

Basic physics

Carbon has two stable, nonradioactive isotopes: carbon-12 (¹²C), and carbon-13 (¹³C). In addition, there are trace amounts of the unstable isotope carbon-14 (¹⁴C) on Earth. Carbon-14 has a half-life of 5730 years, meaning that the amount of carbon-14 in a sample is halved over the course of 5730 years due to radioactive decay. Carbon-14 would have long ago vanished from Earth were it not for the unremitting cosmic ray impacts on nitrogen in the Earth's atmosphere, which create more of the isotope. The neutrons resulting from the cosmic ray interactions participate in the following nuclear reaction on the atoms of nitrogen molecules (N₂) in the atmosphere:

$n + \mathrm{^_N^ \rightarrow \mathrm + p$

The highest rate of carbon-14 production takes place at altitudes of 9 to 15 km (30,000 to 50,000 ft), and at high geomagnetic latitudes, but the carbon-14 spreads evenly throughout the atmosphere and reacts with oxygen to form carbon dioxide. Carbon dioxide also permeates the oceans, dissolving in the water. For approximate analysis it is assumed that the cosmic ray flux is constant over long periods of time; thus carbon-14 is produced at a constant rate and the proportion of radioactive to non-radioactive carbon is constant: ca. 1 part per trillion (600 billion atoms/mole). In 1958 Hessel de Vries showed that the concentration of carbon-14 in the atmosphere varies with time and locality.^[4] For the most accurate work, these variations are compensated by means of calibration curves. When these curves are used, their accuracy and shape are the factors that determine the accuracy of the age obtained for a given sample.

Plants take up atmospheric carbon dioxide by photosynthesis, and are ingested by animals, so every living thing is constantly exchanging carbon-14 with its environment as long as it lives. Once it dies, however, this exchange stops, and the amount of carbon-14 gradually decreases through radioactive beta decay with a half-life of 5,730 ± 40 years.

$\mathrm\rightarrow\mathrm{~^_N^+ e^ + \bar_e$

Carbon-14 was discovered on February 27, 1940, by Martin Kamen and Sam Ruben at the University of California Radiation Laboratory at Berkeley.

Computation of ages and dates

The number of decays per time is proportional to the current number of radioactive atoms. This is expressed by the following differential equation, where N is the number of radioactive atoms and λ is a positive number called the decay constant:

$\frac = -\lambda N.$

As the solution to this equation, the number of radioactive atoms N can be written as a function of time

$N(t) = N_0e^\,$ ,

which describes an exponential decay over a timespan t with an initial condition of N₀ radioactive atoms at $t = 0$ . Canonically, t is 0 when the decay started. In this case, N₀ is the initial amount of ¹⁴C atoms when the decay started.

For radiocarbon dating a once living organism, the initial ratio of ¹⁴C atoms to the sum of all other carbon atoms at the point of the organism's death and hence the point when the decay started, is approximately the ratio in the atmosphere.

Two characteristic times can be defined:

mean- or average-life: mean or average time each radiocarbon atom spends in a given sample until it decays.
half-life: time lapsed for half the number of radiocarbon atoms in a given sample, to decay,

It can be shown that:

$t_ \,$ = $\frac$ = radiocarbon mean- or average-life = 8033 years (Libby value)

$t_\frac \,$ = $t_ \cdot \ln 2$ = radiocarbon half-life = 5568 years (Libby value)

Notice that dates are customarily given in years BP which implies t(BP) = –t because the time arrow for dates runs in reverse direction from the time arrow for the corresponding ages. From these considerations and the above equation, it results:

For a raw radiocarbon date:

$t(BP) = \frac {\ln \frac$

and for a raw radiocarbon age:

$t(BP) = -\frac {\ln \frac$

After replacing values, the raw radiocarbon age becomes any of the following equivalent formulae:

using logs base e and the average life:

$t(BP) = -t_\cdot \ln{\frac$

and

using logs base 2 and the half-life:

$t(BP) = -t_\frac\cdot \log_2 \frac$

Wiggle matching uses the non-linear relationship between the ¹⁴C age and calendar age to match the shape of a series of closely sequentially spaced ¹⁴C dates with the ¹⁴C calibration curve.

Measurements and scales

Measurements are traditionally made by counting the radioactive decay of individual carbon atoms by gas proportional counting or by liquid scintillation counting. For samples of sufficient size (several grams of carbon) this method is still widely used in the 2000s. Among others, all the tree ring samples used for the calibration curves (see below) were determined by these counting techniques. Such decay counting, however, is relatively insensitive and subject to large statistical uncertainties for small samples. When there is little carbon-14 to begin with, the long radiocarbon half-life means that very few of the carbon-14 atoms will decay during the time allotted for their detection, resulting in few disintegrations per minute.

The sensitivity of the method has been greatly increased by the use of accelerator mass spectrometry (AMS). With this technique ¹⁴C atoms can be detected and counted directly vs only detecting those atoms that decay during the time interval allotted for an analysis. AMS allows dating samples containing only a few milligrams of carbon.

Raw radiocarbon ages (i.e., those not calibrated) are usually reported in "years Before Present" (BP). This is the number of radiocarbon years before 1950, based on a nominal (and assumed constant – see "calibration" below) level of carbon-14 in the atmosphere equal to the 1950 level. These raw dates are also based on a slightly-off historic value for the radiocarbon half-life. Such value is used for consistency with earlier published dates (see "Radiocarbon half-life" below). See the section on computation for the basis of the calculations.

Radiocarbon dating laboratories generally report an uncertainty for each date. For example, 3000 ± 30 BP indicates a standard deviation of 30 radiocarbon years. Traditionally this included only the statistical counting uncertainty. However, some laboratories supplied an "error multiplier" that could be multiplied by the uncertainty to account for other sources of error in the measuring process. More recently, the laboratories try to quote the overall uncertainty, which is determined from control samples of known age and verified by international intercomparison exercises.^[5] In 2008, a typical uncertainty better than ±40 radiocarbon years can be expected for samples younger than 10,000 years. This, however, is only a small part of the uncertainty of the final age determination (see section Calibration below).

, the limiting age for a 1 milligram sample of graphite is about ten half-lives, approximately 60,000 years.^[6] This age is derived from that of the calibration blanks used in an analysis, whose ¹⁴C content is assumed to be the result of contamination during processing (as a result of this, some facilities^[6] will not report an age greater than 60,000 years for any sample).

A variety of sample processing and instrument-based constraints have been postulated to explain the upper age-limit. To examine instrument-based background activities in the AMS instrument of the W. M. Keck Carbon Cycle Accelerator Mass Spectrometry Laboratory of the University of California, a set of natural diamonds were dated. Natural diamond samples from different sources within rock formations with standard geological ages in excess of 100 my yielded¹⁴C apparent ages 64,920 ± 430 BP to 80,000 ± 1100 BP as reported in 2007.^[7]