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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 relatively short
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 flux interactions with 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:
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.
[5] 14C can also be produced at ground level at a rate of 1 x 10−4 g−1s−1, which is not considered significant enough to impact on dating without a known other source of neutrons.
[6]
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.[
citation needed] Carbon-14 is stored in different amounts in different reservoirs because there is a
dynamic equilibrium between 14C production and decay.
[6]
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:
As the solution to this equation, the number of radioactive atoms
N can be written as a function of time
,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 number 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:
=
= radiocarbon mean- or average-life = 8033 years (Libby value)
=
= 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:
and for a raw radiocarbon age:
After replacing values, the raw radiocarbon age becomes any of the following equivalent formulae:
using logs base
e and the average life:
and
using logs base 2 and the half-life:
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
The use of accelerator mass spectrometers can improve the sensitivity of radiocarbon dating.
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.
[7] 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).
Samples older than the upper age-limit cannot be dated because the small number of remaining intrinsic 14C atoms will be obscured by 14C background atoms introduced into the samples while they still resided in the environment, during sample preparation, or in the detection instrument. As of 2007
[update], the limiting age for a 1 milligram sample of graphite is about ten half-lives, approximately 60,000 years.
[8] 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
[8] 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
Ma yielded14C apparent ages 64,920 ± 430 BP to 80,000 ± 1100 BP as reported in 2007.
[9]