# Radiocarbon Data & Calculations

In AMS, the carbon or "graphite" derived from a sample is compressed into a small cavity in an aluminum "target" which acts as a cathode in the ion source. The surface of the graphite is sputtered with heated cesium and the ions produced are extracted and accelerated in the AMS system. After acceleration and removal of electrons, the emerging positive ions are magnetically separated by mass and the ^{12}C and ^{13}C ions are measured in Faraday Cups where a ratio of their currents is recorded. Simultaneously the ^{14}C ions are recorded in a gas ionization counter, so that instantaneous ratios of ^{14}C to ^{13}C and ^{12}C are recorded. These are the raw signals that are ultimately converted to a radiocarbon age.

From a contemporary sample, about 150 ^{14}C counts per second are collected. It is expected then, for a 5,570 year (1 half-life) or 11,140 year old (2 half-lives) sample that 75 or 38 counts per second would be obtained. Although one can simply measure older samples for longer times, they are constantly being consumed by the ion source, so there are practical limits to the minimum sample activity that can be measured, depending on how much material you have. At the present time, for a 1 milligram sample of graphite, this limiting age is about ten half-lives, or 60,000 years, if set only by the sample size. However, limiting ages or "backgrounds" are also determined by process blanks which correspond to the method used to extract the carbon from the sample.

» NOSAMS General Statement of ^{14}C Procedures (pdf)

## Process Blanks

The process blanks contain small but measurable amounts of ^{14}C from contamination introduced during chemical preparation, collection or handling. Organic materials, which require the most processing, are limited to younger ages by their corresponding process blank. Since it is always necessary to subtract the counts due to blanks, from the counts due to sample, it may become a statistical limitation for very old samples (small number of ^{14}C atoms) where we are measuring the difference between very small numbers. Thus, ages are limited by the age of the process blanks (more on that below) and by the statistical uncertainty of the ^{14}C measurement.

## Fraction Modern

The Fraction Modern (Fm) is computed from the expression:

$$Fm = \frac{(S - B)}{(M - B)}$$Where B, S and M represent the ^{14}C/^{12}C ratios of the blank, the sample and the modern reference, respectively.

Fraction Modern is a measurement of the deviation of the ^{14}C/^{12}C ratio of a sample from "Modern." Modern is defined as 95% of the radiocarbon concentration (in AD 1950) of NBS Oxalic Acid I normalized to δ^{13}C_{VPDB}=-19 per mil (Olsson, 1970). AMS results are calculated using the internationally agreed upon definition of 0.95 times the specific activity of NBS Oxalic Acid I (SRM 4990B) normalized to δ^{13}C_{VPDB}=-19 per mil. This is equialent to an absolute (AD 1950) ^{14}C/^{12}C ratio of 1.176 ± 0.010 x 10^{-12} (Karlen, et. al., 1968); all results are normalized to -25 per mil using the δ^{13}C_{VPDB} of the sample (see below). The value used for this correction is specified in the report of final results.

## δ^{13}C Correction

In addition to loss through decay of radiocarbon, ^{14}C is also affected by natural isotopic fractionation. Fractionation is the term used to describe the differential uptake of one isotope with respect to another. While the three carbon isotopes are chemically indistinguishable, lighter ^{12}C atoms are preferentially taken up before the ^{13}C atoms in biological pathways. Similarly, ^{13}C atoms are taken up before ^{14}C. The assumption is that the fractionation of ^{14}C relative to ^{12}C is twice that of ^{13}C, reflecting the difference in mass. Fractionation must be corrected for in order to make use of radiocarbon measurements as a chronometric tool for all parts of the biosphere. In order to remove the effects of isotopic fractionation, the Fraction Modern is then corrected to the value it would have if its original δ^{13}C were -25 per mil (the δ^{13}C value to which all radiocarbon measurements are normalized.)

The Fraction Modern corrected for δ^{13}C,** Fm _{δ13C}**,

## Errors

Atoms of ^{14}C contained in a sample are directly counted using the AMS method of radiocarbon analysis. Accordingly, we calculate an internal statistical error using the total number of ^{ 14}C counts measured for each target ( ± √n ). An external error is calculated from the reproducibility of multiple exposures for a given target. For example, we may measure the ^{14}C /^{12}C of a sample up to 9 separate times over the course of a 2-day period. The reproducibility of these measurements gives us a good estimate of the true experimental error. The final error is the larger of the internal or external errors.

Aside from the normal statistical errors intrinsic to the counting of ^{14}C events, there are additional statistical errors associated with the corrections applied to the Fraction Modern that we account for. For example, the δ^{13}C correction, from a stable mass spectrometer has an uncertainty of approximately 0.1‰. The error associated with δ^{13}C is calculated by:

This component of the Fm error is then added in quadrature as follows:

## Radiocarbon Age

Radiocarbon age is calculated from the δ^{13}C-corrected Fraction Modern according to the following formula:

Age = -8033 ln (Fm)

Reporting of ages and/or activities follows the convention outlined by Stuiver and Polach (1977) and Stuiver (1980). Ages are calculated using 5568 years as the half-life of radiocarbon and are reported without reservoir corrections or calibration to calendar years. For freeware programs, we suggest that you look at the following web site for a list of programs that will calibrate radiocarbon results to calendar years (including making reservoir corrections).[ Radiocarbon-Related Information Sources]

The error in the age is given by 8033 times the relative error in the Fm . Therefore a 1% error in fraction-modern leads to an 80 year error in the age. Ages are rounded according to the convention of Stuiver & Polach, shown below.

## Rounding Convention

Age |
Nearest |
Error |
Nearest |

<1000 | 5 | <100 | 5 |

1000-9999 | 10 | 100-1000 | 10 |

10000-20000 | 50 | >1000 | 100 |

>20000 | 100 |

## Limiting Ages

There are two situations that limit an age; the first is that the measured Fm is smaller than that of the corresponding process blank measured in the same suite of samples on the AMS. If this is the case, then the reported age will be quoted as an age greater than the age of the process blank. No age is reported greater than 60,000 years. The typical background age for organic combustions is 48,000 years and for inorganic carbon samples, 52,000 years.

One other situation that limits the age (if not already limited by the background age) is the error of the AMS result. If twice the reported error of the Fraction Modern (let's call this 2sigma) is larger than the sample Fraction Modern, then a limiting age is reported. The limiting age is then calculated as -8033 * ln(2sigma) and rounded according to conventions outlined above.

## Age > Modern

Since Modern is defined as 95% of the ^{14}C activity for AD 1950, as defined by the oxalic acid standard, sample activities can be substantially greater than Modern, and so the ages are reported as > Modern.

## Δ^{14}C

We also report the Δ^{14}C value as defined in Stuiver and Pollach (1977) as the relative difference between the *absolute international standard* (base year 1950) and sample activity corrected for age and δ^{13}C. The Δ^{14}C is age corrected to account for decay that took place between collection (or death) and the time of measurement so that two measurements of the same sample made years apart will produce the same calculated Δ^{14}C result. Collection year must be specified in question 8 of the submittal form in order for Δ^{14}C results to be calculated.

Where lambda is 1/(true mean-life) of radiocarbon = 1/8267 = 0.00012097

Y_{c} is year of collection.

## References

Karlen, I., Olsson, I.U., Kallburg, P. and Kilici, S., 1964. Absolute determination of the activity of two ^{14}C dating standards. *Arkiv Geofysik*, 4:465-471.

Olsson, I.U., 1970. The use of Oxalic acid as a Standard. In I.U. Olsson, ed., Radiocarbon Variations and Absolute Chronology, *Nobel Symposium, 12th Proc.*, John Wiley & Sons, New York, p. 17.

Stuiver, M. and Polach, H.A., 1977. Discussion: Reporting of ^{14}C data. *Radiocarbon*, 19:355-363. (pdf)

Stuiver, M., 1980. Workshop on ^{14}C data reporting. *Radiocarbon*, 22:964-966.

*Last updated: December 4, 2014*