K–Ar dating

Potassium–argon dating, abbreviated K–Ar dating, is a radiometric dating method used in geochronology and archaeology. It is based on the measurement of the product of the radioactive decay of potassium-40 (⁴⁰K) into argon (Ar). Potassium is a common element in many materials, such as feldspars, micas, clay minerals, tephra, and evaporites. In these materials, the decay product ⁴⁰
Ar can escape the liquid (molten) rock but starts to accumulate when the rock solidifies (recrystallizes). The amount of argon sublimation that occurs is a function of the sample's purity, the composition of the mother material, and several other factors. These factors introduce error limits on the upper and lower bounds of dating so that the final determination of age is reliant on the environmental factors during formation, melting, and exposure to decreased pressure or open air. Time since recrystallization is calculated by measuring the ratio of the amount of ⁴⁰
Ar accumulated to the amount of ⁴⁰
K remaining. The long half-life of ⁴⁰
K allows the method to be used to calculate the absolute age of samples older than a few thousand years.

The quickly cooled lavas that make nearly ideal samples for K–Ar dating also preserve a record of the direction and intensity of the local magnetic field as the sample cooled past the Curie temperature of iron. The geomagnetic polarity time scale was calibrated largely using K–Ar dating.

History

Radioactivity was discovered at the end of the 19th century, and by the early years of the 20th century it was known that some minerals were radioactive and that radioactive decay caused some elements to transmute into other elements. In 1905 Ernest Rutherford suggested that since helium was a byproduct of the decay of uranium, it might be possible to determine the age of rocks that contained uranium by measuring how much helium had been produced. In the same year, Bertram Boltwood suggested that radioactive uranium transmuted into lead, and proposed a geological dating mechanism on that basis. Over the next few years it became understood that an element might exist as different isotopes, and during the 1920s two isotopes of potassium were discovered: ³⁹
K and ⁴¹
K. By the late 1930s a third isotope, ⁴⁰
K, had been identified; it was known to be radioactive, decaying to ⁴⁰
Ca, an isotope of calcium.

In 1937 C. F. Von Weizsäcker proposed that ⁴⁰
K might also decay into ⁴⁰
Ar (argon), and that this would explain the abundance of ⁴⁰
Ar in the earth's atmosphere. Attempts were made to prove the theory by looking for excess argon in minerals, but early experiments did not measure the isotopic ratio of the argon that was found, which meant the excess could not be detected. In 1948, L. T. Aldrich and A. O. Nier tested four old minerals and found the ratio of ⁴⁰
Ar to ³⁶
Ar to be higher than the atmospheric ratio in each case. This made it clear that argon isotope ratios could be used to determine the age of rocks.

Physical details

Argon, being a noble gas, does not form compounds, so any argon found in rock comes from the decay of potassium or from atmospheric contamination. Argon atoms, which have a radius of 1.9 Å, are too large to pass through most crystal lattices, so when a rock crystallizes argon stops being able to pass through it. This means that an age determination made on the basis of trapped argon is a measurement of how long it has been since the rock last crystallized. For a rock such as lava, cooling rapidly, the time of crystallization may be only a few hours; for other rocks, such as an intrusive igneous rock, the crystallization may take much longer, up to millions of years. Not all rocks can have their ages determined by K-Ar dating: sedimentary rocks, for example, include grains of older rocks, and a measurement taken on those grains will give the age of the original rock from which the grain was eroded, rather than the age of formation of the sedimentary rock. ` Potassium naturally occurs in three isotopes: ³⁹
K and ⁴¹
K are stable, and ⁴⁰
K is radioactive, with a half-life of 1.266×10⁹ years. ⁴⁰
K is the least abundant of the three, constituting only 0.0117% of present-day potassium. Because ⁴⁰
K is radioactive, its abundance will decrease over time: in 16 million years it will be one percent less abundant than it is today, and it was correspondingly more abundant in the past. There are four different decay paths for ⁴⁰
K. The most common path is via electron emission to ⁴⁰
Ca, which accounts for 89.59% of decay events. The other three paths all produce ⁴⁰
Ar, mainly through electron capture.

The equation governing the decay of a radioactive isotope is

${\displaystyle N=N_{0}\,e^{-\lambda t}\,}$

where N₀ is the number of atoms of the isotope in the original sample and N is the number of atoms of that isotope left after time t. λ is a constant that depends on the particular isotope; for a given isotope it is equal to the reciprocal of the mean-life—i.e. the average or expected time a given atom will survive before undergoing radioactive decay. If an isotope decays into two different isotopes, as ⁴⁰
K does, the number of atoms of the original isotope is equal to the remaining original atoms of the isotope, plus the number of atoms of the decay products. For ⁴⁰
K this means that

${\displaystyle N_{0}=K+A+C}$

where K, A and C represent the number of atoms in the present day sample of ⁴⁰
K, ⁴⁰
Ar and ⁴⁰
Ca. Then the decay equation can be rearranged as$${\displaystyle A+C=K\left[e^{(\lambda _{A}+\lambda _{C})t}-1\right]}$$where λ_A and λ_C are the decay constants associated with the argon and calcium decay paths. For the purposes of calculating geological age, the amount of radiogenic calcium (that is, calcium created by radioactive decay) can be ignored, so the equation becomes

${\displaystyle A=K{\frac {\lambda _{A}}{\lambda _{A}+\lambda _{C}}}\left[e^{(\lambda _{A}+\lambda _{C})t}-1\right]}$

If the values of A and K—the amount of radiogenic ⁴⁰
Ar, and the remaining amount of ⁴⁰
K—are measured for a rock sample, then this equation allows t, the age of the rock, to be calculated.

Substituting in the values for the decay constants, we get

${\displaystyle A=K{\frac {0.5757\times 10^{-10}}{0.5757\times 10^{-10}+4.9548\times 10^{-10}}}\left[e^{(0.5757\times 10^{-10}+4.9548\times 10^{-10})t}-1\right]}$

which simplifies to

${\displaystyle A=0.1041\,K\left[e^{(5.5305\times 10^{-10})t}-1\right]}$

Solving for t:

${\displaystyle t=1.808\times 10^{9}\ln \left[9.607{\frac {A}{K}}+1\right]}$

For small x, ln(1 + x) is close to the value of x, so for young rocks, it is possible to simplify this further. The simplified equation becomes

${\displaystyle t\approx 1.737\times 10^{10}{\frac {A}{K}}}$

The error this introduces increases with age and is less than about 1% for rocks younger than about 40 million years.

Measurements

Argon isotope dilution

The argon to be measured is extracted from the rock sample by fusing the rock in a vacuum, and separating the reactive gases. The usual method of measuring the quantity of argon in the resulting gases is isotope dilution. For an element with a single natural isotope, this method works by adding a known quantity (called a tracer) of an artificial isotope, and measuring the isotope ratio of the resulting mixture with a mass spectrometer. If there are x atoms of the natural isotope, and y atoms of the artificial isotope, and the measured ratio is R, then

${\displaystyle x=yR}$

However, argon has three stable isotopes of argon, ³⁶
Ar, ³⁸
Ar, and ⁴⁰
Ar, which means this form of the equation cannot be used. An additional complication is that the ⁴⁰
Ar in the sample to be measured comes from two sources: the radiogenic ⁴⁰
Ar from the decay of potassium, and some ⁴⁰
Ar from atmospheric contamination, either from the measurement process or the inclusion of some atmospheric argon in the rock at the time it crystallized.

The tracer used for K-Ar dating is nearly pure ³⁸
Ar. The calculations for isotope dilution also require a knowledge of the isotopic composition of atmospheric argon, which is 99.6% ⁴⁰
Ar, 0.063% ³⁸
Ar, and 0.337% ³⁶
Ar. The following three equations are needed. In these equations, the left side, with the subscript "M", represents the quantity of that isotope measured by mass spectrometer.The subscripts "T" and "A" indicate the quantity of argon in the tracer and the atmospheric contamination, respectively. In each case—mass spectrometer measurements, tracer, and atmospheric contamination—the quantity of each isotope is the sum of the amount from atmospheric contamination and from the tracer and, for the mass spectrometer measurement, the amount of radiogenic ⁴⁰
Ar.

${\displaystyle {}^{40}\mathrm {Ar} _{M}={}^{40}\mathrm {Ar} _{rad}+{}^{40}\mathrm {Ar} _{T}+{}^{40}\mathrm {Ar} _{A}}$

1

${\displaystyle {}^{38}\mathrm {Ar} _{M}={}^{38}\mathrm {Ar} _{T}+{}^{38}\mathrm {Ar} _{A}}$

2

${\displaystyle {}^{36}\mathrm {Ar} _{M}={}^{36}\mathrm {Ar} _{T}+{}^{36}\mathrm {Ar} _{A}}$

3

The isotopic compositions of the tracer and of atmospheric argon are known, so the tracer and atmospheric terms in equations 1 and 3 can be written as multiples of the corresponding amounts of ³⁸
Ar:

${\displaystyle {}^{40}\mathrm {Ar} _{M}={}^{40}\mathrm {Ar} _{rad}+({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{T}\,{}^{38}\mathrm {Ar} _{T}+({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{A}\,{}^{38}\mathrm {Ar} _{A}}$

4

${\displaystyle {}^{36}\mathrm {Ar} _{M}=({}^{36}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{T}\,{}^{38}\mathrm {Ar} _{T}+({}^{36}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{A}\,{}^{38}\mathrm {Ar} _{A}}$

5

Dividing equation 2 by equation 5 gives an expression for the measured ³⁸
Ar/³⁶
Ar ratio, which can be solved for the amount of atmospheric ³⁸
Ar:

${\displaystyle {}^{38}\mathrm {Ar} _{A}={}^{38}\mathrm {Ar} _{T}\left[{\frac {1-({}^{38}\mathrm {Ar} /{}^{36}\mathrm {Ar} )_{M}\,({}^{36}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{T}}{({}^{38}\mathrm {Ar} /{}^{36}\mathrm {Ar} )_{M}\,({}^{36}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{A}-1}}\right]}$

6

Similarly, dividing equation 4 by equation 2 gives an expression for the measured ⁴⁰
Ar/³⁸
Ar ratio, which can be solved for the amount of radiogenic ⁴⁰
Ar:

${\displaystyle {}^{40}\mathrm {Ar} _{rad}={}^{38}\mathrm {Ar} _{T}\,({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{M}-{}^{38}\mathrm {Ar} _{T}\,({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{T}-{}^{38}\mathrm {Ar} _{A}\left[({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{A}-({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{M}\right]}$

7

Substituting equation 6 into equation 7 gives the radiogenic ⁴⁰
Ar in terms of the measured isotope ratios, the known isotope ratios of the tracer and of atmospheric argon, and the known amount of ³⁸
Ar in the tracer:

${\displaystyle {\begin{aligned}{}^{40}\mathrm {Ar} _{rad}={}^{38}\mathrm {Ar} _{T}{\Bigg \{}&({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{M}-({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{T}\\&-\left[{\frac {1-({}^{38}\mathrm {Ar} /{}^{36}\mathrm {Ar} )_{M}\,({}^{36}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{T}}{({}^{38}\mathrm {Ar} /{}^{36}\mathrm {Ar} )_{M}\,({}^{36}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{A}-1}}\right]\left[({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{A}-({}^{40}\mathrm {Ar} /{}^{38}\mathrm {Ar} )_{M}\right]{\Bigg \}}\end{aligned}}}$

8

Accuracy

Measurement errors

If an error is made in the measurement of the Ar/K ratio, the resulting error in the calculated age of the sample depends on the age of the sample (see graph).

Similarly, the effect of an error in the decay constants for the argon and calcium decay paths for ⁴⁰
K depends on the age of the sample. Here λ_A is the decay constant for the decay to ⁴⁰
Ar, and λ_C is the decay constant for the decay to ⁴⁰
Ca.

Percent error in K–Ar age for errors in λ_A

Percent error in K–Ar age for errors in λ_C

Extraneous and lost ⁴⁰
Ar

The K-Ar method measures the ratio of ⁴⁰
K in a sample to the ⁴⁰
Ar produced by radioactive decay from the ⁴⁰
K originally in that sample. This method can be rendered inaccurate, either by the presence in the sample of extraneous argon, defined as ⁴⁰
Ar that was not produced by decay of the ⁴⁰
K originally in the rock, or by the loss from the rock of the ⁴⁰
Ar produced by the decay of the ⁴⁰
K.

Applications

K–Ar dating was instrumental in developing the geomagnetic polarity time scale. Although it finds the most utility in geological applications, it plays an important role in archaeology. One archeological application has been bracketing the age of archeological deposits at Olduvai Gorge by dating lava flows above and below the deposits. It has also been indispensable in other early East African sites with a history of volcanic activity such as Hadar, Ethiopia. The K–Ar method continues to have utility in dating clay mineral diagenesis. In 2017, the successful dating of illite formed by weathering was reported. This finding indirectly led to the dating of the strandflat of Western Norway from where the illite was sampled. Clay minerals are less than 2 μm thick and cannot easily be irradiated for Ar–Ar analysis because Ar recoils from the crystal lattice.

In 2013, the K–Ar method was used by the Curiosity Mars rover to date a rock on the Martian surface, the first time a rock has been dated from its mineral ingredients while situated on another planet.

Notes

References

Sources

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Further reading

The Wikibook Historical Geology has a page on the topic of: K-Ar dating

• "K/Ar and ⁴⁰K/³⁹K methodology". New Mexico Geochronology Research Laboratory. Archived from the original on 17 April 2006.
• Michaels, G. H.; Fagan, B. M. (15 December 2005). "Chronological Methods 9: Potassium-Argon Dating". University of California. Archived from the original on 10 August 2010.
• Moran, T. J. (2009). "Teaching Radioisotope Dating Using the Geology of the Hawaiian Islands" (PDF). Journal of Geoscience Education. 57 (2): 101–105. Bibcode:2009JGeEd..57..101M. doi:10.5408/1.3544237.