# 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 an isotope of potassium (K) 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.

## Decay series

Potassium naturally occurs in 3 isotopes: $^{39}$ K (93.2581%), $^{40}$ K (0.0117%), $^{41}$ K (6.7302%). $^{39}$ K and $^{41}$ K are stable. The $^{40}$ K isotope is radioactive; it decays with a half-life of 1.248×10$^9$ years to $^{40}$ Ca and $^{40}$ Ar . Conversion to stable $^{40}$ Ca occurs via electron emission (beta decay) in 89.3% of decay events. Conversion to stable $^{40}$ Ar occurs via electron capture in the remaining 10.7% of decay events.
Argon, being a noble gas, is a minor component of most rock samples of geochronological interest: It does not bind with other atoms in a crystal lattice. When $^{40}$ K decays to $^{40}$ Ar ; the atom typically remains trapped within the lattice because it is larger than the spaces between the other atoms in a mineral crystal. However, it can escape into the surrounding region when the right conditions are met, such as changes in pressure or temperature. $^{40}$ Ar atoms can diffuse through and escape from molten magma because most crystals have melted, and the atoms are no longer trapped. Entrained argon – diffused argon that fails to escape from the magma – may again become trapped in crystals when magma cools to become solid rock again. After the recrystallization of magma, more $^{40}$ K will decay and $^{40}$ Ar will again accumulate, along with the entrained argon atoms, trapped in the mineral crystals. Measurement of the quantity of $^{40}$ Ar atoms are used to compute the amount of time that has passed since a rock sample has solidified.
Despite $^{40}$ Ca being the favored daughter nuclide, it is rarely useful in dating because calcium is so common in the crust, with $^{40}$ Ca being the most abundant isotope. Thus, the amount of calcium originally present is unknown and can vary enough to confound measurements of the small increases produced by radioactive decay.

## Formula

The ratio of the amount of $^{40}$ Ar to that of $^{40}$ K is directly related to the time elapsed since the rock was cool enough to trap the Ar by the equation:
    $t=t_{\frac {1}{2}}\log _{2}\left({\frac {{\ce {K}}_{f}+{\frac {{\ce {Ar}}_{f}}{0.109}}}{{\ce {K}}_{f}}}\right)$,
Where:
* t is time elapsed
* t$_{1/2}$ is the half-life of $^{40}$ K
* K$_f$ is the amount of $^{40}$ K remaining in the sample
* Ar$_f$ is the amount of $^{40}$ Ar found in the sample.
The scale factor 0.109 corrects for the unmeasured fraction of $^{40}$ K which decayed into $^{40}$ Ca ; the sum of the measured $^{40}$ K and the scaled amount of $^{40}$ Ar gives the amount of $^{40}$ K which was present at the beginning of the elapsed period. In practice, each of these values may be expressed as a proportion of the total potassium present, as only relative, not absolute, quantities are required.

## Obtaining the data

To obtain the content ratio of isotopes $^{40}$ Ar to $^{40}$ K in a rock or mineral, the amount of Ar is measured by mass spectrometry of the gases released when a rock sample is volatilized in a vacuum. The potassium is quantified by flame photometry or atomic absorption spectroscopy.
The amount of $^{40}$ K is rarely measured directly. Rather, the more common $^{39}$ K is measured and that quantity is then multiplied by the accepted ratio of $^{40}$ K /$^{39}$ K (i.e., 0.0117%/93.2581%, see above).
The amount of $^{40}$ Ar is also measured to assess how much of the total argon is atmospheric in origin.

## Assumptions

According to McDougall & Harrison (1999, p. 11), the following assumptions must be true for computed dates to be accepted as representing the true age of the rock:
* The parent nuclide, $^{40}$ K , decays at a rate independent of its physical state and is not affected by differences in pressure or temperature. This is a well-founded major assumption, common to all dating methods based on radioactive decay. Although changes in the electron capture partial decay constant for $^{40}$ K possibly may occur at high pressures, theoretical calculations indicate that for pressures experienced within a body the size of the Earth, the effects are negligibly small.
* The $^{40}$ K /$^{39}$ K ratio in nature is constant so the $^{40}$ K is rarely measured directly, but is assumed to be 0.0117% of the total potassium. Unless some other process is active at the time of cooling, this is a very good assumption for terrestrial samples.
* The radiogenic argon measured in a sample was produced by in situ decay of $^{40}$ K in the interval since the rock crystallized or was recrystallized. Violations of this assumption are not uncommon. Well-known examples of incorporation of extraneous $^{40}$ Ar include chilled glassy deep-sea basalts that have not completely outgassed preexisting $^{40}$ Ar *, and the physical contamination of magma by the inclusion of older xenolitic material. The Ar–Ar dating method was developed to measure the presence of extraneous argon.
* Great care is needed to avoid contamination of samples by absorption of non-radiogenic $^{40}$ Ar from the atmosphere. The equation may be corrected by subtracting from the $^{40}$ Ar $_{measured}$ value of the amount present in the air where $^{40}$ Ar is 295.5 times more plentiful than $^{36}$ Ar . $^{40}$ Ar $_{decayed}$ = $^{40}$ Ar $_{measured}$ − 295.5 × $^{36}$ Ar $_{measured}$.
* The sample must have remained a closed system since the event was dated. Thus, there should have been no loss or gain of $^{40}$ K or $^{40}$ Ar *, other than by radioactive decay of $^{40}$ K . Departures from this assumption are quite common, particularly in areas of complex geological history, but such departures can provide useful information that is of value in elucidating thermal histories. A deficiency of $^{40}$ Ar in a sample of a known age can indicate a full or partial melt in the thermal history of the area. Reliability in the dating of a geological feature is increased by sampling disparate areas which have been subjected to slightly different thermal histories.
Both flame photometry and mass spectrometry are destructive tests, so particular care is needed to ensure that the aliquots used are truly representative of the sample. Ar–Ar dating is a similar technique that compares isotopic ratios from the same portion of the sample to avoid this problem.

## Applications

Due to the long half-life of $^{40}$ K , the technique is most applicable for dating minerals and rocks over 100,000 years old. For shorter timescales, it is unlikely that enough $^{40}$ Ar will have had time to accumulate to be accurately measurable. 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 Mars Curiosity 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.