K–Ar dating

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:

${\displaystyle t=t_{\frac{1}{2}}{\log }_2\left(\frac{{\text{K}}_f+\frac{{\text{Ar}}_f}{0.109}}{{\text{K}}_f}\right)}$,

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.

To derive this, note that the equation is equivalent to ${\displaystyle \frac{K_f}{K_{f,t=0}}=2^{-t/t_{1/2}}}$, where ${\displaystyle K_{f,t=0}=K_f+\frac{{Ar}_f}{0.109}}$ is the original amount of K-40.

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 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.