Potassium-Argon Dating

Potassium-Argon dating has the advantage that the argon is an inert gas that does not react chemically and would not be expected to be included in the solidification of a rock, so any found inside a rock is very likely the result of radioactive decay of potassium. Since the argon will escape if the rock is melted, the dates obtained are to the last molten time for the rock. Since potassium is a constituent of many common minerals and occurs with a tiny fraction of radioactive potassium-40, it finds wide application in the dating of mineral deposits. The feldspars are the most abundant minerals on the Earth, and potassium is a constituent of orthoclase, one common form of feldspar.

Potassium occurs naturally as three isotopes. The radioactive potassium-40 decays by two modes, by beta decay to 40Ca and by electron capture to 40Ar. There is also a tiny fraction of the decay to 40Ar that occurs by positron emission. The calcium pathway is not often used for dating since there is such an abundance of calcium-40 in minerals, but there are some special cases where it is useful. The decay constant for the decay to 40Ar is 5.81 x 10-11yr-1. 钾以三种同位素形式自然存在。放射性同位素钾-40通过两种方式衰变，即通过β衰变转化为钙-40，或通过电子捕获转化为氩-40。此外，还有极小比例的钾-40通过正电子发射衰变至氩-40。钙的衰变路径在矿物中钙-40含量非常丰富，因此不常用于测年，但在某些特殊情况下仍具有用途。衰变至氩-40的衰变常数为5.81 x 10^-11 yr^-1。

Even though the decay of ⁴⁰K is somewhat complex with the decay to ⁴⁰Ca and three pathways to ⁴⁰Ar, Dalrymple and Lanphere point out that potassium-argon dating was being used to address significant geological problems by the mid 1950's. The energy-level diagram below is based on data accumulated by McDougall and Harrison.

For a radioactive decay which produces a single final product, the decay time can be calculated from the amounts of the parent and daughter product by

where N₀ and N are the initial and final numbers of the parent isotope, λ is the decay constant and T is the half-life. But the decay of potassium-40 has multiple pathways, and detailed information about each of these pathways is necessary if potassiun-argon decay is to be used as a clock. This information is typically expressed in terms of the decay constants.

The measured amount of radiogenic ⁴⁰Ar* in terms of the current measured amount of ⁴⁰K can be expressed as

This can be solved for the time t

When the values for the decay constants in the table above are used, the expression for the radiometric age becomes

Here, it is useful to make use of the series representation of ln(x+1), which may be approximated by x if x<< 1:

Since the population of ⁴⁰Ar* is usually quite small, the approximation of ln(x+1)≈x gives

Geyh & Schleicher comment that the above approximation results in only 1% error at 10⁷ years.

* The asterisk in ⁴⁰Ar* is a reminder that a valid date is obtained only if all the argon-40 is of radiogenic origin in that particular sample. The assumptions made are

1. When the radiometric clock was started, there was a negligible amount of ⁴⁰Ar in the sample.
   当放射性钟启动时，样品中氩-40的含量非常微小。
2. The rock or mineral has been a closed system since the starting time.
   岩石或矿物自起始时间以来就是一个封闭系统。
3. The closure of the system was rapid compared to the age being determined.
   系统的闭合速度比所测定的年龄要快。

⁴⁰Ar/³⁹Ar Geochronology

Dating with ³⁹Ar and ⁴⁰Ar depends upon the fact that the ³⁹K can be bombarded with neutrons in a nuclear reactor to produce an amount of ³⁹Ar which is proportional to the potassium content of the sample. By then comparing the population of ³⁹Ar and ⁴⁰Ar* atoms in a single sample, you can compute a ⁴⁰Ar*/⁴⁰K ratio and thus an age for the sample. The conventional potassium-argon dating process is technically difficult and usually is carried out by analyzing for potassium in one part of the sample and measuring ⁴⁰Ar in another. The Ar-Ar process can be done on the same small piece of a sample, analyzing for both gases in a mass spectrometer.

The bombarding of a geological sample with neutrons produces a population of 39Ar which is proportional to the 39K content of the sample. The proportionality is related to the probability or "cross-section" for the nuclear interaction. Since the ratio of 40K to 39K has been found to be reproducible in a wide range of environments, this allows the calculation of the 40Ar*/40K ratio. 对地质样品进行中子轰击会产生与样品中39K含量成正比的39Ar群体。这种比例性与核相互作用的概率或'截面'有关。由于40K与39K的比值在广泛环境中已被发现具有可重复性，这使得可以计算40Ar*/40K的比值。

One of the complications that must be monitored is that of the production of ³⁹Ar by neutron scattering from the calcium content of the mineral sample. There are also complications with the atomospheric argon content and various argon contamination scenarios.The details are best pursued in a dedicated text like McDougall and Harrison.

The use of a mass spectrometer to evaluate the populations of ⁴⁰Ar* and ³⁹Ar makes possible the calculation of an age with an expression similar to that in the potassium-argon method.

where the proportionality factor J, sometimes called the "fluence", is determined by using the known age t for the calibration sample to work backwards to find the value for J. This allows the ³⁹Ar population to be used as a proxy for the ⁴⁰K content of the sample to make possible the calculation of the age for the sample.

This simplified conceptual treatment does not give a fair picture of the detailed design and execution of age determinations for a wide variety of types of geological samples. But it hopefully makes the point that Ar-Ar dating can take data from small samples based on mass spectrometry. It has contributed to the vast collection of age data for earth minerals, moon samples and meteorites.

Argon-Argon Dating and the Chicxulub Impact

In the early 1990s there was an intense controversy about the association of the Chicxulub Crater of the Mexican Yucatan Peninsula with the extinction of the dinosaurs in the period about 65 million years ago. The Cretaceous-Tertiary boundary in the geological age scale was associated with an iridium-rich layer which suggested that the layer was caused by an impact with an extraterrestrial object. Because that time period, commonly referred to as the K-T boundary, was associated with the extinction of vast numbers of animals in the fossil record, much effort was devoted to dating it with potassium-argon and other methods of geochronology. The time of 65 million years was associated with the K-T boundary from these studies.

Other large impact craters such as the Manson crater in Iowa (dated to 74 My) were examined carefully as candidates for the cause of the extinction, but none were close to the critical time. Chicxulub was not so obvious as a candidate because much of the evidence for it was under the sea. More attention was directed to the Yucatan location after published work by Alan Hildebrand in 1991 demonstrated the chemical similarity of Chicxulub core samples with material found distributed in the K-T boundary layer. Carl Swisher organized a team to produce three independent measurements of the age of intact glass beads from the C-1 core drill site in the Chicxulub impact area. The measurements were done by the argon-argon method.

Even this extraordinary matching with the age of the K-T boundary was insufficient to convince many geologists. The team proceeded to date spherules of glass found in Haiti to provide another bit of evidence. Many pieces of glass ejecta had been found on Haiti, which is over a thousand miles from the impact point currently. But geologists project a much smaller distance between the points at the time of the impact because of measured sea floor expansion. The Haitian spherules were measured to have age to melting of 65.01 +/- 0.08 My, in extraordinary agreement with the measured ages of the core samples.

A third piece of evidence came from age measurements of shocked zircon crystals which were found in the K-T layer as far away as Colorado and Saskatchewan. Zircon has sometimes produced puzzles in radiometric dating because its melting temperature is so high that the crystals sometimes survive in hot melted minerals, giving different melt dates than the other minerals surrounding them. But in this case the nature of zircon was an advantage. The shocked crystals were partially melted, and when measured by the uranium-lead method method gave two ages, 65 My and 545 My. Since the crustal basement in the Yucatan area was known to have an age in the neighborhood of the older age, this gave some confirmation to the Chicxulub crater as the origin of the K-T boundary layer. According to Frankel, this was the step that had most geologists convinced by 1994 that this impact was the source of the iridium-rich K-T boundary deposit and the extinction of the dinosaurs.

Potassium-Calcium Isochrons

The common potassium-argon dating process makes use of the decay of 40K to 40Ar, even though much more of the 40K decays to 40Ca. The reason is that 40Ca is common in minerals, and sorting out what fraction of that calcium came from potassium decay is not practical. But for special cases where the calcium content of the mineral is very low, less than 1/50th of the potassium content, it is sometimes useful to use potassium/calcium isochrons for dating. 钾-氩测年法通常利用钾-40衰变成氩-40的过程，尽管钾-40大部分衰变成钙-40。原因在于钙-40在矿物中很常见，要确定其中有多少来自钾的衰变是不现实的。但在某些特殊情况下，当矿物中的钙含量非常低，低于钾含量的1/50时，使用钾/钙等时线进行测年有时是有用的。

Following the standard approach for decays by multiple pathways, the expression for the age from the radiogenic ⁴⁰Ca can be written

Using non-radiogenic ⁴²Ca for comparison, the equation for an isochron can be developed.

The slope of the isochron line gives a measure of the radiometric age.

Geyh and Schleicher cite this example and compare to a Rb-Sr isochron age of 1008 +/13 My from Barker, et al in 1976.