K-Ar and 40Ar/39Ar Thermochronology

In this post I'd like to outline some of the more important aspects of Ar/Ar thermochronology. I am going to start with a brief outline of the K-Ar technique, but spend most of the post on the more popular Ar/Ar variant. This post is a long time coming; as far as published reliable radiometric dates, I'd bet that Ar/Ar thermochronology is the most used chronometer in earth science (although I did recently hear Mark Harrison claim he had "age characterized" 100,000 zircons for U-Pb geochronology, so I could be way way off). There are a few primary reasons for the popularity of the method, some I'll discuss later. But first the basics.

Potassium (K) is the 7th most abundant element in the earth's crust, and is a major component of many common rock forming minerals. Very common minerals like biotite and muscovite mica and potassium feldspar contain weight percent levels of K. Naturally occurring K is made of 3 isotopes, ³⁹K (93.258%), ⁴⁰K (0.012%) and ⁴¹K (6.730%). ³⁹K and ⁴¹K are stable, ⁴⁰K is radioactive. Natural Ar is fairly abundant in the earth's atmosphere (~1% of the air we breathe) and is composed of three isotops, ⁴⁰Ar (99.600%), ³⁸Ar (0.063%), and ³⁶Ar (0.337%). ⁴⁰K decay is an interesting process, unlike the U and Th systems I've blogged about earlier which undergo a chain decay, ⁴⁰K has a branched or dual decay scheme. That means that 89.52% of the time, ⁴⁰K decays by beta emission to ⁴⁰Ca. 10.48% of time time, ⁴⁰K decays by extranuclear electron capture to ⁴⁰Ar. The ⁴⁰K to ⁴⁰Ar process is the one we will concern ourselves with in this post. ⁴⁰K - ⁴⁰Ca is rarely used as a geochronometer because Ca is so abundant in nature. It is hard to differentiate in minerals what Ca formed in the mineral from decay of K and what Ca was included in the original mineral structure. Argon, a noble gas, is rarely present in significant amounts within crystals, so we do not have to worry as much about the original ⁴⁰Ar component (although you will see later that the ⁴⁰Ar/³⁹Ar technique lets us deal even with "inherited" ⁴⁰Ar). ⁴⁰K has a half-life of 1.25 billion years (Ga), which makes it perfect for addressing a wide range of geologic problems. The graph below shows the evolution of ⁴⁰K, ⁴⁰Ca, and ⁴⁰Ar abundance versus time.

Like in all graphs that show parents and daughters, the curves intersect at the half-life. Focusing on the ⁴⁰Ar then, we can see that the daughter to parent ratio, or ⁴⁰Ar/⁴⁰K changes over time, like this
We can measure the ⁴⁰Ar and ⁴⁰K in a sample, and from that ratio, calculate an age. The age equation (shown below)

then relates the amount of parent (⁴⁰K), daughter (⁴⁰Ar) and time. Like with all geochronologic systems, actually calculating the age is fairly straightforward. The actual K-Ar age equation is a little more involved because of the branched decay I mentioned earlier, so the final form looks like thiswhere
is the ⁴⁰K decay constant (total)

and

are the decay constants for the two decay paths that produce ⁴⁰Ar, and

is the decay constant for ⁴⁰Ca production from ⁴⁰K. The term in the age equation just before the daughter to parent ratio therefore describes the fraction of ⁴⁰K decays that result in the formation of a ⁴⁰Ar atom.

The K-Ar geochronometer came into widespread use in the 1940's, once the basics of the decay scheme were worked out. There are two main issues with the technique, however, that are kind of a pain, and are the reasons that more often that not the ⁴⁰Ar/³⁹Ar technique is much more popular today. First, you cannot measure K and Ar on the same machine. Typically, samples would first be heated (often using an induction furnace) to very high temperatures (~1400°C). At these temperatures the Ar trapped in the sample is liberated and can then be measured. The ⁴⁰Ar from the sample would be spiked with ³⁸Ar, and then measured on a mass spectrometer. The degassed sample would then be retrieved from the furnace, and analyzed separately for ⁴⁰K. With the amount of ⁴⁰Ar and ⁴⁰K, you culd then calculated an age.
The second problem with the K/Ar technique is determining where the Ar came from. As I mentioned earlier, Ar is fairly abundant in the atmosphere, making up about 1% of what we breathe. It is also fairly common in groundwater and geo-fluids (fluids associated with igneous intrusions and metamorphism). Some of this Ar could become trapped in the crystals, therefore rendering the calculated age kind of meaningless. We know the atmospheric ⁴⁰Ar/³⁶Ar ratio very well, so by measuring the ³⁶Ar we can correct for this "trapped Ar", but we have to assume that the composition is atmospheric. In some cases this is a perfectly reasonable assumption, but with the advent of the ⁴⁰Ar/³⁹Ar technique, we are actually able to evaluate that assumption.

The basis of the ⁴⁰Ar/³⁹Ar technique is the fact that when K is bombarded with neutrons (in a nuclear reactor), some important reactions take place. Namely, ³⁹K is converted into ³⁹Ar. We know the natural ³⁹K/⁴⁰K ratio, so if we can measure the efficiency of this transformation in the reactor, we can use ³⁹Ar as a proxy for ⁴⁰K. This would allow us to measure an age by simply measuring the ⁴⁰Ar/³⁹Ar ratio of a sample. What is even better is that ³⁹Ar does not exist naturally, so we do not have to worry about any inheritance or atmospheric contamination.

³⁹Ar production can be described by this equation

where
is the neutron flux at energy Eis the neutron capture cross section at energy E for the reaction ³⁹K(n,p)³⁹Ar, and
is the duration of the irradiation. In practice, instead of measuring these values directly we figure them out using well defined age standards, producing this equation


where ⁴⁰Ar^* is the ⁴⁰Ar derived from radioactive decay, ³⁹Ar _K is the ³⁹Ar produced in the reactor from ³⁹K, and J is a term that incorporates all of the irradiation variables but can be calculated from standards of known age (determined with other methods), whereThe advent and refinement of the ⁴⁰Ar/³⁹Ar method solved some of the larger problems with the K/Ar technique. First, sample analysis became much more straightforward, measuring different isotopes of the same element is much easier that measuring different elements. Measuring a ratio of two isotopes is also easier that measuring abundances of two elements. The ⁴⁰Ar/³⁹Ar frees us from uncertainty in the spikes used during analysis.

Second, it allows the geochronologist to evaluate the assumption that the "extraneous Ar" in the sample is atmospheric in composition. This is done by progressively degassing the sample at higher and higher temperatures (called step-heating experiments). The first temperature steps are at low temperatures, and tend only to liberate very loosely bound Ar from the crystal, perhaps Ar adsorbed to the surface of the crystal, or held near the surface. With increasing temperature we access and analyze gas that is more tightly bound up in the crystal structure. In the end we then have a series of steps, each with separate isotopic values. If the gas came only from the atmosphere, and was not related to any K, then ³⁹Ar/⁴⁰Ar would equal 0. If the gas was entirely radiogenic, and had no atmospheric input, then ³⁶Ar/⁴⁰Ar would equal 0. In truth, we typically have some mixture of inherited (or atmospheric) Ar and radiogenic Ar. If we plot up these values on an inverse isochron diagram we can determine what the composition of the extraneous Ar reservoir is, and therefore correct our values to determine the real age.

as shown above. If your sample is a mixture of inherited Ar from a single reservoir and radiogenic Ar of a single age, then your individual temperature steps will define a mixing line between those two compositions. We can use the inherited Ar composition to correct our ⁴⁰Ar values, and get the real age. It is also common to see the data displayed as an age spectra, shown below. Here, each box represents an indivual temperature step from a single sample. The y-axis is the age (here in Ma), corrected for extraneous Ar. The height of the box is proportional to the uncertainty of the measurements. The x-axis is the cumulative percent of Ar released. The steps are in order, so the first low temperature steps are on the left, and progressively higher temperatures towards the right. The width of the box is proportional to the amount of radiogenic Ar (⁴⁰Ar^*) released during that step. The large flat area in the middle of the graph is called a plateau and indicates that ages from different temperature steps agree. There is no real physical basis for the definitions of a plateau, and variations abound. Realistically, if our measurement precision was high enough, we would never see plateaus, because natural volume diffusion should round out the profile and make the early temperature steps slightly younger. But, age spectra are very common in literature, so it is important to know how to read them.

The inverse isochron diagram is much more powerful, and I am suspect of Ar papers that do not include inverse isochron diagrams. If the sample is very radiogenic, and there is little ³⁶Ar, then the corrections aren't significant, but still that information should be included. Plateaus and age spectra diagrams are interesting on their own, almost from a Psychology of Science perspective. Even though they are not the most powerful Ar/Ar tool, they are visually pleasing, and telling someone you got a nice plateau usually convinces them of your data quality faster than discussing inverse isochron diagrams. The definitions of a proper plateau usually involve some minimum number of consecutive steps representing some fraction of the total gas that are within error of each other for age and have no statistically discernible slope. You can imagine that if your measurements were kind of crappy, and the age uncertainty on each step high, then getting consecutive steps to agree isn't very difficult. As labs drive for higher and higher precision, and individual step ages have uncertainties less than 1% and official plateau get tougher and tougher to create, even when the age and sample are perfectly acceptable.

⁴⁰Ar/³⁹Ar thermochronology is a very common and useful tool for geologists, and is particulary interesting because it allows the direct examination of a number of the assumptions involved with radiogenic isotope geochronology. Later posts will examine the power of the technique in more detail.

Note - the inverse isochron diagram I used is one I made up just as a teaching tool, the age spectra is real data I collected a few years ago.