### How old is that rock? Part 4: ⁴⁰Ar/³⁹Ar dating diagrams

Many ^{⁴⁰}Ar/^{³⁹}Ar dating publications use age spectrum and isotope correlation diagrams to interpret their data and calculate ages. These can be quite confusing if you don’t know how to interpret them so I have sketched some schematic examples to explain how they work.

First up, let’s look at an Age Spectrum:

In the diagram below I have drawn 2 different age spectra. The bottom, green spectrum is what we would expect to see if we had an ideal sample that has no excess-Ar, and the top, blue spectrum is what we might expect if the sample contained excess-Ar in fluid inclusions. Both of these “experiments” involved 7 heating steps where the temperature increased for each step and the gas released from each step was analysed. The data for each of those 7 steps is represented by one of the 7 boxes on the diagram.

The Y-axis (vertical) shows the ^{⁴⁰}Ar/^{³⁹}Ar age calculated for each step (these are schematic diagrams, so I have not put a scale on them). On an age spectrum, the ages are plotted as boxes to show how big the errors are on each step. On the green diagram I have also drawn age data points and error bars at the end of each box to help you visualise it better. Hopefully you can see that, on the green diagram, all the ages are very similar, but on the blue diagram the first three steps give older Ar-ages.

The X-axis (horizontal) shows the cumulative percentage of ^{³⁹}Ar released in the experiment and each step is plotted sequentially, which means, for stepped heating experiments, temperature increases to the right. If you look at the width of the boxes you can tell which heating steps released the most ^{³⁹}Ar (for the green diagram steps 4 and 6 are the widest, so we know these released the most ^{³⁹}Ar). Remember that ^{³⁹}Ar is the same as K, which is the source of ^{⁴⁰}Ar* and should usually be evenly distributed throughout the crystal, so we can use this to understand something about where in the crystal the gas we analysed came from.

If a sample is well behaved and has no excess-Ar, all of the temperature steps should give the same ^{⁴⁰}Ar/^{³⁹}Ar age and this is what we see in the green diagram – all of the ages overlap within error. In this situation we can use all of the data to calculate a more precise age for the sample – that is represented by the dotted black line.

But what if there are fluid inclusions in the sample that add excess-Ar, like we discussed in the last blog? Well, it is quite common for these inclusions to break down and release their gas at relatively low temperatures.

This means that the ages we calculate from the first few temperature steps will be older than the later steps that release gas from the crystal lattice. You can see how this typically manifests in the blue age-spectrum, where the first 3 steps have older ages than the later steps. In this situation we can just discard the data from the steps contaminated with excess-Ar and calculate an age from the steps that give a nice flat, consistent spectrum. We call this part of the spectrum the plateau, because it is flat.

So, hopefully you now know a bit more about what those strange block diagrams mean. Let’s move onto isotope correlation diagrams. These are basically just data points along mixing lines between 2 or more different things. Let’s start with a non-geological example. I am going to make a creamy chocolate coconut dessert. I’m going make this by alternating layers of ganache, which is a mixture of cream and chocolate, and a pre-made coconut pudding that is a mixture of cream and coconut. Both of these ingredients contain cream. If I know what the proportion of chocolate to cream is in the ganache, and the proportion of coconut to cream in the coconut pudding, then I can calculate how much chocolate, coconut and cream there is in different mixtures of ganache and coconut pudding. I’ve done this in the table here.

So, the ganache is 70% chocolate and 30% cream. The coconut pudding is 50% coconut and 50% cream. If the final dessert has half ganache and half coconut pudding, then it will have 35% chocolate, 25% coconut and 40% cream.

If I calculate some ratios of the ingredients, and then plot them up on some graphs, we can see that the composition of the ganache – coconut pudding mixtures all lie on a straight line between the compositions of the 2 main ingredients.

The x-axis is chocolate / cream (amount of chocolate divided by the amount of cream). That means, as you move to the right of the graph, you either have more chocolate, or less cream (or both). That also means that, if you move to the left of the graph, you have less chocolate or more cream (or both). When you reach zero on the x-axis, that basically means you have no more chocolate <sad face>.

We know that there is no chocolate in the coconut cream, so any data points that have 0 on the x-axis, must be 100% coconut cream, and the value they have on the y-axis (we call this the intercept, because it is where a line intercepts the axis) is the ratio of coconut to cream. We can see from the diagram that this ratio is 1; if you look at the table, you can see that the composition of coconut pudding is 50:50 coconut and cream; 50/50 = 1. Similarly, the value of the x-intercept is the composition of the ganache – we can see from the table that ganache is 70% chocolate and 30% cream; 70 / 30 = 2.3, which is what we can see on the graph.

Next, let’s reverse this. Pretend we don’t know the composition of our 2 basic ingredients – all we have are the compositions for 3 different mixtures of ganache and coconut pudding – they make up the 3 data points in the middle of the graph. If we draw a line through them and extrapolate it to both axes, we can calculate the compositions of our original ingredients.

OK, now let’s use this tool to help work out some ages.

Here are a couple of schematic isotope correlation diagrams that I sketched. They are both the same kind of diagram, but illustrating different types of samples.

On the x-axis, we have ^{³⁹}Ar/^{⁴⁰}Ar and on the Y-axis we have ^{³⁶}Ar/^{⁴⁰}Ar. Recall that ^{³⁹}Ar comes from the K in the sample, ^{³⁶}Ar comes from the atmosphere, and ^{⁴⁰}Ar can come from radiogenic ^{⁴⁰}Ar*, from the atmosphere, or from excess-Ar (or, in many cases, all three). Data points are usually plotted as ellipses, to represent the analytical errors on both the x and y axes. A regression line is calculated to fit through the data points and this also calculates the values on the x and y intercepts (shown as stars on this diagram ). We use this diagram for 2 things: 1) to work out the ratio of ^{⁴⁰}Ar*/^{³⁹}Ar in the sample, so we can calculate an age, and 2) to identify the composition of any trapped Ar – is it atmospheric or is there excess-Ar present?

If we move to the left on this graph, that means we either have less ^{³⁹}Ar or more ^{⁴⁰}Ar. We know that ^{⁴⁰}Ar* is always going to be associated with ^{³⁹}Ar because K is needed to produce the ^{⁴⁰}Ar*. So that means, when we hit the y-axis,^{ ³⁹}Ar = 0 and that must mean all ^{⁴⁰}Ar is non-radiogenic – it is either atmospheric or excess-Ar; the value of the y-intercept is the composition of the trapped Ar in the sample. If the trapped Ar is purely atmosphere, the y-intercept should have a value of 0.0033 (this has been measured on samples of air). If the value is smaller, that means there is more ^{⁴⁰}Ar than we would expect in the atmosphere, which means excess-Ar is present.

So, how do we get the age? ^{³⁶}Ar only comes from the trapped, atmospheric-composition gas in the sample. Moving down the graph means we either have less ^{³⁶}Ar or more^{ ⁴⁰}Ar (or both), and when we hit the x-axis, that means there is no ^{³⁶}Ar, and so no trapped-Ar; All the ^{⁴⁰}Ar on the x-intercept is radiogenic. We simply read off the ^{³⁹}Ar/^{⁴⁰}Ar* value and use this to calculate the age of the sample. Using this method it doesn’t matter whether excess-Ar is present or not, because the calculated age is completely independent of the trapped Ar composition.

In the sketch diagrams I have illustrated a few examples. In the left diagram I am comparing how data would look, for 2 samples that are the same age, but one has excess-Ar (red) and the other doesn’t (blue). You can see that the blue data fall on a mixing line between air (y-axis) and the radiogenic component (x-axis), while the red data fall on a mixing line between the same radiogenic component and a trapped Ar composition with more ^{⁴⁰}Ar than in air. If you were to calculate Ar-ages from each of the individual data points, the blue data would all give the correct age, but the red data would give ages that are too old.

In the right hand diagram I am comparing two samples that do not have excess-Ar but are different ages. This might happen if a volcano erupts explosively and the ash picks up some old crystals from a previous eruption. In this case, the data fall onto 2 mixing lines between air and different radiogenic components. The yellow data have an x-intercept that is a lower value than the green data – this means the yellow data contain more ^{⁴⁰}Ar* and so they are older.

The lines that we calculate and extrapolate between the data points represent a range of isotopic compositions that are all associated with the same age, and so these diagrams are often called “isochron diagrams” instead of isotope-correlation diagrams. There are other ways of plotting and analysing the data to make isochrons, and the example here is actually called an “inverse isochron” to distinguish it from another diagram that plots ^{³⁹}Ar/^{³⁶}Ar against ^{⁴⁰}Ar/^{³⁶}Ar. I focussed on inverse isochrons for this blog entry because they seem to be the most popular way of displaying and analysing Ar-isotope data at the moment, but the principles of mixing between 2 different components (trapped and radiogenic Ar) are the same for all isochron diagrams.

So, I hope that if you now decide to pick up a paper about Ar-dating you’ll be able to understand a little bit about some of the strange diagrams in it.

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