# Historical Geology/Rb-Sr dating

In this article I shall introduce the Rb-Sr dating method, and explain how it works; in the process the reader should learn to appreciate the general reasoning behind the isochron method.

## The isotopes

editThere are three isotopes used in Rb-Sr dating. ^{87}Rb (rubidium-87) is an unstable isotope with a half-life of about 49 billion years. It produces the stable daughter isotope ^{87}Sr (strontium-87) by beta minus decay. The third isotope we need to consider is ^{86}Sr, which is stable and is not radiogenic, meaning that in any closed system the quantity of ^{86}Sr will remain the same.

As rubidium easily substitutes chemically for potassium, it can be found doing so in small quantities in potassium-containing minerals such as biotite, potassium feldspar, and hornblende. (The quantity will be small because there is much more potassium than rubidium in the Universe.)

## Strontium in rocks

editThis means that if we wanted to date a rock, and if there was no ^{87}Sr present initially, and if we could measure the ^{87}Sr/^{87}Rb ratio present today, then it would be easy to derive a formula giving the age of the rock: it would be like that we used for K-Ar, except that as ^{87}Rb has only one decay mode, we could drop the term *c*.

But there is no reason at all to suppose that there was no ^{87}Sr present initially. When we produced the formula for K-Ar dating, it was reasonable enough to think that there was little to no argon present in the original state of the rock, because argon is an inert gas, does not take part in chemical processes, and so in particular does not take part in mineral formation.

Strontium, on the other hand, does take part in chemical reactions, and can substitute chemically for such elements as calcium, which is commonly found in igneous rocks. So we have every reason to think that rocks when they form *do* incorporate strontium, and ^{87}Sr in particular.

## The isochron diagram

editHowever, there is still a way to extract a date from the rock. In the reasoning that follows, the reader may recognize a sort of family resemblance to the reasoning behind step heating in the Ar-Ar method, although the two are not exactly alike.

The reasoning, then, goes like this. When an igneous rock is first formed, its minerals will contain varying concentrations of rubidium and strontium, with some minerals being high in rubidium and low in strontium, others being high in strontium and low in rubidium. We can expect these differences to be quite pronounced, because rubidium and strontium have different chemical affinities: as we have noted, rubidium substitutes for potassium, and strontium for calcium.

Now consider the distribution of the two strontium isotopes ^{87}Sr and ^{86}Sr. Because they are *chemically* indistinguishable, they will appear in the same ratio in every mineral at the time when it is formed: some minerals will have more strontium, some will have less, but all must necessarily have the same ^{87}Sr/^{86}Sr ratio.

The initial state of the rock may therefore be schematically represented by the graph to the right, which shows the initial states of four minerals imaginatively named A, B, C, and D. They have different chemical compositions, and therefore have different ^{87}Rb/^{86}Sr ratios, but they all have exactly the same ^{87}Sr/^{86}Sr ratio, for reasons explained in the previous paragraph. Hence the dotted line connecting the four minerals and extended beyond them must be straight and horizontal, and the point at which it intersects the vertical axis is the initial value of the ^{87}Sr/^{86}Sr ratio.

Now consider what will happen to this system over time, as the ^{87}Rb decays to ^{87}Sr. For each mineral, this will decrease the ^{87}Rb/^{86}Sr ratio and increase the ^{87}Sr/^{86}Sr ratio.

This will have the biggest impact on the ratios of minerals such as D which have high initial ^{87}Rb/^{86}Sr ratios, and the smallest impact on the ratios of minerals such as A, which have low initial ^{87}Rb/^{86}Sr ratios. (If you have difficulty seeing this, try considering the extremal case of a mineral which contains no rubidium at all. Its ^{87}Rb/^{86}Sr ratio will be initially zero and will stay that way.)

The effect of the decay process on the isotope ratios can again be plotted on a graph, as shown to the right.

We shall omit the math, but it happens to work out so that after any given period of time, the minerals will *still* lie on a straight line on the graph, as the diagram shows, and, crucially, the point at which this line intersects the vertical axis is *still* the initial value of ^{87}Sr/^{86}Sr.

So now we can find a date for the rock. What we have to do is take samples from the rock consisting of different minerals, or at least of different mineral composition, so that our samples will all have different ^{87}Rb/^{86}Sr ratios.

For each sample we then measure its ^{87}Rb/^{86}Sr ratio and its ^{87}Sr/^{86}Sr ratio.

We then use the isochron diagram to find the initial value of the ^{87}Sr/^{86}Sr ratio. This one additional piece of information about the *initial* state of the rock allows us to calculate its age.

## Confounding factors

editAs with the other methods we've discussed so far, the Rb-Sr method will only work if nothing but the passage of time has affected the distribution of the key isotopes within the rock. And of course this is not necessarily the case. Hydrothermal or metasomatic events may have added or subtracted rubidium and strontium to or from the rocks since their formation; or a metamorphic event may have redistributed the rubidium or strontium among its constituent minerals, which would also interfere with the method.

However, barring an extraordinary coincidence, the result of such events will be that when we draw the isochron diagram, the minerals will no longer lie on a straight line. A small deviation from a straight line tells us that there is some uncertainty about the date, and this degree of uncertainty can be calculated; and if we get something which is nothing like a straight line, then the method simply doesn't supply us with a date. So just as step heating in Ar-Ar dating protects us from error, so too does the isochron method in Rb-Sr dating: it may not always lead us to the right date, but it is a good safeguard against our accepting one that is wrong.

## Mixing

editThere is, however, one potential source of error which will not show up on the isochron diagram, since it is expected to produce a straight line. Suppose that the original source of the rock was two different magmas (call them X and Y) imperfectly mixed together so that some parts of the rock will be all X, some all Y, some part X and part Y in varying proportions. Then these different parts of the rock, when analyzed for their isotopic composition, will plot in a straight line on the isochron diagram; and the slope of this line, and the point at which it intercepts the vertical axis, will have nothing to do with the age of the rock, and everything to do with the compositions of X and Y.

About half the time this will produce a straight line with negative slope: that is, it will slope down from left to right instead of up. Such a line must necessarily be produced by mixing, since a real isochron will always have positive slope: the rarity of such an occurrence tells us that mixing of this type must itself be rare.

We can also test for mixing using what is known as a **mixing plot**: if we draw up a graph of the composition of our sample in which the ^{87}Sr/^{86}Sr ratio is the vertical axis (as in the isochron diagram) but the horizontal axis represents 1/Sr (the reciprocal of the quantity of both isotopes of strontium taken together) then if the rock was produced by this mixing process, then the points on this graph will lie along a straight line.

It can happen that if we produce a mixing plot for a perfectly good isochron, it will by some statistical fluke produce a straight line on the mixing plot; we would then be throwing out a perfectly good date. However, this is worth it: it would, as I say, require a fluke for this to happen, so if we reject dates based on the mixing plot, then we will be throwing out a hundred bad dates for every good one.