Lake Suigetsu is a floating chronology that is tied to the present by aligning the 14C/12C ratios from fossils in the sediments to the ratios in the tree ring chronologies.
Ooh, I didn't know that. OK, I'll delete the reference to it, which doesn't fit the context any more apparently.
Thanks. As our local dating-things buff, please let me know about anything I should include. I'd never heard of your speleowhatits until you posted about them the other day, and I'm trying to be inclusive as possible --- I even sent Pressie the draft of a thing about amino acid dating the other day, even though much of the article is devoted to explaining why it usually doesn't work. At present my plans are to do that; radiometric dating (obviously); sclerochronology; and the rhythmite analogs to sclerochronology --- and then that's it. But if there's something I'm missing, please let me know.
Well, can you give me an example of a case where a geologist has used it to answer a question in geology, rather than an archeologist using it to answer an archeological question?
So far as I know, it's only been used to date "sedimentary layers" in the most tangential sense, as though you said that a car is used to generate carbon dioxide. What it's actually used for is to date human societies.
If you can give me an example of a strictly geological sense in which it has been applied, then I may consider including it.
As it is, I'm getting pushback from Pressie for talking about amino acid dating. He wants to know if this belongs in a geology textbook, I argue that it does. I can't think of a single argument for the inclusion of thermoluminescence right now.
In this article we shall discuss the principles behind amino acid dating (also known as racemization dating); we shall discuss how it ought to work, and why it often doesn't.
An object is said to have chirality if it is not possible to make it into a mirror-image of itself by turning it round. For example, a shoe is chiral: you cannot turn a left-foot shoe into a right-foot shoe by turning it round or flipping it over. On the other hand, an object such as a table-knife is not chiral: if you have it lying on the table so that the blunt edge is on the right and the serrated edge is on the left, then you can produce the mirror-image of this situation by rotating the knife around its long axis.
Some molecules are chiral. For example, consider the two molecules in the picture below. They both have exactly the same chemical formula, but one is left-handed, and the other is right-handed. They are said to be enantiomers of one another.
When we make chiral molecules using ordinary chemical processes, we usually produce equal quantities of both enantiomers. Such a mixture is said to be racemic.
However, biological processes produce molecules with a distinct chirality: all the amino acids are "left-handed" (with the exception of glycine, which is not chiral) and all the sugars are "right-handed".
So when an organism dies, its amino acids are left-handed. But after its death, the amino acids can spontaneously change their chirality, flipping from being left-handed to right-handed, and indeed back again.
The result of this process is that eventually the amino acids will collectively become racemic: each particular amino acid will have one chirality or another, but collectively the amino acids won't favor one enantiomer over another. This process is known as racemization.
We should note that although the underlying basis for this process is random, and that in principle the amino acids could by some statistical fluctuation become less racemic and more chiral, the laws of statistics ensure that in practice if we are looking at a large enough sample of amino acids, the chances are astronomically remote that such a thing will occur.
So the process of racemization looks like a good candidate for one of nature's clocks. We know that when an organism dies, its amino acids will all be left-handed; and we know that as time progresses the amino acids will become continually more and more racemic.
So it would seem that if we want to know how long it was since an organism died, all we have to do is see how racemic its amino acids are. And this would work, on one proviso. The process of racemization would have to go at a constant rate, and we'd have to know what it was.
And this is where the whole idea breaks down.
How do we know it works?
The problem with racemization is that it depends on chemical processes that are affected by temperature, humidity, and the nature of the original material undergoing racemization. As a result, it isn't possible to say that racemization happens at such-and-such a rate.
However, it does have some applications. Suppose we examine a particular material (let us say tests of the foraminiferan Neogloboquadrina pachyderma) in a particular environment (let us say in mud in Arctic waters) and by comparing it with a dating method we know we can rely on, we establish that under these conditions racemization does happen at a reasonably steady rate.
In that case we could use the foraminiferans to date sediment in places where we aren't able to use radiometric dating. (For it would be strange and anti-scientific to conjecture that the rate of racemization of the shells in the Arctic mud is constant whenever we can check it, but variable when we can't.)
So dating by racemization can have a few applications, but the conditions under which it can confidently be applied are rather rare. What's more, racemization happens quite fast by geological standards, so, like the other methods of absolute dating we have discussed so far, dating by racemization cannot take us far back in geological time.
All this is not to say that the reader should dismiss out of hand results obtained by amino acid dating; but it can be trusted only when the people applying it have taken care to ensure that they are using it in a context in which it is known to work. In early papers, before geologists and archaeologists had learned the pitfalls associated with amino acid dating, inaccurate dates were presented with much more confidence than they deserved, and such papers should not be relied on.
As a prelude to our articles on radiometric dating, it is desirable that the reader should know something about the mechanisms of radioactive decay. This article provides a fairly non-technical explanation of what it is and how it works.
The reader should recall from high school that the nucleus of an atom consists of protons (positively charged particles) and neutrons (uncharged particles which have almost exactly the same mass as protons). The nucleus is surrounded by a cloud of electrons, negatively charged particles having negligible weight. The number of electrons is equal to the number of protons.
The number of protons in an atom is its atomic number, and the sum of the protons and the neutrons gives its atomic weight.
The chemical properties of an atom are determined by the behavior of its electrons, and so are in effect determined by its atomic number. Hence in chemistry atoms are classified into elements according to their atomic number: an element such as carbon, for example, is defined by having an atomic number of 6. However, two atoms can have the same atomic number and different atomic weights. So, for example, 12C (carbon-12) has six protons and six neutrons, whereas 14C (carbon-14) has six protons and eight neutrons. They are both carbon, and they both behave chemically as though they are carbon, but they have a different atomic weight. So they are said to be the same element, namely carbon, but to be different isotopes of carbon. An isotope is defined by its atomic number and its atomic weight.
In the example in the previous paragraph, we have shown the notation used for isotopes. An atom with six protons and eight neutrons is written as 14C. The fact that it has six protons is revealed by the "C", which is the chemical symbol for carbon; by definition, all carbon atoms have six protons. The fact that it has eight neutrons is revealed by the little "14" written above and to the left of the "C": this is the atomic weight of the isotope, and so since the atomic weight is the number of protons plus the number of neutrons, and since all carbon atoms have six protons, this tells us that this isotope of carbon must have eight neutrons.
Radioactive decay may be defined as any spontaneous event which changes the state of the nucleus, emitting energy from the nucleus in the process. With the exception of gamma decay, which need not concern us here, this will involve changing the number of protons, or neutrons, or both, and so also changing the atomic number, the atomic weight, or both.
There are a number of mechanisms by which decay may take place. For our purposes, the most important are:
* Alpha decay. In this form of decay, the nucleus ejects an alpha particle consisting of two neutrons and two protons, reducing the atomic number by two and the atomic weight by four.
* Beta minus decay. In this form of decay, one of the neutrons in the atom is converted to a proton by the atom emitting an electron. Hence, the atomic number goes up by one, while the atomic weight stays the same.
* Beta plus decay. In this form of decay, a proton is converted into a neutron by the emission of a positron (a particle like an electron only positively charged) the result being that the atomic number goes down by one while the atomic weight stays the same.
* Electron capture. In this form of decay, one of the atom's own electrons combines with one of its protons, converting the proton into a neutron. This reduces the atomic number by one while leaving the atomic weight the same.
When decay takes place, the original atom is called the parent atom, and the new isotope produced by decay is called the daughter atom. Those isotopes which are produced by radioactive decay are said to be radiogenic. Not all isotopes undergo decay: those that do are called unstable isotopes (or radioactive isotopes) and conversely those that don't are called stable. So for example 12C is stable and will go on being 12C forever; by contrast 14C is unstable and has a tendency to decay into 14N (nitrogen-14). As we can see from this example, it is perfectly possible for different isotopes of the same element to differ in their stability.
The reader should note that when a parent atom decays to a daughter atom, the daughter is not necessarily stable; sometimes the daughter will undergo further decay. Such a situation is described as a decay chain.
Statistics of radioactive decay
It is important to understand how and why radioactive decay takes place. According to physicists, radioactive decay occurs at random: an atom of (for example) 22Na (sodium-22) will undergo beta decay and produce an atom of 22Ne (neon-22) just because its number has come up.
The age of the atom has nothing to do with it. Consider, by analogy, a man playing Russian Roulette with a six-shooter. Every single time he plays, he has a one-in-six chance of dying, and this is true no matter how long he's been playing. The same is true of radioactive decay. If we have an atom of 22Na, then no matter how old it is, it has a 50% chance of decaying in the next 2.6 years; and if it survives that period of time, then it has a 50% chance of decaying in the next 2.6 years; and so on.
Consider what this means if we have a large sample of 22Na. Because the sample is large, its behavior will closely approximate our statistical expectations for it: so after 2.6 years 50% of the atoms will have decayed to 22Ne; after a further 2.6 years 50% of the remaining22Na will have decayed (leaving only 25% of the original 22Na); after a further 2.6 years 50% of those22Na atoms will have decayed, leaving 12.5% of the original sample; and so forth.
This figure of 2.6 years is known as the half-life of 22Na: that is, the time over which an atom of 22Na has a 50% chance of decaying; or, equivalently the time over which our statistical expectation is that 50% of the sample will have decayed.
Such situations are mathematically well-understood, and can be represented by the equation for exponential decay:
N(t) = N0 × 2-t/2.6
where N0 is the original quantity of 22Na, 2.6 is the half-life in years, t is the time elapsed as measured in years, and N(t) is the quantity of 22Na left in the sample at time t.
The situation can be represented by the graph below.
For ease of exposition, we have used 22Na consistently as an example, but the same rules apply to all radioactive isotopes, the only difference is that the half-lives of different isotopes will be different: for example 107Pd (palladium-107) has a half-life of 6.5 million years.
People are sometimes startled by such a statement: how, they ask, is it possible to say this when we haven't been watching a sample of 107Pd for 6.5 million years to check that this is how long it takes for half of it to decay?
However, this is not really a problem. After all, by analogy, it is not necessary for a police officer to observe your car for an hour to report that you were traveling at 72 kilometers per hour. It is sufficient to observe you traveling at 20 meters in a single second, and then do the math.
In the same way, if we spend just a single year observing a sample of 107Pd and see that over that time only 0.0000106638% of the sample decays, then we can write:
100 - 0.0000106638 = 100 × 2-1/h
and then solving the equation for h will give us the half-life in years.
Invariance of the half-life
Each unstable isotope, then, has its own characteristic half-life. What is more, for each isotope, this half-life is constant: it is a property of the isotope, and virtually unaffected by external circumstances.
How do we know this? In the first place, it should be true in principle: it can be deduced from the underlying laws of quantum mechanics. In the second place it is confirmed by actual observation: shortly after the discovery of radioactive decay, scientists began trying to change the decay rate by subjecting unstable isotopes to heat, pressure, magnetism, and so forth, with negative results.
Some small variations have been observed in certain isotopes depending on which other chemicals they form molecular bonds with. An example is 7Be (beryllium-7) in which the decay rate varies by about 1% according to the chemical environment it's in. There are theoretical reasons why 7Be ought to be particularly susceptible to such effects: it is a very small atom, and it decays by electron capture. Even so, the variation is fairly small. For the same reasons, even smaller variations have been achieved by putting 7Be under intense pressure.
We should also mention the peculiar isotope 187Re. Because the energy emitted from the nucleus when it decays is so very small, it is possible to change its half-life by ionizing it. If it is completely stripped of all its electrons, its half-life falls from 43 billion years to 33 years! However, as this will happen nowhere on Earth outside a physics laboratory, we can take its half-life to be 43 billion years for all geological purposes.
So to really affect the half-life of an isotope one needs to resort to highly artificial methods --- such as dropping it into the core of a nuclear reactor. In nature, and in particular in rocks, there are sound theoretical and observational reasons to conclude that unstable isotopes will have constant or very nearly constant half-lives and so will undergo decay, and will have undergone decay, in a regular and predictable manner.
The astute reader will probably have figured out where this is going. In the decay of unstable isotopes, we have a set of natural processes each of which goes at an utterly predictable rate.
Naively speaking, the idea behind radiometric dating goes something like this. If we have a rock and we know the original quantity of some radioactive isotope that the rock contained when it was first formed, and if we know its half-life, and if we can accurately measure the quantity of the isotope in the rock at the present time, then we can figure out how old the rock is. Or alternatively if we knew the original quantity and the present quantity of the daughter isotope of the radioactive isotope, then again we could figure out the age of the rock. Such a technique of absolute dating is known as radiometric dating.
Only how can we know the original composition of the rock? The answer to this question varies depending on which radioactive isotope we're talking about; so at this point it is time to stop talking in generalities and instead look at particular methods of radiometric dating in detail. These will be the subject of the next few articles.
In this article we shall examine the basis of the K-Ar dating method, how it works, and what can go wrong with it.
Decay of 40K
40K (potassium-40) is rather a peculiar isotope, in that it can undergo decay in three different ways: by beta minus decay into 40Ca (calcium-40); by electron capture into 40Ar (argon-40); and by beta plus decay into 40Ar again. It is possible to measure the proportion in which 40K decays, and to say that about 89.1% of the time it decays to 40Ca and about 10.9% of the time to 40Ar. 40K has a half-life of 1.248 billion years, which makes it eminently suitable for dating rocks.
Potassium is chemically incorporated into common minerals, notably hornblende, biotite and potassium feldspar, which are component minerals of igneous rocks.
Argon, on the other hand, is an inert gas; it cannot combine chemically with anything. As a result under most circumstances we don't expect to find much argon in igneous rocks just after they've formed. (However, see the section below on the limitations of the method.)
This suggests an obvious method of dating igneous rocks. If we are right in thinking that there was no argon in the rock originally, then all the argon in it now must have been produced by the decay of 40K. So all we'd have to do is measure the amount of 40K and 40Ar in the rock, and since we know the decay rate of 40K, we can calculate how long ago the rock was formed. From the equation describing radioactive decay, we can derive the following equation:
t = h × log2(1 + R/c)
* t is the age of the rock in years; * h is the half-life of 40K in years; * c is the proportion of 40K which decays to 40Ar rather than to 40Ca (about 10.9%); * R is the measured ratio of 40Ar to 40K.
Limitations of K-Ar dating
There are a number of problems with the method. One is that if the rocks are recent, the amount of 40Ar in them will be so small that it is below the ability of our instruments to measure, and a rock formed yesterday will look no different from a rock formed fifty thousand years ago. The severity of this problem decreases as the accuracy of our instruments increases. Still, as a general rule, the proportional error in K-Ar dating will be greatest in the youngest rocks.
A second problem is that for technical reasons, the measurement of argon and the measurement of potassium have to be made on two different samples, because each measurement requires the destruction of the sample. If the mineral composition of the two sample is different, so that the sample for measuring the potassium is richer or poorer in potassium than the sample used for measuring the argon, then this will be a source of error.
Another concern with K-Ar dating is that it relies on there being no 40Ar in the rock when it was originally formed, or added to it between its formation and our application of the K-Ar method. Because argon is inert, it cannot be chemically incorporated in the minerals when they are formed, but it can be physically trapped in the rocks either during or after formation. Such argon is known as excess argon.
If the source of this argon is atmospheric contamination, then we can correct for this. The reasoning is as follows: the atmosphere does not only contain 40Ar, but also 36Ar. There is 295 times as much 40Ar as 39Ar in the atmosphere, and there is no reason why an atom of 40Ar should be preferentially incorporated into rocks rather than an atom of 36Ar, or vice versa. So this means that for every atom of 36Ar we find in our sample, we can discount 295 atoms of 40Ar as being atmospheric argon.
However, this only works if all the excess argon did indeed come from the atmosphere. But consider what happens if the argon came from deep within the Earth, where it was formed by 40K decay, and was then trapped in magma or transported into the rock by hydrothermal fluid. Then the excess argon will not have the same 40Ar/39Ar ratio as is found in the atmosphere, and the formula that corrects for atmospheric carbon will not correct for this.
Finally, we must consider the possibility of argon loss. When a rock undergoes metamorphism, some or all of its argon can be outgassed. If all the argon was lost, this would reset the K-Ar clock to zero, and dating the rock would give us the time of metamorphism; and if we recognized the rock as metamorphic this would actually be quite useful. However, we cannot rely on all the argon being lost, and if it is not then when we apply K-Ar dating this will give us an essentially arbitrary date somewhere between the formation of the rock and the metamorphosis event.
For these reasons K-Ar dating has largely been superseded by Ar-Ar dating, which will be the subject of the next article.
I've edited the article on radioactive decay (a) to mention the freaky behavior of 187Re (b) to define the term "radiogenic", as it will make subsequent articles easier to read and write. Anyone who's following this series will want to go back and have a look, as the term will be used and the behavior of 187Re will be briefly alluded to.
Some of the problems of K-Ar dating can be avoided by the use of the related Ar-Ar dating method. In this article we shall explain how this method works and why it is superior to the K-Ar method. The reader should be thoroughly familiar with the K-Ar method before reading any further.
In the previous article we introduced you to 40K, an unstable isotope of potassium which produces the daughter isotope 40Ar by electron capture or beta plus decay.
The Ar-Ar dating method relies crucially on the existence of two other isotopes. 39K is a stable isotope of potassium, which by definition means that it will not spontaneously undergo decay into another isotope. However, if you put it near the core of a nuclear reactor, so that it is bombarded by neutrons, then this will convert it into 39Ar. This isotope of argon is quite unstable, having a half-life of only 269 years. Consequently, the amount of it found in rocks is negligible --- unless you subject them to an artificial neutron source.
A crucial point to note is that because 39K and 40K are isotopes of the same element, they have the same chemical properties. Therefore when the rock first forms, some of the minerals in it will have more potassium in and some less, but all the minerals will have the same initial ratio of 39K to 40K, because since they have identical chemical properties, there is no way that the 40K could preferentially end up in the hornblende and the 39K in the biotite.
First, you take your rock sample and place it near the core of a nuclear reactor. As a result, some of the 39K (a stable isotope of potassium) is converted to 39Ar as a consequence of the neutron bombardment.
Then you heat the rock sample to release the 39Ar and the 40Ar. The first of these, you will recall, is produced by our artificial neutron bombardment of the stable 39K isotope; the second is produced by the natural decay of the unstable 40K isotope in the rock.
So if all has gone well, and if there were no problems with argon loss or excess argon, then the age of the sample would be given by the following formula:
t = h × log2(1 + J × R)
* t is the age of the rock in years; * h is the half-life of 40K in years; * R is the measured ratio of 40Ar to 39Ar.
But what is J? J is a factor which depends on the nature of the neutron bombardment. J is not calculated on theoretical grounds, but is found experimentally; alongside the sample we're interested in, we irradiate and then heat a sample of known age (a standard).
Measuring the 39Ar and 40Ar emitted from the standard, and knowing the time t that it was formed, we can put these figures into the equation above and solve it for J.
So now we know J, and we have measured the R-value of the sample we're actually interested in dating, so we can use these data to solve the equation for t, giving us the age we're looking for.
You will note that this means that we have to be able to date some rocks accurately using some method other than Ar-Ar, so that we can find a standard to use for the determination of J; fortunately we can do this, and geologists have put a lot of effort into identifying rocks which can be accurately dated and used as standards.
Advantages of the Ar-Ar method
So far, all we seem to have done is taken the K-Ar dating method and made it much more complicated for no apparent reason. However, there are advantages to this more complex method.
In the first place, recall that one of the potential problems with the K-Ar method is that it requires two different samples, one to measure the potassium and the other to measure the argon; if the two samples had different chemical compositions when they first formed then this will introduce an error. However, in Ar-Ar dating the two isotopes of argon are both measured from the same sample, and so at least one potential source of error is eliminated.
The other important advantage of Ar-Ar dating is the extra data gained from step heating: instead of heating the irradiated sample to the highest possible temperature all at once, and so releasing all the argon all at once, we can increase the temperature in steps starting at a low temperature.
What's the point of this? Well, different minerals within the rock will give up their argon at different temperatures, so each step will give us a ratio of 40Ar to 39Ar which we can use in the equation to calculate a date. Now, recall that we said that when the rock was first formed, the 39K and 40K from which these are derived must have appeared in the same ratio in each mineral, because both isotopes of potassium have the same chemical properties.
This means that if the rock cooled rapidly enough that all the minerals in it have the same date, and if there has been no argon loss, and if there is no excess argon added to the system, then the dates we calculate at each step of the heating will be the same date.
If we don't get the same date at each step, then we may be able to work out what's going on.
For example, if the date increases at each step, then we are quite possibly looking at a slow-cooling igneous rock in which different minerals crystalized out of the magma at different times, a possibility we can investigate further.
Or if we consistently get one date for the steps below (for example) 400°C, and consistently get another date in the steps above 400°C, then it seems as though argon loss occurred as a result of a metamorphosis at a temperature of about 400°C, with the younger date representing the date of the metamorphosis, and the older date representing the formation of the rock; and we can investigate this clue further by looking for other evidence of the metamorphic event.
And if the dates we get are all over the place, then we are probably looking at excess argon. Now the bad news is that there is no way we can somehow manipulate this data to give us a correct date for the sample. But the good news is that we do know that there's a problem; whereas if we'd analyzed the same rock using the K-Ar method, then it would have supplied us with a date and there'd have been no sign in the K-Ar data of anything wrong with it.
For these reasons Ar-Ar dating has largely superseded K-Ar dating, although the simpler method is still employed in some cases where it is known to be unproblematic or where Ar-Ar is unsuitable for some technical reason.
Hello. very interesting subject. But for me probably will be rather difficult. As I in physics am not so strong, but nevertheless there is a wish to understand àðõè. I will explain why. My purpose of visit of this forum was to find the expert who has skills in definition of radio-carbon dating. And this interest is chosen not casually. At some stage I noticed that the radio-carbon method shows not those data. And if to understand in essence that can be found places where cretaceous layers are shifted by the coal. Though by natural definition it is possible to call with confidence 100% cretaceous prime, and then have to be coal. But radio-carbon measurements for any reason show on a turn. Especially as chalk in itself has to stand as the coal which has faded from time. Other aspect. http://fotki.yandex.ru/users/ibr2020-12/view/674138?page=0 To us this photo it is noticed as limestones any breed blocks still. But in my opinion it is sandstone hardened. I approximately calculated as the hardened breeds of sandstones were formed. And for me big surprise began to see and understand that before sandstones were formed there could be still any life on a planet. Moreover and in such quantity. I certainly don't say goodbye but that more while I won't be able to tell about it. After all I couldn't finish the author at all the picture to show on this district.
In this article we 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.
There are three isotopes used in Rb-Sr dating. 87Rb (rubidium-87) is an unstable isotope with a half-life of about 49 billion years. It produces the stable daughter isotope 87Sr (strontium-87) by beta minus decay. The third isotope we need to consider is 86Sr, which is stable and is not radiogenic, meaning that in any closed system the quantity of 86Sr 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
This means that if we wanted to date a rock, and if there was no 87Sr present initially, and if we could measure the 87Sr/87Rb 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 87Rb has only one decay mode, we could drop the term c.
But there is no reason at all to suppose that there was no 87Sr 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 87Sr in particular.
The isochron diagram
However, 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 87Sr and 86Sr. Because they are chemically indistinguishable, they will appear in the same ratio in every mineral: some minerals will have more strontium, some will have less, but all must necessarily have the same 87Sr/86Sr ratio.
The initial state of the rock may therefore be schematically represented by the graph below, which shows the initial states of four minerals imaginatively named A, B, C, and D. They have different chemical compositions, and therefore have different 87Rb/86Sr ratios, but they all have exactly the same 87Sr/86Sr ratio, for reasons explained in the previous paragraph. Hence the dotted line connecting the four minerals and extended beyond them must be straight, and the point at which it intersects the vertical axis is the initial value of the 87Sr/86Sr ratio.
Now consider what will happen to this system over time, as the 87Rb decays to 87Sr. For each mineral, this will decrease the 87Rb/86Sr ratio and increase the 87Sr/86Sr ratio.
This will have the biggest impact on the ratios of minerals such as D which have high initial 87Rb/86Sr ratios, and the smallest impact on the ratios of minerals such as A, which have low initial 87Rb/86Sr ratios. (If you have difficulty seeing this, try considering the extremal case of a mineral which contains no rubidium at all. Its 87Rb/86Sr ratio will be initially zero and will stay that way.)
The result of the decay process can be represented on our graph as shown below.
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 87Sr/86Sr.
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 87Rb/86Sr ratios.
For each sample we then measure its 87Rb/86Sr ratio and its 87Sr/86Sr ratio.
We then use the isochron diagram to find the initial value of the 87Sr/86Sr ratio. This one additional piece of information about the initial state of the rock allows us to calculate its age.
As 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.
There 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 87Sr/86Sr 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.