# Historical Geology/Other isochron methods

In this article I shall point out some other dating methods which work the same way as the Rb-Sr method. The reader who has not read the article on the Rb-Sr method will find this present article almost completely incomprehensible, and should go back and read it.

## The isochron method generalized

I have introduced the isochron method in the context of rubidium and strontium. But is there anything particularly special about those two elements? Not really. For the isochron method to work, what we need are three isotopes with the following properties.
(1) An unstable isotope. This should have a fairly long half-life if it is to be of any use in dating rocks, but not too long, or it will hardly undergo any decay at all. A figure expressible in billions of years is ideal. In the Rb-Sr method, we used $^{87}$Rb.
(2) A stable daughter isotope of isotope (1). In the Rb-Sr method, we used $^{87}$Sr.
(3) An isotope which is the same element as isotope (2) and which is neither unstable nor radiogenic, so that in a closed system it remains constant in quantity. In the Rb-Sr method, we used $^{86}$Sr.
Given a set of three such isotopes, we can apply exactly the same reasoning as we did for $^{87}$Rb, $^{87}$Sr and $^{86}$Sr, and it will be equally valid.

## The isotopes

The table below shows some sets of three isotopes which can be treated like rubidium and strontium for the purposes of dating; the table also shows the half-life of the parent and its decay mode. The numbers (1) (2) and (3) are as in the section above.
| Method | (1) | half-life | decay mode | (2) | (3) |
| Rb-Sr | $^{87}$Rb | 48×10$^9$ yr | beta minus | $^{87}$Sr | $^{86}$Sr |
| Sm-Nd | $^{147}$Sm | 106×10$^9$ yr | alpha | $^{143}$Nd | $^{144}$Nd |
| Lu-Hf | $^{176}$Lu | 36×10$^9$ yr | beta minus | $^{176}$Hf | $^{177}$Hf |
| Re-Os | $^{187}$Re | 43×10$^9$ yr | beta minus | $^{187}$Os | $^{186}$Os |
| La-Ba | $^{138}$La | 105×10$^9$ yr | electron capture | $^{138}$Ba | $^{137}$Ba |
| La-Ce | $^{138}$La | 105×10$^9$ yr | beta minus | $^{138}$Ce | $^{142}$Ce |
| K-Ca | $^{40}$K | 1.2×10$^9$ yr | beta minus | $^{40}$Ca | $^{42}$Ca |
| U-Pb | $^{238}$U | 4.5×10$^9$ yr | decay chain | $^{206}$Pb | $^{204}$Pb |
| U-Pb | $^{235}$U | 0.7×10$^9$ yr | decay chain | $^{207}$Pb | $^{204}$Pb |

## Notes

* I said that isotope (3) should be stable. $^{186}$Os is not in fact stable, but as it has a half-life of two quadrillion years, it might as well be.
* In the Lu-Hf method, again, $^{144}$Nd is unstable, but has a half-life even longer than that of $^{186}$Os.
* Similarly in the La-Ce method, neither isotope of cerium used is strictly speaking stable, but their half-lives are so enormously long that for all practical purposes they may be treated as stable.
* In using the K-Ca method, we have to make a slight mathematical adjustment to take into account the fact, mentioned in our article on the K-Ar method, that $^{40}$K decays to $^{40}$Ar as well as to $^{40}$Ca.
* Similarly, $^{138}$La can decay two ways, to $^{138}$Ce or $^{138}$Ba. As you can see from the table, both are susceptible to the isochron method.
* We have noted the peculiarities of the half-life of $^{187}$Re in the article on radioactive decay. As we only have to consider how it behaves in rocks, and not in elaborate equipment in physics laboratories, we may take its half-life to be 43 billion years as given in the table.
* There are two entries for U-Pb because there are two parent isotopes we can use, $^{238}$U and $^{235}$U. Each decays to a (different) final stable element of lead by a complex decay chain.
* In practice, the U-Pb decay chain is usually exploited by methods other than isochron dating, for reasons that will be explained in the next article.
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