# Does the relative abundance of radioactive isotopes reflect their half-lives?

In Radiometric Dating: A Christian Perspective, the author argues (persuasively, I think) in favor of the mainstream interpretation of radiometric dating (as opposed to the claims of young earth creationists). He says the fact that short-lives isotopes are generally not found on Earth is evidence showing that the Earth is billions of years old.

Just about the only radioisotopes found naturally are those with very long half-lives of close to a billion years or longer.... The only isotopes present with shorter half-lives are those that have a source constantly replenishing them...

The Earth is old enough that radioactive isotopes with half-lives less than half a billion years decayed away, but not so old that radioactive isotopes with longer half-lives are gone.

I would guess that some of these shorter-lived elements still exist in minute amounts because a decay curve has a long tail, but is the basic statement correct - that shorter-lived, non-replenished radioactive isotopes are absent (or nearly so) from Earth?

Clarification: I didn't make it clear enough, but I'm specifically asking about the relative abundance of non-replenished radioisotopes - those that are not produced on Earth by natural processes. Eg, excluding carbon-14 (produced in the atmosphere by cosmic rays), excluding thorium-230 (produced by the decay of uranium-234), etc.

• Short lived isotopes are constantly being produced by natural processes. – user20217 Jun 14 '20 at 12:19
• From "Promordial nuclides" on Wikipedia: "primordial isotopes, are nuclides found on Earth that have existed in their current form since before Earth was formed...The shortest-lived primordial isotope, 235U, has a half-life of 704 million years, about one sixth of the age of the Earth and the Solar System." - en.wikipedia.org/wiki/… – Nathan Long Jun 15 '20 at 2:20
• Yes, the question is about all unstable isotopes used for dating, but you only look at the primordial ones. Radiocarbon is constantly replenished, it is not primordial any more in a sense "leftover from earth's early days". One can read about this in all texts about absolute physical dating. – user20217 Jun 15 '20 at 14:42
• @a_donda sorry, yes - I wasn't responding to your comment there, I was just noting that the article on primordial nuclides seems to support the idea that shorter-lived nuclides which aren't being replenished are not found on earth, since the shortest-lived primordial nuclide is 235U. – Nathan Long Jun 15 '20 at 14:54
• You also have to take into account just how much of any particular radioisotope originally existed. They weren't all created in equal amounts, some might have segregated into different parts of the original solar nebula, or in different parts of the primordial Earth. E.g. if most of isotope X segregated into the core, how would we know its abundance? – jamesqf Jun 16 '20 at 5:09

Yes, the general idea you have is correct: isotopes with relatively short half-lives are found in quantities so minute as to be marginally detectable, and then only if they are being produced by something else: either as decay products of some long-lived isotope, or by cosmic-ray bombardment.

An example is gadolinium-150. This has a half-life of about 1.8 million years, and I believe simply does not occur naturally at all. Gadolinium-152, with a half-life of about $$10^{14}$$ years, does occur naturally.

So whatever primordial gadolinium-150 there was has gone (in 4 billion years, you get a decrease of a factor of about $$10^{669}$$: the Earth contains around $$10^{50}$$ atoms, so there is a chance of $$1/10^{619}$$ of finding an atom of it on Earth). Unfortunately in order to know how much there was you would need to understand nucleosynthesis in supernovae in considerable detail (gadolinium is heavier than iron and so originated mostly or entirely in supernovae), and I don't. But I think it's safe to say that they probably made at least some, on the grounds that anything we do in the way of crashing particles into things to make isotopes (which we have done to make $$^{150}\mathrm{Gd}$$) supernovae also do, except much, much more. If we assume the Earth was only a million years old, then about $$68\%$$ of the primordial $$^{150}\mathrm{Gd}$$ would still be here: we'd find it.

Now of course, there are two catches here:

• perhaps the $$^{150}\mathrm{Gd}$$ has all gone somewhere secret where we can't find it;
• this all rests on the idea of supernova nucleosynthesis.

So to answer the first: if this was true it would be utterly extraordinary. In particular we spend a lot of time (really a lot of time) trying to isolate different isotopes of elements: people go to really enormous efforts to distinguish between different isotopes of uranium for instance. This is something you can't do chemically, because the different isotopes have the same chemical properties. Instead you have to do things like spin them in very high-speed centrifuges, or use mass spectrometers. These things just don't exist naturally on Earth, so, since we do find other isotopes of gadolinium (including $${}{150}\mathrm{Gd}$$, which is not stable), how come we don't find this one? Well, we don't find it because it's not there, of course.

The second problem is perhaps more severe: people who believe the Earth is a million years old probably have no truck with supernovae at all, or are happy to just argue that, well, supernovae don't produce some isotopes because they just don't. I don't think there's a useful argument against that.

While a long half-life, hundreds or millions of years or more, favors a radioisotope being relatively common or even occurring in nature at all, it is not the only consideration. Whether and how effectively the isotope can be regenerated also figures in.

Niobium. occurs as only one isotope, $$_{41}^{93}\text{Nb}$$. All other isotopes decay away much too quickly, including $$_{41}^{94}\text{Nb}$$ with a half-life of only about 20,300 years -- not enough to survive in the billions of years since primordial times when the Earth formed.

Now look at carbon. The radioisotope carbon-14, $$_{6}^{14}\text{C}$$, has a half-life of only 5730 years so surely it, too, is all gone, meeting the same end as $$_{41}^{94}\text{Nb}$$. In fact, as many readers know, carbon-14 is still with us (even if it's only trace amounts) and in use as a probe for dating organic matter. The difference, of course, is there is a readily available mechanism for regenerating this isotope through the action of cosmic rays high above Earth. The availability of this mechanism means carbon-14 need not have enough half-life to survive billions of years since the Earth's formation, only enough to survive weeks of circulation time to get to the ground.

We can correlate half-life to abundance only when comparing isotopes that are formed by similar mechanisms.