1 {photo of a chain rusting from one end}
1 Caption: Chains of Decay

The Alchemist, by Joseph Wright. Public domain image from Wikimedia Commons.

For centuries, alchemists sought the means to turn other substances into gold. Philosophically, this seemed like an achievable task. They were working more or less from the concept of matter as passed down from the Ancient Greeks. In this view, matter was composed of the four elements: earth, air, water, and fire. Each element was a fundamental substance that could not be broken down or transmuted into a different element. Other substances were mixtures of the elements, in varying proportions. Gold would have been primarily earth, because it was heavy, with some specific amounts of the other elements mixed in.^[1] Other metals would be similar, differentiated by the precise proportions of the elements or how they were combined.

So conceptually the idea of turning, say, lead, which has a similar heaviness, into gold didn't seem that far-fetched. You might only need to extract a touch of water and add a smidge of fire, for example. The exact recipe was unknown though, so the alchemists had to approach the problem by trial and error. They found they could in fact turn many substances into other substances by the actions of heat, or adding and mixing various things. But try as they might, they never managed to turn anything into gold.

Slowly alchemy transitioned into chemistry, following the work of Antoine Lavoisier (who we've met before) and those who succeeded him. With this better understanding of the true principles of matter and how chemical reactions occurred, it soon became clear that there were not four elements, but dozens of them.^[2] And most unfortunately for the alchemists, gold was in fact one of the elements. This meant it could be combined with other elements, or extracted from ores in which gold was already present, but it could not be created from other elements.

As the years rolled by, evidence mounted that matter was made of atoms, and it was the types of atoms that defined the elements. Concepted as indivisible, atoms were the nail in the coffin of the alchemist's dream of creating gold from other substances.

Alpha decay. Public domain image from Wikimedia Commons.

But then something even more wonderful happened. Electricity provided the necessary hints. The work of J.J. Thomson, Marie Curie, Ernest Rutherford, James Chadwick, and their contemporaries established that atoms were in fact composed of smaller particles: electrons, protons, and neutrons. Furthermore, radioactive decay showed that by rearranging these particles atoms of one element could turn into atoms of a different element. This was in fact what radioactive decay was. The early researchers named three different types of radioactive decay, based on their properties. They chose the first three letters of the Greek alphabet to refer to them: alpha, beta, and gamma.

Alpha decay is when an atom emits an energetic alpha particle, an alpha particle being composed of two protons and two neutrons bound together. This is actually the same thing as the nucleus of a helium atom. But what's left behind is an atom with two fewer protons and two fewer neutrons than it started with. And an element is defined by the number of protons in the nucleus of its atoms, also called the atomic number.

So, for example, let's start with an atom of uranium, which has 92 protons. In particular, let's use the most common isotope of uranium, with 146 neutrons, known as uranium-238 (where 238 is the total number of protons plus neutrons, which is called the mass number). This atom is unstable, and will eventually decay by radioactivity, although it will probably take a very long time to do so. When it decays, it emits an alpha particle. This changes the atom's atomic number by -2, and the atom now has only 90 protons (and 144 neutrons, since it's lost two of those as well). This makes it an atom of the element known as thorium, specifically the isotope thorium-234.

Beta decay. Public domain image from Wikimedia Commons.

Thorium-234 is also unstable, and a lot more so. It decays within a matter of days. But it does so by a different method: beta decay. The thorium atom emits a beta particle, which is actually just an electron. But this electron is not one of the electrons that surround the nucleus. It is actually ejected from within the nucleus. But there are no electrons in the nucleus, so where did it come from? The answer is that the (electrically neutral) neutrons themselves are inherently unstable particles, and can decay into a (positively charged) proton plus a (negatively charged) electron, plus a third particle which I haven't talked about before, called a neutrino. Neutrinos are electrically neutral, extremely low in mass, and actually not that important for what I'm talking about today, but I wanted to mention them for completeness.

The important thing is that when a neutron inside a nucleus decays, causing beta decay, the produced electron (and the neutrino) gets ejected with high energy, while the new proton sticks in the nucleus. The result is that the atom loses a neutron and gains a proton, which changes its atomic number by +1. The element with atomic number 91 is protactinium. So thorium-234 beta decays and turns into protactinium-234. (The mass number remains the same.)

Protactinium-234 is also unstable and can decay by beta decay. This increases the atomic number from 91 to 92, which is uranium again! But fear not, we have not entered a vicious cycle, because the total number of protons and neutrons is now 234. We have an atom of uranium-234, not the uranium-238 we started with. Overall, we've lost four neutrons along the way, though they were ejected as two protons and two neutrons (the alpha decay), plus two electrons (the two beta decays).

Decaying chain.

Uranium-234 decays by alpha decay, forming thorium-230, which decays by alpha to form radium-226. Another alpha decay takes us to radon-222, then another to polonium-218, and yet another which produces lead-214. Even this is not the end of the road, as lead-214 decays by beta decay into bismuth-214, and from there it pinballs around a bunch of other radioactive isotopes before we finally reduce the mass number enough and end up at lead-206. This, at long last, is the end of the line, as lead-206 is completely stable and will not undergo radioactive decay.

The chain of isotopes from uranium-238 to lead-206 is known as a decay chain. You may have noticed that the mass number only ever decreases by 4 (alpha decay) or stays the same (beta decay). So if we instead start with the different radioactive isotope uranium-235, we will never land on any of the same isotopes. We go down a completely different chain, which ends in lead-207. And there are two other radioactive decay chains, corresponding to the other two remainders left over when you divide the mass number by 4. Every heavy radioactive element isotope belongs to one of these four chains.

A complication of both alpha and beta decay is that often the newly produced nucleus is left in an unstable energy state itself. It's a bit like taking a block out of a game of Jenga. With all the pre-existing protons and neutrons in an atom, it's nice and solid. Toss out an alpha or beta particle, and what's left can be a bit rickety. The leftover protons and neutrons might need to rearrange themselves a bit to get comfortable and stable again. Just as in Jenga, the collapse is triggered because there's an excess of potential energy that can't be supported by the structure. A Jenga tower gets rid of the excess energy by collapsing loudly. A nucleus gets rid of the excess energy as it relaxes by emitting it in the form of electromagnetic radiation - the nucleus produces a photon of light. The energy involved is quite high though, and the resulting photon has a lot more energy than visible light. It belongs to the high-energy part of the electromagnetic spectrum now known as gamma rays, because it corresponds to the third type of radiation named by early radioactivity researchers.

Typically these gamma rays are emitted within a second or so of the alpha or beta decay for any given atom. The atom might then wait for some time before undergoing another alpha or beta decay and continuing down its decay chain.

Pitchblende, a uranium ore. Creative Commons Attribution-NonCommercial image by Rui Costa.

The amounts of time that all of these radioactive decays take is random for any given atom. But there are isotopes that tend to decay faster, such as thorium-234 which typically decays within a few days, and isotopes that decay more slowly, such as uranium-238 which typically lasts billions of years before decaying. These decay times can be characterised by what's known as the half-life. Each isotope has a particular half-life, which is an amount of time. For example, the half-life of thorium-234 is 24.10 days. The half-life of uranium-238 is 4.468 billion years (American billions, 4.468×10⁹ years).

What does a half-life mean in practice? It's usually stated that the half-life is the amount of time it takes for exactly half of the atoms in a sample of a radioactive isotope to decay. It can also be stated that for any given individual atom, its half-life is the amount of time you'd need to wait in order for the atom to have a 50% probability of decaying.

Say we are given a single atom of thorium-234 as a New Year's gift at midnight when the fireworks go off (by some sort of mad scientist, presumably). After 24.1 days (at about 2:24 am on 24 January), it will have had a 50% chance of decaying - it might have decayed, or it might not, with equal probability. If it has decayed (a 50% chance), we're done, and we can curse our mad scientist friend for giving us such a shoddy gift. If it has not decayed yet (a 50% chance), we keep waiting. After another 24.1 days (4:48 am on 17 February), it will have had another 50% chance of decaying. The overall chance that it has decayed by now is the 50% for the first 24.1 days, plus another 50% of 50%, or 25%, for a total of 75%. The overall chance that you still have your gift is now down to 25%. If you're lucky, you can keep cherishing your thorium-234 atom a bit longer. After another 24.1 days (7:12 am on 13 March (or 12 March in a leap year)), it will have had yet another 50% chance of decaying. The overall chance that it has decayed by now is the 75% for the first 48.2 days, plus another 50% of 25%, or 12.5%, for a total of 87.5%. The overall chance that you still have your gift is now down to 12.5%. And so on.

Sample of thorium. Creative Commons Attribution-NonCommercial image by Rui Costa.

You can see that for every period of 24.1 days that you wait, your chance of still having an undecayed thorium-234 atom reduce by half. It is possible for it to last a year or more, but the odds gradually become extremely small. Now let's extend this to a large sample of a radioactive isotope.

Say your mad scientist friend gives you a block of thorium-234. A whole kilogram! Besides thinking that he might be trying to kill you with radiation poisoning, you'd be delighted. But after 24.1 days, every single atom of thorium-234 in the block has had a 50% chance of decaying. Since there are zillions of atoms in there, very close to half of them will in fact have decayed. Think of it this way: each atom is given a coin to flip. If it comes up heads, that atom will decay at some time within the first 24.1 days, if it comes up tails, it survives. Toss zillions of coins, and by far the most likely result is that very close to half of them will come up heads. Yes, it's possible that they'll all come up heads, but the odds of that are stupendously small.

The result is that very close to half of your original kilo of thorium-234 has now decayed. You now have half a kilo of thorium-234, plus or minus a very tiny bit due to the random fluctuation of the exact numbers of heads and tails. In practice, the variation is almost always so tiny as to be negligible.

So now you have half a kilo of thorium-234. After another 24.1 days, the remaining atoms have done their coin toss again, and very close to half of them will have come up heads and decayed, while the other half came up tails and survived. So an extra quarter of a kilo of thorium has decayed, and you're left with (very close to) a quarter of a kilo of thorium-234 remaining. And so on: after every 24.1 day period the amount of thorium-234 you have left is reduced by (very close to) half. By 13 March you'll only have one-eighth of a kilo. That's how half-life works.

Thorium is in fact named after the Norse God of Thunder. How cool is that? Public domain image from Wikimedia Commons.

But where do the decayed thorium-234 atoms go? Well, they don't go anywhere. They turn into protactinium, then into uranium-234, and so on down the decay chain until they eventually become lead-206. Some of these steps take a long time, however. In fact, uranium-234 has a half life of 245500 years, so pretty much all of the decayed thorium will stick around as uranium-234 for as long as you're alive. However, if you waited a quarter of a million years, half of the uranium-234 would have decayed. And no other isotope further down the decay chain has a longer half-life, so most of it will have decayed all the way down into lead-206. There will be a bit of thorium-230, with a half-life of 75380 years, but virtually no atoms at all of tellurium-210, which has a half-life of 1.3 minutes.

All of this is slightly complicated by the fact that more thorium-234 is decaying all the time, launching new atoms down from the top of the decay chain. But you can account for this mathematically, and develop a complete list of the ratios of amounts of different atoms in the decay chain, at any given time after the sample of thorium-234 was generated. Go back a step, up the chain, beginning with the extremely long-lived uranium-238 (half-life 4.468 billion years, so there's actually quite a bit of left on Earth since the planet was formed 4.5 billion years ago - very close to half of it), and you have a list of isotope ratios that depends on the amount of time since the uranium-238 was formed.

So when a geologist finds a rock and wants to know how long it has been since the uranium-238 in it formed, they can measure the amounts of various isotopes in the decay chain, and figure it out! In practice, the only isotopes you need to measure are the uranium-238 and lead-206, since all the intermediate isotopes have comparatively much smaller half-lives and will be present only in trace amounts. The ratio of the amount of uranium-238 and lead-206 tells you, essentially, how old the rock is.

An even cooler thing is that this method, known as radiometric dating, has a built-in error checking mechanism when used on ancient rocks. Remember that the decay chain starting with uranium-238 is only one of four possible decay chains. Another chain (mentioned briefly above) starts with uranium-235, and ends with lead-207. But uranium-235 has a half-life of 0.704 billion years, significantly shorter than uranium-238. This means the ratio of uranium-235 to lead-207 in an old rock sample will be different to the ratio of uranium-238 to lead-206. And these two ratios provide independent measures of how old the rock sample is! So measuring these four isotopes gives you the age of the rock, and an error check to make sure nothing has gone wrong. If the two ages agree, then you have nailed the age of the rock with a high degree of certainty. This method is known as uranium-lead dating.

Carbon-14 dating. Creative Commons Attribution-NonCommercial image by Flickr user Travis S.

Other things we might be interested in learning the age of, such as mammal fossils or human cultural relics like pottery, either have negligible uranium in them or are much too young for any measurable amount of the uranium to have decayed, or both. To date such objects, we can use a different decay chain. There are only four decay chains starting with heavy elements (atomic numbers higher than lead), but there are other decay chains involving lighter elements. A particularly important one is the chain beginning with carbon-14. This "chain" only has one link, a beta decay to nitrogen-14, which is stable. Carbon-14 has a half-life of 5730 years, so it is good for dating objects from a few years old up to about 60,000 years before the quantities of atoms involved become too small to measure.

So-called radiocarbon dating is a bit more complicated than uranium-lead dating. Firstly, nitrogen-14 can escape from the sample, and can also enter the sample, since this is by far the most common isotope of nitrogen, and in fact makes up around 78% of the air you're breathing right now. So getting a measure of the amount of nitrogen-14 in a sample is basically useless. Also, and you may be wondering this already, with a half-life of under 6000 years, how is there any carbon-14 around in the first place?

The answer to that is that carbon-14 is generated at a steady rate in our upper atmosphere, by the action of cosmic rays on nitrogen. Cosmic rays, which come from distant sources such as exploding stars and giant black holes, include rapidly moving neutrons. When one of these hits a nitrogen-14 atom, it can be absorbed and knock out a proton, converting the nitrogen-14 into carbon-14. Since this happens at a steady rate, it reaches an equilibrium where the rate of production of carbon-14 equals the rate at which the produced carbon-14 decays. And the atmosphere mixes up, so that there is a constant supply of carbon-14 in the air all around us.

Plants, animals, and bacteria all take in this carbon-14 and use it to build the cells in their bodies. Chemically, carbon-14 behaves pretty much the same as regular, stable carbon-12. So the ratio of carbon-14 to carbon-12 in a living body is exactly the same as the ratio in the atmosphere. But when the organism dies, it no longer accumulates carbon. The carbon-14 decays, while the carbon-12 just stays there. After 5730 years, the ratio of carbon-14 to carbon-12 in the dead organism is now half of what it was when it died - in other words, half the ratio seen in the atmosphere. So if we find, say, a mummified body, and we measure that it has half the carbon-14 to carbon-12 ratio as in the atmosphere, we know the mummy is about 6000 years old! if it has a quarter of the atmospheric ratio, it must be closer to 12,000 years old. And so on. Using this method, we can determine to within a few decades how old a biological sample is.

Uranium-lead and radiocarbon are only two of many forms of radiometric dating. Several other decay chains give us access to dating mechanisms that span several of the time periods we are interested in measuring. So radioactivity gives us a way of establishing the age of the Earth, of human culture, and of other things like fossils and meteorites.

It also gives us the hint that elements can change into other elements. They do so all by themselves! Perhaps... perhaps we could play with the protons and neutrons in atoms and turn one element into another deliberately. In fact, if we could just remove three protons from lead atoms, we could turn them into gold...

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[1] I don't know exactly how alchemists would have considered gold to be comprised. Presumably a bit of fire for its golden lustre, maybe some water because as a metal it is cool to the touch. I'm not sure how you'd justify any air, but maybe there was supposed to be a little of that too.

[2] And in fact neither air, earth, water, nor fire were actually elements.