## Why does radioactive decay have to be exponential?

A rather interesting question came up on Physics Stack Exchange (semi-)recently: How do we know that $${}^{14}\mathrm{C}$$ decay is exponential and not linear? This question addresses something that confuses a lot of people when they’re first learning about radioactivity, namely the use of a half-life to describe the different rates at which different kinds of radioactive atoms decay. When you first hear that, say, carbon-14 has a half life of 5700 years, you might wonder why we don’t just say that it has a lifetime of twice that, or 11400 years? If half the sample is gone in the first 5700 years, won’t the other half be gone after the next 5700 years?

The model suggested by that statement is called linear decay, because the number of atoms remaining decreases linearly with time. Of course, we know from experiments that radioactive decay is not linear, it’s exponential. But you can also use simple physical reasoning to convince yourself that radioactive decay wouldn’t be described with a half-life if it were linear. Here’s a little thought experiment to show that:

1. Take a billion radioactive carbon-14 atoms, put them in a box, and wait around for one half-life (5700 years). Now you have 500 million $${}^{14}\mathrm{C}$$ atoms left.
2. Take a different sample of 500 million atoms and put it in another box. Then wait 5700 years, again.
3. Open both boxes. What’s left?

Clearly, box #2 would still have 250 million atoms; after all, you started with 500 million, and you waited for one half-life, exactly long enough for half of those atoms to decay. Box #1, on the other hand, should have no radioactive atoms left according to the linear decay model, since you waited for two half-lives after sealing it up.

But wait, that doesn’t make sense! When you sealed up the second box, it was exactly identical to the first box at that time. Each box contained 500 million radioactive carbon-14 atoms, and the radioactive atoms in box #1 don’t “remember” that they came from an earlier sample of a billion atoms. So the contents of box #1 should behave exactly the same as the contents of box #2. In particular, they must have decayed identically, so when you open up both boxes in step 3, you should find the same contents in both. You can’t have one with no $${}^{14}\mathrm{C}$$ and the other with 250 million atoms of it, as the linear model would predict. So something about this explanation has to break: either linear radioactive decay doesn’t work, or there’s no such thing as a half-life because the lifetime of half your sample depends on its size.

As I mentioned earlier, experiments show that it’s the linear decay model that is wrong. In fact, there are theoretical reasons why exponential decay is the only model that really makes sense. The idea is that each radioactive atom’s decay is unaffected by its environment — that is, the probability that the atom will decay in a short interval of time is independent of, say, what other atoms are around it. Consequently, if you have a large sample of $$N$$ radioactive atoms, the mathematical expectation for the number of decays in that short interval of time should be proportional to $$N$$. You can write this as

$$\ud{N}{t} = -\lambda N$$

This is an easily solvable differential equation: the solution is

$$N(t) = N_0 e^{-\lambda t}$$

in other words, the exponential decay curve, with $$N_0$$ being the initial number of radioactive atoms.

If this is enough to pique your interest, I’ll have more to say about radioactive decay in an upcoming post. Stay tuned!