In the popular media, you often hear the phrase “the ‘something-or-other’ has grown exponentially.” Quite often in such situations, the word ‘exponentially’ is used imprecisely to mean ‘quite a lot’ or ‘rather fast’, but in science it has a very specific meaning.
Exponential changes happen when the rate of change of a quantity is proportional to the value of that quantity. They are characterised by equal fractional changes of a quantity in equal time intervals. An exponential change can be an increase (such as unlimited population growth) or a decrease (such as the number of radioactive nuclei remaining in a sample as the sample decays), just so long as equal time intervals result in equal fractional changes.
In fact, it is this latter example – radioactivity – that will occupy us in this post.
Many of us will have seen a radioactive process being modelled using a set of many (like, hundreds of) dice. The dice in the model represent radioactive nuclei. Each dice has one face (out of six) marked somehow (I’m not so pedantic that I’m calling one of these things a ‘die’). Then all the dice are rolled in one go. The dice rolling models the passage of a certain amount of time – let’s be orthodox, and call it a ‘second’. All the dice that fall marked-side-up are deemed to have decayed in that second. They are still there, but they are now, typically, nuclei of a different element, and take no further part in the ‘game’. Then the ‘surviving nuclei’ (marked-side-NOT-up dice) are rolled again, representing the passage of the next second. This continues until either (a) there are no surviving nuclei, or (b) everyone has gone home.
This model can be used to exemplify all sorts of things in radioactivity, like:
(a) the fact that it is a random process, in the sense that we can predict how many might decay, but not which ones.
(b) the meaning of activity
(c) why and how this process is exponential
(d) the meaning of half-life and decay constant
But there’s a really interesting wrinkle to this model that rarely gets discussed (as far as I can tell). So I’ll briefly mention (a) to (d), but do stick around for the wrinkle, as that’s my reason for writing this in the first place!
Some real data (honest)
So I got 500 dice (nuclei) and did this, and recorded how many survived after each roll (second). Frankly, I wish I hadn’t bothered – I could have made up the results and you would never have known. Still, here the results are, in table and graph form…
So this graph certainly seems to show a pattern. It’s not perfect, but there’s a pattern (and if I had the time to count millions of dice, rather than 500, it’s overwhelmingly likely that the pattern would have been better). And yet the pattern emerges out of a random process – there’s no way I could predict which dice (nuclei) would land marker-side-up (decay), but it does seem to be possible to predict how many.
So, how many? What would we have predicted? Well, in an ideal world, each roll of the dice (second) would result in one sixth of the number of dice decaying, and this would correspond to the activity of the sample (the number decaying per second). Five sixths would, therefore, survive. In this statement is the reason for this process being exponential (a rate of change being proportional to the quantity). Radioactive decay is exponential because the more nuclei there are, the more decay (and in a strict proportional sense, too), and so the faster the rate of change of how many there are.
So here are the results again, with predicted values added – in the table, each number in the red column is five sixths of the previous one.
There is a pretty good agreement between experimental (blue) and predicted (red) data.
Decay constant and half life
How does this predictability emerge from the randomness? It happens because the probabilities for the dice do not change with time. For any given dice, there is a 1 in 6 chance of ‘decay’ this second. And the next second. And the one after that. This behaviour is not that surprising for dice. But radioactive nuclei also behave like that, and the more you think about that, the weirder it gets. It means that if a nucleus survives this second, it has no ‘memory of its history’ – the probability of decay for the next second is unaltered by how many seconds it has already survived. If our dice were nuclei, the decay constant for this particular isotope would be 1/6 per second – just the probability of each decaying per unit time. (Note that, provided you have enough that they are behaving statistically sensibly, that is the same as the fraction present decaying per second. In a real sample, that’s not a problem – there will be squillions.) The graph can be used to find the half life of this ‘isotope’ of dice, which mean the time taken for half of them to decay. By looking across on the graph from 250 (half of 500), we see that the half life of this isotope is approximately 4 rolls (seconds).
(Apparently the cool kids nowadays do this with Skittles – ‘S’ up = decayed. Then the decay constant would be ½ per second. I think real learning could happen from comparing the behaviour of Skittles and dice, but I have a sneaking suspicion that Skittles are used because they are edible.)
So where’s this wrinkle?
Well, once we have taken our exams at age 16, and decided to carry on with physics, someone shows us the exponential decay law, giving the number \(N\) of nuclei remaining at time \(t\), given a particular decay constant \(\lambda\). This is useful, because before we were at this stage of our education we could only find \(N\) at times equal to multiples of the half life. Here is the exponential decay law, but you really don’t need to worry about it… \(N=N_0{{\mathrm{e}}^{-{\lambda}t}}\). All you need to believe is that it works for radioactive nuclei.
If you plot this function on the same axes as before, for a decay constant of 1/6 per roll (second), you might hope that it would match the predicted (red) data. Here it is in grey.
They don’t match.
The dice model has values number of nuclei remaining that are too low (assuming the exponential decay law works, which it does). The red line is an exponential function, and it’s the right one for dice, but it’s not the right one for radioactive nuclei of decay constant 1/6 per second.
Why do dice not model radioactive decay accurately? Well, our model calculates an activity (number landing marked-side-up) from the number of nuclei, but it applies this activity for the whole second, until the next row, when it recalculates. In reality, the activity changes not in discrete steps, but continually. In each 1-second interval the activity will be continually falling as the number remaining falls, whereas the dice model assumes it to stay at its calculated value for the whole second. This leads the dice model to overestimate the activity at each step, and underestimate the number of nuclei remaining. In other words, there is a flaw in our analogy: dice numbers change with discrete numbers of rolls, whereas the number of nuclei changes with the continuous flow of time.
You can see from this that using smaller time intervals should give a better model, with a better fit to “reality”. You can try this yourself by recreating the spreadsheet with time intervals of, say 0.1 s (ignore the blue experimental data and just muck about with the red and grey columns…). The true exponential function for decay constant = 1/6 will be reached in the limit as the time intervals you use shrink to zero.