# Half-Life

Besides from being a series of popular first-person shooter video games, half-life is an important factor to consider in any applications of radioactivity. It seems quite a weird quantity at first glance; we usually talk about the lifespan of things, rather than half a lifespan. But as you will see, the ‘half-life’ is far more logical, because the lifespan of a radioactive substance is… well, infinite (theoretically).

A radioactive sample consists of many radioactive nuclei. We cannot know which nuclei will decay, nor can we know when they will decay. This process is completely and utterly random.

We can conduct a simple simulation. Imagine we have a ‘radioactive sample’ of 1000 dice. Each die represents a radioactive nucleus that will be rolled multiple times, and we’ll decide any dice that rolls a 6 to have decayed. Before we roll the dice however, we cannot know which dice will show a 6, or after how many number of rolls they will show a 6, because the process is random.

We can assign a value to the probability of a radioactive nucleus decaying, called the decay constant. In this experiment, we know the decay constant is ⅙ (studies have shown that this may not be true, but we’ll leave that for now), but different radioactive substances have different decay constants.

We roll the dice. We expect a sixth of the dice to roll a 6 (that’s about 167 dice), but it will of course vary. We remove these dice that have ‘decayed’, because they are no longer radioactive.

We roll the dice. This time we expect around 28 dice.

We roll the dice again. This time we expect around 5 dice.

We roll the dice again. This time we expect around 1 die.

And it may take several rolls for this last die to show a 6, simply because it is random.

This is what the number of remaining dice looks like over this period of time:

This type of graph, called an exponential decrease, does not reach zero. Even though the number of radioactive nuclei will reach zero in reality, it will still linger for a long time when the sample is very small, making it pointless to measure the full lifespan.

This is where half-life comes in. Half-life is the time taken for half the radioactive nuclei to decay. This is a far more useful method of measuring how quickly a radioactive sample decays, because its value corresponds better to its application in real time.

For example, iodine-131 has a half-life of around 8 days, and is used as a medical tracer in humans. You can easily tell that after 8 days, half the iodine will be gone, and after 16 days, only a quarter of the original sample will be left. It would be pointless to describe the sample in terms of how long it would take for all of the sample to decay, because: one, it would be a ridiculously long time, making it difficult to match the correct radioactive substance to the correct application; and two, the radiative effect of the sample would be negligible at low amounts.

Half-life is not only used in nuclear physics. Because of the abundance of exponentiation, half-life can be found in many areas of science (like your beer for instance, but we’ll get to that another time).

Yanhao