Half Life

Include: mass doesn't halve.

We will consider the simple case of radioactive atoms decaying into stable atoms by the emission of a particle. Unfortunately, there is no way of avoiding drawing lots of diagrams. The writing between each is quite short, and you will find that you quickly get the idea of what is going. It is that passage after the diagrams that matters.

Imagine a jack-in-the box that is permanently shivering. When it is only shaking a little it stays shut, and will stay shut for ever if it is only shaking a little. But every so often there is a rogue shake, and it springs open, releasing an orange ball that flies off into the distance.

• Notice that any given box can only do this once.

• If you have a collection of these boxes, they all shiver independently of one another. Each exists in its own closed little world, entirely ignorant of what is happening to the others.

• This particular design of box shivers in such a way that in any given one-minute period about a quarter of the boxes spring open.

• The fact that a particular box survives for a minute doesn't make it any more or less likely that it will succumb during the next minute

Now consider a tray containing sixteen such boxes.

By the time a minute has gone by, four will have opened and their orange balls will have been ejected. Let's suppose that we managed to catch them all and that we have piled them up beside the tray as trophies.

There are only twelve unopened boxes left now, and over the course of the next minute a quarter of them will spring open. (To help with identification, newly sprung boxes are shown with orange lids.)

Now there are just nine boxes, so only about two (just under a quarter of them) will spring open during the next minute.

Seven boxes remain. Perhaps two will spring open during the next minute? This is a slight over-estimate if we think that a quarter ought to open; but then we slightly under-estimated last time.

Perhaps, with only five boxes left, just one will open during the next minute.

Graphing the data

The left hand graph shows the number of unopened boxes left as a function of time (namely, 16, 12, 9, 7, 5, 4), while the right hand graph shows the number of orange balls ejected as a function of time (namely 4 during the first minute, 3 during the second, and then 2, 2, 1 and 1). Notice that, although the vertical scales are different, both graphs have the same shape.

Here is the left-hand graph on its own, without any blobs.

• It takes about 2.3 minutes for the value to fall from 16 to 8 (i.e. to halve its value). This is shown by the orange dotted line.
• It takes another 2.3 minutes for the value to fall from 8 to 4 (i.e. to halve again). This is shown by the green dotted line.

Summary

The boxes in this model represent atoms. The decay of an atom is represented by a box opening and ejecting a ball (which represents an alpha-particle, a beta-particle or a gamma-ray).