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3. Half life page 11
What is half life?
Radioactive decay is an example of an exponential change. It also has a random element to it – i.e. the curve is not smooth. In this chapter we will explain what these terms mean and how we can understand the measurements in terms of a model of what's happening.

Picture 3.1 shows a plot of the activity of radon-220 (this has been corrected for the background count – see below). We have drawn a smooth curve through the results. The activity decreases and heads down towards the axis. But also, there are random fluctuations within the actual readings. First let's look at the trend of the curve.

Exponential decay graph
Picture 3.1 The decay of radon-220. It has a hlaf life of 52 seconds. In a given time interval, the activity always drops by the same proportion.
Exponential changes
There are a number of features to notice:
  • the line heads towards the x-axis but doesn't cross it (we say the x axis is an asymptote)
  • the line starts on the y-axis (i.e. the y axis is not an asymptote)
  • every 20 seconds, the count rate drops by about 25%.

These are all features of an exponential change. However, the last one is the one that is useful for analysis. We can see how long it takes for the decay to drop by any proportion. It will always be a constant. For example, the activity drops by 25% every 20 seconds.This is true whether we start at the start, at 30 seconds or at 80 seconds. The proportional drop is always 25% in any 20 second interval.

Table of results
Decay graph
Picture 3.2 Graph of decay for protactinium-234. We can correct this for the background count.

Half life
Similarly, we can look at how long it takes the activity to drop by 50%. It's 52 seconds. This is true whether we take the time to drop from 50 counts per second (cps) to 25 cps; or the time to drop from 20 cps to 10 cps. In each case, the activity drops by a half. We call this time period the half life. It is fixed for a given decay. In this case, we can say that radon-220 has a half life of 52 seconds.

The half life is the average time taken for the number of nuclei of a nuclide to halve. It is also the average time for the activity of that nuclide to decay to half of its original value.

Measuring half life
You can measure the half life of some isotopes in the laboratory. For example, protactinium-234 can be generated in the lab and its activity measured using a GM tube plugged into a scaler counter. Table 4 shows some results of this experiment (the count was recorded every 25 seconds). By subtracting each previous count, we can work out the average count rate (in counts per second) for each 25 second interval. We can plot this against time – picture 3.2.

Notice that the asymptote for the count rate is not the x axis. The count rate is actually tending towards a value of 1.2 counts per second. So, even when the protactinium has been used up, there will still be a count of 1.2 counts per second. This is the backgound count. It is activity that is always around us from natural radioactive sources in the ground and air.

To measure the half life of protactinium–234, we need to remove the background count from all our readings. The effect is to move the line down so that the asymtote is the x-axis. We can now take a measurement of how long it takes the activity to drop by a half. We can make three consecutive measurements of the time it takes the value to halve. They are 68 s, 70 s and 72 s. This gives an average of 70 seconds.

What do we mean by used up?
Although, in a mathematical sense, exponential curves never reach the origin, in reality a radioactive isotpe will always get used up. Here are two views as to why:
  • The engineer: it will eventually get so small that we can’t measure it.
  • The physicist: it will eventually get down to a small number of individual radioactive nuclei that each have a probability of decaying. These will, randomly, decay.

Even so, we can’t get a definite measurement of the time it takes to stop entirely. That's why we use the half life – the time it takes to drop to a half of its starting value.

Question 10
a) Radioactive decay is an example of an exponential decay. Give some other examples of exponential changes (they could be growth or decay).
b) Why do we have to take account of the background count?
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