S4.11.3 Chapter 3

Half-life

A sample gives out 800 counts each minute. After 20 minutes it gives out 400. How long until it is down to 0?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Another 20 minutes, because it falls by 400 every 20 minutes.
You picked A. Read on. At the end, the chapter comes back to your guess.
B It is never exactly 0. It halves again every 20 minutes.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Another 40 minutes, because it falls to 0 twice as slowly.
You picked C. Read on. At the end, the chapter comes back to your guess.
D It stays at 400, because decay stops when half has gone.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find the half-life from a decay graph
Model random decay by tossing coins and removing the heads

Chapter 1 said that one nucleus decays at a random time. Yet a big sample behaves in a steady way. Its radiation falls away in a pattern that is the same every time. The pattern is described by one number, the half-life.

The half-life of a nucleus is the time it takes for half of the undecayed nuclei in a sample to decay. Each kind of nucleus has its own half-life. It does not change with heat, pressure or chemistry.

Take 8 nuclei with a half-life of 1 hour. After 1 hour, 4 are left undecayed. After 2 hours, half of those 4 are left, which is 2. After 3 hours, 1 is left. Each half-life halves what is left, and does not take away a fixed amount.

See it three ways

With things

Think of 8 nuclei with a half-life of 1 hour. A coin toss for each one in each hour gives the same kind of result.

As a picture

In words

Each half-life, half of the undecayed nuclei decay. Filled circles have not decayed and open circles have. With so few nuclei, a real sample would wander a little from these numbers.

In symbols

8 → 4 → 2 → 1

The decayed nuclei do not vanish. They have become nuclei of another kind, usually of another element. The sample does not lose half its mass at each half-life. It loses half of its undecayed nuclei, and so half of its radiation, if the new nuclei are stable.

We cannot count nuclei, but a detector counts decays. The number it counts each second or minute is the count rate. When there are half as many undecayed nuclei, half as many decays happen each second. So the count rate halves in the same time.

So we can also say: the half-life is the time for the count rate to fall to half. A graph of count rate against time is called a decay curve. It falls steeply at first, and then more and more slowly.

Each dashed pair of lines shows one more half-life. The count rate halves every 20 min.

Worked example

The graph above is the decay curve of a sample. Find its half-life.

The count rate at the start is 800 counts/min. Half of that is 400 counts/min.
Find 400 on the up axis. Go across to the curve, then straight down to the time axis.
The time axis reads 20 min.
The half-life is 20 min. Check: from 400 the curve falls to 200 in another 20 min.

Answer: 20 min

Read the half-life from any part of the curve and you get the same answer. Start at 400 and find where it falls to 200. Start at 200 and find where it falls to 100. Each takes the same time. Only a curve of this kind does that.

Worked example

A sample of iodine 131 has a half-life of 8 days. Its count rate is 640 counts/min. What will the count rate be after 24 days?

Number of half-lives = time ÷ half-life.
Number of half-lives = 24 ÷ 8 = 3.
Halve the count rate 3 times: 640 → 320 → 160 → 80.
The count rate will be 80 counts/min.

Answer: 80 counts/min

After 1 half-life, one half is left. After 2, one quarter. After 3, one eighth. After 4, one sixteenth. The fraction left is one half multiplied by itself once for each half-life. It gets smaller and smaller, but it is never exactly 0.

A wrong idea many people have

After 2 half-lives, all of the radioactive nuclei have gone.

Why it feels right: One half-life takes away half, so it seems that a second half-life takes away the other half.

The test that breaks it: Start with 8 nuclei. After 1 half-life, 4 are left. The second half-life takes half of those 4, which is 2. It does not take all 4.

What is true: Each half-life takes away half of what is left. After 2 half-lives, one quarter is left. After 3, one eighth. A sample never reaches 0 in a set number of half-lives.

Half-lives are very different. Some nuclei last a tiny fraction of a second, and some last billions of years. A short half-life means a lot of decay now, but the sample soon runs down. A long half-life means weak decay that goes on for a very long time.

Some half-lives, which are rounded. Po is polonium, I iodine, Co cobalt, C carbon and U uranium.

A real detector also counts background radiation, which is always there. Take the background reading away before you look at the sample. Only the corrected count rate halves.

Worked example

A detector gives 260 counts/min at the start, 195 after 5 min, 140 after 10 min and 107 after 15 min. The background is 20 counts/min. Find the half-life.

Take away the background: corrected start = 260 − 20 = 240 counts/min.
Half of the corrected start = 240 ÷ 2 = 120 counts/min.
At 10 min the reading is 140. The corrected rate = 140 − 20 = 120 counts/min.
The corrected rate has halved at 10 min, so the half-life is 10 min.

Answer: 10 min

Be careful about one idea. Half-life describes a large number of nuclei. For one nucleus, it gives the chance. In one half-life, it has an even chance of decaying. With few nuclei the halving is rough. You can see this with coins.

Try it

Model decay with coins

You need: 32 coins, or 32 paper discs marked with a cross on one side, a cup, a table, paper and a pencil.

Be safe: Keep small coins away from young children, who may put them in their mouths. Pick up every coin when you finish.

Put all 32 coins in the cup. Write 0 and 32 on your paper.
Shake the cup and tip the coins onto the table.
Take away every coin that shows heads. These have "decayed". Count the coins that are left.
Write the toss number and the number left. Put the left coins back in the cup.
Repeat until you have done 6 tosses, or no coins are left.
Draw a graph of coins left against toss number.

What you will see: About half of the coins go each time, so the numbers roughly halve, for example 32, 16, 8, 4, 2, 1. Yours will not match exactly, because each coin is random. Do it again and you will get different numbers. The pattern is the same. In this model, one toss is one half-life.

If you cannot try it, this is what happens.

A wrong idea many people have

A radioactive sample with a half-life of 1 hour will be safe after 2 hours.

Why it feels right: It feels like 2 half-lives should finish the job, because half and half make a whole.

The test that breaks it: Work out what is left: half of the sample after 1 hour, and a quarter after 2 hours. A quarter of the original radiation is still there.

What is true: After 2 half-lives, a quarter of the radiation remains. How long it takes to be safe depends on how strong the sample was and on what level is safe.

Key words

half-life: the time for half of the undecayed nuclei to decay, or for the count rate to fall to half.
count rate: the number of decays a detector counts each second or minute.
decay curve: a graph of count rate against time. It halves again and again.
background: radiation that is always around us. Take it away from a reading.

Back to your guess

The answer is B: It is never exactly 0. It halves again every 20 minutes.

The count rate halves every 20 minutes: 800, 400, 200, 100, 50 and so on. Each step takes away half of what is left, so it gets very small but never reaches exactly 0.

Keep it

Copy this into your exercise book.

Half-life: the time for half of the undecayed nuclei, or the count rate, to halve. After 1, 2, 3 half-lives: one half, one quarter, one eighth. Never exactly 0. Take away background first. One nucleus is random.

Draw: A decay curve from 800 counts/min, with dashed lines to 400, 200 and 100 and the times 20, 40 and 60 min.