# Half-thickness of gamma rays in lead

## The gamma photon has no memory of how far it's already travelled

Imagine a gamma photon travelling through some lead. It interacts once and then disappears, passing on its energy to an electron or nucleon.

For each millimetre that it travels through the lead there is a constant chance that it will be absorbed. This chance doesn’t depend on how much lead it has already travelled through.

## A dice analogy

We can use dice to model the random absorption.

In this simulation if a six is rolled the photon is absorbed. So at each position there is a one in six chance of this happening. It’s important to understand that the chances of rolling a six don’t depend AT ALL on what’s been rolled before.

As the photon gets further into the lead it has to get past more dice. So the chances of seeing a six *somewhere* increase. But the chances of any *given* dice showing a six are always the same.

Imagine sitting on dice 4 (strictly ‘die’ 4). For the photon to get to you it will have to NOT be absorbed 3 times i.e. by dice 1, 2 and 3. If you repeated the experiment lots of times you’d see that about 60% of photons will make it to dice 4.

What proportion of these remaining photons will then make it to dice 7?

Again, any photon that makes it to dice 7 will have to NOT have been absorbed by three dice: numbers 4, 5 and 6.

## The nub of the matter

And here we get to a key point. We know that about 60% of photons can get past three dice. But it doesn’t matter *where* those three dice are. If the photon gets as far as the first one it has a 60% chance of getting past the third.

60% make it to dice 4, 60% of what’s left make it to dice 7, 60% of what’s left make it to dice 10 and so on…

This is a feature of an ‘exponential’ relationship. A fixed change in one thing (number of dice) gives a fixed PROPORTIONAL change in another (number of photons getting that far).

## The gamma photon may pass straight through the lead

Any given gamma photon can be absorbed anywhere in the lead or even pass straight through. The gamma photon behaves as if there is a fixed chance of absorption for every unit of distance travelled.

For example there is the same chance that the photon will get absorbed each millimetre it travels through the lead. It doesn’t matter how many millimetres of lead the photon has already gone through.

## Same change in thickness gives same proportional reduction

For this energy of gamma photons what thickness of lead did you have to go through to reduce the number getting through by a half?

In this case it’s always 4.2 mm. Every 4.2 mm the gamma photons travel through, half of them get absorbed.

## Half-thickness

We call 4.2 mm the ‘half-thickness’ of these particular gamma photons in lead.

The ‘half-thickness’ tells us the thickness of a given material needed to absorb half the incident photons from a particular source.

If you have more of the gamma emitter it will emit more photons per second. We call this a higher ‘intensity’ source. No matter how many photons are emitted, half of them will always get absorbed in the same length.

The half-thickness depends on both the energy of the photons (i.e. type of source) and the material of the absorber. Half-thickness increases for higher energy photons and for lower density absorbers, e.g. steel.

## An exponential relationship

So we’ve seen that absorption of gamma rays in a given thickness of material is an exponential relationship. This relationship can be expressed as: ‘For any given thickness the same fraction will always make it through (or get absorbed).’

This is called the ‘constant ratio’ property. Seeing if there is a ‘half-thickness’ is really just testing for this constant ratio.

## There's nothing special about a half

But there’s nothing particularly special about half-thickness. Half is just a convenient fraction.

You could choose the thickness needed to go from 90% to 60% to 40% of the original number of photons, giving a ‘two-thirds-thickness’. Or from 80% to 20% to 5%, giving the 'one-quarter-thickness'.

We’ll come across this ‘exponential’ relationship again when we look at how radioactivity changes with time.

## Finding half-thickness in the lab

In reality it would be hard to devise an experiment to find out where each photon was absorbed in a thick piece of lead. It’s easier to change the thickness of the lead and count the photons that get through with a Geiger counter.