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Radioactive decay involves one isotope changing into another (and often, another and another and another...).  This is called a decay chain.

## A natural sample of radioactive rock contains many generations of isotopes

So a typical radioactive sample actually consists of lots of different isotopes all decaying from one to another.  The long natural decay chains were made in supernovae billions of years ago and so were present when the Earth was formed from their dust.

When a decay chain has had aeons to settle down, the proportion of an isotope is proportional to its half-life.  In other words if an isotope has a short half-life you won't find very much of it.

This is because all the isotopes in the decay chain are in radioactive equilbrium.  In other words they're created as fast as they decay.

## Radioactive equilibrium happens when there's something to aim at

When a bath empties, the speed the water leaves depends on how deep the bath is.  The deeper the bath the faster water flows out of the plug hole.

If you run water into the bath from the tap you'll initially find the tap fills the bath quicker than it empties.  But as the depth of the bath increases there'll be a point when the depth stays constant because it's just deep enough for water to flow down the plug hole at the same rate as it's coming in from the tap.

Now the thing about filling the bath using the tap is that no matter how long you run the tap for, the rate of water coming out of it is always the same.  So the rate of water emptying out of the plug hole has this rate to aim at.  At some point it'll always find it because the target rate from the tap doesn't move.

As soon as this target rate is hit then the bath just stays at the same depth forever.  We could call this the equilibrium depth and the rate the equilibrium rate.

The equilibrium rate doesn't depend on the size of the plug hole.  It must always equal the rate that the tap is working at.

But the equilibrium depth does depend on the size of the plug hole.  If the plug hole is small the equilibrium depth will be deep.  If the plug hole is big it will be shallow.

The depth of the bath is like the amount of an isotope there is.  The rate at which water flows through the plug hole is like the decay rate and the size of the plug hole is like the half-life (except a big plug hole means a short half-life).

The tap is like the parent isotope.  Remember we got a steady, equilibrium depth if the tap was kept at a steady rate.  Isotopes that don't change their decay rate very quickly have to have a long half-life.  So if you want to have radioactive equilibrium then your decay chain has to be headed by an isotope with a very long half-life.

The longer the half-life of the daughter isotope, the smaller the plug hole so the bigger the equilibrium amount.

## If the parent's in equilibrium its half-life seems infinite

It turns out that you only need one daughter isotope to be in equilibrium for the equilibrium to cascade down the decay chain regardless of the half-life of the other daughters.

The cunning thing is that by definition an isotope in equilibrium is produced as quickly as it decays.  If the amount stays constant then this equilibrium rate stays constant.  So as far as its daughter is concerned it has an infinite half-life because the rate never changes.

So the daughter has this steady rate to aim at and will eventually be in equilibrium too and so on down the chain.

## Ingrowth: starting a decay chain from scratch, increasing total radioactivity

There are a couple of things you can do to start a decay chain from scratch.  Neither of them involve stopping the decay but it means you start with a clean slate.

The first is to remove the daughter products, by for example chemically treating a sample of uranium-bearing rock so you're only left with the uranium.

The second thing you can do is to synthesise the parent, perhaps by splitting a bigger nucleus to create two smaller radioactive ones.

In either case, the equilibrium has to establish itself again.

Imagine it starts off with the parent decaying at a rate that's fairly constant because it has a long half-life.  Initially there isn't very much of the first daughter so the first daughter doesn't add much in the way of radioactivity.

But as the amount of the first daughter builds up its radioactivity becomes more and more significant until it's the same as the parent's.  So eventually the total radioactivity will be almost double what it was.

All the time the first daughter's decaying the second daughter is building up and it's radioactivity is becoming more and more significant and so on.

This is called ingrowth and it means you have to plan quite carefully for safe levels of radioactivity if you're starting a decay chain from scratch.

## Ingrowth ends with equilibrium or decay

Eventually the whole decay chain will have reached the equilibrium rate of the parent isotope (assuming it has a long enough half-life to cause equilibrium).  After this, the total radioactivity just falls slowly as the parent isotope gradually becomes less and less radioactive.

If the parent isotope doesn't have a long enough half-life compared with the first daughter then you may get ingrowth without equilibrium, resulting in a definite peak followed by obvious reduction in radioactivity.

back to Lesson 3: Half-life part 1