The Radioactive The Radioactive Decay model.Decay model.

In this presentation we will learn:In this presentation we will learn:

1.1. What is meant by the termsWhat is meant by the terms

Probability of decay per Probability of decay per secondsecond

ActivityActivity

Radioactive decay constantRadioactive decay constant

BecquerelBecquerel

2. A simple mathematical model of how 2. A simple mathematical model of how radioactive elements decay.radioactive elements decay.

The decay of a radioactive element is The decay of a radioactive element is totally random. (Rutherford found this hard totally random. (Rutherford found this hard to believe. He tried heating, pressure, to believe. He tried heating, pressure, cooling – to no avail).cooling – to no avail).

A particle has a small probability of being A particle has a small probability of being emitted from a nucleus in a given time.emitted from a nucleus in a given time.

There is just one rule for each nucleus:There is just one rule for each nucleus:

It can decay with a fixed probability in It can decay with a fixed probability in a given time.a given time.

Predict a result when decay occurs.Predict a result when decay occurs.

What happens to the population?What happens to the population?

It declines……..It declines……..

………….the number left to decay gets less and .the number left to decay gets less and less.less.

But…the number decaying (proportional to But…the number decaying (proportional to the number left behind) also gets less. the number left behind) also gets less. The population decline gets slower and The population decline gets slower and slower.slower.We’ll be dealing with a RATE of decline, We’ll be dealing with a RATE of decline, which means using small time-chunks (which means using small time-chunks (t).t).

It’s important to understand the idea of It’s important to understand the idea of PROBABILITY of decay and PROBABILITY of PROBABILITY of decay and PROBABILITY of decay PER SECOND. decay PER SECOND.

If each nucleus has a probability If each nucleus has a probability pp to decay, to decay, and there are N nuclei at any moment,and there are N nuclei at any moment,

Number decaying = Number decaying = pp NN

The longer you wait, The longer you wait, the more chance there is the more chance there is for a decay.for a decay.

pp tt

pp = = tt

is called the is called the DECAY CONSTANT in sDECAY CONSTANT in s-1-1

This may seem confusing, so think of the This may seem confusing, so think of the decay constant as THE CHANCE OF decay constant as THE CHANCE OF DECAY IN ONE SECOND.DECAY IN ONE SECOND.

ActivityActivityThe activity of a radioactive source is The activity of a radioactive source is THE NUMBER OF NUCLEI DECAYING PER THE NUMBER OF NUCLEI DECAYING PER SECOND. (UNITS: Bq or Bequerels)SECOND. (UNITS: Bq or Bequerels)

We can now link activity to probability.We can now link activity to probability.

Activity a = Activity a = NN

tt

no. decayingno. decaying

per secondper second

But But pp N = N = t Nt N

So,So,

Activity a = Activity a = t Nt N

tt

And so And so a = a = N N

If you know the activity of the source and If you know the activity of the source and the number of nuclei (using Avogadro’s the number of nuclei (using Avogadro’s number) you can find the decay constant for number) you can find the decay constant for any radioactive material. This is a very any radioactive material. This is a very useful constant, as we shall find out later.useful constant, as we shall find out later.

Fading away…….Fading away…….

The number of nuclei present at The number of nuclei present at any moment decreases at a rate any moment decreases at a rate

equal to the activity.equal to the activity.

NN

tt== - - N N

N = N = activityactivity

decadecayy

Differential Differential of N wrt timeof N wrt time

This equation captures the This equation captures the SMOOTH behaviour of SMOOTH behaviour of decay.decay.

It works for many different types of It works for many different types of decay – a very useful MODEL.decay – a very useful MODEL.

This sort of equation, involving the This sort of equation, involving the RATE of CHANGE of a quantity, is called RATE of CHANGE of a quantity, is called a DIFFERENTIAL EQUATION.a DIFFERENTIAL EQUATION.

It works for GROWTH as well as DECAY; It works for GROWTH as well as DECAY; just use a + instead of a – sign.just use a + instead of a – sign.